# AIRO 2015 BOOK OF ABSTRACTS

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AIRO 2015 BOOK OF ABSTRACTS

AIRO 2015 45th Annual Conference of the Italian Operational Research Society Optimization for Energy, Environment and Sustainability Pisa, September 7-10, 2015 BOOK OF ABSTRACTS AIRO 2015 Committees Conference Chair Maria Grazia Scutellà (Università di Pisa) Scientific Committee Edoardo Amaldi (Politecnico di Milano) Elisabetta Allevi (Università di Brescia) Marida Bertocchi (Università di Bergamo) Giancarlo Bigi (Università di Pisa) Riccardo Cambini (Università di Pisa) Paola Cappanera (Università di Firenze) Laura Carosi (Università di Pisa) Antonio Frangioni (Università di Pisa) Giorgio Gallo (Università di Pisa) Manlio Gaudioso (Università della Calabria) Bernard Gendron (CIRRELT, Canada) Martine Labbé (ULB, Belgium) Federico Malucelli (Politecnico di Milano) Massimo Pappalardo (Università di Pisa) Mauro Passacantando (Università di Pisa) Ulrich Pferschy (TU Graz, Austria) Giovanni Rinaldi (IASI - CNR) Fabrizio Rossi (Università dell’Aquila) Anna Sciomachen (Università di Genova) Maria Grazia Scutellà (Università di Pisa) Paola Zuddas (Università di Cagliari) Organizing Committee Mauro Passacantando - Chair (Università di Pisa) Daniela Ambrosino (Università di Genova) Riccardo Cambini (Università di Pisa) Laura Carosi (Università di Pisa) Paola Cappanera (Università di Firenze) Antonio Frangioni (Università di Pisa) Laura Galli (Università di Pisa) Maddalena Nonato (Università di Ferrara) Maria Grazia Scutellà (Università di Pisa) Program at a glance SALA GERACE SALA SEMINARI EST SALA SEMINARI OVEST SALA RIUNIONI EST SALA RIUNIONI OVEST 9:00-14:00 14:00-14:15 14:15-15:15 15:15-15:30 Mon 7 Registration Opening session AULA MAGNA – Plenary lecture – Luís Eduardo Neves Gouveia, Load(Time)-Dependent and Layered graph approaches for the Traveling Salesman Problems break Equilibrium Models: Theory and Exact Methods for Routing Problems Health Care 1 (invited by Cappanera, Optimization for Water Resources 15:30-17:00 Application Features (invited by Graphs & Networks (contributed) (invited by Gambella, Vigo) Tanfani) Management (invited by Zuddas) Daniele, Scrimali) 17:00-17:30 17:30-19:00 20:30 9:00-10:30 10:30-11:00 11:00-13:00 Tue 8 13:00-14:15 14:15-15:15 15:15-15:30 15:30-17:00 17:00-17:30 17:30-19:00 19:30-20:30 9:00-10:30 10:30-11:00 11:00-13:00 Wed 9 13:00-14:15 14:15-15:15 15:15-15:45 15:45-19:00 20:00 20:30 9:00-10:30 coffee break AULA MAGNA – CIRO Meeting & CORAC Meeting Welcome party Data Envelopment Analysis (invited VRPs with Backhauls (invited by Di Health Care 2 (invited by Cappanera, Optimization for Energy Applications Logistics (contributed) by Carosi) Francesco) Tanfani) (invited by Vespucci) coffee break Real Time Management in Public Nonlinear Programming 1 (invited by Transportation: with MAIOR we Health Care 3 (invited by Cappanera, Optimization in Finance and Packing & Cutting (contributed) Palagi) stand, divided we fall (invited by Tanfani) Insurance (invited by Consigli) MAIOR) lunch time AULA MAGNA – Plenary lecture – Claudia Sagastizábal, Modern Lagrangian Relaxation, or how to navigate the passage between Scylla and Charybdis break Nonlinear Programming 2 (invited by Applications of Vehicle Routing Numerical Methods for Complex Optimization in Logistics (invited by LASSIE (invited by Frangioni) Palagi) (invited by Gambella, Vigo) Networks (invited by Fenu) Bertazzi) coffee break Nonlinear Programming 3 (invited by Optimization for Sustainable Public Optimization for Energy & Classification & Forecasting Palagi) Transport (invited by Galli) Environment (contributed) (contributed) Social event @ Museo degli Strumenti del Calcolo: 1st AIRO e-Sports competition “Killing Polyhedral Models” Nonlinear Programming 4 (invited by Optimization Applications Health Care 4 (invited by Cappanera, Railway Optimization (contributed) Palagi) (contributed) Tanfani) coffee break Variational Problems & Equilibria 1 Optimization for Search & Networks Sportello Matematico (invited by Routing (contributed) (contributed) (contributed) Sgalambro) lunch time AULA MAGNA – Plenary lecture – Leo Liberti, Measuring smart grids coffee break AULA MAGNA – AIRO meeting Tourist train to San Rossore Social dinner Optimization & Classification (invited Maritime Logistics 1 (invited by Health Care 5 (invited by Cappanera, Network Optimization (invited by Green ICT (invited by Mattia) by Gaudioso) Ambrosino, Monaco, Zuddas) Tanfani) Guerriero) 10:30-11:00 Thu 10 11:00-13:00 Variational Problems & Equilibria 2 (contributed) 13:00-14:15 14:15-15:45 Game Theory (contributed) 15:45-16:00 16:00-17:00 coffee break Applications in Economics & Finance Scheduling (contributed) (contributed) lunch time Maritime Logistics 2 (invited by AIRO Prizes Ambrosino, Monaco, Zuddas) break AULA MAGNA – Closing session Optimization Methodologies (contributed) Contents PLENARY LECTURES 6 INVITED SESSIONS Applications of Vehicle Routing (Gambella, Vigo) . . . . . . . . . . . . . Data Envelopment Analysis (Carosi) . . . . . . . . . . . . . . . . . . . . Equilibrium Models: Theory and Application Features (Daniele, Scrimali) Exact Methods for Routing Problems (Gambella, Vigo) . . . . . . . . . . Green ICT (Mattia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Health Care 1 (Cappanera, Tanfani) . . . . . . . . . . . . . . . . . . . . Health Care 2 (Cappanera, Tanfani) . . . . . . . . . . . . . . . . . . . . Health Care 3 (Cappanera, Tanfani) . . . . . . . . . . . . . . . . . . . . Health Care 4 (Cappanera, Tanfani) . . . . . . . . . . . . . . . . . . . . Health Care 5 (Cappanera, Tanfani) . . . . . . . . . . . . . . . . . . . . LASSIE (Frangioni) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) . . . . . . . . . . . . Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) . . . . . . . . . . . . Network Optimization (Guerriero) . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming 1 (Palagi) . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming 2 (Palagi) . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming 3 (Palagi) . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming 4 (Palagi) . . . . . . . . . . . . . . . . . . . . . Numerical Methods for Complex Networks (Fenu) . . . . . . . . . . . . . Optimisation for Sustainable Public Transport (Galli) . . . . . . . . . . . Optimization and Classification (Gaudioso) . . . . . . . . . . . . . . . . Optimization for Energy Applications (Vespucci) . . . . . . . . . . . . . Optimization for Water Resources Management (Zuddas) . . . . . . . . . Optimization in Finance and Insurance (Consigli) . . . . . . . . . . . . . Optimization in Logistics (Bertazzi) . . . . . . . . . . . . . . . . . . . . . Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR) . . . . . . . . . . . . . . . . . . . . . . . . Sportello Matematico (Sgalambro) . . . . . . . . . . . . . . . . . . . . . VRPs with Backhauls (Di Francesco) . . . . . . . . . . . . . . . . . . . . 21 23 29 35 41 48 55 62 68 77 83 90 95 102 109 116 123 129 135 140 147 154 161 166 173 178 186 194 196 CONTRIBUTED SESSIONS 202 Applications in Economics & Finance . . . . . . . . . . . . . . . . . . . . 204 Classification & Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . 213 4 Game Theory . . . . . . . . . . . . . . . Graphs & Networks . . . . . . . . . . . . Logistics . . . . . . . . . . . . . . . . . . Optimization Applications . . . . . . . . Optimization for Energy & Environment Optimization for Search & Networks . . Optimization Methodologies . . . . . . . Packing & Cutting . . . . . . . . . . . . Railway Optimization . . . . . . . . . . Routing . . . . . . . . . . . . . . . . . . Scheduling . . . . . . . . . . . . . . . . . Variational Problems & Equilibria 1 . . . Variational Problems & Equilibria 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 226 232 239 245 252 261 269 278 285 293 301 309 AIRO PRIZES 315 Best Application-Oriented Paper . . . . . . . . . . . . . . . . . . . . . . 316 OR for the society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Author index 318 5 PLENARY LECTURES 6 Plenary Lecture - Monday 7, 14:15-15:15 - Aula Magna 7 Load(Time)-Dependent and Layered graph approaches for the Traveling Salesman Problems1 Luis Eduardo Neves Gouveia∗ Departamento de Estatı́stica e Investigação Operacional, Faculdade de Ciências da Universidade de Lisboa, Portugal, [email protected] Abstract. We focus on the use of time or load dependent models for the traveling salesman Problem (TSP) and two variants, a pick up and delivery variant and the precedence constrained TSP. With respect to the TSP we start by showing one way of improving the Picard and Queyranne formulation, which might be considered the basic time-dependent TSP formulation and also be viewed as a model cast in a layered graph. The improved model is one of the compact formulations with tightest bounds known from the literature. We also address the one-to-one multi-commodity pickup and delivery TSP (m-PDTSP) which can be seen as a capacitated generalization of the TSP with precedence constraints. For this problem we start by presenting a model that might be viewed as close to the previous model to the TSP. Although leading to very tight gaps, the model is difficult to use due to the number of variables and constraints. In order to derive a model, that is still strong, but more manageable, we start by making use of a result showing that (m-PDTSP) is equivalent to a single commodity variant WITH adequate precedence constraints. This second equivalent variant is much more attractive to be modelled by computationally feasible formulations. In particular, we consider layered graph models for the capacity constraints and introduce new valid inequalities for the precedence relations. Computational results show that the new model, when compared with models known from the literature, produces best results for tightly capacitated instances with a large number of commodities. As a side result, for the uncapacitated variant, which is the TSP with precedence constraints, the model produces best results for several ATSP instances, and in particular some instances are solved for the first time. 1 Joint research with Teresa Godinho, CIO-Dep. de Matemática, Instituto Politécnico de Beja, Beja, Portugal; Pierre Pesneau, Institut des Mathématiques de Bordeaux, Université Bordeaux 1, France; Mario Ruthmair, Dynamic Transportation Systems, Mobility Department, AIT Austrian Institute of Technology, Austria. Plenary Lecture - Tuesday 8, 14:15-15:15 - Aula Magna 8 Modern Lagrangian Relaxation, or how to navigate the passage between Scylla and Charybdis Claudia Sagastizábal∗ IMPA, Rio de Janeiro (visiting researcher), [email protected] Abstract. The classical Lagrangian relaxation is a useful tool to solve non-linear mixed integer problems, especially those that are large-scale with separable structure. The approach, however, provides a solution to a problem that is bidual to the original one, and as such does not guarantee primal feasibility. To address this issue, extended Lagrangians such as the as socalled “Sharp” and “Proximal” can be used, but there is a loss of separability. To avoid having to choose between two evils, we combine the recent on-demand accuracy bundle methods with those extended Lagrangians in a manner that offers a good compromise between separability and primal convergence. We outline basic ideas and computational questions, highlighting the main features and challenges with examples from energy optimization. Credit to various co-authors will be given during the presentation. What the talk will be about Real-life optimization problems often depend on data subject to unknown variations that can be due to imprecise measurements or to the stochastic nature of the data itself. In general, modelling such problems involves many variables and constraints, typically yielding a large nonlinear programming problem with mixed 0-1 variables (MINLP). For complex problems arising in the energy sector, it is crucial to design solution methods that provide primal and dual solutions of good (excellent) quality. This is easily understood by recalling the following two distinctive features of electric energy: • electricity cannot be stored in large quantities. It must be generated on the fly, as “needed” (keeping in mind that there are network losses and demand is uncertain). This makes it important to have tight generation plans, which amounts to have accurate primal solutions of the MINLP at hand. • The price that consumers pay depends on “signals” given by certain multipliers. This makes it important to have accurate dual solutions for the MINLP, especially when considering the huge economical impact of a few cent difference in the price calculation. For illustration, consider the following example from [5], describing a problem faced by a company owing two power plants, one thermal and one hydraulic, when the manager needs to decide the generation plan for the next day. To supply the requested demand of energy, say d ∈ <, decisions must be taken every half hour in a manner that optimally combines the technological limitations of each plant. For thermal (hydraulic) generation variables pT (pH ) with respective generation costs and technological constraint sets denoted by CT and PT (CH and PH ), the manager Plenary Lecture - Tuesday 8, 14:15-15:15 - Aula Magna solves the optimization problem below: min CT (pT ) + CH (pH ) s.t. pT ∈ PT , pH ∈ PH v(d) := pT + pH = d , 9 (1) where we denoted the optimal value v(d) to emphasize the dependence of the output on the given demand. Since this data is an estimate, determining the impact of demand variations is often a concern for the company. To quantify such an impact, the manager can use Lagrangian relaxation. Specifically, associating a multiplier x ∈ < to the demand constraint gives the Lagrangian L(pT , pH , x) := CT (pT ) + CH (pH ) + x(d − pT + pH ) . (2) If (1) is a convex problem, by duality arguments, min pT ∈ PT pH ∈ PH max L(pT , pH , x) = max min L(pT , pH , x) x∈< x ∈ < pT ∈ PT pH ∈ PH (without convexity, the equality is replaced by “≥”.) The optimal value of the rightmost problem, called dual to (1), equals v(d) and has the form min CT (pT ) − xpT min CH (pH ) − xpH v(d) = max xd + + . (3) s.t. pT ∈ PT s.t. pT ∈ PH x∈< When seen as a function of d, and for each fixed dual variable x, the argument in the maximum is the sum of a linear term (with slope x), and a constant term (the two minima over pT and pH ). As a result, the function v(d) is convex piecewise affine, and, by Danskin’s theorem [2], Chapter VI.4.4, any x̄ solving (3) satisfies v(d0 ) ≥ v(d) + x̄(d0 − d) for all d0 ∈ < . Since v(d) is the cost for the company to satisfy the given demand, this inequality tells the manager that if demand was underestimated in, say, 1%, for the plants to produce more than the scheduled amounts p̄T and p̄H , the company would have to spend at least additional .01x̄d. Also, if the manager was to set the unit selling price below x̄, the company would lose money. Solving the dual problem (3) gives the manager the marginal cost x̄, which can be used as a guide when setting the company selling prices. The interpretation of Lagrange multipliers as marginal costs (or shadow prices in the parlance of Linear Programming) has plenty of useful applications, for instance, to set the value of green certificates, delivered by the government to the power companies as a compensation for respecting environmental constraints related to carbon emissions. Our simple example (1) illustrates the importance of computing with high accuracy optimal dual variables: changing electricity prices in the cents has a tremendous impact on society. For simple problems, x̄ can be obtained directly from solving (1). For more realistic problems, the direct method is not possible and the dual approach (3), separating the optimization problem over PT from the one over PH , is the only one applicable. The reason is that for realistic problems the sets PT Plenary Lecture - Tuesday 8, 14:15-15:15 - Aula Magna 10 and PH describe very different technologies, and no optimization package can solve the problem directly (typically, PT uses 0-1 variables and PH is described by nonconvex constraints). In (3), by contrast, the optimization problem in variable pT is decoupled from the one in variable pH , and separate off-the-shelf solvers can be employed for their respective solution. Of course, the price to pay is that now there is one more optimization layer, the maximization over the dual variable x. The nonsmooth concave dual problem (3) needs to be handled by algorithms that are efficient and accurate, like bundle methods [1], Part II. Taking a detour through the dual problem (3) was needed because the direct solution of (1) was not possible, due to the MINLP complexity. A negative outcome of the dual detour is a loss of primal feasibility. If the manager was only interested in pricing the electricity generation, the obtained output is highly satisfactory, as bundle methods ensure the numerical solution is very precise, thanks to its reliable stopping criterion. If the manager is also interested in determining a production plan, some additional calculations might be needed when there is a duality gap. This happens often in energy optimization problems, because for thermal power plants the set PT involves binary variables. The duality gap arises because only the convex hull of PT can be seen with the dual approach. In particular, when those variables represent units to be switched on and off, this means that the optimization process may advice the manager to turn on only a fraction of the corresponding unit! On the other hand, if the manager decides to generate electricity on the basis of feasible primal variables associated with the optimal price x̄, the company may end up producing more than the requested demand. This trap, forcing to choose between Scylla and Charybdis, is typical in classical Lagrangian relaxation. The classical Lagrangian (2) is a particular case of certain general dualizing parameterizations, described in Chapter 11.K in [4], devoted to augmented Lagrangians and nonconvex duality. We consider two options therein, namely the Proximal and Sharp Lagrangians, which eliminate the duality gap. These Lagrangians are no longer separable, and suitable approximations need to be put in place to ensure a satisfactory primal-dual convergence. We shall discuss how to build such approximations in a manner that the recent on-demand-accuracy bundle methods [3] can be put in place. References [1] Bonnans, J., Gilbert, J., Lemaréchal, C., and Sagastizábal, C. Numerical Optimization. Theoretical and Practical Aspects. Universitext. Springer-Verlag, Berlin, 2006, 2nd. edition, xiv+423 pp. [2] Hiriart-Urruty, J.-B. and Lemaréchal, C., Convex Analysis and Minimization Algorithms. Number 305-306 in Grund. der math. Wiss. Springer-Verlag, 1993. [3] Oliveira, W., Sagastizábal, C., and Lemaréchal, C. Convex proximal bundle methods in depth: a unified analysis for inexact oracles. Mathematical Programming, 148(1-2):241–277, 2014. [4] Rockafellar, R. and Wets, R.-B. Variational Analysis. Number 317 in Grund. der math. Wiss. Springer-Verlag, 1998. [5] Sagastizábal, C., On Lagrangian Decomposition for Energy Optimization, Proceedings of the ICIAM, Beijing, China (2015). Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 11 Measuring smart grids2 Leo Liberti∗ CNRS LIX, Ecole Polytechnique, 91128 Palaiseau, France, [email protected] Claudia D’Ambrosio Pierre-Louis Poirion Sonia Toubaline CNRS LIX, Ecole Polytechnique, 91128 Palaiseau, France, dambrosio,poirion,[email protected] Abstract. The SO-grid project aims to develop a smart grid pilot for the French electricity distribution network. Our workpackage is concerned with the efficient placement of measuring devices on the network. We give an overview of our progress so far: combinatorial problems and variants, complexity, inapproximability, polynomial cases, single- and bilevel mathematical programming formulations, row-generation algorithms, and generalization to a large class of bilevel MILPs. Introduction The three defining properties of a smart grid are reliability, sustainability and value. Reliability is the main motivation for making electrical grids smart: when it was still possible to increase network capacity to make it reliable, electrical grids were not termed smart. As soon as capacity increase became infeasible, we started noticing unstable behaviours, such as large-scale blackouts due to a single point of failure, or to a short but high demand peak. The only possible way out, when capacity is maximum, is to address the demand. By installing an Advanced Metering Infrastructure (AMI), it became possible to define new policies based on Demand Response (DR), mainly by predictive and prescriptive analytics based on the continuing measurements of electrical quantities such as voltage and current. One of the most important DR tools these analytics can help improve is pricing. In parallel, renewable source of energies such and sun and wind power were integrated in the power grid, addressing some sustainability issues. And finally, in capitalistic societies the private sector’s only motive for investing into such a costly transformation of the power grid is increased value (another feature which pricing can help leverage). The type of analytics we discuss here is prescriptive, and concerns the optimal placement of certain measuring devices on the electrical grid. Although we are not at liberty to disclose the exact specification or nature of such devices, for the purposes and the intended audience of this paper we shall assume such devices to be Phasor Measurement Units (PMU), which measure the voltage at the node v of the grid they are installed at, as well as at a varying number of nodes in its neighbouring set N (v) [1]. Ohm’s law applied to a link {u, v} ∈ E of the grid (represented by a graph G = (V, E)), Vv − Vu = R Iuv , 2 This work was carried out as part of the SO-grid project (www.so-grid.com), co-funded by the French agency for Environment and Energy Management (ADEME) and developed in collaboration between participating academic and industrial partners. Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 12 allows the measure of the current along the link if the voltage is known at the endpoints. Moreover, by Kirchoff’s law, X ∀v ∈ V Iuv = 0, {u,v}∈E it follows that, if Vv is known and, for all u in the neighbourhood N (v) except for at most one (call it w), Vu is also known, then Vu is known for all u ∈ N (v), including w. It suffices to exploit Kirchoff’s law to compute the current along the link {v, w}, and then Ohm’s law to compute Vw . We call observed the nodes with known voltage and the links with known currents. As shown above, nodes and links can be observed directly (because of PMUs) or indirectly, through the application of Ohm’s and Kirchoff’s laws. Since PMUs can be expensive, the general problem we are concerned with is to find the placement of the smallest number of PMUs which allows the grid to achieve the state of full observability, i.e. such that all nodes and links of the grid are observed. PMUs may have a limited number k of observation channels, meaning that if a PMU is placed at v ∈ V , it measures the voltage at v and at up to k nodes in N (v). When k = 1, a PMU installed at v can measure the voltage at v and at a single neighbour u. Since, by Ohm’s law, the current on {u, v} is known, this situation is logically equivalent at placing the PMU on the edge {u, v} and measuring the current. In the rest of this paper, we mostly focus on PMUs with k = 1. Observability As mentioned above, the aim is to find the smallest P such that the set of observed nodes Ω is equal to V . If we use PMUs with k = 1, by Ohm’s law we have: R1 ∀{u, v} ∈ P (u, v ∈ Ω). Moreover, by Kirchoff’s law (as above) we also have: R2 ∀v ∈ V (v ∈ Ω ∧ |N (v) r Ω| ≤ 1 → N (v) ⊆ Ω). Rules R1 and R2 are known as observability rules, and yield an iterative procedure to achieve full observability of the grid: from a given P , one applies R1 first (to all links in P ) and R2 for as many times as possible. If Ω = V , P is a feasible placement, and otherwise it is infeasible. Combinatorial problems All problem variants aim at minimizing |P | in order to achieve Ω = V . The name of the problem and its exact definition depends on k. For unlimited k, the problem is known as Power Dominating Set (PDS) [2], [3]. For bounded k, the problem is known as k-PMU placement problem. The 1-PMU problem is also known as Power Edge Set (PES), and the 0-PMU problem is also known as Power Vertex Set (PVS). While PDS, k-PMU and PES can all be applied to the setting of electrical Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 13 grids, the PVS has an application to influence spreading in social networks [4]. For all variants but the PES, P ⊆ V , whereas for PES (which is the case we shall consider) we have k = 1 and consequently P ⊆ E, as discussed above. The PDS is known to be NP-complete on bipartite graphs, planar bipartite graphs and cographs, and √polynomial on trees, meshes, block graphs and bounded treewidth graphs. An O( n) approximation algorithm is known for the PDS (where 1−ε n = |V |), but it is hard to approximate it to within a factor 2log n . Hardness and inapproximability We show that PES is NP-hard by reduction from the Vertex Cover problem on 3-regular graphs (3VC). We transform a given graph instance G0 of 3VC into a graph G00 where each vertex v of G0 is mapped to a 10-vertex gadget in G00 : the hardness proof is long but elementary. A similar proof shows hardness of approximation to within a factor 1.12 − ε for any ε > 0 (we can also prove NP-hardness for the PVS). Moreover, we show that the PES is polynomial on trees (by reduction to Path Cover on trees) and meshes. An iteration-indexed Mathematical Program Our first test towards solving PES instances was to model the iterative observability procedure mentioned above within a Mixed-Integer Linear Program (MILP) using binary indicator variables: • suv = 1 iff {u, v} ∈ P • ωvt = 1 if v ∈ V enters the set Ω during iteration t of the observability procedure. Note that ωvt are iteration-indexed P variables, and considerably increase the problem size. The objective function suv minimizes |P |, and one of the constraints {u,v}∈E requires that Ω = V at the last iteration Tmax (which is O(n)). The constraints implement R1 and R2 in a straightforward way. This formulation turned out to be computationally unsatisfactory using CPLEX 12.6 on graphs with up to 30 nodes. Fixed point reformulation Although PES observability is naturally modelled by an iterative procedure, it is nonetheless of a monotonic nature. A node is either observed or not; once in Ω, no node can exit the set during the iteration procedure. This idea led us to describe the observability iteration applied to a vertex v ∈ V by means of the following function of ω, parametrized on s (the indicator vector of the PMU placement): X X s θv (ω) = max suv , max ωu + ωw − (|N (u)| − 1) . (4) u∈N (v) u∈N (v) w∈N (u) w6=v Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 14 Note that in Eq. (4) ω is independent of the iteration index t: essentially, it describes the evolution of ω at a given iteration: if ω is the indicator vector of Ω before the iteration takes place, θs (ω) = (θvs (ω) | v ∈ V ) is the indicator vector of Ω after the iteration. By monotonicity, we can express the end of the iteration procedure by means of a convergence to a fixed point: ω = θs (ω). The iteration-indexed MILP can now be re-written as the following nonlinear problem (where m = |E|): P minm suv s∈{0,1} {u,v}∈E ω∈{0,1}n P (5) ωv = n v∈V θs (ω) = ω, which has fewer decision variables than the MILP described above since there are no iteration indexed variables. Note that the first constraint expresses Ω = V , and the nonlinearity arises because of the function θs defined in Eq. (4). Next, we remark that the fixed point condition on θs expresses, by means of the ω variables, the smallest set of vertices obeying rules R1 (encoded in the left term of the outer maximum in Eq. (4)) and R2 (encoded in the right term of the outer maximum in Eq. (4)). As such, it can be written as the following MILP, where the variables s are now to be considered as problem parameters: P min n ωv ω∈{0,1} v∈V ∀{u, v} ∈ E ωv ≥ suv P (6) ∀v ∈ V, u ∈ N (v) ωv ≥ ωu + ωw − (|N (u)| − 1). w∈N (u) w6=v Finally, we replace the fixed point condition in Eq. (5) by the MILP in Eq. (6) to obtain the following bilevel MILP (BMILP): P min m suv s∈{0,1} {u,v}∈E f (s) = n P min ω v n ω∈{0,1} v∈V ∀{u, v} ∈ E ωv ≥ suv P f (s) = ∀v ∈ V, u ∈ N (v) ω ≥ ω + ω − (|N (u)| − 1). v u w w∈N (u) w6=v (7) Single-level reformulation In general, BMILPs are practically even harder to solve than MILPs, so Eq. (7), by itself, is not much of a gain. On the other hand, we can prove, by induction on the steps of the observability iteration procedure, that the integrality constraints in the lower-level problem can be relaxed to ω ≥ 0 without changing the optimal solution. Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 15 This implies that the lower-level problem can be replaced by the KKT conditions [5] of its continuous relaxation to the nonnegative orthant, yielding the following single-level MINLP: P minm suv s∈{0,1} {u,v}∈E λ,µ≥0 P (suv µuv + (1 − |N (u)|)λuv ) ≥ n (8) {u,v}∈E P P ∀u ∈ V (µuv + λuv − λvu − λwv ) ≤ 1, w∈N (u) v∈N (u) w6=v where µ, λ are dual variables of the relaxed lower-level problem. We can also prove that the variables µ are bounded above (though the bound M is exponentially large), so the bilinear products suv µuv appearing in Eq. (17) can be linearized exactly using Fortet’s reformulation: each product is then replaced by an additional variable puv . Finally, this yields the following single-level MILP: P minm suv s∈{0,1} λ,µ∈[0,M ] {u,v}∈E P (puv + (1 − |N (u)|)λuv ) ≥ n {u,v}∈E P P ∀u ∈ V (µuv + λuv − λvu − λwv ) ≤ 1 (9) w∈N (u) v∈N (u) w6=v ∀{u, v} ∈ E puv ≤ M suv ∀{u, v} ∈ E puv ≤ µuv ∀{u, v} ∈ E puv ≥ µuv − M (1 − suv ). Eq. (9) yields computational improvements of around 1-2 orders of magnitude w.r.t. the iteration-indexed MILP. Its weakest point is that the lower-level problem Eq. (6) cannot easily be changed, lest we should lose its integrality property, which is key to reformulating the bilevel problem to a single level one. The SO-grid application, however, requires us to impose some robustness constraints Υω ≤ ξ to the lower level problem. This means we can no longer reformulate Eq. (7) to Eq. (17); and, ultimately, that we need a different solution approach. Solving the bilevel problem directly It is well known that every subset S of vertices of the hypercube (in any dimension) can be described linearly, i.e. the convex hull of the vertices in S does not contain any other hypercube vertex aside from those in S. We apply this fact to the feasible region F = {s ∈ {0, 1}m | f (s) = n} of the bilevel problem Eq. (7), obtaining the single level problem: X min suv | s ∈ conv(F) ∩ {0, 1}m . {u,v}∈E Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 16 Since we do not know a polyhedral description of F, we look for a polyhedron P having the same intersection with the hypercube as F, and aim at solving: X (10) min suv | s ∈ P ∩ {0, 1}m {u,v}∈E using a row generation algorithm [6], [7]. Inequalities for P Here we write the set Ω(s) of observed nodes in function of the PMU placement encoded by the indicator vector s. Suppose a placement s is given such that Ω(s) ( V . To reach full observability we must install a PMU onto an unobserved link, obtain a new s, run the observability iteration procedure, and repeat until Ω(s) = V . This yields a sequence (s0 , . . . , sk ) with two properties: [1] Ω(sh ) ( |V | for h < k and Ω(sk ) = V P h [2] for h < k, sh+1 is minimally larger than sh , i.e. (sh+1 uv − suv ) = 1. {u,v}∈E Property [1] implies that s0 , . . . , sk−1 are all infeasible placements for Eq. (7). Our polyhedron P, which approximates the feasibile region F of Eq. (7), must therefore be defined by inequalities which cut off all of these infeasible placements. Since we can obtain them all very efficiently by repeatedly running observability iterations, we can simply adjoin no-good cuts to P for each sh for h < k. To do this, for all h < k we define αh = sh xor 1 as the Boolean complement of sh , so that αh sh = 0, which implies that: αh s ≥ 1 (11) separates sh from P, as desired. The reason why this idea works well in practice is that, by Property [2], we can prove that αk−1 s ≥ 1 dominates all other no-good cuts Eq. (11) for h < k − 1. We can intuitively see why this holds by asking when the inequality Eq. (11) is tightest whilst still being valid. This occurs when the LHS is (nontrivially) as small as possible, i.e. when as many as possible of the components of αh are zeros, which, by definition of Boolean complement, means that as many as possible of the components of sh must be ones. Note that the algorithm which generates the placements sh is: [1] iterate observability using R1, R2 from sh [2] if Ω(s) ( V , pick {u, v} 6∈ P such that installing a PMU on {u, v} yields a larger |Ω(s)|, set shuv = 1, increase h and repeat from Step [1]. Hence, by Property [2], the iteration index h corresponding to the infeasible placement with as many ones as possible is the last before termination, namely h = k −1. A row generation algorithm Note that the no-good cuts of Eq. (11) are defined by means of PMU placements s̄ that are infeasible (Property [1] above) and ≤-maximally dominant (Property [2] Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 17 above). Let F̄ be the set {0, 1}m r F of placements that are infeasible w.r.t. the constraint f (s) = n, F¯max the set of ≤-maximal elements of F̄, and ζ(s̄) = {e ∈ E | s̄e = 0} be the set of component indices where s̄ is zero. We define: X P = s ∈ [0, 1]m | ∀s̄ ∈ F¯max se ≥ 1 . e∈ζ(s̄) The row generation algorithm we employ dynamically generates facets of P using the combinatorial algorithm described above in order to find maximally infeasible placements sk−1 , and then solves the PMU minimization subproblem defined on P ` ∩ {0, 1}m , where P ` is the MILP relaxation of P consisting of all the cuts (Eq. (11)) found up to the `-th iteration. This algorithm terminates when the cut generation procedure is unable to find a new cut. The current placement s is then optimum because of the minimization direction of the subproblems. Note that every iteration requires the solution of a MILP, which, in practice, we can solve fairly efficiently using CPLEX 12.6. Compared to solving the two MILPs described above with CPLEX, this row generation approach is the only one which is able to solve the PES over the medium scale networks required by the SO-grid project (up to around 120 nodes and 180 links). Generalization to arbitrary bilevel MILPs It turns out that the algorithmic framework described above is generalizable to a large class of bilevel MILPs for which practically viable solution methods do not appear to have been found yet: min n χx x∈{0,1} Ax ≥ b f (x) ≥ ( c − γx (12) min β(x)y y∈Y f (x) = y ∈ Θ(x), where χ, γ ∈ Qn , A ∈ Qmn , b ∈ Qm , c ∈ Q, β : Qn → Qq , Θ(x) is a polyhedron in Rq for all x ∈ Rn , and Y is a mixed-integer set in Rq . Note that the bilevel formulation Eq. (7) of the PES is the special case of Eq. (12) given by A = 0, b = 0, γ = 0, c = n, f (·) = |Ω(·)|, χ, β = 1, and Θ(·) given by the θs function defined in Eq. (4). Dominance assumptions In order to simplify the presentation, we make two assumptions, which we shall show how to relax later on: [1] γ ≥ 0 [2] ∀x0 ≤ x00 satisfying Ax ≥ b, we have Θ(x0 ) ⊇ Θ(x00 ). Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 18 Note that these two assumptions are satisfied by the bilevel formulation Eq. (7) of PES: γ = 0 and, for a PMU placement s00 dominating a given placement s0 , the R1 constraints ωv ≥ suv ensure that the feasible region of the lower level problem in Eq. (6) becomes smaller. Mutatis mutandis3 , the method outlined above for the PES formulation goes through essentially unchanged, with F = {x ∈ {0, 1}n | Ax ≥ b ∧ f (x) ≥ c − γx} F̄ = {x ∈ {0, 1}n | Ax ≥ b ∧ f (x) < c − γx}, aside from the cut generation step, since the one discussed above is based on the structure of the PES problem. Next, we outline the generalization of the cut generation subproblem. The generalized cut generation subproblem Recall the cut generation step in the PES setting: we start from an infeasible placement s, compute the observed nodes, then add one PMU to s and repeat until all nodes are observed; we then take the maximal infeasible placement sk−1 at the last iteration, before full observability is achieved, and separate it with a no-good cut. In the generalized setting we lack the combinatorial relationship between ω and s given by Ω(s) and defined using the propagation rules R1 and R2. However, given an infeasible solution x0 of the upper level problem, we can find the set X ∗ of all x that ≤-minimally dominate x0 , replace x0 by all of the elements of X ∗ (in turn) that are also in F̄, stopping when X ∗ ⊆ F. When this happens, x0 is in F¯max (i.e. ≤-maximally infeasible), and can be used to generate a facet of P similar to Eq. (11): if α0 is the complement of x0 in {0, 1}n , the cut is α0 x ≥ 1. Given x0 ∈ F̄, we compute X ∗ by finding all solutions to the following cut generation MILPs (for all j ≤ n): P min xi n x∈{0,1} i≤n Ax ≥ b CGj = (13) 0 x ≥ x (†) xj ≥ 1, (‡) where (†) enforces nonstrict domination w.r.t. x0 , with strictness on at least one coordinate direction ej being enforced by (‡) and the objective function. Relaxing the dominance assumptions The first dominance assumption γ ≥ 0 is mostly a technical detail which we shall not discuss here; its relaxation follows because of the new methods employed to deal with the second assumption, which intuitively states that, given x0 ∈ F̄, in the direction of the negative orthant pointed at x0 we can never find a feasible solution. 3 The Italian translation avendo cambiato le mutande is a courtesy of a few friends from high school — those very same friends who created the eternal motto of our high school banners, namely schola mutanda est. Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 19 We can relax this assumption by simply replacing the negative orthant with an infeasible cone pointed at x0 . For this purpose, we write Θ(x) explicitly as: Θ(x) = {y ∈ Rq | By ≥ d + Cx} for appropriately sized B, d, C, and define a (C, γ)-dominance by means of a cone C(x0 ) = {x ∈ Rn | Cx ≤ Cx0 ∧ γx ≤ γx0 } pointed at x0 , which allows us to state: x ≤C x0 ⇔ x ∈ C(x0 ). One complication introduced by this generalized dominance is that it prevents us from using no-good cuts, since a no-good for a maximally infeasible solutions might fail to dominate no-goods from lesser infeasible solutions; moreover, no-good cuts may be invalid in this setting. For any x ∈ {0, 1}n , we define a distance ∆x0 (x) from x to the infeasible cone C(x0 ): minn 1 (e + f ) ∀x ∈ Rn e,f ∈R+ ∆x0 (x) = C(x − e + f ) ≤ Cx0 γ(x − e + f ) ≤ γx0 , which is obtained as the sum of slacks needed to make the above Linear Program (LP) feasible, and replace the no-good cut of Eq. (11) by the nonlinear cut ∆x0 (x) > 0. Since strict inequalities are not allowed in MP, we need to solve an auxiliary MILP to find the maximum scalar δx0 such that ∆x0 (x) ≥ δx0 (14) is a valid cut for x0 . The cut in Eq. (14), however, is only nonlinear because ∆x0 (x) is computed via an LP. This LP can simply be replaced by its KKT conditions, which are a nonlinear system of equations and inequalities in function of primal and dual variables. As with the single-level reformulation Eq. (17), the nonlinearities are bilinear products between binary and continuous variables, which can be linearized exactly. This ultimately yields feasibility-only MILP which we shall refer to as RC(x0 ). Since RC(x0 ) is feasibility only, it simply consists of new constraints and new variables, which can be appended to any MP. We shall therefore take it as the output of the cut generation subproblem in this generalized setting without assumptions. Due to the introduction of the dual variables at each iteration, the row generation algorithm now becomes a row-and-column generation algorithm, where the block RC(x0 ) is added to the master problem at each itereation. The cut generation subproblem is somewhat complicated by the replacement of the negative orthant with the infeasible cone. More precisely, we can no longer define auxiliary subproblems for each coordinate direction ej for j ≤ n as in CGj , since these span an orthant. Instead, we heuristically find a few rays of a Hilbert basis H [8] of the intersection of the negative of the infeasible cone with an appropriately chosen orthant, and define an equivalent subproblem (not altogether dissimilar from CGj ) for each ray in H, which, taken together, replace the coordinate directions ej . Of course Hilbert bases can have exponential size, so we use this tool heuristically: we shall not find facets of P, but simply cuts. Plenary Lecture - Wednesday 9, 14:15-15:15 - Aula Magna 20 Incredibly, for all this heavy use of worst-case exponential time algorithms, the whole scheme appears to be surprisingly efficient in practice. These techniques allow us to solve bilevel MILPs with matrices in the order of 1 to 10 rows and 20 to 30 columns in a few seconds to a few minutes of CPU time of a modern laptop. References [1] N. Manousakis and G. Korres and P. Georgilakis, Taxonomy of PMU placement methodologies, IEEE Transactions on Power Systems 27 (2012), 1070–1077. [2] A. Aazami and M. Stilp, Approximation algorithms and hardness for domination with propagation, in M. Charikar and K. Jansen and O. Reingold and J. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques, volume 4627 of LNCS, pages 1–15, New York, 2007. Springer. [3] D. Brueni and L. Heath, The PMU placement problem, SIAM Journal on Discrete Mathematics 162 (2005), 744–761. [4] P. Domingos and M. Richardson, Mining the network value of customers, In F. Provost and R. Srikant, editors, International Conference on Knowledge Discovery and Data Mining, volume 7 of ACM-SIGKDD, pages 57–66, New York, 2001. ACM. [5] S. Dempe, Foundations of bilevel programming, Kluwer, Dordrecht, 2002. [6] B. Colson, P. Marcotte, and G. Savard, Bilevel programming: a survey, 4OR 3 (2005), 87–107, 2005. [7] A. Lodi, T. Ralphs, and G. Woeginger, Bilevel programming and the separation problem, Mathematical Programming A, to appear. [8] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester, 1986. INVITED SESSIONS 21 Applications of Vehicle Routing (invited by Gambella and Vigo) Tuesday 8, 15:30-17:00 Sala Seminari Est 22 Applications of Vehicle Routing (Gambella, Vigo) 23 An Open Source Spreadsheet Solver for Vehicle Routing Problems Güneş Erdoğan∗ School of Management, University of Bath, UK, [email protected] Abstract. Assembling the data sources, solution algorithms, and visual representation of the results of a Vehicle Routing Problem (VRP) into a single platform is a problem on its own, due to the mix of software being utilized. Most academics develop optimization algorithms in C++ and JAVA. Distance and driving time data have to be retrieved from a Geographical Information Systems (GIS) database. However, Microsoft Excel is widely accepted as the standard quantitative analysis software for small to medium scale businesses. In this talk, we present an open source VBA code embedded into an Excel file, which can retrieve data from public GIS, solve the VRP instances, and visualize the result. Familiarity of the business world with Excel as well as the public availability of the code facilitate the widespread use of the solver, and hold the promise for enabling greener logistics practice. Our experience with logistics specialists from various companies has revealed that most of them have very limited information about routing algorithms and have a pessimistic view of the possibility to implement a routing software. To the best of our knowledge, there is no unified format for stating, solving, and visualizing Vehicle Routing Problems (VRPs). Without a unified format, every VRP should be tackled on its own, with the results getting lost in the ocean of information that is the Internet. Hence, there is an apparent need to for wide accessible, transparent, and easy to use VRP software. On one hand, the standard software for small to medium scale quantitative analysis for businesses has been established as, arguably, Microsoft Excel. On the other hand, most academics develop solution algorithms in C++ and the resulting codes are not for the faint of heart. Distance and driving time data have to be retrieved from a Geographical Information Systems (GIS) database, which requires investment. The results of the algorithms are usually represented as a single value, the total cost, and it can only mean so much. It is not straightforward to manually find a solution of a VRP, much less so to compute its cost or to visualize it. Hence, constructing a unified format for the data sources, solution algorithms, and visual representation of the results is a problem on its own. Microsoft Excel is being taught at the management / business schools of many universities as of the time of this writing. VBA is a programming language that is embedded within Excel. On the upside, VBA provides all the basic functionality and flexibility of a high level programming language, as well as access to all the data you are storing within the Excel workbook. On the downside, VBA cannot possibly compete with C++ in terms of efficiency. To the best of our knowledge, there is no formal benchmarking study to compare the two languages, but our experiments have showed that VBA is at least 3 times lower than C++. However, it is possible to use C++ within VBA to improve the efficiency. The capabilities of public GIS have significantly increased in the past few years. Although there are many more, three functions of the public GIS systems suffice to retrieve the data to solve a VRP instance. Geocoding is the function that converts Applications of Vehicle Routing (Gambella, Vigo) 24 and address into the corresponding Latitude / Longitude values. Directions is the function that returns the distance and driving time between two points in addition to the directions. Finally, Static maps is the function that returns image files, which are defined by their centre point, zoom level, and their size. Used in the given order, it is possible to get the coordinates of the locations, the driving distances and duration, and visualize the routes on a map. As the result of this study, we have developed an open source VBA code embedded into an Excel file, which can retrieve data from public GIS, solve the VRP instances, and visualize the result. The code is capable of solving VRP instances up to size 200. Familiarity of the business world with Excel as well as the public availability of the code facilitate the widespread use of the solver, and hold the promise for enabling greener logistics practice. The code has also been used in various universities for teaching purposes. Applications of Vehicle Routing (Gambella, Vigo) 25 Applications of the vehicle routing problem with time period constraints Giuseppe Stecca∗ Istituto di Analisi dei Sistemi ed Informatica, Consiglio Nazionale delle Ricerche, Italia, [email protected] Lucio Bianco Stefano Giordani Dipartimento Ingegneria dell’Impresa, Università di Roma “Tor Vergata”, Italia, [email protected] [email protected] Pasquale Carotenuto Instituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Italia, [email protected] Abstract. This work describes two applications of the vehicle routing problem (VRP) to the design of fixed and periodic routes. The first application is an industrial case in the field of touristic cruise planning where points of interest should be visited within exactly one of multiple time windows on a weekly time basis. The second application is in retail distribution of fuel oils where petrol stations must be refueled with given fuel oil amounts periodically within a given time horizon. The work studies the VRP models for the two described applications evaluating intersections between them and studying possible unified modeling formulation. Application case studies Vehicle routing has several and different applications which involve distribution of goods and implementation of services. The role of time period constraints is fundamental in several real problems and can be used for their characterization and for the solution strategies. The first case studied in this work is related to the planning of touristic routes. The routes must contain a set of points of interest, each one being visited within one of multiple time windows. Other interesting requirements are the fixed number of vehicles to be used, not mandatory visit of all points of interest, variable traveling time, and additional time constraints. The second case studied refers to the final distribution of fuel oil from central depots to petrol stations (customers) [1]. In this particular application the total demand qitot of customer i may be split non-equally among a subset of the week days: in particular, the delivery to customer i can be done in one of given refueling patterns, where pattern πi is a list of possible visit-period combinations along with the specific demands. The algorithm developed for this application is similar to the hybrid genetic algorithm HGSADC proposed in [2], tailored to solve particular real instances where the ratio between truck capacity and average customer daily demands is very small, which means that solutions are characterized by a large number of daily routes with few visited customers per route. VRP models The analysis of the two cases has been focused on the characteristics of the time period constraints and can be considered as special cases of Periodic Vehicle Routing Problem (PVRP), fixed routes (FR), and VRP with multiple time windows Applications of Vehicle Routing (Gambella, Vigo) 26 (VRPMTW). In PVRP a planning horizon of t days is considered [3]; each customer i has a service frequency and a pattern of availabilities in days. In fixed routes applications [4], vehicle routes must be defined on a daily basis. For many reasons fixed routes may offer several advantages. Moreover the customer daily may vary, resulting in a stochastic problem. As a consequence the determination of fixed routes should for example minimize expected operational costs. Fixed routes are related to VRPMTW, described in [5]. For each customer i a number of alternative time windows pi is given. Both the aforementioned problems involve time period issues and constraints. The difference between PVRP and VRPMTW is that, while for the former problem tour durations and customer time windows are similar in length, in the latter tour durations are typically much larger than time windows lengths. We analyze special cases which belong to both problems in order to study possible unified modeling formulation and extensions. References [1] Carotenuto, P., Giordani, S., Massari, S., Vagaggini, F., Periodic capacitated vehicle routing for retail distribution of fuel oils, 18th Euro Working Group on Transportation, submitted to EWGT 2015, 14-16 July 2015, Delft, The Netherlands. [2] Vidal, T., Crainic, T.G., Gendreau M., Lahrichi, N., Rei, W., A hybrid genetic algorithm for multi-depot and periodic vehicle routing problems, Operations Research 60:3 (2012), 611-624. [3] Cordeau, J-F., Gendreau M., and Laporte G., A tabu search heuristic for periodic and multidepot vehicle routing problems, Networks 30 (1997), 105–119. [4] Erera A.L., Savelsbergh M., and Uyar E., Fixed routes with backup vehicles for stochastic vehicle routing problems with time constraints Networks 54:4 (2009), 270–283. [5] Belhaiza, S., Hansen P., and Laporte G., A hybrid variable neighborhood tabu search heuristic for the vehicle routing problem with multiple time windows, Computers & Operations Research 52(2014), 269–281. Applications of Vehicle Routing (Gambella, Vigo) 27 Designing Granular Solution Methods for Routing Problems with Time Windows Daniele Vigo∗ Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione “Guglielmo Marconi”, Università di Bologna, Italia, [email protected] Michael Schneider Fabian Schwahn Logistics Planning and Information Systems, TU Darmstadt, Germany, [email protected] [email protected] Abstract. The use of granular neighborhoods is one way to improve the run-time of localsearch-based metaheuristics for combinatorial optimization problems without compromising solution quality. So-called sparsification methods (SMs) are applied to reduce the neighborhoods to include only elements which are likely to be part of high-quality solutions. The goal of this work is to provide insights about the design of effective and efficient granular solution methods for routing problems with time windows. In extensive numerical experiments with a granular tabu search (GTS) for the vehicle-routing problem with time windows (VRPTW), we find that SMs using reduced-cost values based on the solution of a linear relaxation of the original problem outperform standard SMs. We also investigate other useful issues such as including additional depot arcs into the restricted arc set, inserting the arcs involved in the best move selected and those of the current best-known solution. Finally, dynamically altering the size of the restricted arc set can be used to successfully diversify and intensify the search, which has a significant positive effect on solution quality. The usefulness of the gained insights about the design of granular solution methods is demonstrated by the performance of the developed GTS for VRPTW, which obtains state-of-the-art results and reaches a considerable computational efficiency. More precisely, with an average runtime of three seconds on a standard desktop computer, our GTS proves to be the fastest method in the literature that is able to find the best-known cumulative number of vehicles of 405 on the well-known Solomon VRPTW instances. References [1] G. Desaulniers, O. Madsen, and S. Røpke. The vehicle routing problem with time windows. In P. Toth and D. Vigo, editors, Vehicle routing: Problems, methods, and applications, volume 18 of MOS-SIAM Series on Optimization, chapter 5, pages 119–159. SIAM, 2nd edition, 2014. [2] P. Toth and D. Vigo. The granular tabu search and its application to the vehicle-routing problem. INFORMS Journal on Computing, 15(4):333–346, 2003. Data Envelopment Analysis (invited by Carosi) Tuesday 8, 9:00-10:30 Sala Gerace 28 Data Envelopment Analysis (Carosi) 29 Global public spending efficiency in Tuscan municipalities Giovanna D’Inverno∗ IMT Institute for Advanced Studies Lucca, Italy, [email protected] Laura Carosi Department of Economics and Management, University of Pisa, Italy, [email protected] Letizia Ravagli IRPET - Istituto Regionale per la Programmazione Economica della Toscana, Italia, [email protected] Abstract. In this paper, a Data Envelopment Analysis is performed to study the efficiency of Tuscan municipalities’ public expenditure. For five strategic functions of Tuscan municipalities, a non-aggregate analysis is first run and then a composite indicator is proposed to analyze the overall expenditure composition of each municipality and to evaluate the global spending efficiency. As the municipal size has been one of the most discussed issues in the political debate, this study aims at investigating whether the efficiency of public expenditure is really affected by the municipal dimension. References [1] Afonso, A., Schuknecht, L., Tanzi, V., Public Sector Efficiency: An International Comparison, Public Choice, 123 (3-4), (2005), pp. 321-347. [2] Afonso, A., Fernandes, S., Assessing and explaining the relative efficiency of local government, Journal of Socio-Economics, 37 (5), (2008), pp. 1946-1979. [3] Athanassopoulos, A. D., Triantis, K., Assessing aggregate cost efficiency and the policy implications for Greek local authorities, INFOR: Information Systems and Operational Research, 36 (3), (1998), pp. 66-84. [4] Balaguer-Coll, M. T., Prior, D., Tortosa-Ausina, E., On the determinants of local government performance: a two-stage nonparametric approach, European Economic Review, 51 (2), (2007), pp. 425-451. [5] Banker, R. D., Charnes, A., Cooper, W. W., Models for the estimation of technical and scale inefficiencies in Data Envelopment Analysis, Management Science, 30 (9), (1984), pp. 1078-1092. [6] Bowlin, W., Measuring performance: An introduction to data envelopment analysis (DEA), Journal of Cost Analysis, Fall 1998, (1998), pp. 3-27. [7] Charnes, A., Cooper, W., Rhodes, E., Measuring the efficiency of decision making units, European Journal of Operations Research, 2 (6), (1978), pp. 429-444. [8] Da Cruz, N. F., Marques, R. C., Revisiting the determinants of local government performance., Omega, 44, (2014), pp. 91-103. [9] De Borger, B., Kerstens, K., Cost efficiency of Belgian local governments: a comparative analysis of FDH, DEA, and econometric approaches, Regional Science and Urban Economics, 26 (2), (1996), pp. 145-170. [10] Dyson, R. G., Allen, R., Camanho, A. S., Podinovski, V. V., Sarrico, C.S., Shale, E. A., Pitfalls and protocols in DEA, European Journal of Operational Research, 132, (2001), pp. 245-59. Data Envelopment Analysis (Carosi) 30 [11] Farrell, M., The measurement of productive efficiency, Journal of the Royal Statistical Society. Series A (General) 120, (1957), pp. 253-281. [12] Lo Storto, C., Evaluating technical efficiency of Italian major municipalities: a Data Envelopment Analysis model, Procedia Social and Behavioral Sciences, 81, (2013), pp. 346-350. [13] Loikkanen, H. A., Susiluoto, I., Cost efficiency of Finnish municipalities in basic service provision 1994-2002, Urban Public Economics Review, 4, (2005), pp. 39-64. [14] Nijkamp, P., Suzuki, S., A generalized goals-achievement model in data envelopment analysis: an application to efficiency improvement in local government finance in Japan, Spatial Economic Analysis, 4 (3), (2009), pp. 249-274. [15] Prieto, A. M., Zofı́o, J. L., Evaluating effectiveness in public provision of infrastructure and equipment: the case of Spanish municipalities, Journal of Productivity Analysis, 15 (1), (2001), pp. 41-58. [16] Sampaio de Sousa, M., Stosic, B., Technical efficiency of the Brazilian municipalities: Correcting nonparametric frontier measurements for outliers, Journal of Productivity Analysis, 24, (2005), pp. 157-181. [17] Worthington, A. C., Cost efficiency in Australian local government: a comparative analysis of mathematical programming and econometric approaches, Financial Accountability & Management, 16 (3), (2000), pp. 201-223. [18] Zhou, P., Ang, B. W., Zhou, D. Q., Weighting and aggregation in composite indicator construction: a multiplicative optimization approach, Social Indicators Research 96 (1), (2010), pp. 169-181. Data Envelopment Analysis (Carosi) 31 Standing on the Shoulders of Giants: Benchmarking the Water Industry Rui Cunha Marques∗ CESUR-IST, University of Lisbon, Portugal, [email protected] Francisco Silva Pinto CESUR-IST, University of Lisbon, Portugal, [email protected] Abstract. There are several studies that aim at evaluating the performance of those utilities that provide water and wastewater services. However, none has assessed the largest (in terms of customers served) utilities in the world. This paper quantifies the efficiency of more than 70 of the largest water and wastewater utilities around the world. Relative efficiency scores obtained by Data Envelopment Analysis (DEA) indicate that sub-optimal performance is evident. Therefore, an aim of this work is to obtain benchmarks for inefficient utilities. Introduction Since the development of the first Data Envelopment Analysis (DEA) models (arguably in the late 70s) that this type of methods, using linear programming for assessing the productive efficiencies of operating units has been widely used [1]. The same applies to the number of studies that aim at evaluating the performance of those utilities that provide water and wastewater services. However, none has assessed the largest (in terms of customers served) utilities in the world. Indeed, mostly in regulated water industries, different benchmarking approaches (e.g., yardstick competition, sunshine regulation) have been applied for different purposes [2]. This paper quantifies the efficiency of more than 70 of the largest water and wastewater utilities around the world, including countries, such as Australia, Brazil, Chile, Germany, France, Japan, Spain, United Kingdom, and the United States. Relative efficiency scores obtained by Data Envelopment Analysis (DEA) indicate that sub-optimal performance is evident in some utilities. Therefore, an aim of this work is to determine the efficiency of each unit in order to obtain benchmarks for inefficient utilities. The weights provided for particular inputs and outputs are used to find the contributions of particular criteria to the achieved score, enabling to determine the critical points of the assessed utilities. We attempt to identify determinants of performance, so these also have implications for yardstick comparisons. Methodology In this paper we selected multiple inputs and outputs in order to generate a pool from which we could build several models. Additionally, the models used vary not only in the inputs and outputs selected, but also in the returns to scale assumptions, meaning that both variable and constant returns to scale were employed. Nonetheless, due to the water supply and wastewater services nature, only input oriented models were used. Data Envelopment Analysis (Carosi) 32 The main inputs used were labor (nr. of employees), operating and maintenance costs (e), labor costs (e), capital costs (e), water supply and wastewater network length (kilometer) and the undesirable output leakage (m3 ). The main outputs used were volume of treated wastewater (m3 ), volume of water supplied (m3 ), water connections (nr.), wastewater connections (nr.), volume of water billed (m3 ). Furthermore, important explanatory factors, or in a different way characteristics, were considered as yje population density, the management model, the water sources and energy costs. Conclusions Due to the diversity of models run, this work was able to assess important determinants of performance, which by having important implications in the water sector allows it to be policy relevant. However, by comparing the largest utilities from different contexts, some cautions were taken, and important explanatory factors considered. References [1] Emrouznejad, A., Parker, B. R., & Tavares, G. (2008). Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA. Socio-economic planning sciences, 42(3), 151-157. [2] Marques R.C., Simões P., & Pires J.S. (2011). Performance benchmarking in utility regulation: the worldwide experience. Polish Journal of Environmental Studies, 20(1),125-132. Data Envelopment Analysis (Carosi) 33 Identifying cost and efficiency drivers in WWTPs through a DEA model with undesirable outputs Laura Carosi∗ Department of Economics and Management, University of Pisa, Italy, [email protected] Andrea Guerrini Department of Management, University of Verona, Italy, [email protected] Fabrizio Mancuso Ingegnerie Toscane srl, Italia, [email protected] Giulia Romano Department of Economics and Management, University of Pisa, Italy, [email protected] Abstract. Many scholars have focused on studying the performance of water companies (e.g. (e.g. Anwandter and Ozuna, 2002; Byrnes et al., 2010; Saal and Parker 2001). However, few have analyzed the efficiency and its determinants of wastewater treatment plants (WWTPs) (but see Rodriguez-Garcia et al. 2011; Hsiao et al., 2012). The current paper observes the main efficiency drivers of 139 WWTPs located in Italy, through a two stage DEA method. The applied DEA model accounts also for some undesirable outputs. The main provided evidences demonstrate that some environmental and operating variables exert a relevant impact on efficiency. References [1] Anwandter, L., Ozuna T. Jr. Can public sector reforms improve the efficiency of public water utilities?. Environment and Development Economics 7.04 (2002): 687-700. [2] Byrnes, J., Crase, L., Dollery, B., R. Villano, The relative economic efficiency of urban water utilities in regional New South Wales and Victoria. Resource and Energy Economics 32.3 (2010): 439-455. [3] Hernández-Sancho, F., M. Molinos-Senante, and R. Sala-Garrido. Techno-economical efficiency and productivity change of wastewater treatment plants: the role of internal and external factors. Journal of Environmental Monitoring 13.12 (2011): 3448-3459. [4] C.K. Hsiao, C.C. Yang, Performance measurement in wastewater control? Pig farms in Taiwan, WIT Transactions on Ecology and the Environment 103 (2007): 467-474. [5] Masotti L., Depurazione delle acque. Tecniche ed impianti per il trattamento delle acque di rifiuto, (2001), Calderini. [6] Metcalf & Eddy, Ingegneria delle Acque Reflue - Trattamento e riuso 4/ed, McGraw-Hill (2006), ISBN: 9788838661884 [7] Saal, D. S., Parker D., Productivity and price performance in the privatized water and sewerage companies of England and Wales. Journal of Regulatory Economics 20.1 (2001): 61-90. Equilibrium Models: Theory and Application Features (invited by Daniele and Scrimali) Monday 7, 15:30-17:00 Sala Gerace 34 Equilibrium Models: Theory and Application Features (Daniele, Scrimali) 35 Variational inequality approach to random Cournot-Nash principle Annamaria Barbagallo∗ Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Italy, [email protected] Paolo Mauro Department of Mathematics and Computer Science, University of Catania, Italy, [email protected] Abstract. In this talk, we study a stochastic variational inequality that models the oligopolistic market equilibrium problem in conditions of uncertainty. In particular, a random CournotNash equilibrium condition is given and the equivalence with the stochastic variational inequality is proved. Moreover, some existence results for the random equilibrium solutions are showed. Making use of the infinite dimensional Lagrange duality theory, the existence of Lagrange multipliers associated to the problem is obtained. At last, a numerical example is discussed. References [1] Barbagallo, A., Mauro, P., A stochastic variational inequality for random Cournot-Nash principle, submitted. Equilibrium Models: Theory and Application Features (Daniele, Scrimali) 36 A variational inequality model for a natural gas supply chain Giorgia Oggioni∗ Department of Economics and Management, University of Brescia, Italy, [email protected] Elisabetta Allevi Department of Economics and Management, University of Brescia, Italy, [email protected] Abstract. Gas is sold according to two main methods in today markets: long-term contracts (LTCs) and hub pricing system. LTCs, developed at the origin of the industry, are linked to crude or oil products. In the more recent hub pricing system, gas is traded, every day, on a spot market that determines prices and volumes on the short term. In this paper, we propose a variational inequality model for a natural gas supply chain where producers, mid-streamers and consumers can sell and buy gas through LCTs or/and on spot market. The model accounts for the seasonality of gas demand and storage. Application of variational inequality theory to gas market Spatial equilibrium models describe complex spatially distributed systems where many players interact all together (see e.g. [6]). The theory of variational inequalities (see [4] and [5]) facilitates the formulation of equilibrium problems that describe the interactions of several agents whose choices are subject to technical and economic constraints. The spatial control of such systems requires to consider not only the reactions of separate markets (or agents), but also technical constraints such as production capacity and network limits. Variational inequalities have been developed and adopted to study the equilibrium behavior of decision makers in many research fields such as spatial equilibrium models, supply chain management, financial and transportation networks. In this paper, taking as reference the equilibrium models already developed in the literature (see [1], [2] and [3]), we propose a variational inequality model to describe the natural gas supply chain. The supply chain for natural gas starts with producers that extract gas from reservoirs. The next step, operated by mid-streamers, is the gas transportation from production sites to either citygate or directly to the consumption sectors that can be classified into residential, industrial sector and power generation. In this supply chain, we assume that gas can stored at the citygate level. In particular, storage operators take advantage of seasonal arbitrage by buying and injecting gas into storage in the low demand season (summer) and then selling it to consumers in the high demand season (winter). In order to provide a realistic representation of the natural gas market, we account for both long-term contracts (LTCs) and hub pricing system that represent the two methods adopted to sell gas in today markets. LTCs, developed at the origin of the industry, are characterized by the Take or Pay (TOP) quantity clause that obligates the buyer to take a certain quantity of gas or to pay for it. The price of LTCs is indexed to the price of some crude or oil products. On the other side, the hub pricing approach, introduced in the nineties in the US and UK, is now developing in Europe. In Equilibrium Models: Theory and Application Features (Daniele, Scrimali) 37 this system, gas is traded on a spot market that, every day, determines prices and volumes on the short term. The objective of the paper is to develop a variational inequality model that can describe the interactions of the players operating in the described gas supply chain taking into account different assumptions on price systems and demand levels. References [1] Abada, I., Ehrenmann, A., Smeers, Y., Endogenizing long-term contracts in gas market models, CORE Discussion Paper 36 (2014). [2] Egging, R.G., Gabriel, S.A., Examining market power in the European natural gas market, Energy Policy 34 (2006) 2762–2778. [3] Egging, R., Gabriel, S.A., Holz, F., Zhuang, J., A complementarity model for the European natural gas market, Energy Policy 36 (2008), 2385–2414. [4] Facchinei, F., Pang, J.S., Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, Berlin (two volumes), 2003. [5] Nagurney, A., Network Economics: A Variational Inequality Approach. Kluwer, Dordrecht, 1999. [6] Samuelson, P.A., Spatial Price Equilibrium and Linear Programming, The American Economic Review 42 (1952), 283-303. Equilibrium Models: Theory and Application Features (Daniele, Scrimali) 38 Evolutionary variational inequalities and coalitional games in sustainable supply chain networks Laura Scrimali∗ Dipartimento di Matematica e Informatica, Università di Catania, Italia, [email protected] Abstract. In this paper an equilibrium model of a sustainable supply chain coalitional game is developed. The supply chain network structure consists of three layers of decision-makers, in the case where prices and shipments evolve in time. Equilibrium conditions corresponding to a Cournot-Nash equilibrium are derived and an evolutionary variational inequality formulation of the coalitional game is established. The existence of solutions is also discussed. Introduction Supply chains have become an essential element to operational efficiency of companies. Supply chain management encompasses not only the design and planning of supply chain activities, but also integration with marketing; product and process engineering; accounting and finance. Many decisions are made at a different level of the supply chain and affect all the levels. Therefore, the challenge is how planning strategic decisions in multi-echelon supply chains. In the increasingly competitive economic environment, the basic question is then whether it is better making a choice independently or integrating with some (or all ) levels in multi-echelon supply chains. Decision-making in a supply chain network can be performed in a decentralized or a centralized way. In the decentralized case suppliers set its own price, manufacturers decide the order quantity and the wholesale price, while retailers charge the final price to the consumers; in the centralized or integrated case, there exists a central authority responsible for decision-making (see [1]). Inspired by [2], our research considers a three-echelon supply chain coalitional game in a duopolistic market consisting of suppliers, manufacturers and retailers in an time-dependent setting. Our paper focuses on the formation of intra-chain coalition and resulting value allocation. Supply chains are mutually competing in the market; hence the value of a coalition will depend on the outcome of inter-chain competition. The outcome allocation may then induce members of a coalition to deviate and join another coalition that promises higher payoff. The formation and deviation of chain coalitions will continue until a stable Nash equilibrium is reached. We study the supply chain model in the case when prices and shipments evolve in time (see [3]). We observe that the physical structure of the supply chain network may remain unchanged, but the phenomena which occur in the network change over time. The connection with previous papers describing static supply chain models can be made considering that the static configuration only represents a fixed moment of an evolving real phenomenon. Therefore, studying the static case can be considered only a first step to understand reality; whereas the evolutionary framework studies the evolution in time and provides curves of equilibria that allow us to grasp further features of the model. Equilibrium Models: Theory and Application Features (Daniele, Scrimali) 39 The concern for environmental quality has become one of the most important issues for companies, also due to global warming and associated security risks regarding energy supplies. Thus, we assume that suppliers, manufacturers and retailers are multicriteria decision-makers with the environmental criteria weighted distinctly by the different decision-makers. Starting from the decentralized model, we give the equilibrium conditions and the multicriteria decision-making problems of all levels of the supply chains. Then, we study the coalition formation providing the equilibrium conditions and present an evolutionary variational inequality formulation of the coalitional game. The existence of solutions is also discussed. Numerical examples are presented for illustration purposes. References [1] Mahdiraji, H.-A., Govindan, K., Zavadskas, E.K., Hajiagha, S.H.R, Coalition or decentralization: a game-theoretic analysis of a three-echelon supply chain networks, Journal of Business Economics and Management 15 (3) (2014), 460–485. [2] Lin, C.-C., Hsieh, C.-C., A cooperative coalitional game in duopolistic supply-chain competition, Networks and Spatial Economics 12 (2012), 129–146. [3] Daniele, P., Evolutionary variational inequalities and applications to complex dynamic multilevel models, Transportation Research Part E 46 (2010), 855–880. Exact Methods for Routing Problems (invited by Gambella and Vigo) Monday 7, 15:30-17:00 Sala Seminari Est 40 Exact Methods for Routing Problems (Gambella, Vigo) 41 An exact algorithm for the mixed capacitated general routing problem with time windows Claudio Ciancio∗ Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy, [email protected] Demetrio Laganà Roberto Musmanno Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy, [email protected] [email protected] Francesca Vocaturo Department of Economics, Statistics and Finance, University of Calabria, Italy, [email protected] Abstract. In general routing problems over mixed graphs, the aim is to determine a leastcost traversal of a subset of edges, arcs and nodes. In this field, we analyse the problem under vehicle capacity and time windows constraints. It can be transformed into an equivalent node routing problem and solved exactly. Problem Statement The general routing problem (GRP) arises in arc routing contexts where it is necessary not only to traverse some arcs or edges of the graph representing the street network, but also some isolated nodes. Many authors focused on the uncapacitated GRP (see, e.g., [2]). However, the capacitated counterpart is attracting more and more contributions from the scientific community (see, e.g., [3]). We deal with a capacitated GRP defined over a mixed graph. Specifically, we tackle the Mixed Capacitated General Routing Problem with Time Windows (MCGRPTW). It consists of determining a set of least-cost vehicle routes that respect some requirements. Each route starts and ends at the depot, which is the node at which a fleet of homogeneous vehicles is based. There exists a subset of nodes, arcs and edges with strictly positive demand: they must be serviced exactly once while respecting prespecified time windows. Finally, the total demand collected by a vehicle cannot exceed its capacity. Solution Approach The transformation of routing problems in equivalent ones defines a solution approach quite often used in the scientific literature (see [1] for an example). In this work, we transform the MCGRPTW into an equivalent node routing problem over a directed graph. Thus, we solve the equivalent problem by using a branch-priceand-cut algorithm. Branch-price-and-cut is a variant of branch-and-bound, with bounds obtained by solving linear programs and performing column-and-cut generation. Our algorithm, which uses a set-partitioning-based formulation, combines beneficial ingredients from the literature. In order to evaluate the performance of the solution approach, we derived MCGRPTW instances from the Capacitated Arc Exact Methods for Routing Problems (Gambella, Vigo) 42 Routing Problem with Time Windows (CARPTW, [4]-[5]). In a series of preliminary tests, we solved both CARPTW and MCGRPTW instances. Numerical results show that the approach is effective. References [1] Blais, M., Laporte, G., Exact solution of the generalized routing problem through graph transformations, Journal of the Operational Research Society 54 (2003), 906–910. [2] Corberán, A., Mejı́a, G., Sanchis, J.M., New results on the mixed general routing problem, Operations Research 53 (2005), 363–376. [3] Irnich, S., Laganà, D., Schlebusch, C., Vocaturo, F., Two-phase branch-and-cut for the mixed capacitated general routing problem, European Journal of Operational Research, 243 (2015), 17–29. [4] Labadi, N., Prins, C., Reghioui, M., GRASP with path relinking for the capacitated arc routing problem with time windows, in: Advances in Computational Intelligence in Transport, Logistics, and Supply Chain Management (Fink, A., Rothlauf, F., Eds), Studies in Computational Intelligence Vol. 144, 111–135. Springer, 2008. [5] Wøhlk, S., Contributions to Arc Routing, Ph.D. dissertation. Syddansk Universitetsforlag (University Press of Southern Denmark), 2006. Exact Methods for Routing Problems (Gambella, Vigo) 43 The probabilistic orienteering problem Carlo Filippi∗ Dipartimento di Economia e Management, Università di Brescia, Italia, [email protected] Enrico Angelelli Claudia Archetti Michele Vindigni Dipartimento di Economia e Management, Università di Brescia, Italia, [email protected] [email protected] [email protected] Abstract. The probabilistic orienteering problem is a stochastic variant of the orienteering problem where each node is available for visit only with a certain probability and a recourse action, which consists in skipping the absent nodes, is considered. We discuss the relevance of the problem and formulate it as a linear integer stochastic model. We develop a branch-and-cut approach and several matheuristic methods. Results of extensive computational tests on instances with up to 100 nodes are given. Problem description We consider a time-constrained, selective, stochastic routing problem. Consider a graph where nodes represent customers; a service courier, located at a distinguished origin, visits a subset of customers and ends up to a distinguished destination. For every visited customer a prize is collected; a time is needed for every traversed arc and a maximum total time is given. The classic orienteering problem (OP) consists in finding a path from the origin to the destination that maximizes the sum of collected prizes while respecting the maximum time constraint [1]. In our problem, each customer requires service only with a specified probability, and the sequence of decisions and information is as follows. In the first stage, an a priori path from the origin to the destination, covering a subset of customers within the specified total time, has to be designed. In the second stage, information about the customers to be visited is available, and the customers are visited in the same order as they appear in the a priori path by simply skipping the absent ones. The profit of the a priori path is defined as the difference between the sum of collected prizes among the visited customers and the cost of travelling the path skipping the absent customers, where we assume that time and cost are proportional and satisfy the triangle inequality. Notice that the actual profit is revealed only after the design of the a priori tour. The probabilistic orienteering problem (POP) consists in finding an a priori path of maximum expected profit with respect to all possible subsets of nodes to be visited and such that the total time of the path cannot exceed the maximum time limit. The POP is both a stochastic version of the OP and an extension of the probabilistic travelling salesman problem [2]. Model and results At an abstract level, the POP can be formulated as min{C · E[T (p)] − E[P (p)] : p ∈ P}, Exact Methods for Routing Problems (Gambella, Vigo) 44 where P is the set of all elementary paths from the origin to the destination of total duration not exceeding a deadline Dmax ; C is the unit time cost; E[T (p)] is the expected time required by path p (depending on the recourse action); E[P (p)] is the expected prize collected along path p. Set P and function E[P (p)] can be expressed using linear relations on binary variables. Function E[T (p)] is piecewise linear and can be modelled by properly defined optimality cuts [3]. This leads to a two-index, deterministic equivalent MILP formulation for the POP, with an exponential number of constraints. We adapt and extend to the POP the branch-and-cut method developed in [4] for the probabilistic TSP. In particular, a basic version and three variants based on different branching rules are considered. Moreover, we develop a heuristic scheme for the POP which is based on the exact branch-and-cut algorithm, where heuristic branching and variable fixing are used to exclude search regions where an optimal solution is unlikely to exist. We tested the exact method and the matheuristic variants on a set of 264 instances derived from the TSPLIB95 library, with up to 100 nodes. The time limit has been set to 7200 seconds for all algorithms. The best exact method is able to prove optimality in 75% of the instances, with an average optimality gap of 3.70% and an average computing time of 1986 seconds. The matheuristics have an overall average error (with respect to the best known solution) below 1%, with an average computing time of 710 seconds, and an average time to the best solution of 143 seconds. References [1] Vansteenwegen, P., Souffriau, W., van Oudheusden, D., The orienteering problem: A survey, European Journal of Operational Research 209 (2011), 1–10. [2] Jaillet, P., A priori solution of a traveling salesman problem in which a random subset of the customers are visited, Operations Research 36 (1988), 929–936. [3] Laporte, G., Louveaux, F.V., The integer L-shaped method for stochastic integer programs with complete recourse, Operations Research Letters 13 (1993), 133–142. [4] Laporte, G., Louveaux, F.V., Mercure, H., A priori optimization of the probabilistic traveling salesman problem, Operations Research 42 (1994), 543–549. Exact Methods for Routing Problems (Gambella, Vigo) 45 A rich maritime extension of the Vehicle Routing Problem Alberto Santini∗ DEI, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy, [email protected] Stefan Røpke DTU Management Science, Produktionstorvet 424, 2800 Kgs. Lyngby, Denmark, [email protected] Christian E. M. Plum Maersk line, Esplanaden 50, 1098 København, Denmark, [email protected] Abstract. We present the Feeder Network Design Problem, a rich extension of the Vehicle Routing Problem that arises in maritime applications, when designing the schedules of liner container vessels serving regional markets as feeders to/from a central hub, picking up and delivering cargo to peripheral ports. We devise a branch-and-price algorithm and present the results of computational experiments based on real-life data. Container liner shipping is the main freight transportation service used to move large quantities of cargo over long distances. As opposed to tramp shipping — where individual ships operate in order to fulfil the transportation requests available at a certain time, and decide which one is more convenient for them to accept — liner ships operate along fixed routes and according to a published schedule. The Feeder Network Design Problem (FNDP) arises when planning the routes of liner container vessels in regional feeder networks. Intercontinental container routes are operated by big vessels that only call the main ports. These ports are called hubs and are characterised by an extensive infrastructure that makes them suitable to operate with large quantities of containers and to efficiently load and unload extremely big vessels. Networks of smaller vessels load the containers delivered at the hubs and deliver them to smaller ports in a particular region. Similarly, they collect containers at the small ports and unload them at the hubs, where they will later be shipped to their final destination on the intercontinental routes. In short, liner shipping is organised in hub-and-spoke networks. As the feeder routes start and end at the main regional hub, forming a closed loop, we refer to them as rotations. Since the operator issues a public timetable that includes the ports each vessel will call and the corresponding day and time, it is clearly convenient that such a schedule be periodic of a multiple of one week, so that each port is called on a certain day of the week that does not change. This requirement strongly constrains the length of a rotation. The main aim of the FNDP is then to come up with a certain number of rotations such that each port in the region is visited by one or more vessels and its demand is fulfilled. The constraints the operator is faced with are the limited capacity and number of vessels available, the fact that the rotations must have a fixed length of time, that ports observe certain closing time windows (e.g. most ports are closed for operations at night), that certain goods might have a maximum time span during which they can travel (e.g. perishable goods) and that ports have a maximum draught and therefore not every vessel can enter every port. Exact Methods for Routing Problems (Gambella, Vigo) 46 The quantity to minimise is the total cost of rotations. Costs include port taxes and calling fees, and the vessel’s bunker cost. This latter cost is particularly important, since it is greatly impacted by the steaming speed of a vessel, with the relationship between the two being approximately cubic: s 3 cost(s) = ∗ · cost(s∗ ) s where s is the steaming speed and s∗ and cost(s∗ ) are the design speed and cost at design speed, which are known in advance. A solution must therefore establish an optimal speed profile for each rotation. We give a MIP formulation of the problem and then develop a branch-and-price algorithm, based on the decomposition of the MIP formulation, where the master problem is a set partitioning problem and the pricing subproblem is an elementary shortest path problem with resource constraints on a time-expanded graph. At each node, a relaxation of the subproblem is first solved heuristically with constructive heuristics and various labelling algorithm that either work on a reduced graph, or employ the state-space relaxation technique introduced in [1]. If the heuristic algorithms fail to generate promising solutions, the pricing problem is solved to optimality with a label-setting algorithm. The columns introduced by solving the relaxation of the subproblem might be infeasible, and are therefore eliminated during branching. We tested our approach on the Linerlib instances [2], which are based on reallife data provided by the largest container shipping operator in the world. When running the computational experiments, we also consider variations of our problem in which not all ports need to be served (and a penalty is paid for not serving a port) and where vessels can be chartered out. Computational results on the Baltic and West Africa scenarios show that our approach is competitive in terms of computational time and finds very high quality solutions. References [1] Nicos Christofides, Aristide Mingozzi, and Paolo Toth. State-space relaxation procedures for the computation of bounds to routing problems. Networks, 11(2): 145-164, 1981. [2] Berit D Brouer, J Fernando Alvarez, Christian EM Plum, David Pisinger, and Mikkel M Sigurd. A base integer programming model and benchmark suite for liner-shipping network design. Transportation Science, 48(2): 281-312, 2013. Green ICT (invited by Mattia) Thursday 10, 9:00-10:30 Sala Riunioni Ovest 47 Green ICT (Mattia) 48 Energy-aware joint management of networks and Cloud infrastructures Bernardetta Addis∗ LORIA (CNRS UMR 7503) - INRIA Nancy Grand Est, Université de Lorraine, France, [email protected] Danilo Ardagna Antonio Capone Giuliana Carello Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] [email protected] [email protected] Abstract. Geographical distribution of Service Centers (SCs) offers many opportunities for reducing enviromental impact. Assigning workload to DCs in different locations allows to exploit different renewable energy availability and different energy prices. Data application are moved through the network, whose energy consumption must be accounted for, as well. We present an optimization model which jointly optimizes workload allocation and network routing, taking into account the use of brown and green energy and the required quality of service, so as to to minimize the energy costs. Introduction Cloud services have been massively adopted in the recent years. As a consequence, their CO2 emissions are rapidly increasing and their resources must be carefully managed to meet an ever increasing demand. So far, Service Centers (SCs) and communication networks have been managed independently, but the new generation of Cloud systems are based on a strict integration of SCs and networking infrastructures, and they may greatly benefit from geographical distribution both from the resource allocation and the energy cost perspective. The geographical distribution of the computing facilities offers many opportunities for optimizing energy consumption and costs by assigning the computational workload to different sites, taking into account different availability of renewable energy sources and different time zones and hourly energy prices. The network is therefore needed to move the application related data and its energy consumption must be accounted for, as well. We present an optimization model which jointly optimizes workload allocation and network routing, taking into account the use of brown and green energy and the required quality of service. The goal is to minimize the overall energy impact. Problem description and contribution We consider a provider operating multiple SCs distributed over multiple physical sites. Physical sites are connected by a wide area network. The SCs must support the execution of customer applications, classified in request classes, based on their computing demands, workload intensities, and bandwidth requirements. The applications are served by Virtual Machine (VMs). Each SC has a set of different types of VMs that can serve incoming requests. Each request has bandwidth and CPU requirements, while VMs and SCs have capacity limits. Requests change along time. VMs can be switched off to save energy. Each VM is characterized by Green ICT (Mattia) 49 en energy consumption for running and an energy required for turning it on or off. The application workload is collected at one SC but it can be forwarded to another SC so as to exploit cheaper or green energy where it is available. However, this has an impact on the network energy consumption. Further, network connections are capacitated. The availability of renewable resources and different energy prices in different sites are considered. Given the application workload profiles, the problem consists in assigning application workload to SCs and in switching on/off VMs in order to jointly minimize SCs and network energy costs. We proposed a MILP model for the problem, which is solved on a daily basis: a prediction of the application workload is considered and requests are assigned to SCs for the next 24 hours. The considered 24 hours horizon is discretized and divided into time bands. We present a set of numerical results on a realistic case study that considers the real worldwide geographical distribution of SCs of a Cloud provider and the variations of energy cost and green energy availability in different world regions. Moreover, we show some results to characterize the scalability of the proposed optimization model and the sensitivity to workload and green energy prediction errors. We propose two representations for the network, a high level approximated model and a detailed one. We show that the energy consumption obtained with the approximated version is very close to the one obtained with the detailed network representation and that the approximated version requires significantly smaller computational effort. Green ICT (Mattia) 50 Heuristics for an energy-aware management problem in cellular networks Giuliana Carello∗ Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] Bernardetta Addis LORIA (CNRS UMR 7503) - INRIA Nancy Grand Est, Université de Lorraine, France, [email protected] Mattia Raco Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] Abstract. Energy consumption is nowadays a global concern. Telecommunication networks have a twofold impact on energy issues. They may help in reducing the consumption of other human activities, but they are highly energy consuming. As they are usually overdimensioned to deal with traffic peaks, their power consumption can be reduced exploiting traffic variations without losing quality of service. We consider an energy aware problem in wireless networks which accounts for both long term decisions and operational ones. We developed and evaluate several heuristics, which are tested on a large set of instances. Problem description The concern on global warming and carbon footprint, has been constantly increasing in recent years. Information and Communication Technology (ICT) has a twofold impact on environmental issues: on the one hand it may help in reducing energy consumption of other human activities, on the other it is itself highly energy consuming. Among ICT services, telecommunication networks contribute significantly to the environmental impact. However, as telecommunication networks are usually overdimensioned to face traffic peaks, their energy consumption can be reduced via energy-aware planning and management, exploiting traffic variations. Network device power can be tuned to meet the real traffic demand and devices can be switched off when unused to reduce network consumption in low traffic periods. We consider a problem of jointly planning and managing an energy-aware wireless network [1], [2]. From the long term point of view, a set of Base Stations (BSs) must be installed among a set of candidate sites and equipped with devices in order to serve the traffic demand of a given set of Test Points (TPs). Each device has a different cost and a different coverage area. The TP traffic demand varies cyclically on a daily basis. The time horizon is divided into time slots. Thus, from the operational point of view, each TP must be assigned to a BSs in each time period, but it must be assigned to the BS that offers the highest quality of signal. Device emitted power levels must be tuned in each time period: different levels correspond to different coverage areas, but also to different energy consumptions. The aim is to guarantee total area coverage and to serve TPs’ demand at minimal overall cost. The cost takes into account both deployment and installation expenditure (CAPEX) and network operating and maintenance costs (OPEX). The OPEX cost accounts for the Green ICT (Mattia) 51 energy consumption of the BSs. Indeed, significant energy consumption reduction in operating the network can be achieved if operational energy issues are taken into account in the network design phase. The problem shares many features with dynamic facility location problems. Furthermore, coverage and closest assignment constraints are present. Proposed approaches We propose several heuristic approaches for the problem. Different greedy approaches exploiting the covering structure of the problem have been developed and tested. A local search step is proposed to improve the solution: the neighborhood is based on BSs location, while assignments and power tuning are decided solving an ILP subproblem. Further, constructive approaches exploiting ILP formulation of the problem have been developed. A heuristic concentration approach is proposed which solves the ILP formulation on a subset of candidate sites selected by the greedy algorithms. An alternative subset of candidate sites, on which the overall model is solved, is selected through a geographical decomposition based approach which exploits the position of candidate sites and TPs to decompose the problem into subproblems, each defined on a subarea. The local search refining phase is then applied on the obtained solutions. All the proposed approaches have been tested on realistic instances and their behavior is compared. References [1] Boiardi, S., Radio planning and management of energy-efficient wireless access networks, Ph.D. Thesis, Politecnico di Milano - Polytechnique de Montreal, 2014. [2] Boiardi, S., Capone, A., Sansò, B., Radio planning of energy-aware cellular networks Computer Networks 57(13) (2013), 2564 – 2577. Green ICT (Mattia) 52 Energy aware survivable networks Sara Mattia∗ Istituto di Analisi dei Sistemi ed Informatica, Consiglio Nazionale delle Ricerche, Italia, [email protected] Bernardetta Addis LORIA (CNRS UMR 7503) - INRIA Nancy Grand Est, Université de Lorraine, France, [email protected] Giuliana Carello Dipartimento di Elettronica, Informatica e Bioingegneria, Politecnico di Milano, Italia, [email protected] Abstract. We consider the problem of minimizing the energy consumption of a telecommunications network by selecting the edges to be activated. When a fault occurs, all edges can be temporarily activated and the consumption is negligible. A survivable routing of the demands is ensured by shared protection. We discuss, study and test different formulations of the problem. Introduction Telecommunications networks face two conflicting challenges: survivability and sustainability. Due to their growing importance in everyday life, they must guarantee service even in case of faults. To handle failure situations, they are usually overdimensioned with respect to the standard amount of traffic. This results in many network devices that consume energy, but are not used for most of the time, as they are needed only in case of failures. To have a sustainable energy consumption in a survivable network, one can consider to switch off the unused devices and to activate them temporarily only when needed. Moreover, a careful protection policy, that may require less backup capacity, can help limiting the capacity needed to ensure the routing, thus further reducing the energy consumption. Contribution We study the problem of managing survivable telecommunications networks, switching edges on and off in order to minimize the energy consumption. Unplittable flows, single edge failure scenarios and end-to-end shared protection are considered. That is, a set of point-to-point traffic demands must be routed using a single path for each demand. When a fault occurs, one edge of the network becomes unavailable and then some of the demands must be rerouted. Therefore, for every traffic demand two paths are selected: a working (or primary) path, which is normally used, and an edge-disjoint backup (or secondary) path, which is used when some edge of the working path fails. Backup capacity can be shared by demands not affected by the same failure and the energy consumption only depends on the edges supporting the primary paths. In fact, edges used only by secondary paths can be switched off and are temporarily activated only in case of faults. Since faults are unlikely to occur and they are supposed to be solved quickly, such edges will be switched off most of the time and their contribution to the energy consumption is negligible Green ICT (Mattia) 53 [1]. Despite its relevance, very few papers address this challenging problem [2] and most practical approaches rely on heuristics [3]. We investigate two formulations of the problem: a complete and a projected formulation, discussing valid inequalities and presenting a branch-and-cut algorithm. References [1] Addis, B., Capone, A., Carello, G., Gianoli, L.G., Sansò B., On the energy cost of robustness and resiliency in IP networks, Computer Networks 75, Part A (2014), 239 – 259. [2] Addis, B., Carello, G., Mattia, S., Energy aware survivable networks, Electronic Notes in Discrete Mathematics, Proceedings of the 7th International Network Optimization Conference (2015), to appear. [3] Haahr, J.T., Stidsen, T.R., Zachariasen, M., Heuristic methods for shared backup path protection planning, Proceedings of the 4th International Workshop on Reliable Networks Design and Modeling (2012), 712–718. Health Care 1 (invited by Cappanera and Tanfani) Monday 7, 15:30-17:00 Sala Seminari Ovest 54 Health Care 1 (Cappanera, Tanfani) 55 A simulation–based multiobjective optimization approach for health care services planning and management Massimo Roma∗ Dipartimento di Ingegneria Informatica, Automatica e Gestionale, SAPIENZA - Università di Roma, Italia, [email protected] Stefano Lucidi Dipartimento di Ingegneria Informatica, Automatica e Gestionale, SAPIENZA - Università di Roma, Italia, [email protected] Massimo Maurici Luca Paulon Dipartimento di Biomedicina e Prevenzione Laboratorio di Simulazione e Ottimizzazione dei Servizi del SSN Università degli Studi di Roma “Tor Vergata”, Italia, [email protected] [email protected] Francesco Rinaldi Dipartimento di Matematica, Università di Padova, Italia, [email protected] Abstract. In this work we propose a simulation–based optimization model for hospital services resource allocation. In particular, a discrete event simulation model has been constructed reproducing all the processes of interest. Then it is combined with a derivative–free multiobjective optimization method in order to obtain the optimal setting. The results obtained on the obstetrics ward of an Italian hospital are reported. This study also represents a starting point to investigate alternative settings, different from hospitalization, to care pregnant woman during the natural childbirth. Introduction In the last years, controlling health care costs while providing the best possible health outcomes became a more and more critical issue (see e.g. [1]). Hence the necessity to make available to hospitals top managers procedures for determining optimal resources allocation. Moreover, recently in many National Health Services (NHS), hospital financing has changed from budget oriented system to fee-for-service system. As a consequence of this change, hospitals strongly need an optimal planning of resources. In this framework, we consider in this work a simulation–based optimization model for resource planning of the emergency room and a specific ward of an hospital, aiming at optimizing the performances from both an economical and a clinical point of view, taking into account suitable restrictions. Discrete event simulation methods have been widely used over the last decade for modelling and analyzing health care systems performance (see e.g. [2]). More recently such methods have been often combined with heuristic optimization techniques in order to determine the optimal setting, namely the best “scenario” according to some performance criteria. In order to get better results, and due to the “black–box” nature of the problem in hand, in this work we propose the use of a Derivative–Free Multiobjective Optimization method for solving the resulting Mixed Integer Nonlinear Programming problem and we experiment it on a real– world problem arising from the management of the obstetrics ward of an Italian Health Care 1 (Cappanera, Tanfani) 56 hospital. In particular, the case study considers the optimal resource allocation of the emergency room and obstetrics ward of the Fatebenefratelli San Giovanni Calibita Hospital in Rome. It is one of the most important hospital of the Italian NHS in terms of number of childbirth cases. The study was carried out within the project “Business Simulation for Healthcare” (BuS-4H ). The services under study are the caesarean section without complications or comorbidities and the vaginal childbirth without complications (for a description of the case study see [3], where the single objective case is considered). The efficient management of the hospitalizations in an obstetrics ward is greatly important. In particular, the choice of the resources (number of beds, gynecologists, nurses, midwives and so on) to be employed strongly affects the management costs and the income as well as the quality of the services. The sources of the costs are several and mainly due to staff salaries and management of medical equipments and consumable goods. The income derives from the refunds through the NHS of the services delivered, namely caesarean sections and vaginal childbirth. In the allocation of the resources several constraints must be taken into account. They are structural constraints or deriving from clinical and regulatory needs. A crucial role is played by the rate of caesarean sections with respect to the overall childbirth. Indeed, due to the higher risk for mother or child in the case of caesarean delivery, the rate of caesarean sections should be as lowest as possible while the profit should be maximized. These are contrasting goals since the income for a cesarean section is considerably higher than the one for natural childbirth. Therefore the problem can be stated as a two–objectives Mixed Integer Nonlinear Programming problem (see also [4]). References [1] J. Y. Kim, P. Farmer, and M. E. Porter, Redefining global health-care delivery, Lancet, 382 (2013), 1060–1069. [2] M.M. Günal and M. Pitt, Discrete event simulation for performance modelling in health care: a review of the literature, Journal of Simulation, 4 (2010), 42–51. [3] S. Lucidi, M. Maurici, L. Paulon, F. Rinaldi, M. Roma, A derivative–free approach for a simulation–based optimization problem in healthcare, SAPIENZA–University of Rome, Department of Computer, Control and Management Engineering “Antonio Ruberti” Technical Reports, 15 (2014) 1-17. To appear in Optimization Letters. [4] L. Paulon, Mathematics in Health Care with Applications, Phd thesis, SAPIENZA–Università di Roma, 2013. Health Care 1 (Cappanera, Tanfani) 57 Assistance to palliative pediatric patients: simulation model Giorgio Romanin-Jacur∗ Dept. of Management and Engineering, University of Padova, Italy, [email protected] Arturo Liguori Francesca Menegazzo Dept. of Paediatrics, University of Padova, Italy, [email protected] Giorgia Mondin Dept. of Management and Engineering, University of Padova, Italy, [email protected] Abstract. We discuss a simulation model describing paediatric palliative assistance network behaviour and its use by patients. The model is absolutely general, but has been tested on Veneto Region (Italy), currently equipped with a network including paediatric hospital departments, a paediatric hospice and home integrated assistance. Different assistance politics were examined. As expected, politics increasing hospice and home integrated assistance for palliative patients, improve orderly patients assistance in hospitals and permit a remarkably reduced total cost. World Health Organization defines Paediatric Palliative Cares as an approach which improves patients’ and their families’ life quality in front of the problem connected with life-threatening and life-limiting diseases, through prevention and relief of suffering thanks to a rapid identification, an accurate evaluation and the treatment of pain and other physical, psycho-social and spiritual problems. Remark that, differently from adults, paediatric palliative cares may last longer and face alternative phases; paediatric subjects bear hospitalization with pain. Whenever a pediatric subject is stricken by an incurable disease, the patient prefers to live at home a life as normal as possible; therefore hospital admissions should be reduced; in the same time, many types of non clinical assistance should be provided, in coordination to obtain the best effect. The two residential admission types are: Home Integrated Assistance, where general practitioner, hospital specialized doctor, nurses and social assistants collaborate, and Hospice, where the family is admitted for short times and both specialized and palliative care personnel are present. A model, describing paediatric palliative assistance behaviour and its use by patients has been built. It reports about a general situation, in order to evidence critical points and possible deficiencies in the assistance network, to suggest remedial work and evaluate consequent effects. Once considered uncertainty an evaluative instead than an optimization model has been chosen. In the resulting stochastic simulation model palliative patients enter the system and move among resources at disposition until his/her death, according to adopted policy; orderly and palliative patients compete against one another to take hospital places. The simulation model is implemented by means of tool Arena, whose graphical representation is easily readable also by non expert people, like health professionals. The model has 40 blocks of the Create, Decide, Assign and Process type. The model is absolutely general, but has been tested on Veneto Region (Italy), currently equipped with a palliative paediatric assistance network including paediatric hospital departments, a paediatric hospice with 4 places and a home integrated assistance for 40 patients, in a period of 5 years, since all data about patients were at disposition. Three different assistance politics were examined: the first one addressing 50% patients to Health Care 1 (Cappanera, Tanfani) 58 hospitals, 40% to integrated home assistance and 10% to hospice; the second one addressing 10% patients to hospitals, 80% to integrated home assistance and 10% to hospice; the third future hypothetical politic addressing patients like the second one, but in the case resources were increased up to 8 hospice places and 80 home integrated assistance places. The simulation permitted to obtain the following results: number of orderly patients who do not succeed in entering a hospital department, number of palliative patients who do not succeed in entering a hospital department, number of palliative patients who do not succeed in entering the hospice and number of palliative patients who do not succeed in entering home integrated assistance. As expected, politics increasing hospice and home integrated assistance use for palliative patients improve orderly patients assistance in hospitals and permit a remarkably reduced total cost. Health Care 1 (Cappanera, Tanfani) 59 A Hybrid Simulation Framework to Manage Emergent and Elective Patient Flows and Hospital Bed Capacity Michele Sonnessa∗ Dipartimento di Economia, Università di Genova, [email protected] Paolo Landa Elena Tanfani Angela Testi Dipartimento di Economia, Università di Genova, [email protected] [email protected] [email protected] Abstract. This paper introduces a hybrid simulation framework able to assess the impact of organizational strategies intended to allocate inpatient beds among emergent and elective flows inside a hospital. The framework, based on a System Dynamics (SD) model and a Discrete Event Simulation (DES) model linked in a hybrid fashion, is able to forecast the hospital resource allocation over time-fluctuation of patient arrivals in the Emergency Department (e.g. in winter). Indeed, it can help to individuate improving strategies to allocate bed capacity among different departments, in order to reduce ED overcrowding. Problem description Emergency Department (ED) overcrowding together with waiting lists for elective surgery are two of the main topical issues in many publicly funded health systems. They are regularly reported in the newspaper headlines: it seems that health resources are not organized in a manner consistent with the true needs of citizens. What is still worse is that citizen’s needs are put in competition: elective patients, who are entitled to an intervention, could delay their intervention due to unpredictable excessive arrivals of patients to ED which need an emergent hospital admission to be further needed. Besides, the boarding of emergent patients in ED waiting to be admitted in hospital inpatient wards is a major reason of ED overcrowding. In recent years, two approaches to face the ED overcrowding have been suggested. The first is intended to reduce the number of patients addressing the ED inappropriately, while the second is directed to facilitate early discharges from inpatient wards and/or block the elective patient arrivals to smooth the emergent admissions in some peak periods [1], [2]. The problem related to emergent patients who could safely be treated in more appropriate care settings is well known in almost all countries. However, these improper accesses (in particular “white” urgency codes), while constituting about 30% of the total accesses in Italy [3], are managed quickly and undertake only less than 15% of the total hours of the ED staff, hiding usually more a social than a health emergency. Hybrid simulation framework Thanks to the collaboration with the Local Health Authority of the Liguria region, an observational analysis was conducted based on data collected over a one-year period in a large city’s health district. As outlined by the literature hybrid simulation can be profitability used for modeling complex healthcare systems [4],[5]. Health Care 1 (Cappanera, Tanfani) 60 In the presented framework two simulation models have been developed to interact each other and manage the overall ED/hospital system over time based fluctuations of incoming patient flows. The first is a System Dynamics (SD) model able to reproduce at a high level the whole system and the relationships and causal effects between the emergent and elective patient flows. The model can be used to evaluate the impact resulting from exogenous variations of the flows dynamics. As an example an increase of the rate of arrival at the Emergency Department , can trigger a reinforcing loop increasing elective waits which in turns results on further overcrowding of ED. The second model developed is a Discrete Event Simulation (DES) model which can be utilised to evaluate corrective organizational strategies mainly aimed at re-allocating bed capacity between emergent and elective flows. The effects of alternative scenarios can be compared with respect to a set of performance metrics able to capture the performance of both ED and hospital wards, such as waiting times of emergent patients to be admitted in hospital, proportion of misallocated patients, number of trolleys in EDs, inpatient bed occupancy rates and elective patients postponed. References [1] Landa, P., Sonnessa, M., Tanfani, E., Testi, A., A Discrete Event Simulation model to support bed management, In Proceedings of the 4th International Conference On Simulation And Modeling Methodologies Technologies And Applications, SIMULTECH, pp. 901-912,(2014). [2] Bagust, A., Place, M., Posnett, J., Dynamics of bed use in accommodating emergency admissions: stochastic simulation model. British Medical Journal, 310 (7203), 155-158 (1999). [3] Agenas (National Agency for Health Care Services), Monitor. In Special Issue on the Emergency System. Rome (Italy) (2012) Available via website AGENAS, http://www.agenas. it/pubblicazioni/monitor-rivista-agenas Cited 15 Feb 2015. [4] Chahal, K., and Eldabi, T., Towards the holy grail: Combining system dynamics and discreteevent simulation in healthcare, In B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yucesan (Eds.) Proceedings of the 2010 winter simulation conference, pp. 2293-2303,(2010). [5] Cooper, K., Brailsford, S.C., Davies, R.,Choice of modelling technique for evaluating health care interventions. Journal of the Operational Research Society, 58 (2), 168-176, (2007). Health Care 2 (invited by Cappanera and Tanfani) Tuesday 8, 9:00-10:30 Sala Seminari Ovest 61 Health Care 2 (Cappanera, Tanfani) 62 An Iterated Greedy algorithm for planning elective surgeries with detailed bed leveling Alessandro Agnetis∗ Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Italia, [email protected] Marco Pranzo Simone Sbrilli Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Italia, [email protected] [email protected] Abstract. We present a model to integrate surgical planning with bed management in a hospital. Given the waiting lists of different surgical disciplines, our model accounts for different levels of intensity of care required by the patients, number of beds in the wards, post-surgical expected length of stay and patient gender. We illustrate a metaheuristic algorithm based on the Iterated Greedy principle. Experiments on real data from an Italian public hospital show the practical viability of our approach. Integrating surgical planning and bed management The Operating Theatre (OT) and ward beds are among the most critical resources in hospitals and determine the perceived quality of service of healthcare providers. Typically, an OT is composed of several Operating Rooms (OR), possibly having different characteristics, and an OT is usually shared among different surgical disciplines. In the so-called block scheduling approach, OR time is divided into OR sessions, i.e., time intervals during which an OR is allocated to a single discipline, and the weekly plan associating OR sessions to surgical disciplines is known as the Master Surgical Schedule (MSS). The typical pathway for the post-surgical patient is to either be admitted to an intensive care unit (ICU) if his/her health conditions are critical, or to be moved directly to a ward delivering the needed level of care. Specifically, we consider two different care levels: Day Surgery (or Low-Intensity, where the patient typically stays before being discharged home on the same day of the surgery) and High-Intensity. The ward is organized in rooms, each containing a given number of beds. Rooms may have different sizes and at any given time patients assigned to a room must be of the same gender. In this context, the so called Surgical Case Assignment Problem (SCAP) deals with the assignment of elective surgeries to OR sessions [1] on a short time horizon (usually a week). In the literature, most of the SCAP approaches focus only on OT operations. However, operating rooms may not be the only bottleneck in the surgical pathway, since also post-surgery beds in wards may be a scarce resource thus hindering the patients’ flow. In this talk we present a model for elective surgical planning that accounts for both OT activities and detailed post-surgical beds in the wards. Health Care 2 (Cappanera, Tanfani) 63 Iterated greedy Due to the intrinsic complexity of the problem, we propose a metaheuristic algorithm based on the Iterated Greedy (IG) framework to quickly generate good quality solutions even for large instances. Iterated Greedy ([2], [3]) is a conceptually very simple metaheuristic approach. It is designed to improve the performance of a “black box” greedy heuristic by means of the iteration of a Destruction-Construction cycle. More specifically, at each iteration the algorithm destroys a portion of the incumbent solution by removing some elements, thus obtaining a partial solution. Then, starting from the partial solution, a new complete solution is built by the greedy heuristic. This solution is evaluated and possibly accepted as a new incumbent. To test our approach, we considered data from a medium-size OT of an Italian hospital and we carried out tests on instances corresponding to different scenarios. The algorithms were run for several ward configurations. The results obtained are compared with those produced by a mathematical formulation for the problem. The experiments show that by appropriately tuning the metaheuristic parameters, it is possible to obtain high quality solutions in less than 1 minute of computation. References [1] Testi, A., Tanfani E., Torre G.C., A three phase approach for operating theatre schedules, Health Care Management Science 10 (2007), 163–172. [2] Cesta, A., Oddi, A., Smith, S.F., Iterative flattening: A scalable method for solving multicapacity scheduling problems, Proceedings of the National Conference on Artificial Intelligence, AAAI Press (2000), 742–747. [3] Ruiz, R., Stützle, T., A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem, European Journal of Operational Research 177 (2007), 2033– 2049. Health Care 2 (Cappanera, Tanfani) 64 Hierarchical Reoptimization Approach for solving Inpatient Bed Scheduling Problems Domenico Conforti∗ Dipartimento di Ingegneria Meccanica, Energetica, Gestionale, Università della Calabria, Italia, [email protected] Rosita Guido Maria Carmela Groccia Dipartimento di Ingegneria Meccanica, Energetica, Gestionale, Università della Calabria, Italia, [email protected] [email protected] Abstract. The patient-bed scheduling problem consists in assigning elective patients to beds by satisfying hard and soft constraints. High complexity and need of tools to assist admission scheduler in making fast decisions motivate researchers to design suitable approaches. In this work, we formulate two optimization models to support hospital managers in the bed assignment decision-making process. In order to provide efficient solution approaches, we propose a framework for solving the problem by achieving optimal or sub-optimal solutions at rather low computational cost. The proposed solution framework, based on a hierarchical rescheduling phases, has been tested on benchmark instances available from the relevant literature. We get improvements on the best-known results in the patient-bed assignment literature for nine out of thirteen instances. Health Care 2 (Cappanera, Tanfani) 65 OR planning and bed leveling of a surgery department: a Variable Neighborhood Search approach Paolo Landa∗ Dipartiimento di Economia, Università degli studi di Genova, Italia, [email protected] Roberto Aringhieri Dipartimento di Informatica, Università degli studi di Torino, Italia, [email protected] Elena Tanfani Dipartiimento di Economia, Università degli studi di Genova, Italia, [email protected] Abstract. This work deals with the problem of Operating room (OR) scheduling over a given planning horizon. A set of elective surgery patients belonging to different surgical specialties are waiting for being scheduled over a set of available operating room block times and a set of beds is available for each surgical specialty. The objective is to define an ORs schedule able to level the post-surgery ward bed occupancies during the days of the planning horizon, allowing a regular and smooth workload in the ward. Indeed, a certain quantity of beds, that will be used by emergent patients (usually represented by the concurrent flow of patients from Emergency Department to hospital wards) must be reserved. In this work we exploited the flexibility of the Variable Neighborhood Search developing a solution framework that can be easily adapted to different hospital operative contexts of OR Scheduling. In order to validate the framework, preliminary results reported are tested on a set of real based instances of a Surgery Department of a large Italian public hospital. Problem description Operating Rooms (ORs) planning is a critical activity which has a great financial impact for hospital organitation settings. In italian public hospitals the demand for surgery usually overwhelms supply, causing long waiting list and waiting times and, as consequence, reducing patients quality of life [5]. During ORs planning, surgeons and hospital managers have to allocate OR capacity in order to improve hospital efficiency and productivity. The literature review of operations research applied to health care management clearly reveals an increasing interest of researchers towards OR planning and scheduling problems [3]. In ORs planning usually are defined three levels of decisions: strategic (long term), tactical (medium term) and operational (short term). These levels better characterize planning or scheduling problems even if there are no clear and universally accepted definitions [3] and level of integration [1] of the three decision levels. The operational decisions represented by the short term period are generally distinguished into “advance scheduling” and “allocation scheduling”. The first one, usually referred to Surgical Case Assignment Problem (SCAP), which is the assignment of a surgery date and OR block to each patient over the considered planning horizon, can range from one week to one month. Health Care 2 (Cappanera, Tanfani) 66 We consider the planning decisions concerning the advance scheduling problem SCAP in a block scheduling setting under different operative contexts. In this work we consider the criteria dealing with the ward stay bed levelling, a planning able to develop a smooth (without peaks) stay bed occupancies, that will determine a smooth workload in the ward and, at the end, an improved quality of care provided to patients. The workload balance is a challenging problem [2]. Variable Neighborhood Search methodology is exploited in order to develop a solution framework that can be adapted to different hospital operative contexts of OR Scheduling. References [1] Aringhieri R., P. Landa, P. Soriano, E. Tanfani, A. Testi, A two level Metaheuristic for the Operating Room Scheduling and Assignment Problem, Computers & Operations Research 54 (2015), 21–34. [2] Beliën, J., E. Demeulemeester, B. Cardoen, Building cyclic master surgery schedules with levelled resulting bed occupancy: A case study, European Journal of Operational Research 176 (2007), 1185–1204. [3] Cardoen B., E. Demeulemeester, J. Beliën , Operating room planning and scheduling: A literature review, European Journal of Operational Research 201 (2010), 921–932. [4] Guerriero F., R. Guido, Operational research in the management of the operating theatre: a survey, Health Care Management Science 14 (2011), 89–114. [5] Oudhoff, J., D. Timmermans, D. Knol, A. Bijnen, G. van der Wal, Waiting for elective general surgery: impact on health related quality of life and psychosocial consequences, BMC Public Health 7 (2007). Health Care 3 (invited by Cappanera and Tanfani) Tuesday 8, 11:00-13:00 Sala Seminari Ovest 67 Health Care 3 (Cappanera, Tanfani) 68 Tactical and operational optimization of Operating Theaters: a multi-level decision support system Paolo Tubertini∗ DEI, Università di Bologna, Italia, [email protected] Valentina Cacchiani Andrea Lodi DEI, Università di Bologna, Italia, [email protected] [email protected] Matteo Buccioli ORM Research Group, [email protected] Abstract. The high costs of health care push national and regional health services as well as local authorities to constantly improve management performances in order to obtain more efficient organization of hospitals activities and provide patients with the best possible care. We present a multi-level decision support approach composed by two optimization models, a tactical one and an operational one, that are integrated by a discrete event simulator. The proposed approach is focused on a multispecialty Operating Theater for Emilia-Romagna region. Extended Abstract The high costs of health care push national and regional health services as well as local authorities to constantly improve management performances in order to obtain more efficient organization of hospitals activities and provide patients with the best possible care. Tactical and operational planning decisions in Italy are classified as separate problems both in terms of time frame and in terms of decision makers. The tactical planning process is performed, on a monthly or quarterly basis, by the hospital management board and defines the assignment of operating room time slots to medical specialties. Examples of tactical planning can be found in [1], [2] and [3].The operational planning process is performed, on a weekly basis, by the specialty head physician and defines the scheduling of patients in the assigned operating room time slots. Examples of operational planning model can be found in [4], [5], and [6]. We present a multi-level decision support approach focused on a multispecialty Operating Theater for Emilia-Romagna region. First, we present a tactical optimization model that calculates, on a monthly or quarterly basis, the assignment of operating room time slots to medical specialties in order to minimize: (i) the length of specialties waiting lists weighted by their relative importance for patients safety, (ii) the cost overrun, and (iii) the gap between the negotiated case mix and the final one. The final objective is a tradeoff between (i) and (ii), (iii). Health Care 3 (Cappanera, Tanfani) 69 Second, we present an operational optimization model that calculates, on a weekly basis, the subset of patients in the waiting list that will be scheduled for surgery treatment in order to: (a) comply with the regional guidelines related to maximum waiting time per pathology, and (b) significantly reduce the violation of time slots (overtime) and the misuse of surgical time (under-utilization). Finally, we present a simulation model that integrates the tactical and the operational optimization models evaluating their effectiveness on a monthly or quarterly planning horizon. References [1] Testi, A., Tanfani, E., and Torre, G., A three-phase approach for operating theatre schedules, Health Care Manag. Sci.,(2007), 163–172 [2] Chaabane, S., Meskens, N., Guinet, A., and Laurent, M., Comparison of two methods of operating theatre planning: Application in belgian hospital, In Service Systems and Service Management, 2006 International Conference on, volume 1, (2006), pp 386–392 [3] Guinet, A. and Chaabane, S., Operating theatre planning, International Journal of Production Economics, 85(1), (2003), 69–81 [4] Roland, B., Di Martinelly, C., Riane, F. and Pochet, Y., Scheduling an operating theatre under human resource constraints, Comput. Ind. Eng., 58(2), (2010), 212–220 [5] Riise, A. and Burke, E. K., Local search for the surgery admission planning problem, J. Heuristics, 17(4), (2011), 389–414 [6] Marques, I., Captivo, M., and Vaz Pato, M., An integer programming approach to elective surgery scheduling, OR Spectrum, 34(2), (2012), 407–427 Health Care 3 (Cappanera, Tanfani) 70 Tactical and operational decisions for sharing resources among surgical clinical pathways Roberto Aringhieri∗ Dipartimento di Informatica, Università di Torino, Italia, [email protected] Davide Duma Dipartimento di Informatica, Università di Torino, Italia, [email protected] Abstract. In our previous work the need of optimization approaches for the performance improvement of a single surgical clinical pathway has been proven. Performances are measured defining a set of patient– and facility– centred indices. In this paper, we extend that work to deal with the management of shared resources among two or more surgical clinical pathways. Problem description The current development of the health care systems is aimed to recognize the central role of the patient as opposed to the one of the health care providers. In this context, Clinical Pathways (CPs) shift the attention from a single health benefit to the health care chain that starts to resolve the illness episode. They can be defined as “healthcare structured multidisciplinary plans that describe spatial and temporal sequences of activities to be performed, based on the scientific and technical knowledge and the organizational, professional and technological available resources” [3]. A CP can be conceived as an algorithm based on a flow chart that details all decisions, treatments, and reports related to a patient with a given pathology, with a logic based on sequential stages [4]. For this reason, they can be considered an operational tool in the clinical treatment of diseases, from a patient-focused point of view [5]. The aim of a care pathway is to enhance the quality of care by improving patient outcomes, promoting patient safety, increasing patient satisfaction, and optimizing the use of resources as stated by the European Pathway Association. Moreover, while many papers show that, appropriately implemented, CPs have the potential to increase patient outcome, reduce patient length of stay and limit variability in care, thereby yielding cost savings [6], little attention has been dedicated to study how CP can optimize the use of resources as reported in [1] to which the reader can refer to deepen this topic. In [1], we proved the need of optimization approaches in order to improve the performance of a single surgical CP. Performance is measured defining a set of patient– and facility– centred indices. Among the several optimization approaches, we showed that the most effective is the operating room Real Time Management (RTM) determining a general improvement of all the performance indices. It consists in a sort of centralized surveillance system whose main task is to supervise the execution of the planning and, in the case of delays, to take a decision regarding the patient cancellation or the assignment of the overtime, that is the extra operating time available to be assigned when a surgery can not be finished within the normal operating time. Health Care 3 (Cappanera, Tanfani) 71 In this paper, we extend the previous work to deal with the management of shared resources among two or more surgical CPs. We propose a hybrid simulation and optimization model: simulation is used to generate real patient flows while offline and online optimization is used to determine the more rational decision on the use of shared resources. Shared Resources The shared resources taken into account are the Operating Rooms (OR) and the overtime. The Master Surgical Schedule (MSS) defines the specific assignment of OR session – an OR session identifies an OR and the day in which the OR is available – to be shared to specialties. The MSS must be updated whenever the total amount of OR time changes or when the make up of some specialties change. This can occur not only as a response to long term changes in the overall OR capacity or fluctuations in staffing, but also in response to seasonal fluctuations in demand. The objective is to have a fair assignment of the OR sessions to specialties in such a way to guarantee a minimum number of OR sessions to each specialty. When some delay occurs and the overall delay could determine the exceeding of the jth OR session duration Sj , RTM should deal with the problem of postponing a surgery or using a part of the available overtime. Such a decision poses the problem of evaluating the impact of consuming overtime or to have a cancellation. When the overtime is a shared resources, the online decision of using the overtime or to cancel a surgery should take into account a fairness criterion. References [1] Aringhieri, R., B. Addis, E. Tànfani, and A. Testi, Clinical pathways: Insights from a multidisciplinary literature survey, In Proceedings of ORAHS 2012. [2] Aringhieri R. and Duma D., A hybrid model for the analysis of a surgical pathway. In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications 2014, 889–900. Best paper award. [3] Campbell, H., N. Bradshaw, and M. Porteous, Integrated care pathways, British Medical Journal 316 1998, 133–144. [4] De Bleser, L., R. Depreitere, K. De Waele, K. Vanhaecht, J. Vlayen, and W. Sermeus, Defining pathways, Journal of Nursing Management 14 2006, 553–563. [5] Panella, M., S. Marchisio, and F. Stanislao, Reducing Clinical Variations with Clinical Pathways: Do Pathways Work?, International Journal for Quality in Health Care 15 2003, 509– 521. [6] Rotter, T., L. Kinsman, E. James, A. Machotta, H. Gothe, J. Willis, P. Snow, and J. Kugler, Clinical pathways: effects on professional practice, patient outcomes, length of stay and hospital costs (review), The Cochrane Library 7 2010. Health Care 3 (Cappanera, Tanfani) 72 A mixed offline and online approach to manage elective and non-elective patients Davide Duma∗ Dipartimento di Informatica, Università di Torino, Italia, [email protected] Roberto Aringhieri Dipartimento di Informatica, Università di Torino, Italia, [email protected] Abstract. In this paper, we propose a mixed offline and online approach to improve the management of elective and non-elective patients. The solution framework is built on a DES methodology in order to model the patient flows and to deal with the inherent stochasticity of the problem. Further, we will address the analysis of the trade-offs between the use of dedicated operating rooms and a flexible policy. Problem description At the operational decision level, the Operating Room (OR) management is also called “surgery process scheduling” and is generally separated into two sub-problems referred to as “advanced scheduling” and “allocation scheduling” [5]. Usually, the two sub-problems have different and conflictual objectives, that is to maximize the OR utilization and to minimize the number of patient delayed or cancelled, respectively [3]. The surgery process scheduling is also characterized by three main sources of uncertainty, that is the variability of patient operating times, the variability of patient length of stays and the arrival of patients. Among the three sources of uncertainty, the more interesting and difficult from a methodological point of view is that regarding the arrival of patients and, in particular, the problem of dealing with the joint arrival of elective patients and non-elective patients. Elective patients are those waiting for a planned surgery and usually listed on a long waiting list. On the contrary, non-elective patients are those arriving, for instance, from the Emergency Department, not allowing their ex-ante planning but only ex-post decisions. Note that the majority of the literature deals with elective patients (see, e.g.,[7]) while contributions dealing with non-elective patients are really limited as reported in [4]. The management of non-elective patients is a really complex task: actually, delaying an urgent non-elective surgery may increase the risk of postoperative complications and morbidity. Therefore, the speed at which an OR is available for that surgery, is the crucial factor to guarantee a positive final outcome. A common approach is to reserve OR capacity since it is believed to increase the responsiveness. This approach poses a question, that is if it is better to have dedicated ORs or, alternatively, to reserve capacity in the elective ORs. We discuss the problem of dealing with a joint flow of elective and non-elective patients within a surgical pathway. In literature, different solutions (dedicated operating rooms vs. flexible policy) have been proposed determining opposed results. Furthermore, to the best of our knowledge, online optimization is never been applied to the context of the Operating Room Planning. Health Care 3 (Cappanera, Tanfani) 73 In this paper, we propose a mixed offline and online approach to improve the management of elective and non-elective patients. The solution framework is built on a DES methodology in order to model the patient flows and to deal with the inherent stochasticity of the problem. Further, we will address the analysis of the trade-offs between the use of dedicated operating rooms and a flexible policy. This paper summarizes the work presented in [6]. Optimization approaches To properly generate the elective and non-elective patient flows we take into account a generic surgical clinical pathway. In this pathway, four different and critical moments determine four optimization problems to deal with, that is, the (a) advance and the (b) allocation scheduling, the (c) Real Time Management (RTM) and the (d) rescheduling: the optimization problems (a), (b) and (d) are solved exploiting a set of offline optimization approaches partially inspired by the work reported in [2] while an online optimization approach is discussed for the RTM extending the work presented in [1]. RTM consists in a sort of centralized surveillance system whose main task is to supervise the execution of the OR planning and, in the case of unexpected events, to take a decision to ensure the minimal disruption of the current planning. Possible unexpected events could be: a delay caused by the variability of patient operating times which can determine a patient cancellation or an overtime assignment; the arrival of non-elective patients to be operated within a predefined time limit which can require the complete re-planning of the surgery sequence determining delays. References [1] Aringhieri R. and Duma D., A hybrid model for the analysis of a surgical pathway. In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications 2014, 889–900. Best paper award. [2] Aringhieri R., Landa P., Soriano P., Tànfani E. and Testi A., A two level Metaheuristic for the Operating Room Scheduling and Assignment Problem, Computers & Operations Research, 54 2015, 21–34. [3] Beaulieu I., Gendreau M. and Soriano P., Operating rooms scheduling under uncertainty, in Tànfani E. and Testi A., Advanced Decision Making Methods Applied to Health Care, International Series in Operations Research & Management Science 173 2012, 13–32. [4] Cardoen B., Demeulemeester E. and Belien J., Operating room planning and scheduling: A literature review, European Journal of Operational Research 201 2010, 921–932. [5] Magerlein, J.M., and Martin J.B., Surgical demand scheduling: A review, Health Services Research 13 1978, 418–433. [6] Duma D. and Aringhieri R., Optimal management of elective and non-elective patients. EURO Summer Institute (ESI XXXII) on Online Optimization 2015. [7] Guerriero F. and Guido R., Operational research in the management of the operating theatre: a survey, Health Care Management Science 14 2011, 89–114. Health Care 3 (Cappanera, Tanfani) 74 Medium term operating room scheduling with patient release dates Elena Tanfani∗ Dipartimento di Economia, Università di Genova, Italia, [email protected] Giuliana Carello Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] Bernardetta Addis LORIA (CNRS UMR 7503) - INRIA Nancy Grand Est, Université de Lorraine, France, [email protected] Abstract. Surgery departments have a great impact on hospital efficiency and costs, thus operating room planning activities have been gaining an increasing attention. Different levels of decision must be addressed when managing operating rooms: at the operational level the set of patients to be scheduled and their operating date must be decided. This work addresses the problem of scheduling a subset of patients from an elective surgery list with the aim of improving the perceived quality of service. Introduction In recent years, hospital organizations have been facing a strong pressure to improve their health care delivery processes and to increase their productivity and operational efficiency. In most hospitals, surgical departments contribute significantly to the total expenditure; besides, they have a great impact on the quality of service provided and on waiting times. The crucial role of surgery departments results in an increasing number of research studies about Operating Rooms (ORs) planning and scheduling problems. Recent literature reviews are reported in [1] and [2], where several variants of the problem are considered, which deal with different decision levels. This work focuses on the operational level and addresses the operating room surgery process scheduling assuming a block scheduling policy, namely each specialty can use a set of a priori determined OR blocks [3], [4], [5]. An elective waiting list, a set of operating room blocks, and a planning horizon are given. Due to the length of the waiting list, not all the waiting patients can be operated on during the considered planning horizon. Thus the problem decisions are to determine the subset of patients to be scheduled and their surgery date. The patient-centered objective function aims at guaranteeing a good quality of service provided to patients. Problem description and approach The problem consists in determining the set of patients to be operated on and their scheduling over a medium term planning horizon up to three months. We consider an elective waiting list: each patient in the waiting list is characterized by a waiting time, a surgery duration and an urgency level. Based on the patient Health Care 3 (Cappanera, Tanfani) 75 urgency level, a maximum waiting time before treatment is also given for each patient (deadline). Beside, we consider new elective patients who join the waiting list during the considered planning horizon. New patients are characterized by an urgency level, a surgery duration, a deadline and the date in which they are registered in the waiting list (release date). A set of available operating blocks in the planning horizon is given, each characterized by a surgery time capacity. In order to provide the best possible quality of service from the patient point of view, the surgery must be performed before deteriorating patient clinical conditions. Thus the objective function accounts for waiting time and tardiness with respect to the deadline of patients. Different MILP formulations and MILP-based heuristics are proposed. All the proposed methods are tested and compared on a set of reallife based instances. Their behavior, with respect to both computational time and quality of the obtained solutions, is evaluated. References [1] Cardoen B., Demeulemeester E. , Beliën J., Operating room planning and scheduling: A literature review. European Journal of Operational Research 210(2010), 921–932. [2] Guerriero F., Guido R., Operational research in the management of the operating theatre: a survey. Health Care Management Science 14(2011), 89–114. [3] Aringhieri R., Landa P., Soriano P., Tanfani E., Testi A., A two level metaheuristic for the Operating Room Scheduling and Assignment Problem. Computers & Operations Research 54(2015), 21–34. [4] Addis B., Carello G., Grosso A., Tanfani E., Operating room scheduling and rescheduling: a rolling horizon approach. Flexible Services and Manufacturing Journal (2015), 1–27. [5] Agnetis A., Coppi A., Corsini M., Dellino G., Meloni C., Pranzo M., A decomposition approach for the combined master surgical schedule and surgical case assignment problems. Health Care Management Science 17(2014), 49–59. Health Care 4 (invited by Cappanera and Tanfani) Wednesday 9, 9:00-10:30 Sala Seminari Ovest 76 Health Care 4 (Cappanera, Tanfani) 77 Balancing the Production of Blood Bags from Donation through Appointment Scheduling Ettore Lanzarone∗ CNR-IMATI, Milano, Italy, [email protected] Seda Baş Zeynep Ocak Industrial and Systems Engineering Dept., Yeditepe University, Istanbul, Turkey, [email protected] [email protected] Giuliana Carello Dip. di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy, [email protected] Semih Yalçındağ HEC Montréal and CIRRELT, Dept. of Decision Sciences, Montréal, Canada, [email protected] Abstract. Blood is fundamental in several care treatments and surgeries, thus playing a crucial role in the healthcare system. It is a limited resource, with a short shelf life, provided by donors through the Blood Donation (BD) system. Managing donors’ appointments is one of the main BD issues, as it may impact the whole blood supply chain. We address a blood donation scheduling problem with the aim of providing a balanced supply of blood bags to transfusion centres and hospitals along days. Introduction Blood plays a crucial role in the health care system, because it is fundamental in several care treatments and surgeries. On the other hand, it is a limited resource, as it can be produced only by human donors, and its shelf life is short. Blood bags are provided to transfusion centres and hospitals through the Blood Donation (BD) System [1]. Many papers deal with the BD supply chain management (see [2] for a recent survey). However, not all phases have been equally addressed in the literature [3]; for instance, the management of blood storage and distribution has been widely addressed, whereas the donation phase and in particular donors’ appointment scheduling have been only marginally considered. Nevertheless, the blood collection phase is highly relevant and may become the bottleneck of the overall system if not appropriately managed. Indeed, a bad donation scheduling cannot meet the temporal pattern of blood demand and, at the same time, may worsen the treatment quality perceived by donors, with a consequent reduction of donors’ willingness to return. Optimizing the donors appointment scheduling is therefore fundamental for adequately feeding the entire BD system and guaranteeing a good treatment to donors. Problem description and approach We address an appointment scheduling problem for the donations in a blood collection center. The goal is guaranteeing a well-balanced production of blood bags Health Care 4 (Cappanera, Tanfani) 78 among days over a planning horizon, in order to provide a quite constant feeding for the BD system. The proposed approach consists of providing pre-allocated slots per each day, which are reserved to donors based on their blood types. Such pre-allocated slots are then used when scheduling the real requests for donation appointments. Then, the pre-allocation decision is refreshed at fixed frequency to account for the newly scheduled appointments. The approach considers medical staff capacity, and both booking donors and donors arriving at the centre without reservation are included. The slot allocation problem is modeled as an Integer Linear Programming model. The approach has been applied to the real case of the Milan Department of the Associazione Volontari Italiani Sangue (AVIS), i.e., one of the main BD associations in Italy. Results on some test cases based on the AVIS scenario validate the approach and confirm its applicability and effectiveness. References [1] Pierskalla, W.P. Supply chain management of blood banks (Chapt. 10). In: Brandeau, M.L., Sainfort, F., Pierskalla, W.P. (eds.): Operations research and health care, Springer International Series in Operations Research & Management Science (2004). [2] Belien, J., Forcé, H. Supply chain management of blood products: A literature review. European Journal of Operations Research 217(2012), 1–16. [3] Baş, S., Carello, G., Lanzarone, E., Ocak, Z., Yalçındağ, S. Management of blood donation system: Literature review and research perspectives. In: Proceedings of the Second International Conference on Health Care Systems Engineering (HCSE 2015), in press. Health Care 4 (Cappanera, Tanfani) 79 Patient Scheduling policies for Pre-Admission Testing with Cross-Trained Resource Constraints Paola Cappanera∗ Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Firenze, Firenze, Italy, [email protected] Maddalena Nonato EnDIF, Department of Engineering, University of Ferrara, Ferrara, Italy, [email protected] Saligrama Agnihothri School of Management, Binghamton University, SUNY, Binghamton, NY, USA, [email protected] Filippo Visintin Dipartimento di Ingegneria Industriale, Università degli Studi di Firenze, Firenze, Italy, [email protected] Abstract. We address a planning problem at a medical center that administers pre-admission tests to patients before their scheduled surgery date. Tests are performed by multiple crosstrained/dedicated nurses (servers) with shared resources. The aim of the study is to determine a scheduling policy that maximizes the system efficiency by reducing waiting time for patients inside the exam rooms and idle time for nurses. Network flow based integer programming models are proposed for the deterministic case and validated on realistic data-sets. The structure of the problem We address a pre-admission testing (PAT) problem, where patients are grouped according to the type of tests they have to undergo. For each patient the required tests can be done in any order. Tests are performed by multiple nurses in one of the several exam rooms. Scheduling is not preemptive, i.e. once a patient enters an exam room he/she remains there until all the required tests are completed. In addition to test times, patients may be idle in the exam room waiting for a nurse to perform a specific test they require. At the same time, a nurse may be idle waiting for resources to perform the tests. This study aims at determining a scheduling policy that maximizes the system efficiency by reducing idle times. In the machine scheduling framework, each patient in PAT corresponds to a job while pre-admission tests correspond to tasks. Nurses and medical equipment such as X-ray and EKG scanners correspond to machines. Some tasks can be performed on a specific machine in the ”machine set” while other tasks can be performed on any machine in the pool. No precedence constraints linking tasks of the same job exist thus classifying the problem addressed as an open shop problem. In summary, PAT presents several features ([3]) that have been addressed separately in the literature ([1], [2] and [4] ) while, to the best of our knowledge, their simultaneous treatment has never been studied before and gives rise to a new problem in machine scheduling. In the following, we present a brief sketch of the building blocks leading to a formulation of PAT as a network flow problem. Let P be the set of patients, T the set of tests, T p ⊆ T, ∀p ∈ P the subset of tests patient p has to undergo, N the set of operators, R the set of exam rooms. PAT is formulated on a properly defined Health Care 4 (Cappanera, Tanfani) 80 graph. Specifically, each task is identified by a color. For each patient p ∈ P the network contains the following nodes: (i) a set of colored nodes associated with the set of tasks T p ; (ii) dummy origin and destination nodes, denoted by op and dp respectively, representing the beginning and the end of the service for patient p. For each of the nurses, the network contains two dummy nodes, on and dn representing respectively the beginning and the end of the related workday. Finally an extra depot node is added to the network. The main decisions characterizing PAT are: (i) the assignment of patients to exam rooms; (ii) the sequencing of tasks for each patient; and (iii) the assignment of patients to nurses. Each of these decisions gives rise to a network flow problem while the linking between these problems is modeled by temporal variables. The sub-problems can be explained as follows: -assignment of patients to exam rooms, modeled via a Capacitated Vehicle Routing Problem (CVRP) where the exam rooms correspond to vehicles in the VRP jargon and the underlying network contains the origin-destination pairs (op , dp ∀p ∈ P ) plus the depot node. The arcs in a tour connect consecutive patients in a room. Exam rooms do not need to be individually distinguished, thus a two index formulation for CVRP might be used. -sequencing of tasks for each patient, modeled as a Hamiltonian Path problem. For each patient p the path starts from op , ends in dp and visits all the colored nodes in T p . -assignment of patients to nurses, modeled as a CVRP where nurses correspond to vehicles. For each nurse n ∈ N , the tour from on to dn represents the sequence of tasks he/she performs. Specifically, each nurse is characterized by a set of skills, each corresponding to the capability of performing a task (a color) and a nurse can visit only task nodes colored compatible with the skills he/she possess. Different nurses might have different working hours, and thus they need to be individually distinguished. Consequently a three index formulation for CVRP is required. The problems above mentioned are time constrained and the temporal variables associated with the nodes of the network play a crucial role in linking the blocks of constraints. Variants of the model are proposed for the deterministic case and validated on realistic data-sets. References [1] Agnetis, A., Flamini, M., Nicosia, G., and Pacifici, A., A Job-shop Problem with One Additional Resource Type, Journal of Scheduling 14 (3) (2011), 225–237. [2] Agnetis, A., Murgia, G., and Sbrilli, S., A Job-shop scheduling problem with human operators in handicraft production, International Journal of Production Research 52 (13) (2014), 3820– 3831. [3] Agnihothri, S., Banerjee, A., and Thalacker, G., Analytics to improve service in a preadmission testing clinic, in Proceedings of the 48th Hawaii International Conference on System Science, (2015). [4] Sethi, S.P., Sriskandarajah, C., Van De Velde, S., Wang, M., and Hoogeveen, H., Minimizing makespan in a pallet-constrained flowshop, Journal of Scheduling 2 (3) (1999), 115–133. Health Care 4 (Cappanera, Tanfani) 81 On a metaheuristic approach for the INRC-II nurse scheduling competition Federica Picca Nicolino∗ Dipartimento di Ingegneria dell’informazione, Università di Firenze, Italia, [email protected] Francesco Bagattini Luca Bravi Niccolò Bulgarini Alessandro Galligari Fabio Schoen Dipartimento di Ingegneria dell’informazione, Università di Firenze, Italia, [email protected] [email protected] [email protected] [email protected] [email protected] Abstract. In this paper we will present our approach to the solution of a multiperiod nurse scheduling problem. The metaheuristic method we developed was used for the international competion INRC-II devoted to nurse scheduling. In this talk we will present our approach as well as the results obtained in the competition. Although based on a quite standard local searchbased method, our approach includes several specific strategies which signficantly improved its performance. Introduction Nurse rostering concerns the problem of finding the optimal assignment and scheduling of nurses to working shifts. Taking into account that each nurse must have only one assignment per day and that each of them has fixed skills, the problem we dealt with for the competition is also subject to other hard and soft constraints and has a multiperiod, fixed, planning horizon of 4 or 8 weeks. The multi-stage nature of the problem has to be faced (due to the competition rules [1]) splitting the overall planning horizon in single week optimization problems, so each week is considered as a standalone problem that only uses information coming from the previous one as history data for the single week scheduling problem. For each week we have been provided of three different kind of data files, one with a scenario that is common to every week, one with history data needed for the first week optimization problem and 4 or 8 week data files with specific requirement data of the single weeks, e.g. daily coverage requirements. It should be observed that the competition is a blind one (no one knows the result of the others) and time constrained (every algorithm must return a solution in a limited amount of CPU time). So, because of the nature of the problem, we present a metaheuristic approach mainly based on local-search adapted to this one and to the time limit imposed by the competition rules. Since the competition is not run out, results and detailed optimization techniques will be presented after the competion expiration date. References [1] Ceschia, S., Dang, N. T. T., De Causmaecker, P., Haspeslagh, S., Schaerf, A., Second International Nurse Rostering Competition (INRC-II) - Problem Description and Rules, http://mobiz.vives.be/inrc2/, 2015 Health Care 5 (invited by Cappanera and Tanfani) Thursday 10, 9:00-10:30 Sala Seminari Ovest 82 Health Care 5 (Cappanera, Tanfani) 83 Metaheuristic algorithms for a multi-depot dial-a-ride problem arising in healthcare Garazi Zabalo Manrique de Lara∗ Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, University of Siena, Italy, [email protected] Paolo Detti Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, University of Siena, Italy, [email protected] Francesco Papalini Azienda Sanitaria di Firenze, Italia, [email protected] Abstract. In this work, a dial-a-ride problem arising from a real-world healthcare application, concerning the non-emergency transportation of patients, is addressed. Variable Neighborhood Search and a Tabu Search algorithms are proposed able to tackle all the characteristics of the problem. Computational results on large real world and simulated instances are presented. Introduction and problem definition In this work, a transportation problem arising from a real-world healthcare application is addressed. Namely, the non-emergency transportation of patients in an Italian region [1], [3], Tuscany. Non-emergency transportation is an essential service for many patients attending hospital clinics for medical treatments, exams or therapies (e.g., see [1], [2], [5]), who can not use ordinary bus service or private transportation. Non-emergency transportation also include transferrals and discharges by equipped vehicles or ambulances. In this context, a transportation service consists in transporting a patient from a pickup location (e.g., patient’s home, hospital, etc.) to a delivery location, and, eventually, vice versa. The transportation services are performed by non-profit organizations, by means of a heterogeneous fleet of vehicles (e.g., ambulance, bus, car, etc.), located in geographically distributed depots. Other constraints concern arrival and departure times, patients-vehicles compatibility, patients’ preferences and quality of service issues. More precisely, (i) patients have to attend on time their healthcare services, (ii) a set of quality of service requirements on the length of the travels and the waiting times of the patients must be fulfilled, (iii) patients must be only transported by vehicles that are compatible with their conditions (e.g., patients on a wheel chair or a stretcher have to use equipped vehicles or ambulances) and, eventually, belonging to given non-profit organizations (user’s preferences). The objective is to minimized the total transportation cost while fulfilling the quality of service requirements. From a modeling point of view, the problem can be formulated as a Dial-a-Ride problem (DARP) [4], [5], with additional constraints and features. In this work, heuristic algorithms based on the tabu search and the variable neighborhood search techniques are proposed. The algorithms have been tested and compared on real-life and simulated instances. The computational results highlights the effectiveness of the proposed approaches and also show the benefits of coordination and central dispatching versus Health Care 5 (Cappanera, Tanfani) 84 the current system, in which the non-profit organizations collect the own patient’ appointments and route their own vehicles. Computational results on real data The algorithms have been tested on real and simulated instances. The real-life instance arises from data collected by the Health Care System of Tuscany, an Italian region, and includes all the transportation services performed in a day by a Local Health Care Agency (ASL 11) and all the vehicles (distributed in several depots) available at that day. In the instance, there are 246 transportation services and 313 vehicles of different typologies, distributed over 29 depots. The real cost paid by ASL 11 for all the transportation services was 10276.1 EUR. On this instance, TS and VNS attained a solution with cost 9819.63 and 9210.93 , respectively, in less than three hours of computation, with a saving of 4.45% and 10.37%, respectively, with respect to the real solution. References [1] Agnetis, A., Coppi, A., De Pascale, G., Detti, P., Raffaelli, J., Chelli, P., Colombai, R., Marconcini, G., Porfido, E., Applicazione di tecniche di operations management per minimizzare il costo di trasporto di pazienti, MECOSAN, Italian Quart. of Health Care Management Economics and Politics, 20 (2012), 51–62. [2] Bowers, J., Lyons, B., Mould, G., Developing a resource allocation model for the Scottish patient transport service, Operations Research for Health Care, 1 (2012), 84–94. [3] Coppi, A., Detti, P., Raffaelli, J., A planning and routing model for patient transportation in healthcare, Electronic Notes in Discrete Mathematics, 41 (2013), 125–132. [4] Cordeau, J.-F., Laporte, G., The dial-a-ride problem: models and algorithms, Annals of Operations Research, 153 (2007), 29–46. [5] Melachrinoudis, E., Ilhana, A. B., Min, H., A dial-a-ride problem for client transportation in a health-care organization, Computers & Operations Research, 34 (2007), 742–759. Health Care 5 (Cappanera, Tanfani) 85 Forecasting Demand in Health Care Organizations Aleksandra Marcikic∗ Faculty of Economics Subotica, University of Novi Sad, Serbia, [email protected]f.uns.ac.rs Boris Radovanov Faculty of Economics Subotica, University of Novi Sad, Serbia, [email protected] Abstract. In health care organizations, ability to predict demand is an issue of great importance. A highly complicating, but challenging factor that influences demand of health care services is randomness. In order to obtain the most accurate forecast of the number of ambulance rides per day, we will use and compare different forecasting methods. The data set provides us information about the number of ambulance rides per day for the three years period. Seasonal ARIMA models, Holt-Winters exponential smoothing methods and multiple regression models have been applied and compared. Also practical relevance and usage of those models is discussed. Introduction Ability to predict demand is of paramount importance (4) for decision making process in health care organizations, but this area has seen little systematic study. A highly complicating but challenging factor that influences the demand of ambulance services is randomness. Call arrival patterns tend to be highly time and location dependent. Also the availability of ambulance vehicles and personnel is random due to uncertainty in their occupancy times. Mostly, demand forecast for ambulance services can be divided into two categories, first dealing with spatial distribution of demand and second that investigates how demand evolves over time. In this paper we will examine if there are some patterns and regularities that explain variations of demand for ambulance services over time. In this paper we investigated data provided to us by Ambulance Service in Subotica, city of Vojvodina province, Serbia. Demand for its services is presented as daily number of calls that arrived to the ambulance station. The data set which we use in this research provides us information about the daily number of received calls for three years period, from February 15th 2012 until February 15th 2015. We used the data from February 15th 2012 to January 21st 2015 to estimate models. The rest of the data was used as a test set to validate models and forecasts. The preliminary data analysis confirmed the assumption that data of the number of ambulance requests show correlations and typical yearly and weekly arrival patterns. Thus, we need to perform forecasting methods which include seasonality. In previous researches (2), (5), (8), (10) both univariate and multivariate models have been used for forecasting demand of medical services, so in our paper we decided to estimate two groups of univariate models: seasonal ARIMA models and HoltWinters exponential smoothing methods (both additive and multiplicative types of modeling seasonal effects). Since our preliminary statistical analysis confirms the presence of the month of year and day of week effects, we also estimated multiple regression to model behavior of number of ambulance rides per day. We used multiple explanatory variables, month of the year and day of the week, which we modeled in a linear manner. Health Care 5 (Cappanera, Tanfani) 86 In order to obtain adequate short-term forecast for daily number of ambulance rides we analysed results of all estimated models: two regression models (model with all introduced parameters and secondly the model with only significant parameters), two types of Holt-Winters exponential smoothing models (additive and multiplicative) and seasonal ARIMA model. To decide which of the models is the most accurate to forecast the ambulance calls per day we look at forecasts generated by our models and compare them to our test set. Extrapolation has been done to generate forecast for the next 14 days. There are different ways to measure the exactness of a forecast and in this research we used three indicators: Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE). From all examined models, we can conclude that the best forecast has been obtained by multiple regression model with all parameters. References [1] Brandeau M.L., Sainfort F., Pierskalla W.P., Operations research and health care: a handbook of methods and applications. Springer Science + Business Media, Inc., 2005. [2] Channouf N., LEcuyer P., Ingolfsson A., Avramidis A. N., The application of forecasting techniques to modeling emergency medical system calls in Calgary, Alberta, Health Care Manage Science 10 (2007), 25–45. [3] Diebold F.X., Elements of forecasting. Thomson South-Western, 2004. [4] Goldberg J.B., Operations research models for the deployment of emergency services vehicles, EMS Management Journal 1 (2004), 20–39. [5] Matteson D.S., McLean M.W., Woodard D.B., Henderson S.G., Forecasting emergency medical service call arrival rates, Annals of Applied Statistics 5 (2011), 1379–1406. [6] Moeller B.,Obstacles to measuring emergency medical service performance, EMS Management Journal 3 (2004), 8–15. [7] Ozcan Y.A. Quantitaive methods in health care management: techniques and applications. John Wiley Sins Inc., San Francisco, 2009. [8] Taylor J. W.,A comparison of univariate time series methods for forecasting intraday arrivals at a call center, Management Science 54 (2008), 253–265. [9] Teow, K.L.,Practical operations research applications for healthcare managers, Annals Academy of Medicine Singapore (2009), 564–566. [10] Zuidhof, G.M. Capacity Planning for Ambulance Services: Statistical Analysis, Forecasting and Staffing. Master thesis, Vrije Universiteit Amsterdam,Centrum Wiskunde en Informatica Amsterdam, 2010. Health Care 5 (Cappanera, Tanfani) 87 Pharmaceuticals Optimization for ICUs Andrea Manno∗ Dipartimento di Ingegneria dell’Informazione, Università degli Studi Firenze, Italia, [email protected] Paola Cappanera Dipartimento di Ingegneria dell’Informazione, Università degli Studi Firenze, Italia, [email protected] Maddalena Nonato Dipartimento di Ingegneria, Università degli Studi di Ferrara, Italia, [email protected] Abstract. Pharmaceuticals used in Intensive Care Units (ICUs) do constitute one of the most relevant cost entries of a hospital. An optimized management of pharmaceuticals in ICUs could result in a significant reduction of the healthcare costs. However the structural features of ICUs make it a very difficult task, that has been not deepened in literature. In this work we present a combination of different optimization approaches to support physicians operating in ICUs in the management of pharmaceuticals. Pharmaceutical Optimization Literature and the ICUs Context Several studies have been proposed in literature about optimization of pharmaceuticals in healthcare [3]. They mainly deal with the pharmaceutical supply chain and inventory management [1],[2],[5] for relatively high dimensional structures (like hospital pharmacies or pharmaceutical suppliers), operating on a large amount of aggregated and structured data. Most of these works result in optimization models for replenishment and inventory policies. These models cannot be directly applied to the optimization of pharmaceuticals in ICUs as these units are characterized by low dimensions, great dynamism and uncertainty. However pharmaceuticals of ICUs are generally very expensive and have a great impact on healthcare costs of a hospital. This paper focuses on a new optimization approach for the complex ICUs case, that, by integrating classical pharmaceutical optimization models with simulation-based [4] and machine learning techniques [6], tries to capture the key factors of ICUs. In particular machine learning is used to cluster patients on the basis of what kind of pharmaceutical therapy they will be subject to; statistical and simulation-based analysis of historical data are used to extract information about the general behavior of the ICU system. All these information are inserted in an optimization model that minimizes costs related to pharmaceuticals considering all the constraints of the ICU structure. This new approach could represent a useful tool to support physicians and nurses operating in ICUs in critical decisions concerning the management of pharmaceuticals. References Health Care 5 (Cappanera, Tanfani) 88 [1] Uthayakumar, R., Priyan, S., Pharmaceutical supply chain and inventory management strategies: Optimization for a pharmaceutical company and a hospital, Operations Research for Health Care, 2(3), 52-64 (2013). [2] Kelle, P., Woosley, J., Schneider, H., Pharmaceutical supply chain specifics and inventory solutions for a hospital case, Operations Research for Health Care, 1(2), 54-63 (2012). [3] de Vries, J., Huijsman, R.,Supply chain management in health services: an overview, Supply Chain Management: An International Journal, 16(3), 159-165 (2011). [4] Vila-Parrish, A. R., Ivy, J. S., King, R. E., Abel, S. R., Patient-based pharmaceutical inventory management: a two-stage inventory and production model for perishable products with Markovian demand, Health Systems, 1(1), 69-83 (2012). [5] Baboli, A., Fondrevelle, J., Tavakkoli-Moghaddam, R., Mehrabi, A., A replenishment policy based on joint optimization in a downstream pharmaceutical supply chain: centralized vs. decentralized replenishment, The International Journal of Advanced Manufacturing Technology, 57(1-4), 367-378 (2011). [6] Bishop, Christopher M. Pattern recognition and machine learning, Vol. 4. No. 4. New York: springer (2006). LASSIE (invited by Frangioni) Tuesday 8, 15:30-17:00 Sala Riunioni Est 89 LASSIE (Frangioni) 90 Revisiting the use of Robust Optimization in unit commitment problems under market price uncertainty Fabio D’Andreagiovanni∗ Dept. of Mathematical Optimization, Zuse Institute Berlin (ZIB), Berlin, Germany, DFG Research Center MATHEON, Technical University Berlin, Berlin, Germany, Einstein Center for Mathematics (ECMath), Berlin, Germany, Institute for Systems Analysis and Computer Science, Consiglio Nazionale delle Ricerche (IASI-CNR), Roma, Italy, [email protected] Giovanni Felici Institute for Systems Analysis and Computer Science, Consiglio Nazionale delle Ricerche (IASI-CNR), Roma, Italy, [email protected] Fabrizio Lacalandra Quantek s.r.l., Bologna, Italy, [email protected] Abstract. In recent years, increasingly attention has been given to uncertain versions of the classical Unit Commitment Problem (UC - see [1] for a recent survey), in order to tackle the issues associated with the presence of data uncertainty. In this work, we focus on a price-uncertain version of the UC, managed through a Robust Optimization (RO) approach. We review central references available in literature about the topic, pointing out their limits and we then show how RO should be correctly applied, highlighting the improvements that can be obtained on realistic instances. References [1] Tahanan, M., van Ackooij, W., Frangioni, A., Lacalandra F. Large-scale Unit Commitment under uncertainty: a literature survey. Technical Report 14-01, Dipartimento di Informatica, Università di Pisa, 2014. LASSIE (Frangioni) 91 Lagrangian Relaxation for the Time-Dependent Combined Network Design and Routing Problem Enrico Gorgone∗ Département d’Informatique, Université Libre de Bruxelles, Belgium, [email protected] Bernard Fortz Département d’Informatique, Université Libre de Bruxelles, Belgium, [email protected] Dimitri Papadimitriou Alcatel-Lucent Bell Labs, Antwerp, Belgium, [email protected] Abstract. In communication networks, the routing decision process remains decoupled from the network design process. To keep both processes together, we propose a distributed optimization technique aware of distributed nature of the routing process by decomposing the optimization problem along same dimensions as (distributed) routing decision process. We propose a Lagrangian approach for computing a lower bound by relaxing the flow conservation constraints. The approach is more robust than any LP solvers. LASSIE (Frangioni) 92 A multiplicative weights update algorithm for real-world nonconvex MINLPs Luca Mencarelli∗ CNRS LIX, École Polytechnique, Paris, France, [email protected] Leo Liberti CNRS LIX, École Polytechnique, Paris, France, [email protected] Youcef Sahraoui CNRS LIX, École Polytechnique, Paris, France and OSIRIS, EDF R&D, Clamart, France, [email protected] Abstract. We describe a general adaptation of the well-known Multiplicative Weights Update algorithm for Mixed-Integer Nonlinear Programming. This algorithm is an iterative procedure which updates a distribution based on a gains/costs vector at the preceding iteration, samples decisions from the distribution, and updates the gains/costs vector based on the decisions. Our adaptation relies on the concept of “pointwise reformulation”, which is expressed in function of some parameters θ and has the property that, for a certain value of θ, the optimum of the reformulation is an optimum of the original problem. The Multiplicative Weights Update algorithm is used to find good values for θ. We applied the Multiplicative Weights Update framework to four real-world problems very difficult to solve, namely the Distance Geometry Problem, the Pooling Problem, a nonconvex Portfolio Selection and the hydro-power short-term Unit Commitment Problem. Introduction The Multiplicative Weights Update (MWU) algorithm (see the excellent survey [1]) is a stochastic iterative heuristic with a twist: it has a “built-in” (weak) performance guarantee. After T iterations, the overall solution error EMWU , which is a weighted no sum of many stochastic decisions ψ t = (ψ1t , . . . , ψqt ) for t ≤ T , is asymptotically P worse than a piecewise linear function of the smallest cumulative error t ψit over all i ≤ q. Given further information on the problem structure, this guarantee can sometimes make the MWU algorithm a proper approximation algorithm, as shown in [1]. The MWU algorithm iteratively updates a probability distribution pt = (pt1 , . . . , ptq ) over q “advisors”. The value of the advice given by the i-th advisor at iteration t ≤ T is ψit ∈ [−1, 1], with −1 being a gain for minimization or a cost for maximization, and vice versa. At each iteration t, the MWU elects to follow the advice of advisor i ≤ q with probability proportional to pti . The errors ψ t are used to update the probability distribution according to: ∀t ≤ T r {1}, i ≤ q where pti = wit P `≤q w`t and η ≤ 1 2 wit = wit−1 (1 − ηψit−1 ), (15) is given. One can then show [1] that: ! EMWU , X t≤T ψ t pt ≤ min i≤q X t≤T ψit + η X t≤T |ψit | + ln q . η (16) LASSIE (Frangioni) 93 A Mixed-Integer Nonlinear Programming (MINLP) problem is usually cast in the general form: minn f (x) x∈R (17) ∀` ≤ m g` (x) ≤ 0 [P ] ∀j ∈ Z xj ∈ Z, where Z ⊆ {1, . . . , n} is given. Pointwise reformulation Our strategy is to exploit a pointwise reformulation ptw(P, θ) of Eq. (17), parametrized on a parameter vector θ, with the following properties: • ptw(P, θ) can be solved efficiently (or sufficiently efficiently) in practice — most often this will mean that ptw(P, θ) is either a Linear Program (LP), or a convex NLP, or perhaps even a Mixed-Integer Linear Program (MILP); • if P is feasible, then there must exist values θ∗ such that the optima of ptw(P, θ∗ ) are also optima of P . This can be achieved, for example, by turning some of the decision variables of P into parameters θ. Pointwise reformulations can be exploited by algorithms such as MWU, since it alternately decides parameter values based on gains/costs, which can be computed by solving ptw(P, θ), and, more in general, by any alternating projection algorithm [2]. We apply the MWU framework to four real-world MINLP problems very difficult to solve, both from a theoretical and a practical viewpoint, namely the Distance Geometry Problem [3], the pooling problem [4], a nonconvex variant of the Markowitz Portfolio Selection and the hydro-power short-term EDF Unit Commitment Problem. The problem structure of each of those problems is exploited to “manually” construct a pointwise reformulation. References [1] Arora, S., Hazan, E. and Kale, S., The Multiplicative Weights Update Method: A Metaalgorithm and Applications, Theory of Computing 8 (2012), 121–164. [2] Deutsch, F., Approximation Theory, Spline, Functions and Applications, pages 105–121. Kluwer, Dordrecht, 1992. [3] Liberti, L., Lavor, C., Maculan, N. and Mucherino, A., Euclidean Distance Geometry and Applications, SIAM Review, 56 (2014) 3–69. [4] Misener, R. and Floudas, C.A., Advances for the Pooling Problem: Modeling, Global Optimization, and Computational Studies, Applied and Computational Mathematics 8 (2009) 3–22. Maritime Logistics 1 (invited by Ambrosino, Monaco and Zuddas) Thursday 10, 9:00-10:30 Sala Seminari Est 94 Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) 95 Sizing a non-mandatory Truck Appointment System for dealing with congestion issues in seaport container terminals Claudia Caballini∗ DIBRIS-Department of Informatics, BioEngineering, Robotics and Systems Engineering. University of Genova, Italy, [email protected] Daniela Ambrosino Lorenzo Peirano DIEC-Department of Economics and Business Studies. University of Genova, Italy, [email protected] [email protected] Simona Sacone DIBRIS-Department of Informatics, BioEngineering, Robotics and Systems Engineering. University of Genova, Italy, [email protected] Abstract. The present work faces with the critical issue of congestion in seaport container terminals in relation to the truck cycle. Truck Appointment Systems (TAS) can help container terminals in properly managing vehicles with the goal of increasing terminal productivity and efficiency and maintaining a high service level for trucks. A non-mandatory TAS is here considered and modeled through a network flow. An integer programming formulation is proposed; different scenarios are analyzed and related results will be shown during the Conference. Introduction The phenomenon of naval gigantism is changing the operativeness of container terminals, forcing them to load and unload huge quantities of containers to/from big vessels and to manage great quantities of trucks concentrated in short time intervals, in correspondence of ship arrivals. If not properly managed, truck arrivals may determine long queues of vehicles and congestion issues both at the terminal gate and inside the terminal; this affects truck service times and terminal productivity, as well as the mobility related to urban areas adjacent to ports. For these reasons, advanced methodologies should be investigated in order to limit negative impacts determined by vehicles congestion [1],[2]. In this regard, Truck Appointment Systems may help container terminals in adequately managing truck flows for import and export purposes. Problem description and proposed model Generally, each truck arrives at the terminal gate in a certain time window with a preferred arrival time and is characterized by a given number of tasks to be executed (up to a maximum of four in case of two import containers to be picked up and two export to be delivered in the terminal). Once approached the terminal, the vehicle may queue at the gate and, after having executed the gate-in activities, it can enter the terminal and go to the indicated block(s) where it may wait again for the terminal handling means to serve it. Finally, once it has executed its unload/pick up operations, it can leave the terminal through the gate out. In order to minimize the time spent in queue by trucks a non compulsory TAS is investigated. In particular, Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) 96 the focus is on the problem of sizing the TAS in terms of number of vehicles that should make a booking and number of both gate lanes and handling means to dedicate to booked trucks. For this purpose, inspired by [3], a non compulsory TAS is modeled through a network flow. Moreover, an Integer Programming model is proposed for solving the problem of sizing the non mandatory TAS. A certain number of constraints are considered in the model such as the maximum number of vehicles that the terminal can host in each time interval, the maximum productivity of both gate lanes and terminal equipment and, also, the maximum allowed length of queues. Preliminary results The proposed model is implemented in C# programming language and solved through the commercial solver Cplex 12.5. A certain number of scenarios are randomly generated and used to test the proposed model. Preliminary results will be presented during the Conference. References [1] Chena, G., K. Govindanb, Yangc, Z., Managing truck arrivals with time windows to alleviate gate congestion at container terminals, International Journal of Production Economics. (2013), 141 (1): 179-188. [2] Zhang, X., Zeng, Q., Chen, W. Optimization Model for Truck Appointment in Container Terminals. Intelligent and Integrated Sustainable Multimodal Transportation Systems Proceedings. (2013), 96: 1938-1947. [3] Zehendner, E.,Feillet, D.Benefits of a truck appointment system on the service quality of inland transport modes at a multimodal container terminal. European Journal of Operational Research. (2014), 235: 461-469. Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) 97 An optimization model for evaluating the manpower planning policy of a real transshipment container terminal Nuria Diaz-Maroto Llorente∗ Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] Massimo Di Francesco Simone Zanda Paola Zuddas Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] [email protected] [email protected] Abstract. We look at a case study of the manpower planning problem in which personnel shifts and vessel services do not overlap. It is investigated by an integer linear programming model. The optimal solutions of the model are compared to the decision policy of a real transhipment container terminal (TCT). The experimentation shows when the terminal policy is effective or there is room for optimization. Modeling and evaluating the manpower planning problem The hub-and-spoke topology of maritime networks results in a critical role for transshipment container terminals (TCTs), owing to the consolidation of flows among them. In addition, since few ports adopt completely automated systems, manpower is a key resource for TCTs, particularly in those with high labour costs. In the manpower planning problem TCTs must assign each operator to shifts, tasks and vessel services, while avoiding both personnel undermanning and overmanning [1]. Unlike [2], we investigate a case study of the manpower planning problem in which personnel and vessel services do not overlap. It is modelled by an integer linear programming model, which is optimally solved by a freeware solver in a few minutes. The optimal solutions of the model are compared to the decision policy of a real TCT, which ranks vessel services in a priority list and assigns operators to them starting from the topmost task. A key criticality in this policy is the unexploited option of flexible shifts, because operators are assigned to shifts in such a way to provide a uniform manpower supply in each day. The motivation of this choice is the simplicity of implementation for the TCT, which has no planning tools to evaluate alternative manpower configurations. The experimentation shows that the terminal policy is optimal in the case of low variance in the daily manpower demand, as the same solution of the optimization model is obtained. In case of medium and high variance, the model outperforms significantly the terminal policy, because it is able to increase the manpower in the peak periods of the manpower demand. References [1] Legato, P., Monaco, M.F., Human resources management at a marine container terminal, European Journal of Operational Research 156 (2004), 769–781. Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) 98 [2] Di Francesco, M., Fancello, G., Serra, P., Zuddas, P., Optimal Management of Human Resources in Transhipment Container Ports, Maritime policy & management 42 (2015), 127–144. Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) 99 Optimal management of the equipment maintenance at a container terminal Maria Flavia Monaco∗ Dipartimento di Ingegneria Informatica, Modellistica Elettronica e Sistemistica, Università della Calabria, Italia, [email protected] Antonino Chiarello Luigi Moccia Sammarra Marcello Istituto di Calcolo e Reti ad Alte Prestazioni, CNR, Italia, [email protected] [email protected] [email protected] Manlio Gaudioso Dipartimento di Ingegneria Informatica, Modellistica Elettronica e Sistemistica, Università della Calabria, Italia, [email protected] Abstract. In this talk we describe an optimization model for the periodic preventive maintenance of the handling equipments at a maritime container terminal. Inspired by the real context of the Gioia Tauro port, we discuss the problem, propose a decomposition in two phases, based on different planning horizons, and focus on the short-term planning problem. Some preliminary numerical results on real instances are presented. Problem description The main activity of a maritime container terminal consists in loading/unloading containers into/from the berthed vessels, by means of quay cranes. The container moves from the yard storage areas to the quay, and vice versa, are performed by different transportation means, like straddle carriers, AGV’s, or trailers. There is a wide agreement on the fact that the reliability and availability of the cargo handling equipment play a very relevant role to guarantee a high quality service. Actually, the most significant indicators of the global terminal performance (throughput, ship turnaround time) are related to the vessel service time, which, on turn, is strongly influenced by the continuous availability and full functionality of the technical resources. Nevertheless, the planning and management of the equipment repair and maintenance has not received, so far, enough attention in the scientific literature. In order to introduce the problem, it is worthy to underline that most of the maintenance activities require the downtime of the involved equipment; therefore it is not easy to allocate suitable servicing time windows without affecting the terminal operative continuity. For this reason, the planning of the maintenance activities follows a two-phase process. On the basis of the weekly berthing plan, the service records and the forthcoming deadlines for the preventive maintenance activities of the equipments, the first phase (long-term planning) is aimed at selecting the equipments to inspect. Then, given the set of equipments to be serviced, the second phase (short-term planning) consists of scheduling all the maintenance activities, while taking into account the availability of the workers responsible for the maintenance. Here we describe an optimization model for the periodic preventive maintenance of the container handling equipments. Inspired by the real context of the Gioia Tauro container terminal, we discuss the problem, and focus on the short-term Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) 100 planning problem. Some preliminary numerical results on real instances are presented. Maritime Logistics 2 (invited by Ambrosino, Monaco and Zuddas) Thursday 10, 14:15-15:45 Sala Seminari Est 101 Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) 102 The Multi Port Stowage Planning problem: extensions, new models and heuristics Daniela Ambrosino∗ Dept. of Economics and Business Studies, University of Genova, Italy, [email protected] Massimo Paolucci Dept. of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Italy, [email protected] Anna Sciomachen Dept. of Economics and Business Studies, University of Genova, Italy, [email protected] Abstract. In this paper we extend the Master Bay Plan Problem (MBPP), to the MultiPort Master Bay Plan Problem (MP-MBPP) that is the problem of determining stowage plans for containers into a ship having to visit a given number of ports in its circular route. Many types of containers are included in the analysis: 20’ and 40’ standard containers, reefer and open top containers. Moreover, new stability conditions are investigated. These extensions are faced by new MIP models and a MIP heuristic. Some tests have been executed for ships with increasing capacity up to 18000 TEUs and preliminary results are shown. Introduction Shipping companies choose to serve the maritime transportation demand by even larger containerships and, in this new scenario, the ports became very crucial nodes in the logistics systems. Unloading and loading operations strongly impact on the efficiency of the terminal [1] and in the recent literature a greater number of contributes are devoted to the optimization of the stowage plans for containerships. The ship coordinator has a view of the whole trip of the ship and he receives the o-d transport demands. He define a stowage plan for each port of the trip of the ship and update it all time he receives (and accepts) a new transport demand for that ship [2]. He has to solve many times the MP-MBPP. For this reason it is necessary to be able to find good stowage solutions in short amount of time. The problem under investigation and solution approaches In the MP-MBPP the whole route of the ship is investigated: at each port of the route different sets of containers must be loaded for being shipped to a next port. Thus, the MP-MBPP consists in determining how to stow a given set of containers, split into different groups, according to their size, type, class of weight and destination, into locations, either on the deck or in the stow, of the bays of a containership. Some structural and operational constraints, related to the containers, the ship and the maritime terminals, have to be satisfied. Many types of containers are included in the analysis: 20’ and 40’ standard containers, reefer and open top containers. Moreover, new stability conditions are investigated. Here, we propose new MIP models and a MIP heuristic for the MP-MBPP with the main aim of minimizing the total berthing time of the ship. Unproductive Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) 103 movements are included in the analysis, as well as the workload of the quay cranes used in each port visited by the ship The main idea of this MIP heuristic is to solve a partial linear relaxation of a MIP model (MPR) in order to obtain a starting solution and some bounds exploited during the search for integer feasible solutions. Then, the proposed heuristic iterates the solution of the MIP model, whose integer variables are partially fixed to the integer values found MPR, in order to obtain an effective solution for the MP-MBPP. The procedure is called ”progressive random fixing” since, during the iterations, it tries to solve the MIP model having fixed a randomly chosen subset of integer variables whose cardinality is progressively reduced. The proposed MIP heuristic permits to find good stowage solutions in short amount of time and thus to include the model in a DSS that can help the ship coordinator for defining the planning in accordance with the new transport demand. Some tests have been executed for ships with increasing capacity up to a very large ship with a capacity of 18000 TEUs. Preliminary results show the effectiveness of the proposed method . References [1] Rashidi H., Tsang EPK. , Novel constraints satisfaction models for optimization problems in container terminals, Applied Mathematical Model 37, (2013), 3601–3634. [2] Ambrosino, D., Paolucci, M., Sciomachen, A., Experimental evaluation of mixed integer programming models for the multi-port master bay plan problem, Flexible Services and Manufacturing, (2013), 1–22. Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) 104 The train load planning problem with multiple cranes in seaport terminals: a multiobjective optimization approach Silvia Siri∗ Dept. of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Italy, [email protected] Daniela Ambrosino Dept. of Economics and Business Studies, University of Genova, Italy, [email protected] Luca Bernocchi Dept. of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Italy, [email protected] Abstract. In this paper the multi-crane train load planning problem arising in seaport terminals is studied. This problem has a multiobjective nature, since the assignment of the containers to the wagons on which they are loaded and to the cranes which perform the loading operations is realized by considering the simultaneous optimization of several objectives. In this work, different multiobjective optimization techniques are applied and compared, and preliminary experimental results are shown. Introduction Due to the high competition among seaport container terminals, terminal operators rely on information technology and automated control technology in order to be more and more efficient and to optimize the activities related to container handling and forwarding (see for e.g. [1], [2]). This work considers a very crucial activity in seaport terminals, i.e. the train load planning. In particular, the considered problem extends the one described in [3], [4], by accounting for the case of multiple cranes working simultaneously to sequentially load containers on a train. Train load planning: a multiobjective optimization problem The train load planning problem with multiple cranes can be briefly summarized as follows. Given a set of containers in a storage area (characterized by different lengths, weights, commercial values and specific positions in the storage area), a train with a set of wagons (characterized by different lengths, load configurations and weight constraints), and a set of cranes for loading the train, the problem is to determine how to assign containers to wagon slots and which cranes have to load such containers. The assignment is done by taking into account constraints regarding the load configurations of wagons, the sequencing of operations of the cranes and different weight constraints for slots, wagons and for the whole train. In the present study the load planning problem involves the simultaneous optimization of several objectives: maximization of the total commercial value of containers loaded on the train, minimization of the number of reshuffles in the storage area, minimization of the number of pin changes in the wagons to be loaded, Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) 105 minimization of the distance covered by the handling equipment between the storage area and the wagons, minimization of the unbalance among the number of operations realized by the different cranes. For this reason, the train load planning problem with multiple cranes can be regarded as a multiobjective optimization problem. Different multiobjective optimation approaches (both of a priori and a posteriori type) have been applied to solve the multi-crane train load planning problem. First of all, the weighted sum method have been considered, according to which the cost function is a weighted sum of different objectives to be minimized. Then, the goal-programming method has been adopted, so that the objective function is the minimization of the absolute deviations of the objectives from given target values. Finally, the -constraint method is applied in order to generate Pareto optimal solutions, i.e. a set of alternatives with different trade-offs among the objectives. The multiobjective optimization approaches adopted for the multi-crane train load planning problem have been tested and compared trough an experimental analysis based on realistic data coming from an Italian case study. References [1] Steenken, D., Voss, S., Stahlbock, R., Container terminal operation and operations research - a classification and literature review, OR Spectrum 26 (2004), 3–49. [2] Günther, H.-O., Kim, K.-H., Container terminals and terminal operations, OR Spectrum 28 (2006), 437–445. [3] Ambrosino, D., Siri, S., Models for train load planning problems in a container terminal, in: Freire de Sousa, J., Rossi, R. (Eds.), Computer-Based Modelling and Optimization in Transportation, Advances in Intelligent Systems and Computing, 262, Springer (2014), 15– 25. [4] Ambrosino, D., Siri, S., Comparison of solution approaches for the train load planning problem in seaport terminal, Transportation Research E 79 (2015), 65–82. Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) 106 Optimization of simulation parameters for stochastic empty container repositioning Paola Zuddas∗ Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] Massimo Di Francesco Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] Alexei Gaivoronski NTNU - Norwegian University of Science and Technology, Trondheim, Norway, [email protected] Abstract. To correct directional imbalances in the flows of loaded containers, empty containers must be repositioned from ports where they are in surplus to ports where they are in shortage. A relevant characteristic of empty container repositioning is data uncertainty. We face this problem by an algorithm optimizing the parameters of a discrete event simulation model. The discrete event simulation model Discrete event simulation is a common way to model problems by a discrete sequence of events occurring at a particular instant in time and making a change in the current system state. In the proposed problem, the system state at time t is represented by the position of vessels at time t, the number of empty containers carried by vessels at time t and the inventory of empty containers in ports at time t. The state transformation rules depend on the current system state, the observation of random parameters entering the system description at time t and the parameters of the control strategy, selected to govern the system. This control strategy defines when and where the loading containers on the ships and unloading of containers at ports will take place at time t. The following strategy is considered: • If the amount of empty containers exceed an upper threshold, the excess containers are prepared for loading on vessels arriving in the next day, as long as there is sufficient space for empty containers on vessels. • If, instead, the number of empty containers in the port of arrival falls below a lower threshold, the port inventory of empty containers is replenished to the value of threshold, provided there are sufficient containers on arriving ship. Hence, the control parameters are the lower and upper inventory thresholds of each port. At each time step, the simulation logic updates the system state depending on observed demands, loading and unloading decisions, and checks inventory levels against inventory thresholds. At each time step, the overall system performance is computed by a linear function where the unitary costs of inventory, transportation, loading, unloading and shortage are multiplied by the corresponding decisions in terms of number of empty containers. Although this model can be straightforwardly understood and utilized in practice, the performance of the repositioning process largely depends on the values of inventory thresholds. More Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) 107 important, determining the inventory thresholds minimizing the overall system performance means performing an impractical number of simulation runs. Therefore, it is crucial to determine the best (or a highly performing) setting of inventory thresholds. An optimization algorithm is proposed to address this challenge. Optimization of inventory thresholds In this phase, we look for the inventory thresholds optimizing the average system performance with respect to all observations of random parameters entering the system description in each period of the planning horizon. These random parameters include the demand for containers at ports, weather-dependent travel times, ship loading and unloading times. In order to obtain the optimal values of thresholds, the simulation model is combined with an optimization algorithm, which belongs to the family of stochastic gradient methods. In order to evaluate the optimization of the simulation model, it is compared to a different stochastic programming approach. It consists in describing uncertainty by a finite (and relatively small) number of scenarios organized in scenario tree. With additional modelling simplifications, it is possible to model this problem as multistage stochastic linear problem with recourse [1]. Next, this problem is transformed in its deterministic equivalent, which is a large-scale integer problem of special structure [2]. A commercial software for mixed-integer programming is applied to solve this problem. Preliminary results show that the simulation of the optimization model produces similar or superior solutions to the scenario tree approach in small instances and solves large realistic instances within reasonable computing times. References [1] Birge, J.R., Louveaux, F. Introduction to Stochastic Programming. Springer, New York, 2011. [2] Di Francesco, M., Crainic, T.G., Zuddas, P. The effect of multi-scenario policies on empty container repositioning, Transportation Research part E 45 (2009), 758–770. Network Optimization (invited by Guerriero) Thursday 10, 9:00-10:30 Sala Riunioni Est 108 Network Optimization (Guerriero) 109 The resource constrained spanning tree problem: an optimal solution approach Luigi Di Puglia Pugliese∗ Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy, [email protected] Francesca Guerriero Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy, [email protected] José Luis Santos CMUC, Department of Mathematics, University of Coimbra, Portugal, [email protected] Abstract. We address a variant of the spanning tree problem. In particular, we assume that each edge consumes a certain amount of resource w. The aim is to obtain a spanning tree with the smallest cost and the total resource consumption must not exceed a fixed value W . An optimal solution approach is devised. The main idea is to construct a graph such that each path is a spanning tree in the original graph. Fathoming procedures and dominance relations are defined for the problem at hand. Problem definition The spanning tree problem (STP ) aims at finding the shortest undirected paths from a pre-determined source node to each non-source node in a given graph. Many solution approaches have been proposed in the scientific literature to address the STP . For a detailed description and a computational study of these methods, the reader is referred to [1]. The STP is defined over an undirected graph G(N, E), where N = {1, . . . , n} is the set of nodes and E = {e1 , . . . , em } is the set of edges. Each edge ek , k ∈ {1, . . . , m}, is a 2-element subset of N , thus, edge ek can be represented by an unordered pair [i, j] with i, j ∈ N , and consequently, [i, j] = [j, i]. A cost cij is associated with each edge [i, j]. A tree PT is a connected subgraph of G containing n nodes and n−1 edges. Given c(T ) = [i,j]∈T cij , representing the cost of a tree T , the aim of the STP is to obtain a tree T with the minimum cost c(T ). Here we study the Resource Constrained STP (RCSTP ). Beside theP cost cij , a resource consumption wij is associated with each edge [i, j]. Let w(T ) = [i,j]∈T wij be the resource consumed by tree T . The objective of the RCSTP is to find a minimum cost tree T such that w(T ) ≤ W , where W is the maximum amount of available resource w. Optimal solution approaches are defined based on branchand-bound scheme (e.g., [2] among others), whereas, [3] presents an approximation procedure. We define a path p = hv0 , e1 , v1 , . . . , e` , v` i between s, t ∈ N as a sequence of nodes and edges such that vi ∈ N , ∀i ∈ {0, . . . , `}; s = v0 and t = v` ; and ei = [vi−1 , vi ] ∈ E, ∀i ∈ {1, . . . , `}. Network Optimization (Guerriero) 110 In order to solve to optimality the RCSTP , we propose a new solution approach dealing with the construction of a graph GT , where each path p in GT is associated with a unique tree T in G. Solution approach Given an undirected network G = (N, E, c, w), the built network GT = (N T , AT , cT , wT ) is a directed network (with multiple-arcs) defined in the following way: • N T = {{1} ∪ X : X ⊆ N \{1}} is the set of nodes (|N T | = 2n−1 ); • AT = {(X, X ∪ {j})[i,j] : X ∈ N T , i ∈ X, j 6∈ X, [i, j] ∈ E} is the set of arcs; • cT is the vectorial cost function, where cT ((X, X ∪ {j})[i,j] ) = cij . • wT is the vectorial resource function, where wT ((X, X ∪ {j})[i,j] ) = wij . Note that GT is a layered network, where the layer i contains the nodes of N T with i + 1 elements (that is, with i + 1 nodes of N ) and the arcs link nodes between two consecutive layers. In order to recover the information of the initial network G, we define the function e : AT → E such that e( (X, X ∪ {j})[i,j] ) = [i, j]. Let p = hX0 , a1 , X1 , . . . , an−1 , Xn−1 i be a path in GT . Then, the set of edges associated with p, that is, T p = {e(a1 ), . . . , e(an−1 )} is a tree in G. Additionally, T p is the unique tree of G represented by p. Proposition Based on Proposition above, we design an algorithm that builds the network GT by considering only non-dominated and feasible paths. At the end, among all paths ∗ of GT that with minimum cost p∗ is selected and T p is retrieved. References [1] Bazlamacc, C.F. and Hindi, K.S., Minimum-weight spanning tree algorithms A survey and empirical study. Computers & Operations Research 28 (2001), 767–785. [2] Aggarwal, V. and Aneja, Y.P. and Nair, K.P.K., Minimal Spanning Tree subject to a side Constraint. Computers & Operations Research 9 (1982), 287–296. [3] Ravi, R. and Goemans, M.X., The constrained minimum spanning tree problem. Algorithm Theory – SWAT’96 Lecture Notes in Computer Science Volume 1097 (1996), 66–75. Network Optimization (Guerriero) 111 A robust version of the green vehicle routing problem with uncertain waiting time at recharge station Giusy Macrina∗ Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy, [email protected] Luigi Di Puglia Pugliese Francesca Guerriero Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy, [email protected] [email protected] Michael Poss UMR CNRS 5506 LIRMM, Universite de Montpellier, 161 rue Ada, 34392 Montpellier Cedex 5, France, [email protected] Abstract. We address the problem of routing a fleet of electrical vehicles in an urban area. The vehicles must collect demands from a set of customers within predefined time windows. Recharge stations are present in the area. The time spent at the recharge stations can be seen as the sum of recharge times and waiting times. We model here the waiting times as uncertain parameters. A two-stage robust formulation is given and we propose a solution approach to provide a robust routing. Problem definition Let G(R ∪ N , A) be a complete graph, where N is the set of customers, R is the set of recharging stations and A is the set of arcs. A particular node in R is the depot, that is the vertex where the routes of the vehicles start and terminate. Each customer i ∈ N is characterized by a demand qi and a service time si . All nodes can be visited only once. A time window [ei , li ] is associated with each node i ∈ N . The vehicles have limited transportation capacity and battery capacity. A distance dij and a travel time tij are associated with each arc (i, j) ∈ A. All recharge stations i ∈ R are characterized by the recharge speed ρi and the waiting time wi . The energy consumption is assumed to be proportional to the distance traveled. We assume that w ∈ R|R| is not known withP precision and belongs to the budgeted i ≤ Γ, wi ≥ 0}, where ŵi is the uncertainty set from [2], U Γ = {w ∈ R|R| : i∈R w ŵi maximum waiting time at node i. Let P be a set of feasible routes p with respect to the capacity and the avoidingout-of-battery constraints. Given a route p, c(p) defines the traveling cost, whereas g(p) represents the recharge cost. The traveling cost c(P) and the recharge cost g(P) of a set P are defined as the sums of the traveling costs and the sums of the recharge costs of all routes p ∈ P, respectively. Let P̄ be the set of all feasible solution P. The problem can be stated as follows: min c(P) + g(P) s.t. ei ≤ τi (w, P) ≤ li , P ∈ P̄ Γ ∀i ∈ N , w ∈ U , (18) (19) Network Optimization (Guerriero) 112 where τi (w, P) is the arrival time at node i starting from the depot at time zero and using the set of routes P. Time τi depends on the specific value of w since the vehicle may have to recharge its battery along its routes. Notice that the dependency of τi (w, P) on P can easily be modeled by linear constraints using classical formulations (e.g. [1]). Solution approach Our problem is closely related to the vehicle routing problem studied in [1] so that we can adapt their approach to solve it. Their main idea is to relax all constraints ([2]) but those in a finite subset U ∗ . Then, the variables and constraints that correspond to elements in U Γ \ U ∗ are added on the fly in the course of a row-and-column generation algorithm. We can check the feasibility of a route in polynomial time by applying the dynamic programming (DP ) algorithm from [1] on an extended version of graph G(R ∪ N , A). The DP either returns a violated element w∗ ∈ U Γ \ U ∗ or asserts the feasibility of P. The steps of the proposed approach are devised in Algorithm 1. Algorithm 1 1: U ∗ = ∅, stop=false 2: while !stop do 3: stop=true 4: Solve problem ([1])–([2]) with U ∗ obtaining P ∗ 5: for p ∈ P ∗ do 6: perform DP on p 7: if p is infeasible then 8: U ∗ ← U ∗ ∪ {w∗ } 9: stop=false 10: end if 11: end for 12: end while 13: P ∗ is the optimal robust routing References [1] Agra, A. and Christiansen, M. and Figueiredo, R. and Hvattum, L.M. and Poss, M. and Requejo, C., The robust vehicle routing problem with time windows. Computers & Operations Research 40(3) (2013), 856-866. [2] Bertsimas, D. and Sim, M., Robust discrete optimization and network flows. Mathematical Programming 98 (2003), 49–71. Network Optimization (Guerriero) 113 A Lagrangian relaxation for the Link Constrained Steiner Tree Problem Giovanna Miglionico∗ DIMES, Unical, Italy, [email protected] Luigi Di Puglia Pugliese Francesca Guerriero DIMEG, Unical, Italy, [email protected] [email protected] Manlio Gaudioso DIMES, Unical, Italy, [email protected] Abstract. We consider the undirected Steiner Tree Problem (USTP ) obtained from the classical Steiner Tree problem by adding a constraint on the total number of edges in the tree. This problem is known to be NP-hard. We present a Lagrangian relaxation of the problem and develop a multipliers update procedure aimed at solving the Lagrangian dual. We also introduce a heuristic procedure to build some feasible solutions of the USTP problem. We present numerical results on test problems drawn from the literature. A Lagrangian relaxation for the Undirected Steiner Tree Problem Let G be a network with a set of nodes V , a set of edges E and a set of edge costs {ce |e = (i, j) ∈ E}. Let T be a subset of V . The undirected Steiner tree problem (USTP ) aims at finding a minimum cost subtree that contains a path between node s and every member of T . The cost of a tree is the sum of the costs for arcs in the tree. In this paper we consider a variant of the USTP in which the number of edges of the Steiner tree has to be less than or equal to a given threshold K. We call this variant Link Constrained USTP (LCUSTP ). A directed version of this problem is discussed in [1] together with the definition of local search strategies, that can be used for improving solutions obtained by some heuristic algorithms. A possible formulation of the LCUSTP is the following: min X ce x e (20) e∈E if i = s 1 −1 if i = k ∀k ∈ T /{s} i ∈ V fijk − fjik = 0 otherwise {j∈V } {j∈V } X X fijk ≤ xe ∀e = (i, j) ∈ E, k ∈ T /{s} X xe ≤ K e∈E fijk ≥ 0 ∀(i, j) ∈ E, k ∈ T xe binary (21) (22) (23) (24) (25) where xe is a binary variable indicating whether or not arc e is included into the solution. The variable fijk is the amount of commodity k (the amount of flow Network Optimization (Guerriero) 114 between nodes s and k) on arc (i, j). Equations (21) are the classical flow constraints from node s to each k ∈ T . Constraints (22) state that flow on an arc is allowed only if the arc is included in the solution. A Lagrangian relaxation of problem (20)–(24) can be obtained by associating to constraints (22) the multipliers µek ≥ 0 and to constraint (23) multiplier λ ZLR (µ, λ) = min X ce x e − e∈E = XX (k) µek fij e∈E k∈T XX X xe − K) µek (xe − fijk ) + λ( e∈E k∈T + X (ce + λ − e∈E e∈E X µek )xe k∈T s.t. (21), (24), (25) { We observe that ZLR (µ, λ) can be viewed as composed of X two distinct subproblems: LR(1) = X X (k) (2) min µek fij , s.t. (21), (24)} and LR = {min (ce + λ − µek )xe , s.t. (25)}. Given a vector X k∈T e∈E e∈E k∈T of Lagrangian multipliers µ, solving LR(1) requires the solution of |T | minimum path problems, P whereas LR(2) can be solved by inspection by setting xe = 1, if ce + λ − k∈T µek < 0, xe = 0, otherwise. A heuristic to solve the Lagrangian dual To solve the Lagrangian dual max zLR (λ, µ) of our problem we propose a dual ascent heuristic λ,µ based on updating one multiplier at time. Our heuristic makes also use of some sub-gradient iterations whenever the multiplier update procedure is unable to generate a significant increase of the Lagrangian dual objective. At each iteration we also calculate an upper bound on the USTP by adjusting the infeasibility of the obtained solutions with respect to the original problem. We tested our method on a set of well known instances drawn from the literature. References [1] Burdakov, O., Doherty, P., Kvarnström, J., Local Search Heuristics for Hop-constrained Directed Steiner Tree Problem, Technical Report LiTH-MAT-R 2014/10 SE, Department of Mathematics, Linkoping University, Sweden, 2014 Nonlinear Programming 1 (invited by Palagi) Tuesday 8, 11:00-13:00 Sala Gerace 115 Nonlinear Programming 1 (Palagi) 116 A new active set method for bound constrained optimization problems Andrea Cristofari∗ Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Sapienza Università di Roma, Italia, [email protected] Marianna De Santis Fakultät für Mathematik, TU Dortmund, Germany, [email protected] Stefano Lucidi Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Sapienza Università di Roma, Italia, [email protected] Francesco Rinaldi Dipartimento di Matematica, Università di Padova, Italia, [email protected] Abstract. We propose a feasible active set method for solving a bound constrained optimization problem {min f (x) : l ≤ x ≤ u}. In each iteration, first we estimate the active variables and set them to the corresponding bounds. Then we minimize f only with respect to the free variables, employing a gradient-related direction. In particular, a truncated-Newton method is applied. The method is well-suited for large-scale problems, since the minimizations involve only a subset of the original variables. The convergence is proved under usual assumptions. Introduction In this paper we propose a new method for bound constrained optimization problems. Starting from the results provided by [1] and [2], we develop a new active set algorithm, which exploits multipliers functions for estimating KKT multipliers. They also allow estimating the active variables. The main feature of our algorithm is a variables decomposition carried out at each iteration. We start estimating the active variables at a given point, and we set them to the corresponding bounds, giving rise to a decrease of the objective function. Then, we estimate the KKT multipliers at the new point, and we minimize only with respect to the variables estimated non-active, by employing a truncated-Newton algorithm. We repeat these steps until convergence, which is proved under usual assumptions. Finally, preliminary numerical results are provided. References [1] Bertsekas, D.P., Projected Newton Methods for Optimization Problems with Simple Constraints, SIAM Journal on control and Optimization 20.2 (1982), 221–246. [2] De Santis, M., Di Pillo, G., Lucidi, S., An active set feasible method for large-scale minimization problems with bound constraints, Computational Optimization and Applications 53.2 (2012), 395–423. Nonlinear Programming 1 (Palagi) 117 A Global Heuristic for Solving the Standard Quadratic Problem Umberto Dellepiane∗ Department of Computer, Control and Management Engineering Antonio Ruberti, Sapienza University of Rome, Italy, [email protected] Laura Palagi Department of Computer, Control and Management Engineering Antonio Ruberti, Sapienza University of Rome, Italy, [email protected] Abstract. The Standard Quadratic Problem (StQP) is an NP-hard problem with many local minimizers. We propose a new heuristic based on the use of ellipsoidal relaxation of the StQP and the eigenvalue decomposition of the quadratic term. The main idea is to find a heuristic that allows to escape from a non-global solution. Although this heuristic may lead to an infeasible point of the StQP, this may be used as a starting point for a further local procedure. We test our method on StQP problems arising from the Maximum Clique Problem on a graph. Introduction The Standard Quadratic Problem (StQP) is defined as follows: 1 (26) max{f0 (y) = y > Ay : y ∈ ∆} 2 where ∆ denotes the standard simplex in n-dimensional Euclidean space IRn , namely ∆ = {y ∈ IRn : e> y = 1, y ≥ 0} , (27) where A = [aij ] ∈ IRn×n is a symmetric matrix not positive semidefinite, e is the n-vector of all ones and the apex > denotes the transposition. The problem is non-convex and we look for a global solution y ∗ , i.e. a point y ∗ such that f0 (y ∗ ) ≤ f0 (y) for all y ∈ ∆. It is well known that StQP is an NP-hard problem and that many local minimizers or stationary points that are not global exist. We consider the following relaxation of the StQP 1 max{f0 (y) = y > Ay : y ∈ H} 2 where H denotes the ellipsoid 1 ∆ ⊆ H = {y ∈ IRn : e> y = 1, (y − y 0 )> H(y − y 0 ) ≤ } , 2 (28) (29) where y 0 ∈ IRn and H 0 is symmetric matrix. Solving problem (28), which can be formulated as a polynomial time trust region subproblem, gives us a bound to the optimal solution of the StQP. In [1], [2] unconstrained formulations of the constrained StQP, which lead to different heuristics, have been proposed. Nonlinear Programming 1 (Palagi) 118 Using the spectral decomposition of the matrix Q we are able to obtain starting points for the unconstrained procedure that exploit in some way the non-convexity of the problem and allow us to escape from a local minimizer. Some computational results obtained on the problems arising from the DIMACS challenge are reported and discussed. References [1] Bomze, I.M. and Palagi, L., Quartic formulation of standard quadratic optimization problems, Journal of Global Optimization 32 (2005), 181–205. [2] Bomze, I.M. and Grippo, L. and Palagi, L., Unconstrained formulation of standard quadratic optimization problems. TOP, 2012 Nonlinear Programming 1 (Palagi) 119 Derivative Free Methodologies for Circuit Worst Case Analysis Vittorio Latorre∗ Department of Computer, Control and Management Engineering, University of Rome Sapienza, Italy, [email protected] Husni Habal Helmut Graeb Institute for Electronic Design Automation, Department of Electrical Engineering and Information Technology, Technische Universität München, Germany, [email protected] [email protected] Stefano Lucidi Department of Computer, Control and Management Engineering, University of Rome Sapienza, Italy, [email protected] Abstract. In this work we propose a new derivative free algorithm for minimizing a black box objective function in the case where some of the variables must satisfy a spherical constraint and some other box constraints. The proposed algorithm is used to solve a particular problem of circuit yield optimization. The design of analog circuits needs to compute the values of some parameters like transistor lengths and widths in such a way that the performance specifications are fulfilled. Due to the statistical variations of the manufacturing process the worst cases of the performance specification is considered. This process is called Worst Case Analysis (WCA). In the field of Computer Aided Design the WCA is generally modeled by a particular black box optimization problem. Such a problem is characterized by: • a derivative free objective function whose values are obtained by a simulation code (possibly affected by noise), therefore the first order derivatives cannot be neither computed nor approximated; • a spheric constraint on a subset of the decision variables, called process variables; • box constraints on another subset of the decision variables, called operating variables. In this work we propose a derivative free optimization algorithm that tries to exploit as much as possible the particular proprerties of the constraints. The presented approach derives in an efficient way from the approaches reported in [1, 2]. The performances of the proposed algorithm are compared with the optimization methods in Wicked, a suite for circuit analysis, modeling, sizing, optimization and surrogate model generation that is widely used in the industrial sector for circuit analysis. References Nonlinear Programming 1 (Palagi) 120 [1] Lucidi, Stefano, and Marco Sciandrone. 2002. “A derivative-free algorithm for bound constrained optimization.” Computational Optimization and applications 21 (2): 119–142. [2] Lucidi, Stefano, Marco Sciandrone, and Paul Tseng. 2002. “Objective-derivative-free methods for constrained optimization.” Mathematical Programming 92 (1): 37–59. Nonlinear Programming 1 (Palagi) 121 A two-objectives optimization of liner itineraries for a cruise company Gianni Di Pillo∗ Dipartimento di Ingegneria informatica automatica e gestionale A. Ruberti, Sapienza Università di Roma, Italia, [email protected] Marcello Fabiano Stefano Lucidi Massimo Roma Dipartimento di Ingegneria informatica automatica e gestionale A. Ruberti, Sapienza Università di Roma, Italia, [email protected] [email protected] [email protected] Abstract. We consider the problem of a cruise company which manages a liners fleet, and aims to optimizing the cruise itineraries in a given maritime area in order to minimize the itinerary costs due to fuel and port costs, and to maximize some attractiveness index of the itinerary. The problem turns out to be a bi-objective optimization problem. As to the first objective, the fuel consumption depends nonlinearly on the speed of the cruise liner; the speed depends on the distances between the ports and the deadlines for entering and leaving the port; the port cost depends on the port and on the services provided by the port. As to the second objective, it is evaluated by giving a rating to each port. The constraints of the problem are due to the cruise duration, to the minimum and maximum numbers of ports to be visited, to the allowable time window for the stay in the port, to the time windows for arrival and departure in the port, to the fact that for each liner only a subset of the ports in the maritime area is feasible. The decision variables specify the ports to be visited by each liner in each day, the arrival and departure times, which determine the itinerary of each cruise, and the navigation speed of each liner between two given ports. As far as we are aware, this problem appears to be tackled for the first time, and no references on the subject seem to be available in the literature. The problem is of great interest for the a major cruise company which supports the development of a software application to be used in planning the deployment of its fleet. The problem has been modelled as a MILP with two objective, whose solution gives the Pareto efficient frontier in the space costs-attractiveness. We present some computational results, obtained using the real data provided by the cruise company. Nonlinear Programming 2 (invited by Palagi) Tuesday 8, 15:30-17:00 Sala Gerace 122 Nonlinear Programming 2 (Palagi) 123 Finding all solutions of Nash equilibrium problems with discrete strategy sets Simone Sagratella∗ Dipartimento di Ingegneria Informatica Automatica e Gestionale “Antonio Ruberti”, Sapienza Università di Roma, Italia, [email protected] Abstract. We present a new method for finding all solutions of Nash equilibrium problems with integrality constraints. This method mimics the branch and bound paradigm for discrete optimization problems. It consists of a systematic enumeration of candidate solutions by means of strategy space search. We propose a new procedure to fix variables during the tree exploration by exploiting the continuous relaxation of the problem. Numerical results show that our method outperforms other existing methods both when computing one single solution and when computing the whole solution set. Problem definition We consider a Nash equilibrium problem (NEP) with N players and denote by xν ∈ Rnν the vector representing the ν-th player’s strategy. We further define the 0 n ν −ν vector x−ν := (xν )N ν6=ν 0 =1 and write R 3 x := (x , x ), where n := n1 + · · · + nN . Each player solves an optimization problem in which the objective function depends on the other players’ variables, while the feasible set is defined by convex constraints depending on the player’s variable only, plus integrality constraints: minxν θν (xν , x−ν ) g ν (xν ) ≤ 0 (30) xν ∈ Znν , where θν : <n → < and g ν : <nν → <mν . We call this problem as discrete Q ν NEP. Let us define X ν := {xν ∈ <nν : g ν (xν ) ≤ 0} and X := N X , a point ν=1 n x∗ ∈ X ∩ Z is a solution (or an equilibrium) of the discrete NEP if, for all ν, xν∗ is an optimal solution of problem (30) when one fixes x−ν to x−ν ∗ , that is: ν −ν θν (xν∗ , x−ν ∗ ) ≤ θν (x , x∗ ), ∀ xν ∈ X ν ∩ Znν . Discrete NEPs include a variety of realistic models, such as market-clearing auctions in electric power markets and network equilibria in energy markets, see e.g. [2]. By its discrete nature, reformulations for a discrete NEP by using KKT conditions or variational inequalities are impossible. However it is easy to see that relaxing integrality and then solving the resulting continuous NEP may produce an integer solution, which then is also a solution for the original discrete problem, see [2] and [3]. However, in general, these favorable solutions are only a subset of the solution set of a discrete NEP. Moreover very often a discrete NEP can have multiple solutions but none favorable ones. As an example, let us consider a toy Nonlinear Programming 2 (Palagi) 124 problem in which there are two players each controlling one variable: 9 2 2 9 1 2 1 2 1 1 2 1 2 2 1 2 min θ min θ 2 (x , x ) = (x ) + 7x x − 72x 1 (x , x ) = (x ) + 7x x − 72x x2 x1 2 2 0 ≤ x1 ≤ 9 0 ≤ x2 ≤ 9 x1 ∈ Z x2 ∈ Z This discrete NEP has the following equilibria: (3, 6), (4, 5), (5, 4) and (6, 3). By relaxing integrality the resulting continuous NEP has the unique solution (4.5, 4.5) which is not integer. Finding all solutions of a discrete NEP is a challenging task, and it is not trivial how to effectively exploit its continuous relaxation. A branching method for finding all equilibria We present a new method for finding all solutions of the discrete NEP (30). This method mimics the branch and bound paradigm for discrete and combinatorial optimization problems. It consists of a systematic enumeration of candidate solutions by means of strategy space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, and during the enumeration of the candidate solutions a procedure is used in order to reduce the strategy set, in particular by fixing some variables. This procedure exploits the continuous relaxation of the problem and solves the resulting continuous NEP, see e.g. [1]. It is important to say that bounding strategies, which are standard for discrete and combinatorial optimization problems, in general cannot be easily applied for discrete NEPs because of the difficulty to compute a lower bound for each branch. Numerical results show that our method outperforms other existing methods when the task is computing one equilibrium of the discrete NEP, but also when we want to compute the whole solution set. References [1] Dreves, A., Facchinei, F., Kanzow, C., Sagratella, S., On the solution of the KKT conditions of generalized Nash equilibrium problems, SIAM J. Optim. 21(3) (2011), 1082–1108. [2] Gabriel, S.A., Conejo, A.J., Ruiz, C., Siddiqui, S., Solving discretely constrained, mixed linear complementarity problems with applications in energy, Comput. Oper. Res., 40(5) (2013), 1339–1350. [3] Yang, Z., On the solutions of discrete nonlinear complementarity and related problems, Math. Oper. Res., 33(4) (2008), 976–990. Nonlinear Programming 2 (Palagi) 125 A Linesearch Derivative-free Method for Bilevel Optimization Problems Stefania Renzi∗ Department of Computer, Control and Management Engineering Antonio Ruberti, Sapienza University of Rome, Italy, [email protected] Stefano Lucidi Department of Computer, Control and Management Engineering Antonio Ruberti, Sapienza University of Rome, Italy, [email protected] Abstract. In this work we consider bilevel optimization problems with a supposed strictly convex lower level program. In this framework we propose a Derivative Free Non-Smooth algorithm that, for each explored trial point, computes the solution of the lower level problem. The results of the preliminary numerical experience seem to be promising from a practical point of view. Introduction Bilevel optimization problems are programs with a particular nested structure. “ min ” x,y F (x, y) s.t. x ∈ X y ∈ S(x), where S(x) = arg min z s.t. z ∈ Z, f (x, z) (31) (32) with X and Z feasible sets composed by general constraints, F : <nx ×ny 7→ < (upper level objective) and f : <nx ×ny 7→ < (lower level objective) continuously differentiable functions. In such a structure a problem, called lower lever or inner Problem (32), is nested inside the feasible region of another problem, called upper lever or outer Problem (31). When, for any fixed upper level choice, the lower level optimal solution is not uniquely determined then the outer minimization can be illdefined. To address this pathology there are two main approaches: Optimistic and Pessimistic. In order to overcome this possible issue, we focused on bilevel programs with a supposed strictly convex lower level problem. Based on this assumption, we propose a new linesearch derivative-free resolution algorithm. Derivative Free (or Direct Search) methods do not required knowledge or calculation of derivative and for this reason they are widely used to solve real world problems. The idea underlying this work is to use a Derivative Free Non-Smooth algorithm that, for each trial point which it explores, will compute the solution of the lower level problem. Roughly speaking, at each iteration of the algorithm an appropriate set of directions is explored in order to find a new point where the objective function value is sufficiently lower than the one assumed in the current optimal point. Under suitable assumptions it is possible to prove that an accumulation point of the sequence produced by the algorithm is a stationary point of the considered problem. With the aim to evaluate the performance of the proposed algorithm different numerical tests have been performed showing a possible practical interest in the procedure. Nonlinear Programming 2 (Palagi) 126 A Fast Branch-and-Bound Algorithm for Non-convex Quadratic Integer Optimization Subject To Linear Constraints Using Ellipsoidal Relaxations Laura Palagi∗ Dipartimento di Ingegneria informatica automatica e gestionale A. Ruberti, Sapienza Università di Roma, Italia, [email protected] Christoph Buchheim Marianna De Santis Fakultät für Mathematik, Technische Universität Dortmund, Germany, [email protected] [email protected] Abstract. We propose two exact approaches for non-convex quadratic integer minimization subject to linear constraints where lower bounds are computed by considering ellipsoidal relaxations of the feasible set. In the first approach, we intersect the ellipsoids with the feasible linear subspace. In the second approach we penalize exactly the linear constraints. We investigate the connection between both approaches theoretically. Experimental results show that the penalty approach significantly outperforms CPLEX on problems with small or medium size variable domains. We address quadratic integer optimization problems with box constraints and linear equality constraints, min q(x) = x> Qx + c> x s.t. Ax = b (33) l≤x≤u x ∈ Zn , where Q ∈ Rn×n is assumed to be symmetric but not necessarily positive semidefinite, c ∈ Rn , and w.l.o.g. A ∈ Zm×n and b ∈ Zm . Moreover, we may assume l < u and l, u ∈ Zn . Problems of this type arise, e.g., in quadratic min cost flow problems, where the linear equations model flow conservation, l = 0 and u represents edge capacities. Problems of type (33) are very hard to solve in theory and in practice. In general, the problem is NP-hard both due to the integrality constraints and due to the nonconvexity of the objective function. Few exact algorithms have been proposed in the literature so far, most of them based on either linearization or convexification [1],[5] or on SDP-relaxations [3]. For the variant of (33) containing only box constraints, but no other linear constraints, Buchheim & al. in [2] recently proposed a branch-and-bound algorithm based on ellipsoidal relaxations of the feasible box. More precisely, a suitable ellipsoid E containing [l, u] is determined and q(x) is minimized over x ∈ E. The latter problem is known as the trust region subproblem and can be solved efficiently, thus yielding a lower bound in our context. Besides many other improvements, this branch-and-bound algorithm mostly relies on an intelligent preprocessing technique that allows solving the dual of a trust region problem in each node of Nonlinear Programming 2 (Palagi) 127 the enumeration tree in a very short time, making it possible to enumerate millions of nodes in less than a minute. Our aim is to adapt this method to the presence of linear equations Ax = b. For this, we propose two different, but related approaches. In the first approach, we intersect the ellipsoid E with the subspace given by Ax = b. This leads, by considering an appropriate projection, to a trust region type problem that, in principle, can still be solved efficiently. In the second approach, we instead lift the constraints into the objective function by adding a penalty term M ||Ax − b||2 drawing inspiration from the augmented Lagrangians theory [4]. A finite and computable real value M̄ > 0 exists such that the resulting quadratic problem with only the simple constraints [l, u] ∩ Zn is equivalent to (33). Thus the branch-and-bound algorithm defined in [2] which uses an ellipsoidal relaxation E of the feasible set can be used in a straighforward way. References [1] Billionnet, A., Elloumi, S., Lambert, A., A branch and bound algorithm for general mixedinteger quadratic programs based on quadratic convex relaxation, Journal of Combinatorial Optimization 28 (2), 2014, 376–399 [2] Buchheim, C., De Santis, M., Palagi, L., Piacentini, M., An exact algorithm for nonconvex quadratic integer minimization using ellipsoidal relaxations, SIAM Journal on Optimization 23, (2013) 1867–1889. [3] Buchheim, C., Wiegele, A., emidefinite relaxations for non-convex quadratic mixed-integer programming, Mathematical Programming 141 (1-2), 2013, 435–452 [4] Poljak, S., Rendl, F., Wolkowicz, H., A recipe for semidefinite relaxation for (0,1)-quadratic programming, Journal of Global Optimization 7 (1), (1995), 51–73. [5] Sherali, H. D., Adams, W. P., A reformulation-linearization technique for solving discrete and continuous nonconvex problems, (1999), Vol. 31 of Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht. Nonlinear Programming 3 (invited by Palagi) Tuesday 8, 17:30-19:00 Sala Gerace 128 Nonlinear Programming 3 (Palagi) 129 Stochastic Dual Newton Ascent: Theory and Practical Variants Luca Bravi∗ Dipartimento dell’Ingegneria dell’Informazione, Università di Firenze, Italia, [email protected] Peter Richtárik School of Mathematics, University of Edinburgh, UK, [email protected] Abstract. In this work we focus on stochastic algorithms for regularized empirical loss minimization problems. A Stochastic Dual Newton Ascent (SDNA) algorithm is shown and several variants of the method aimed at speeding up its performance are described. Experiments both on real and artificial datasets are shown. Introduction In the big data age new algorithmic challenges have arisen. In many applications among which machine learning is prominent, problems of unprecedented sizes are increasingly common. Classical algorithms are not designed for such problems and hence new approaches are needed. First-order stochastic algorithms have shown to outperform classical algorithms. Recently, there have been several attempts to design fast and scalable methods combining randomization with a cheap but effective use of curvature information. In this talk I will describe Stochastic Dual Newton Ascent (SDNA) [1], which is a natural extension of the celebrated Stochastic Dual Coordinate Ascent (SDCA) [2] method to a setting where cheap second order information is used to speedup convergence. This effect is both theoretical and practical. SDNA at each iteration picks a random subset of dual variables, which corresponds to picking a minibatch of examples in the primal problem, utilizing all curvature information available in the random subspace. Even relatively small minibatch sizes lead SDNA to require fewer iterations to convergence, but this improvement does not come for free: as the minibatch size increases the subproblems solved at each iteration involve a larger portion of the Hessian. In this work this problem is faced, in particular I will describe several variants of the algorithm aimed at speeding up its practical performance. Results on artificial and real dataset are reported to show the effectiveness of the proposed solution. References [1] Qu Z., Richtárik P., Takác̃ M., Fercoq O., SDNA: Stochastic Dual Newton Ascent for Empirical Risk Minimization, arXiv preprint arXiv:1502.02268. (2015). [2] Shalev-Shwartz S., Zhang T., Stochastic dual coordinate ascent methods for regularized loss. The Journal of Machine Learning Research, 14(1), 567-599, (2013). Nonlinear Programming 3 (Palagi) 130 Exploiting conjugacy for defining approximate inverse preconditioning strategies, in large scale unconstrained optimization Andrea Caliciotti∗ Dipartimento di Ingegneria Informatica, Automatica e Gestionale, SAPIENZA - Università di Roma, Italia, [email protected] Giovanni Fasano Dipartimento di Management Università Ca’Foscari Venezia, Italia, [email protected] Massimo Roma Dipartimento di Ingegneria Informatica, Automatica e Gestionale, SAPIENZA - Università di Roma, Italia, [email protected] Abstract. In this work we propose the use of Conjugate Gradient (CG)–based methods for constructing preconditioners, in the framework of large scale unconstrained optimization. The basic idea of our approach draws its inspiration from Approximate Inverse Preconditioners, which have proved, to large extent, to be efficient when solving linear systems. This work has several purposes, and investigates different issues related to preconditioning in a matrix–free framework. Contents The basic idea of our approach draws inspiration from Approximate Inverse Preconditioners [1] for linear systems. These methods claim that in principle, on several classes of problems an approximate inverse of system matrix should be computed, and used as a preconditioner. However, in practice it might be difficult to ensure that the resulting preconditioner collects enough information on the inverse matrix, while preserving positive definiteness and using matrix–free techniques. To the latter purpose we use the information obtained as by product from CG– based methods, in order to both exploit information on system matrix and preserve a specific representation of the preconditioner. Then, further technicalities are introduced, so that positive definiteness along with spectral analysis can be pursued. In particular, we analyze a couple of different frameworks where either any Krylov– based method can be adopted or the Preconditioned Nonlinear CG (PNCG) is applied. In the first part, we propose a parameter dependent class of preconditioners for a linear system, in order to capture the structural properties of the system matrix. Our proposal predicts the shift of some eigenvalues of the preconditioned system matrix to a specific value. We can apply any Krylov-subspace method (or possibly L-BFGS) to build our preconditioners, needing to store just a small tridiagonal matrix, without requiring any product of matrices. As we collect information from Krylov-subspace methods, we assume that the necessary information is gained by matrix-free procedures, namely routines which provide the product of the matrix times a vector. Note that, typically, this approach allows fast parallel computing, which is another possible advantage of our proposal in large scale settings. The general preconditioners proposed in this work depend on several parameters, whose Nonlinear Programming 3 (Palagi) 131 effect is substantially that of exalting the information on the system matrix collected by the Krylov subspace method. In the second part we focus on the PNCG, which can be seen [2] as a natural extension of the Preconditioned CG (PCG) to general functions. Although the PNCG method has been widely studied and often performs very efficiently on large scale problems, a key point for increasing its effectiveness is the use of a smart preconditioning strategy, especially when dealing with non–convex ill–conditioned problems. On this guideline, this work is also devoted to combine quasi–Newton methods and PCG, since we investigate the use of quasi–Newton updates as preconditioners for the NCG. In particular, we want to propose a class of preconditioners, which possibly inherit the effectiveness of the L–BFGS update, and satisfy the current secant equation. This represents an attempt to improve the efficiency of the NCG method, by conveying information collected from a quasi–Newton approach in the PNCG. In particular, we study new symmetric low–rank updates of our preconditioners. It is worth noting that there exists a close connection between BFGS and NCG, and on the other hand, NCG algorithms can be viewed as memoryless quasi–Newton methods [2], [3]. The idea of using a quasi–Newton update as a preconditioner within CG or NCG algorithms is not new. As an example, in [4] a scaled memoryless BFGS matrix is used as preconditioner for the NCG. Moreover, an automatic preconditioning strategy based on a limited memory quasi–Newton update for the linear CG is proposed in [5], within Hessian–free Newton methods, and is also extended to the solution of a sequence of linear systems. References [1] M. Benzi, and M. Tuma, A comparative study of sparse approximate inverse preconditioners, Applied Numerical Mathematics, 30 (1999), 305–340. [2] J. Nocedal, and S. Wright, Numerical Optimization (Springer Series in Operations Research and Financial Engineering - Second Edition), Springer, New York (2000). [3] R. Pytlak, Conjugate Gradient Algorithms in Nonconvex Optimization, Springer, Berlin (2009). [4] N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optimization Methods and Software, 22 (2007), 561–571. [5] J.L. Morales and J. Nocedal, Automatic preconditioning by limited memory quasi–Newton updating, SIAM Journal on Optimization, 10 (2000), 1079–1096. Nonlinear Programming 3 (Palagi) 132 Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces, in Optimization Frameworks Giovanni Fasano∗ Dipartimento di Management, Università Ca’Foscari Venezia, Italia, [email protected] Riccardo Gusso Raffaele Pesenti Dipartimento di Management, Università Ca’Foscari Venezia, Italia, [email protected] [email protected] Abstract. We recall results from the theory of Polarity, for quadratic hypersurfaces in homogeneous coordinates. The latter is then used to recast geometric properties of Conjugate Gradient (CG)-based methods. Our analysis draws its inspiration from [1], by Hestenes and Stiefel, and has three main theoretical purposes: we exploit Polarity for the CG, we suggest extensions of Polarity to treat conjugacy for indefinite linear systems, then we extend Polarity to study Planar-CG methods. Introduction We consider the Conjugate Gradient methods (CG-based methods) for the solution of linear systems [1], within the framework of the theory of Polarity. Polarity allows to approach CG-based methods from a different standpoint than a pure algorithmic one. We introduce this perspective as we believe that it can foster a new line of research on CG-based methods. Indeed, Polarity may provide a geometric framework to describe the properties of these methods [2]. In turn, the geometric framework motivates the extension of the use of CG-based methods for the solution of indefinite linear systems and, possibly, for the search of stationary points of polynomial functions. Furthermore, Polarity can be used to recast the Planar-CG methods [3], [4] for the solution of indefinite linear systems. We also conjecture, given the relation existing between the CG and BFGS or LBFGS methods, that this perspective may suggest a possible geometric insight for the Quasi-Newton updates. To the best authors’ knowledge, the use of Polarity to detail the CG was just hinted by Hestenes and Stiefel in [1], but since then little can be found in the literature. In fact, most of the literature on the CG devotes great effort to its algebraic formalization, in order to improve and exploit more and more its computational characteristics. Nowadays one prefers to see CG as a method which permits the transformation of a matrix in tridiagonal form. Although this line of research is certainly of great importance, it is out of the scope of the present work. Here, we focus on some theoretic characteristics of CG-based methods. Specifically, on the ones that justify the use of CG-based methods as solvers for linear systems, in particular within optimization algorithms for large scale optimization problems. In this last framework, CG-based methods are used to solve either positive definite and indefinite linear systems whose approximate solutions, on one side, define gradient-related directions that assure the global convergence of the optimization algorithms; on the other side, guarantee also a fast convergence to a stationary Nonlinear Programming 3 (Palagi) 133 point. Specifically, for unconstrained problems, a proper application of CG-based methods may often yield superlinear convergence. In this work, we initially introduce homogeneous coordinates and basic concepts of projective geometry [5] that are exploited by Polarity. Then, we show how the main characteristics of the CG, that hold in a natural fashion in Rn , have a counterpart in homogeneous coordinates. We show also that Polarity justifies the application of CG-based methods to the solution of linear systems both in the definite and indefinite case. In the latter situation we exploit the properties of the asymptotic cone of quadratic hypersurfaces. Finally, we observe that fundamental theorems of the theory of Polarity, such as the Permutability Theorem, Reciprocity Theorem and Section Theorem [5], apply to general algebraic hypersurfaces and then may possibly suggest further guidelines to study the geometry underlying the Nonlinear CG-based methods and BFGS. References [1] M. Hestenes, and E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards, 49 (1952), 409–435. [2] M. Hestenes, Conjugate Direction Methods in Optimization, Springer Verlag, New York, Heidelberg, Berlin (1980). [3] D. Luenberger, Hyperbolic pairs in the Method of Conjugate Gradients, SIAM Journal on Applied Mathematics, 17 (1969), 1263–1267. [4] G. Fasano, Planar-Conjugate Gradient algorithm for Large Scale Unconstrained Optimization, Part 1: Theory, Journal of Optimization Theory and Applications, 125 (2005), 543– 558. [5] M. Beltrametti, E. Carletti, D. Gallarati, and G. Monti Bragadin, Lectures on Curves, Surfaces and Projective Varieties - A Classical View of Algebraic Geometry, European Mathematical Society (2009). Nonlinear Programming 4 (invited by Palagi) Wednesday 9, 9:00-10:30 Sala Gerace 134 Nonlinear Programming 4 (Palagi) 135 An optimization-based method for feature ranking in nonlinear regression problems Veronica Piccialli∗ Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università degli studi di Roma Tor Vergata, Italia, [email protected] Luca Bravi Marco Sciandrone Dipartimento di Ingegneria dell’Informazione, Università di Firenze, Italia, [email protected] [email protected] Abstract. We consider the feature ranking problem consisting in associating a score with the features in order to assess their relevance. We focus on regression problems and give a formal definition of relevance of a feature by introducing a minimum zero-norm inversion problem of a neural network that leads to a smooth, global optimization problem to be solved in order to assess the relevance of the features. Computational experiments show the effectiveness of the proposed method. Feature ranking via inversion of a neural network In supervised machine learning a set of training instances is available, where each instance is defined by a vector of features and a label. Classification and regression are learning techniques to build predictive models using the available training data. In a classification task the labels belong to a finite set, while in a regression task the labels are continuous. Reducing the high dimensionality of real-world data has become increasingly important in machine learning and, in this context, feature selection plays a crucial role. The problem of feature selection is that of determining a subset of features in such a way that the accuracy of the predictive model built on the training data, containing only the selected features, is maximal. In this work we consider regression tasks and we focus on the feature ranking problem which consists in computing a score for each feature (a high score indicates that the feature is highly “relevant”). We are interested in developing a multivariate ranking method that assesses the relevance of the features processing them simultaneously. The training data are used, first, to build an analytical model of the process using all the features. Then, the training set and the built mathematical model are employed together to measure the “relevance” of the features. In both the two phases the original features are considered simultaneously. The attempt of the paper is that of providing a feature ranking technique starting from a formal notion of relevance of a variable. The peculiarity of our approach is to state the notion in terms of a well-defined optimization problem involving both the mathematical model underlying the data and the training instances. The ranking is built by solving multiple instances of a global optimization problem. We test the method by comparing it on both artificial and real datasets, and we compare it with other existing packages representative of different classes of feature selection methods. The numerical results show the effectiveness of the method that increases with the goodness-of-fit of the built model to the data. Nonlinear Programming 4 (Palagi) 136 Decomposition algorithms for traffic assignment problems Alessandro Galligari∗ Dipartimento di Ingegneria dell’Informazione, Università di Firenze, Italia, [email protected] Marco Sciandrone Niccolò Bulgarini Dipartimento di Ingegneria dell’Informazione, Università di Firenze, Italia, [email protected] [email protected] Abstract. In this work we focus on continuous optimization problems underlying traffic assignment models. We present decomposition algorithms both for computing a user-equilibrium solution with elastic demand and for the calibration of the network parameters. A theoretical convergence analysis is performed. The numerical results of an extensive set of computational experiments are presented. Introduction Traffic assignment concerns the problem of forecasting the loadings on a transportation network where users choose routes from their origins to their destinations. Assignment can be performed using one of the two alternative principles enunciated by Wardrop: • the user-optimal principle, which leads to a network equilibrium where the travel costs of all the routes actually used are equal to or less than those on nonutilized routes; • the system-optimized principle, stating that the choices of the users are such that a system aggregate criterion is optimized. We focus here on the traffic assignment problem based on the user-optimal principle. The network is modelled by a direct graph, whose nodes represent origins, destinations, and intersections, and arcs represent the transportation links. There is a set of node pairs, called Origin/Destination (OD). The most standard traffic assignment model assumes fixed demand and separable cost functions. We consider here the more realistic case of elastic demand, that is, the demand for each OD pair depends on the cheapest route cost. Once a solution is found, it represents the traffic assignment for the network with an estimation of the transportation demand for each OD pair. However, in real cases, network parameters are not known at priori and are generally estimated starting from measurements in some points of the network such as speed traps, vehicles counters, satellite tracking systems, ecc. Henceforth, our goal is providing a general optimization framework to deal with a bilevel programming formulation of the traffic assignment in which network parameters have to be estimated. The framework can be summarized in two main points: Nonlinear Programming 4 (Palagi) 137 • let P the given set of network parameters, a novel decomposition algorithm is performed for the traffic assignment problem with elastic demand that extends the one described in [1]; • let W (P) a black-box function that returns the network cost of the assignment as a function of P, global optimization with derivative-free algorithms is performed for the estimation of P in a decomposition way, as is described in [3]. References [1] Di Lorenzo D., Galligari A., Sciandrone M. , A convergent and efficient decomposition method for the traffic assignment problem, Computational optimization and applications (2014). [2] Seungkyu Ryu, Anthony Chen, Keechoo Choi , A modified gradient projection algorithm for solving the elastic demand traffic assignment problem, Computers & Operations Research, Volume 47, (2014). [3] Ubaldo M. Garcı́a-Palomares, Ildemaro J. Garcı́a-Urrea, On sequential and parallel nonmonotone derivative-free algorithms for box constrained optimization, Optimization Methods and Software, (2013). Nonlinear Programming 4 (Palagi) 138 Feasible methods for nonconvex nonsmooth problems: Decomposition and applications Francisco Facchinei∗ Dipartimento di Ingegneria informatica automatica e gestionale Antonio Ruberti, Università di Roma La Sapienza, Italia, [email protected] Lorenzo Lampariello Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Università di Roma La Sapienza, Italia, [email protected] Gesualdo Scutari Department of Electrical Engineering, State University of New York (SUNY) at Buffalo, USA, [email protected] Abstract. We propose a general feasible method for nonsmooth, nonconvex constrained optimization problems. The algorithm is based on the (inexact) solution of a sequence of strongly convex optimization subproblems, followed by a step-size procedure. Two key new features of the scheme are: i) it preserves feasibility of the iterates for nonconvex, nonsmooth problems with nonconvex constraints; and ii) it naturally leads to parallel/distributed implementations in settings that are not covered by other methods, even in the case of smooth problems. We illustrate the application of the method to an open problem in the distributed cross-layer design of multihop wireless networks. Numerical Methods for Complex Networks (invited by Fenu) Tuesday 8, 15:30-17:00 Sala Seminari Ovest 139 Numerical Methods for Complex Networks (Fenu) 140 Edge modification techniques for optimizing network communicability Francesca Arrigo∗ Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Italia, [email protected] Michele Benzi Department of Mathematics and Computer Science, Emory University, USA, [email protected] Abstract. We show how to add, delete, and rewire edges in a network in order to change its ability to diffuse information among the nodes according to certain goals. We propose different edge selection techniques which can be efficiently implemented and are applicable in a wide variety of situations. Our techniques select the modifications so as to tune the total communicability of the network (see [3], [2]), which is an index that measures the ease of spreading information across the network. The choice of the edges to be modified involves the use of centrality measures which can be computed using fast sparse matrix algorithms. This talk is based on the papers [1] and [2]. References [1] Arrigo, F. and Benzi, M., Updating and downdating techniques for optimizing network communicability, arXiv: 1410.5303 (2014). [2] Arrigo, F. and Benzi, M., Heuristics for optimizing the communicability of digraphs, in preparation. [3] Benzi, M. and Klymko, C., Total communicability as a centrality measure, J. Complex Networks 1(2) (2013), pp. 124–149. Numerical Methods for Complex Networks (Fenu) 141 The effectiveness of spectral methods in community detection Dario Fasino∗ Dipartimento di Chimica, Fisica e Ambiente, Università di Udine, Italia, [email protected] Francesco Tudisco Department of Mathematics and Computer Science, Saarland University, Saarbrücken, Germany, [email protected] Abstract. Most popular techniques for community detection are based on the maximization of a merit function called modularity, which in turn is based upon the quadratic form associated with a real symmetric modularity matrix. Various modularity-type matrices have been proposed in the recent literature on community detection, which share a common underlying structure. We revise here certain spectral properties of these matrices, mainly addressing their eigenvalue distributions and certain features of leading eigenvectors, which provide theoretical foundation to the effectiveness of various numerical methods in community detection. Community detection via modularity maximization A relevant aim in complex network analysis is the detection of tightly intraconnected subnetworks, usually referred to as communities. Community detection problems differ from graph partitioning in that communities are not always delimited by small graph cuts. Moreover, the number and size of the sought subgraphs are generally not apriori specified. Instead it is assumed that the graph is intrinsically structured into communities or groups of vertices which are more or less evidently knit, the aim being to reveal the presence and the consistency of such groups. Although there is no clear or universally accepted definition of community in a graph, almost all recent definitions and community detection methods are based on a quadratic merit function related with a special graph matrix, called modularity matrix. Hence, the localization of leading communities, or the partitioning of a given graph into (possibly overlapping) communities, is takled by the combinatorial optimization of such modularity-type functionals over subsets or partitions of the given graph. Modularity matrices and spectral methods Spectral methods in community detection basically arise as continuous relaxations of the optimization problems related to modularity-based functions. They exploit eigenvalue distributions and level sets of the leading eigenvectors of modularity matrices in order to localize principal communities, or construct graph partitionings with optimal modularity. Despite the wealth of different modularity matrices that appeared in the recent literature on community detection, all these matrices share a common additive structure. Indeed, they are built as a rank-one correction of the adjacency matrix Numerical Methods for Complex Networks (Fenu) 142 of a graph, possibly endowed by loops and weighted edges, which is derived from the original graph where community detection is considered. The common structure shared by all modularity matrices implies certain spectral properties which provide theoretical support to the apparent effectiveness of spectral methods in community detection. In particular, we can prove a relationship between the number of communities in a graph and the number of leading, positive eigenvalues; connectivity properties of the nodal domains of leading eigenvectors; and certain algebraic properties of the greatest eigenvalue. These properties can be also traced in modularity-type matrices arising from certain Markov chains whose states are graph edges, rather than vertices. References [1] D. Fasino, F. Tudisco. An algebraic analysis of the graph modularity. SIAM J. Matrix Anal. Appl., 35 (2014), 997–1018. [2] D. Fasino, F. Tudisco. Generalized modularity matrices, Linear Algebra Appl., to appear (2015). Numerical Methods for Complex Networks (Fenu) 143 Fast Computation of Centrality Indices Caterina Fenu∗ Department of Computer Science, University of Pisa, Italy, [email protected] James Baglama Department of Mathematics, University of Rhode Island, USA, [email protected] David Martin Lothar Reichel Department of Mathematical Sciences, Kent State University, USA, [email protected] [email protected] Giuseppe Rodriguez Department of Mathematics and Computer Science, University of Cagliari, Italy, [email protected] Abstract. One of the main issues in complex networks theory is to find the “most important” nodes. To this aim, one can use matrix functions applied to its adjacency matrix. We will introduce a new computational method to rank the nodes of both directed and undirected (unweighted) networks according to the values of these functions. The algorithm uses a low-rank approximation of the adjacency matrix, then Gauss quadrature is used to refine the computation. The method is compared to other approaches. Important Nodes Graphs and Complex Networks are used to model interactions between various entities in real life applications, e.g. in computer science, sociology, economics, genetics, epidemiology. A graph is a pair of sets G = (V, E), with |V | = n and |E| = m. The elements of V are called nodes or vertices and those of E are known as edges or arcs. If the edges can be travelled in both directions the network is said to be undirected, directed otherwise. The adjacency matrix corresponding to an unweighted graph is the matrix A ∈ Rn×n such that Aij = 1 if there is an edge from node i to node j, and Aij = 0 if node i and j are not adjacent. This kind of matrix is binary and, in general, nonsymmetric. One of the main issues in Complex Networks Theory is to find the “most important” nodes within a graph G. To this aim, various indices (or metrics) have been introduced to characterize the importance of a node in terms of connection with the rest of the network. The simplest and most classical ones are the in-degree and the out-degree, that is, the number of nodes that can reach one node or that can be reached from that node, respectively. These metrics do not give global information on the graph, since they only count the number of neighbors of each node. We will focus on indices that can be computed in terms of matrix functions applied to the adjacency matrix of the graph. We can define a class of indices starting from a matrix function f (A) = ∞ X cm Am , cm ≥ 0. m=0 Since [Am ]ij gives the number of paths of length m starting from the node i and ending at node j, [f (A)]ij is a weighted average of all the paths connecting i to j, and describes the ease of travelling between them. We refer to [f (A)]ii as the Numerical Methods for Complex Networks (Fenu) 144 f -centrality of node i, and [f (A)]ij as the f -communicability between node i and node j. In the literature, particular attention A has been reserved to the exponential function. In [4], [5], the authors refer to e ii as the subgraph centrality of node i and to eA ij as the subgraph communicability between node i and node j, in the case of an undirected graph. Recently, the notion of hub centrality and authority centrality has been introduced [3] in the case of a directed graph. Benzi and Boito [2], following the techniques described by Golub and Meurant [8], employed quadrature formulas to find upper and lower bounds of bilinear forms of the kind uT f (A)v (with u and v unit vectors) in the case of a symmetric adjacency matrix. If we assume that [f (A)]ii is a measure of the importance of node i, then we can identify the m most important nodes as the m nodes with the largest centrality. In order to do this using Gauss-type quadrature rules, we may apply this method with u = ei , for i = 1, . . . , n. Since a complex network is generally very large, this approach may be impractical. Low-rank approximation We will describe a new computational method to rank the nodes of both undirected and directed unweighted networks, according to the values of the above matrix functions. The idea is to reduce the cardinality of the set of candidate nodes in order to apply Gauss-type quadrature rules to a smaller number of nodes. The first part of the resulting algorithm, called hybrid method, is based on a low-rank approximation of the adjacency matrix. If the network is undirected a partial eigenvalue decomposition is used [6], while if the network is directed we make use of a partial singular value decomposition [1]. We will compare the hybrid algorithm to other computational approaches, on test networks coming from real applications, e.g. in software engineering, bibliometrics and social networks. We will also present a block algorithm to compute the entries of the matrix exponential for both symmetric and nonsymmetric matrices, which is particularly efficient on computers with a hierarchical memory structure. This algorithm is based on block Gauss and anti-Gauss quadrature rules. In the case of a nonsymmetric matrix the block approach is necessary to avoid breakdowns during the computation [7], [9]. References [1] J. Baglama, C. Fenu, L. Reichel, G. Rodriguez, Analysis of directed networks via partial singular value decomposition and Gauss quadrature, Linear Algebra Appl. 456 (2014), 93– 121. [2] M. Benzi, P. Boito, Quadrature rule-based bounds for functions of adjacency matrices, Linear Algebra Appl. 433 (2010), 637–652. [3] M. Benzi, E. Estrada, C. Klymko, Ranking hubs and authorities using matrix functions, Linear Algebra Appl. 438 (2013), 2447–2474. [4] E. Estrada, N. Hatano, Communicability in complex networks, Phys. Rev. E 77 (2008), 036111. Numerical Methods for Complex Networks (Fenu) 145 [5] E. Estrada, J. A. Rodrı́guez-Velázquez, Subgraph centrality in complex networks, Phys. Rev. E 71 (2005), 056103. [6] C. Fenu, D. Martin, L. Reichel, G. Rodriguez, Network analysis via partial spectral factorization and Gauss quadrature, SIAM J. Sci. Comput. 35 (2013), A2046–A2068. [7] C. Fenu, D. Martin, L. Reichel, G. Rodriguez, Block Gauss and anti-Gauss quadrature with application to networks, SIAM Journal on Matrix Analysis and Applications 34(4) (2013), 1655–1684. [8] G. H. Golub, G. Meurant, Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton, 2010. [9] D. R. Martin, Quadrature approximation of matrix functions, with applications, PhD Thesis, 2012. Optimisation for Sustainable Public Transport (invited by Galli) Tuesday 8, 17:30-19:00 Sala Seminari Est 146 Optimisation for Sustainable Public Transport (Galli) 147 A railway timetable rescheduling approach for handling large scale disruptions Valentina Cacchiani∗ Department of Electrical, Electronic and Information Engineering “Guglielmo Marconi”, University of Bologna, Italy, [email protected] Lucas P. Veelenturf School of Industrial Engineering, Eindhoven University of Technology, The Nederlands, [email protected] Martin P. Kidd Department of Management Engineering, Technical University of Denmark, Denmark, [email protected] Leo G. Kroon Rotterdam School of Management, Erasmus University, The Netherlands and Process quality & Innovation, Netherlands Railways, The Nederlands, [email protected] Paolo Toth Department of Electrical, Electronic and Information Engineering “Guglielmo Marconi”, University of Bologna, Italy, [email protected] Abstract. We focus on timetable rescheduling on a macroscopic level in a railway network, when large scale disruptions (e.g. the unavailability of a railway track) occur. We propose an Integer Linear Programming model to minimize the number of canceled and delayed train services. Train rerouting as well as all the stages of the disruption management process are considered. Computational tests on a heavily used part of the Dutch network show that the model finds optimal solutions in short computing times. An Integer Linear Programming Model A sustainable railway system must take into account several principles such as being customer-driven, i.e. to recognize the customers as the “heart” of the whole system: customers mainly require an efficient and reliable system, which provides high punctuality services. However, on a daily basis, large scale disruptions cause delays and train service cancellations, and require infrastructure managers and railway operators to reschedule their railway timetables together with their rolling stock and crew schedules. Due to its complexity, the recovery problem is usually decomposed into phases that are solved in sequence. The main phases consist of timetable rescheduling, rolling stock rescheduling and crew rescheduling (see e.g. [1]). In this work, we focus on the timetable rescheduling for passenger train services, thereby taking into account constraints from the rolling stock rescheduling phase in order to increase the probability of obtaining a feasible rolling stock schedule during the second phase. We study timetable rescheduling at a macroscopic level, i.e. with high level constraints disregarding detailed information on signals and routes inside stations or junctions, in order to deal with a complex real-world railway network in short computing times. We consider large-scale disruptions related to blockages of one or more railway tracks between stations for a certain period of time (e.g. two hours). Disruptions of Optimisation for Sustainable Public Transport (Galli) 148 this kind are very hard to manage by railway operators and infrastructure managers, and occur very frequently. The main contribution of this work consists of proposing an Integer Linear Programming (ILP) formulation for the timetable rescheduling problem under large scale disruptions on a real-world network. The ILP model is based on an EventActivity Network, represented by a directed graph N = (E, A), where E is the set of vertices (events) and A the set of arcs (activities). Graph N is associated with a set of train services T and an original timetable for these train services. The set E = Etrain ∪ Einv of events consists of a set Etrain of train events, and a set Einv of inventory events. Each train event e ∈ Etrain represents either a departure or an arrival at a certain station, and is associated with a set of resources (tracks in open track sections, tracks in stations, and rolling stock compositions) which it uses at the moment the event takes place. An inventory event e ∈ Einv represents the resource inventory of an open track section (i.e. the number of tracks in the open track section), a station (i.e. the number of tracks at the station), or the shunting yard at the start of the day (i.e. the number of available rolling stock compositions). Activities represent the sequential use of a resource (track of an open track section, track of a station or rolling stock composition) by two events. More precisely, an activity a = (e, f ) ∈ A directed from event e ∈ E to event f ∈ E denotes that event f uses one of the resources occupied by e, after e has taken place. Each activity a = (e, f ) ∈ A has an associated minimum duration La which is necessary for the specific resource used by e to become available for use by f . In summary, the activities determine the possible orders (and the corresponding time durations) in which the resource units are used for the events. Our work extends the ILP model presented in [2], in which two instances of a double track line were considered and a partial or full track blockage was taken into account. With respect to their work, we consider the complete disruption management process, i.e. from the start of the disruption to the time at which the normal situation is restored. Another new feature consists of dealing with a real-world railway network instead of one double track line. Furthermore, we take into account the possibility of rerouting train services along alternative geographical paths in the network in order to reduce the number of train services that are canceled or delayed. The proposed ILP model is solved to optimality by a general purpose solver on a set of real-world instances of Netherlands Railways (the major railway operator in the Netherlands) in short computing times. This makes our approach suitable for practical usage. References [1] Cacchiani, V., Huisman, D., Kidd, M., Kroon, L., Toth, P., Veelenturf, L., Wagenaar, J., An overview of recovery models and algorithms for real-time railway rescheduling, Transportation Research Part B: Methodological 63 (2014), 15–37. [2] Louwerse, I., Huisman, D., Adjusting a railway timetable in case of partial or complete blockades, European Journal of Operational Research 235(3) (2014), 583–593. Optimisation for Sustainable Public Transport (Galli) 149 Sustainable planning for service quality in public transport Massimo Di Francesco∗ Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] Benedetto Barabino Technomobility S.r.l., Cagliari, Italia, [email protected] Abstract. This study proposes a practical, simple and holistic framework for the involvement of all stakeholders in the stages of characterization, measurement and management of quality monitoring in public transport. These stages are investigated, integrated at different planning levels and discussed. The framework can plug-in optimization methods, whose inclusion guarantees a sustainable service planning. Motivations of this study Since service quality is important for all the stakeholders in the public transport industry [1], there is a strong interest in the old idea of the user perspective to drive service improvement and attain quality parameters in service contracts for possible award of financial bonuses. The application of a fully user-oriented approach to service quality monitoring in public transport requires consideration of different time frames, and the characterization, measurement and management of quality, which can be represented as quality blocks. Often bus operators use quality parameters which neglect passenger desires; they may ignore experts and use parameters, which are not immediately understandable by the average user, resulting in conclusions too focused on their perspective as a service provider. Bus operators usually collect and analyze data at system level, aggregating data and presenting results in terms of several outputs, but rarely evaluating the impact on passengers. As a result, the management of the process is typically blunt and bus-operator-dependent. In contrast transit quality theorists usually address quality monitoring from the user point of view, involving users and experts in the selection of quality parameters. However, because quality theorists may not have a detailed knowledge of specific services, they propose high-level parameters, which may not capture service performance on the ground. Quality theorists sometimes work on subjective data and present outputs as average grades. Even in this case, the management seems to be blunt, because passenger opinions may be too heterogeneous. Unsurprisingly, both bus operators and quality theorists have recently investigated the possible integration of their modus operandi. Therefore, it may be interesting to adopt a framework to bridge the gap between bus operators and users. Objectives of this study The goal of this paper is to provide a practical tool for monitoring service quality across a transit network and/or a route using selected parameters. The tool moves from some existing approaches to the characterization of service quality (e.g. [2]), Optimisation for Sustainable Public Transport (Galli) 150 the measurement and the management of quality (e.g. [3]; [4];[5]) and integrates them into a single framework. To sum up, our main objectives are: • To integrate characterization, measurement and management into a single framework, which also takes account of the time dimension. This integration represents an important achievement because, 1) it considers the characterisation of quality at strategic level, involving all stakeholders in the selection of parameters; 2) it addresses in detail users desires and the bus operators’ targets at tactical level, in order to collect at operational level objective and subjective data on relevant quality parameters; it takes account the difference between hard and soft parameters; 3) it addresses, albeit at operational level, the management process by focusing on gap analysis and prioritising problems and actions. • To present outputs in terms of percentages of passengers. This feature enables the evaluation of the impact of the service from passenger and bus operators side and provides an output, which can be easily understood by all stakeholders. • To close the quality loop by the calculating the magnitude of the four quality gaps and providing an integrated interpretation of them. This is a significant improvement in the state-of-art, which included many disjointed analyses, partial gap calculations and presentation of multiple outputs. References [1] CEN/TC 320. Transportation - Logistics and services, European Standard EN 15140: Public passenger transport - Basic requirements and recommendation for systems that measure delivered service quality, European Committee for Standardization, Brussels. Technical Report 2006. [2] Dell’Olio, L., Ibeas A., Cecin P., The quality of service desired by public transport users. Transport Policy 18 (2011), 217–227. [3] Liekendael, J.C., Furth, P.G., Muller, T.H.J., Service quality certification in Brussels, Belgium. A quality process with teeth. Transportation research record 1955 (2006), 88–95. [4] Stradling, S., Anable, J., Carreno, M., Performance, importance and user disgruntlement: a six method for measuring satisfaction with travel modes. Transportation Research Part A 41 (2007) 98–106. [5] Eboli, L., Mazzulla, G., A methodology for evaluating transit service quality based on subjective and objective measures from the passengers point of view. Transport Policy 18 (2011) 172–181. Optimisation for Sustainable Public Transport (Galli) 151 Risk mitigation and shipping costs in hazmat routing: the issue of carrier fairness Maddalena Nonato∗ Dipartimento di Ingegneria, Università di Ferrara, Italia, [email protected] Paola Cappanera Dipartimento di Ingegneria dell’Informazione, Università di Firenze, Italia, [email protected] Abstract. The Gateway Location Problem is the core of a new strategy for routing vehicles carrying hazardous materials. Each vehicle is assigned a compulsory gateway it has to cross on its way form origin to destination. Carriers pursue cost minimization while the Authority in charge of locating and assigning the gateways is risk aware. Any strategy devoted to reroute vehicles diverting them from their risky shortest path, achieves risk mitigation to the detriment of shipping costs. We focus on this compromise when handling individual carrier cost increase as an independent variable. Trading cost for risk mitigation The Gateway Location Problem (GLP) is at the core of a new risk mitigation strategy for routing vehicles carrying hazardous materials on a road network. In a free scenario, each vehicle would travel from origin to destination along its minimum cost path. This itinerary can be quite risky when, for example, crossing urban areas or driving on heavy traffic road segments. To divert vehicles away from their risky shortest path, the Governmental Authority in charge of hazmat shipment regulation in its jurisdiction can enforce a set of directives: carriers modify their choices to adapt to those rules. One policy tool is banning the use of certain road segments by hazardous vehicles, yielding hazmat network design problems that were first introduced by Kara and Verter in 2004 [6]. The alternative policy that we envision is based on the GLP, according to which each vehicle is assigned a compulsory gateway to be crossed on its way form origin to destination. Carriers pursue shipping cost minimization in traveling on the shortest path from origin to the assigned gateway and again, from that gateway to destination, while the Authority locates the gateways on the territory and assigns them to each shipment in a risk aware manner. This gives raise to a bi-level optimization problem in which the Authority takes into account drivers reaction to gateway assignments, as first discussed in [2]. Previous works experimentally assessed the risk mitigation potential of this new strategy even when very few gateways are used, selected from a limited set of candidate sites [3], as well as the ability of retaining full efficacy in case of variable demand if allowed to periodically relocate very few gateways [1]. However, any strategy devoted to reroute vehicles away from their shortest path, achieves risk mitigation to the detriment of shipping costs [5]. Since carriers collaboration is of paramount importance for the successful implementation of any risk mitigation policy and the transport sector must be kept economically viable, not only should carriers cost deterioration be carefully monitored but also possibly treated as an independent decision variable in the model. In the framework of hazardous network design, Verter and Kara in [7] proposed a path-based formulation Optimisation for Sustainable Public Transport (Galli) 152 in which itineraries not economically viable for carriers are simply left out of the model. The carriers point of view is represented at the cost of an explicit enumeration of the viable paths, which hardly scales well. Back to the GLP based strategy, in [4] it is shown that, in a system perspective, the set of computed itineraries lies very close to the risk/cost Pareto front of the non dominated solutions relating total risk to total cost, i.e., relating the sum for each vehicle of the shipping demand for the risk of the chosen itinerary to the sum for each vehicle of the shipping demand for the cost of the chosen itinerary. This means that, as a whole, GLP is able to obtain almost the maximum risk mitigation that is achievable for that cost increase. However, despite of the fact that on benchmark instances cost increase is rather low, there is no a priori guarantee on the maximum cost deterioration faced by individual carriers, so that some can be much more affected than others. To fill this gap and ensure fairness among carriers, in this study we take the individual carrier perspective. In particular, we analyze how the GLP optimization model can be generalized to include a custom set threshold on the percentage cost increase faced by individual carriers, and we experimentally evaluate on realistic data the impact on risk mitigation. This paves the way for more sophisticated systems in which carriers can be subsidized according to a personalized policy, on the basis of their availability to facing cost increases up to a predetermined and individually fixed threshold. References [1] Cappanera, P, Nonato, M., and F. Visintin, Routing hazardous material by compulsory check points in case of variable demand Proceedings of INOC 2015, to appear. [2] Bruglieri, M., P. Cappanera, A. Colorni, and M. Nonato, Modeling the gateway location problem for multicommodity flow rerouting, in Network Optimization: 5th International Conference, J. Pahl, T. Reiners and S. Voss (eds), LNCS 6701 (2011), 262–276, SpringerVerlag Berlin Heidelberg. [3] Bruglieri M., P. Cappanera, and M. Nonato, The Gateway Location Problem: Assessing the impact of candidate site selection policies, Discrete Applied Mathematics 165 (2014), 96-111. [4] Cappanera, P., and M. Nonato, The Gateway Location Problem: A Cost Oriented Analysis of a New Risk Mitigation Strategy in Hazmat Transportation, Procedia - Social and Behavioral Sciences 111 (2014), 918–926, . [5] Erkut, E., and O. Alp, Designing a road network for dangerous goods shipments, Computers and Operations Research 34(5) (2007) 1389–1405. [6] Kara, B.Y., and V. Verter, Designing a road network for hazardous materials transportation, Transportation Science 38(2) (2004), 188–196. [7] Verter, V., and B.Y. Kara, A path-based approach for hazmat transport network design, Management Science 54(1), (2008) 29-40. Optimization and Classification (invited by Gaudioso) Thursday 10, 9:00-10:30 Sala Gerace 153 Optimization and Classification (Gaudioso) 154 Spherical classification for detecting malicious URL Annabella Astorino∗ ICAR-CNR, c/o University of Calabria, 87036 Rende (CS), Italy, [email protected] Antonino Chiarello ICAR-CNR, c/o University of Calabria, 87036 Rende (CS), Italy, [email protected] Manlio Gaudioso Antonio Piccolo DIMES, University of Calabria, 87036 Rende (CS), Italy, [email protected] [email protected] Abstract. For computer security it is useful to have some tools for detecting malicious URL. To this aim we introduce a binary classification method based on Spherical Separation, which handles information both on the URL syntax and its domain properties. In particular we adopt a simplified model which runs in O(t log t) time (t is the number of training samples) and thus it is suitable for large scale applications. We test our approach using different sets of features and report the results in terms of testing correctness according to the well-established ten-fold cross validation paradigm. The problem A useful resource to prevent risks in computer security is provided by the so called black lists, which are data bases containing a typically large number of IP addresses, domain names and related URL’s for suspicious sites in terms of generation of threats. A rich literature is available on the creation and usage of such lists (see e.g. [7]). If a URL is comprised into a black list, it is convenient to deviate the network traffic from it and, in fact, many Internet Service Providers (ISP) simply block all messages coming from it. The users who detect anomalies in messages or activities they consider suspicious often transfer the related information to appropriate web sites devoted to risk analysis. Anther possible way to compile black lists is the use of certain spam trap addresses which are diffused in the aim of being contacted by crawler spiders, typically used by phishers. As soon as one of such site address is contacted, the calling site is included into the black list Although black lists are rather useful, we cannot expect that they are exhaustive of all possible threats, either because the number of potentially dangerous site is extremely high or because the system is highly dynamic and it is almost impossible to keep any black list sufficiently updated. Every time there exists any suspect about the malicious nature of a site the Whois service is able to provide some useful information in terms of IP, domain name and other characteristics related to it. Whois registers are publicly available and there exist online services providing upon request such information, in an appropriate form. The very basic idea of [4] is to use the information available about a given set of URL’s, in connection to the related Whois, to design a classifier based on some machine learning technique. In particular one of the tools adopted is the well known Optimization and Classification (Gaudioso) 155 SVM paradigm which, being suitable for supervised classification, requires that a sufficiently rich training set is available in advance. Such training set is constituted by a list of URL’s labeled in the form malicious-non malicious. A set of both qualitative and quantitative features is defined and each URL is associated to a string of possible values of the features. Of course different sets of features can be adopted for classification purposes and in next section we will describe in details the ones we have considered. The differences of our approach w.r.t. the one previously cited [4] are twofold. As we are aimed at providing a methodology suitable for very large datasets too, we have confined ourselves to a limited number of features (e.g. we have used no “bag of words”, which in general requires an explosion in the size of the sample space) and, on the other hand, we have adopted a low complexity algorithm, accepting in advance the possibility of obtaining less accurate classification performance w.r.t. the SVM approach which requires solution of a structured quadratic programming problem ([6], [5]). Following [4] we take into account in our classification model both lexical features of the URL and host information, as those provided by the Whois. As a classification tool, we adopt the spherical separation paradigm ([2], [1]), which differs from SVM basically because separation in the feature space is not pursued by means of a hyperplane, instead by a spherical surface. In addition we adopt a simplified approach to spherical separation [1] which allows us to calculate the classifier in O(t log t), where t is the size of the training set. References [1] Astorino, A., Gaudioso, M., A fixed-center spherical separation algorithm with kernel transformations for classification problems, Computational Management Science 6 (3) (2009), 357–372. [2] Astorino, A., Fuduli, A., Gaudioso, M., DC models for spherical separation, Journal of Global Optimization 48 (4) (2010), 657–669. [3] Astorino, A., Fuduli, A., Gaudioso, M., Margin maximization in spherical separation, Computational Optimization and Applications 53 (2) (2012), 301–322. [4] Ma J., Saul L. K., Savage S., Voelker, G. M., Beyond Blacklists: Learning to Detect Malicious Web Sites from Suspicious URLs, KDD09 (2009), 1245–1253. [5] Palagi L., Sciandrone M., On the convergence of a modified version of SV M light algorithm, Optimization Methods and Software 20 (2-3) (2005), 317–334. [6] Vapnik, V., The Nature of the statistical learning theory, Springer Verlag, New York, 1995. [7] Zhang, J., Porras, P., Ullrich, J., Highly Predictive Blacklisting, USENIX Security Symposium 2008 - usenix.org. Optimization and Classification (Gaudioso) 156 A semisupervised approach in spherical separation Antonio Fuduli∗ Dipartimento di Matematica e Informatica, Università della Calabria, Rende, Italia, [email protected] Annabella Astorino ICAR-CNR, c/o DIMES, Università della Calabria, Rende, Italia, [email protected] Abstract. We embed the concept of spherical separation of two finite disjoint set of points into the semisupervised framework. This approach appears appealing since in the realworld classification problems the number of unlabelled points is very large and labelling data is in general expensive. We come out with a model characterized by an error function which is nonconvex and nondifferentiable, that we minimize by means of a bundle method. Numerical results on some small/large datasets drawn from literature are reported. The proposed approach The objective of pattern classification is to categorize data into different classes on the basis of their similarities. The application field in this area is very vast: see for example text and web classifications, DNA and protein analysis, medical diagnosis, machine vision and many others. The mathematical programming approaches for pattern classification are based on separation of sets, an interesting field which has become increasingly relevant in the last years. In particular, in binary classification, the separation problem consists of finding an appropriate surface separating two discrete point sets. Here we focus on separating two finite disjoints sets of points by means of a sphere. Spherical classifiers have been used so far only in the supervised framework (see for example the recent works [1] [2]), where the error function is calculated only on the basis of the labelled samples, while, in the applications, often happens that both labelled and unlabelled samples are available. On the opposite, there exist also many approaches aimed at clustering the data by means of information coming only from the unlabelled points. This is the case of the unsupervised learning. The semisupervised classification [3] is a relatively recent approach consisting in classifying the data by learning from both the labelled and unlabelled samples. It is something halfway between the supervised and unsupervised machine learning. The main advantage is that in practical cases most of the data are unlabelled and then it could be appealing to entirely exploit the available information. A possible drawback is that, in general, differently from the supervised case, the error function may become more difficult to minimize. In this work we propose a semisupervised classification model based on spherical separation and characterized by a nonsmooth nonconvex error function. In particular, analogously to the TSVM technique (which is the semisupervised version of the well known SVM approach), we provide a separating sphere staying in the middle between two “support spheres” playing the same role as the support Optimization and Classification (Gaudioso) 157 hyperplanes in SVM. More precisely, we embed in our approach the idea coming from TSVM, taking into account the minimization of the number of unlabelled points in the margin zone, i.e. the area comprised between the two support spheres. Finally some numerical results are given. References [1] Astorino, A., Fuduli, A., Gaudioso, M., Margin maximization in spherical separation, Computational Optimization and Applications 53 (2012), 301–322. [2] Le Thi, H. A., Le, H. M., Pham Dinh, T., Van Huynh, N., Binary classification via spherical separator by DC programming and DCA, Journal of Global Optimization 56 (2013), 1393– 1407. [3] Chapelle, O, Schölkopf B., Zien A., Semi-supervised learning. MIT Press, Cambridge, 2006. Optimization and Classification (Gaudioso) 158 Edge detection via clustering Manlio Gaudioso∗ Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Università della Calabria, Italia, [email protected] Annabella Astorino Istituto di Calcolo e Reti ad Alte Prestazioni-C.N.R., Rende, Italia, [email protected] Pietro D’Alessandro ICT Sud, Italia, [email protected] Walaa Khalaf Computer and Software Engineering Dept., College of Engineering, AlMystansiriya University, Baghdad, Iraq, [email protected] Abstract. We introduce a method for edge detection based on clustering of the pixels representing any given digital image into two sets (the edge pixels and the non-edge ones). The process associates with each pixel a vector representing the differences in luminosity w.r.t. the surrounding pixels. Clustering is driven by the norms of such vectors and we adopt a parsimonious optimization algorithm to detect the required two clusters. We use DC (Difference of Convex) decomposition of a classic nonsmooth nonconvex model function for clustering. The results for some benchmark images are reported. The model Over the history of digital image processing a variety of edge detectors have been devised which differ in their mathematical and algorithmic properties. The approach used in our work to obtain a distinction between all the pixels that are part of a contour from all remaining pixels is based on the following idea. Each pixel is analysed in terms of brightness on the basis of its eight neighbours, taking a portion of the original matrix of size 3x3. Thus a vector of eight elements is associated with each pixel, each component being the difference in brightness between the central pixel and its eight neighbours. Each of such vectors enters into our computational model being represented by its euclidean norm, which we refer to as the sample. We formulate the edge detection problem as a clustering one, where, in particular, two clusters (the edge and the non-edge) samples are to be identified. A continuous formulation of the clustering problem, once the number K of clusters is pre-defined, is the following min x(k) ∈Rn , k=1,...,K m X i=1 min kai − x(k) k, k=1,...,K (34) where the set A = {a1 , . . . , am } is the set of sample points in Rn to be clustered and the points x(k) ∈ Rn , k = 1, . . . , K are the centers of the clusters to be determined. The above problem is both nonsmooth and non convex and it has been tackled as a DC (Difference of Convex) optimization program in [1]. A method to get a suboptimal solution is the well known K-means method. In our application the points ai , i = 1, . . . , m, are the scalar samples previously defined and K = 2. Letting z1 and z2 the abscissas of the two centers, we rewrite Optimization and Classification (Gaudioso) 159 the problem as: min m X amin ≤z1 ,z2 ≤amax min{|ai − z1 |, |ai − z2 |}, (35) i=1 4 4 where amin = min1≤i≤m ai and amax = max1≤i≤m ai . Using a quite standard transformation, problem (35) can be rewritten in DC (Difference of Convex) functions as follows: min amin ≤z1 ,z2 ≤amax where functions 4 f (z1 , z2 ) = m X f (z1 , z2 ) − h(z1 , z2 ) (36) (|ai − z1 | + |ai − z2 |) i=1 and 4 h(z1 , z2 ) = m X max{|ai − z1 |, |ai − z2 |} i=1 are both convex and piecewise affine. To tackle problem (36) we adopt the DCA approach introduced in [1], which we tailor for our low dimension environment. The approach is tested on a number of digital images widely used for validating effectiveness of edge detection algorithms. In particular we cluster the samples representing the images and then we apply a refinement technique aimed at thinning the edges. References [1] Le Thi, H.N. and Le Hoai, M. and Pham Dihn, T., New and efficient DCA based algorithms for minimum sum-of-squares clustering, Pattern Recognition 47 (2014), 388–401. Optimization for Energy Applications (invited by Vespucci) Tuesday 8, 9:00-10:30 Sala Riunioni Est 160 Optimization for Energy Applications (Vespucci) 161 A Bilevel Programming approach to solve the European ahead market clearing problem with blocks and complex orders Federica Davò∗ Department of Management, Economics and Quantitative Methods, University of Bergamo, Italy, [email protected] Dario Siface RSE S.p.A., Italia, [email protected] Paolo Pisciella Maria Teresa Vespucci Department of Management, Economics and Quantitative Methods, University of Bergamo, Italy, [email protected] [email protected] Abstract. We propose an exact linear bilevel model developed to solve the problem associated with the day-ahead power markets coupling in the Price Coupling of Regions (PCR), with nonconvexity arising from block orders and complex orders, e.g. Minimum Income Condition orders, giving rise to a MINLP problem. The lower level represents the standard market clearing problem in a primal-dual formulation, and the upper level imposes the constraints which determine the acceptance of complex orders. Optimization for Energy Applications (Vespucci) 162 A MILP Approach for the EdF Unit Commitment Problem Kostas Tavlaridis-Gyparakis∗ LIX, École Polytechnique, France, [email protected] Antonio Frangioni Department of Computer Science, University of Pisa, Italy, [email protected] Wim van Ackooij EDF R&D. OSIRIS, France, [email protected] Abstract. Our work focuses on the development of tight MILP-formulations and on the implementation of Lagrangian Heuristics for solving large scale Unit Commitment Problems, and more specifically for the case of EdF, one of the main electrical operators in France. In our approach we take into account the highly complicated Unit Commitment model of EdF and aim to explore different possible MILP formulations that will provide the tightest possible formulation. The Proposed Approach The Unit Commitment (UC) Problem in electrical power production is in general a very difficult problem to solve, as it requires to co-ordinate the operations of a large set of rather different generators over a given time horizon to satisfy a prescribed energy demand. Our work focuses on the specific UC of EdF, which is the main electrical operator in France and one of the major operators worldwide. This is a particularly challenging problem for several reasons. Some are pretty typical for UC, i.e., the very large scale of the problem due to a very large number of generating units with very different operational characteristics. Above and beyond these difficulties, the EdF UC has specific load and safety requirements (the latter mainly represented as reserve constraints) that couples the generating units together more than in usual UC problems and substantially complicates the description of the operating rules (and, therefore, the mathematical models) of the variables. In addition, some specific types of units (in particular, nuclear ones) have uncommon operational constraints which further complicate matters. All of the above already result to a highly challenging problem; furthermore, the operating constraints demand a solution approach that can provide good feasible solutions in “unreasonably” small time with respect with the problem size and complexity. The mathematical description of the EdF, UC due to their complex operational constraints, offers a variety of modeling approaches. More specifically, a significant modeling decision in the Edf UC is the treatment of the commitment variables. In the extended literature about MILP-models for the UC, in most of the cases the binary commitment variables of the thermal units simply denote if the thermal unit is on or off during each time step of the optimization period, which leads to a two-index variable. Yet again the complicated relationships that describe the primary production (i.e., power produced to satisfy active demand) together with the reserved production (i.e., power kept available for emergencies) makes it interesting to consider a 3-index commitment variables describing the different production states as well. On top of this, the EdF thermal units require a very specific treatment of changes in power production (the so-called “modulation”), which again makes it possible to consider the addition of an extra index for the commitment Optimization for Energy Applications (Vespucci) 163 variables that will describe the different states. All the above give us a variety of different possible model formulations for the EdF UC based on the treatment of the binary commitment variables. Our first aim is to examine these different formulations in terms of quality of the obtained gaps, cost of solving the continuous relaxation, and overall solution time. As the direct use of general-purpose MILP solvers is unlikely to be efficient enough for the EdF UC Problem, we will also consider Lagrangian relaxation techniques, that have been successfully used in the past for tackling UC problems, and are also the current solution scheme in EdF (e.g. [1],[6]). Relaxing the demand constraints results to a separable problem that can be decomposed into a set of independent hydro and thermal sub-problems, one for each production unit, that can then be more easily solved. We investigate whether the newly proposed formulations for UC, in particular for the thermal subproblems, lead to efficient Lagrangian schemes, either by solving them with general-purpose solvers or by adapting specialized DP algorithms for simpler formulations of the problem [4]. We believe that the effort of investigating different MILP formulation for the UC will lead to an improvement on the already state-of-the-art solution process followed by EdF [3]. For this to happen we will have to exploit the already available sophisticated solution algorithms for the solution of the Lagrangian dual problem (inexact disaggregated Bundle methods) in order to construct Lagrangian heuristics, based on the recent succesful Langrangian Heuristics based on “simplified” versions of UC (e.g.[2],[5]). References [1] F. Bard “Short-term scheduling of thermal-electric generators using Lagrangian Relaxation” Operations Research 36, 756–766, 1988 [2] A. Borghetti, A. Frangioni, F. Lacalandra and C. A. Nucci “Lagrangian Heuristics Based on Disaggregated Bundle Methods for Hydrothermal Unit Commitment” IEEE Transactions on Power Systems 18(1), 313–323, 2003 [3] G. Doukopoulos, S. Charousset, J. Malick and C. Lemaréchal “The short-term electricity production management problem at EDF” Optima - Newsletter of the Mathematical Optimization Society 84, 2–6, 2010 [4] A. Frangioni, C. Gentile “Solving Nonlinear Single-Unit Commitment Problems with Ramping Constraints” Operations Research 54(4), 767–775, 2006 [5] A. Frangioni, C. Gentile and F. Lacalandra “Solving Unit Commitment Problems with General Ramp Contraints” International Journal of Electrical Power and Energy Systems 30, 316–326, 2008 [6] M. Tahanan, W. van Ackooij, A. Frangioni and F. Lacalandra “Large-scale Unit Commitment under uncertainty: a literature survey” Technical Report 14-01, Dipartimento di Informatica, Universit di Pisa, 2014 Optimization for Energy Applications (Vespucci) 164 Optimal operation of power distribution networks Maria Teresa Vespucci∗ Dipartimento di Scienze aziendali, economiche e metodi quantitativi, Università di Bergamo, Italia, [email protected] Paolo Pisciella Dipartimento di Scienze aziendali, economiche e metodi quantitativi, Università di Bergamo, Italia, [email protected] Diana Moneta Giacomo Viganò RSE - Ricerca sul Sistema Energetico S.p.A. - Milano, Italia, [email protected] [email protected] Abstract. A medium-voltage AC network with distributed generation and storage devices is considered for which set points are assigned in each time period of a given time horizon on the basis of forecasted values of some parameters. When realized values differ from forecasts, new set points need to be determined in order to restore feasibility. We propose a NLP model that minimizes distributor’s dispatching costs, while ensuring service quality and satisfying security requirements as well as local control constraints. An interior point algorithm is developed that exploits the problem structure. A Distribution System Operator (DSO) will be in charge of operating power distribution networks, in order to compensate generation-load imbalances with respect to a previously determined scheduling, while guaranteeing constraints on currents in lines for security and voltages at nodes for power quality. Internal (i.e. owned by DSO) regulation resources will be electricity storage devices and on-load tap changers. DSO’s external regulation resources (i.e. owned by third parties) will be the dispatch of active and reactive power of generation plants and the exchange of active and reactive power with the high voltage transmission network. Costs associated to the use of internal regulation resources reflect device deterioration; costs associated to the use of external regulation resources are to be defined by the Regulator, so as to allow a technically efficient operation of the network. The optimal redispatch minimizes the total costs for using internal and external resources, constrained by power flow equations, balance equation for the batteries and local control constraints. Active losses are also considered and penalized in the objective function. The problem is modeled by using a non linear sparse formulation and solved using a primal-dual interior point method. The procedure allows finding efficient configurations of the network and can be used as a simulation tool by the Regulator to analyze the impact of different costs associated to external regulation resources. References [1] Bosisio, A., Moneta, D., Vespucci, M.T., Zigrino, S., A procedure for the optimal management of medium-voltage AC networks with distributed generation and storage devices, Procedia Social and Behavioral Sciences, Operational Research for Development, Sustainability and Local Economies 108 (2014), 164–186. Optimization for Water Resources Management (invited by Zuddas) Monday 7, 15:30-17:00 Sala Riunioni Est 165 Optimization for Water Resources Management (Zuddas) 166 Global minimization using space-filling curves Daniela Lera∗ Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] Yaroslav Sergeyev Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, University of Calabria, Italia, [email protected] Abstract. In this paper the global optimization problem of a multiextremal function satisfying the Lipschitz condition over a hyperinterval is considered. To solve it we propose algorithms that use Peano-type space-filling curves for reduction of dimensionality. The knowledge of the Lipschitz constant is not required. Local tuning on the behavior of the objective function and a new technique, named local improvement, are used in order to accelerate the search. Convergence conditions are given. Numerical experiments show quite promising performance of the new technique. Introduction Many decision-making problems arising in various fields of human activity (technological processes, economic models, etc.) can be stated as global optimization problems (see, e.g., [2, 3, 7, 9]). Objective functions describing real-life applications are very often multiextremal, nondifferentiable, and hard to eval- uate. Numerical techniques for finding solutions to such problems have been widely discussed in the literature (see, e.g., [1, 3, 4, 7]). In this paper, the Lipschitz global optimization problem is considered. This type of optimization problem is sufficiently general both from theoretical and applied points of view. Mathematically, the global optimization problem considered in the paper can be formulated as minimization of a multidimensional multiextremal black-box function that satisfies the Lipschitz condition over a domain [a, b] ⊂ RN with an unknown constant L, i.e., finding the value F ∗ and points y ∗ such that F ∗ = F (y ∗ ) = min{F (y) : y ∈ [a, b]}, |F (y 0 ) − F (y 00 )| ≤ Lky 0 − y 00 k, y 0 , y 00 ∈ [a, b], (37) (38) with a constant L, 0 < L < ∞, generally unknown. In the literature, there exist numerous methods for solving the problems (37), (38), see, for example, [1, 3, 5, 6, 7, 9]. In this paper, we consider an approach that uses numerical approximations of space-filling curves to reduce the original Lipschitz multidimensional problem to a univariate one satisfying the Hölder condition [8]. These curves, first introduced by Peano (1890), fill in the hyperinterval [a, b] ⊂ RN , i.e., they pass through every point of [a, b]. More precisely, a space-filling curve is a parameterized function which maps a unit line segment to a continuous curve in the N-dimensional unit hypercube, which gets arbitrarily close to any point in the unit hipercube as the parameter increases (for a detailed description see [8]). It has been shown by Strongin (see [9]) that, by using space filling curves, the multidimensional global minimization problem (37), (38) is turned into a one-dimensional problem. In particular, Strongin has proved that finding the global minimum of the Lipschitz function F (y), y ∈ RN , Optimization for Water Resources Management (Zuddas) 167 over a hyperinterval is equivalent to determining the global minimum of the function f (x): f (x) = F (p(x)), x ∈ [0, 1], (39) where p(x) is the Peano curve. Moreover, the Hölder condition |f (x0 ) − f (x00 )| ≤ H|x0 − x00 |1/N , holds for the function f with the constant √ H = 2L N + 3, x0 , x00 ∈ [0, 1], (40) (41) where L is the Lipschitz constant of the multidimensional function F (y). Thus, we can solve the problem (37), (38) by using algorithms proposed for minimizing functions in one dimension but it is required to use Hölder metric. References [1] Dixon, L.C.W., Szegö G.P., Towards Global Optimization. Vol.2, North-Holland, Amsterdam, 1978. [2] Floudas, C.A., Pardalos P.M., Adjiman C., Esposito W., Gümüs Z., Harding S., Klepeis J., Meyer C., Schweiger C., Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. [3] Horst, R., Pardalos P.M., Handbook of of Global Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. [4] Horst, R., Tuy H., Global Optimization Deterministic Approaches. Springer-Verlag, Berlin, 1993. [5] Lera, D., Sergeyev Ya.D., Lipschitz and Hölder global optimization using space-filling curves, Applied Numerical Mathematics, 60(1-2) (2010), 115–129. [6] Lera, D., Sergeyev Ya.D., An information global minimization algorithm using the local improvement technique, Journal of Global Optimization, 48(1) (2010), 99–112. [7] Pinter, J., Global Optimization in Action (Continuous and Lipschitz Optimization Algorithms, Implementations and Applications). Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. [8] Sergeyev, Ya.D., Strongin R.G., Lera D., Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York, 2013. [9] Strongin, R.G., Sergeyev Ya.D., Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. Optimization for Water Resources Management (Zuddas) 168 Scenario-Optimization of the Pumping Schedules in Complex Water Supply System Considering a Cost-Risk Balancing Problem Jacopo Napolitano∗ Department of Civil and Environmental Engineering and Architecture, University of Cagliari, Italy, [email protected] Giovanni M. Sechi Department of Civil and Environmental Engineering and Architecture, University of Cagliari, Italy, [email protected] Paola Zuddas Department of Mathematic and Informatics, University of Cagliari, Italy, [email protected] Abstract. The optimization of water pumping plants activation schedules is a significant issue when managing emergency and costly water transfers under drought risk. This study analyzes an optimization problem using the scenario analysis approach. The model searches for the identification of optimal decision rules by balancing the risk of water shortages and the operating cost of pumping stations. Scenario optimization provides ’barycentric’ values defining activation threshold comparing hydrologic synthetic series results. Objectives of this study In this paper we consider the optimization of the pumping schedules in complex multi-reservoirs supply systems, considering different scenarios alternatives and defining activation of emergency and costly water transfers under drought risk. As well known [1] [4], the context of complex multi-sources water system management under scarcity conditions requires application of specifically derived mathematical optimization models defining decision rules. The effectiveness of emergency transfers alleviating droughts requires early warning and activation, but, on the other hand, high operating costs of pumps stations stress system managers to the need of a robust approach defining activation rules. Particularly, treating the effectiveness of early warning and emergency transfers alleviating droughts, the operating costs required by pump stations activation stress the system managers to the need of a robust approach defining rules. An optimization procedure has been developed based on scenario analysis [2]. The model allows identification of the optimal decision rules by balancing the risk of water shortages under different hydrological scenarios and the cost of pumping stations operating [3]. Scenario analysis optimization provide the resource management authority with information defining optimal activation thresholds for pumping stations assuring the water demand level fulfillment for users and activities (irrigational, civil, industrial). Furthermore, a general energy cost minimization and a reduction of damages caused by deficits in the water supply system has been developed in the optimization model. A multiobjective approach is also required in order to balance energy-cost minimization requirements and reduction of damage needs, as can be caused by water shortages. Consequently, a scenario-optimization has been developed considering the multiobjective and cost-risk balancing problem. The optimization model is implemented Optimization for Water Resources Management (Zuddas) 169 using the software GAMS-IDE [5], specifically designed for modelling mixed integer optimization problems. A model application has been developed optimizing water management and energy costs in a real water system with shortage risks in the South-Sardinia (Italy) region. References [1] Sechi, G.M., Sulis, A., Water system management through a mixed optimization-simulation approach, ASCE, Journal of Water Resources Planning and Management (2009), 135(3), 160-170. [2] Pallottino, S., Sechi, G.M., Zuddas, P., A DSS for water resources management under uncertainty by scenario analysis., Environmental Modeling and Software (2004), 20:1031-1042. [3] Gaivoronski, A., Sechi, G.M., Zuddas, P., Balancing cost-risk in management optimization of water resource systems under uncertainty, Physics and Chemistry of the Earth (2012); 42-44:98-107. [4] Napolitano, J., Sechi, G.M., Zuddas, P., Scenario Analysis for Optimization of Pumping Schedules in Complex Water Supply Systems Considering a Cost-Risk Balancing Problem, WDSA 2014 Conference, Bari, Procedia Engineering (2014); 89:565-572. [5] GAMS Development Corporation, A User’Guide. Washington, DC, USA, 2008. Optimization for Water Resources Management (Zuddas) 170 Path Relinking for a Team Scheduling Problem Andrea Peano∗ Dipartimento di Ingegneria, Università di Ferrara, Italia, [email protected] Maddalena Nonato Dipartimento di Ingegneria, Università di Ferrara, Italia, [email protected] Abstract. We apply Path Relinking to a real life simulation-optimization problem, concerning the scheduling of technicians due to manually activate devices located at different sites on a water distribution network, in order to reduce the consumed contaminated water during contamination events. Handling contamination events in hydraulic networks Once contaminant has been injected in a Water Distribution System (WDS), it quickly spreads through the network and population alerting strategies may not entirely ward off users’ water consumption. Therefore contaminant must be diverted from high consumption sectors to mitigate population harm. An effective way of altering contaminated flow is by activating some of the devices which are part of the system. In particular, opening hydrants can expel contaminated water, while closing valves can divert the water flows [2]. The time at which each device is operated heavily impacts on water flow and contaminant consumption, but its effects are not additive so that a schedule (a vector of activation times) must be evaluated as a whole. Moreover, due to the highly non linear functional dependencies that link water flow and the time at which a given device is operated, no analytical description of the objective function is available; the volume of consumed contaminated water is then a black box function depending on the activation times and can be computed by EPANET, a discrete event-based simulator which represents the state of the art in Hydraulic Engineering.4 Regarding the feasibility, a schedule tF is feasible provided that there is an assignment of the Ndev devices to the Nteam teams available and, for each team, a route on the street network visits its devices so that if device j follows device i the difference between the respective activation times in the schedule is equal to τij , i.e., the travelling time from the location of i to the location of j, given that teams gather at the mobilization point at a given time after the alarm is raised. Thus, the feasible region of our problem is well described by the open m-Travelling Salesman Problem [1] (mTSP). EPANET is given a feasible schedule, the description of the hydraulic network, the expected water demand, and a contamination scenario, and it yields, in about 500 , the volume of contaminated water. In this simulation-optimization problem [4], simulation is computationally intensive and it is the bottleneck of any solution approach. Moreover, common sense inspired criteria such as “the sooner the better” lead to low quality solutions, so no further knowledge can drive the search. 4 EPANET 2 users manual: http://nepis.epa.gov/Adobe/PDF/P1007WWU.pdf Optimization for Water Resources Management (Zuddas) 171 These features motivated the choice of a Genetic Algorithm (GA), as described in detail in [5]. Although we could improve by far and large the best solutions available for our case study, that solution approach has a typical GA flaw, that is, it converges to a local optimum which depends on the starting population. So, parallel GAs yield a set of different, high quality solutions. We developed two Path Relinking [3] (PR) variants to take advantage from the knowledge of these local optima, and in particular to explore the trajectories among them, looking for better solutions. Both PRs take the initial (r) and the guiding (g) solutions from the final populations of a number of parallel independent GAs. PRr represents the solutions by way of the routes, each one visiting a subset of devices; the neighbourhood is built up by considering any device having a different predecessor in r wrt g. PRh is instead a hybrid procedure: it represents solutions as vectors of activation times, and it exploits a Mixed Integer Linear Programming solver to compute feasible neighbours. Solving methodology. Experiments were performed on the Ferrara’s hydraulic network, which supplies water to about 120, 000 inhabitants. 20 contamination scenarios have been tested. 3 teams of technicians were considered to be available to activate 13 among valves and hydrants. PRr and PRh were started from the final populations of 10 independent GA runs. We tested for each scenario how many times and how much PRr and PRh can improve the best solution (s∗c ) of the c-th run of 10 GAs. Since GAs are not deterministic, any experiment was ran 10 times; so we also tested whether PRs can improve the global best solution s∗ = minc {s∗c }. Results reports that PRr and PRh are able to improve s∗ on 4 and 11 scenarios, respectively. PRr improves s∗c at least once on 14 scenarios of 20; whereas PRh does it at least twice on every scenario. Over all the scenarios, on average, PRr improves s∗c 3 times, whereas PRh 7 times; the averaged improvement in terms of volume of contaminated water is 38 litres for PRr and 76 litres for PRh. However, since PRr outperforms PRh in the 25% of scenarios, there is no a real dominance, which suggests to integrate both techniques in future. Results. References [1] Bektas, T, The multiple traveling salesman problem: an overview of formulations and solution procedures, Omega, 34(3):209 – 219, 2006. [2] Guidorzi, M., Franchini, M., and Alvisi, S., A multi-objective approach for detecting and responding to accidental and intentional contamination events in water distribution systems. Urban Water, 6(2):115–135, 2009 [3] Glover, F., Laguna, M., and Martı́, M., Fundamentals of scatter search and path relinking. Control and Cybernetics, 39:653–684, 2000. [4] April, J., Glover, F., Kelly, J. P., and Laguna, M., Simulation-based optimization: Practical introduction to simulation optimization. Proceedings of the 35th Conference on Winter Simulation: Driving Innovation, WSC ’03, pp. 71–78. Winter Simulation Conference, 2003. [5] Gavanelli, M., Nonato, M., Peano, A., Alvisi, S., and Franchini, M., Scheduling countermeasures to contamination events by genetic algorithms. AI Communications, 28(2):259–282, 2015 Optimization in Finance and Insurance (invited by Consigli) Tuesday 8, 11:00-13:00 Sala Riunioni Est 172 Optimization in Finance and Insurance (Consigli) 173 Institutional ALM and risk control after the recent sovereign crisis Mohammad Mehdi Hossaindazeh∗ Department of Mathematics, Statistics and Computer Science, University of Bergamo, Italy, hosseinzadeh [email protected] Abstract. During the past decades, financial markets went through a soared fluctuation. With increasing instability on equity as well as fixed income even sovereign markets, investment risks increased for the financial institutions such as pension funds which had long-term horizon plans. In this paper, we consider a dynamic asset-liability management (ALM) problem for the long-term horizon with financial, regulatory and liability constraints for the investment manager of a defined benefit (DB) pension fund. A multistage stochastic programming problem has been formulated with different investment opportunities. The dynamic ALM model defined is discrete time with portfolio rebalancing over a long time horizon and it has focused on the design of pension funds to support the investors by giving a minimum guaranteed return. The mathematical formulation of the objective function is based on a trade-off between long and short-term targets while simultaneous controlling the possible funds’ shortfall by applying the optimization approach [4] & [5]. We consider a 10 years multistage dynamic stochastic optimization program for a DB pension fund manager facing stochastic liabilities. The investment universe includes government bonds, equity and commodities asset classes. For each such investment opportunity a dedicated statistical model has been adopted similar to the models implemented in [3] to generate future price and return scenarios for describing the uncertainty the investment manager is facing over time. The results are generated through a set of software modules combining MATLAB as the development tool, GAMS as the model generator and solution method provider and Excel as the input and output data collector. Numerical results are presented for specifications of the dynamic optimization problem over a long-term horizon with several decision stages. The resulting evidence underlines the value of the dynamic solution and the positive impact of a risk capital control. References [1] Carino et al., The Russel-Yasuda-Kasai model: An asset-liability model for a Japanese insurance company using multistage stochastic programming, Interfaces, 24, 1994, 29–49. [2] Consigli & Dempster , Dynamic stochastic programming for asset-liability management. Annals of Operations Research, 81, 1998, 131-162. [3] Consigli et al., Retirement planning in individual asset-liability management. IMA Journal of Management Mathematics, 23, 2012, 365-396. [4] Dempster et al., Global Asset Liability Management. British Actuarial Journal, 9, 2003, 137-195. [5] Dempster et al., Designing minimum guaranteed return funds, in M. Bertocchi, G. Consigli, and M. Dempster, editors, Handbook on Stochastic Optimization Methods in Finance and Energy. Fred Hillier International Series in Operations Research and Management Science, Springer USA, chapter 2, 2011, 21-42. [6] Dempster et al., Risk Proling DB Pension Schemes, Journal of Portfolio Management, Summer. Optimization in Finance and Insurance (Consigli) 174 Behaviouralizing Black-Litterman model: Behavioural biases and expert opinions in a diffusion setting Sebastien Lleo∗ University of Reims, France, [email protected] Abstract. This paper proposes a continuous time version of the Black-Litterman model that accounts for, and correct, some of the behavioural biases that analysts may exhibit. The starting point is an implementation of the Black-Litterman in Continuous Time model proposed in Davis and Lleo (2013), with market data from 11 ETFs. We propose two methods to obtain the mean of the prior distribution, calibrate analyst views, discuss and show how to mitigate the impact of four behavioural biases. Finally, we compare the results of six dynamic investment models to explore the effect that views and biases have on the asset allocation. We find that the views have a modest impact on the Kelly portfolio, but that the confidence intervals around the views have a large impact on the intertemporal hedging portfolio. Overall, the role of analyst views in the portfolio section process appears more about providing extra scenarios that are not reflected in historical data, rather than providing accurate forecasts. Optimization in Finance and Insurance (Consigli) 175 Optimal Investment Policy in Pension Fund Sebastiano Vitali∗ University of Bergamo, Italy, [email protected] Abstract. We present the definition of an optimal policy decision for a pension fund. Starting from the analysis of the existing members of the fund in order to identify a set of representative contributors, we focus on an individual optimal portfolio allocation in a Pension Plan prospective. In particular, for each representative member, we propose a multistage stochastic program (MSP) which includes a multi-criteria objective function. The optimal choice is the portfolio allocation that minimizes the Average Value at Risk Deviation of the final wealth and satisfies a set of wealth targets in the final stage and in an intermediate stage. Stochasticity arises from investor’s salary process and assets return. The stochastic processes are assumed to be correlated. Numerical results show optimal dynamic portfolios with respect to investor’s preferences. Optimization in Finance and Insurance (Consigli) 176 Solvency II-compliant dynamic risk control or a global P/C insurance portfolio Giorgio Consigli∗ University of Bergamo, Italy, [email protected] Abstract. We consider a 10 year nonlinear multistage stochastic program for a portfolio manager facing stochastic liabilities from the property and casualty (P/C) business and risk capital constraints associated with an evolving regulatory framework (e.g. Solvency II). The investment universe includes liquid (Treasuries on different maturity buckets, corporates, equity, indirect real estate) and illiquid (private equity, renewables, direct real estate, infrastructures) asset classes. From a mathematical viewpoint, the elements of the optimization problems are a dynamic decision policy – the control –, a multidimensional probability space and a multi-criteria objective function with several financial and regulatory constraints. The ALM model captures the key elements of a real-world development and the risk capital constraints are studied under alternative assumptions on the assets correlation matrix leading to a set of inequalities and bounds relevant to infer the effectiveness of an optimal ALM strategy on the consumption of the allocated risk capital. Numerical results are presented for specifications of the dynamic optimization problem over a long term horizon with non-homogeneous decision stages. The gap between a 1-year standard risk capital model and the dynamic risk capital consumption is analyzed under different risk factors correlation matrices. The resulting evidence underlines the value of the dynamic solution and the positive impact of an endogenous risk capital control. Optimization in Logistics (invited by Bertazzi) Tuesday 8, 15:30-17:00 Sala Riunioni Ovest 177 Optimization in Logistics (Bertazzi) 178 The Generalized Bin Packing Problem: a link between Logistics and Maintenance Optimization Mauro Maria Baldi∗ Dipartimento di Automatica ed Informatica, Politecnico di Torino, Turin, Italy, [email protected] Franziska Heinicke Axel Simroth Fraunhofer IVI, Dresden, Germany, Roberto Tadei Dipartimento di Automatica ed Informatica, Politecnico di Torino, Turin, Italy, Abstract. The Generalized Bin Packing Problem (GBPP) is a novel packing problem recently introduced in the literature. While its theoretical importance is the capability to gather a number of knapsack and bin packing problem, its applications mainly arise in the field of Transportation and Logistics. Nevertheless its formulation and methodology can also be exploited in Maintenance Optimization, which integrates with Logistics. In this paper we want to present the GBPP as an important link between these two settings. The origin of Generalized Bin Packing Problem in Logistics and Transportation Logistics blends in with a number of important branches like Transportation, Stocking of goods and Packaging. The Generalized Bin Packing Problem (GBPP) is a novel packing problem recently introduced in the literature [1], [2] which was conceived as an evolution of the Bin Packing Problem in order to model all those applications arising in the aforementioned settings. The GBPP consists of a set of bins, with capacity and cost, and a set of items, with volume and profit. Moreover items can be compulsory or non-compulsory (like in the knapsack problem). The aim of the GBPP is to assign the compulsory items and profitable non-compulsory items to appropriate bins in order to minimize the overall costs, while satisfying capacity constraints. The GBPP addresses a number of applications, in particular in the field of Logistics and Transportation. The strong impact in the field of Logistics is due to the introduction of item profits. In this way, bin costs represent transportation costs paid by a logistic firm to deliver its products (i.e., the items), each with its profit. Further applications in Logistics and Maintenance optimization In this paper, we introduce the GBPP and its variants which allow us to address further applications. In particular, the stochastic [3] and on-line [4] variant are also addressed. A further extension will consist of a bin dependence of item profits [5]. This allows to address more logistic settings, like the trade-off among logistic firms and transportation companies, and the last-mile logistics. Optimization in Logistics (Bertazzi) 179 Finally, we wish to present the GBPP also as an important link between Logistics and Maintenance Optimization. In fact, another important branch which integrates with logistics is Maintenance Optimization, in order to ensure a good quality of the service of the logistic operator. Recently, the multi-attribute features of the GBPP, the methodology and the heuristics were exploited to address a stochastic problem dealing with maintenance optimization [6], [7]. References [1] Baldi M. M., Crainic T. G., Perboli G., Tadei R., The generalized bin packing problem, Transportation Research Part E 48 (2012), 1205–1220. [2] Baldi M. M., Crainic T. G., Perboli G., Tadei R., Branch-and-price and beam search algorithms for the variable cost and size bin packing problem with optional items, Annals of Operations Research 222 (2014), 125–141 [3] Perboli G., Tadei R., Baldi M. M., The stochastic generalized bin packing problem, Discrete Applied Mathematics 160 (2012) 1291–1297 [4] Baldi M. M., Crainic T. G., Perboli G., Tadei R., Asymptotic results for the generalized bin packing problem, Procedia - Social and Behavioral Sciences 111 (2013), 663–671 [5] Baldi M. M., Gobbato L., Perboli G., Tadei R., The generalized bin packing problem with bin-dependent item profits, submitted [6] Baldi M. M., Tadei R., Heinicke F., Simroth A., New heuristics for the stochastic tactical railway maintenance problem, submitted [7] Heinicke F., Simroth A., Tadei R., On a novel optimisation model and solution method for tactical railway maintenance planning. In Proceedings of the 2nd International Conference on Road and Rail Infrastructure. Department of Transportation, Faculty of Civil Engineering, University of Zagreb (2012), 421–427. Optimization in Logistics (Bertazzi) 180 A Note on the Ichoua et al (2003) Travel Time Model Emanuela Guerriero∗ Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Italia, [email protected] Gianpaolo Ghiani Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Italia, [email protected] Abstract. In this talk we discuss some properties of the travel time model proposed by Ichoua et al (2003), on which most of the current time-dependent vehicle routing literature relies. Firstly, we prove that any continuous piecewise linear travel time model can be generated by an appropriate Ichoua et al (2003) model. We also show that the model parameters can be obtained by solving a system of linear equations for each arc. Then such parameters are proved to be nonnegative if the continuous piecewise linear travel time model satisfies the FIFO property, which allows to interpret them as (dummy) speeds. Finally, we illustrate the procedure through a numerical example. As a by-product, we are able to link the travel time models of a road graph and the associated complete graph over which vehicle routing problems are usually formulated. Main contribution Most of the literature on time-dependent vehicle routing relies on the stepwise speed model proposed by Ichoua, Gendreau and Potvin in 2003 (IGP model, in the following). The main point in their model is that they do not assume a constant speed over the entire length of a link. Rather, the speed changes when the boundary between two consecutive time periods is crossed. This feature guarantees that if a vehicle leaves a node i for a node j at a given time, any identical vehicle leaving node i for node j at a later time will arrive later at node j (no-passing or first-infirst-out (FIFO) property). In this talk, we prove that any continuous piecewise linear travel time model can be generated by an appropriate IGP model, and show how to compute the model parameters. We also prove that such parameters can be interpreted as speeds if the time model satisfies the FIFO property. These results allow us to link the travel time models of a road graph and the associated complete graph over which vehicle routing problems are usually formulated. This is quite interesting because, while the hypothesis of instantaneous speed variation over an arc is quite realistic for the arcs of the road graph (at least if the corresponding streets are not too long), it is not so intuitive that this assumption may be reasonable for the associated complete graph as well. The literature on Time-Dependent Vehicle Routing is fairly limited and can be divided, for the sake of convenience, into four broad areas: travel time modeling and estimation; the Time-Dependent Shortest Path Problem (TDSPP); the TimeDependent Traveling Salesman Problem (TDTSP) and its variants; and the TimeDependent Vehicle Routing Problem (TDVRP). Here we focus on the first research stream. In [1], Hill and Benton proposed a model for time-dependent travel speeds and several approaches for estimating the parameters of this model. In [2], Ichoua Optimization in Logistics (Bertazzi) 181 et. al. proposed a travel time modeling approach based on a continuous piecewise linear travel time function (the IGP model). Later, in [3] Fleischmann et al. investigated the assumptions that this function must satisfy to ensure that travel times satisfy the FIFO property. They also described the derivation of travel time data from modern traffic information systems. In particular, they presented a general framework for the implementation of time-varying travel times in various vehiclerouting algorithms. Finally, they reported on computational tests with travel time data obtained from a traffic information system in the city of Berlin. In [4], Horn investigated exact and approximate methods for estimating time-minimizing vehicular movements in road network models where link speeds vary over time. The assumptions made about network conditions recognize the intrinsic relationship between speed and travel duration and are substantiated by elementary methods to obtain link travel duration. The assumptions also imply a condition of FIFO consistency, which justifies the use of Dijkstra’s algorithm for path-finding purposes. This talk is organized as follows. Firstly, we gain some insight into a constant stepwise travel speed model with constant distances and illustrate a procedure for deriving any continuous piecewise linear travel time model from a suitable IGP model. We also show that the model parameters can be obtained by solving a system of linear equations for each arc. Secondly we exploit the relationship between the travel time models of a road graph and the associated complete graph over which vehicle routing problems are usually formulated. References [1] Hill A.V. and Benton, W.C. , Modelling Intra-City Time-Dependent Travel Speeds for Vehicle Scheduling Problems. The Journal of the Operational Research Society 43 (1992), 343–351. [2] Ichoua S., Gendreau, M. and Potvin, J.-Y., Vehicle dispatching with time-dependent travel times. European Journal of Operational Research 144 (2003), 379-396. [3] Fleischmann, B., Gietz, M. and Gnutzmann, S. , Time-Varying Travel Times in Vehicle Routing. Transportation Science 38 (2004), 160–173. [4] Horn, M.E.T. , Efficient modeling of travel in networks with time-varying link speeds. Networks 36 (2000), 80–90. [5] Cordeau, J. F., Ghiani, G. and Guerriero, E. Analysis and Branch-and-Cut Algorithm for the Time-Dependent Travelling Salesman Problem, Transportation Science 48 (2014), 46–58. Optimization in Logistics (Bertazzi) 182 A Transportation Problem under uncertainty: Stochastic versus Robust Optimization solution approaches Francesca Maggioni∗ Dipartimento di Scienze Aziendali, Economiche e Metodi Quantitativi, Università degli Studi di Bergamo, [email protected] Marida Bertocchi Dipartimento di Scienze Aziendali, Economiche e Metodi Quantitativi, Università degli Studi di Bergamo, [email protected] Florian A. Potra Department of Mathematics & Statistic, University of Maryland, Baltimore County, U.S.A., [email protected] Abstract. We consider a supply planning problem to optimize vehicle-renting and transportation activities to satisfy demand in several destinations out of several origins. Uncertainty on demands and on cost of extra vehicles is considered. The problem is to determine the number of vehicles to book to replenish the good at each factory in order to minimize the total cost for the producer, given by the sum of the transportation costs and buying cost from external sources in extreme situations. The problem is addressed using two modeling approaches dealing with uncertainty: stochastic programming and robust optimization. A scenario-based framework for fair comparisons of stochastic versus robust solution approaches is presented. Numerical results on a real case instance are provided showing the advantages and disadvantages of the two modeling approaches. Introduction and Problem Description The problem of transporting goods or resources from a set of supply points (production plants) to a set of demand points (destination factories or customers) is an important component of the planning activity of a manufacturing firm. Critical parameters such as customer demands, row material prices, and resource capacity are quite uncertain in real problems. An important issue is then represented by the decision on quantities to acquire and store at each destination factory before actual demands reveal themselves. This is involved in the tactical planning of the firm supply chain operations. The significance of uncertainty has prompted a number of works addressing random parameters in tactical level supply chain planning involving distribution of raw material and products (see for example [1], [5], [4], [2], [3] and [6]). In this talk we analyze the effect of two modelling approaches, stochastic programming and robust optimization, to a real case of a transportation problem under uncertainty. To the best of our knowledge, a direct comparison between SP and RO on such a class of problems has not been addressed yet in literature. Moreover, robust optimization is relatively new concept and there is very little work applying it in a logistic setting. Stochastic Programming (SP) and Robust Optimization (RO) are considered two alternative techniques to deal with uncertain data both in a single period and in a multi-period decision making process. Optimization in Logistics (Bertazzi) 183 The transportation problem considered, is inspired by a real case of gypsum replenishment in Italy. The logistic system is organized as follows: a set of suppliers, each of them composed of a set of several plants (origins) have to satisfy the demand of a set of factories (destinations) belonging to the same company producer. The weekly demand is uncertain. We assume a uniform fleet of vehicles with fixed capacity and allow only full-load shipments. Shipments are performed by capacitated vehicles which have to be booked in advance, before the demand is revealed. When the demand becomes known, there is an option to discount vehicles booked but not actually used from different suppliers. The cancellation fee is given as a proportion of the transportation costs. If the quantity shipped from the suppliers using the booked vehicles is not enough to satisfy the demand of the factories, residual product is purchased from an external company at a higher price, which is also uncertain. The problem consists in determining the number of vehicles to book, at the end of each week, from each plant of the set of suppliers, in order to minimize the total cost, given by the sum of the transportation costs from origin to destinations (including the discount for vehicles booked but not used) and the cost of buying units of product from external sources in case of inventory shortage. Solution Approaches We solve the problem both via a two-stage stochastic programming and robust optimization models with different uncertainty sets. For the former the goal is to compute the minimum expected cost based on the specific probability distribution of the uncertain demand at the factories and buying cost from external sources based on a set of possible scenarios. However, a reliable forecast and reasonable estimates of demand probability distributions and buying costs are difficult to obtain. This is the main reason that lead us to consider also robust optimization approaches. First we consider static approaches with uncertainty parameters respectively belonging to boxes, ellipsoidal uncertainty sets or mixture of them, and secondly dynamic approaches, via the concept of affinely adjustable robust counterpart. The main advantage of the RO formulations considered, is that they can be solved in polynomial time and have theoretical guarantees for the quality of the solution which is not the case with the aforementioned SP formulations. A robust solution at the tactical level allows to find a feasible solution for the operational planning problem for each possible realization of demand in the uncertainty set considered. In order to make a fair comparison with the stochastic programming methodology, a scenario-based framework has been provided. Numerical experiments show that the adjustable robust approach result in around 30% larger objective function values with respect to RP solutions due to the certitude of constraints satisfaction. Conversely, the computational complexity is higher for the stochastic approach. In conclusion, the SP approach allows the company to reach higher profits, even if the computational effort is expensive expecially due to the scenario generation procedure. On the other hand RO forces the firm to consider an higher cost solution which is strongly dependent on an arbitrarily chosen uncertainty set, but with Optimization in Logistics (Bertazzi) 184 probability guarantee of costraints satisfaction. References [1] R.K. Cheung and W.B. Powell, “Models and Algorithms for Distribution Problems with Uncertain Demands”, Transportation Science 30, 43-59 (1996). [2] L. Cooper and L.J. LeBlanc, “Stochastic transportation problems and other network related convex problems”, Naval Research Logistics Quarterly 24, 327-336 (1977). [3] T.G. Crainic and G. Laporte, “Planning models for freight transportation”, European Journal of Operational Research 97 (3), 409-438, (1997). [4] W.B. Powell and H. Topaloglu, “Stochastic Programming in Transportation and Logistics”, in Handbooks in Operations Research and Management Science 10, 555-635 (2003). [5] H. Van Landeghem and H. Vanmaele, “Robust planning: a new paradigm for demand chain planning”, Journal of Operations Management 20(6), 769-783 (2002). [6] C. Yu and H. Li, “A robust optimization model for stochastic logistic problems”, International Journal of Production Economics 64(1-3), 385-397 (2000). Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (invited by MAIOR) Tuesday 8, 11:00-13:00 Sala Seminari Est 185 Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)186 All for one, MAIOR for all Leopoldo Girardi∗ M.A.I.O.R. Srl, Italia, [email protected] Abstract. The difficult cooperation between Industry and Research is still a key issue in Italy that has never come to a satisfactory conclusion. In this speech, we briefly describe how the combined skills of people, private companies, public institutions and research organizations can result in successful projects and experiences: mutual respect and recognition of the relevance of each player’s goals are the ingredients to ensure quality results. MAIOR, OR, Transportation This is a personal reflection on the role of the industry, of its relationship with Research and the community as an introduction of the session: “Real time management in Public Transportation: with MAIOR we stand, divided we fall”. MAIOR operates in the public transit sector, providing IT solutions for the planning and management of transport services. The company is the Italian market leader and is lucky to base its business, as well as the labour of its employees, on creativity, in all of its forms: from the daily routine of the software development to the creation and management of complex projects, to the research of sophisticated solutions that often require the use of Operations Research optimization techniques and the deployment of research projects. On one side, being able to do and promote research is a privilege, but on the other the implementation of such research into applications is the founding element of the obligations and responsibilities of MAIOR as a social and economic player of the Italian context. SINTESI is an example of how this happens. Two research hubs like the University of Pisa and the Politecnico of Milan, various Italian clients that participated by clarifying the expectations and operations, a main client (ATM) that was the main partner. These actors allowed MAIOR to propose a project that, aside from the actual results, will provide a consistent growth and improvement to the software solutions provided to the market by the company. Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)187 Why would a public transport company need Operations Research? Duccio Grandi∗ M.A.I.O.R. Srl, Italia, [email protected] Fabrizio Ronchi ATM Milano, Italia, [email protected] Abstract. We describe how and why a public transport company needs solutions from Operations Research. Our virtuous example is ATM Milano and its interest in managing disruptions in real time. Every big public transport company uses an AVL system providing real time information about vehicles locations. Today all these data are used by Operational Control Centre experts in order to manage manually problems of delay and queuing. ATM’s goal is building automated real time tools for improving offered service and company’s result with the Municipal Authority. Public transport company problems Public transport problems have always been important Operations Research tasks. MAIOR together with ATM has addressed and solved lots of problems of resources optimizations. Most of them have been related to optimizations of planned shifts, either vehicles shifts or drivers shifts. So after years of analysis and work about planned shifts optimization, ATM decided to address a new type of problems: those which arise during daily execution of offered service. ATM Milano is a public transport company that moves almost 700 millions of persons yearly, with 135 different lines and more than 2000 vehicles. Obviously ATM faces every day lots of small and big “unexpected” events like delays, car jams, accidents and vehicles failures. The sum of all these events can result in disruptions, i.e. when the observed service is exceedingly different from the planned one. Every single event of disruption probably leads to a “malfunction” for the users of public transport system. Besides it leads to a decrease of some indexes that a Municipal Authority uses to measure the effective service offered by the transport company. So our transport company has immediately two different problems born from an event of disruption. Every big company like ATM has an AVL system that controls continuously all its vehicles. This system provides real time information about vehicles locations. All this data can be used to calculate every possible indexes in order to compare planned and effective time tables, headways, delays, and so on. So an Operational Control Centre can be full of experts users, monitors, computers, indexes of all types, but it lacks of intelligent and quick solutions proposed by an automatic algorithm. Very often the disruptions induced problems are identified by drivers, communicated to the Operational Control Centre by phone, solved or mitigated by impromptu interventions. Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)188 Public transport company goals ATM Milano involved MAIOR immediately in order to address these real time disruptions problems. Its main goals can be resumed as: analysis and development of automatic algorithms and tools to solve or mitigate the effects of these unexpected events on the effective offered service. In order to reach these goals, some issues have been analyzed and well defined in order to become the foundations of a following work of Operations Research: • how to use real time (big) data from AVL systems. Today these data are often underused, especially in real time management; • how to calculate an index of regularity or an index of compare between planned and effective service. These indexes would become part of an objective function; • how to identify disruption events as soon as possible; • how to define possible actions on vehicles and drivers duties usable in real time; • how to define new “light” rules and constraints used to rebuild correct drivers shifts in real time. All this work has been necessary to start a project like SINTESI, described in other presentations. SINTESI aims at providing a Decision Support System capable of assisting public transport operators during disruptions. In SINTESI we find all Operations Research study and development, but its foundations are described here. Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)189 Delay Management in Public Transportation: Service Regularity Issues and Crew Re-scheduling Emanuele Tresoldi∗ Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] Samuela Carosi M.A.I.O.R., Italia, [email protected] Stefano Gualandi AntOptima, Switzerland, [email protected] Federico Malucelli Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] Abstract. In this paper, we propose a decision support tool to assist a local public transportation company in tackling service delays and small disruptions. We discuss about different ways to assess and improve the regularity of the service, and we propose a simulation based optimization system that can be effectively used in a real-time environment taking into account both vehicles and drivers shifts. As a case study, we will analyze the urban management of surface lines of Azienda Trasporti Milanese (ATM) of Milan. Delay Management in Public Transportation In this paper, we elaborate, in the context of local public transportation services, on the design and the development of optimization algorithms that can assist the operators in facing different types of disruptions with the ultimate objective of increasing the quality of service of public transportation or, at least, to limit the perception of inconvenience on passengers. As a case study, we will analyze the urban management of surface lines (busses, trolleybuses and trams) of Azienda Trasporti Milanese (ATM) of Milan. We discuss different ways to assess the regularity of the service evaluating pros and cons (see [1]). This is one of the most critical points since, from the service provider point of view and, also, from the municipality or the agency monitoring the service perspective, the regularity of the service should be measured in the simplest and most intuitive way. However, the measure should be also of help when actions, intended to recover the regularity or improve it in the presence of disruptions, must be taken and their definition demanded to a decision support system. In this regard, we present and analyze different types of functions that can be used to effectively evaluate the regularity of the service in a real-time environment. Furthermore, we discuss the necessity of a simulation based evaluation system to automatically estimate the effect of detours and other changes on the regularity of the service (as already describer in [2]). Such system can help the operations central officers in quickly and objectively assessing the impact of different alternative decisions taken to recover the regular service. We present a description of an integrated decision support system that includes in a uniform environment both the simulation and optimization aspects of the problem. Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)190 Finally, we analyze the mathematical aspects underlying the decisional process required in defining the optimal curse of action to promptly react to short-term disruptions. A detailed description of the online algorithms implemented to reoptimize on the fly both vehicles and drivers scheduling is given. In particular, the vehicles re-optimization is tackled with a tabu-search based procedure (see [4]) while the consequential drivers rescheduling is addressed using a standard column generation approach (see [3]). The algorithms were extensively tested on real world case studies provided by ATM and the experimental results obtained are reported and carefully analyzed. References [1] Barabino, B. and Di Francesco, M. and Mozzoni, S., Regularity diagnosis by automatic vehicle location raw data, Public Transport 4 (2013), 187-208. [2] S. Gualandi and F. Malucelli and E. Tresoldi, Disruption Management in Public Transport: Mathematical Analysis of Service Regularity Issues. Technical Report, Politecnico di Milano, 2014. [3] Desaulniers, Guy and Desrosiers, Jacques and Solomon, Marius M., Column Generation, Springer US, 2005. [4] F. Glover and M. Laguna, Tabu Search, Kluwer Academic Publishers, 1997. Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)191 Real-time Integrated Timetabling and Vehicle Scheduling in Disruption Management Benedetta Pratelli∗ M.A.I.O.R. Srl, Italia, [email protected] Antonio Frangioni Laura Galli Dipartimento di Informatica, Università di Pisa, Italia, [email protected] [email protected] Leopoldo Girardi M.A.I.O.R. Srl, Italia, [email protected] Abstract. This work describes how an integrated approach for Timetabling and Vehicle Scheduling, originally developed for line planning, can be used in real time for managing disruptions in Urban Public Transportation. This is done after local emergency decisions have been taken and the acute phase of the disruption is over; yet, these decisions may cause several problems in the medium term, and in particular at the end of the workday, that can be avoided by a global phase trying to restore the originally planned frequencies. Results from real cases confirm the validity of the approach. The problem This work is a part of the SINTESI project, described in other presentations, for real time management in public transportation. In particular, the project aims at providing a Decision Support System capable of assisting public transport operators during disruptions, i.e., when the observed service is exceedingly different from the planned one. The SINTESI project manages a disruption with a two-step approach. In the short term phase (a few hours), the disruption is detected and managed with emergency decisions that are aimed at solving the main issues that are going to occur in the short term. These decisions allow to quickly solve the “acute phase” of the disruption, but since they are chosen only looking at their immediate effects, they may cause several problems in the medium term period, in particular at the end of the workday. In fact, towards the end of the service it is important that: • the drivers reach the relief point in time, in order to prevent exceeding the maximum established driving time and the consequent interruption of the trip; • certain “critical” trips in low-frequency periods, like at the end of the day, are guaranteed, since canceling them may have too heavy consequences for customers. Therefore, in the SINTESI project a specific optimization model is used that covers all the remaining part of the work day after the disruption is over, and tries to prevent these problems by restoring as much as possible the originally planned frequency. This has to take into account not only the schedule of the buses, but also (albeit indirectly), that of the drivers. Our work focuses on this second part of the project. Real Time Management in Public Transportation: with MAIOR we stand, divided we fall (MAIOR)192 The algorithmic approach The approach exploits and extends a model for integrated timetable planning and vehicle scheduling developed in cooperation between MAIOR S.r.l. and the University of Pisa. As urban public transport systems are extremely complex, due to the large size of transportation networks, the conventional planning process follows a sequential approach whereby trips are decided first in order to obtain the desired frequencies (usually different for different moments of the day), and then the optimal schedule of buses to cover the given trips is sought for. Instead, our approach integrates timetabling decisions (when trips are made) with vehicle scheduling (how buses cover the service). This is done by a mathematical model basically divided into two parts: a set of TT (timetabling) subproblems, modeled as shortest paths on appropriate auxiliary graphs, and a BS (bus scheduling) subproblem modeled as a Min-Cost Flow [1]. These subproblems are related by linking constraints, which are relaxed using a Lagrangian approach [4]; a Lagrangian heuristic is then used to obtain good-quality solutions. The approach for the planning problem can be extended to the disruption setting, but the changes are nontrivial. While in the first case there is no need to distinguish vehicles, and therefore they are simply represented by flow units in the BS subproblem, the disruption setting requires that a specific bus (which is driven by a specific driver) must be at the relief point at a certain time. This makes the BS problem a Multicommodity MCF, which fortunately is still well suitable for Lagrangian approaches [5]. Furthermore, constraints ensuring that a specific bus must necessarily perform one of a specific (small) set of consecutive trips must be added; fortunately, these can be incorporated into the graph structure without making the Lagrangian problems more difficult. On top of all this, unlike for planning, in order to be useful in the disruption setting the approach must give solutions in a very limited amount of time. We report experiences on real-world cases showing that the proposed methodology is actually capable of improving the system mid-term response to disruption with a running time compatible with an operational setting. References [1] A.A. Bertossi, P. Carraresi, G. Gallo On some matching problems arising in vehicle scheduling models, Networks 17(3) (1987), 271–281. [2] A. Frangioni About Lagrangian Methods in Integer Optimization, Annals of Operations Research 139 (2005), 163–193. [3] A. Frangioni, G. Gallo A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min Cost Flow Problems, INFORMS Journal On Computing 11(4) (1999), 370–393. Sportello Matematico (invited by Sgalambro) Wednesday 9, 11:00-13:00 Sala Riunioni Est 193 Sportello Matematico (Sgalambro) 194 Sportello Matematico per l’Industria Italiana: la Matematica in rete per l’Innovazione e la Società Antonino Sgalambro∗ Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Italia, [email protected] Abstract. Lo Sportello Matematico per l’Industria Italiana è un progetto attivo dal 2012 presso l’Istituto per le Applicazioni del Calcolo “M. Picone” del CNR, che riunisce in una grande rete inclusiva e dinamica i centri di ricerca italiani di Matematica Applicata ed Industriale, favorendone l’incontro con il mondo delle imprese per promuovere l’innovazione e la valorizzazione della ricerca. Durante l’intervento verranno chiariti gli obiettivi dello Sportello Matematico, le modalità di lavoro, i rapporti con le imprese ed i centri di ricerca. Saranno presentate le relazioni internazionali ed analizzati alcuni casi di successo, alla luce dei risultati positivi e delle criticità emerse durante la fase di avvio del progetto. VRPs with Backhauls (invited by Di Francesco) Tuesday 8, 9:00-10:30 Sala Seminari Est 195 VRPs with Backhauls (Di Francesco) 196 An adaptive guidance meta-heuristic for the vehicle routing problem with splits and clustered backhauls Maria Battarra∗ School of Management, University of Bath, UK, [email protected] Michela Lai Massimo Di Francesco Paola Zuddas Dept. of Land Engineering, University of Cagliari, Italy, [email protected] [email protected] [email protected] Abstract. We present the case study of an Italian carrier, which provides freight transportation services by trucks and containers. Trucks deliver container loads from a port to import customers and collect container loads from export customers to the same port. All import customers in a route must be serviced before all export customers, customers can be visited more than once by different trucks and containers are never unloaded or reloaded from the truck chassis along routes. We model the problem using an ILP formulation and propose an Adaptive Guidance metaheuristic. Our computational experiments show that the metaheuristic is capable of determining good quality solutions in many instances of practical interest for the carrier within 10 minutes. This paper aims at modelling and optimizing the inland distribution activities of Grendi Transporti Marittimi. This logistics company delivers the content of containers arriving at the port of Vado Ligure to import customers and collects the content of containers directed to the same port from export customers. All containers have the same dimensions and loading/unloading requirements. Containers are solely stored at the port; a fleet of identical and capacitated trucks is also based at the port. Importers and exporters accept multiple visits (the load can be split), but importers have to be visited before exporters within a route. Containers are never unloaded from the truck: the content of import containers is unloaded at the customer under the driver’s supervision; containers arrive empty at exporters and are loaded at the customer’s premises. The objective is to minimize the overall routing cost, while performing the distribution operations. This problem is therefore a Vehicle Routing Problem with Clustered Backhauls and Splits and will be called hereafter Split Vehicle Routing Problem with Clustered Backhauls (SVRPCB). An Integer Linear Programming (ILP) model is presented to address small-sized problems. In order to solve larger instances, we propose a metaheuristic which exploits existing algorithms for simpler SVRPCB subproblems and guides them toward the construction of good SVRPCB solutions. More precisely, the meta-heuristic constructs a feasible SVRPCB solution by first decomposing the SVRPCB into two SVRPs, where the first subproblem involves only importers and the second only exporters. These problems are solved by the Tabu Search (TS) of [1]. Next, importer and exporter routes are paired and merged by solving an assignment problem. This two-stage constructive heuristic is the building block for the proposed meta-heuristic. VRPs with Backhauls (Di Francesco) 197 The importer routes and exporter routes built by the TS do not result in good SVRPCB solutions. Therefore, at each iteration of the proposed algorithm, critical properties of the current SVRPCB solution are detected. Guidance mechanisms are implemented by perturbing the data of the two SVRPs, in order to discourage the TS in creating routes having undesired characteristics. The proposed metaheuristic is utilized to evaluate scenarios of larger transportation capacity for the case study of Grendi Trasporti Marittimi. In Italy trucks with trailers are allowed to be 18.75 m (61.5 foot) long and 44.0 t heavy, thus they are allowed to carry up to two 24.5-foot containers per truck. In our case study, vehicles carry either one or two containers. The carrier is interested in adopting a fleet of identical vehicles, all of which can carry two containers. Moreover, regulations may soon be updated in accordance to the practice in other European countries, where trucks with larger capacity are allowed. Thus, Grendi Trasporti Marittimi aims to quantify the transportation costs achievable by new vehicle fleet configurations using optimization algorithms. The metaheuristic algorithm has been tested on instances with similar characteristics to the real problems. The adaptive guidance metaheuristic proved to find high quality solutions within 10 minutes of computing time. Guidance mechanisms are tested and their performance assessed. Finally, savings achievable using trucks with different capacities are presented. References [1] Archetti, C., Speranza, M.G., Hertz, A., A tabu search algorithm for the split delivery vehicle routing problem, Transportation Science 40 (2006), 64–73. VRPs with Backhauls (Di Francesco) 198 Set-Covering formulation for routing of heterogeneous trucks with container loads Ali Ghezelsoflu∗ Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] Massimo Di Francesco Paola Zuddas Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] [email protected] Antonio Frangioni Dipartimento di Informatica, Università di Pisa, Italia, [email protected] Abstract. This paper proposes a new Set-Covering model for a routing problem, where container loads must be shipped from a port to importers and from exporters to the port by trucks carrying one or two containers, without separating trucks and containers during customer service. The Problem Statement Consider a fleet of trucks and containers based at a port. Some trucks carry one container, while other trucks can carry up to two containers. Trucks and containers are used to service two types of transportation requests: the delivery of container loads from the port to importers and the shipment of container loads from exporters to the same port. Typically customers need to ship or receive more than one container load. Therefore, usually each customer must be serviced by multiple containers and must be visited by more than one truck. A relevant characteristic of this problem is the impossibility to separate trucks and containers. As a result, when importers receive container loads by trucks, containers are emptied and moved away by the same trucks used for providing container loads. Similarly, when exporters are served by empty containers, containers are filled and moved away by the same trucks used for providing empty containers. According to the carrier’s policy, importers must be serviced before exporters. As a result, routes may consist in the shipment of container loads from the port to importers, the allocation of empty containers from importers to exporters, and the final shipment of container loads from exporters to the port. Hence, trucks with one container can service up to two customers in a route (one importer and one exporter). Trucks with two containers can service up to four customers in a route (two importers and two exporters). Every pair of containers can be shipped in a truck, all containers leaving from importers can be used to service exporters and no incompatibility occurs between customers and trucks, which can service almost any customer. It is important to note that the number of container loads to be picked up and delivered is generally different. When the number of container loads delivered to importers is larger than the number of container loads shipped by exporters, a number of empty containers must be moved back to the port. When the number of container loads delivered to importers is lower than the number of container loads shipped by exporters, a number of empty containers must be put on trucks leaving VRPs with Backhauls (Di Francesco) 199 from the port, in order to service all customers. The movement of trucks generate routing costs. The objective is to determine the routes of trucks in order to minimize routing costs, such that customers are serviced as requested, trucks capacity containers hold, and importers are serviced before exporters. Modeling details This problem was investigated and modeled in [3] by a node-arc formulation. It was solved by a heuristic method based on a sequence of local search procedures. In this study we aim to investigate exact approaches for the problem using a setcovering formulation. The closest paper from the methodological side is [1], who proposed a set-partitioning formulation for the vehicle routing problem with backhauls. Unlike [1], in our problem setting the fleet of trucks is heterogeneous in terms of transportation capacities. In the talk will also investigate scenarios of larger transportation capacities, which increase the difficulty of this routing problem, because the underlying packing problem becomes more difficult to solve. In this case, exact methods based on column generation will be discussed. Preliminary results will be provided, as well as comparisons to solutions obtained by metaheuristics. References [1] A. Mingozzi, S. Giorgi, and R. Baldacci, An exact method for the vehicle routing problem with backhauls, Journal of Transportation Science, Volume 33 Issue 3, March 1999, Pages 315 - 329 [2] Lai, Michela, Crainic Teodor Gabriel, Di Francesco Massimo, Zuddas Paola, An heuristic search for the routing of heterogeneous trucks with single and double container loads. Transportation Research Part E: Logistics and Transportation Review, Volume 56, September 2013, Pages 108-118 VRPs with Backhauls (Di Francesco) 200 A GRASP Metaheuristic for the Vehicle Routing Problem with Backhauls Gianfranco Fadda∗ Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, gianfranco− [email protected] Gavina Baralla Massimo Di Francesco Luca Meloni Simone Zanda Paola Zuddas Dipartimento di Matematica e Informatica, Università di Cagliari, Italia, [email protected] [email protected] [email protected] [email protected] [email protected] Simona Mancini Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Italia, [email protected] Abstract. The Vehicle Routing Problem with Clustered Backhauls (VRPCB) is an extension of the classical VRP in which all linehaul customers must be visited before any goods can be picked up from backhaul ones. To solve this problem, we propose a GRASP meta-heuristic, which consists of an ad-hoc suited constructive phase, based on a greedy randomized algorithm, combined with a local search. The VRPCB In the VRPCB two sets of customers are involved: linehauls, to whom goods have to be delivered and backhauls, whose goods have to be picked up and transported back to the depot. The VRPCB can be stated as the problem of determining a set of minimum-cost vehicle routes visiting all customers, such that (i) each vehicle performs one route; (ii) each route starts and finishes at the depot; (iii) each vehicle load does not exceed its capacity throughout its route; (iv) on each route all backhaul customers are visited after all linehaul customers. The motivation of the last restriction is the avoidance of cargo rearranging in rear-loaded vehicles. The VRPCB was investigated by both exact and heuristic approaches [1]. Several metaheuristics were recently proposed for this problem [2], [3]. A GRASP for the VRPCB The GRASP is a multistart metaheuristic which consists of a constructive procedure, based on a greedy randomized algorithm, combined with a local search. In the constructive phase, the last linehaul and the first backhaul are determined and, next, two open routes from these nodes to the depot are constructed, so that the resulting routes are feasible for the VRPCB. At each step of the constructive phase, the next node to be included in the route is randomly chosen within a Restricted Candidates List (RCL) obtained by the distance from the previous node in the route, the distance from the depot and the residual capacity of the vehicle. The local search consists of a node-relocation procedure with first improvement. We compare results to the best-known solutions for a large set of benchmark instances. VRPs with Backhauls (Di Francesco) 201 References [1] Toth, P., Vigo, D., The Vehicle Routing Problem, SIAM, Philadelphia, 2002. [2] Ropke, S., Pisinger, D. A unified heuristic for a large class of vehicle routing problems with backhauls, European Journal of Operational Research 171 (2006), 750–775. [3] Zachariadis, E., Kiranoudis, C. An effective local search approach for the vehicle routing problem with backhauls, Expert Systems with Applications 39 (2012), 3174–3184. CONTRIBUTED SESSIONS 202 Applications in Economics & Finance Thursday 10, 11:00-13:00 Sala Seminari Est 203 Applications in Economics & Finance 204 Portfolio Selection by Reward-Risk Ratio Optimization with CVaR Risk Measure Wlodzimierz Ogryczak∗ Institute of Control and Computation Engineering, Warsaw University of Technology, Poland, [email protected] Michal Przyluski Tomasz Śliwiński Institute of Control and Computation Engineering, Warsaw University of Technology, Poland, [email protected] [email protected] Abstract. In several problems of portfolio selection the reward-risk ratio criterion is optimized to search for a risky portfolio offering the maximum increase of the mean return, compared to the risk-free investment opportunities. We analyze such a model with the CVaR type risk measure. Exactly the deviation type of risk measure must be used, i.e. the so-called conditional drawdown measure. We analyze both the theoretical properties (SSD consistency) and the computational complexity (LP models). Reward-CVaR Ratio Optimization Model Portfolio selection problems are usually tackled with the mean-risk models that characterize the uncertain returns by two scalar characteristics: the mean, which is the expected return, and the risk - a scalar measure of the variability of returns. In the original Markowitz model the risk is measured by the standard deviation or variance. Several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz-type) models. While the original Markowitz model forms a quadratic programming problem, many attempts have been made to linearize the portfolio optimization procedure (c.f., [2] and references therein). The LP solvability is very important for applications to real-life financial decisions where the constructed portfolios have to meet numerous side constraints (including the minimum transaction lots, transaction costs and mutual funds characteristics). A risk measure can be LP computable in the case of discrete random variables, i.e., in the case of returns defined by their realizations under specified scenarios. Several such risk measures have been applied to portfolio optimization [2]. Typical risk measures are deviation type. Some of them, like the mean absolute semideviation (MAD model) can be combined with the mean itself into optimization criteria (safety or underachievement measures) that remain in harmony with the Second order Stochastic Dominance (SSD). Some, like the conditional value at risk (CVaR) [5] having a great impact on new developments in portfolio optimization, may be interpreted as such a combined functional while allowing to distinguish the corresponding deviation type risk measure. Having given the risk-free rate of return r0 , a risky portfolio x may be sought that maximizes ratio between the increase of the mean return µ(x) relative to r0 and the corresponding increase of the risk measure %(x), compared to the riskfree investment opportunities. Namely, a performance measure of the reward-risk ratio is defined (µ(x) − r0 )/%(x) to be maximized. The optimal solution of the corresponding problem is usually called the tangency portfolio as it corresponds to the tangency point of the so-called capital market line drawn from the intercept Applications in Economics & Finance 205 r0 and passing tangent to the risk/return frontier. For the LP computable risk measures the reward-risk ratio optimization problem can be converted into an LP form [1]. The reward-risk ratio is well defined for the deviation type risk measures. Therefore while dealing with the CVaR risk model we must replace this performance measure (coherent risk measure) with its complementary deviation representation. The deviation type risk measure complementary to the CV aRβ representing the tail mean within the β-quantile takes the form of ∆β (x) = µ(x) − CV aRβ (x) called the (worst) conditional semideviation or drawdown measure. The measure expresses mean semideviation from the mean within the β quantile and opposite to belowtarget mean deviations (lower partial moments) it is risk relevant. Taking advantages of possible inverse formulation of the reward-risk ratio optimization as ratio ∆β (x)/(µ(x) − r0 ) to be minimized, we show that (under natural assumptions) this ratio optimization is consistent with the SSD rules (similar to the standard CVaR optimization [3]), despite that the ratio does not represent a coherent risk measure. Further, while transforming this ratio optimization to an LP model, we take advantages of the LP duality to get a model formulation providing higher computational efficiency. In the introduced model, similar to the direct CVaR optimization [4], the number of structural constraints is proportional to the number of instruments while only the number of variables is proportional to the number of scenarios, thus not affecting so seriously the simplex method efficiency. The model can effectively be solved with general LP solvers even for very large numbers of scenarios (like the case of fifty thousand scenarios and one hundred instruments solved less than a minute). Acknowledgements The research was partially supported by the National Science Centre (Poland) under the grant DEC-2012/07/B/HS4/03076. References [1] Mansini, R., Ogryczak, W., Speranza, M.G., On LP solvable models for portfolio optimization, Informatica 14 (2003), 37–62. [2] Mansini, R., Ogryczak, W., Speranza, M.G., Twenty years of linear programming based portfolio optimization, European Journal of Operational Research 234 (2014), 518–535. [3] Ogryczak, W., Ruszczyński, A., Dual stochastic dominance and related mean-risk models, SIAM Journal on Optimization 13 (2002), 60–78. [4] Ogryczak, W., Śliwiński, T., On solving the dual for portfolio selection by optimizing Conditional Value at Risk, Computational Optimization and Applications 50 (2011), 591–595. [5] Rockafellar, R.T., Uryasev, S., Conditional value-at-risk for general loss distributions, Journal of Banking & Finance 26 (2002), 1443–1471. Applications in Economics & Finance 206 Sunspot in Economic Models with Externalities Beatrice Venturi∗ Department of Economics and Business, University of Cagliari, Italy, [email protected] Alessandro Pirisinu Department of Economics and Business, University of Cagliari, Italy, [email protected] Abstract. In this paper we construct sunspot equilibria in a deterministic general class of endogenous growth two sector models with externalities (see Mulligan B. and X. Sala-i-Martin (1993), Nishimura and Shigoka (2006)). The optimal control model possesses stochastic characteristics which arise from indeterminate equilibrium and cycles (Hopf cycles, closed to the steady state) (see Chiappori and Guesnerie, 1991, Benhabib, Nishimura, and Shigoka 2006, Slobodyan 2009). As applications of our analysis. we consider a natural resource optimal model. The model undergoes to Hopf bifurcations in a parameters set (see Bella, 2010). We construct a stationary sunspot equilibrium near the closed orbit. We show that the stochastic approach suggest a way out to the poverty enviroments trap. References [1] C. Azariadis (1981), Self-fulfilling prophecies, Journal of Economic Theory 25, 380-396. [2] G. Bella (2010), Periodic solutions in the dynamics of an optimal resource extraction model Environmental Economics, Volume 1, Issue 1, 49-58. [3] G. Bella, P..Mattana, B. Venturi, (2013). The double scroll chaotic attractor in the dynamics of a fixed-price IS-LM model. Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 4(1), p. 1-13 [4] Benhabib, J., Nishimura, K., Shigoka, T. (2008). Bifurcation and sunspots in the continuous time equilibrium model with capacity utilization. International Journal of Economic Theory, 4(2), 337–355. [5] Nishimura, K., Shigoka, T. Makoto Yano (2006). Sunspots and Hopf bifurcations in continuous time endogenous growth models. International Journal of Economic Theory, 2, 199–216. [6] J. Benhabib, and R. Perli (1994), Uniqueness and indeterminacy: On the dynamics of endogenous growth, Journal of Economic Theory 63, 113-142. [7] J. Benhabib, and A. Rustichini (1994), Introduction to the symposium on growth, fluctuations, and sunspots: confronting the data, Journal of Economic Theory 63, 1-18. [8] D. Cass, and K. Shell (1983), Do sunspots matter? Journal of Political Economy 91, [9] P. A. Chiappori, and R. Guesnerie (1991), Sunspot equilibria in sequential market models, in: W. Hildenbrand, and H. Sonnenschein, Eds., Handbook of mathematical economics, Vol.4, North-Holland, Amsterdam, 1683-1762. [10] J. L. Doob (1953), Stochastic processes, John Wiley, New York. [11] J. P. Drugeon, and B. Wigniolle (1996), Continuous-time sunspot equilibria and dynamics in a model of growth, Journal of Economic Theory 69, 24-52. [12] R. E. A. Farmer, and M. Woodford (1997), Self-fulfilling prophecies and the busyness cycle, Macroeconomic Dynamics 1, 740-769. [13] J. M. Grandmont (1986), Stabilizing competitive business cycles, Journal of Economic Theory 40, 57-76. Applications in Economics & Finance 207 [14] J. Guckenheimer, and P. Holmes (1983), Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York. [15] R. Guesnerie, and M. Woodford (1992), Endogenous fluctuations, in: J. J. Laffont, Ed., Advances in economic theory, sixth world congress, Vol. 2, Cambridge University Press, New York, 289-412. [16] S. Itô (1963), An introduction to Lebesgue integral, Shohkaboh, Tokyo (in Japanese). [17] R. E. Lucas (1988), On the mechanics of economic development, Journal of Monetary Economics 22, 3-42. [18] P. Mattana (2004), The Uzawa-Lucas endogenous growth model, Ashgate Publishing Limited Gower House, Aldershot, England. [19] P. Mattana, and B. Venturi (1999), Existence and stability of periodic solutions in the dynamics of endogenous growth, International Review of Economics and Business 46, 259284. [20] U. Neri , B. Venturi (2007). Stability and Bifurcations in IS-LM economic models. International Review of Economics, 54, p. 53-65.. [21] J. Peck (1988), On the existence of sunspot equilibria in an overlapping generations model, Journal of Economic Theory 44, 19-42. [22] P. Romer (1990), Endogenous technological change, Journal of Political Economy 98, S71S102. [23] K. Shell (1977), Monnaie et Allocation Intertemporelle, mimeo. Seminarie d’Econometrie Roy-Malinvaud, Paris. [24] T. Shigoka (1994), A note on Woodford’s conjecture: constructing stationary sunspot equilibria in a continuous time model, Journal of Economic Theory 64, 531-540. [25] S. E. Spear (1991), Growth, externalities, and sunspots, Journal of Economic Theory 54, 215-223. [26] B. Venturi (2014) Chaotic Solutions in non Linear Economic - Financial models Chaotic Modeling and Simulation (CMSIM) 3: 233-254. Applications in Economics & Finance 208 Integration of Stock Markets in South East European Countries Boris Radovanov∗ Faculty of Economics Subotica, University of Novi Sad, Serbia, [email protected] Aleksandra Marcikic Faculty of Economics Subotica, University of Novi Sad, Serbia, [email protected] Abstract. This study examines short and long run dynamics in the relationships between six emerging South East European stock markets using a Markov Switching Vector Error Correction Model (MS-VECM). The long run dynamics are confirmed by a cointegrating vector which relates the stock indices of observed markets. The nonlinearity of the estimated model is justified by the Log-likelihood Ratio test and the information criteria. The MS-VECM with two regimes provides an adequate illustration of the financial policy effectiveness as markets become more exposed to global conditions. Introduction Recent trends in internationalization of financial markets have intrigued numerous studies to offer confirmation on international stock market convergence. As capital becomes borderless, the improvements in allocation efficiency and reductions in volatility have elevating the benefits of economic growth. The stable equilibrium relationships conceived by the cointegration methodology are not appropriate for modeling the dynamics of stock market integration. The traditional linear model does not allow the parameters to adjust for the structural changes. Therefore, many previous works have adopted Markov switching framework in analyzing the relationships between variables. Allowing the involvement of regimes, the mentioned methodology is well suited to domestic and international cycling shifts that affect domestic market. Moreover, it allows for changing relationships among the stock market variables across different time phases. Linked with the basic Markov switching vector autoregression model created by Hamilton (1989), a Markov Switching Vector Error Correction Model (MS-VECM) has recommended in case of unknown regime shifts of stock market prices. In other words, the model introduce state variable generated by the Markov chain, where the market state of tomorrow is driven only by the present market state (Krolzig, 1997). This paper studies the relationships among six emerging stock markets of South East Europen Countries (Bulgaria, Bosnia and Hercegovina, Croatia, Romania, Serbia and Slovenia). Since the stock markets in this region remain small in terms of capitalization, turnover and liquidity compared to developed countries, the main motivation of this paper is to examine the short and long run integration of markets in mentioned region. Scheicher (2001) found that some selected Eastern European stock markets were affected by both regional and global influences, where global integration of those markets is limited while there is higher regional integration. Similar findings are covered by Maneschiold (2006) with examination of Baltic stock markets and international capital markets. Applications in Economics & Finance 209 In order to achieve mentioned goal of this paper, the following methodology is introduced: 1) Augmented Dickey-Fuller test of unit root, 2) Johansen cointegration test, 3) estimating MS-VECM and 4) the Log-likelihood Ratio test and the information criteria in testing nonlinearity of the estimated model. While comparing linear and nonlinear model the paper provide robustness in final results. Similar to Keneurgios and Samitas (2011) our results have confirmed the existence of time-varying cointegrating relationship among South East European stock markets. Moreover, the final results help to understand the dynamic effects in relationships among selected markets by considering the impacts of regime switches. Also, using MS-VECM it is possible to simultaneously estimate short and long run dynamics. The study document that two regimes with changing intercept and variance yields as a good description of mentioned stock market changes. Finally, the results have implications regarding market efficiency hypothesis which impacts international or regional portfolio diversification and the long term economic growth prosperities. References [1] Hamilton, J.D.,A New Approach to the Econometric Analysis of Nonstationary Time Series and Business Cycle, Econometrica 57 (1989), 357–384. [2] Kenourgios, D., Samitas, A. Equity Market Integration in Emerging Balkan Markets, Research in International Business and Finance 25 (2011), 296–307. [3] Krolzig, H.M. Markov Switching Vector Autoregression Modelling, Statistical Inference, and Application to Business Cycle Analysis. Berlin: Springer verlag, 1997. [4] Maneschiold, P.Integration between the Baltic and International Stock Markets, Emerging Markets Finance and Trade 42 (2006), 25–45. [5] Scheicher, M.The Co-movements of Stock Markets in Hungary, Poland and Czech Republic, International Journal of Finance and Economics 6 (2001), 27–39. Applications in Economics & Finance 210 Joining risk diversification and utility maximization for portfolio selection Fabio Tardella∗ Dipartimento MEMOTEF, Sapienza Università di Roma, Italia, [email protected] Francesco Cesarone Dipartimento di Studi Aziendali, Università degli Studi Roma Tre, Italia, [email protected] Andrea Scozzari Facoltà di Economia, Università degli Studi Niccolò Cusano - Telematica, Italia, [email protected] Abstract. Utility maximization has been the main approach to portfolio selection since the pioneering work by Markowitz. However, due to the difficulty of obtaining good estimates of the expected return of a portfolio, several authors have devoted their work to the problem of minimizing or, at least, diversifying risk. Here, we describe a new framework for portfolio selection that aims at unifying the utility maximization and the risk diversification approaches by also taking into account an optimal assets selection step, possibly with limitations on the number of assets required in the selected portfolio. This new framework leads to several hard MINLP models. Risk diversification and utility maximization Starting with the seminal work by Markowitz, a large number of optimization models have been proposed to find an ideal allocation of capital among several available assets to achieve the investor’s objectives. Most of the models aim at maximizing some kind of utility of the investor usually based on appropriate risk and return measures. However, due to the difficulty of obtaining good estimates of the expected return of a portfolio, several authors have devoted their work to the problem of minimizing or, at least, diversifying risk. A straightforward approach to diversify the risk of a portfolio seems to be that of using a Equally Weighted (EW) portfolio. However, if the market contains assets with very different intrinsic risks, then this leads to a portfolio with limited total risk diversification [6]. Nonetheless, the EW portfolio seems to have very good performance in practice [4]. A further naive approach, frequently used in practice, to achieve approximately equal risk contribution of all assets, is to use weights proportional to 1/σi , where σi is the volatility of asset i. A more thorough approach to risk diversification requires to formalize the notion of risk contribution of each asset and then to manage it by a model. The Risk Parity approach has been formalized in a model which aims at making the total risk contributions of all assets equal among them [6]. This might be particularly useful when the assets have very different risk levels. This model can be solved by solving a nonlinear convex optimization problem or a system of equations and inequalities [6],[9]. The risk measure commonly used in the RP approach is volatility. However alternative risk measures can also be considered (see, e.g., [2] and the comprehensive new Applications in Economics & Finance 211 monograph by Roncally on Risk Parity and Risk Budgeting [8]). Other approaches to risk diversification have been recently proposed in [1], [3], [5], [7]. Here we propose a new framework for portfolio selection that aims at unifying the utility maximization and the risk diversification approaches by also taking into account an optimal assets selection step, possibly with limitations on the number of assets required in the selected portfolio. This new framework leads to several hard MINLP models, including some (black box) nonlinear pseudoBoolean optimization problems. We present some empirical, theoretical, and computational results for the solution of the proposed models and we evaluate their performance in real-world markets. References [1] Bera, A.K., Park, S.Z., Optimal Portfolio Diversification Using Maximum Entropy, Econometric Reviews, 27 (2008), 484–512. [2] Boudt, K., Carl, P., Peterson, B., Asset allocation with Conditional Value-at-Risk budgets, J Risk 15 (2013), 39–68. [3] Choueifaty, Y., Coignard, Y., Toward Maximum Diversification, The Journal of Portfolio Management, 34 (2008), 40–51. [4] DeMiguel, V., Garlappi, L., Uppal, R., Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?, Rev Financial Studies 22 (2009), 1915–1953. [5] Lohre, H., Neugebauer, U., Zimmer, C., Diversified Risk Parity Strategies for Equity Portfolio Selection, The Journal of Investing, 21 (2012), 111–128. [6] Maillard, S., Roncalli, T., Teiletche, J., The Properties of Equally Weighted Risk Contribution Portfolios, J Portfolio Management 36 (2010), 60–70. [7] Meucci, A.U., Managing Diversification, Risk, 22 (2009), 74–79. [8] Roncalli, T., Introduction to risk parity and budgeting. Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, FL, 2014. [9] Spinu, F., An algorithm for computing risk parity weights. Available at SSRN: http://ssrn.com/abstract=2297383 (July 30, 2013). Classification & Forecasting Tuesday 8, 17:30-19:00 Sala Riunioni Est 212 Classification & Forecasting 213 Comparative Analysis of Forecasting Methods for Energy Production in a Photo-Voltaic plant Renato Mari∗ Istituto per le Applicazioni del Calcolo ”M. Picone”, Consiglio Nazionale delle Ricerche, Bari, Italy, [email protected] Gabriella Dellino Teresa Laudadio Nicola Mastronardi Istituto per le Applicazioni del Calcolo ”M. Picone”, Consiglio Nazionale delle Ricerche, Bari, Italy, {g.dellino,t.laudadio,n.mastronardi}@ba.iac.cnr.it Carlo Meloni Silvano Vergura Dipartimento di Ingegneria Elettrica e dell’Informazione, Politecnico di Bari, Bari, Italy, {carlo.meloni,silvano.vergura}@poliba.it Abstract. In this study we propose a forecasting method for energy production in a PhotoVoltaic plant based on transfer function models, which are designed to take the effect of irradiance and temperature under consideration. A set of real data coming from a Photo-Voltaic plant are used to compare the proposed method with alternative approaches available in the literature and to evaluate forecasting quality by standard statistical indicators. An accurate daily forecast is necessary to setup the whole distribution network and results are discussed. Overview The problem under investigation concerns the forecasting of energy production for a Photo-Voltaic (PV) plant. The production rate of this kind of renewable energy source is strongly influenced by two exogenous factors related to environmental conditions: irradiance level and operating temperature. The Distribution System Operator has difficulties to manage an electrical system with stochastic generation, thus an accurate forecast is required for an efficient grid planning. In this study we focus on daily forecasting of energy production and propose a forecasting approach based on transfer function (TF) models [3], which are designed to include the effect of exogenous variables, such as irradiance level and operating temperature [1] on the forecasting process. More precisely, we develop a TF method based on the assumption that the relation between time series of energy production and irradiance can be modeled by a TF (to be estimated) plus an error vector described by an Autoregressive Integrated Moving Average (ARIMA) [3]. Formally, given time series of energy production zt and irradiance xt , zt = ω(B)B b xt + nt , δ(B) where B is the backward shift operator defined by Bzt = zt−1 , the transfer function is defined by s zeros, r poles and a delay b, with ω(B) = ω0 − ω1 B − ω2 B 2 . . . − ωs B s δ(B) = 1 − δ1 B − δ2 B 2 . . . − δr B r , and the vector of errors nt is described by an ARIMA model. Therefore, the TF method requires a reliable forecasting of irradiance for the next day. Parameters of both transfer function and ARIMA model are estimated by a tuning procedure Classification & Forecasting 214 based on statistical indicators on a predefined training set [1]. We refer to this approach as TFbase . We also propose an alternative approach to compute the TF, which makes use of an advanced linear algebra tool known as Singular Value Decomposition (SVD) [5], especially suitable when the only available information is represented by the recorded energy production data. In general, given a data matrix A ∈ Rm×n , its SVD is defined as A = U ΣV T , where U ∈ Rm×m , V ∈ Rn×n are orthogonal matrices and Σ ∈ Rm×n is a diagonal one with diagonal entries σi , i = 1, . . . , min{m, n}, and σ1 ≥ σ2 ≥ · · · ≥ σmin{m,n} . It is well known that the best rank–k approximation of P A, with k ≤ min{m, n}, is given by Ã = ki=1 σi ui viT , where ui and vi are the ith column of U and V , respectively [5]. In the present application the energy production data are arranged into a matrix, which turns out to be well approximated by its best rank–one approximation. Therefore, a reliable TF can be retrieved by computing only σ1 , u1 and v1 [4]. It will be denoted by TFSV D . A final step will consist in embedding the forecasting model within an algorithmic framework operating as a decision support system for PV energy production management [2]. We compare the performance of the forecasting model using the two aforementioned TF estimations, i.e., TFbase and TFSV D . To this aim we adopt a well known statistical indicator, the Mean Absolute Percentage Error (MAPE), which Pnamely n 1 zt −z¯t is defined as 100 n t=1 | z¯t | where z¯t is the observed value at time t and n is the length of the forecasting horizon. Preliminary results on a set of real data are reported in the following table showing MAPE for seven different scenarios obtained by changing the training set for parameters’ tuning: scenario 1 TFbase 3.76% TFSV D 2.95% 2 3.81% 3.75% 3 13.82% 16.71% 4 3.78% 5.96% 5 8.92% 10.45% 6 7 8.60% 4.29% 8.10% 3.58% The following remarks are highlighted: the irradiance level is crucial for computing the TF and obtaining a reliable forecast of energy production; nevertheless, in case of missing information about the irradiance, it is still possible to compute the TF by means of the SVD approach and attain an accurate forecast. References [1] Dellino, G., Laudadio, T., Mari, R., Mastronardi, N., Meloni, C., Sales Forecasting Models in the Fresh Food Supply Chain, in Proc. of the Int. Conf. on Operations Research and Enterprise Systems ICORES-2015 (2015), 419–426. [2] Dellino, G., Meloni, C., Mascolo, S., A Linear Physical Programming Approach to Power Flow and Energy Storage Optimization in Smart Grids Models, in Proc. of the Int. Conf. on Operations Research and Enterprise Systems ICORES-2015 (2015), 224–231. [3] Makridakis, S., Wheelwright, S.C., Hyndman, R.J., Forecasting Methods and Applications. Wiley India Pvt. Limited, 3rd edition, 2008. Classification & Forecasting 215 [4] Mastronardi, N., Tyrtyshnikov, E., Van Dooren, P., A fast algorithm for updating and downsizing the dominant kernel principal components, SIAM J. Matrix Anal. Appl. (2010) 31, 2376–2399. [5] Golub, G.H., Van Loan, C.F., Matrix Computations. The Johns Hopkins University Press: Baltimore, 4th edition, 2013. Classification & Forecasting 216 Electre Tri-Machine Learning Approach to the Record Linkage Valentina Minnetti∗ Department of Statistic Science, Sapienza, University of Rome, Italy, [email protected] Renato De Leone School of Sciences and Technology, University of Camerino, Italy, [email protected] Abstract. In this short paper, the Electre-Tri Machine Learning Approach is proposed for solving the Record Linkage Problem. This last problem can be viewed as ordinal sorting problem (i.e. classification problem considering preference information) and solved efficiently using the Electre Tri method. The proposed application highlights the good performances of the classification model found by Electre Tri. Record Linkage and Classification The aim of this paper is to use a multi-criteria based approach classification method to solve the record linkage problem. Record linkage is a solution to the problem of recognizing those records in two files which represent identical persons, objects, or events. Fellegi and Sunter [1] introduced a mathematical foundation for record linkage. Formally, two files A and B have to be matched. The pairs (a, b) ∈ Γ = A × B have to be classified into true matches and true nonmatches. The odds ratios of probabilities is: R= P r(γ ∈ Γ | M ) P r(γ ∈ Γ | U ) where γ is an arbitrary agreement pattern in the comparison space Γ. The decision rule helps to classify the pairs, as the following: • if R > U pper, then the pair (a, b) is the designated matches • if Lower ≤ R ≤ U pper, then the pair (a, b) is the designated potential matches • if R < Lower, then the pair (a, b) is the designated nonmatches In the decision rule, three sets are created: the designated matches, the designated potential matches, the designated nonmatches. They constitute the partition of the set of all the records in the product space Γ in three subsets C3 (matches), C2 (potential matches) and C1 (nonmatches), whose intersections are empty sets. The goal is to solve the record linkage problem as a multi-criteria based classification problem, whose a priori defined classes are the subsets of partition. The proposed application wants to find a classification model, found by Electre Tri method to solve record linkage problem, assigning each record in the space Γ to one of the three categories C1 , C2 and C3 , following the two phases procedure formulated by De Leone and Minnetti [2]. Classification & Forecasting 217 The input data, used in the application, were proposed by Winkler in Simulated list of people from American Census. The performances of the classification model, varying in particular two parameters, were from 99.09% to 99.89%, as well accounted for with much more details in [3]. References [1] I. P. Fellegi and A. B. Sunter. A theory for record linkage. Journal of the American Statistical Association, 64:1183–1210, 1969. [2] R. De Leone and V. Minnetti. The estimation of the parameters in multi-criteria classification problem: The case of the electre tri method. In Donatella Vicari, Akinori Okada, Giancarlo Ragozini, and Claus Weihs, editors, Analysis and Modeling of Complex Data in Behavioral and Social Sciences, Studies in Classification, Data Analysis, and Knowledge Organization, pages 93–101. Springer International Publishing, 2014. [3] V. Minnetti. On the parameters of the Electre Tri method: a proposal of a new two phases procedure. PhD thesis, Department of Statistics, Sapienza, University of Rome, 2015. Classification & Forecasting 218 Dealing with mixed hard/soft constraints via Support Constraint Machines Marcello Sanguineti∗ DIBRIS, Università di Genova, Italia, [email protected] Giorgio Gnecco IMT - Institute for Advanced Studies, Lucca, Italia, [email protected] Marco Gori Stefano Melacci DIISM - Università di Siena, Italia, [email protected] [email protected] Abstract. A learning paradigm is presented, which extends the classical framework of learning from examples by including hard pointwise constraints, i.e., constraints that cannot be violated. In applications, hard pointwise constraints may encode very precise prior knowledge coming from rules, applied, e.g., to a large collection of unsupervised examples. The classical learning framework corresponds to soft pointwise constraints, which can be violated at the cost of some penalization. The functional structure of the optimal solution is derived in terms of a set of “support constraints”, which generalize the classical concept of “support vectors”. They are at the basis of a novel learning parading, that we called “Support Constraint Machines”. A case study and a numerical example are presented. Introduction In recent research, the classic framework of learning from examples has been remarkably enriched to better model complex interactions of intelligent agents with the environment. Efforts have been made to re-frame the learning process in a context described by a collection of constraints, which clearly incorporates learning from examples. The addition of constraints representing prior knowledge on the learning problem under study improves the generalization capability of the learned model, compared to the case in which the same constraints are not taken into account in the problem formulation. An in-depth analysis of learning from constraints emphasizes the need for a theory to better devise effective algorithms for such a learning framework. First results in this direction are contained in [1], [2], [3], [4], and [7]. Main results The framework considered here is a prototype for learning problems in which hard and soft constraints are combined. The distinction between hard and soft constraints is in the way they are embedded into the problem formulation: in the hard case, they restrict the set of feasible solutions, whereas in the soft case their violation is penalized through terms containing a loss function, which are included in the objective of the optimization problem. We focus on pointwise constraints (i.e., constraints defined on a finite set of examples, where each element of the set is associated with one such constraint). Classification & Forecasting 219 They model very general knowledge and are often encountered in machine learning problems. Hard pointwise constraints encode very precise prior knowledge coming from rules, applied, e.g., to a large collection of unsupervised examples (usually, sampled independently according to a possibly unknown probability distribution). Instead, soft pointwise constraints are associated with less reliable knowledge, corresponding, e.g., to examples more heavily corrupted by noise. The emphasis on the unsupervised examples associated with hard pointwise constraints is motivated by the fact that often they are more widely available than supervised ones, due to a possibly high cost of supervision. We formulate the problem of learning from examples with the presence of mixed hard and soft pointwise constraints and investigate the existence and uniqueness of optimal solution. We derive the structure of the optimal solution, introduce the concepts of “constraint reaction” and “support constraint” and discuss computational issues. The results are illustrated via a case study and a numerical example. The talk presents improvements over the results discussed in [5] and [6]. References [1] Gnecco, G, Gori, M, Melacci, M., and Sanguineti, M., Learning with Mixed Hard/Soft Pointwise Constraints, IEEE Transactions on Neural Networks and Learning Systems, to appear (DOI: 10.1109/TNNLS.2014.2361866). [2] Gnecco, G, Gori, M, Melacci, M., and Sanguineti, M., Foundations of Support Constraint Machines, Neural Computation 27 (2015), 388–480. [3] Gnecco, G, Gori, M, Melacci, M., and Sanguineti, M., Learning as Constraint Reactions, Artificial Neural Networks: Methods and Applications. P. Koprinkova-Hristova, V. Mladenov, and N. Kasabov, Eds. Springer Series in Bio/Neuroinformatics. Springer (2015), 245-270. [4] Gnecco, G, Gori, M, Melacci, M., and Sanguineti, M., A Theoretical Framework for Supervised Learning from Regions, Neurocomputing 129 (2014), 25-32. [5] Gnecco, G, Gori, M, Melacci, M., and Sanguineti, M., A Machine-Learning Paradigm that Includes Pointwise Constraints , 20th Conference of the International Federation of Operational Research Societies (IFORS) Barcelona, 13-18 July 2014. [6] Gnecco, G, Gori, M, Melacci, M., and Sanguineti, M., Learning with mixed hard/soft constraints by support constraint machines, 3rd Italian Workshop on Machine Learning and Data Mining (MLDM), Pisa, 10-11 December 2014. [7] Gnecco, G, Gori, M, and Sanguineti, M., Learning with Boundary Conditions, Neural Computation 25 (2012), 1029-1106. Game Theory Thursday 10, 14:15-15:45 Sala Gerace 220 Game Theory 221 The Inverse Problem and the Core of cooperative TU games Irinel Dragan∗ Department of Mathematics, University of Texas, USA, [email protected] Abstract. In an earlier work [1], we stated and solved what we called the Inverse Problem for the Shapley Value. A TU game (N,v) has the Shapley Value L, which shows how to allocate the win of the grand coalition to the individual players. The Inverse Problem is stated as follows: given L, find out the set of cooperative TU games with the set of players N, for which the Shapley Value equals L. A formula providing this set of games, called the Inverse Set relative to the value, was given. Now, assuming that L does not belong to the Core of the game, which is the minimal stability requirement, or sometimes the Core is even empty, we show how could be found in the Inverse Set a new game, with the Shapley Value L, that belongs to its Core. The given procedure, derived from results of Linear Algebra, is offering fomulas for the computation of such a game. Of course, even for an initial game with an empty Core, we find a game with a nonempty Core that has inside the Shapley Value L. In a subsequent paper we discussed also the similar problem for Semivalues, where the Core is replaced by the Power Core, using the ideas introduced in another paper [2]. References [1] Dragan I., The potential basis and the weighted Shapley Value, Libertas Matematica 11 (1991), 139-146. [2] Dragan I., Martinez-Legaz J.E., On the Semivalues and the Power Core of cooperative TU games, IGTR 3 (2001), 127-139. Game Theory 222 Majority judgment and strategy-proofness Stefano Vannucci∗ Dipartimento di Economia politica e statistica, Università di Siena, Italia, [email protected] Abstract. Majority judgment as recently formulated and advocated by Balinski and Laraki in their influential book (Majority Judgment, 2010) is a method to aggregate profiles of judgments on a common bounded linearly ordered set of grades. It is shown that majority judgment is strategyproof but not coalitionally strategy-proof on a very general single peaked preference domain. Introduction Majority judgment (MJ) boils down to the following protocol: fix a linear order Λ as a common language of grades and ask the n voters to assign Λ-grades to all the m alternative candidates, then assign to each candidate a grade given by the median of the grades received if n is odd, and the lower middlemost of the grades received if n is even. Thus, MJ amounts to a function f : ΛY ×N → ΛY i.e. to an n-ary operation that maps grading profiles into gradings, and satisfies Anonymity, Unanimity (i.e. Idempotence) and Monotonicity as defined in the obvious way. In their outstanding book, Balinski and Laraki (2010) strongly advocate majority judgement mentioning that -inter alia- that MJ is strategy-proof on a certain single peaked domain, and mentioning that its single-candidate components are also ‘group strategy-proof’ on such domain. In the present note we shall confirm that MJ is strategy-proof on the full locally unimodal domain (a generalized single peaked domain) showing at the same time that -on the contrary- MJ is coalitionally manipulable on full single peaked domains. Results Definition 1 A topped total preorder <∈ TX -with top outcome x∗ -is locally unimodal (with respect to BX ) if and only if, for each x, y, z ∈ X, z ∈ BX (x∗ , ., y) implies that either z < x or z < y (or both). Let UX ⊆ TX denote the set of all locally unimodal total preorders with respect to BX , and UXN the corresponding set of all N -profiles of locally unimodal total preorders or full locally unimodal domain. Similarly, UX∗ , SX ⊆ TX denote the set of all unimodal (locally strictly unimodal, respectively) total preorders with respect to BX , and UX∗N , SXN the corresponding full unimodal and full locally strictly unimodal domains. A voting rule for (N, X) is a function f : X N → X. The following properties of a voting rule will play a crucial role in the ensuing analysis: Definition 2 A voting rule f : Πi∈N Xi → X is BX -monotonic if and only if for all xN = (xj )j∈N ∈ X N , i ∈ N and x0i ∈ X: f (xN ) ∈ BX (xi , ., f (x0i , xN r{i} ). Game Theory 223 Definition 3 For any i ∈ N , let Di ⊆ UX such that top(<) ∈ Yi for all <∈ Di . Then, f : Πi∈N Yi → X is (individually) strategy-proof on Πi∈N Di ⊆ UXN if and only if, for all xN ∈ Πi∈N Xi , i ∈ N and x0 ∈ Xi , and for all < = (<j )j∈N ∈ Πi∈N Di , not f (x0 , xN r{i} ) <i f (top(<i ), xN r{i} ) . Definition 4 For any i ∈ N , let Di ⊆ UX such that top(<) ∈ Yi for all <∈ Di . Then, f : Πi∈N Yi → X is coalitionally strategy-proof on Πi∈N Di ⊆ UXN if and only if for all xN ∈ Πi∈N Yi , C ⊆ N and x0C ∈ Πi∈C Yi , and for all < = (<j )j∈N ∈ Πi∈N Di , there exists i ∈ C such that not f (x0C , xN rC ) <i f (xN ). A generalized committee in N is a set of coalitions C ⊆ P(N ) such that T ∈ C if T ⊆ N and S ⊆ T for some S ∈ C (a committee in N being a non-empty generalized committee in N which does not include the empty coalition). A generalized committee voting rule is a function f : Πi∈N Xi → X such that, for some fixed generalized committee C ⊆ P(N ) and for all xN ∈ Πi∈N Xi , f (xN ) = ∨S∈C (∧i∈S xi ). A prominent example of a generalized committee voting rule is of course the majority rule f maj defined as follows: for allo xN ∈ Πi∈N Xi , n maj maj f (xN ) = ∨S∈C maj (∧i∈S xi ) where C = S ⊆ N : |S| ≥ b |N2|+2 c . The majority judgment rule is precisely the majority rule as applied to X = Y Λ where Y is the set of alternative candidates and Λ is a bounded linear order that denotes a common grading language. If in particular Λ = ({0, 1} , ≤) the majority judgment rule amounts to majority approval. The following Lemma and Theorem extend some previous results concerning strategy-proofness on trees and on distributive lattices (see Danilov (1994), Vannucci (2012)). Lemma 5 Let X = (X, 6) be a distributive lattice, and f : X N → X a voting rule for (N, X). Then, the following statements are equivalent: (i) f is BX -monotonic; (ii) f is strategy-proof on UXN . Theorem 6 Let X = (X, 6) be a bounded distributive lattice and BX its latticial betweenness relation. Then, a generalized committee voting rule f : X N → X is BX -monotonic. Corollary 7 Let X = (X, 6) be a bounded distributive lattice and BX its latticial betweenness relation. Then, f maj : X N → X is strategy-proof on UXN . In particular, the majority judgment rule f maj : ΛY ×N → ΛY is strategy-proof on UΛNY . Proposition 8 The majority judgment rule f maj : ΛY ×N → ΛY is not coalitionally N N strategy-proof on UΛ∗N Y , SΛY , or UΛY . Game Theory 224 Robust Screening under Distribution Ambiguity Mustafa Pinar∗ Bilkent University, Turkey, [email protected] Can Kizilkale University of California, Santa Barbara, USA, [email protected] Abstract. We consider the problem of screening where a seller puts up for sale an indivisible good, and a buyer with a valuation unknown to the seller wishes to acquire the good. We assume that the buyer valuations are represented as discrete types drawn from some distribution, which is also unknown to the seller, i.e., we abandon the assumption frequent in the economics literature that the seller uses a subjective expected utility representation. The seller is averse to possible mis-specification of types distribution, and considers the unknown type density as member of an ambiguity set and seeks an optimal pricing mechanism in a worst case sense. We consider first two choices for the ambiguity set, and derive the optimal mechanism for both whereas for a third choice of ambiguity representation a result showing that a posted price mechanism remains optimal in a worst-case sense for every perturbation of the prior confined to a ball around a reference prior is obtained. For our first case of box ambiguity, we proved that a posted-price mechanism is optimal whereas for the case when the seller has an idea about the mean valuation it is optimal to adopt an ascending price mechanism. Numerical examples show that the optimal mechanisms lead to improved expected revenues compared to the case where any distribution is allowed, or when a simpler mechanism is used. Summary Mechanism design is an area of economics where optimization is ubiquitous. The present paper aims to contribute to the interface of economics/mechanism design and optimization by investigating optimal pricing mechanisms for an indivisible object in a single seller/single buyer context where buyer valuations and their distribution, assumed to be discrete, are unknown to the seller. In a departure from the common priors assumption pervasive in the literature (i.e., the common priors assumption states that the buyer valuations, while not known to the seller, are drawn from a known distribution) we assume the distribution of valuations is not known to the seller. However, the seller, while risk neutral, is averse to any imprecision in the distribution in the spirit of Gilboa and Schmeidler. In other words, the seller evaluates each action by its minimum expected revenue across all priors. He/she considers a set of distributions, and wishes to design a mechanism that is optimal in a worst-case sense, i.e., he/she maximizes the worst-case expected revenue with respect to nature/adversary’s choice of distribution. We derive the optimal robust mechanism under two different specifications of ambiguity in the prior distribution. Graphs & Networks Monday 7, 15:30-17:00 Sala Riunioni Ovest 225 Graphs & Networks 226 A review of recent heuristic algorithms for the Critical Node Problem Pierre Hosteins∗ Dipartimento di Informatica, Università di Torino, Italia, [email protected].unito.it Bernardetta Addis LORIA, Université de Lorraine, France, [email protected] Roberto Aringhieri Andrea Grosso Dipartimento di Informatica, Università di Torino, Italia, roberto.aringhieri[email protected] [email protected] Rosario Scatamacchia DAUIN, Politecnico di Torino, Italia, [email protected] Abstract. We will review several heuristic algorithms we proposed recently to solve different versions of the Critical Node Problem (CNP), i.e. the maximal fragmentation of a graph G given a connectivity measure, that improve the solution quality greatly over existing competitors. Problem Definition The solution of the CNP is defined by a set S of K nodes belonging to a graph G(V, E), whose removal will disconnect the graph according to a precise connectivity measure f (S). A classic version of the CNP is the minimisation of connectivity function: f (S) = |{i, j ∈ V \ S : i and j are connected by a path in G[V \ S]}| with K fixed, i.e. obtaining a subgraph G[V \ S] where the minimum number of pair of nodes (i, j) are still connected by a path in the graph. Other connectivity measures have been introduced, such as the size of the largest connected components in G[V \ S] (that we try to minimise) or the number of connected components (that we try to maximise) but fewer algorithms exist to tackle these versions. Greedy-based algorithms In [2] we propose efficient and effective heuristics based on greedy rules, partly based on the work of [1]. Starting from an existing set S, we can extend or reduce it through two greedy rules: • Greedy rule 1. Remove from S a node i such that i = arg min{f (S \ {i}) − f (S)}. • Greedy rule 2. Add to S a node i ∈ V \ S such that i = arg max{f (S) − f (S ∪ {i})}. We combined the use of both rules in order to iteratively extend |S| above K or reduce it below K so as to perturb the solution given by a simple greedy procedure and give it a flavour of local search. The algoritms proposed are very fast and Graphs & Networks 227 obtain results of higher quality than some competing metaheuristics. Results are presented in [2]. Local search metaheuristics The two preceding greedy rules can be advantageously used to perform a fast search of the simpler neighbourhood of a solution S, i.e. the exchange of two nodes u ∈ S and v ∈ V \ S. If |S| = K, a complete search of such a neighbourhood has a complexity proportional to K(|V | − K) while the use of smartly implemented greedy rules reduces it to a proportionality of min(K, |V | − K), which in some cases highly speeds up the local search phase. We devised two classes of algorithms, based on VNS and ILS schemes that allowed to further improve the results obtained in [2] for the most difficult graphs, although such metaheuristics are not able to tackle instances as big as the greedy-like algorithms described in the previous section. The results have been partly published in [3] and will be published with full details in [4]. A genetic algorithm for different connectivity measures One can exploit the preceding greedy rules for implementing a very general genetic algorithm that is able to tackle many different versions of the CNP. The core of the algorithm uses a reproduction scheme that generates a new solution from the union of the parents’ sets S1 ∪ S2 and then applies Greedy Rule 1 until necessary. Greedy Rule 2 can also be used in a mutation phase for perturbing the solution. The use of these rules allows to focus and speed up the search of the parameter space towards better quality solutions. Ad hoc greedy rules for alternate connectivity measures allow to apply the algorithm to versions of the CNP for which fast and good quality heuristics do not yet exist. Tests are being performed and will be submitted for publication in [5]. References [1] Arulselvan, A. and Commander, C. W. and Elefteriadou, L. and Pardalos, P. M., Detecting Critical Nodes in Sparse Graphs, Computers & Operations Research 36 (2009), 2193–2200. [2] B. Addis and R. Aringhieri and A. Grosso and P. Hosteins, Hybrid Constructive Heuristics for the Critical Node Problem, Submitted for publication to ANOR. [3] R. Aringhieri and A. Grosso and P. Hosteins and R. Scatamacchia, VNS solutions for the Critical Node Problem, Electronic Notes in Discrete Mathematics 47 (2015), 37–44. [4] R. Aringhieri and A. Grosso and P. Hosteins and R. Scatamacchia, Local Search Metaheuristics for the Critical Node Problem, Accepted for publication in Networks. [5] R. Aringhieri and A. Grosso and P. Hosteins and R. Scatamacchia, A Genetic Algorithm for different classes of the Critical Node Problem, work in progress. Graphs & Networks 228 A Computational Comparison of Master Problem Formulations for Multicommodity Flows James Hungerford∗ M.A.I.O.R., Italy, [email protected] Alberto Caprara University of Bologna, Italy, Antonio Frangioni University of Pisa, Italy, [email protected] Tiziano Parriani OptIt, Italy, [email protected] Abstract. We analyze some variants of the Dantzig-Wolfe/Column Generation approach for solving multicommodity flow problems where we vary the level of aggregation in the master problem. Other than the two known extremes in which commodities are either totally aggregated or each treated separately, we allow intermediate cases corresponding to partial aggregation and we study the effect on the performance of the algorithm. We also compare this to the structured Dantzig-Wolfe approach in which a (restriction of) the standard node-arc formulation is used as the master problem. Computational results are provided for instances from a variety of applications, with a special emphasis on vehicle scheduling problems. Introduction The Multicommodity Min-Cost Flow problem (MMCF) arises when several commodities must be routed through a network while competing for capacities along its arcs. MMCF has a large variety of applications [1], including communication networks and vehicle scheduling. Furthermore, it is the archetype for optimization problems with block-decomposable structure. MMCF is a large-scale LP, which can be difficult to solve in practice as the number of variables can be huge. Furthermore, when integrality restrictions are added MMCF is NP-hard (unless there is only one commodity and capacities and demands are integer [2]). Thus, the continuous relaxation has to be repeatedly solved with exact or heuristic approaches. When the mutual capacity constraints are relaxed, MMCF decomposes into as many Min-Cost Flow problems as there are commodities, each of which can be solved efficiently. This motivates the use of price decomposition techniques, i.e., the Dantzig-Wolfe (DW) decomposition algorithm and its variants. These are based on first reformulating the problem into one with as many columns as there are vertices of the polyhedron, and then solving it by Column Generation. It is well-known [4] that this amounts to solving the corresponding Lagrangian Dual with a Cutting Plane algorithm, or one of its variants [3][5]. Since the feasible region of the subproblem decomposes, it is also well-known that one can add to the master problem as many columns as there are commodities at each iteration; when commodities are origin-destination pairs this amounts to solving the arc-path formulation by Column Generation. Reformulating the problem so that it has more commodities, and therefore so that the master problem Graphs & Networks 229 acquires more columns, has been shown to be beneficial [1]. On the other hand, completely aggregating all columns into one at each iteration allows to reduce the master problem size (and to use specialized approaches [5]), and hence its computational cost. In other words, a nontrivial trade-off exists between a larger and more costly master problem which yields faster convergence, and a nimbler master problem which however implies that more iterations are needed. In this talk we systematically explore this trade-off by considering the effect of varying the degree of aggregation of the commodities. Given the set of user-defined commodities, the degree of aggregation is increased by arbitrarily partitioning the set into subsets and aggregating all commodities in a subset. Conversely, in some cases the degree of aggregation can be decreased by splitting a commodity into subcommodities, each associated with a different set of destinations. Interestingly, this need only happen in the master problem: the original set of user-defined commodities need not be changed (as [1] might seem to imply), and therefore the subproblem also remains the same. We complement this analysis of the impact of master problem formulation on the overall efficiency of price decomposition methods by also comparing the partially aggregated models to a structured Dantzig-Wolfe decomposition [6], in which the master problem is replaced by a restriction of the node-arc formulation of MMCF. In this approach, the pricing subproblem is the same as in the standard DW algorithm, but the solution to the pricing problem is incorporated into the master problem in a more flexible way: instead of taking convex combinations of extreme flows, each flow is broken down into its constituent arcs, which are then un-blocked in the node-arc master problem. Hence, the restricted master problem solves MMCF over a subgraph of the original network. We compare the performance of the above approaches on a collection of test instances from various applications, with a special emphasis on vehicle scheduling problems. References [1] K. Jones, I. Lustig, J. Farvolden, W. Powell Multicommodity network flows: the impact of formulation on decomposition, Mathematical Programming 62 (1993), 95–117. [2] S. Even, A. Itai, A. Shamir On the complexity of timetable and multicommodity flow problems, SIAM Journal on Computing 5(4) (1976), 691-703. [3] A. Frangioni Generalized Bundle Methods, SIAM Journal on Optimization 13(1) (2002), 117–156. [4] A. Frangioni About Lagrangian Methods in Integer Optimization, Annals of Operations Research 139 (2005), 163–193. [5] A. Frangioni, G. Gallo A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min Cost Flow Problems, INFORMS Journal On Computing 11(4) (1999), 370–393. [6] A. Frangioni, B. Gendron A Stabilized Structured Dantzig-Wolfe Decomposition Method, Mathematical Programming 140 (2013), 45–6. Graphs & Networks 230 On the virtual network embedding problem Edoardo Amaldi∗ DEIB, Politecnico di Milano, Milano, Italy, [email protected] Stefano Coniglio Arie Koster Martin Tieves Lehrstuhl II für Mathematik, RWTH Aachen University, Aachen, Germany, [email protected] [email protected] [email protected] Abstract. Network virtualization techniques, which have been attracting considerable attention in the recent networking literature, allow for the coexistence of many virtual networks jointly sharing the resources of an underlying substrate network. We address the offline version of the Virtual Network Embedding problem (VNE) which arises when looking for the most profitable set of virtual networks to embed onto a substrate. Given a graph representing the substrate (or physical) network with node and edge capacities and a set of virtual networks with node capacity demands and node-to-node traffic demands, the problem is to decide how to embed (a subset of) the virtual networks onto the substrate network so as to maximize the total profit while respecting the substrate node and edge capacities. We study the computational complexity of VNE, presenting a polynomial-time reduction implying strong NP-hardness for VNE even for very special classes of graphs and a strong inappproximability result for general graphs. We also discuss some special cases obtained when fixing one of the dimensions of VNE to one. We conclude by investigating an Integer Linear Programming formulation of the problem and comparing it to previously proposed ones. Logistics Tuesday 8, 9:00-10:30 Sala Riunioni Ovest 231 Logistics 232 Rapid transit network design with modal competition Luigi Moccia∗ Istituto di Calcolo e Reti ad Alte Prestazioni, Consiglio Nazionale delle Ricerche, Rende, Italy, [email protected] Gabriel Gutiérrez-Jarpa School of Industrial Engineering, Pontificia Universidad Católica de Valparaı́so, Valparaı́so, Chile, gabriel.g[email protected] Gilbert Laporte HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, Canada H3T 2A7, [email protected] Vladimir Marianov Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile, [email protected] Abstract. We present a mixed integer linear program for the rapid transit network design problem with static modal competition. Previous discrete formulations cannot handle modal competition for realistic size instances because of the complexity of modeling alternatives for each flow in the network. We propose a multi-objective model to address effectiveness, efficiency, and equity concerns. A case study carried out for a metro proposal in Concepción, Chile, shows the suitability of the method. Introduction In recent years, several models and algorithms have been put forward for the design of metro networks (see e.g. [1], [7], [2], and [5]). Here we extend the rapid transit network design problem (RTNDP) of [3] by introducing modal competition and by enriching its multi-objective framework. In that reference an origin-destination flow is considered as captured by rapid transit if some stations are sufficiently close to both the origin and the destination of the flow. We observe that by maximizing the captured traffic using this criterion results in improving access, i.e. the number of commuters that could benefit from the rapid transit network for their daily trips. This is indeed a relevant goal in urban transit, but on its own it does not adequately reflect modal choices. In this paper we consider a traffic flow as captured if the travel time (or equivalently the generalized cost) by rapid transit is less than by car, i.e. an “all or nothing” criterion. This feature has been neglected in most previous discrete mathematical programs because considering origin-destination flows results in models that are too large for realistic instances. As observed by [8], considering traffic flows requires a multi-commodity formulation, where each flow is considered as a distinct commodity. This was the approach taken by [4], but it only allowed the solution of very small instances. The scientific contribution of this paper is to introduce a methodology that overcomes this difficulty by exploiting a pre-assigned topological configuration. As explained by [1], a pre-assigned topological configuration is in itself a positive feature for planners since it incorporates their knowledge of the traffic flows in cities and corresponds to what is often done in practice. Equity concerns relevant to passenger Logistics 233 transportation (see [9] for a review) are also taken into account. We show that our approach can be applied to realistic situations and we illustrate it on data from Concepción, Chile. References [1] Bruno, G. and Laporte, G. (2002). An interactive decision support system for the design of rapid public transit networks. INFOR, 40(2):111–118. [2] Canca, D., De-Los-Santos, A., Laporte, G., and Mesa, J. (2014). A general rapid network design, line planning and fleet investment integrated model. Annals of Operations Research, In Press:1–18. [3] Gutiérrez-Jarpa, G., Obreque, C., Laporte, G., and Marianov, V. (2013). Rapid transit network design for optimal cost and origin-destination demand capture. Computers & Operations Research, 40(12):3000–3009. [4] Laporte, G., Marı́n, A., Mesa, J. A., and Perea, F. (2011). Designing robust rapid transit networks with alternative routes. Journal of Advanced Transportation, 45(1):54–65. [5] Laporte, G. and Mesa, J. A. (2015). The design of rapid transit networks. In Laporte, G., Nickel, S., and Saldanha da Gama, F., editors, Location Science, pages 581–594. Springer, Berlin, Heidelberg. [6] Laporte, G., Mesa, J. A., and Ortega, F. A. (1994). Assessing topological configurations for rapid transit networks. Studies in Locational Analysis, 7:105–121. [7] Laporte, G., Mesa, J. A., Ortega, F. A., and Sevillano, I. (2005). Maximizing trip coverage in the location of a single rapid transit alignment. Annals of Operations Research, 136(1):49– 63. [8] Marı́n, Á. and Garcı́a-Ródenas, R. (2009). Location of infrastructure in urban railway networks. Computers & Operations Research, 36(5):1461–1477. [9] Perugia, A., Cordeau, J.-F., Laporte, G., and Moccia, L. (2011). Designing a home-towork bus service in a metropolitan area. Transportation Research Part B: Methodological, 45(10):1710–1726. Logistics 234 Simulation of supply chains via alternative numerical methods Luigi Rarità∗ Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università di Salerno, Italia, [email protected] Matteo Gaeta Stefania Tomasiello Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università di Salerno, Italia, [email protected] [email protected] Alfredo Vaccaro Dipartimento di Ingegneria, Università del Sannio, Benevento, Italia, [email protected] Abstract. In this talk, we discuss two numerical approaches for the simulation of a continuous model for supply networks. The first one considers Upwind and explicit Euler methods. The second approach foresees Differential Quadrature (DQ) rules and a Picard–like recursion. Some test cases are used to prove that DQ allows better approximations. Moreover, possible applications of the numerical approaches in energetic systems are considered. General overview Industrial applications, and in particular supply systems, represent an important topic, as the principal aim is often the control of unwished phenomena, such as bottlenecks, dead times, and so on. Suppliers, manufacturers, warehouses and stores are components of supply chains and networks, whose dynamics is studied via different models: some are discrete, i.e. they are based on discrete-event simulation; others are continuous and deal with differential equations. In this talk, we focus on two numerical schemes for a continuous model of supply networks, proposed in [2] and [3] and based on partial differential equations for densities of goods on arcs and ordinary differential equations for the dynamics of queues among arcs. The first one considers Upwind and explicit Euler methods, see [1]. The second scheme uses Differential Quadrature (DQ) rules and a Picard–like recursion (see [5], [6] for applications to ordinary differential equations), obtaining a final non–recursive scheme, which uses matrices and vectors, with obvious advantages for the determination of the local error. Comparisons with the first classical scheme are made in some test cases dealing with real examples for assembling car engines. The obtained results prove that DQ rules allows optimizing the approximations degree of densities and queues. Finally, considering the approximation scheme based on DQ, a discussion for losses modelling in energetic networks (see [4] for details) is made. References [1] Cutolo A., Piccoli B., Rarità L., An Upwind-Euler scheme for an ODE-PDE model of supply chains, Siam Journal on Scientific Computing 33(4) (2011), 1669–1688. [2] Göttlich S., Herty M., Klar A., Network models for supply chains, Communication in Mathematical Sciences 3 (2005), 545–559. Logistics 235 [3] Göttlich S., Herty M., Klar A., Modelling and optimization of Supply Chains on Complex Networks, Communication in Mathematical Sciences 4 (2006), 315–330. [4] Krause, T., Andersson, G., Fröhlich, K., Vaccaro, A. Multiple-energy carriers: Modeling of production, delivery, and consumption. Proceedings of the IEEE 99(1) (2011), 15–27. [5] Tomasiello S. Some remarks on a new DQ-based method for solving a class of Volterra integro-differential equations. Applied Mathematics and Computation 219 (2012), 399–407. [6] Tomasiello S. A note on three numerical procedures to solve Volterra integro-differential equations in structural analysis. Computers and Mathematics with Applications 62 (2011), 3183–3193. Logistics 236 Multi-stage location of flow intercepting portable service facilities Antonio Sforza∗ Department of Electrical Engineering and Information Technology, University of Naples “Federico II”, Italy, [email protected] Annunziata Esposito Amideo Claudio Sterle Departement of Electrical Engineering and Information Technology, University of Naples “Federico II”, Italy, [email protected] [email protected] Abstract. Multi-stage flow intercepting facility location problems (MS-FIFLP ) deal with portable service facilities location on a network whose conditions vary on a multi-stage time horizon. To the best of authors’ knowledge MS-FIFLP has never been treated before. For its solution we propose two ILP based approaches, a sequential and an integrated one. Both approaches have been tested on a network derived from a real case-study, related to the location and re-location of portable VMSs in an urban area. The multi-stage flow intercepting facility location problem Literature on facility location problems (FLP ) and related variants, differing for application fields and solving approaches, is very broad. However research activity on flow based demand service facility location problems, also referred to as flow intercepting facility location problem (FIFLP ), is still limited with respect to the widely treated point based demand facility location problems (p-median, simple plant and related variants). FIFLP s are optimization problems where the service facilities to be located in the nodes or on the links of a network do not generate/attract flows but intercept them along their pre-planned paths from origins to destinations ([1], [3], [3]). Addressing the time component by taking into account the evolution of a network over a certain time horizon for flow intercepting facilities location is particularly relevant in many application fields, such as, among others, urban traffic management, network monitoring and control, urban security, where it is fundamental to consider the uncertain nature of the phenomena under investigation. To the best of authors’ knowledge, all the previous works about FIFLP concern the single stage case while the multi-stage case (MS-FIFLP ) has never been treated in literature. Indeed the only contributions on multi-stage FLP s are devoted to the point based demand case, which has been largely studied with reference to logistic and emergency facilities ([4], [5] and [6] ). The multi-stage flow intercepting facility location problem (MS-FIFLP ) is a tactical and operational decision problem arising when portable flow intercepting facilities have to be dynamically located and re-located (repositioned) in the nodes or on the links of a network according to the network conditions, varying on a multi-stage time horizon, expressed in terms of flow traversing the network. The objective of the MS-FIFLP is either to both maximize the flow intercepted by the facilities and minimize the relocation cost associated to them, having fixed the number of facilities that can be used on the network or both minimize the number of used facilities and the relocation cost associated to them, having fixed a certain Logistics 237 percentage of the total flow traversing the network as the minimum amount of flow to be intercepted. We propose a sequential and an integrated ILP based approach for both variants of the MS-FLFLP. It is important to underline that the two approaches are not alternative. The choice between them has to be done considering the specific features of the problem under investigation. If the problem under investigation is characterized by a great uncertainty and a real time service facility repositioning is required on the basis of a flow distribution at a given stage of the network, then the problem could be more effectively and suitably tackled by the sequential approach. On the other side, if the problem under investigation is not characterized by uncertainty and good/trustworthy off-line data are available (based on historical series), then the optimal locations of the service facilities and their repositioning for the whole time horizon could be effectively and suitably addressed by the integrated approach. The two approaches have been experienced on test networks derived from a real case-study, with reference to the location and re-location of portable variable message signs (VMSs) in a urban area. The emerging results confirm the effective usage of the proposed approaches for several flow intercepting facility location problems encountered in real applications. References [1] Berman, O., Hodgson, M.J., Krass, D., Flow interception problems. In Drezner Z (ed) Facility Location: A survey of Applications and Methods. Springer New York, (1995), 389– 426. [2] Boccia, M., Sforza, A., Sterle, C., Flow Intercepting Facility Location: Problems, Models and Heuristics. Journal of Mathematical Modelling and Algorithms, 8, 1, (2010), 35–79. [3] Zeng, W., Castillo, I., Hodgson, M.J., A generalized model for locating facilities on a network with flow-based demand. Network and Spatial Economics, 10, 4, (2010), 579–611. [4] Gendreau, M., Laporte, G., Semet, F., A dynamic model and parallel tabu search heuristic for real-time ambulance relocation. Parallel Computing, 27, (2001), 1641–1653. [5] Brotcorne, L., Laporte, G., Semet, F., Ambulance location and relocation models. European Journal of Operational Research, 147, (2003), 451–463. [6] Melo, M.T., Nickel, S., Saldanha da Gama, F., Dynamic multi-commodity capacitated facility location: a mathematical modeling framework for strategic supply chain planning. Computers and Operations Research, 33, (2005), 181–208. Optimization Applications Wednesday 9, 9:00-10:30 Sala Seminari Est 238 Optimization Applications 239 A p-Median approach for predicting drug response in tumour cells Enza Messina∗ Department of Informatics Sistems and Communication, University of Milano-Bicocca, Italy, [email protected] Elisabetta Fersini Department of Informatics Sistems and Communication, University of Milano-Bicocca, Italy, [email protected] Abstract. The complexity of biological data related to the genetic origins of tumour cells, originates significant challenges to glean valuable knowledge that can be used to predict therapeutic responses. In order to discover a link between gene expression profiles and drug responses, a computational framework based on Consensus p-Median clustering approach is proposed. The main goal is to simultaneously predict (in silico) anticancer responses by extracting common patterns among tumour cell lines. Description of the proposed approach The public available datasets composed of genomic data and drug responses offer the opportunity to reveal valuable knowledge about the hidden relationships between gene expression and drug activity of tumor cells, pointing out the conditions that bring a patient to be more responsive than others to a given therapeutic agent. Although data collection provides the baseline to enable a better understanding of cancer mechanisms, data integration and interpretation is still an open issue. Mathematical and statistical models of complex biological systems play a fundamental role in building drug response prediction models given a patient genomic profile. We propose a computational framework based on the assumption that groups of cell lines homogeneous in terms of both gene expression profile and drug activity should be characterized by a subset of genes that explains the drug responses. The problem of simultaneously predicting the response of several therapeutic compounds given the patient genomic profile is addressed by a computational framework composed of three main building blocks: 1. The creation of homogeneous groups of tumor cell lines by means of p-Median formulations. In particular, a novel Consensus p-Median formulation is proposed and compared with traditional state of the art approaches, i.e. k-Means, STVQ [1] and Relational k-Means [2] and Probabilistic D-Clustering [3]. 2. The selection of relevant genes able to predict the response of hundreds of drugs. We explore the potential of the solutions determined by solving the above mentioned p-Median problem formulation for identifying a subset of genes that characterizes each cluster, i.e. those subsets of genes that could be responsible of drug responses. To accomplish this task two main feature selection policies have been investigated, i.e. Information Gain and Correlation-based Feature Subset Evaluation (CFS). 3. The simultaneous prediction of different drug responses by exploiting the potential of Bayesian Networks. Establishing a straightforward dependency structure Optimization Applications 240 of the Bayesian Network, we explore the ability of the selected genes to predict a panel of drug responses given the genomic profiles of patients. The proposed computational framework exploits the well known dataset provided by the U.S. National Cancer Institute. The dataset consists of 60 cell lines from 9 kinds of cancers, all extracted from human patients, where the tumors considered in the panel derive from colorectal, renal, ovarian, breast, prostate, lung and central nervous system as well as leukemia and melanoma cancer tissues. Computational results show that the proposed Consensus p-Median, combined with gene selection and BN inference engine, yields homogeneous clusters while guaranteeing good predictive power for inferring drug responses for a new cell line. This is also confirmed by the biological evaluation performed on the selected genes: according to the existing literature the set of genes used to train the BNs, which has been selected by using the groups of cell lines obtained by the proposed Consensus p-Median, has shown to be biologically relevant from an oncological point of view. References [1] Burger M, Graepel T, Obermayer K, Phase transitions in soft topographic vector quantization, in Artificial Neural Networks-ICANN, Edited by Gerstner W, Germond A, Hasler M, Nicoud JD. Springer Berlin Heidelberg, New York (1997), 619–624. [2] Fersini E, Messina E, Archetti F, Manfredotti C, Combining gene expression profiles and drug activity patterns analysis: A relational clustering approach. J Math Modelling Algorithms (2010), 275–289. [3] Iyigun C, Ben-Israel A, A generalized weiszfeld method for the multi-facility location problem. Oper Res Lett 2010, 38(3) (2010), 207–214. Optimization Applications 241 Lagrangian-Based Local Improvement Heuristic for Large Scale Generalized Assignment Problems Salim Haddadi∗ LabSTIC, University 8 Mai 1945, Algeria, [email protected] Fatima Guessoum Meryem Cheraitia LabSTIC, University 8 Mai 1945, Algeria, fatima [email protected] [email protected] Abstract. In this contribution, we consider the generalized assignment problem whose aim is to find a minimum cost assignment of n tasks to m agents. A standard subgradient method is used to solve the Lagrangian dual obtained by dualizing the knapsack constraints. A local improvement heuristic is applied to the current best assignment at each iteration of the subgradient method. It consists of solving a special generalized assignment which is converted to a monotone 0-1 integer program with n variables and m constraints. Afterward, we show how to heuristically reduce the size of the problem, in order to speed up the heuristic. Computational experiments on large benchmark instances show the robustness of the local improvement heuristic, and the fastness and efficiency of the overall algorithm. Comparison with existing heuristics shows that our method is superior or competitive. Extended abstract We have n tasks that should be assigned to m agents. Let bi be the resource availability of agent i, let aij be the amount of resource required by agent i to perform task j and let cij be the cost so assigned. We assume that every thing is non-negative and integer. Each task must be assigned to one agent without exceeding his resource availability. The generalized assignment problem (GAP) is to find a minimum cost assignment. The mathematical model is XX min cij xij i∈I j∈J X aij xij ≤ bi , i∈I X j∈J (42) j∈J xij = 1, i∈I xij ∈ {0, 1} , i ∈ I, j ∈ J where xij is a binary variable that indicates whether task j is assigned to agent i, and where I = {i1 , i2 , ..., im } and J = {j1 , j2 , ..., jn }. The GAP is known to be strongly NP-hard, and even the problem of whether these exists a feasible assignment is NP-complete. It has a great practical significance. For this reason, there have been many attempts to design exact algorithms. However, because they need large computing times and memory requirements, these cannot address the large size instances arising in practice. The other alternative is to develop fast algorithms capable of quickly finding good assignments. The literature is rich of such methods. There exist a few Lagrangian heuristics of average efficiency, while the metaheuristics seem to be the most effective. Optimization Applications 242 Our goal in this contribution is to show that simple Lagrangian heuristics are yet capable of competing sophisticated metaheuristics. The key contribution is the introduction of a new and powerful local improvement heuristic which, given the best current feasible assignment of cost c∗ , defines a special GAP (called 2FGAP) which is always feasible, and whose resolution provides a feasible assignment of cost no less than c∗ . Problem 2FGAP is proven to be NP-hard, and transformed to a monotone 0-1 IP with m constraints and n variables. Because of its special structure and sparsity (two nonzeros per column), it is relatively easy to solve (or precisely to approximate to within a small time-limit). The whole algorithm consists of iteratively, at each iteration of a subgradient method applied to the Lagrangian relaxation of the knapsack constraints (1), defining and solving problem 2FGAP. The definition of the successive 2FGAP problems is driven by the subgradient method. Next, we introduce a new heuristic approach for reducing the size of the GAP by variable fixing, in order to speed up the heuristic. This idea turns out to be useful and promising as we shall see. The comparison of the heuristic with existing methods shows that is next to best from the point of view of the quality of the solutions found, but the fastest among all. References [1] Haddadi, S., Guessoum, F., Cheraitia, M., Lagrangian-Based Local Improvement Heuristic for Large Scale Generalized Assignment Problems, submitted to European Journal of Operational Research. Optimization Applications 243 Improved integer programming formulations for the Closest String Problem Claudio Arbib∗ Dipartimento di Ingegneria/Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Italy, [email protected] Mara Servilio Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italy, [email protected] Paolo Ventura Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, CNR, Italy, [email protected] Abstract. Recently, integer linear programming (ILP) formulations have been successfully applied within efficient and effective heuristics for the Closest String Problem (CSP). We consider two ILPs for the binary and general (non-binary) CSP that improve previous ones, and solve them by branch-and-cut. Our method is based on 0, 1/2-Gomory cuts, and can either be used stand-alone to find optimal solutions, or as a plug-in to improve the performance of heuristics that require the exact solution of reduced problems. A computational experience shows the benefit of the proposed approach. Optimization for Energy & Environment Tuesday 8, 17:30-19:00 Sala Seminari Ovest 244 Optimization for Energy & Environment 245 The Thermal self-scheduling Unit Commitment problem: an extended MIP formulation Claudio Gentile∗ Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, Consiglio Nazionale delle Ricerche, Italia, [email protected] Antonio Frangioni Dipartimento di Informatica, Università di Pisa, Italia, [email protected] Abstract. In this work we present the first MIP formulation that describes the convex hull of the feasible solutions for the Single-Unit Commitment problem in electrical power production when ramping constraints are considered. The formulation is derived from a DP algorithm and has a polynomial number of both variables and constraints. From a DP algorithm to a MIP formulation The Unit Commitment (UC) problem in electrical power production requires to optimally operate a set of power generation units over a short time horizon (from one day to one week). Operational constraints depend on the type of the generation units (e.g., thermal, hydro, nuclear, . . . ), and can be rather complex [6]. Among these, the most “basic” ones are minimum and maximum power output, minimum up- and down-time, and ramp-up and -down constraints. The Single-Unit Commitment (1UC) problem is the restriction of (UC) that considers only one unit. In a deregulated system, the self-scheduling Unit Commitment problem requires to optimize the production of a price-taker generation company to maximize its revenue; this problem can be decomposed into as many (1UC) subproblems as the number of generation units. Furthermore, (1UC) is useful when decomposition methods are applied to (multi-units) UC [3]. Recently, the development of “tight” MILP formulations for (UC) has received considerable interest, since—with the right formulation—the advances in off-theshelf MILP solvers have for the first time made it possible to approach (UC) with these methods with an efficiency compatible with the demanding requests of the actual operating environments. Crucial in this effort is the development of tight formulations describing the convex hull of the feasible solutions to the (1UC) problem. In this line of work, Lee, Leung and Margot [5] first found a polyhedral description of (1UC) considering only minimum and maximum power output and minimum up- and down-time constraints. This formulation uses only commitment variables, but requires an exponential number of constraints. Rajan and Takriti [7] proposed an equivalent extended formulation with a polynomial number of both variables and constraints. The latter results were extended by Gentile, Morales and Ramos [4] who provided a characterization the convex hull of (1UC) when start-up and shut-down limits are also considered. In this work we present the first MIP formulation that describes the convex hull of the feasible solutions of (1UC) formulations that also include ramp-up and -down constraints. Our formulation has a polynomial number of both variables Optimization for Energy & Environment 246 and constraints and it is based on the efficient Dynamic Programming algorithm proposed in [2]. This new formulation gives us a description of the convex hull of the feasible solutions for the self-scheduling (UC) under a more realistic set of constraints, and can serve as the basis for tight formulations for the multi-unit (UC). Since the resulting formulations may end up having a rather large size, (Stabilized) Structured Dantzig-Wolfe approaches [1] may be appropriate for their solution. References [1] A. Frangioni, B. Gendron A Stabilized Structured Dantzig-Wolfe Decomposition Method, Mathematical Programming 140 (2013), 45–76. [2] A. Frangioni, C. Gentile Solving Nonlinear Single-Unit Commitment Problems with Ramping Constraints, Operations Research 54 (2006), 767–775. [3] A. Frangioni, C. Gentile, F. Lacalandra Solving Unit Commitment Problems with General Ramp Contraints, International Journal of Electrical Power and Energy Systems 30 (2008), 316–326. [4] C. Gentile, G. Morales-Espana, A. Ramos A tight MIP formulation of the unit commitment problem with start-up and shut-down constraints, Technical Report IIT-14-040A (2014). [5] J. Lee, J. Leung, F. Margot Min-up/min-down polytopes, Discrete Optimization 1 (2004), 77–85. [6] M. Tahanan, W. van Ackooij, A. Frangioni, F. Lacalandra Large-scale Unit Commitment under uncertainty, 4OR, to appear (2015). [7] D. Rajan, S. Takriti Minimum Up/Down polytopes of the unit commitment problem with start-up costs, Research Report RC23628, IBM (2005). Optimization for Energy & Environment 247 A Parallel Multi-Start Non-dominated Sorting Particle Swarm Optimization Algorithm for the Minimization of the Fuel Consumption Multiobjective Vehicle Routing Problem Yannis Marinakis∗ School of Production Engineering and Management, Technical University of Crete, Chania, Greece, [email protected] Iraklis - Dimitrios Psychas Magdalene Marinaki School of Production Engineering and Management, Technical University of Crete, Chania, Greece, [email protected] [email protected] Abstract. In this paper, three Multiobjective Fuel Consumption Vehicle Routing problems are solved using a new version of the Particle Swarm Optimization Algorithm, the Parallel MultiStart Non-dominated Sorting Particle Swarm Optimization algorithm, which uses more than one initial populations of solutions. A Variable Neighborhood Search algorithm for the improvement of each solution separately is used in the algorithm. The results of the algorithm are compared with the results of a Parallel Multi-Start NSGA II algorithm in a number of modified Vehicle Routing Problem instances. General Description The Vehicle Routing Problem is one of the most important problems in the field of Supply Chain Management, of Logistics, of Combinatorial Optimization and, in general, of Operational Research. The interest in this problem has been recently increased both from theoretical and practical aspect. The Vehicle Routing Problem (VRP) or the Capacitated Vehicle Routing Problem (CVRP) is often described as the problem in which vehicles based on a central depot are required to visit geographically dispersed customers in order to fulfill known customer demands. The problem is to construct a low cost, feasible set of routes - one for each vehicle. A route is a sequence of locations that a vehicle must visit along with the indication of the service it provides. The vehicle must start and finish its tour at the depot. The objective function is to minimize the total distance traveled [2],[6]. In recent years, a significant growth of publications in Green Vehicle Routing Problems (GVRPs) has been realized. In these problems, the main target is the reduction of fuel and energy consumption or the reduction of the pollution caused by the CO2 emissions. In this paper, three Multiobjective Fuel Consumption Vehicle Routing Problems (MFCVRPs) are described, formulated and solved using a Multiobjective Particle Swarm Optimization Algorithm. The variant of the Vehicle Routing Problem that is modified in order to produce the energy efficient models is the one with simultaneous pick-ups and deliveries. Thus, three different problems are formulated, the Multiobjective Delivery Fuel Consumption Vehicle Routing Problem (MDFCVRP), the Multiobjective Pick-up Fuel Consumption Vehicle Routing Problem (MPFCVRP) and the Multiobjective Delivery and Pick-up Fuel Consumption Vehicle Routing Problem (MDPFCVRP). Each problem uses two competitive objective functions. The first objective function corresponds to the Optimization for Energy & Environment 248 optimization of the time needed for the vehicle to travel between two customers or between the customer and the depot and the second objective function corresponds to the Fuel Consumption of the vehicle. Two different versions of the second objective function are used. The one version formulates the fuel consumption based on the weight of the vehicle and the distance traveled between two customers or the distance traveled between a customer and the depot and the other version formulates the fuel consumption based on the previous factors and in addition the slope of the road, the speed and the direction of the wind and the driver’s behavior are taken into account. In this paper, three Parallel Multi-Start Non-dominated Sorting Particle Swarm Optimization algorithms (PMS-NSPSOs) are proposed for the solution of the new VRP variants. Particle Swarm Optimization (PSO) is a population-based swarm intelligence algorithm that was originally proposed by Kennedy and Eberhart [4] and simulates the social behavior of social organisms by using the physical movements of the individuals in the swarm. In this algorithm, a number of populations are optimized in parallel each one using a different method to produce the initial solutions and a Variable Neighborhood Search (VNS) [3] algorithm in order to improve the quality of each solution separately. The results of the algorithms are compared with the results of the Parallel Multi-Start Non-dominated Sorting Genetic Algorithm II (PMS-NSGA II). PMS-NSGA II is a modified version of NSGA II (Non-dominated Sorting Genetic Algorithm) [1] and it was originally proposed at [5] for the solution of Multiobjective Vehicle Routing Problems. The results proved the efficiency of the proposed algorithms for this kind of problems. References [1] Deb, K., Pratap, A., Agarwal, S., Meyarivan T., A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6(2), (2002), 182197. [2] Golden, B.L., Raghavan, S., Wasil, E.A. (eds) The vehicle routing problem: Latest advances and new challenges, Operations Research/Computer Science Interfaces Series, 43, Springer LLC, (2008). [3] Hansen, P., Mladenovic, N., Variable neighborhood search: Principles and applications, European Journal of Operational Research, 130, (2001), 449-467. [4] Kennedy, J., Eberhart, R., Particle swarm optimization, Proceedings of 1995 IEEE International Conference on Neural Networks, 4, (1995), 1942-1948. [5] Psychas, I.D., Marinaki, M., Marinakis, Y., A parallel multi-start NSGA II algorithm for multiobjective energy reduction vehicle routing problem, 8th International Conference on Evolutionary Multicriterion Optimization, EMO 2015, Part I, LNCS 9018, Springer International Publishing Switzerland 2015, Gaspar-Cunha, A. et al. (Eds.), (2015), 336-350. [6] Toth, P., Vigo, D. (eds) The vehicle routing problem, Monographs on Discrete Mathematics and Applications, Siam, Philadelphia, (2002). Optimization for Energy & Environment 249 Optimization of Large-Scale Waste Flow Management at HerAmbiente Matteo Pozzi∗ OPTIT srl, Italia, [email protected] Angelo Gordini Lorenzo Ravaglia Fabio Lombardi Tiziano Parriani OPTIT srl, Italia, [email protected] [email protected] [email protected] [email protected] Adriano Guarnieri Fabrizio Salieri Herambiente S.p.A., Italia, [email protected] [email protected] Daniele Vigo DEI, University of Bologna, Italy, [email protected] Abstract. During the last decades, the solid waste management increased its already substantial influence on a variety of factors impacting on the entire society. In this paper we propose mixed integer linear formulations, and relative resolution methods, for problems arising in the context of waste logistic management, with an application on a real world case. In response to the needs of an important Italian waste operator we introduce and formalize aspects commons in this field. During the last decades, solid waste management has constantly increased its influence on a variety of factors impacting on the entire society, especially for what concerns economical and environmental issues. Waste logistic networks became articulated and challenging because the traditional source-to-landfill situation switched to multi-echelon networks in which waste flows generally go through more than one preliminary treatment before reaching their final destinations (see Fig. 1). Complex optimization problems arises in this context, with the objective of maximizing the overall profit of the service. We describe the business case of the leading Italian Waste Management Company, and how their issue was tackled to develop an innovative solution to support strategic, tactical and operations planning using mixed integer linear formulations. The model, formulated and solved using state-of-the-art of commercial software, aims at maximizing the overall margin of the system (logistics costs + treatment costs/revenues) under a set of constraints defined considering several different aspects of the problem (e.g. treatment plants capacities, limits of the law, waste composition in and out the facilities, etc.). The actual challenge, though, is more on the actual capability to manage all network configurations and resolution strategies in a way that could be useful for the business’ purposes, provided that: • the complete value chain counts almost a hundred types of waste; • several hundreds of waste sources are used, with more than 4000 constraints on output flows; Optimization for Energy & Environment 250 Figure 1: A diagram representing the typical waste facilities network. SMW is for Sorted Municipal Waste, ND is for Non-Dangerous, MBT is for Mechanical Biological Treatment, PBT is for Phisiochemical Biological Treatment, WtE is for Waste to Energy, T&EE is for Termal and Electrical Energy, Env. Eng. is for Environmental Engineering, Pre.Tr. is for preliminary treatments • several millions of tons of waste are transported each years (counting hundreds of thousands of trips, with several logistics constraints); • a few hundreds of treatment plants (80 of which are owned by HerAmbiente) are subject to several hundreds flow and mix constraints; • the sheer granularity of the scenario time horizon, that ranges from 4 years for an industrial plan down to 54 weeks used for a tactical planning (budget execution). This was achieved through an enterprise web-based solution that allows central planners manage the strategic module, while over 40 peripheral users (i.e. plant and logistic coordinators) manage short term planning (i.e. week-ahead) through a tactical module, achieving significant return on the project’s investment. Optimization for Search & Networks Wednesday 9, 11:00-13:00 Sala Seminari Ovest 251 Optimization for Search & Networks 252 A relational clustering approach for community detection in social networks Elisabetta Fersini∗ Department of Informatics Sistems and Communication, University of Milano-Bicocca, Italy, [email protected] Daniele Maccagnola Enza Messina Department of Informatics Sistems and Communication, University of Milano-Bicocca, Italy, [email protected] [email protected] Abstract. An important task of social network analysis is to discover communities of users who share some common interests. Most of the research on this topic has been focused either on the network topology, by searching for densely connected nodes in undirected networks, or on content similarity disregarding the network structure. In this talk we present a novel scalable method that accounts simultaneously for edge directions and user generated contents associated with nodes. Description of the proposed approach The wide diffusion of social network has enabled people to create and publish their own content and share their experiences, ideas and opinions. With the pervasive usage of instant messaging systems and the fundamental shift in the ease of publishing content, social network researchers and graph theory researchers are now concerned with inferring community by analyzing the connection patterns among users and their common interests. The community detection problem can be viewed as an uncapacitated location problem, where users (nodes) have to be associated to communities (clusters). For solving this problem, two alternative approaches are traditionally provided: the first one aims at maximizing the similarity between nodes belonging to the same community, where the similarity is computed based on the text messages posted by the users [1]. The second approach aims at maximizing the number of connections (friendship, approval) among users belonging to the same cluster [2, 3, 4, 5] regardless the user generated contents. However, addressing the community detection problem by taking into account both nodes’ contents and network structure can unhide latent connections among users and enable the discovery of novel and meaningful community structures, the prediction of missing content information and the optimization of gathering and diffusion of information across the network. In this work we propose a novel method that accounts simultaneously for edge directions and user generated contents associated with nodes. In particular we propose a p-median based model, able to take into account strength and orientation of social relationships, by defining the clustering problem on an heterogeneous network model - named Approval Network - that encloses directed and weighted relationships between users to model the strength of interest-based social ties between people. Finally we propose a heuristic, based on [6], where the user cluster assignment based on contents - is smoothed according to the dependencies of first order neighbors taking into account the strength of the node connections. Optimization for Search & Networks 253 The proposed approach is validated on real data and compared against traditional baseline community detection methods. References [1] Liu J., Comparative analysis for k-means algorithms in network community detection, in Advances in Computation and Intelligence. Springer Berlin Heidelberg. Springer Berlin Heidelberg, (2010), 158–169. [2] Fortunato S., Community detection in graphs. JPhysics Reports (2010) [3] IXie J., Kelley S., Szymanski B.K. Overlapping community detection in networks: the state of the art and comparative study. ACM Computing Surveys, (2013). [4] Girvan, M., Newman, M.E. Community structure in social and biological networks. Proceedings of the National Academy of Sciences 99(12), (2002) 7821–7826 [5] Ng, A.Y., Jordan, M.I., Weiss, Y. On spectral clustering: Analysis and an algorithm. Advances in neural information processing systems 2 (2002) 849–856. [6] Fersini, E. Messina, E., Archetti, F. A probabilistic relational approach for web document clustering. Information Processing and Management Vol. 46, no 2, p. 117–130, 2010 Optimization for Search & Networks 254 An outreach experience: assignment problem and stable marriage in a speed-date context Stefano Manzionna∗ School of Sciences and Technology, Università degli studi di Camerino, Italia, [email protected] Renato De Leone Ilaria Giancamilli Alessandra Renieri School of Sciences and Technology, Università degli Studi di Camerino, Italia, [email protected] [email protected] [email protected] Abstract. The aim of our experience is to raise the awareness of the pervasiveness of OR and Mathematics in everyday life. In this talk we discuss the result of an evening event we organized at the University of Camerino’s Cafeteria where we applied an assignment model in a speed-date context. Fifty students (25 per gender) compiled a questionnaire collecting their preferences about movies, books, musics, religion, politics . . . Each person expressed an evaluation on everyone they talked with. Based on those two criteria the assignment problem was created and solved using Excel. At the end the 25 best couples where nominated. Introduction According to the European Charter for Researchers “All researchers should ensure...that the results of their research are disseminated and exploited, e.g. communicated, transferred into other research settings or, if appropriate, commercialised...” [1]. Thus, it is part of the researchers’mission to raise the general public awareness of science, especially of Mathematics (the Science which is less popular among people). With this in mind the following question raises quite spontaneously: how is it possible to communicate Math, both theoretical and applied? Starting from this, a group of researchers (students, PhDs, Professors) has decided to establish a research group, with the aim of starting a new Matematita Centre [2], that is a Interuniversity research centre for the communication and the informal learning of Mathematics. Since 2005 and until now there are four Matematita branches in Italy: Milano Bicocca, Milano Statale, Trento and Pisa. Within a short space of time, a fifth branch will be created at the University of Camerino. Our group is made up of students and PhDs that are trying to form themselves in this field collecting practical and theoretical experiences (like Scientific fairs, International Conferences, classes on Scientific Communication), but also professors that have already a great experiences (Sissa Masters on Science Communication,...) participate to this new Centre. The experience that we will report here is one of our realizations. Math & Speed Dating Mathematics and Operation Research (OR) are pervasive in everyday life. The methods and techniques of OR are often used, but in almost all cases general public is unaware of that. To show how OR methodologies can be used in a social context, we organized an evening event at the University of Camerino’s Cafeteria consisting Optimization for Search & Networks 255 in a mathematical speed-date. We collect all important data and constructed an assignment problem which was solved via standard techniques [3], in order to find a matching that maximizes the global affinity. In general, a speed-date consist in a meeting of people who have the chance to talk for a restricted time (generally 200 seconds). Based on that, they must decide if he/she is interested in meeting again the person they talked to. At the end, when all possible couples have encountered, the organizers will check whether the interest is mutual: if so they communicate the number and the name to the couples. We modified this method in several ways. First, we asked each participant to assign a score between 0 and 10, instead of a simple YES/NO, based on how much they appreciated the person they talked to. Moreover, we decided that the final decision cannot be only based on the few minutes that a person had the opportunity to spend with the possible matching. Therefore, we constructed a questionnaire in order to collect a priori preferences. The questions regard some aspects we consider useful for a good couple life (i.e., movies, books, religion, politics,...). To every question a score between 1 and 10 was assigned, to quantify how much the specific topic is important for each participant. In assigning a score to a pair, we took into account not only if both consider important a specific topic (eg, movies) but also if they prefer, for example, the same kind of movies (action movie, comedy, drama, ...). 50 students (25 per gender) have participated at the event. During the evening various breaks were scheduled and few “pills” on Math and Love were presented. In particular we talked about the cardioid - the mathematician’s heart-, the twin primes , the equations of the hormones’s flow and the stable marriage problem. The data collected - via the questionnaires and after talking to each partner - were used to create an assignment problem solved with standard optimization techniques in Excel. Finally the 25 “best couples” were announced. In the end, the event was not only a source of fun but also, and overall, an evening of scientific communication. Every participant had the opportunity of appreciate and be surprised of how Math can be useful to. References [1] The European Charter for Researchers, http://ec.europa.eu/euraxess/index.cfm/rights/ europeanCharter [2] http://www.matematita.it [3] A.E. Roth, U.G. Rothblum, J.H Vande Vate, Stable Matching, Optimal Assignments and Linear Programming, Mathematics of Operations Research, vol. 18(4), 803- 828, 1993 Optimization for Search & Networks 256 Modeling Signals and Learning Effects on the Information Acquisition Incentives of Decision Makers in Sequential Search Processes Francisco J. Santos Arteaga∗ School of Economics and Management, Free University of Bolzano, Italy, [email protected] Debora Di Caprio Department of Mathematics and Statistics, York University, Canada, [email protected] Madjid Tavana Business Systems and Analytics Department, La Salle University, USA, [email protected] Abstract. Consider a rational decision maker (DM) who must acquire a finite amount of information sequentially from a set of products whose characteristics are grouped in two differentiated categories. The DM receives signals on the characteristics’ distribution and updates his expected search utilities following Bayesian and subjective learning rules. We build two real-valued functions that determine the decision of how to allocate each available piece of information and provide numerical simulations illustrating the information acquisition incentives defining the behavior of the DM. The current paper studies the information acquisition behavior of a rational DM when gathering sequentially n ∈ N observations from a set of products whose attributes have been grouped in two differentiated characteristics. This simplification is applied in several disciplines, which concentrate on a small number of characteristic categories when describing the products available to the DMs. For example, consumer choice analysts utilize quality and preference (Lee and Lee, 2009), economists consider performance and cheapness (Malerba et al., 2007), and operational researchers concentrate on variety and quality (Bohlmann et al., 2002). The defining quality of our paper relies on the formalization of the information acquisition process of the DM, which will be defined for each observation available on the following variables: • the values of all the characteristics observed previously, and • the number and potential realizations of all the remaining observations. These requisites altogether prevent the use of standard dynamic programming techniques in the design of the information acquisition algorithm. In particular, the information acquisition incentives of DMs will be based on • the number of remaining observations available, and • the subjective probability that these observations allow the DM to observe a product that delivers a higher utility than the best among the ones observed. Optimization for Search & Networks 257 These incentives must be redefined each time an observation is acquired. We synthesize the information acquisition incentives of the DM within two real-valued functions that determine the expected utility derived from the potential use given to the remaining observations. Given the value of the realizations observed and the number of observations remaining to be acquired, the information acquisition behavior of the DM will be determined by • a continuation function describing the utility that the DM expects to obtain from continuing acquiring information on a partially observed product • a starting function defining the expected utility the DM expects to obtain from starting checking the characteristics of a new product. Moreover, firms will be allowed to issue signals on the characteristics defining their products. These signals lead the DM to update his expected search utilities following both Bayesian and subjective learning rules. Each time an observation is acquired, the DM has to modify the probability of improving upon the products already observed with the remaining observations available and account for the distributional implications derived from the signal. We will illustrate how the characteristic on which the signals are issued plays a fundamental role determining the information acquisition incentives of the DM. This will be the case even if the signals are considered reliable by the DM and indicate the improvement of one of the characteristics of the product. In particular, issuing signals on the second characteristic may have the opposite effect of that intended by the firm. Given a number of observations expected to be acquired by the DM, our model allows firms to forecast the information acquisition behavior of the DM as well as the probability of having their products inspected and considered for purchase. This possibility introduces an important strategic component, particularly in the formalization of online search environments and the subsequent design of decision support tools to guide the information acquisition process of the DM. References [1] J.D. Bohlmann, P.N. Golder, D. Mitra, Deconstructing the pioneer’s advantage: examining vintage effects and consumer valuations of quality and variety, Management Science 48 (2002) 1175-1195. [2] J. Lee, J.-N. Lee, Understanding the product information inference process in electronic word-of-mouth: an objectivity-subjectivity dichotomy perspective, Information & Management 46 (2009) 302-311. [3] F. Malerba, R. Nelson, L. Orsenigo, S. Winter, Demand, innovation, and the dynamics of market structure: the role of experimental users and diverse preferences, Journal of Evolutionary Economics 17 (2007) 371-399. Optimization for Search & Networks 258 ERGMs and centrality measures for the analysis of mythological networks Giulia De Santis∗ School of Sciences and Technology, Università degli Studi di Camerino, Italia, [email protected] Renato De Leone Frederique-Anne Willem School of Sciences and Technology, Università degli Studi di Camerino, Italia, [email protected] [email protected] Abstract. Exponential Random Graph Models are models of graphs whose probability distribution is directly proportional to an exponential function. They are a powerful instrument to study social networks, proceeding as follows: • the observed network is represented through a network (i.e., a graph endowed with attributes); • a model of random graphs with an exponential probability distribution is created; • A check of the goodness-of-fit establishes if the created model represents well the observed network. In this paper, ERGMs and centrality measures are used to study elements of truth in different mythological tales. Applications of ERGMs to mythological networks Exponential Random Graph Models (ERGMs) ([1]) are models of graphs with a probability distribution f (x) which is directly proportional to an exponential funcT tion: f (x) ∝ eθ z(x) . More in detail, given a graph G, its probability can be expressed by Pθ (G) = c eθ1 z1 (G)+θ2 z2 (G)+...+θp zp (G) where c is a normalizing constant and the exponent of e is a sum of weighted network statistics. For a graph G, the network statistics are counts of configurations useful to explain the structure of G, such as: • number of edges; • number of triangles (i.e., three vertices, each of them connected with the other two: given the nodes i, j and k, a triangle is {(i, j); (j, k); (k, i)}) ; • number of 2-stars (i.e., two edges which share a node); • number of k–triangles (i.e., two connected nodes are also jointly connected to k other distinct nodes); • bow-ties (i.e., two triangles which share a node); • ... Optimization for Search & Networks 259 Properties of a graph can be collected also using Centrality Measures [3], which try to identify the most important or central vertices of a graph. There are many possible definitions of importance, and, correspondingly, many centrality measures: • Degree centrality; • Closeness centrality; • Betweenness centrality; • Eigenvector centrality; • Pagerank centrality. The aim of this talk is to show the advantages of both using results form Exponential Random Graph Models and from centrality measures to the study of networks, both social and from mythological literature. In particular, following and expanding the results in [2], ERGMs and centrality measures are used to identify elements of truth in mythological tales. References [1] Lusher, D.; Koskinen, J.; Robins, G.; Eponential Random Graph Models for social networks: Theory, Methods and Applications, Cambridge University Press, 2012. [2] Mac Carron, P.; Kenna, R.; Universal properties of mythological networks. EPL, 99, 2012. [3] Freeman, L. C.; Centrality in social networks: Conceptual clarifcation. Social Networks, January 1979. Optimization Methodologies Thursday 10, 11:00-13:00 Sala Riunioni Est 260 Optimization Methodologies 261 MIP neighborhood synthesis through semantic feature extraction and automatic algorithm configuration Emanuele Manni∗ Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Italia, [email protected] Tommaso Adamo Gianpaolo Ghiani Antonio Grieco Emanuela Guerriero Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Italia, [email protected] [email protected] [email protected] [email protected] Abstract. The definition of a suitable neighborhood structure on the solution space is a key step when designing heuristics for Mixed-Integer Programming (MIP). In this talk, we review some recent algorithms which take advantage of the peculiar features of a MIP model to automatically design efficient neighborhoods, without any human analysis. We show computational results on compact formulations of well-known combinatorial optimization problems and compare the automatically-constructed neighborhoods with state-of-the-art domain-independent neighborhoods. Introduction The definition of a suitable neighborhood structure on the solution space is a key step when designing several types of heuristics for Mixed Integer Programming (MIP). Typically, in order to achieve efficiency in the search, the neighborhood structures need to be tailored not only to the specific problem but also to the peculiar distribution of the instances to be solved (reference instance population), the imposed time limit and the hardware at hand. Nowadays, this is done by human experts through a time-consuming process comprising: (a) problem analysis, (b) literature scouting and (c) experimentation. In this paper, we extend the results given in [3], developing an Automatic Neighborhood Design algorithm that mimics steps (a) and (c) and derives automatically “good” neighborhood structures from a given MIP model of the problem under consideration. The ultimate goal is to automatically generate neighborhood structures that may provide human competitive results [5]. Our work is related to other model-derived neighborhoods, such as Local Branching [2] and Relaxation-Induced Neighborhood Search [1]. Moreover, our work is also related to Automatic Algorithm Configuration [4]. Contribution Given a combinatorial optimization problem P , a neighborhood structure associates a set of feasible solutions N (s) to each feasible solution s of P . In this paper, we assume that the user supplies a MIP model. Since the “best” neighborhood structure has to be tailored not only to the specific problem but also to the peculiar distribution of the instances to be solved, the user also provides a training set representative Optimization Methodologies 262 of the reference instance population. Furthermore, we require that the MIP model is written through an algebraic modelling language (AMPL, GAMS, . . . ) in terms of a set of entities (e.g, a set of customers, a set of commodities, . . . ). Entities can be classified as derived or fundamental (derived entities are subsets or cartesian products of other sets). With this assumption, each variable and constraint is tagged by one or more fundamental entities. For instance, in a network design model, the design variable yij of a link will be tagged by vertices (fundamental entities) i and j. We define a neighborhood structure N (s) as the set of feasible solutions that can be obtained by first destroying and then repairing the fragment of the current solution s tagged by a subset F of fundamental entities. The repair is done with an off-the-shelf solver (e.g., CPLEX with a given parameter setting). The identification of F can be done on the basis of m semantic features fi (i = 1, . . . , m) extracted from the MIP model. Indeed, the choice of the “best” subset F depends on the relative weight wi given to each feature fi (i = 1, . . . , m). The weight vector w = (w1 , . . . , wm ) as well as the neighborhood size (expressed by the number of free variables) are then determined by using an Automatic Algorithm Configuration procedure on the training set. When assessed on several well-known combinatorial optimization problems, our automa-tically-generated neighborhoods outperform state-of-the-art model-based neighborhoods with respect to both scalability and solution quality. References [1] Danna, E., E. Rothberg, C. Le Pape, Exploring relaxation induced neighborhoods to improve MIP solutions. Mathematical Programming A 102 (2005), 71–90. [2] Fischetti, M., A. Lodi, Local branching. Mathematical Programming B 98 (2003), 23–47. [3] Ghiani, G., G. Laporte, E. Manni, Model-based automatic neighborhood design by unsupervised learning. Computers & Operations Research 54 (2015), 108–116. [4] Hoos, H.H., Programming by optimization. Communications of the ACM 55 (2012), 70–80. [5] Koza, J.R., Human-competitive results produced by genetic programming. Genetic Programming and Evolvable Machines 11 (2010), 251–284. Optimization Methodologies 263 Carousel Greedy: A Generalized Greedy Algorithm with Applications in Optimization and Statistics Carmine Cerrone∗ Dipartimento di Matematica, Università di Salerno, Italia, [email protected] Raffaele Cerulli Dipartimento di Matematica, Università di Salerno, Italia, [email protected] Bruce Golden Robert H. Smith School of Business, University of Maryland, USA, [email protected] Abstract. In this paper, we introduce carousel greedy, an enhanced greedy algorithm which seeks to overcome the traditional weaknesses of greedy approaches. We have applied carousel greedy to a variety of well-known problems in combinatorial optimization such as the minimum label spanning tree problem, the minimum vertex cover problem, the maximum independent set problem, and the minimum weight vertex cover problem. We also compared carousel greedy against the widely used statistical technique of Stepwise Regression. In all cases, the results are very promising. Since carousel greedy is very fast, it can be used to solve very large problems. In addition, it can be combined with other approaches to create a powerful, new metaheuristic. Our goal in this paper is to motivate and explain the new approach and present extensive computational results. Optimization Methodologies 264 A Binarisation Approach to Non-Convex Quadratically Constrained Quadratic Programs Laura Galli∗ Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy, [email protected] Adam N. Letchford Department of Management Science, Lancaster University, Lancaster LA1 4YW, UK, [email protected] Abstract. The global optimisation of non-convex quadratically constrained quadratic programs is a notoriously difficult problem, being not only N P-hard in the strong sense, but also very difficult in practice. We present a new heuristic approach to this problem, which enables one to obtain solutions of good quality in reasonable computing times. The heuristic consists of four phases: binarisation, convexification, branch-and-bound and local optimisation. Computational results, on box-constrained and point packing instances, are encouraging. Non-Convex QCQP A quadratically constrained quadratic program (QCQP) is an optimisation problem that can be written in the following form: inf xT Q0 x + c0 · x s.t. xT Qk x + ck · x ≤ hk (k = 1, . . . , m) x ∈ Rn , (43) (44) (45) where the Qk are symmetric matrices of order n, the ck are n-vectors and the hk are scalars. (We write ‘inf’ rather than ‘min’ because it is possible that the infimum is not attainable.) If all of the Qk are positive semidefinite (psd) matrices, then QCQPs are convex optimization problems, and therefore can be solved efficiently (see, e.g., [3]). Non-convex QCQPs, on the other hand, are N P-hard. In fact, just checking unboundedness of a quadratic form over the non-negative orthant is N P-complete in the strong sense [6]. Solving QCQPs to proven optimality is very hard in practice as well in theory (e.g., [2]). One obvious source of this difficulty is the possibility of an exponentially large number of local minima. Another, less obvious, source of difficulty is that variables can take irrational values at the optimum, which means that one must either be content with solutions that are optimal only up to some given accuracy, or somehow represent solutions implicitly (for example, via the KKT conditions, as in [4]). On the other hand, QCQPs are a powerful modelling tool. A folklore result (see, e.g., [5]) is that mixed 0-1 linear and quadratic programs can be modelled as QCQPs, since the condition xi ∈ {0, 1} is equivalent to the quadratic constraint xi = x2i . Also, Shor [7] pointed out that any optimisation problem in which the objective and constraint functions are polynomials can be transformed to a QCQP, by adding suitable variables and constraints. (For example, one can replace a cubic Optimization Methodologies 265 term xi xj xk with the quadratic term xi xjk , where xjk is a new variable, if one also adds the quadratic equation xjk = xj xj .) Several other applications of QCQPs can be found in, e.g., [1]. In this paper, we present a new heuristic for QCQPs, which is aimed mainly at “loosely-constrained” QCQPs, where it is fairly easy to find a feasible solution. The heuristic exploits the fact that there now exist very good software packages for solving convex QCQPs with binary variables. It consists of four phases: binarisation, convexification, branch-and-bound and local optimisation. In our computational experiments, on standard test instances, the heuristic typically found solutions of very good quality in reasonable computing times. References [1] F.A. Al-Khayyal, C. Larsen & T. Van Voorhis A relaxation method for non-convex quadratically constrained quadratic programs, J. Glob. Optim. 6 (1995), 215–230. [2] K.M. Anstreicher, Unified approach to quadratically convergent algorithms for function minimization, Journal of Optimization Theory and Applications 136 (2012), 233–251. [3] S. Boyd & L. Vandenberghe, Convex Optimization, Cambridge: Cambridge University Press [4] S.A. Burer & D. Vandenbussche, A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations, Math. Program. 113 (2008), 259–282. [5] C. Lemarechal & F. Oustry, SDP relaxations in combinatorial optimization from a Lagrangian viewpoint, Advances in Convex Analysis and Global Optimization. Dortrecht: Kluwer. [6] K.G. Murty & S.N. Kabadi, Some N P-complete problems in quadratic and nonlinear programming, Math. Program. 39 (1987), 117–129. [7] N.Z. Shor, Quadratic optimization problems, Soviet J. Comput. Syst. Sci. 25 (1987), 1–11. Optimization Methodologies 266 Constrained domatic bipartition of a tree Luigi De Giovanni∗ Dipartimento di Matematica, Università di Padova, Italia, [email protected] Giovanni Andreatta Carla De Francesco Dipartimento di Matematica, Università di Padova, Italia, [email protected] [email protected] Paolo Serafini Dipartimento di Matematica e Informatica, Università di Udine, Italia, [email protected] Abstract. The Constrained Domatic Bipartition Problem has to determine a bipartition of the nodes of an undirected graph into two dominating sets, one of which has a given cardinality. We show that the problem is NP-complete on general graphs and we provide an integer programming model that has the integrality property for trees, if the cardinality constraint is relaxed. Finally, we propose a polynomial dynamic-programming algorithm for trees which simultaneously solves the problem for all possible cardinalities. The constrained domatic bipartition problem Given an undirected graph G = (V, E), a subset of nodes is dominating if each node of G is either in the subset or adjacent to some node in the subset. Given G and an integer H, the H-Domatic Partition Problem asks whether there exists a partition of V into H dominating subsets. Problems related to the domatic partition of a graph have been the object of several studies (see e.g. [1]), showing, among other results, that the H-Domatic Partition Problem is NP-complete for H ≥ 3 on general graphs [2]. It is easy to see that the H-Domatic Partition Problem with H = 2 has always a positive answer [1], assuming G has no isolated nodes. In this paper we address the 2-Domatic Partition Problem with additional constraints on the cardinality of the two dominating sets. This is the Constrained Domatic Bipartition Problem (CBDP) and can be formally defined as follows. Given G and an integer p (1 ≤ p ≤ |V |), we want to determine if V can be partitioned into two disjoint dominating sets V 0 and V 00 , such that |V 0 | = p and |V 00 | = |V | − p, and, in case of positive answer, find such a partition. By reduction from the Node Cover Problem, it is possible to prove the following Theorem. CDBP is NP-complete. A possible Integer Linear Programming formulation for CDBP considers binary variables xi for all nodes i ∈ V to denote if the node is in the first dominating set (xi = 1) or in the second one (xi = 0), and the following constraints 1≤ X xj ≤ d(i) ∀i∈V (46) ∀i∈V ∀i∈V (47) (48) j∈S(i) 0 ≤ xi ≤ 1 xi ∈ Z X xi = p i∈V (49) Optimization Methodologies 267 where S(i) is the star in G centered in i (including i itself) and d(i) the degree of i in G. Inequalities (46) are called star inequalities and state that, for every star, at least one node but not all the nodes, must be in the first dominating set. Inequalities (47) and (48) state that the variables xi are binary, and (49) is the cardinality constraint. Let P (G) ⊆ Rn be the polytope associated to inequalities (46) and (47). Notice that P (G) does not consider the cardinality constraint. If G is a tree, it is possible to prove the following result, even if the related constraint matrix is not totally unimodular. Theorem Given a tree T , the vertices of P (T ) are integer. We remark that the integrality property does not hold for the polytope obtained by intersecting P (T ) with the cardinality constraint (49). A possible extension of CBDP considers a weight associated to each node and asks if V can be partitioned into two disjoint dominating sets, one of which has a given total weight. This problem is NP-Hard, even for trees, by reduction from the Number Partition Problem. A polynomial-time algorithm for trees If G is a tree, CDBP can be solved in polynomial time by adopting a dynamic programming approach. The proposed algorithm arbitrarily chooses a root and stores, for each node i, all the integer values p for which the subtree rooted in i allows a domatic bipartition, and one of such partitions. We notice that this information is different depending on the hypothesis on which of the two dominating sets node i and its father belong to. Hence, four different cases are considered and the corresponding information is computed and stored for each node. A bottom-up approach is adopted, since it is straightforward to determine the required information for leaves. Then, the information for the nodes whose sons have been already examined is recursively computed: the algorithm determines the values of p for which the subtree rooted in the current node admits a constrained domatic bipartition by conceptually combining the stored domatic bipartitions of its sons. When the root node is reached, the information about all possible p for which CBDP admits a solution and one possible constrained domatic bipartion for each of such p is available. The algorithm, whose details will be given during the conference, runs in O(|V |3 ). References [1] Cockayne, E.J. and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977), 247–261. [2] Garey, M.R. and D.S. Johnson, Computer and intractability: A guide to the theory of NPcompleteness. W.H. Freeman & Co., San Francisco, 1979. Packing & Cutting Tuesday 8, 11:00-13:00 Sala Riunioni Ovest 268 Packing & Cutting 269 Exact algorithms for the 0–1 penalized knapsack problem Rosario Scatamacchia∗ Politecnico di Torino, Italia, [email protected] Federico Della Croce Politecnico di Torino, Italia, [email protected] Abstract. We consider a generalization of the 0–1 Knapsack problem, that is the Penalized Knapsack Problem. We devise an improved dynamic programming algorithm and a new integer linear formulation of the problem. Further, we propose an exact approach that exploits the peculiarities of the new ILP formulation. The approach turns out to be very effective in solving hard instances and compares favorably to both the state of art solver CPLEX 12.5 and the algorithms available in literature. Introduction We study the Penalized Knapsack Problem (PKP) as introduced in [1], that is a generalization of the classical 0–1 Knapsack Problem in which each item has a profit, a weight and a penalty. The problem calls for maximizing the sum of the profits minus the greatest penalty value of the selected items in the solution. Among the applications of interest, PKP arises as a pricing sub–problem in branch–and–price algorithms for the two–dimensional level strip packing problem in [2]. The Penalized Knapsack Problem is N P–hard in the weak sense since it contains the standard Knapsack as special case, namely when the penalties of the items are equal to zero, and it can solved by a pseudo–polynomial algorithm. In [1], a dynamic programming approach is recalled with time complexity O(n2 × W ), with n and W being the number of items and the capacity of the knapsack respectively. Also, an exact algorithm is presented and tested on various instances with up to 10000 variables. In this work, we devise a simple dynamic programming algorithm with a lower complexity O(n × W ). Then, we propose a new ILP formulation of the problem and a method which properly works out the peculiarities of this formulation. We investigate the effectiveness of our approach on a large set of instances generated according to the literature and involving different type of correlations between profits, weights and penalties. The proposed approach turns out to be very effective in solving hard instances and compares favorably to both the state of art solver CPLEX 12.5 and the exact algorithm in [1]. References [1] Ceselli, A., Righini, G., An optimization algorithm for a penalized knapsack problem, Operations Research Letters 34 (2006), 394–404. [2] Lodi, A., Martello, S., Monaci, M., Two-dimensional packing problems: a survey, Eur. J. Oper. Res. 141 (2002), 241–252. Packing & Cutting 270 Combinatorial Benders’ Decomposition for the Orthogonal Stock Cutting Problem Maxence Delorme∗ DEI “Guglielmo Marconi”, University of Bologna, Italy, [email protected] Manuel Iori DISMI, University of Modena and Reggio Emilia, Italy, [email protected] Silvano Martello DEI “Guglielmo Marconi”, University of Bologna, Italy, [email protected] Abstract. We study the orthogonal Stock Cutting Problem whose aim is to pack, without overlapping, a set of rectangular items into a strip of fixed width using the minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90◦ is allowed. We propose a method based on Benders’ decomposition that showed quite encouraging preliminary results for the standard instances. Contribution Given n rectangular items of width wj and height hj (j = 1, . . . , n) and a strip of fixed width w, the orthogonal Stock Cutting Problem (SCP) consists in packing all the items into the strip without overlapping, using the minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90◦ is allowed. The counterpart of the SCP in which rotation is not allowed is known in the literature as the Strip Packing Problem (SPP). Both the SCP and SPP are important because they have many real world applications, especially in wood, paper, glass, and metal industries, just to cite some. A recent approach proposed by Côté, Dell’Amico and Iori [1] uses the so-called Combinatorial Benders’ decomposition to solve the SPP and shows interesting results. The main idea of this method, originally introduced by Benders [2], is to solve a difficult problem by means of the iterative solution of two subproblems, the master problem and the slave problem. The master problem, which is usually a relaxed version of the original difficult problem, is in this case a one-dimensional bin packing with contiguity constraints (CBP). In the CBP, that was studied by Martello, Monaci, and Vigo [3], each item is horizontally cut into slices, and the aim is to pack all items into the minimum number of bins of capacity w so that slices belonging to the same item are packed into contiguous bins. The aim of the slave problem is to check if the optimal solution provided by the master is feasible for the original problem. If it is not, a valid cut is added to the master to prevent such a solution to be regenerated. The process is iterated until the optimal solution found by the master is validated by the slave. Note that in the original approach by Benders [2] the subproblem is a continuous linear program, but later on a number of algorithms treated the case where the subproblem is an integer (possibly difficult) program (see, e.g., Hooker [4] and Codato and Fischetti [5]). The aim of this paper is to continue this line of research and propose a decomposition method that takes into account the rotation of the items and solves the SCP to optimality. We propose a new integer linear programming model for solving the Packing & Cutting 271 CBP, based on an extension of the well-known ARCFLOW formulation introduced by Valério de Carvalho [6], and a new way to solve the slave problem based on constraint programming. Preliminary results show that our approach usually requires larger computational time than other existing methods, e.g., Kenmochi, Imamichi, Nonobe, Yagiura, and Nagamochi [7] and Arahori, Imamichi, and Nagamochi [8] on zero-waste instances, but it has interesting results for the benchmark instances where some waste occurs. Indeed, it solves for the first time to proven optimality instance “gcut02” (with rotation allowed), which has been an open problem for quite a long time. References [1] Côté, J.F, Dell’Amico, M., and Iori, M., Combinatorial Benders’ cuts for the strip packing problem, Operations Research (2014), 643–661. [2] Benders, J.F., Partitioning procedures for solving mixed-variables programming problems, Numerische mathematik (1962), 238–252. [3] Martello, S., Monaci, M., Vigo, D., An exact approach to the strip-packing problem, INFORMS Journal on Computing (2003), 310–319. [4] Hooker, J. N., Planning and scheduling by logic-based benders decomposition, Operations Research (2007), 588–602. [5] Codato, G. and Fischetti, M., Combinatorial Benders’ cuts for mixed-integer linear programming, Operations Research (2006), 756–766. [6] Valério de Carvalho, J.M., Exact solution of bin packing problems using column generation and branch and bound, Annals of Operations Research (1999), 629–659. [7] Kenmochi, M., Imamichi, T., Nonobe, K., Yagiura, M., and Nagamochi, H. Exact algorithms for the two-dimensional strip packing problem with and without rotations, European Journal of Operational Research (2009), 73–83. [8] Arahori, Y., Imamichi, T., Nagamochi, H., An exact strip packing algorithm based on canonical forms , Computers & Operations Research (2012), 2991–3011. Packing & Cutting 272 Maximum lateness minimization in one-dimensional Bin Packing problems Fabrizio Marinelli∗ Università Politecnica delle Marche, Dipartimento di Ingegneria dell’Informazione, Italia, [email protected] Claudio Arbib Università degli Studi dell’Aquila, Dipartimento di Ingegneria-Scienze dell’Informazione e Matematica, Italia, [email protected] Abstract. In the One-dimensional Bin Packing problem (1-BP) items of different lengths must be assigned to a minimum number of bins of unit length. Regarding each item as a job that requires unit time and some resource amount, and each bin as the total (discrete) resource available per time unit, the minimization of the number of bins corresponds to the minimization of the makespan. We here generalize the problem to the case in which each item is due by some date: our objective is to minimize a convex combination of makespan and maximum lateness. We propose a time indexed ILP formulation to solve the problem. The formulation can be decomposed and solved by column generation, in which case single-bin packing is relegated to a pricing problem: therefore, extensions to s-dimensional problems can be dealt with independently. We show how to couple the formulation with quite simple bounds to (individual terms of) the objective function, so as to get very good gaps with instances that are considered difficult for the 1-BP. Packing & Cutting 273 Bin Packing Problem With General Precedence Constraints Mauro Dell’Amico∗ DISMI, University of Modena and Reggio Emilia, Italy, [email protected] Manuel Iori Raphael Kramer DISMI, University of Modena and Reggio Emilia, Italy, [email protected] [email protected] Abstract. In this paper we study the bin packing problem with general precedence constraints, in which a set of weighted items has to be packed in the minimal number of capacitated bins, while satisfying precedence relationships among pair of items. The problem generalizes the well-known Simple Assembly Line Balancing problem, and models relevant real-world situations. To solve the problem we propose a series of lower and upper bounding techniques, including an iterated local search algorithm. Preliminary computational results show the efficiency of the proposed approach in solving complex instances. Introduction and Problem Description We study the Bin Packing Problem With General Precedence Constraints (BPPGP), which is a generalization of the well-known Bin Packing Problem (BPP). In the BPPGP we are given of set N of n items, each of weight wj , a set M of m identical bins, each having capacity C, and a set A of precedence relationships, j ≺ k, of values tjk ≥ 0, giving the minimum distance required between the bin allocating item j and the bin allocating item k. The BPPGP asks for minimizing the number of used bins that can allocate all items. Let the bins be numbered from 1 to m. By introducing a first binary variable yi that takes value 1 if bin i is used and 0 otherwise, and a second binary variable xij that takes value 1 if item j is allocated to bin i, 0 otherwise, the BPPGP can be modeled as the following Integer Linear Program (ILP). X min yi (50) i∈M s.t. X wj xij ≤ Cyi i∈M (51) j∈N (52) (j, k) ∈ A (53) xik = 0 k ∈ N : ∃tjk > 0 (54) yi ≥ yi+1 yi ∈ {0, 1} xij ∈ {0, 1} i = 1, . . . , m − 1 i∈M i ∈ M, j ∈ N. (55) (56) (57) j∈N X xij = 1 i∈M X ixik ≥ i∈M tjk X X ixij + tjk i∈M i=1 Packing & Cutting 274 The objective function (50) seeks to minimize the number of used bins. Constraints (51) guarantee that if a bin is used, then the sum of the weights of the items allocated to that bin does not exceed the bin capacity. Equation (52) states that all items must be assigned to a bin. Constraints (53) ensure that, if there is precedence between items j and k, then the index of the bin where k is packed must be greater than or equal to the index of the bin receiving item j plus the value of the precedence relationship. Constraints (54) prohibit that an item k for which there exists a precedence relationship (j, k) ∈ A of value tjk > 0 is allocated to one of the first tjk bins. Constraints (55) impose that a bin cannot be used if its predecessor is not also used. Finally, constraints (56) and (57) impose variables to be binary. The BPPGP is important because model several real-world situations in which the aim is to assign tasks to workstations such that all precedence constraints are fulfilled and the station time does not exceed the cycle time in any station. In the BPP notation, the workstations correspond to the bins, the jobs to the items and the cycle time to the capacity of the bins. In particular, the BPPGP model those situations where the distance between two operations must be large enough to guarantee structural properties of the products (see, e.g., [5]). Notably, the BPPGP generalizes the Single Assembly Line Balancing Problem-1 (SALBP-1), where tjk = 0 for all (j, k) ∈ A, and the Bin Packing Problem with Precedence Constraints (BPP-P), where tjk = 1 for all (j, k) ∈ A. For the literature on the SALBP-1 we refer the interested reader to the surveys by [1] and [4]. For the BPP and the BPP-P, we refer, respectively, to the survey by [6] and to the recent work by [2]. Solution Algorithms and Preliminary Computational Results The ILP model (50)-(57) introduced in the previous section is not able to solve large size instances. Thus, for the solution of the BPPGP we developed a set of lower and upper bounding techniques. In terms of lower bounds, we generalized those proposed by [2] for the BPP-P, by taking into account the new precedence constraints. These lower bounds are either based on classical techniques originally developed for the BPP, including dual feasible functions, or on computation of the longest path on the acyclic graph imposed by the precedences. They also make use of combinations of these two topics (BPP and longest path) that are tailored for the BPPGP. In terms of upper bounds, we developed an Iterated Local Search (ILS) algorithm, that starts with a solution obtained by two constructive heuristics. The first constructive algorithm is a “first-fit”, that assigns one item at the time to the first feasible bin, if any, or opens a new bin otherwise. This process stops when all items are allocated. The second algorithm consists in a “Max Subset Sum”, which looks for maximising the load in each bin. That is, of all possible assignments it chooses the one for which the resulting bin load is a maximum. The Max Subset Sum heuristic iteratively adds new bins to the partial solution until all items have been assigned. At each iteration it adds a new bin by invoking an ILP model to solve the problem of the load maximization. The best solution obtained by the two heuristics is optimized by means of a Packing & Cutting 275 local search procedure. To this aim, we have developed three operators: “Move” attempts to move an item from its current bin to another; “Swap” attempts to exchange the position of two items; “Push” moves an item from a first bin to a second bin, and, if this exceeds the capacity, selects another item in the second bin to be moved to a third bin. All operators try to minimize the minimum load among the bins involved, with the aim of producing the most unbalanced situation. If the minimum load of the last bin becomes equal to zero, then the best solution is updated. Our implementation sorts all possible movements, tacking into accounts the value of the improvements for all operators (Move, Swap and Push) and then updates the current solution starting from the best movement. If the solution found is not proven to be optimal by comparison with the lower bound, then we start the ILS. This algorithm iteratively destroys the current solution by (i) emptying completely a subset of bins, and (ii) randomly removing some items from the residual subset of bins. The solution is then re-completed by re-inserting the removed items using alternatively two policies: (i) a variant of the first fit heuristic, and (ii) a mathematical model based on the ILP (50)-(57). The solution obtained in this way is optimized by means of the local search procedure. The process is iterated until a proven optimal solution is found or a maximum time limit is reached. To test our algorithms, we ran them on a new set of SALBP-1 instances (2100 in total) which were generated by [3]. All algorithms were implemented in C++ and run with a time limit of 5 minutes on an Intel Xeon E5530 at 2.4 GHz with 23.5 GB of memory, under the Linux Ubuntu 12.04.1 LTS 64-bit operating system. In Table 1 we report, for each class of instance size (label n), the number of unsolved instances in literature (label uns. lit.) accordingly to [3], and, for ILS and ILP, respectively, the number of unsolved instances (label uns.), the percentage gap of the unsolved instances (label gap% ) computed as (UB-LB)/LB*100, and the average running time in CPU seconds. The label n.a. indicates that no solution was obtained by ILP within the given time limit. The last line give the total number of unsolved instances from the literature, by ILS and by ILP. ILS and ILP can easily solve all the 525 small instances, including the 4 instances that have not been solved before in the literature. For medium instances (n = 50) ILS solves three more instances than ILP with a computing time which is two orders of magnitude smaller. ILS finds 73 new solutions, with respect to the existing literature. The gap of the unsolved instances is around 4% for both methods. For large instances (n = 100) ILS improves upon the literature by finding 36 new optimal solutions. The gap of the unsolved instances is smaller than 4%. ILP fails in 57 cases and the computing time remains two order of magnitude larger than that of ILS, while the gap of the unsolved solutions is double of ILS. For the very large instances (n = 1000) ILP was not able to find any solution, while ILS finds 89 new optimal solutions in less than 70 seconds, on average. The gap of the unsolved solutions is rather small: 1.71%. Packing & Cutting 276 Table 1: Preliminary computational results on SALBP-1 instances. ILS n 20 50 100 1000 tot uns. lit. uns. 4 0 99 26 170 134 339 250 610 412 sec. gap% avg. 0.00 0.00 4.14 0.99 3.67 6.89 1.71 69.03 ILP uns. gap% 0 0.00 29 4.26 191 7.78 525 n.a. 745 sec. avg. 0.15 77.25 263.95 n.a. Conclusions In this paper we have considered the Bin Packing Problem with General Precedence Constraints, which generalized the other combinatorial optimization problems including the Simple Assembly Line Balancing Problem of type 1. We proposed a mathematical model and we developed an Iterated Local Search heuristic algorithm. Preliminary computational experiments with a new benchmark of 2100 SALBP-1 instances, from the literature, show that a commercial general purpose Integer Linear Programming solver using our mathematical model, is able to improve upon the existing results for small and medium size instances. Its performance rapidly deteriorate with the size of the instance and it is definitely stuck for very large instances. The ILS heuristic is able to solve 198 previously unsolved SALBP-1 instances, within small computing times. It also solves one half of the very large instances. Moreover the gap of the unsolved instances, is limited to some percents. These promising results encourage us to continue the research by generating and testing ILS on new BPP-P and BPPGP instances obtained by modifying the SALBP-1 benchmark instances, adding precedences values greater than 0 in several ways. References [1] Becker, C. and Scholl, A. (2006). A survey on problems and methods in generalized assembly line balancing. European Journal of Operational Research, 168, 694–715. [2] Dell’Amico, M., Diaz-Diaz, J., and Iori, M. (2012). The bin packing problem with precedence constraints. Operations Research, 60(6), 1491–1504. [3] Otto, A., Otto, C., and Scholl, A. (2011). SALBPGen a systematic data generator for (simple) assembly line balancing. Technical report, School of Economics and Business Administration. https://ideas.repec.org/p/jen/jenjbe/2011-05.html. [4] Scholl, A. and Becker, C. (2006). State-of-the-art exact and heuristic solution procedures for simple assembly line balancing. European Journal of Operational Research, 168, 666–693. [5] Tirpak, T. (2008). Developing and deploying electronics assembly line optimization tools: A motorola case study. Decision Making in Manufacturing and Services, 2, 63–78. [6] Valério de Carvalho, J. (2002). LP models for bin packing and cutting stock problems. European Journal of Operational Research, 141, 253–273. Railway Optimization Wednesday 9, 9:00-10:30 Sala Riunioni Est 277 Railway Optimization 278 RECIFE-MILP for real-time railway traffic optimization: main results and open issues Paola Pellegrini∗ Univ. Lille Nord de France, IFSTTAR, COSYS, LEOST, France, [email protected] Grégory Marlière Joaquin Rodriguez Univ. Lille Nord de France, IFSTTAR, COSYS, ESTAS, France, [email protected] [email protected] Abstract. When railway traffic is perturbed by unexpected events, an appropriate train routing and scheduling may be applied to minimize delay propagation. RECIFE-MILP is a mixedinteger programming-based heuristic which has been studied and tested in various traffic situations. In this paper, we summarize the main results obtained on real case-studies and we discuss the main open questions which need to be answered in future research. RECIFE-MILP Railway traffic is often perturbed by unexpected events which cause primary delays. These primary delays may cause the emergence of conflicts: when a train does not respect its original schedule, it may claim a track section in concurrence with another train; one of them must then slow down, or even stop, to ensure safety. Hence, one of these trains will suffer a secondary delay due to the traffic perturbation. The emergence of conflicts is particularly remarkable at junctions, that is, at locations where multiple lines cross. Here, the infrastructure capacity is often fully used, at least at peak-hours, and even slight delays may quickly propagate to several trains due to a snowball effect. Nowadays, conflicts are manually solved by railway dispatchers who may decide to re-schedule trains to minimize delay propagation, that is, to change the originally planned train order at critical locations. At junctions where several routes are available for connecting origin-destination pairs, also train routes may possibly be changed with respect to the originally planned ones (re-routing). A noticeable number of academic studies have been devoted to finding effective algorithms for real-time railway traffic management. Among this studies, we proposed a mixed-integer linear programming (MILP) formulation for solving to optimality the problem of routing and scheduling trains in case of railway traffic perturbation [4]. This formulation is the basis of the heuristic algorithm RECIFEMILP [5], in which several performance boosting methods are applied to the MILP solver (here CPLEX), which is run with a fix time limit. Since its proposal, we analyzed and tested RECIFE-MILP in several case-studies with very different characteristics, always being in close contact with the involved European railway infrastructure managers. In particular, we applied RECIFEMILP in the context of the two FP7 European projects ON-TIME [1] and CAPACITY4RAIL [2], and of the French national project SigiFRET [3]. RECIFE-MILP successfully tackled traffic perturbations in railway infrastructures as the Iron Ore line in Norway and Sweden [1], [2], a portion of the East Coast Main Line in the UK Railway Optimization 279 [1], a portion of the Dutch network in the Netherlands [1], and two critical nodes on the line between Paris and Le Havre in France [3]. Despite these promising tests, several questions need to be answered before an actual deployment of optimization tools as RECIFE-MILP in practical railway traffic management. These questions concern technical, algorithmic and process design issues. For example, from the technical side, how can an optimization algorithm be deployed in practice? What kind of software architecture fits the specific application? Or, from the algorithmic perspective, is it possible to include the consideration of the uncertainty which necessarily characterize the railway traffic? Finally, from the process design point of view, how often shall the optimization decision be re-assessed and until when can they be modified? In this paper, after discussing the performed applications of RECIFE-MILP, we will try to shed some light on some of these open questions. References [1] On-Time Consortium Optimal networks for train integration management across Europe – ON-TIME, http://www.ontime-project.eu/ (2014). [2] Capacity4Rail Consortium Increasing Capacity 4 Rail networks through enhanced infrastructure and optimised operations http://http://www.capacity4rail.eu/ (2015). [3] Marlière, G., Pellegrini, P., and Rodriguez, J. Simulation of an innovative management of freight trains In International Symposium of Transport Simulation - ISTS (2014). [4] Pellegrini, P., Marlière, G., and Rodriguez, J. Optimal train routing and scheduling for managing traffic perturbations in complex junctions Transportation Research Part B (2014), 59:58–80. [5] Pellegrini, P., Marlière, G., Pesenti, P., and Rodriguez, J. RECIFE-MILP: An effective MILP-based heuristic for the real-time railway traffic management problem Intelligent Transportation Systems, IEEE Transactions on (2015), to appear. Railway Optimization 280 Real-time near-optimal train scheduling and routing in complex railway networks Marcella Samà∗ Dipartimento di Ingegneria, Università Roma Tre, Italia, [email protected] Andrea D’Ariano Dario Pacciarelli Dipartimento di Ingegneria, Università Roma Tre, Italia, [email protected] [email protected]ma3.it Francesco Corman Maritime and Transport Technology, Delft University of Technology, The Netherlands, [email protected] Abstract. This paper focuses on the development of new algorithms for the computation of upper and lower bounds for the real-time train scheduling and routing problem. A method for the computation of a good lower bound is discussed as well as its use for the computation of heuristic solutions of the problem. Promising computational results are presented for practical-size railway instances. Introduction This work addresses the Real-Time Train Scheduling and Routing Problem (RTTSRP), i.e., the problem of computing in real time a conflict-free schedule for a set of trains circulating in a network within a given time window W = [t, t + δ], given the position of the trains at time t and the status of the network in W . The objective function is the minimization of train delays. A schedule is conflict-free if it satisfies the railway traffic regulations, which prescribe a minimum separation between consecutive trains on a shared resource in order to ensure the safety of train movements and to avoid deadlock situations in the network. The alternative graph of Mascis and Pacciarelli [1] is among the few models in the literature that incorporate, within an optimization framework, the microscopic level of detail that is necessary to ensure the fulfillment of traffic regulations. This model generalizes the job shop scheduling model in which each operation denotes the traversal of a resource (block/track section or station platform) by a job (train route). A big-M MILP formulation of the RTTSRP problem can be easily obtained from the alternative graph by introducing a binary variable for each train ordering decision (i.e. alternative pair) and a binary variable for each routing decision [6]. The resulting problem is strongly NP-hard [1]. This paper reports on recent improvements implemented in the state-of-the-art optimization solver AGLIBRARY [3]. The solver includes a branch and bound algorithm for scheduling trains with fixed routes [5] and several metaheuristics for re-routing trains [4], [2]. Previous research left open two relevant issues. The first issue is how to certify the quality of the solutions produced by the metaheuristics by means of effective lower bounds. This issue is made difficult due to the poor quality of the lower bounds based on the linear relaxation of the big-M MILP formulation of the problem. Railway Optimization 281 The second issue concerns with the computation of effective upper bounds through the development of new solution methods. Both these issues motivate this paper, whose contribution consists of the following algorithms. A first algorithm is proposed for the computation of a lower bound for the RTTSRP. This is obtained by the construction of a particular alternative graph for a relaxed RTTSRP in which each job is composed by two types of components: (i) real operations that are in common with all alternative routes of the associated train; (ii) fictitious operations that represent the shortest path between two resources of the network that are linked by some alternative routes of the associated train. For the latter type of component, no train ordering decision is modeled, disregarding the potential conflicts between trains. The resulting alternative graph is then solved to optimality by the branch and bound algorithm in [5]. A second algorithm is a constructive metaheuristic proposed in order to optimize the selection of default routes. This heuristic starts from the optimal solution obtained for the alternative graph of the relaxed RTTSRP problem and iteratively replaces the fictitious components of each job with a particular routing alternative. The selection of the routing alternative is based on the evaluation of the insertion of various train routes via the construction of the corresponding alternative graph and the computation of train scheduling solutions via fast heuristics. A third algorithm is a variable neighborhood search proposed for the train rerouting problem, based on systematic changes of a combination of neighborhood structures. The core idea is to improve the scheduling solution via re-routing multiple trains. Various neighbourhood definitions and explorations are proposed to search for better train routing combinations. Computational experiments are performed on practical-size railway instances. The new algorithms often outperform state-of-the-art algorithms developed in previous versions of AGLIBRARY and a commercial MILP solver in terms of smaller computation times and better quality upper and lower bounds to the optimal RTTSRP solutions. References [1] Mascis, A., Pacciarelli, D. (2002) Job shop scheduling with blocking and no-wait constraints. European Journal of Operational Research 143 (3), 498–517. [2] Corman, F., D’Ariano, A., Pacciarelli, D., Pranzo, M. (2010) A tabu search algorithm for rerouting trains during rail operations. Transportation Research Part B 44 (1), 175–192. [3] D’Ariano, A. (2008) Improving Real-Time Train Dispatching: Models, Algorithms and Applications. PhD Thesis, TRAIL Thesis Series T2008/6, The Netherlands. [4] D’Ariano, A., Corman, F., Pacciarelli, D., Pranzo, M. (2008) Reordering and local rerouting strategies to manage train traffic in real-time. Transportation Science 42 (4), 405–419. [5] D’Ariano, A., Pacciarelli, D., Pranzo, M. (2007) A branch and bound algorithm for scheduling trains in a railway network. European Journal of Operational Research 183 (2), 643– 657. [6] D’Ariano, A., Samà, M., D’Ariano, P., Pacciarelli, D. (2014) Evaluating the applicability of advanced techniques for practical real-time train scheduling. Transportation Research Procedia, 3 279–288. Railway Optimization 282 Rail Yield Management at Trenitalia Alessandra Berto∗ Trenitalia, Roma Tre University, Italy, [email protected] Stefano Gliozzi IBM Italia, stefano [email protected] Abstract. Since 2005 Trenitalia operates a rail Yield Management System (YMS) developed by IBM for the rail industry, which (i) provides the unconstrained prediction of the demand, (ii) optimizes the capacity allocation per origin-destination during the booking horizon, based on a fare family clustered structure, and a set of defined business rules, through a stochastic approach, (iii) simulates the effects of the new set of inventory controls, (iv) monitors the presence of spill, spoilage and stifling. Description Yield management (YM) is an umbrella term for a set of strategies that enable capacity-constrained service industries to realize optimum revenue from operations, providing “the right service to the right customer at the right time for the right price”. In particular, Rail (YM) aims at maximizing revenues on each given combination train/date of departure by optimally managing the seats availability per OriginDestination (O&D)—possibly grouped—or per leg, at each price level through the booking horizon. Within the Transportation YM problem, the Rail YM differs slightly from the Airline YM because, while the perishable and fixed offer is constrained by leg, here the demand is O&D related. Moreover, given the social impact of the rail transportation system, prices fluctuations need to be limited, while the fares availability has to be perceived as “fair” by the customers. Since 2005 Trenitalia, the main Italian train operating company, operates a YM System (YMS) developed by IBM. The system has been further adapted to a railcompany needs, and integrated with the Sales & Reservation System, with a cooperation between IBM and the ”Demand Analysis & Revenue Management” structure of Trenitalia; both the authors where part of the project since its inception in 2004. The YMS is able to optimize the capacity allocation per O&D based on a fixed fare family clustered structure. Starting from a defined set of sound business rules, the YMS (i) provides the forecast of the potential demand—additive with unconstraining and multiplicative corrections—at each point of the ‘load curve’, (ii) optimizes the capacity allocation through a stochastic optimization approach, (iii) simulates the effects of the new set of inventory controls—resilient with distinct orders of arrival—in the context of a partial nesting among O&D and fares, (iv) monitors the presence of spill, spoilage and stifling and the results achieved both of the YMS and the analysts through performance indicators and a revenue opportunity estimation. Competitor and price/time sensitivity information integrate a full automated system. Protection levels are set against dilution, with a partial nesting technique, which uses a variable nesting order. The variable nesting order is computed using the Railway Optimization 283 opportunity costs from the stochastic optimization instance. In 2014 the YMS managed dynamically around 230 trains average/day which carried more than 50 million passengers. The system optimized approximatively 4 Million instances of the model, leading to nearly 120 Billion of train-date-classO&D-fare-quantity decisions! The system provided satisfactory results through a crucial decade for the former monopolist Trenitalia, with the opening of High Speed lines, and the competition on Italian high-speed routes with the newcomer Nuovo Trasporto Viaggiatori entering the market in 2012. The Stochastic Optimization Model The scenario-based stochastic model at the very core of the system is fairly simple in its logic, and is represented as a linear program: XXX X X max(z) = X̌ris ws (1 − α)vris + αΛr + U` ε − O` ζ (58) r i s ` ` subject to: X̌ris ≤Kri ∀ r, i, s (59) ∀` (60) ! X r δr` X Kri + U` =C` + O` i with bounded real variables: ~ ris ≤ X̌ris min (Xri , X −∞ ≤ Kri ~ ris ≤X ≤∞ ∀ r, i, s (61) ∀ i, r (62) Where r, i, s and `, are the indices used over respectively O&D, fares, scenarios, and legs; K is the protection level: the decision variable of the first stage; X̌ is the second stage decision variable, i.e. the number of selected passengers in each ~ we designate scenario; w is the weight assigned to each scenario wile with X and X respectively the already booked seats and the forecast of potential passengers ; v is the unit value; Λ the O&D length; α a user defined weight; C is the seat capacity by leg; δ is the generic element of the incidence matrix between O&D and legs; U , O, with their respective weights , ζ, are used to ensure feasibility and minimum K value under any operational circumstance. References [1] Gliozzi S., Marchetti A. M., A new Yield Management Approach for Continuous MultiVariable Bookings: The Airline Cargo case, Kluwer Academic Publisher, in: Operations Research in Space and Air, ISBN/ISSN 1402012187, 2003 [2] I.B.M., US Patent: US 7,430,518 B2 (Sept. 30, 2008) and US 8,117,055 B2 (Feb. 14 2012). US Patent Office 2008-2012. [3] Williamson, E. L., Airline Network Seat Control. Ph. D. Thesis, MIT, Cambridge, MA, 1992. Routing Wednesday 9, 11:00-13:00 Sala Seminari Est 284 Routing 285 Upper and Lower Bounds for the Close Enough Traveling Salesman Problem Francesco Carrabs∗ Dipartimento di Matematica, Università di Salerno, Italia, [email protected] Carmine Cerrone Raffaele Cerulli Dipartimento di Matematica, Università di Salerno, Italia, [email protected], [email protected] Manlio Gaudioso Dipartimento di Elettronica Informatica e Sistemistica, Università della Calabria, Italia, [email protected] Abstract. This paper addresses a variant of the Euclidean traveling salesman problem in which the traveler visits a node if it passes through the neighborhood set of that node. The problem is known as the close-enough traveling salesman problem (CETSP). We introduce a mixed-integer programming model based on our new adaptive discretization schema and our graph reduction algorithm. This last algorithm is able to significantly reduce the size of the problem making easier the identification of the optimal solution for the discretized graph. From this solution we derive an upper and lower bound for the CETSP optimal tour. We evaluate the effectiveness and the performance of our approach on benchmark instances provided in the literature. The computational results show that our algorithm is very fast and often it is more effective than the other exact approaches for this problem. Routing 286 Exact and hybrid approaches for the vehicle routing problem with multiple deliverymen Pedro Munari∗ Industrial Engineering Department, Federal University of Sao Carlos, Brazil, [email protected] Aldair Alvarez Reinaldo Morabito Industrial Engineering Department, Federal University of Sao Carlos, Brazil, [email protected] [email protected] Abstract. In this work, we address the vehicle routing problem with time windows and multiple deliverymen, a variant of the vehicle routing problem that has been recently proposed in the literature. We develop a branch-price-and-cut (BPC) method to solve this problem. Also, meta-heuristics are combined with the BPC method in order to find good integer solutions at an earlier stage. Experiments indicate that the proposed methodology is able to find good solutions for instances available in the literature. Introduction In this study, we address the vehicle routing problem with time windows and multiple deliverymen (VRPTWMD), a variant of the vehicle routing problem with time windows [7] that, besides typical routing and scheduling decisions, includes the decision of how many deliverymen should be assigned to each route. Thus, service times are also a function of the number of deliverymen assigned to the route rather than fixed for a given request. This variant is particularly relevant to situations with long service times when compared with traveling times. This problem models situations faced by many companies in practice, particularly by those which must deliver and/or collect goods in very busy urban areas, such as beverage, dairy and tobacco industries, for which daily requests must be delivered on the same day, the total operation cannot be completed within the maximum routing time and violations to the latter are highly undesirable. The VRPTWMD was recently introduced in [2] and [5], where the authors present a vehicle flow formulation based on standard 2-index flow formulations of the VRPTW [1]. As the current general-purpose MIP solvers are not effective to solve such types of formulations in reasonable time, even for relatively small instances, the authors in [2] propose several heuristic and meta-heuristic approaches to solve the problem. Also, [6] propose two meta-heuristic approaches to obtain obtain good feasible solutions for the VRPTWMD. To the best of our knowledge, no exact methodology has been proposed specifically for this problem in the literature up to this moment. We aim at closing this gap by proposing a branch-price-and-cut method for the VRPTWMD. Solution methods In this study we propose a set partitioning formulation and develop a branch-priceand-cut method to solve the VRPTWMD. This method is based on central primal- Routing 287 dual solutions that are obtained by an interior point method. The advantage of using an interior point method is that they are able to offer well centered solutions naturally, without the need of any artificial resource such as variable bounds or penalty costs [4]. Also, a strong branching strategy is incorporated in the method in order reduce the number of nodes exploited and we use the subset row (SR) inequalities [3] in order to improve the overall performance of the method. To be able to solve large-scale instances, we combine a metaheuristic approach with the branch-price-and-cut method in a cooperative scheme. This meta-heuristic is based on large neighborhood search and helps to find good integer solutions at an earlier stage. Results We have test the proposed solution strategy using sets of instances based on the classic Solomon instances. The computational results indicate that the method finds optimal solutions of instances that were previously unsolved. In addition, it obtains relatively good solutions quicker than other methods proposed in the literature, which shows the advantages of combining meta-heuristics and exact solution methodologies for solving challenging problems. References [1] Desaulniers, G., Madsen, O.B. and Ropke, S., The vehicle routing problem with time windows. In Toth, P. and Vigo, D. editors, Vehicle routing: Problems, methods, and applications, MOS/SIAM Ser Optim, 2014, pages 119–159. [2] Ferreira, V.O. and Pureza, V., Some experiments with a savings heuristic and a tabu search approach for the vehicle routing problem with multiple deliverymen, Pesquisa Operacional, 2012, 32:443–463. [3] Jepsen, M., Petersen, B., Spoorendonk, S. and Pisinger, D., Subset-row inequalities applied to the vehicle routing problem with time windows, Operations Research, 2008, 56(2):497-511. [4] Munari, P. and Gondzio, J., Using the primal-dual interior point algorithm within the branchprice-and-cut method. Computers & Operations Research, 40(8):2026–2036, 2013. [5] Pureza, V., Morabito, R. and Reimann, M., Vehicle routing with multiple deliverymen: Modeling and heuristic approaches for the VRPTW, European Journal of Operational Research, 2012, 218(3):636–647. [6] Senarclens, De Gancy, G. and Reimann, M., Vehicle routing problems with time windows and multiple service workers: a systematic comparison between ACO and GRASP, Central European Journal of Operations Research, 2014. [7] Toth, P. and Vigo, D., Vehicle Routing: Problems, Methods and Applications. MOS-SIAM Series in Optimization, Second edition, 2014. Routing 288 A new location routing problem: ILP formulation and solution approach Claudio Sterle∗ Departement of Electrical Engineering and Information Technology, University of Naples “Federico II”, Italy, [email protected] Maurizio Boccia Departement of Engineering , University of Sannio, Italy, [email protected] Teodor Gabriel Crainic CIRRELT and Dèpartement de management et technologie, Ècole des sciences de la gestion, University of Quebec in Montrèal, Canada, [email protected] Antonio Sforza Departement of Electrical Engineering and Information Technology, University of Naples “Federico II”, Italy, [email protected] Abstract. In this work we present a new location-routing problem where location decisions are tackled considering the facilities as flow intercepting facilities (FIFLOR). This choice allows to overcome some drawbacks which arise when classical location-routing problems (LRP ) are used to tackle the problem of designing a urban freight distribution system. We propose an original ILP formulation for the problem and we present a branch-and-cut algorithm for its solution. Results on several instances, differing for network topology, are finally discussed. The flow intercepting facility location routing problem City Logistics (CL) issues are incumbent for all the national and local area governments aimed at achieving highly shared environmental targets such as the reduction of congestion, air and noise pollution, energy and work time consumption, damages and deterioration of infrastructures and of the historical centers. As widely recognized these externalities are mainly due to the great amount of commercial vehicles (large and in many cases not environmental friendly) performing the last mile distribution within congested urban areas. On this basis, many CL measures have been adopted to tackle this phenomenon. In this context great interest has been addressed to the design of new and more effective and efficient single and multiechelon solutions for the enhancement of the urban freight distribution systems. These systems are aimed at preventing the penetration of a large number of commercial vehicles in the city center intercepting them at facilities located along t he pre-planned paths towards the destinations, and providing then an efficient “shared transportation service” to perform the last leg of the distribution [1]. In other words large commercial vehicles coming from the outskirts and directed to the center can be forced to park their own vehicles at intermediate logistic platforms, where the freights of different carriers addressed to more destinations are deconsolidated, transferred and consolidated on smaller and green vehicles more suitable to perform the last mile distribution. In the last ten years, this problem has been widely treated in literature approaching it as an integrated location-routing problem (LRP ), where design decisions concern the size, number and position of one or more types of logistic facilities whereas routing decisions concern the definition of dedicated or multi-stop routes Routing 289 of homogeneous or heterogeneous vehicle fleets. A complete review of the main contributions in the field can be found in [2]. In LRP s, location decisions have been always tackled considering the logistic facilities as flow generating/attracting facilities. Hence, classical facility location problems, e.g., p-median and uncapacitated facility location problems (PMP and UFLP ), have been integrated with routing decisions. However this choice, in some cases, has several drawbacks which do not allow designing a cost-effective logistic system and do not allow taking into account specific features of the design problem under investigation. More precisely, in many cities the installation of the logistic facilities should performed just in points located along the main entrance roads or near a pre-existing road system, so avoiding additional design cost. Moreover, if the goal of the system is the traffic congestion reduction, the logistic facilities should be placed to capture the private commercial vehicles early en route to their destination. Finally, nowadays the good/service demand i n an urban area is highly customized (due also to the arising e-commerce solutions), since freights of more suppliers, coming from different origins, have to be distributed to the customers requiring them. This makes the problem under investigation inherently a multi-commodity problem. LRP s proposed in literature are instead generally based on a single-commodity idea, which well fits to situations where the freights of a single supplier have to be distributed to several customers, for which their origin is of no importance. In order to overcome these drawbacks, in this work we propose a new locationrouting model (FIFLOR) for the design of a city logistic system, where location decisions are tackled considering the logistic facilities as flow intercepting facilities. Hence we integrate flow interception facility location (FIFL) models [3], with multi depot vehicle routing decisions. To the best of our knowledge these two problems have never been treated together. The only work addressing a similar problem is the one tackled in [4], where the flow interception approach is used for the location of rail park-and-ride facilities. The problem is formulated by an original integer linear programming (ILP ) model and solved by a branch and cut algorithm which exploits valid inequalities derived from the literature. Finally, the proposed approach is experienced on several original instances differing for their sizes and for the network topology. References [1] Crainic, T.G, Ricciardi, N., Storchi, G., Advanced freight transportation systems for congested urban areas. Transportation Research Part C: Emerging Technologies, 12, 2, (2004), 119–137. [2] Prodhon, C., Prins, C., A survey of recent research on location-routing problems. European Journal of Operational Research, 238, 1, (2014), 1–17. [3] Boccia, M., Sforza, A., Sterle, C., Flow Intercepting Facility Location: Problems, Models and Heuristics. Journal of Mathematical Modelling and Algorithms, 8, 1 (2010), 35–79. [4] Horner, M.W., Groves, S., Network flow-based strategies for identifying rail park-and-ride facility locations. Socio-Economic Planning Sciences, 41, 3, (2007), 255–268. Routing 290 A particular vehicle routing and scheduling problem in last-mile logistics Maurizio Bruglieri∗ Dipartimento di Design, Politecnico di Milano, Italia, [email protected] Alberto Colorni Dipartimento di Design, Politecnico di Milano, Italia, [email protected] Federico Lia Consorzio Poliedra, Politecnico di Milano, Italia, [email protected] Abstract. We address a particular optimization problem arising in last-mile logistics where freight is delivered to the customers through intermediate depots, so called Urban Satellite Platforms (USPs), meeting points between long-haul carriers (LHCs) and last-mile carriers (LMCs). Unlike the two-echelon vehicle routing problem (2E-VRP), we are not interested to determine the routes of the LHCs but only to assign their parcels to the USPs. Vice versa we consider features as delivery time windows, optional deliveries, heterogeneous fleet and LHCs/LMCs’ preferences that are not usually considered in the 2E-VRP. Introduction The growth in the volume of freight traffic, as well as environmental and traffic congestion reasons, has led in recent years to a switch of distribution strategy from the direct shipping to systems where freight is delivered to the customers through intermediate depots, so called Urban Satellite Platforms (USPs). In the USPs, the packages, that previously traveled on trucks, are disaggregated and reassigned to lighter and environmentally sustainable vehicles (e.g. electric vans and bicycles) for urban center deliveries. In this context we addressed inside the project OPTILOG (OPTImal and sustainable LOGistics in urban areas), financed by Lombardia Region (Italy), a particular optimization problem arising in last-mile logistics. A set of long-haul carriers (LHC) and a set of last-mile carriers (LMC) are given. At an early accreditation stage, each LHC company must declare the preferred USPs and the preferred LMC companies. In addition, some LHC companies may allow their products to travel in light vehicles along with those of the other LHCs, while others may require an exclusive transport service. Each LHCs’ parcel is characterized by a destination address, a volume, a weight and a delivery time window (TW). Some deliveries can be optional in order to model parcels that must be delivered within a long period (e.g. a week): in this way, a multi-period delivery planning is avoided, by making the shipping optional until its final delivery day is reached, becoming mandatory after that. In each PSU a fleet of heterogeneous vehicles belonging to different LMC companies, are present. Each vehicle is characterized by typology and capacity. Moreover each LMC has a favorite zone, i.e., a zone of the urban center that he better knows on the basis of earlier made deliveries. The problem that we want to solve is to decide how to assign the parcels to the USPs, and within each USP how to route and schedule each LMC taking into account the following constraints: Routing 291 1. each LMC route must start and end in the belonging USP; 2. each LMC can ship only the parcels assigned to his USP; 3. vehicle capacity must not be exceeded; 4. delivery TWs must be respected; 5. route durations must not to exceed the duty time; 6. preferences of both LHCs and LMCs must be respected. Multiple objectives need to be optimized at the same time: minimize the LMCs employed, minimize the total length of the vehicle routes, balance them as much as possible and maximize the preferences of both LHCs and LMCs. We obtain this, through the optimization of a single objective given by the weighted sum of the utility functions associated with each criterion. Although our problem owns some similarities with the two-echelon vehicle routing problem (2E-VRP) [1], [2], it differs from it since we are not interested to determine the routes of the LHCs but only to assign their parcels to the USPs. Moreover, features as delivery TW, optional deliveries, heterogeneous fleet and LHCs/LMCs’ preferences are not usually considered in the 2E-VRP. The latter feature can instead be modeled as incompatible loading constraint [3]. For this problem we present a Mixed Integer Linear Programming (MILP) formulation. Moreover, we also develop a heuristic approach in order to face real world alike instances derived from the Milan road network. References [1] Hemmelmayr, V.C., Cordeau, J.F., Crainic, T.G. An adaptive large neighbourhood search heuristic for Two-Echelon Vehicle Routing Problems arising in city logistics, Computers & Operations Research 39 (2012), 3215–3228. [2] Perboli, G., Tadei, R., Vigo, D., The two-echelon capacitated vehicle routing problem: models and math-based heuristics, Transportation Science 45 (2011), 364–380. [3] Wang, Z., Li, Y., Hu, X. A heuristic approach and a tabu search for the heterogeneous multitype fleet vehicle routing problem with time windows and an incompatible loading constraint, Computers & Industrial Engineering, to appear. Scheduling Thursday 10, 11:00-13:00 Sala Seminari Ovest 292 Scheduling 293 Hybrid of Genetic and Simulated Annealing Algorithm for Permutation Flow-Shop Scheduling Problem Omer Faruk Yilmaz∗ Department of Industrial Engineering, Technical University of Istanbul, Turkey, [email protected] Abstract. This article presents a hybrid genetic and simulated annealing algorithm (HGSA) to solve permutation flow-shop scheduling problem. The permutation flow shop scheduling problem (PFSP) is known to be an NP-hard combinatorial optimization problem and one of the most extensively-studied problem. Nature of this problem is appropriate to handle via evolutionary algorithms (EAs). Genetic algorithm (GA) is mostly known effective population based metaheuristic (P-metaheuristic) of this category and has a powerful diversification property. On the other hand, simulated annealing (SA) is one of the most using single solution based metaheuristic (S-metaheuristic) and has a powerful convergence property. Taking all above mentioned point into consideration, hybrid GA&SA algorithm is developed to handle FSP problem in this research. A comparison was made between GA and HGSA in terms of solving the same problem. Simulation results show that the HGSA has more rapid convergence speed and better searching ability to find an appropriate solution in a reasonable amount of time. The algorithm use GA and SA operators, which makes the search-for-optimal-solution process much more effective and efficient. The proposed hybrid GA&SA algorithm is described as follows. 1. Initial population is randomly generated and temperature is initialized. 2. After the fitness function evaluations, roulette wheel selection, one point crossover and swap mutation operators are applied to obtain an offspring population. Selection and crossover operators are applied on the whole population. 3. Neighborhood strategies: to conduct a better localized search, SA is employed to generate a new offspring population. If the new individual is better than the old one then accept the new individual and put it in the place of the old one. If the new individual is not better than the old one then calculate accepting probability of the new individual. If this probability is greater than random number then accept the new individual for old offspring population. 4. Temperature is decreased and if ending temperature or maximum iteration number is reached then algorithm is ended. The algorithm is applied to an example and compared with GA using the same example for permutation flow-shop scheduling. References [1] Huang, H.Y., The complexity of flow shop and job shop scheduling, Mathematics of Operations Research 1 (1976), 17-129. [2] Deb, K., et al., A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions Evolutionary Computation 6 (2002), 182-197. [3] Yang, Y., et al., Combining local search into non-dominated sorting for multi-objective linecell conversion problem, International Journal of Computer Integrated Manufacturing 26 (2013), 316-326. [4] K. Baker, Introduction to sequencing and scheduling. Wiley, 1974. Scheduling 294 [5] Yenisey, M.M and Yagmahan, B., Multi-objective permutation flow shop scheduling problem: Literature review, classification and current trends, Omega 45 (2014), 119-135. [6] Zameer, A., et al., Core loading pattern optimization of a typical two-loop 300 MWe PWR using Simulated Annealing (SA), novel crossover Genetic Algorithms (GA) and hybrid GA(SA) schemes, Annals of Nuclear Energy 65 (2014) 122-131. Scheduling 295 Energy cost optimization for single machine scheduling with earliness and tardiness costs and controllable processing times Federica Laureri∗ Dipartimento di Informatica, Bioingegneria, Robotica e Ingegneria dei Sistemi, Università di Genova, Italia, [email protected] Simone Chiaverini Davide Giglio Massimo Paolucci Riccardo Minciardi Dipartimento di Informatica, Bioingegneria, Robotica e Ingegneria dei Sistemi, Università di Genova, Italia, [email protected], [email protected], [email protected], [email protected] Abstract. This work considers an extension of single machine scheduling problem with distinct due dates and earliness and tardiness costs. In particular, as the energy costs vary from peak to off-peak time periods and job durations can be controlled, also the energy costs for job execution and the costs for compressing the job processing times must be minimized. A MILP formulation and a metaheuristic approach are proposed and analyzed considering a set of random generated instances. The scheduling problem and the proposed approach The increased energy demand in emerging markets, dwindling oil reserves and the planned nuclear phaseout in various nations, causes energy prices to increase significantly. In addition, take into account, in a combined way, economic efficiency of manufacturing production with environmental issues, is considered more and more essential [1]. Thus, energy efficiency (EE) has recently considered as a complementary factor to productivity. The increased need of companies to account for energy consumption costs in order to sustain their competitiveness then stimulates a growing interest in the recent literature for scheduling and control strategies in manufacturing aiming at energy saving [2]. The reduction of energy costs can be pursued both increasing EE of production processes and implementing new operational management approaches. This work is focused on the second kind of approach. This work considers the problem of non–preemptive scheduling a set of jobs on a single machine in order to minimize the sum of their weighted earliness and tardiness (ET) penalties, taking also into account variable energy prices during the scheduling horizon (e.g., the day). In particular, the scheduling horizon is assumed partitioned into a set of time intervals characterized by a different unitary energy cost (e.g, peak periods with high costs and off–peak periods with low costs). This aspect is considered in [3] where jobs can be shifted from peak to off-peak periods to take advantage from reduced energy prices, and where the possibility of turning off the machine during idle times is also evaluated. A further feature is the possibility of decreasing the processing times of the jobs paying a given processing compression cost: a nominal (or maximal) and minimal processing time is associated with the jobs, so that the job duration can be con- Scheduling 296 trolled. In [4] the total cost due to earliness, tardiness and controllable processing time is minimized for a single machine scheduling with no inserted idle times. The problem faced in this work then consists in finding the schedule of the jobs on a single machine such that the overall cost, given by the sum of the ET penalties, the job compression costs and the energy consumption costs, is minimized. To the best author’s knowledge, in literature there is no previous approach for this kind of problem, which is apparently NP–hard as it generalizes the single machines ET problem with distinct due dates [5]. This work proposes both a MILP formulation and a metaheuristic algorithm based on a greedy randomized adaptive search procedure (GRASP). The considered objective function is non-regular and corresponds to the sum of both convex linear functions, associated with the ET penalties, and non-convex piecewise linear functions due to the variable energy costs. In addition, the costs of the decisions relevant to the job durations, that ideally are modelled with continuous variables, add a further difficulty to the timing algorithm that is repetitively invoked within the metaheuristics to determine the optimal start times for a given sequence. Therefore, a discretization is considered for the feasible ranges of the job processing times, modelling such decisions as combinatorial ones. The computational tests, performed on a set of random generated instances, are presented and the obtained results are discussed. References [1] Despeisse, M., Ball, P.D., Evans, S., Levers, A., Industrial ecology at factory level –a conceptual model Journal of Cleaner Production 31 (2012) 30–39. [2] Langer, T., Schlegel, A., Stoldt, J., Putz, M., textitA model–based approach to energy–saving manufacturing control strategies Procedia CIRP 15 (2014) 123–128. [3] Shrouf, F., Ordieres–Mer, J., Garca–Sanchez, A., Ortega-Mier, M., Optimizing the production scheduling of a single machine to minimize total energy consumption costs Journal of Cleaner Production 67 (2014) 197–207. [4] Kayvanfar, V., Mahdavi, I., Komaki, GH.M, Single machine scheduling with controllable processing times to minimize total tardiness and earliness Computers & Industrial Engineering 65 (2013) 166–175. [5] Garey, M. Tarjan, R., Wilfong, G. One–processor scheduling with symmetrical earliness and tardiness penalties Mathematics of Operations Research 13 (1988) 330–348. Scheduling 297 Single machine scheduling problems to minimize total earliness-tardiness with unavailability period Gur Mosheiov∗ School of Business Administration, The Hebrew University, Jerusalem, Israel, [email protected] Enrique Gerstl School of Business Administration, The Hebrew University, Jerusalem, Israel, [email protected] Abstract. We study several versions of a single-machine scheduling problem, where the machine is unavailable for processing for a pre-specified time period. A common due-date for all the jobs is assumed, which can be either prior to, or after, or within the unavailability period. In the basic problem, the objective function is minimizing total earliness-tardiness, and we consider first the setting that no idle times are allowed. We then extend the problem to general earliness and tardiness cost functions, to the case of job-dependent weights, and to the setting that idle times are allowed. All these problems are known to be NP-hard. We introduce in all cases pseudopolynomial dynamic programming algorithms, indicating that these problems remain NP-hard in the ordinary sense. Our numerical tests indicate that the proposed solution algorithms can solve medium size problems (of hundreds of jobs) in reasonable computation time. Description The basic paper dealing with scheduling with unavailability period is that of [1], who studied this setting on a number of machine settings, and considered various objective functions. The most relevant papers dealing with Just-In-Time scheduling are [3] who focused on minimizing earliness-tardiness around a common restrictive (i.e. sufficiently small) due-date, and [2] who solved the case of minimum total weighted earliness-tardiness. Our study combines and extends the settings considered in these papers. References [1] Lee, C-Y., Machine Scheduling with an Availability Constraint, Journal of Global Optimization 9 (1996), 395–416. [2] Hall, N.G., Posner, M., Earliness-tardiness scheduling problems, I: weighted deviation of completion times about a common due date, Operations Research 39 (1991), 836–846. [3] Hall, N.G., Kubiak, W., Sethi, S.P., Earliness-tardiness scheduling problems, II: derivation of completion times about a restrictive common due date, Operations Research 39 (1991), 847–856. Scheduling 298 A branch-and-reduce exact algorithm for the single machine total tardiness problem Federico Della Croce∗ Politecnico di Torino, Italia, [email protected] Michele Garraffa Politecnico di Torino, Italia, [email protected] Lei Shang Vincent T’kindt Université F. Rabelais de Tours, France, [email protected] [email protected] Abstract. This work deals with the one-machine total tardiness problem, a classical NPHard scheduling problem. Several exact algorithms have been proposed in the past, however, to the authors knowledge, none of them solves the problem in O∗ (cn ) (c being some constant) and polynomial√ space. In this work, we propose a branch-and-reduce algorithm whose time complexity is O∗ ((1 + 2)n ), while its space complexity is polynomial. We also discuss some improvements for the described approach, which allow to improve the worst case complexity. Introduction P We consider the one-machine total tardiness 1|| Tj problem where a jobset N = {1, 2, . . . , n} of n jobs must be scheduled on a single machine. For each job j, we define a processing time pj and a due date dj . The problem calls forP arranging the jobset in a sequence S = (1, 2, . . . , n) so as to minimize T (N, S) = nj=1 Tj = P Pn − dj , 0}, where Cj = ji=1 pi . j=1 max{C Pj The 1|| Tj problem is NP-hard in the ordinary sense [2]. It has been extensively studied in the literature and many exact procedures ([1][4][5][7]) have been proposed. The current state-of-the-art exact method of [7] dates back to 2001 and solves to optimality problems with up to 500 jobs. Here, we study their application in the context of worst-case analysis and look for best results in terms of exponential time complexity as a function of the number of variables (in our case jobs). In this context, classical search tree algorithms are more commonly defined as branch-and-reduce algorithms as typically a branch in the search tree induces the generation of two or more subproblems each with a reduced number of variables with respect to the original problem. In what follows, we express the time complexity of the algorithms with the O∗ notation used in [8], it indicates that we ignore polynomial factors. In the context of exact exponential algorithm, the dynamic programming of [4] cannot P be considered due to the pseudopolynomial time complexity being function of pi . On the other hand, the standard technique of doing dynamic programming across the subsets [8] is applicable also to the total tardiness model and runs with complexity O∗ (2n ) but requires also O∗ (2n ) space, that is it requires both exponential time and space. To the authors knowledge, there is currently no available exact algorithm for the total tardiness problem running in O∗ (cn ) (c being some constant) and polynomial space. The proposed algorithm is based on some theoretical properties ([3][4][5][6]) of the problem. The basic idea of our branch-and-reduce algorithm is to iteratively Scheduling 299 branch on the largest processing time job n, assign it to all possible positions and correspondingly decompose the problem of each single branch according to the decomposition property in [4]. The presentation of the work at the√conference will show how the algorithm can achieve a time complexity of O∗ ((1 + 2)n ). Furthermore, it will be shown how to improve the exponential time complexity by using some simple precomputation strategies. References [1] Della Croce, F., Tadei, R., Baracco, P., Grosso A., A new decomposition approach for the single machine total tardiness scheduling problem, Journal of the Operational Research Society 49 (1998), 1101–1106. [2] Du, J., Leung,J. Y. T., Minimizing total tardiness on one machine is NP–hard, Mathematics of Operations Research 15 (1990), 483–495. [3] Emmons, H., One-machine sequencing to minimize certain functions of job tardiness, Operations Research 17 (1969), 701–715. [4] Lawler,E. L., A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness, Annals of Discrete Mathematics 1 (1977), 331–342. [5] Potts, C. N., Van Wassenhove, L. N., A decomposition algorithm for the single machine total tardiness problem, Operations Research Letters 5 (1982), 177–181. [6] Szwarc, W., Single machine total tardiness problem revisited, Y. Ijiri (ed.), Creative and Innovative Approaches to the Science of Management (1993), Quorum Books, Westport, Connecticut (USA), 407–419. [7] Szwarc, W. , Grosso,A., Della Croce, F., Algorithmic paradoxes of the single machine total tardiness problem, Journal of Scheduling 4 (2001), 93–104. [8] Woeginger, G. J., Exact algorithms for NP-hard problems: a survey. Combinatorial Optimization - Eureka! You shrink!, Lecture Notes in Computer Science 2570 (2003), 185–207. Variational Problems & Equilibria 1 Wednesday 9, 11:00-13:00 Sala Gerace 300 Variational Problems & Equilibria 1 301 A variant of Forward-Backward splitting method for the system of inclusion problem Reinier Dı́az Millán∗ Federal Institute of Education, Science and Technology, Goiânia, Brazil, [email protected] Abstract. In this manuscript, we propose variants of Forward-Backward splitting method for solving the system of splitting inclusion problem. We propose a conceptual algorithm containing three variant, each of them, have a different projection steps. The algorithm consist in two parts. The first and main part of our approach, contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part of the scheme consists in special projection steps. The convergence analysis of the proposed scheme is given assuming monotonicity on both operators, without Lipschitz continuity assumption on the forward operators, improving the known results in the literature. Introduction First, we introduce some notation and definitions. The inner product in Rn is denoted by h·, ·i and the norm induced by the inner product by k · k. We denote by 2C the power set of C. For X a nonempty, convex and closed subset of Rn , we define the normal cone to X at x ∈ X by NX (x), i.e., NX (x) = {d ∈ Rn : n hd, x − yi ≥ 0 ∀y ∈ X}. Recall that an operator T : Rn → 2R is monotone if, for all (x, u), (y, v) ∈ Gr(T ) := {(x, u) ∈ Rn ×Rn : u ∈ T (x)}, we have hx−y, u−vi ≥ 0, and it is maximal if T has no proper monotone extension in the graph inclusion sense. In this paper, we propose a modified algorithm for solving a system of splitting inclusion problem, for the sum of two operators. Given a finite family of pair of operators {Ai , Bi }i∈I , with I =: (1, 2, · · · , `) and ` ∈ N. The system of inclusion problem consist in: Find x ∈ Rn such that 0 ∈ (Ai + Bi )(x) for all i ∈ I, (63) where the operators Ai : Rn → Rn are point-to-point and monotone and the opern ators Bi : Rn → 2R are point-to-set maximal monotone operators. The solution of the problem is given by the interception of the solution of each component of the system, i.e., S∗ = ∩i∈I S∗i , where S∗i is defined as S∗i := {x ∈ Rn : 0 ∈ Ai (x) + Bi (x)}. The problem (63) is a generalization of the system of variational inequalities considering that operators Bi = NCi for all i ∈ I, which have been study in [7], [2], [3], [4]. A generalization of this results have been study in [8], [6], where the hypothesis that all Ai are Liptchitz continuous for all i ∈ I, is assumed for the convergence analysis. In this paper we improve this result assuming only monotonicity for all operators Ai , and maximal monotonicity for the operators Bi . The ideas for this paper comes from the references [1], [5]. Problem (63), have many applications in operations research, mathematical physics, optimization and differential equations. This kind of problem, have been deeply studied and has recently received a lot attention, due to the fact that many Variational Problems & Equilibria 1 302 nonlinear problems, arising within applied areas, are mathematically modeled as nonlinear operator system of equations and/or inclusions, which each ones are decomposed as the sum of two operators. References [1] Bello Cruz, J.Y., Dı́az Millán, R. A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy. Optimization DOI:10.1080/ 02331934.2014.883510(2014). [2] Y. Censor, A. Gibali, and S. Reich. A von Neumann alternating method for finding common solutions to variational inequalities.Nonlinear Analysis Series A: Theory, Methods and Applications 75, (2012) 4596-4603. [3] Y. Censor, A. Gibali, S. Reich, and S. Sabach. Common solutions to variational inequalities. Set-Valued and Variational Analysis 20, (2012) 229-247. [4] Y. Censor, A. Gibali, and S. Reich. Algorithms for the split variational inequality problem. Numerical Algorithms 59, (2012) 301-323. [5] Dı́az Millán, R. On several algorithms for variational inequality and inclusion problems. PhD thesis, Federal University of Goiás, Goiânia, GO, 2015. Institute of Mathematic and Statistic, IME-UFG. [6] Eslamian, M., Saejung, S., Vahidi, J. Common solutions of a system of variational inequality problems. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 77 Iss.1 (2015). [7] Konnov, I.V.: Splitting-type method for systems of variational inequalities. Comput. Oper. Res. 33, (2006) 520-534. [8] Semenov, V.V. Hybrid splitting methods for the system of operator inclusions with monotone operators. Cybernetics and Systems Analysis 50 (2014) 741-749. Variational Problems & Equilibria 1 303 KKT optimality conditions with applications to variational problems Giandomenico Mastroeni∗ Dipartimento di Informatica, Università di Pisa, Italia, [email protected] Fabian Flores Bazán Department of Mathematical Engineering, University of Concepcion, Chile, [email protected] Abstract. We present recent characterizations of Karush- Kuhn-Tucker (KKT) conditions for a differentiable optimization problem with inequality constraints and we consider their extensions to a variational inequality (VI). Such characterizations are related to strong duality of a suitable linear approximation of the given problem and the properties of its associated image mapping, without assuming constraints qualifications. After analyzing in detail the case of a single inequality constraint we present an application to a bilevel programming problem where the feasible set is given by the solution set of a (VI). References [1] Flores-Bazán F., Mastroeni G., Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications, SIAM J. Optim., 25 (2015), 647-676 . Variational Problems & Equilibria 1 304 Game-Theoretic Approach to Joint Admission Control and Capacity Allocation for MapReduce Michele Ciavotta∗ Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] Eugenio Gianniti Danilo Ardagna Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italia, [email protected] [email protected] Mauro Passacantando Dipartimento di Informatica, Università di Pisa, Italia, [email protected] Abstract. The MapReduce framework, alongside its open source implementation Apache Hadoop, is among the main enabling technologies for Big Data applications. In a private cloud setting with multiple user classes, we developed a distributed game-theoretic model to perform admission control and capacity allocation. Our goal is the reduction of power consumption, while meeting due dates defined in the SLAs and avoiding penalties associated with job rejections. Problem statement In this work we developed a distributed formulation of the joint admission control and capacity allocation problem for private clouds hosting Hadoop workloads. Given the capacity of the cluster, our goals are, on one hand, to minimize the overall cluster power consumption and running costs, on the other, to fulfill SLAs, which impose the completion of a number of MapReduce jobs within given due dates. These requirements are the one in contrast to the other and this called for a formalization of the optimization problem so as to determine an optimal configuration that minimizes costs avoiding penalties. According to Verma et al. [2], it is possible to derive from past execution logs a set of invariants that can be exploited to estimate the execution time of MapReduce jobs under varying resource configuration. Moreover, we adopted the more recent approximate formulae on job execution times presented in Malekimajd et al. [1]. We exploited the approach proposed in those works to determine the expected performance with respect to the assigned resources, hence enabling a decision process on the number of jobs to execute and the corresponding resource allocation. Formulation and validation At first, we elaborated a preliminary, centralized mathematical programming model for the problem under study, which showed efficiency issues due to the integrality of some variables. This formulation has been, therefore, relaxed and reduced to a convex continuous problem. Karush-Kuhn-Tucker (KKT) conditions provided a theoretical guarantee on the existence of an optimal solution and allowed to write an analytical relation linking the amount of computational resources allocated to a job class and the obtained concurrency level. Exploiting this result, it was possible to calculate for each class of jobs in closed form the maximum and minimum resource Variational Problems & Equilibria 1 305 demand required to satisfy the SLAs. Despite providing guarantees of optimality, the centralized approach is not suitable to represent real-life scenarios as it assumes that a single entity in the cluster has a global knowledge of the problem parameters; this is in contrast with the needs of private clouds managed in multi-tenancy and the architecture of the resource negotiator released in the latest version of Hadoop, namely YARN. So as to overcome the need for globally available information, we modeled the main entities implemented in YARN as players of a Generalized Nash Equilibrium problem (GNEP). We associated a mathematical programming problem to the Resource Manager (RM), that is the entity in charge of allocating computational resources, and another to the Application Masters (AMs), the application-specific components that request resources based on the application needs. The formulation entails one instance of the RM problem and N instances of the AM problem, where A is the set containing all the AMs and N = |A|. However, such distributed formulation does not allow to apply theoretical results guaranteeing the existence of the equilibrium. Nonetheless, we sought it through a best reply algorithm. Moreover, since we are interested in integer solutions to support runtime cluster management, we devised and realized a heuristic strategy for solution rounding. In order to verify the quality of the proposed approach, we performed a thorough validation. First of all, we compared the solutions obtained with the centralized and the distributed formulations in large experimental campaigns, where we fixed some problem parameters and focusing only on those that most affect performance. Under these circumstances, we verified that both approaches lead to realistic behaviors in three relevant cases: with decreasing cluster capacity, increasing concurrency level, or decreasing duedates. Then, we performed a scalability analysis considering up to 500 AMs and 10 randomly generated instances for each size of the AM set. Furthermore, we studied the sensitivity of the distributed algorithm to the tolerance used in the stopping criterion repeating the scalability experiment by varying the tolerance values. At last, we validated the quality of the solution and of the rounding algorithm comparing the results obtained through our approach and the timings measured on the official simulator, YARN SLS. The average accuracy, showing a relative error below 20%, is in line with the typical expectations of the research area. References [1] Malekimajd, M., Ardagna, D., Ciavotta, M., Rizzi, A. M., Passacantando, M., Optimal Map Reduce Job Capacity Allocation in Cloud Systems, ACM SIGMETRICS Performance Evaluation Review 42 (2015), 50–60. [2] Verma, A., Cherkasova, L., Campbell, R. H., ARIA: Automatic Resource Inference and Allocation for MapReduce Environments, Proceedings of the Eighth International Conference on Autonomic Computing (2011), 235–244. Variational Problems & Equilibria 1 306 A Two-Player Differential Game Model for the Management of Transboundary Pollution and Environmental Absorption Giorgio Gnecco∗ IMT - Institute for Advanced Studies, Lucca, Italy, [email protected] Fouad El Ouardighi ESSEC Business School, Cergy Pontoise, France, [email protected] Konstantin Kogan Bar-Ilan University, Faculty of Social Sciences, Ramat-Gan, Israel, [email protected] Marcello Sanguineti University of Genova, Genova, Italy, [email protected] Abstract. It is likely that the decentralized structure at the level of nations of decision-making processes related to polluting emissions will aggravate the decline in the efficiency of carbon sinks. A two-player differential game model of pollution is proposed. It accounts for a time-dependent environmental absorption efficiency and allows for the possibility of a switching of the biosphere from a carbon sink to a source. The impact of negative externalities from the transboundary pollution non-cooperative game wherein countries are dynamically involved is investigated. The differences in steady state between cooperative, open-loop, and Markov perfect Nash equilibria are studied. For the latter, two numerical methods for its approximation are compared. Introduction With rare exceptions, the problem of declining environmental absorption efficiency has been disregarded in dynamic game models of transboundary pollution. Such models are usually based on the assumption of a linear environmental absorption function. This assumption, which used to hold as long as the pollution effects on the absorption efficiency were reduced, led to the computation of linear (e.g., Van der Ploeg and de Zeeuw, 1992; Long, 1992) as well as non-linear emissions strategies (e.g., Wirl, 2007). Recently, a nonlinear absorption function has been proposed to account for the declining environmental absorption efficiency (e.g., Brock and Dechert, 2008). This formulation, which combines the linear negative influence of the pollution stock with a nonlinear positive feedback related to the release of past accumulated pollution, involves multiple steady state equilibria with different stability properties. That is, below (beyond) a certain pollution stock level, the environmental absorption capacity is positive (exhausted), and pollution is reversible (irreversible). Although this change in perspective allows a more accurate handling of potential irreversible environmental degradation, it relies on the assumption of instantaneous change in the absorption efficiency that precludes from investing in a restoration effort. Variational Problems & Equilibria 1 307 Main results In this work, we show that further results can be derived if these two implausible assumptions, i.e., instantaneous change in the absorption efficiency and absence of restoration effort, are dropped. We suggest a two-player differential game model of pollution that accounts for a time-dependent environmental absorption efficiency that allows for the possibility of a switching of the biosphere from a carbon sink to a source. In the proposed setup, we investigate the impact of negative externalities resulting from the transboundary pollution control non-cooperative game wherein countries are dynamically involved. To do so, we assess differences related to both transient and steady states between cooperative and non-cooperative pollution control and environmental absorption efficiency management (El Ouardighi, Kogan, Gnecco, and Sanguineti, 2015). We compare the outcomes related to an open-loop Nash equilibrium (OLNE), which reflects the situation where polluters commit to a predetermined plan of action, and a Markov perfect Nash equilibrium (MPNE), which corresponds to the case where polluters make decisions contingent on the current state of the biosphere (e.g., Long, 2010). To investigate the main differences between the cooperative, OLNE and approximated MPNE in terms of steady state and transient path, we adopt a quadratic approximation of the value function around a steady state solution and show through alternative methods based on discretization and successive approximations that it provides locally a valid approximation of the MPNE. Using the cooperative solution as a benchmark, we identify which decision rule among OLNE and MPNE most prevents the durable or transient switching of the biosphere from a pollution sink to a source, if any. The results show that the players’ transient behavior has a critical impact on the resulting steady state in terms of pollution stock and environmental absorption efficiency. More importantly, unexpected contrasts are found between the cooperative and non-cooperative steady states. References [1] Van der Ploeg, F., de Zeeuw, A.J., International aspects of pollution control, Environmental and Resource Economics 2 (2), 117–139, 1992. [2] Long, N.V., Pollution control: A differential game approach, Annals of Operations Research 37 (1), 283–296, 1992. [3] Wirl, F., Do multiple Nash equilibria in Markov strategies mitigate the tragedy of the commons?, Journal of Economic Dynamics and Control, 31 (10), 3723–3740, 2007. [4] Brock, W.A., Dechert, W.D., The polluted ecosystem game, Indian Growth and Development Review, 1 (1), 7–31, 2008. [5] El Ouardighi, F., Kogan, K., Gnecco, G., Sanguineti, M., Transboundary pollution control and environmental absorption efficiency management, submitted, 2015. [6] Long, N.V., A Survey of Dynamic Games in Economics. World Scientific, 2010. Variational Problems & Equilibria 2 Thursday 10, 11:00-13:00 Sala Gerace 308 Variational Problems & Equilibria 2 309 A quantization approach to an optimal order execution problem for ETD markets Immacolata Oliva∗ Dept. of Computer Sciences, University of Verona, Italy, [email protected] Michele Bonollo Iason Ltd and IMT Lucca, Italy, [email protected] Luca Di Persio Dept. of Computer Sciences, University of Verona, Italy, [email protected] Abstract. In this paper we want to apply quantization techniques to numerically solve the optimal order execution problem in the case of exchange-traded markets (ETD), taking advantage of the mathematical framework provided by the dynamic programming. Introduction, aim and scope In the financial market analysis, a crucial role is played by understanding how liquidity affects the market participants trade and, consequently, the bid/ask spread. This is referred as the optimal execution problem, that provides for determining the “best” trading strategy, viz., the optimal one, see e.g. [5]. Such a strategy results to be efficient, since minimizes variance for a specified level of the expected cost [13]. In the literature, there exist two optimal trading strategies: the discrete-time and the continuous-time approaches. The former consists in trading at fixed deterministic times [3], at exogenous random discrete times [14] or at discrete times chosen according to an impulse control function, [8],[10],[11]. The latter inheres in trading over a fixed time horizon [0, T ], see e.g. [15], but, although the whole Itô stochastic theory can be applied, it results to be inadequate in real applications. In mathematical terms, a trading strategy is well defined in terms of dynamic programming quasi-variational inequalities, DPQVI from now on. Indeed, DP guarantees to solve variational problems, since it assures the optimality of the solution, a numerical stability and the compliance of hard constraints [10]. Moreover, DP is widely used to get viscosity solutions of suitable Hamilton-Jacobi-Bellman quasivariational inequalities, e.g., in the general multidimensional diffusion-type stochastic control problems, see [1]. In a financial setting, DP was initially adopted as a probabilistic model for pricing and hedging American-type options, by devising a tree-structure vector quantization for spatial discretization, see [2] and references therein, but, in recent years, it results to work also with the description of optimal portfolio liquidation models. By following the way paved in [7], we aim at exploiting the quantization techniques used in [2], in order to numerically evaluate the solutions of optimal control problems, applied in the exchange-traded derivatives (ETD) markets, such as the Eurex Exchange or the London Stock Exchange, namely, derivatives markets characterized by the presence of market makers that promote market liquidity and enhance the efficiency of the price discovery process, as it is well explained in the Variational Problems & Equilibria 2 310 Italian Equity Derivatives Market (IDEM) guidelines, see [4]. In particular, we perform an innovative application, concerning the above derivative markets, that looks very realistic in the description of the financial markets, as well as the employment of quantization procedures is encouraging in numerical performances. References [1] Amini, A.A., Weymouth, T.E., Jain, R.C., Using dynamic programming for solving variational problems in vision, IEEE Trans. Pattern Anal. Mach. Intell., 12(9) (1990) 855–867. [2] Bally, V., Pagès, G., Printemps, J., A quantization tree method for pricing and hedging multi-dimensional American options, Math. Finance, 15(1), (2005) 119–168. [3] Bertsimas, D., Lo, A., Optimal control of execution costs, J. Financ. Mark., 1, (1998) 1–50. [4] Borsa Italiana website, http://www.borsaitaliana.it/derivati/derivati/derivati.htm [5] Busseti, E., Lillo, F., Calibration of optimal execution of financial transactions in the presence of transient market impactJ. Stat. Mech. (2012). [6] Forsyth, P.A., A Hamilton-Jacobi-Bellman approach to optimal trade execution, Appl. Numer. Math., 61(2) (2011) 241–265. [7] Guilbaud, F., Mnif, M., Pham, H., Numerical methods for an optimal order execution problem, preprint (2010). [8] He, H., Mamaysky, H., Dynamic trading policies with price impact, J. Econ. Dyn. Control 29, (2005) 891–930. [9] Hull, J.C., Options, Futures and Other Derivatives (6th edition), Prentice Hall: New Jersey (2006). [10] Kharroubi, I., Pham, H., Optimal Portfolio Liquidation with Execution Cost and Risk, SIAM J. Financial Math, 1(1) (2010) 897–931. [11] Ly Vath, V., Mnif, M., Pham,H., A model of optimal portfolio selection under liquidity risk and price impact, Finance Stoch., 11, (2007) 51–90. [12] Ma, L., Yong, J., Dynamic Programming for Multidimensional Stochastic Control Problems, Acta Math. Sin., 15(4) (1999) 485–506. [13] Obizhaeva, A.A., Wang, J., Optimal trading strategy and supply/demand dynamics, J. Financ. Mark., 16(1), (2013) 1–32. [14] Pham, H., Tankov, P., A model of optimal consumption under liquidity risk with random trading times, Math. Finance, 18, (2008) 613–627. [15] Rogers, L.C.G., Singh, S., The cost of illiquidity and its effects on hedging, Math. Finance 20(4) (2010) 597–615. Variational Problems & Equilibria 2 311 Spillovers effect on the profitability and the social welfare of the joint research lab in Cournot oligopoly with symmetric firms Razika Sait∗ Research Unit LaMOS, Bejaia University, Algeria, [email protected] Abdelhakim Hammoudi National Institute of Agronomic Research-Food and Social Sciences (INRA-ALISS), France, [email protected] Mohammed Said Radjef Research Unit LaMOS, Bejaia University, Algeria, [email protected] Abstract. This study analyzes the impact of technological spillovers and the number of cooperating firms on the formation and the profitability of the R&D cooperation and its effect on the social welfare. This cooperation can arise endogenously in Cournot oligopoly between N symmetric firms. So, a three-stage oligopoly game-theoretic model is developed. At the first, the N firms decide simultaneously whether or not to conduct cost reducing R&D activities in a joint research lab retaining a profit function as d’Aspremont and Jacquemin [1]. At the second one, the joint research lab and the non-member firms engage in a non-cooperative game where they decide simultaneously their innovation efforts which are subject to spillovers. At the third stage, the N firms remain non-cooperative rivals in the product market and compete in quantities. Our results show that the incentive to undertake research in a joint research lab depends on the R&D cooperation size, on the spillovers levels and on the type of research conducted by the cooperating firms. Our contribution allows us to reach an important results in that, for the intermediate values of spillovers, the profitability is significant larger than that of competition strategy only for medium-sized R&D cooperation but it is not socially beneficial. References [1] D’Aspremont, C and Jacquemin, A., Cooperative and noncooperative R&D in duopoly with spillovers, The American Economic Review 5 (1988), 1133-1137. Variational Problems & Equilibria 2 312 Gap functions and descent methods for quasi-equilibria Giancarlo Bigi∗ Dipartimento di Informatica, Università di Pisa, Italia, gian[email protected] Mauro Passacantando Dipartimento di Informatica, Università di Pisa, Italia, [email protected] Abstract. In this talk we focus on the quasi-equilibrium problem find x∗ ∈ C(x∗ ) s.t. f (x∗ , y) ≥ 0, ∀y ∈ C(x∗ ) (QEP ) where the bifunction f : Rn × Rn → R satisfies the equilibrium condition f (x, x) = 0 for any x ∈ Rn and the constraints are given by a set-valued map C : Rn ⇒ Rn such that the set C(x) is closed and convex for any x ∈ Rn . QEPs are modelled upon quasi-variational inequalities (shortly QVIs). Also generalized Nash equilibrium problems (shortly GNEPs) can be reformulated through (QEP) with the NikaidoIsoda aggregate bifunction. It is also worth noting that (QEP) is a natural generalization of the so-called abstract equilibrium problem (shortly EP), i.e., the case in which the set-valued map C is constant. As EP subsumes optimization, multiobjective optimization, variational inequalities, fixed point and complementarity problems, Nash equilibria in noncooperative games and inverse optimization in a unique mathematical model, further “quasi” type models could be analysed through the QEP format beyond QVIs and GNEPs. Unlikely QVI and GNEP, the QEP format did not receive much attention. The goal of the paper is to reformulate (QEP) as an optimization problem through a suitable gap function and develop an ad-hoc descent algorithm, supposing that the set-valued map C can be described by constraining bifunctions. Gap functions have been originally conceived for variational inequalities and later extended to EPs, QVIs, jointly convex GNEPs via the Nikaido-Isoda bifunction and generic GNEPs via QVI reformulations Though descent type methods based on gap functions have been extensively developed for EPs, the analysis of gap functions for QVIs and GNEPs is focused on smoothness properties and error bounds while no descent algorithm is developed. Indeed, the reformulation of (QEP) as an optimization problem brings some difficult issues in devising descent methods which are not met in the EP case: the gap function is not necessarily differentiable even though the equilibrium and the constraining bifunctions are differentiable; the feasible region is given by the fixed points of the set-valued constraining map C and is therefore more difficult to handle; the so-called stationarity property, which guarantees all the stationary points of the gap function to be actually global minimizers and therefore solutions of (QEP), requires monotonicity assumptions both on the equilibrium and constraining bifunctions. These issues are dealt with in the talk. After the gap function has been introduced and the reformulation of (QEP) as an optimization problem shown, the smoothness properties of the gap function are analysed; in particular, an upper estimate of its Clarke directional derivative is given, which provides a key tool in devising the descent method. Furthermore, classes of constraints which allow guaranteeing the stationarity property are identified. The convergence of the descent method is proved under standard assumptions, and finally error bounds are given, which guarantee that the sequence generated by the algorithm is bounded. Variational Problems & Equilibria 2 313 Cutting surface methods for equilibria Mauro Passacantando∗ Department of Computer Science, University of Pisa, Italy, [email protected] Giancarlo Bigi Giandomenico Mastroeni Department of Computer Science, University of Pisa, Italy, [email protected], [email protected] Abstract. We present cutting type methods for solving an abstract Equilibrium Problem via the so-called Minty gap function, relying on lower convex approximations. These methods consist in solving a sequence of convex optimization problems, whose feasible region is refined by nonlinear convex cuts at each iteration. Global convergence is proved under suitable monotonicity or concavity assumptions. Introduction In this talk we focus on the abstract Equilibrium Problem: find x∗ ∈ C s.t. f (x∗ , y) ≥ 0, ∀ y ∈ C, (EP) where C is a closed convex subset of Rn and the bifunction f : Rn × Rn → R is such that f (x, x) = 0 and f (x, ·) is convex for any x ∈ Rn . The abstract equilibrium problem (EP) provides a rather general setting which includes several mathematical models such as optimization, variational inequalities, fixed point and complementarity problems, Nash equilibria in noncooperative games (see [1]). It is well known that if the bifunction f is continuous and pseudomonotone on C, i.e., the following implication f (x, y) ≥ 0 =⇒ f (y, x) ≤ 0 holds for any x, y ∈ C, then (EP) is equivalent to solve the so-called Minty Equilibrium problem: find x∗ ∈ C s.t. f (y, x∗ ) ≤ 0, ∀ y ∈ C. (MEP) The natural merit function [4] associated with (MEP), defined by ψ(x) := sup f (y, x), y∈C allows us to reformulate (MEP) as a convex optimization problem. In fact, ψ is convex since it is the supremum of convex functions; moreover, it is non-negative on C and x∗ solves (MEP) if and only if x∗ ∈ C and ψ(x∗ ) = 0. Therefore, (MEP) is equivalent to the following convex optimization problem: min ψ(x) s.t. x ∈ C. Though ψ is a convex function, it can be difficult to evaluate since this requires to solve nonconvex optimization problems. Variational Problems & Equilibria 2 314 Main results In this talk we present cutting type methods for solving EP via the Minty gap function ψ, relying on lower convex approximations which are easier to compute. These methods actually amount to solving a sequence of convex optimization problems, whose feasible region is refined by nonlinear convex cuts at each iteration. Exploiting duality results in convex optimization, the global convergence of these methods is proved under monotonicity assumptions on f or concavity assumptions on f (·, y). The results of preliminary numerical tests on Nash equilibrium problems with quadratic payoffs, other linear EPs and variational inequalities are also reported. References [1] G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Existence and solution methods for equilibria, European Journal of Operational Research 227 (2013), pp. 1–11. [2] G. Mastroeni, Gap functions for equilibrium problems. Journal of Global Optimization 27 (2003), pp. 411–426. [3] S. Nguyen, C. Dupuis, An efficient method for computing traffic equilibria in networks with asymmetric transportation costs, Transportation Science 18 (1984), pp. 185–202. [4] M. Pappalardo, G. Mastroeni, M. Passacantando, Merit functions: a bridge between optimization and equilibria, 4OR, vol. 12 (2014), pp. 1–33. AIRO PRIZES Thursday 10, 14:15-15:45 Sala Seminari Ovest 315 Best Application-Oriented Paper 316 Best Application-Oriented Paper Optimal deployment of a cruise fleet Gianni Di Pillo∗ Marcello Fabiano Stefano Lucidi Massimo Roma Un sistema per la pianificazione degli interventi sul territorio per la gestione delle utenze della rete gas nell’area di Genova Simona Angelini Monia Gazzano∗ Davide Anghinolfi Massimo Paolucci Fabrizio Zeba An Integrated System for Production Scheduling in Steelmaking and Casting Plants Maria Pia Fanti Walter Ukovich∗ Giuliana Rotunno Gabriella Stecco Stefano Mininel OR for the society 317 OR for the society A derivative-free approach for a simulation-based optimization problem in healthcare Stefano Lucidi Massimo Roma∗ Massimo Maurici Luca Paulon Francesco Rinaldi A multi-depot dial-a-ride problem with heterogeneous vehicles and compatibility constraints in healthcare Francesco Papalini Paolo Detti∗ Garazi Zabalo Manrique de Lara Transportation for Seniors: Planning a Senior Shuttle Service Using GIS and OR Nigel Waters Monica Gentili∗ Muhammad Tubbsum Steve McClure Dennis Nicholas Author index Śliwiński, Tomasz, 204 Adamo, Tommaso, 261 Addis, Bernardetta, 48, 50, 52, 74, 226 Agnetis, Alessandro, 62 Agnihothri, Saligrama R., 79 Allevi, Elisabetta, 36 Alvarez, Aldair, 286 Amaldi, Edoardo, 230 Ambrosino, Daniela, 95, 102, 104 Andreatta, Giovanni, 266 Angelelli, Enrico, 43 Angelini, Simona, 316 Anghinolfi, Davide, 316 Arbib, Claudio, 243, 272 Archetti, Claudia, 43 Ardagna, Danilo, 48, 304 Aringhieri, Roberto, 65, 70, 72, 226 Arrigo, Francesca, 140 Astorino, Annabella, 154, 156, 158 Baş, Seda, 77 Bagattini, Francesco, 81 Baglama, James, 143 Baldi, Mauro Maria, 178 Barabino, Benedetto, 149 Baralla, Gavina, 200 Barbagallo, Annamaria, 35 Battarra, Maria, 196 Benzi, Michele, 140 Bernocchi, Luca, 104 Berto, Alessandra, 282 Bertocchi, Marida, 182 Bianco, Lucio, 25 Bigi, Giancarlo, 312, 313 Boccia, Maurizio, 288 Bonollo, Michele, 309 Bravi, Luca, 81, 129, 135 Bruglieri, Maurizio, 290 Buccioli, Matteo, 68 Buchheim, Christoph, 126 Bulgarini, Niccolò, 81, 136 Caballini, Claudia, 95 Cacchiani, Valentina, 68, 147 Caliciotti, Andrea, 130 Capone, Antonio, 48 Cappanera, Paola, 79, 87, 151 Caprara, Alberto, 228 Carello, Giuliana, 48, 50, 52, 74, 77 Carlo, Meloni, 213 Carosi, Laura, 29, 33 Carosi, Samuela, 189 Carotenuto, Pasquale, 25 Carrabs, Francesco, 285 Cerrone, Carmine, 263, 285 Cerulli, Raffaele, 263, 285 Cesarone, Francesco, 210 Cheraitia, Meryem, 241 Chiarello, Antonino, 99, 154 Chiaverini, Simone, 295 Ciancio, Claudio, 41 Ciavotta, Michele, 304 Colorni, Alberto, 290 Conforti, Domenico, 64 Coniglio, Stefano, 230 Consigli, Giorgio, 176 Corman, Francesco, 280 Crainic, Teodor Gabriel, 288 Cristofari, Andrea, 116 D’Alessandro, Pietro, 158 D’Ambrosio, Claudia, 11 D’Andreagiovanni, Fabio, 90 D’Ariano, Andrea, 280 D’Inverno, Giovanna, 29 Davò, Federica, 161 De Francesco, Carla, 266 318 Author Index De Giovanni, Luigi, 266 De Leone, Renato, 216, 254, 258 De Santis, Giulia, 258 De Santis, Marianna, 116, 126 Dell’Amico, Mauro, 273 Della Croce, Federico, 269, 298 Dellepiane, Umberto, 117 Dellino, Gabriella, 213 Delorme, Maxence, 270 Detti, Paolo, 83, 317 Di Caprio, Debora, 256 Di Francesco, Massimo, 97, 106, 149, 196, 198, 200 Di Persio, Luca, 309 Di Pillo, Gianni, 121, 316 Di Puglia Pugliese, Luigi, 109, 111, 113 Diaz-Maroto Llorente, Nuria, 97 Dragan, Irinel, 221 Duma, Davide, 70, 72 El Ouardighi, Foaud, 306 Erdoğan, Güneş, 23 Esposito Amideo, Annunziata, 236 319 Gentile, Claudio, 245 Gentili, Monica, 317 Gerstl, Enrique, 297 Ghezelsoflu, Ali, 198 Ghiani, Gianpaolo, 180, 261 Giancamilli, Ilaria, 254 Gianniti, Eugenio, 304 Giglio, Davide, 295 Giordani, Stefano, 25 Girardi, Leopoldo, 186, 191 Gliozzi, Stefano, 282 Gnecco, Giorgio, 218, 306 Golden, Bruce, 263 Gordini, Angelo, 249 Gorgone, Enrico, 91 Gori, Marco, 218 Gouveia, Luis Eduardo Neves, 7 Graeb, Helmut, 119 Grandi, Duccio, 187 Grieco, Antonio, 261 Groccia, Maria Carmela, 64 Grosso, Andrea, 226 Gualandi, Stefano, 189 Guarnieri, Adriano, 249 Guerriero, Emanuela, 180, 261 Guerriero, Francesca, 109, 111, 113 Guerrini, Andrea, 33 Guessoum, Fatima, 241 Guido, Rosita, 64 Gusso, Riccardo, 132 Gutiérrez-Jarpa, Gabriel, 232 Fabiano, Marcello, 121, 316 Facchinei, Francisco, 138 Fadda, Gianfranco, 200 Fanti, Maria Pia, 316 Fasano, Giovanni, 130, 132 Fasino, Dario, 141 Felici, Giovanni, 90 Fenu, Caterina, 143 Fersini, Elisabetta, 239, 252 Habal, Husni, 119 Filippi, Carlo, 43 Haddadi, Salim, 241 Flores Bazán, Fabian, 303 Hammoudi, Abdelhakim, 311 Fortz, Bernard, 91 Heinicke, Franziska, 178 Frangioni, Antonio, 162, 191, 198, 228, Hossaindazeh, Mohammad Mehdi, 173 245 Hosteins, Pierre, 226 Fuduli, Antonio, 156 Hungerford, James, 228 Gaeta, Matteo, 234 Gaivoronski, Alexei, 106 Galli, Laura, 191, 264 Galligari, Alessandro, 81, 136 Garraffa, Michele, 298 Gaudioso, Manlio, 99, 113, 154, 158, 285 Gazzano, Monia, 316 Iori, Manuel, 270, 273 Khalaf, Walaa, 158 Kidd, Martin P., 147 Kizilkale, Can, 224 Kogan, Konstantin, 306 Koster, Arie, 230 Author Index Kramer, Raphael, 273 Kroon, Leo G., 147 320 Mauro, Paolo, 35 McClure, Steve, 317 Melacci, Stefano, 218 Meloni, Luca, 200 Mencarelli, Luca, 92 Menegazzo, Francesca, 57 Messina, Enza, 239, 252 Miglionico, Giovanna, 113 Millán, Reinier Dı́az, 301 Minciardi, Riccardo, 295 Mininel, Stefano, 316 Minnetti, Valentina, 216 Moccia, Luigi, 99, 232 Monaco, Maria Flavia, 99 Mondin, Giorgia, 57 Moneta, Diana, 164 Morabito, Reinaldo, 286 Mosheiov, Gur, 297 Munari, Pedro, 286 Musmanno, Roberto, 41 Lacalandra, Fabrizio, 90 Laganà, Demetrio, 41 Lai, Michela, 196 Lampariello, Lorenzo, 138 Landa, Paolo, 59, 65 Lanzarone, Ettore, 77 Laporte, Gilbert, 232 Latorre, Vittorio, 119 Laudadio, Teresa, 213 Laureri, Federica, 295 Lera, Daniela, 166 Letchford, Adam, 264 Lia, Federico, 290 Liberti, Leo, 11, 92 Liguori, Arturo, 57 Lleo, Sebastien, 174 Lodi, Andrea, 68 Lombardi, Fabio, 249 Lucidi, Stefano, 55, 116, 119, 121, 125, Napolitano, Jacopo, 168 316, 317 Nicholas, Dennis, 317 Nonato, Maddalena, 79, 87, 151, 170 Maccagnola, Daniele, 252 Macrina, Giusy, 111 Ocak, Zeynep, 77 Oggioni, Giorgia, 36 Maggioni, Francesca, 182 Ogryczak, Wlodzimierz, 204 Malucelli, Federico, 189 Oliva, Immacolata, 309 Mancini, Simona, 200 Mancuso, Fabrizio, 33 Pacciarelli, Dario, 280 Manni, Emanuele, 261 Palagi, Laura, 117, 126 Manno, Andrea, 87 Paolucci, Massimo, 102, 295, 316 Manzionna, Stefano, 254 Papadimitriou, Dimitri, 91 Marcikic, Aleksandra, 85, 208 Papalini, Francesco, 83, 317 Mari, Renato, 213 Parriani, Tiziano, 228, 249 Marianov, Vladimir, 232 Passacantando, Mauro, 304, 312, 313 Marinaki, Magdalene, 247 Paulon, Luca, 55, 317 Marinakis, Yannis, 247 Peano, Andrea, 170 Marinelli, Fabrizio, 272 Peirano, Lorenzo, 95 Marlière, Grégory, 278 Pellegrini, Paola, 278 Marques, Rui Cunha, 31 Pesenti, Raffaele, 132 Martello, Silvano, 270 Picca Nicolino, Federica, 81 Martin, David, 143 Piccialli, Veronica, 135 Mastroeni, Giandomenico, 303, 313 Piccolo, Antonio, 154 Mastronardi, Nicola, 213 Pinar, Mustafa, 224 Mattia, Sara, 52 Pinto, Francisco Silva, 31 Maurici, Massimo, 55, 317 Author Index Pirisinu, Alessandro, 206 Pisciella, Paolo, 161, 164 Plum, Christian E. M., 45 Poirion, Pierre-Louis, 11 Poss, Michael, 111 Potra, Florian, 182 Pozzi, Matteo, 249 Pranzo, Marco, 62 Pratelli, Benedetta, 191 Przyluski, Michal, 204 Psychas, Iraklis - Dimitrios, 247 Røpke, Stefan, 45 Raco, Mattia, 50 Radjef, Mohammed Said, 311 Radovanov, Boris, 85, 208 Rarità, Luigi, 234 Ravagli, Letizia, 29 Ravaglia, Lorenzo, 249 Reichel, Lothar, 143 Renieri, Alessandra, 254 Renzi, Stefania, 125 Richtarik, Peter, 129 Rinaldi, Francesco, 55, 116, 317 Rodriguez, Giuseppe, 143 Rodriguez, Joaquin, 278 Roma, Massimo, 55, 121, 130, 316, 317 Romanin-Jacur, Giorgio, 57 Romano, Giulia, 33 Ronchi, Fabrizio, 187 Rotunno, Giuliana, 316 Sacone, Simona, 95 Sagastizábal, Claudia, 8 Sagratella, Simone, 123 Sahraoui, Youcef, 92 Sait, Razika, 311 Salieri, Fabrizio, 249 Samà, Marcella, 280 Sammarra, Marcello, 99 Sanguineti, Marcello, 218, 306 Santini, Alberto, 45 Santos Arteaga, Francisco J., 256 Santos, José Luis, 109 Sbrilli, Simone, 62 Scatamacchia, Rosario, 226, 269 Schneider, Michael, 27 321 Schoen, Fabio, 81 Schwahn, Fabian, 27 Sciandrone, Marco, 135, 136 Sciomachen, Anna, 102 Scozzari, Andrea, 210 Scrimali, laura, 38 Scutari, Gesualdo, 138 Sechi, Giovanni M., 168 Serafini, Paolo, 266 Sergeyev, Yaroslav, 166 Servilio, Mara, 243 Sforza, Antonio, 236, 288 Sgalambro, Antonino, 194 Shang, Lei, 298 Siface, Dario, 161 Silvano, Vergura, 213 Simroth, Axel, 178 Siri, Silvia, 104 Sonnessa, Michele, 59 Stecca, Giuseppe, 25 Stecco, Gabriella, 316 Sterle, Claudio, 236, 288 T’kindt, Vincent, 298 Tadei, Roberto, 178 Tanfani, Elena, 59, 65, 74 Tardella, Fabio, 210 Tavana, Madjid, 256 Tavlaridis-Gyparakis, Kostas, 162 Testi, Angela, 59 Tieves, Martin, 230 Tomasiello, Stefania, 234 Toth, Paolo, 147 Toubaline, Sonia, 11 Tresoldi, Emanuele, 189 Tubbsum, Muhammad, 317 Tubertini, Paolo, 68 Tudisco, Francesco, 141 Ukovich, Walter, 316 Vaccaro, Alfredo, 234 Van Ackooij, Wim, 162 Vannucci, Stefano, 222 Veelenturf, Lucas P., 147 Ventura, Paolo, 243 Venturi, Beatrice, 206 Author Index Vespucci, Maria Teresa, 161, 164 Viganò, Giacomo, 164 Vigo, Daniele, 27, 249 Vindigni, Michele, 43 Visintin, Filippo, 79 Vitali, Sebastiano, 175 Vocaturo, Francesca, 41 Waters, Nigel, 317 Willem, Frederique-Anne, 258 Yalçındağ, Semih, 77 Yilmaz, Omer Faruk, 293 Zabalo Manrique de Lara, Garazi, 83, 317 Zanda, Simone, 97, 200 Zeba, Fabrizio, 316 Zuddas, Paola, 97, 106, 168, 196, 198, 200 322