AIRO 2015 BOOK OF ABSTRACTS

Transcription

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)
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)
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
(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) . . . . . . . . . . . . . . . . . . . .
LASSIE (Frangioni) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maritime Logistics 1 (Ambrosino, Monaco, Zuddas) . . . . . . . . . . . .
Maritime Logistics 2 (Ambrosino, Monaco, Zuddas) . . . . . . . . . . . .
Network Optimization (Guerriero) . . . . . . . . . . . . . . . . . . . . .
Nonlinear Programming 1 (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
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
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.
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
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
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.
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
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]
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 ).
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.
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.
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
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.
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
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.
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.
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.
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.
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.
33
Identifying cost and efficiency drivers in WWTPs
through a DEA model with undesirable outputs
Laura Carosi∗
Andrea Guerrini
Department of Management, University of Verona, Italy, [email protected]
Fabrizio Mancuso
Ingegnerie Toscane srl, Italia, [email protected]
Giulia Romano
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
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.
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
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.
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.
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
[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
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.
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},
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.
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.
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∗
[email protected]
Bernardetta Addis
[email protected]
Mattia Raco
[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
[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
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
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.
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
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.
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].
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
Tuesday 8, 9:00-10:30
Sala Seminari Ovest
61
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.
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.
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.
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.
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
Tuesday 8, 11:00-13:00
Sala Seminari Ovest
67
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).
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
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
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.
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.
72
A mixed offline and online approach to manage
elective and non-elective patients
Davide Duma∗
Roberto Aringhieri
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.
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.
74
Medium term operating room scheduling with
patient release dates
Elena Tanfani∗
Dipartimento di Economia, Università di Genova, Italia,
[email protected]
Giuliana Carello
[email protected]
Bernardetta Addis
[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
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
Wednesday 9, 9:00-10:30
Sala Seminari Ovest
76
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
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.
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
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.
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
Thursday 10, 9:00-10:30
Sala Seminari Ovest
82
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
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.
85
Forecasting Demand in Health Care
Organizations
Aleksandra Marcikic∗
Faculty of Economics Subotica, University of Novi Sad, Serbia, [email protected]
Boris Radovanov
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.
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.
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
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.
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,
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.
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
[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,
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.
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
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
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
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,
Anna Sciomachen
[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
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.
104
The train load planning problem with multiple
cranes in seaport terminals:
a multiobjective optimization approach
Silvia Siri∗
Daniela Ambrosino
[email protected]
Luca Bernocchi
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,
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.
106
Optimization of simulation parameters for
stochastic empty container repositioning
Paola Zuddas∗
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
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∗
[email protected]
Francesca Guerriero
[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, . . . , `}.
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.
111
A robust version of the green vehicle routing
problem with uncertain waiting time at recharge
station
Giusy Macrina∗
[email protected]
Luigi Di Puglia Pugliese
Francesca Guerriero
[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)
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.
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
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,
Marianna De Santis
Fakultät für Mathematik, TU Dortmund, Germany, [email protected]
Stefano Lucidi
Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Sapienza Università di Roma,
Francesco Rinaldi
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.
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
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.
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
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,
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
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.
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.
(invited by Palagi)
Tuesday 8, 15:30-17:00
Sala Gerace
122
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
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.
125
A Linesearch Derivative-free Method for Bilevel
Optimization Problems
Stefania Renzi∗
Stefano Lucidi
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.
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
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.
(invited by Palagi)
Tuesday 8, 17:30-19:00
Sala Gerace
128
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).
130
Exploiting conjugacy for defining approximate
inverse preconditioning strategies, in large scale
unconstrained optimization
Andrea Caliciotti∗
Giovanni Fasano
Dipartimento di Management
Università Ca’Foscari Venezia, Italia, [email protected]
Massimo Roma
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
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.
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
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).
(invited by Palagi)
Wednesday 9, 9:00-10:30
Sala Gerace
134
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.
136
Decomposition algorithms for traffic assignment
problems
Alessandro Galligari∗
[email protected]
Marco Sciandrone
Niccolò Bulgarini
[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:
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).
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.
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
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).
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
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.
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
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.
149
Sustainable planning for service quality
in public transport
Massimo Di Francesco∗
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]),
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.
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
[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
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
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.
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
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.
158
Edge detection via clustering
Manlio Gaudioso∗
Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Università della
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
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,
Dario Siface
RSE S.p.A., Italia, [email protected]
Paolo Pisciella
Maria Teresa Vespucci
Department of Management, Economics and Quantitative Methods, University of Bergamo,
[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.
162
A MILP Approach for the EdF Unit
Commitment Problem
Kostas Tavlaridis-Gyparakis∗
LIX, École Polytechnique, France, [email protected]
Antonio Frangioni
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
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
164
Optimal operation of power distribution networks
Maria Teresa Vespucci∗
Dipartimento di Scienze aziendali, economiche e metodi quantitativi, Università di Bergamo,
Paolo Pisciella
Dipartimento di Scienze aziendali, economiche e metodi quantitativi, Università di Bergamo,
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∗
Yaroslav Sergeyev
Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, University of
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 evaluate. 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 ,
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.
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,
Giovanni M. Sechi
Department of Civil and Environmental Engineering and Architecture, University of Cagliari,
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
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.
170
Path Relinking for a Team Scheduling Problem
Andrea Peano∗
Maddalena Nonato
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
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.
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.
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.
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.
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.
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
[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
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.
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.
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
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.
Why would a public transport company need
Operations Research?
Duccio Grandi∗
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.
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.
Delay Management in Public Transportation:
Service Regularity Issues and Crew Re-scheduling
Emanuele Tresoldi∗
[email protected]
Samuela Carosi
M.A.I.O.R., Italia, [email protected]
Stefano Gualandi
AntOptima, Switzerland, [email protected]
Federico Malucelli
[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.
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 Integrated Timetabling and Vehicle
Scheduling in Disruption Management
Benedetta Pratelli∗
Antonio Frangioni
Laura Galli
Dipartimento di Informatica, Università di Pisa, Italia,
[email protected]
[email protected]
Leopoldo Girardi
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.
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
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.
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.
198
Set-Covering formulation for routing of
heterogeneous trucks with container loads
Ali Ghezelsoflu∗
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
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
200
A GRASP Metaheuristic for the Vehicle Routing
Problem with Backhauls
Gianfranco Fadda∗
gianfranco− [email protected]
Gavina Baralla
Luca Meloni
Simone Zanda
Paola Zuddas
[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.
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
Thursday 10, 11:00-13:00
Sala Seminari Est
203
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
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.
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.
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.
208
Integration of Stock Markets in South East
European Countries
Boris Radovanov∗
Aleksandra Marcikic
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.
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.
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
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).
Tuesday 8, 17:30-19:00
Sala Riunioni Est
212
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,
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
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.
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.
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].
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.
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).
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∗
Bernardetta Addis
LORIA, Université de Lorraine, France, [email protected]
Roberto Aringhieri
Andrea Grosso
[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, [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
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
[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
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.
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.
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.
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.
Tuesday 8, 17:30-19:00
Sala Seminari Ovest
244
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,
Antonio Frangioni
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
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).
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
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).
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;
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
252
A relational clustering approach for community
detection in social networks
Elisabetta Fersini∗
[email protected]
Daniele Maccagnola
Enza Messina
[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.
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
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
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
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.
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.
258
ERGMs and centrality measures for the analysis
of mythological networks
Giulia De Santis∗
[email protected]
Renato De Leone
Frederique-Anne Willem
[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);
• ...
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.
Thursday 10, 11:00-13:00
Sala Riunioni Est
260
261
MIP neighborhood synthesis through semantic
feature extraction and automatic algorithm
configuration
Emanuele Manni∗
[email protected]
Tommaso Adamo
Gianpaolo Ghiani
Antonio Grieco
Emanuela Guerriero
[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
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.
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
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.
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
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.
266
Constrained domatic bipartition of a tree
Luigi De Giovanni∗
Giovanni Andreatta
Carla De Francesco
[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)
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
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
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
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.
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]
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.
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.
[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.
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
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∗
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∗
Michele Garraffa
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.
Wednesday 9, 11:00-13:00
Sala Gerace
300
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
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.
303
KKT optimality conditions with applications to
variational problems
Giandomenico Mastroeni∗
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 .
304
Game-Theoretic Approach to Joint Admission
Control and Capacity Allocation for MapReduce
Michele Ciavotta∗
[email protected]
Eugenio Gianniti
Danilo Ardagna
[email protected]
[email protected]
Mauro Passacantando
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
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.
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.
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.
Thursday 10, 11:00-13:00
Sala Gerace
308
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
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.
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.
312
Gap functions and descent methods for
quasi-equilibria
Giancarlo Bigi∗
Mauro Passacantando
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.
313
Cutting surface methods for equilibria
Mauro Passacantando∗
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.
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

AIRO 2015 BOOK OF ABSTRACTS

Transcription

Similar documents

starter service

Types Of Video For Gaming Operators

Global Optimization Datasheet

AMendoza-BCP Group - ARC Advisory Group

Chitarre Acustiche d`Italia

Quality Management System

WE GUARANTEE ThE hiGhEST AVAiLABiLiTY OF iT

Contents of this book

Fredrik Strauss: Automated Optimization of Online Advertising

Supercharching_Category_Management_with_the_P