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orbit
# 19
September 2012
50 års jubilæum
Tilbageblik • Tutorial on smart markets • Sustainable Course • Optimering for Transrail Sweden AB
leder
ORbit
medlemsblad for
Dansk Selskab
for Operationsanalyse
og
Svenska Operationsanalysföreningen
Leder
Den 29. Maj 2012 modtog jeg en mail fra DORS formanden Tor Justesen, hvori
and anmodede mig om at varetage posten som chefredaktør for ORbit. Jeg er
meget beæret over den tillid bestyrelsen herved viste mig, og tog men glæde
imod udfordringen.
Redaktion:
Ansv. Sanne Wøhlk (sw)
Tor Fog Justesen (tfj)
Jesper Larsen, DORS (jla)
Tomas Gustafsson, SOAF (tg)
ORbit er et fantastisk blad. Det bringer populærvidenskabelige artikler om OR
fra dan danske forskningsfront og fra erhvervslivet og bringer nyheder fra det
store udland. Den tidligere redaktion har gjort et kæmpe arbejde med at samle
materiale og få udgivet bladet. Det skal lyde en stor tak til Jesper Larsen, og i de
seneste år, Natalia Rezanova, for deres store indsats for ORbit.
DORS
DTU Management, bygn. 424
Danmarkt Tekniske Universitet
DK-2800 Kgs. Lyngby
Telefon: +45 4525 3385
Fax:
+45 4588 2673
E-mail: [email protected]
Næste deadline:
1. Marts 2013
Hvad er så min vision med ORbit? ORbit skal fortsat være et medie for formidling af OR relateret stof i et
letfordøjeligt sprog. Det er mit håb at de der arbejder med OR i hverdagen, hvad enten det drejer sig om
akademisk forskning eller gennem en mere praktisk tilgang i erhvervslivet, vil genfinde lysten til både at
læse og bidrage til ORbit.
Var det det? Nej. ORbit er et medlemsblad. Derfor er det af yderste vigtighed, at vi kan bruge bladet til at
annoncere både foreningernes egne arrangementer og andre arrangementer, som er af generel interesse
for foreningens medlemmer. For at sikre dette, er det målet at ORbit fremadrettet skal udkomme: fast midt i
foråret og midt i efteråret. Så krydser jeg fingre for at skrivelysten bland medlemmerne er tilstrækkelig stor
Tryk:
Print Provider Aps
til at vi alle kan få et lækkert blad at læse.
Oplag: 370
ISSN 1601-8893
semplar bugner af spændende artikler. Oli B.G. Madsen fortøæller om sin introduktin til operationsanalyse
Det første blad er færdigt. ORbits klassiske røde design er udskiftet med et grønt, og dette jubilæumseki 1964 og John F. Raffensperger giver en introduktion til smart markets. I bladet findes også to DORSpris
artikler og meget meget mere. God læselyst!
Forside: colourbox.com
Sanne Wøhlk
Ansvarshavende redaktør
Aktuelt om DORS
Medlemsskab
Kontingentsatser for 2012
Personlige medlemmer
(incl. ph.d.-studerende): 270 kr./år
Studerende: 60 kr./år
Firmamedlemmer:
3200 kr./år
Institutmedlemmer:
1800 kr./år
2
ORbit 19
Sekretariat
DORS
DTU Management
Bygning 424
Danmarks Tekniske Universitet,
2800 Kgs. Lyngby
e-mail: [email protected]
Internet: www.dorsnet.dk
indhold
Bästa SOAF-medlemmar!
Indhold
Redaktøren har ordet
2
SOAF 2012
3
gått så känns det som att aktivitetsnivån i SOAF för närvarande är hög, och
Tillykke fra DORS formanden
4
att väldigt mycket har hänt! Under våren har SOAF bland annat startat två nya
Dantzigs besøg 1976
5
Those were the days
6
Beslutningsstøtte ved containerfragt
9
Hösten närmar sig så smått, och med den börjar aktiviteterna i SOAF komma
igång igen efter sommaruppehållet. När jag tänker tillbaka på halvåret som har
intressegrupper: Underhållsgruppen, som leds av Michael Patriksson, hade
sitt första möte den 6 mars på Volvo Aero i Trollhättan, och hälso- och sjukvårdsgruppen (ledd av Marie Persson) träffades den 16:e januari på Karolinska
Institutet i Stockholm. SOAF har också haft årsmöte den 13 mars i Karlskrona,
då undertecknad valdes till ordförande efter Martin Joborn, som glädjande nog
stannar kvar i styrelsen tills vidare. Glädjande var också att Ximena Karlsson på MTR valdes in som ny
SOAF Exjobbspris
12
DORS prisen
13
A tutorial on smart markets
14
Om AOO 2012
20
A sustainable course?
22
Optimering for Transrail
24
SOAF har också under sommaren varit representerat vid EURO-konferensen i Vilnius. I dag samlas mycket
EURO summer institute
26
stora datamängder in i närmast alla branscher och sektorer, och detta innebär stora möjligheter, men också
Strategic balanced scorecard
simulation
29
identifiering av mönster i datamängder, samt att rekommendera beslut utifrån dessa. Styrelsen kommer att
Om optimal tildeling af KBUforløb
34
fortsätta diskutera denna trend under hösten. SOAF har också uppdaterat webbsidan under året, och vi
High school planning
38
adjungerad styrelsemedlem. På årsmötet föreslogs det också att vi ska fortsätta fokusera på intressegrupper, OR-utbildning samt eventuellt öka vårt internationella arbete genom EURO och IFORS. Vinnare av
exjobbspriset 2011 är Ferran Mach med examensarbetet »Optimization analysis of the number and location
of holding control stops - A simulation-based evaluation of line number 1, Stockholm”.
Den 9 maj anordnade SOAF, tillsammans med SICS och Trafikverket, en järnvägskonferens på SICS i
Kista. Arrangemanget var mycket uppskattat av deltagarna och kommer få en efterföljare under hösten!
utmaningar, för operationsanalysen. Då jag i sommar också kunnat besöka AAAI, en av de stora konferenserna inom AI-området, så är det tydligt att det också där finns ett stort intresse för ”analytics”, d.v.s.
kommer under hösten gå igenom och uppdatera informationsmaterialet på denna.
Jag ser fram emot en aktiv höst med mycket nya och spännande aktiviteter!
Markus Bohlin
Ordförande SOAF
Aktuellt om SOAF
Företagsmedlemmar:
• Blekinge Tekniska Högskola
• Cambio Healthcare Systems AB
• FOI
• Green Cargo
• Högskolan i Skövde
• Industrial Optimizers
• Jeppesen
• Kungliga Tekniska Högskolan
• Linköpings Universitet
• Optimal Solutions AB
• Preference
• Riiplan
• Scania
• SICS
• SJ
• Systecon
• Trafikverket
• Transrail Sweden AB
• Uppsala Universitet
• Vattenfall
• Vectura
• Volvo Aero
SOAF Medlemsavgifter 2012:
• Individuella medlemmar (inkl. ORbit):
150 kr
• Juniormedlem (exkl. ORbit): 75 kr
• Akademiska institutioner
(3 ORbit): 1500 kr
• Företag med 1-5 intressenter
(2 ORbit): 1500 kr
• Företag med 6-10 intressenter
(4 ORbit): 3000 kr
• Företag med fler än 10 intressenter
(6 ORbit): 4500 kr
Betala in på postgiro: 19 94 48-2
(Svenska Operationsanalysföreningen)
e-mail: [email protected]
Internet: www.soaf.se
ORbit 19
3
Hip hip hurra
Dansk Selskab for Operationsanalyse blev stiftet
21. august 1962 og kan således i år fejre sit 50
års jubilæum. 
Ifølge vedtægterne er selskabets formål er at fremme den operationsanalytiske forskning og anvendelse af operationsanalytiske metoder i Danmark samt at formidle kontakten med tilsvarende foreninger og organisationer, ved i videst mulige omfang
at etablere forbindelse med andre landes operationsanalytiske forskere, organisationer og sammenslutninger, og ved at foranstalte medlemmernes deltagelse i møder, kongresser, seminarer og lignende om operationsanalytiske spørgsmål.
For at alle disse gode hensigter skal kunne føres ud i livet er der behov for hænder. Selskabet er drevet udelukkende ved
frivillig arbejdskraft, og foreningens har kun kunnet overleve i alle disse år, takket være utallige personers lyst og overskud til
at bruge utallige timer og aftener på at arrangere seminarer, workshops, virksomhedsbesøg, indkræve kontingent, opdatere
medlemsdatabase, opsætte og tilrette ORbit, osv. Der skal således lyde en stor TAK til alle tidligere bestyrelser og øvrige personer der alle har arbejdet aktivt for selskabet, hvad enten det har drejet sig om at hjælpe til med det praktiske i forbindelse
med et arrangement, eller det har drejet sig om en omlægning af klubbens medlemsdatabase og -håndtering.
DORS er til for sine medlemmer, men omvendt ville DORS ikke eksistere, hvis ikke det netop var for medlemmerne; jeres
støtte af foreningen er uvurderlig! Uden at jeg kan sige hvordan fremtiden bliver, så vil jeg love, at både den nuværende
bestyrelse og de kommende bestyrelser vil gør alt hvad de kan for at sikre, at DORS forbliver kilden til et godt netværk for
operationsanalytikere.
Med disse ord vil jeg med stolthed ønske DORS et kæmpe stort TILLYKKE med de 50 år. Og så vil jeg glæde mig til de næste
mange år, som forhåbentlig vil byde på endnu flere spændende OR-relaterede oplevelser!
Tor Fog Justesen
Formand for DORS
DORS’ firma- og institutmedlemmer
Firmamedlemmer
Institutmedlemmer
• Afdeling for Anvendt Matatematik og Statis- • Datalogisk Institut, Københavns Universitet
• Institut for Virksomhedsledelse og
tik, Københavns Universitet
• Afdelingen for Operationsanalyse, Aarhus
Universitet
• Center for Research in the Foundations of
Electronic Markets
• CORAL, Handelshøjskolen, Aarhus Universitet
Økonomi, SDU
• Institut for Planlægning, Innovation og
• Ange Optimization
• A.P. Møller – Mærsk
• DONG Energy
• DSB
Ledelse (DTU Management), Danmarks
• DSB S-tog
Tekniske Universitet
• Jeppesen
• Institut for Transport (DTU Transport),
Danmarks Tekniske Universitet
• Københavns Lufthavne
• MOSEK
• Novo Nordisk (CMC Clinical Supplies)
• Rapidis
• Transvision
Hermes Traffic Intelligence
4
ORbit 19
Denne artikel af William Cauchi blev først publiceret i Politiken 8. September 1976. Artiklen er gentrykt her med tilladelse. Redaktionen
takker Politiken for deres velvillighed.
ORbit 19
5
artikel
Af Oli B.G. Madsen
Those were the days
- erindringer fra min OR debut
I anledning af DORS’s 50 års jubilæum har
redaktionen bedt professor Oli B.G. Madsen fra
DTU dele mine minder fra operationasnalysens
spæde ungdom i Danmark. Oli fortæller:
Efter studentereksamen begyndte jeg
på elektroingeniørstudiet på DTU den
1. september 1961. Det foregik i DTU’s
gamle lokaler på Sølvtorvet, men allerede i 1962 flyttede undervisningen til
Lyngby, hvor vi sammen med årgang
1962 som de første tog de nye bygninger på den nuværende DTU campus i
brug. Elektroingeniørstudiet valgte jeg
nok primært, fordi det var et civilingeniørstudium, og fordi man lige havde
omlagt hele studiet og havde lagt mere
matematik ind i studieplanen. Endvidere
var det nye studium det sværeste at
komme ind på i 1961, og man skulle til
Lyngby til en helt ny campus. Der var en
masse pionerånd over det.
at have bestået første del sommeren
1964 ud af, at jeg kunne ”specialisere”
mig i anvendt matematik. Det forgik på
den måde, at man kunne læse nogle
ekstra fag mod at droppe ét elektrofag.
Det var lige noget for mig, selvom det
gav et ekstra pres på studiet. Blandt
de ekstra fag var numerisk analyse,
funktionalanalyse,
Markoffprocesser,
videregående matematisk statistik, videregående kompleks funktionsteori samt
operationsanalyse fag.
Mit første møde med operationsanalysen fandt sted foråret 1964 ved en elektronikforelæsning, hvor en af mine medstuderende sad og bladede i en bog.
Jeg spurgte om, hvad han læste. Det
viste sig at være
Studieplanerne var
dengang helt faste. »Det var min introduktion bogen »Introduction to Operations
Man skulle som til operationsanalyse, og
af
svagstrømsingeni- det påvirkede ret drastisk Research«
min
fremtid.«
Churchman, Ackoff
ørstuderende tage
og Arnoff. Jeg
nogle bestemte fag
i bestemte semestre, og eksaminerne bladede i bogen og fandt ud af, at det
skulle bestås samlet. Studiet bestod af så meget spændende ud. Det var min
en treårig første del opdelt i 3 årsprøver, introduktion til operationsanalyse, og det
en etårig forprøve og en halvanden årig påvirkede ret drastisk min fremtid.
slutprøve, i alt 5 ½ år. Hvis man ikke
bestod, så skulle alle fag hørende til den Operationsanalyseundervisningen forgik
på det i 1963 oprettede Institut for Matepågældende prøve tages om.
matisk Statistik og Operationsanalyse (i
Jeg fandt hurtigt ud af, at de elektrotek- 1967 opfandt min daværende kollega
niske fag ikke kunne bringe mit blod i Flemming Rasmussen og jeg det kortere
kog, men fysik og især matematik var navn IMSOR). Instituttet var udsprunget
inspirerende. Heldigvis fandt jeg efter af Laboratoriet for Anvendt Matematik,
6
ORbit 19
som var ledet af den navnkundige professor Richard Petersen (1894-1969),
i populær omtale kaldet lille p. IMSOR
var ledet af professor Arne Jensen
(1920-2008). Han var oprindelig aktuaruddannet i 1944 og var meget optaget af Erlangs køteoretiske arbejder;
noget ret naturligt da Arne Jensen var
knyttet til KTAS, Københavns Telefon
Aktie Selskab. Han var tillige knyttet til
Københavns Universitet (KU) i perioden
1946-1963 og blev i 1954 dr.phil. på en
afhandling om stokastiske processer.
Derudover var han tidligt opmærksom
på operationsanalysens betydning efter
et ophold i USA som Rockefeller stipendiat. Han underviste i 1950erne i lineær
programmering (LP) på KU, og blev i
1962 lektor på DTU, hvor han året efter
blev professor.
Foruden på DTU var der også andre
steder i Danmark, hvor man interesserede sig for OR. På Økonomisk Institut
på KU underviste professor Sven Danø
(1922-1998) i OR. Danø udgav bogen
»Linear Programming in Industry
– Theory and Applications« i 1960. Han
var en af de få danskere, som George
B. Dantzig kendte til. Jeg husker, at
Dantzig fortalte mig om Danø’s cases
fra en iscreme fabrik. På Århus Universitet underviste professor i driftsøkonomi
Svend Fredens i lineær programmering,
simulation, lagerteori og køteori. På
Handelshøjskolen i København (HHK)
artikel
underviste Erik Johnsen (f. 1928) i OR. forudsætninger formuleres som et LP tid. Til sidst kom der en høj stabel papir
Han havde opholdt sig på Princeton i problem, men med 4090 restriktioner og ud af printeren. Der var ingen, der rigtig
1955/56. Blev derefter knyttet til HHK 45056 variable. Da den mest effektive kendte til hvordan man anvendte LP/90,
og blev professor der i 1969. Han var LP løser i Danmark på dette tidspunkt, så det måtte man selv finde ud af.
tillige på det tidspunkt knyttet til IMSOR LP/90, kun kunne håndtere op til 1023
som ekstern underviser. Johnsen udgav restriktioner og én million koefficienter Det var en meget spændende periode,
i 1962 bogen »Introduktion til Opera- forskellig fra nul, måtte der foretages en hvor mange emner var i sin vorden.
tionsanalyse«. På den første danske aggregering af modellen. LP/90 kørte Dantzig og Ramser havde skrevet om
’The Truck Dispatching Problem’ i 1960.
it-virksomhed Regne»IBM 7090 havde en på Danmarks dengang
Dantzig og Wolfe havde præsenteret
centralen stiftet i 1955
ordlængde på 36 bit største computer en
og producent af de
IBM 7090 mainframe, deres dekompositionsmetode i 1960,
og et lager på 32 K.
første danske compusom stod på DTU, og men den blev ikke rigtig brugt til noget
Den kostede i 1960
tere DASK (1958) og
som var en gave fra IBM bortset fra Gilmore og Gomory’s løsning
som ny 2.898.000
GIER (1961) var der
i 1965. IBM 7090 havde af ’The Cutting Stock Problem’ i 1961US$ svarende til
også et blomstrende
en ordlængde på 36 bit 1963. Land og Doig ’opfandt’ branchomkring 21-22 milliand-bound (BB) i 1960. BB blev anvendt
OR-miljø. Der arbejoner danske kroner« og et lager på 32 K. Den
dede man bl.a. med
kostede i 1960 som ny på the travelling salesman problem
anvendelsen af OR
i produkti- 2.898.000 US$ svarende til omkring 21- (TSP) af Little, Murty, Sweeney og Karel
onsplanlægning. Jeg arbejdede 1965- 22 millioner danske kroner (til sammen- i 1963, men var kun kendt af nogle få
1967 på Regnecentralen for at tjene til ligning var startlønnen i 1960 for en civil- i Danmark. Clarke og Wright’s Savings
studierne og var meget betaget af den ingeniør omkring 1900 kr. om måneden algoritme var lige præsenteret i 1964 og
inspirerende atmosfære, der var der.
før skat). På dette tidspunkt fandtes der heltalsprogrammering var i sin spæde
ikke noget modelleringssprog som f.eks. vorden. Gomory’s cutting planes fra
Undervisningen i OR i 1964-1967 var GAMS eller OPL, så man hurtigt kunne 1950erne var kendt, men de fungerede i
noget anderledes end i dag. Vi havde generere
1964 endnu ikke
en
»Dantzig
og
Wolfe
havde
præf.eks. to semestre med dynamisk pro- for computeren
i praksis.
grammering (med Peter Mark Pruzan læsbar model. senteret deres dekompositionsProblemet med
som underviser). Derimod lærte vi om Jeg måtte selv metode i 1960, men den blev
at løse større
lineær programmering som en biting konstruere en ikke rigtig brugt til noget bortset
fra...«
LP
problemer
i faget investeringsplanlægning (med relativt generel
fascinerede mig
Inge Thygesen som underviser). Pruzan modelgenera(f. 1936) kom fra USA, hvor han var tor som en del af specialet. Program- så meget, at jeg valgte dette som emne
uddannet på Harvard, Princeton og koden blev programmeret i FORTRAN for mit ph.d.-studium, som startede lige
Case-Western
Reserve
University. IV, og program og grunddata blev efter, at jeg var blevet civilingeniør den
Thygesen (f. 1935) var ligesom Arne fodret ind i computeren via hulkort. Da 1. februar 1967. Det hed dengang et
Jensen aktuaruddannet og havde bl.a. beregningstiden (Blot for at finde opti- licentiatstudium. Emnet var dekomposiopholdt sig et år på Massachusetts mum eksklusive følsomhedsanalyser) tion og matematisk programmering.
Institute of Technology i Boston, USA.
Både Jensen, Pruzan og Thygesen var
således velbevandrede i den nyeste
udvikling inden for OR fra deres tid i
USA. Arne Jensen selv underviste ikke
meget.
Mit speciale, som jeg arbejdede på apriloktober 1966, handlede om et kombineret produktionsplanlægnings- og
transportproblem baseret på en virkelig
problemstilling i et korn- og foderstofkompagni. Problemet kunne under visse
var temmelig lang, kunne jeg kun køre
programmet i weekenderne, hvor det var
muligt at reservere computeren udelukkende til eget brug. Som eksempler på
beregningstider kan nævnes 140 min
for en 512x5632 LP model, 133 min
for 512x3616 og 99 min for 318x1640.
Man fodrede computeren med stabler
af hulkort. Så stod den og læste den
inverse basis frem og tilbage mellem to
magnettapebåndstationer. En runde for
hver iteration. Det var især det, der tog
Under ph.d.-studiet deltog jeg i en række
internationale konferencer, hvor der blev
opbygget gode kontakter til internationalt kendte forskere. Eksempelvis deltog
jeg sommeren 1969 i NATO Advanced
Institute on Integer and Nonlinear Programming. Konferencen foregik på Île
de Bendor, en lille ø, der ligger ud for
Frankrigs middelhavskyst mellem Marseilles og Toulon. Det var en rigtig øjenåbner med deltagelse af mange af tidens
store kanoner, f.eks. George Dantzig,
ORbit 19
7
artikel
Philip Wolfe, R. Fletcher, M.J.D. Powell, skulle her holde et foredrag om dekomMartin Beale, Richard Cottle, David position og var godt nervøs ved at have
Gale, Ralph Gomory, Fred Glover, Egon Dantzig og Wolfe som tilhørere. Det gik
Balas, G.W. Graves, Claude Berge, dog over al forventning.
samt lovende yngre forskere som John
Tomlin, Gautam Mitra og Hans Jürgen NATO’s research foundation var den
Zimmerman. Jeg fik lært nogle af dem gang sponsor for mange videnskabelige møder,
godt at kende
»DORS,
som
jo
var
blevet
stifsom
ikke
på de 12 dage,
tet
i
1962,
var
i
denne
periode
nødvendigvis
som konferenogså et aktivt forum for OR
havde noget
cen varede og
interesserede.«
med krig at
fik samtidig en
gøre.
Forfin opdatering af
uden ovennævnte møde på Île de
state-of-the-art inden for området.
Bendor, deltog jeg også i 1972 i et 12
Året efter deltog jeg i den internationale dages møde i Cambridge omhandlende
konference ’Optimisation Methods’, som ’Decomposition of Large Scale Systems’
foregik på Hotel Marienlyst i Helsingør. og i 1974 i et 12 dages møde i Versailles
Her var en del af ovennævnte personer om ’Combinatorial Optimisation’. Disse
til stede foruden Arthur Geoffrion, Peter ret lange møder på mindre steder var
Hammer, Michael Held, Ellis L. Johnson, meget værdifulde, da man virkelig fik tid
Leon Lasdon, Allan Manne, William til at lære folk at kende.
Orhard-Hays og Harvey Wagner. Jeg
Arc Routing Workshop
Siden da er der sket utrolig meget både
med hensyn til undervisning, computere,
modeller, algoritmer og anvendelser, en
udvikling, der bestemt ikke er slut, men
som accelererer mere og mere.
Oli B.G. Madsen
professor,
dr.techn., er
ansat ved Institut
for Transport på
DTU.
DTU Management annoncerer følgende foredrag:
CORAL, Aarhus university is responsible for organizing a
workshop on arc routing.
• Thursday 6/9 Jørgen Haahr “Heuristic Planning of Shared
Backup Path Protection”.
Venue: Hotel Scandic Copenhagen
• Thursday 20/9 Margret Otterstedt “Statistics and OR
(CART & SP) for New Mortgage Refinancing Strategies”
Date: 22-24 May 2013
• Thursday 4/10 Joakim Juhl “An ethnography on mathe
matical models - Producing science and technology”
Organizing committee:
• Ángel Corberan, Universidad de Valencia, Spain
• Bruce Golden, University of Maryland, US
• Geir Hasle, SINTEF, Norway
• Richard Eglese, Lancaster University Management
School, UK
• Sanne Wøhlk, Aarhus University, Denmark
More information will follow on Aarhus Universities
webpage.
8
DORS, som jo var blevet stiftet i 1962,
var i denne periode også et aktivt forum
for OR interesserede. Der blev bl.a.
hvert år arrangeret et meget fornøjeligt og inspirerende weekendmøde,
hvor OR interesserede fra hele landet
mødtes. Jeg husker møder i Holstebro,
Svendborg og Nordborg, som var meget
succesfulde.
ORbit 19
• Thursday 1/11 Trine Krogh Boomsma “TBA”
• Thursday 15/11 Michael Pascal Simonsen Nielsen “TBA”
• Thursday 29/11 Thomas Stidsen “TBA” Line Blander
Reinhardt “Liner shipping”
• Thursday 6/12 Tor Justesen “Aircraft Stand Allocation with
Asso ciated Resource Scheduling”
• Thursday 20/12 Agnieszka Konicz “TBA” Jonas Christen
sen, Karen Arntoft “Patient Admission Scheduling”
Se DTU Management’s hjemmeside for detaljer.
DORS pris
Af Jakob Dirksen
Beslutningsstøtte til håndtering af forsinkelser i
containerfragt
DORS prisen 2010/2011 går til Jakob Dirksen fra
DTU management. Bedømmelses kommiteen
skriver: »DORS Prisen 2010 - 2011 går
således til et speciale, hvor Operationsanalyse
kombineret med skarp, analytisk tankegang leder
til værdifulde og praktisk anvendelige løsninger;
en pointe, der bekræftes af, at projektets
virksomhedspartner har ytret ønske om at gå
videre med det arbejde som er blevet grundlagt
af Jakob Dirksen.« Redaktionen sender et stort
tillykke til Jakob og hans vejleder David Pisinger.
Motivation for at håndtere forsinkelser
Skibstransport af containere er den mest brugte og energieffektive metode til fragt af store mængder gods på tværs af
kloden. Eksempelvis foregår en væsentlig del af verdens
samlede transport via de utallige ruter fra fabrikkerne i Østasien til forbrugerne i Nordamerika og Europa. Den enkelte
container håndteres typisk af et liner shipping firma. Firmaet
Figur 1: Mærsk skib fyldt med gods på vej ud af en asiatisk havn.
opererer et netværk af skibe, som sejler med regelmæssige
afgange efter et fastlagt skema. Det er shippingfirmaet, der
beslutter, hvordan den enkelte container transporteres fra
oprindelsen til destinationen. For at få fyldte skibe og opnå
economies-of-scale vælger liner shipping firmaer oftest et
netværksdesign, som medfører, at den enkelte container skal
skifte skib en eller flere gange.
Et firma som Mærsk Line, der opererer mere
end 600 containerskibe og hver dag flytter
millioner af containere, oplever konstante
forstyrrelser af deres globale transport netværk. Ifølge et studie oplever 70-80 % af
alle containerskibe forsinkelser i minimum én
havn igennem en rundtur. De typiske årsager
er dårlige vejrforhold, køer i vigtige passager
eller ekstraordinære ventetider i havne, men
også pirateri, medarbejderstrejker og politisk
ustabilitet kan påvirke sejladserne.
De mange forsinkelser af sejladserne påvirker direkte pålideligheden, men medfører
også store økonomiske tab for liner shipping
firmaerne. En forsinkelse af et enkelt skib
kan betyde, at flere tusinder containere bliver
forsinkede og i værste fald misser forbindel-
ORbit 19
9
DORS pris
skulle besluttes,
bedst fortsatte.
hvordan
skibet
De to løsninger der blev overvejet,
var at indhente forsinkelsen ved at
øge hastigheden over Stillehavet
eller ved at sejle udenom Yokohama. En øget fart ville betyde, at
omkostningerne til brandstof steg
voldsomt1, imens det at springe
Yokohama over ville betyde, at en
række containere fra en vigtig kunde
blev forsinkede en fuld uge2. Beslutningen blev støttet af ad hoc udregninger i et regneark, men afgørelsen
Figur 2: Ruten for stillehavskrydseren Mærsk Sarnia.
blev i sidste ende truffet på baggrund af intuition opbygget
ser med andre skibe. Ekstra omkostninger opstår ved dette, igennem mange års erfaring.
idet containerne ikke følger, den optimale plan, men derimod
bruger ekstra tid på land og måske optager pladsen på det Mærsk valgte at sejle til Yokohama, krydse Stillehavet med let
næste skib fra andre containere. Kunderne oplever ofte også øget fart, og først rigtig indhente forsinkelsen på tilbageturen
tab ved forsinkelser, og liner shipping firmaerne risikerer over Stillehavet. Beslutningen endte som et politisk komproderfor, at kunderne vælger en konkurrent ved næste fragt. Det mis, hvor der blev lagt vægt på dels at medtage containerne
er derfor utrolig vigtigt for liner shipping firmaerne at håndtere fra den vigtige kunde men samtidig begrænse stigningen i
brændstofudgifterne. Den valgte sejlads for Mærsk Sarnia
forsinkelser bedst muligt.
medførte desværre at en betydelig del af containerne missede
forbindelser og derved blev forsinket en fuld uge.
En konkret case fra Mærsk Line (1 af 2)
Space [ports]
På grund af en stor storm
i Østasien blev stillehavskrydseren Mærsk Sarnia
forsinket i alt 39 timer på
Time Space network
vej ud af Kwangyang (Sydkorea). Skibet var plansat
Mother vessel
Feeder vessel
S2
til at stoppe i Yokohama
Delayed vessel
(Japan) og derfra fortsætte
fuld af containere fra Asien
Port1
til
Mellemamerika
(se
figur 2). Ved ankomst til
den store havn i Balboa
Port2
(Panama) skulle størstedelen af containerne skifte
Port3
til andre skibe, som ville
transportere godset det
sidste stykke til aftagerene
S1
langs Amerikas østkyst.
Forsinkelsen betød, at en
0
20
40
60
80
100
120
140
stor del af containerne ville
Time [hours]
misse deres forbindelser. Figur 3: Eksempel på graf for simpelt problem. Problemet dækker 3 skibe, som alle tillades at sejle med
Der var derfor travlhed justeret hastighed. Det forsinkede skib tillades endvidere at justere rækkefølgen for havnekald og springe
på Esplanaden, hvor det havn 2 over.
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ORbit 19
DORS pris
Matematisk model til beslutningsstøtte
Videre arbejde
Forsinkelserne i containerfragten har en klar parallel i luftfartsindustrien. Flyselskaber oplever konstant forsinkelser og er
nødsaget til hurtigt at træffe beslutninger om, hvordan driften
bedst muligt genoprettes. Matematisk modellering danner
fundamentet for disse beslutninger, da avanceret software
bygget på operationsanalyse straks kan give operatørerne
overblik over et bredt udvalg af mulige håndteringer, samt
hvilke konsekvenser og omkostninger hver enkelt vil medføre.
Der forestår fortsat meget arbejde før Mærsk Line, og andre
liner shipping firmaer, har adgang til et real-time beslutningssystem, som løbende kan håndtere deres forsinkelser. Beregninger viser dog tydeligt, at emnet har et stort potentiale, og
at det med relativt simple modeller er muligt at finde stærke
løsninger. Liner shipping systemet er så komplekst, at en
tilstrækkelig grundig manuel vurdering af hver løsningsmulighed simpelthen ikke kan foretages. Grundlæggende bør
udviklingen på området gå i to parallelle spor, hvor den ene er
akademisk forskning i udbygning af den foreslåede model, og
I projektet blev den grundlæggende model til håndtering af den anden er en praktisk implementering og yderligere test af
forsinkelser i luftfartsindustrien tilpasset situationen i liner den foreslåede model.
shipping3. Modellen er baseret på et time-space netværk,
hvor hver knude svarer til et havnekald på et givet tidspunkt, Referencer
og hver kant er en mulig sejlads i mellem to havne med en
given fart. Startende med skibenes udgangsposition kan det • Disruption Management in Liner Shipping af Jakob Dirksen
fulde løsningsrum udspændes (se figur 3). I grafen inkluderes
de forskellige måder at håndtere en forsinkelse, herunder • The Vessel Schedule Recovery Problem (VSRP) - a MIP
justere sejlfarten imellem havne, ændre sekvensen af havne, model for handling disruptions in liner shipping af Berit D.
Brouer, Jakob Dirksen, David Pisinger, Christian E. M. Plum,
der besøges og lade containernes forbindelse vente.
og Bo Vaaben (ikke publiceret endnu)
På baggrund af grafen defineres en heltals flow-model.
Begrænsningerne sikrer, at skibene starter korrekt, at de
gennemløber en lovlig sejlads, og at containere gennemfører
en lovlig rejse. Størrelsen af problemer afgrænses ved kun
at lade en delmængde skibe justere deres sejlads og ved at
ignorere re-flowing af containere der ikke kommer frem. Målfunktionen bliver at minimere sejlomkostninger samt en strategisk straf for containere, der henholdsvis bliver forsinkede
eller misser en forbindelse. Ved test af fire faktiske problemer
fra Mærsk er observeret gode køretider og løsninger.
Arbejdet ville ikke have været mulig uden støtte fra Mærsk
Line (ML) og Danmarks Tekniske Universitet (DTU). Der skal
således gå en stor tak til Daniel Bruun (ML), Mikkel M. Sigurd
(ML), Steffen Conradsen (ML), Berit D. Brouer (DTU), David
Pisinger (DTU), Christian E.M. Plum (DTU) og Bo Vaaben
(DTU).
Noter
1 Friktionen i vand er et tredje grads polynomium af hastigheden
2 På ruten over Stillehavet har Mærsk Line en ugentlig afgang
3 Luftfart og liner shipping har mange ligheder, men også
En konkret case fra Mærsk Line (2 af 2)
forskelligheder. Forskellighederne inkluderer, at containerskibe
sejler i døgndrift, at farten kan justeres betydelig mere, at det
Håndteringen af Mærsk Sarnias forsinkelse i Sydkorea kan
næste aldrig er effektivt at bytte et forsinket skib ud med et
andet og at gods i gennemsnit skifter skib mere end 2 gange.
simpelt modelleres med den beskrevne metodik. Det samForskellene er med til at øge kompleksiteten
lede problem, udspændt i grafen med løsningsrummet, har
7106 variable og 1706 begrænsninger. Målfunktionen for
problemet er baseret på Mærsk Lines reelle og strategiske
omkostninger. Heltals-problemet kan løses på dette virkelige
Jakob Dirksen er civilingeniør fra
data i CPLEX i løbet af sekunder. Den resulterende optimale
DTU. Han har specialiseret sig i
løsning er at sejle udenom Yokohama. På trods af en straf for
operationsanalyse, men arbejder
at ignorere de vigtige containere, er den samlede omkostninnu som managementkonsulent for
gen for Mærsks faktiske håndtering er 24 % højere end for
McKinsey&Company i København.
denne løsning. Løsningen er efterfølgende blevet verificeret
med Mærsk Line, som på baggrund af de tilgængelig informationer i udregningerne er enige i, at det ville have været bedre
at springe Yokohama over.
ORbit 19
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nyhed
Svenska Operationsanalysföreningen
(SOAF)
OA-föreningens Exjobbspris i Operationsanalys 2012
SOAF’s årliga exjobbspris i Operationsanalys instiftades 2010 för att uppmuntra studenter inom
området och hjälpa till att sprida högkvalitativa uppsatser.
För att kunna deltaga ställs två krav:
1. En lyckad tillämpning på ett praktiskt problem (hos ett företag, myndighet e.d.).
Lyckad innebär att studien har gjort nytta, d.v.s. att företaget (eller motsv.) har fått ett beslutsunderlag av värde.
2. Att man har använt en vetenskapligt sund metod, utan krav på att den ska vara metodmässigt revolutionerande.
Priset kommer i formen av ett diplom och en premie på 3000 kr per exjobb.
Pristagarna förväntas vidare presentera sitt exjobb på ett SOAF-seminarium.
För att delta i tävlingen ska exjobbet vara godkänt under perioden från 1 november 2011 till 1 november 2012. Exjobbet
skall sändas i elektronisk form samt som en papperskopia. Det ska åtföljas av ett brev från handledaren eller examinatorn där det motiveras varför just detta exjobb ska få priset.
Deadline för insändande av exjobb för 2012 års pris är 15 november.
Kontakt:
Michael Patriksson (ordf.)
styrelseledamot SOAF
P O Lindberg
styrelseledamot SOAF
[email protected]@kth.se
12
ORbit 19
nyhed
Dansk operationsanalyseselskab
(DORS)
DORS pris 2012
DORS - Dansk Selskab for Operationsanalyse - beder hermed om indstillinger til Danmarks bedste
speciale i operationsanalyse 2012.
Et speciale indstilles af vejlederen, og vi skriver til dig, fordi det måske er dine studerende, der har skrevet det allerbedste
projekt i perioden. Invitationen er åben, så send den endelig videre til de vejledere, vi måtte have glemt.
Et speciale kan indstilles hvis:
 - Specialet er skrevet ved et dansk universitet og involverer operationsanalyse.
 - Specialet er på dansk eller engelsk.
 - Specialet er afleveret i 2012.
 - Specialet er forsvaret i 2012, eller skal forsvares i januar 2013. 
(I sidstnævnte tilfælde eftersendes specialkarakteren til [email protected] inden udgang af januar.)
Ved uddeling af DORS-prisen lægges vægten på analyse og løsning af praktiske problemer.
Et speciale indstilles til DORS prisen ved at vejlederen senest 31. december 2012 sender en email til
[email protected] (Tor Justesen,formand for DORS) med følgende indhold:
 - Specialet. Vedhæftes som PDF.
 - Begrundelse for, at specialet bør vinde prisen. Max. 1 side. Skrives af vejlederen. Vedhæftes som PDF.
 - Karakteren for specialet (med undtagelse af specialer forsvaret i januar 2013, se ovenfor).
 - Email-adresser og telefonnumre på alle specialets forfattere og på vejlederen.
DORS-prisen uddeles ved generalforsamlingen for DORS til april 2013. Prisen er på 5000 kroner til deling mellem specialets forfattere. Det er en forudsætning for udbetalingen, at der laves en kort artikel til ORbit om specialet.
Skriv til [email protected] hvis I har spørgsmål.
ORbit 19
13
tutorial
By John F. Raffensperger
A tutorial on smart markets
Over the past thirty years, governments and companies have
implemented smart markets for a huge range of goods and
services. This essay gives a brief tutorial in how they work.
Introduction. First example: transportation
markets
from Chicago to Peoria on Thursday.« Then the giant carrier
could solve an optimization problem to maximize its profit, and
could post provisional results to let buyers change their bids.
I used to work for the US Postal Service. They move par- Batch sales from a big seller is a one-sided auction. When
cels and mail between hubs via trucks and airplanes. How cleared with an optimization model, it’s a smart market.
should USPS purchase its transportation services? USPS
could solve an optimization problem to minimize the cost of What if many companies sell transportation services, and
transportation, with constraints to ensure that the transpor- many companies need to buy such services? We could open
tation satisfied demand for material flow between hubs. The a small office with a web page which invites combinations of
optimization model needs the cost data, so USPS has to ask bids, some from carriers and some from firms needing carrier
the carrier companies for it. »How much would you charge services. The optimization model matches supply to demand,
to move 10 truckloads from Chicago to Peoria this Thursday by truck type, route, and any other complication you can imagine. The solution to the optimization model would be a good
afternoon?«
way to move materials. What should be our objective? We
We can think of this problem as an auction. USPS could set up should maximize the money we get from the buyers, minus
a web page where carrier companies enter bids, »I will charge the money we have to pay to the sellers. That would maxi$500/truckload to ship from Chicago to Peoria…« Then USPS mize our profit. We would not accept a money-losing deal.
would have the data to solve the optimization, and the results The solution which maximizes our profit would also maximize
of the optimization would tell USPS which bids to accept. benefit to the region, assuming a competitive market (lots of
USPS could even do a little better if they posted provisional buyers and sellers). This two-sided auction, with many buyers
results, and let the carriers change their bids. They could do and many sellers, is a smart market, too. The optimization
better still if they let the carriers put in multiple complex bids, model can find solutions that would be impossible if done
such as »I will charge $500/truckload to ship from Chicago to manually, or if done only between pairs of people.
Peoria. But I will charge only $750 for the round trip«. Allowing
bidders flexibility in how they bid can let the optimization take This isn’t a mere thought experiment – people are already
making money with the idea – see for example Moore (1991),
advantage of the bidder’s economies of scope and scale.
Brewer and Plott (2002), Ledyard et al. (2002), Andres FiglioSo batch procurement from a big buyer can be viewed as a zzi et al. (2003), Caplice and Sheffi (2003), Song and Regan
one-sided reverse auction. When cleared with an optimization (2003), and Sheffi (2004).
model, it’s a smart market. Simply sorting bids from lowest to
highest wouldn’t solve the problem at all, any more than we Key characteristics
could solve an integer program only by sorting the variables
Definition
by their objective coefficients.
of a smart market.
The problem could be turned around. A big carrier could be A smart market is an auction cleared by optimization.
selling transportation to small companies. The big carrier
could post a web page where the small companies would bid • The smart market is operated by a market manager, who
to buy services, »I will pay you $480 to move one truckload clears the market with an optimization model. EBay.com and
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Qxl.dk don’t need optimization models A general formulation
to clear their auctions, so those are not
This formulation assumes that the
smart markets by this definition.
market has both buyers and sellers.
• Trades are not between pairs of
people, but rather to or from the market Each trader i can have multiple bids,
manager. EBay.com and Qxl.dk match
pairs of buyers and sellers, so those Indices: i traders, b bids.
web sites are not smart markets by this
definition either.
Data: Advantages of a smart market.
The smart market simplifies trading
complex goods and services. Buyers
and sellers know where to go to find
each other, and are usually assured of
competitive prices. The market manager
can set and enforce rules on who is
allowed to trade, so traders can be more
confident that the trade will actually
happen. So the market is orderly and
transparent. When something goes
wrong, rules are in place to resolve the
problem (McCabe, Rassenti & Smith
1991).
BuyPricei,b BuyStepi,b SellPricei,b SellStepi,b Initiali (A, d) modeling a piecewise linear demand
function (the idea that one piece of cake
is great, a second piece of cake is okay,
and I don’t really need the third piece of
cake).
= bid price of trader i for step b.
= additional quantity that trader i would buy at price BuyPricei,b.
= offer price of trader i for step b.
= additional quantity that trader i would sell at price SellPricei,b.
= starting quantity for trader i.
= general technology matrix.
You can think of BuyPricei,b as the value per serving, and BuyStepi,b as the serving
size.
If the price is BuyPricei,1 = $1, I will buy up to BuyStepi,1 = 5 units.
If the price is BuyPricei,2 = $0.76, I will buy an additional BuyStepi,2 = 3 units.
If the price is BuyPricei,3 = $0.50, I will buy an additional BuyStepi,3 = 2 units.
Model SM
Smart markets are now possible due to 1. Maximize net benefit: ∑i∑b BuyPricei,bbuyqtyi,b – ∑i∑b SellPricei,bsellqtyi,b,
the combination of optimization and the
internet. Increased computation power subject to
lets the market manager solve problems
that before would have been impossible. 2. Calculate net trades: Initiali + ∑b buyqtyi,b – ∑b sellqtyi,b = finalqtyi, for all i.
The internet lets traders bid easily.
3. Bids are piecewise linear: buyqtyi,b ≤ BuyStepi,b, for all i, b,
Most of us can accept the idea of a big
company using an auction to purchase
services like transportation from other
businesses. Some of us do not like the
idea of using markets for services such
as fresh water (which I’ll discuss below).
As operations researchers, we specialize in finding good allocations of resources. Well-regulated honest markets
allocate resources extremely well! Smart
markets tend to be highly regulated, as
they are centralized, and the market
manager usually pays a lot of attention
to the behaviour of participants.
4. Technology constraints: A∙finalqty ≤ d,
5. Non-negativity, integrality: buyqtyi,b ≥0, sellqtyi,b ≥ 0, possibly integral.
6. Avoid degeneracy: finalqtyi free, possibly integral.
Why is this a maximization? Imagine if
you were the market manager. Buyers
pay you, and you pay sellers. To
maximize your profit, you would want
to take the high bids from the buyers,
but pay out only the small bids to the
sellers.
For
the
transportation
problem
mentioned above, the (A, d) matrix corresponds to demand for transport across
the transportation network, possibly with
conservation of flow constraints, truck
capacity constraints, and so on. The bid
subscripts b could be from-to pairs of
hubs rather than serving quantities.
ORbit 19
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tutorial
Model SM is written as a net pool market, in which the model
has the initial rights as an input, and calculates all the buying
and selling. Surprisingly, perhaps, we can usually write a
simpler formulation for model SM, in which we pretend that
everyone is a buyer (a gross pool market). Then we just
calculate trades based on the initial positions after we know
the final allocation. We have to assume that participants are
willing to trade sufficient quantities to allow feasibility. The
gross pool market will tend to be mathematically feasible, but
could have an unacceptably high cost in the optimal objective
value, should buy bids be too low compared to sell bids. The
difference between these two formulations is usually only
algebraic, as the two market formulations are economically
equivalent by the Coase theorem (that the final optimal
allocation doesn’t usually depend on the initial rights).
Pricing
as the value of additional capacity.
A combinatorial auction is a smart market in which goods
are indivisible. Those markets must be cleared with integer
programs. If model SM is an integer program, then we have
the elementary problem that integer programs, like all nonconvex models, do not give good dual price information. We
could use the price-as-bid approach. Alternatively, Rassenti,
Smith and Bulfin (1982) found ranges of good prices by solving pseudo-dual models. O’Neill et al (2005) first solve the
integer program, and then solve an auxiliary linear program
to get good prices. Pricing in non-convexity will always be
difficult, which helps keep operations researchers in business!
An entire issue of Management Science (vol. 51, no. 3, March
2005) was devoted to this topic. See also Pekec and Rothkopf
(2003).
Initial rights
Given the primal result to model SM, at what price should we
charge or pay trader i? We have some options, depending on Model SM above has the parameter Initiali. To have a market,
traders need to know who owns the right at the start of the
the problem.
auction. For a company buying transport services, of course
Most obviously, we can charge them their price as bid. Trader i the carrier firms own the rights, and the buyer has to pay for
offered to buy up to 5 units at $1/unit, and the optimal solution the service. That makes complete sense. You could also imais to give her 4 units, so we can reasonably charge her $4. In gine that the buyer pays the carrier for work to be done next
this case, trader i will be nervous about bidding. She will be month, and then the buyer changes her mind and sells that
tempted to bid a little lower than her true reservation price, contract to someone else.
hoping to save some money, in which case she may not get as
much as she really wants. She will also be tempted to sneak But initial ownership is not always clear. A few years ago, the
around the other bidders, trying to find out what they would U.S. government tried to implement a trading system for takebid. This problem has been studied thoroughly in the auction off and landing slots at Kennedy Airport in New York. Implementation would have solved a lot of problems and made
literature.
everyone better off! But as far as I can tell, they failed because
Many smart markets allocate divisible goods such as they could not get agreement on the initial rights. The airlines
electricity and natural gas, and can be cleared with ordinary said, »We’ve been taking off and landing all this time, so those
linear programming. If our smart market model SM is a linear rights are ours.« The Port Authority said, »We own the land,
program, we can use the dual prices pi on constraint 2 to so we own the rights.« The Federal Aviation Administration
charge or pay trader i the amount pi(∑b buyqtyi,b – ∑b sellqtyi,b). said, »We actually give permission to take off and land, so
If trader i buys more than she sells then she pays; otherwise we own the rights.« What a mess! The problem is even more
she gains. (We need finalqtyi to be free to avoid degeneracy, complicated when the quantity available is uncertain, as with
to make sure pi is correct.) This marginal cost pricing pro- fresh water.
vides many nice economic results due to the LP optimality
conditions. For example, facing price pi, trader i will be satis- Financial trading
fied with finalqtyi. Trader i is assured that she will be given
a truly market price, which everyone faces. So she is more Most smart markets are spot markets, in which goods or serlikely to feel comfortable bidding her true reservation price. vices are traded and delivered immediately.
Every seller is guaranteed to receive at least as much as was
bid and possibly more. Every buyer is guaranteed to pay no A spot market could be cleared by simple pair-wise trading
more than was offered, and possibly less. We also get useful (as for a grocery store). A spot market could be cleared by a
resource prices from the technology constraints (A, d), such simple many-to-many auction (as for some water markets in
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Australia). A spot market could also be smart and multi-lateral whole area around Copenhagen. So this could be a set
(as for a modern electricity market).
packing problem, with the rows corresponding to regions: the
sum of the accepted bids for Copenhagen must be less than
A smart market could be run as a one-off for long-term rights, or equal to 1.
such as radio spectrum. I don’t think this would be considered a spot market, simply because it isn’t active enough. By Furthermore, I might offer the government a premium to get
contrast, modern smart markets for electricity are run as spot licenses in several big cities, and I might even want to put
markets, with trades clearing every few minutes, and power in some complicated and/or bids like, “I’ll bid X kroner for
generated and delivered immediately.
Copenhagen and Aarhus, or Y kroner for Copenhagen and
Odense.« This is easily done with integer programming.
Much more interestingly, someone could observe a spot
market operating for a while, and be fearful that the price may Electricity
spike in the wrong direction. So this person could make a side
deal, outside of the smart market, as insurance or hedging, High-voltage power, which goes across large regions, requires
based on the spot price sometime in the future. Of course, a lot of coordination to ensure that generation matches
anyone can make a deal like this based on any market. Such demand and that power flows do not exceed network line
capacities. The security and reliability requirements are very
deals are especially frequent in the electricity markets.
high. This coordination can be done extremely well by linear
A big factory may worry that the price of power might go high, programming. Before the smart market, this coordination
so the factory manager could make a deal with a generator was done with linear programming, but the line operator,
to “buy” power at a fixed price. “Factory F will pay generator generators, and distributors were usually part of a monopoly.
G a price of $0.50/megawatt for 7 megawatts of power on 1 With a smart market, these could be different parties, thus
Sep 2011.” Of course, the generator cannot really sell power creating improvements through competition. The first market
directly to the factory, because G’s electrons scatter to all parts of this type was implemented in New Zealand in 1996 (Alvey
of the power grid as soon as G generates them, and because et al 1998, Hogan et al 1996), and this design is now copied all
the spot trade must go through the smart market. But on 1 Sep over the world. These markets clear every few minutes, with a
2011, factory F can demand 7 megawatts from the market lot of money changing hands each time.
manager, and pay the going price, say, $0.60/megawatt. To
resolve the deal, G now has to pay F $0.10/megawatt for The modern electricity market is an important example of a
7 megawatts. If in fact the spot price of power were only two-sided smart market. Generators are the sellers, offering
$0.55/megawatt, then G gets to keep $0.05/megawatt for the to supply power at a range of prices. Wholesale power firms
7 megawatts. Consequently, G has an incentive to keep the (those that sell to homes) and big industrial firms are the
spot price of power low, thus stabilizing prices. Meanwhile, buyers, who bid to buy power at a range of prices. To clear the
the factory manager F got what she wanted – lower risk from market, the market manager solves a linear program in which
the decision variables are how much power to accept from
more reliable prices.
each generator, the flow of power on each line, and how much
More examples of smart markets
power to provide to each distributor.
Radio frequency auctions
The spectrum auction is a one-sided smart market cleared
by an integer program (McCabe et al 1991, Chakravorti et al
1995). Participants purchase radio spectrum licenses from
government. These combinatorial auctions are cleared as bid,
rather than at prices based on dual variables.
The problem is combinatorial due to radio spectrum
interference. If I get the license for frequency 980MHz in
Copenhagen, I don’t want the government to give someone
else the license for 980MHz in nearby Lyngby. I want the
The flow-on-the-power-line variables appear in constraints
which ensure balance of flow at each power line connection
(node), like ∑node i flowi,j – ∑node k flowj,k = 0 for each node j. Picture one of those industrial-looking places with lots of transformers and high tension wires – that is probably a node in this
model. The dual price pj on this constraint is the nodal price
at j. If a generator were willing to put in another megawatt of
power there, the market manager should pay the generator
pj. If a wholesaler or factory were to take another megawatt of
power there, the market manager should charge them pj. By
clearing the market based on the dual prices, participants are
charged on marginal values, rather than as bid.
ORbit 19
17
tutorial
Natural gas
Schrage 1993). The system ensures that the class seats go
to those students who most want them, while ensuring that
Natural gas, transported over cross-country pipe networks, the number of students in each class stays within the room
is sometimes managed with a smart market (McCabe, Ras- capacity.
senti & Smith 1990), as in Australia. The system operator
serves as the market manager. The market manager has to Smart markets are now being proposed for environmental
match gas supply to demand, while ensuring that flows do not services, including water (Murphy et al. 2000, Raffensperger
exceed pipe capacities. Gas suppliers offer to sell a range of et al. 2009).
quantities at a range of prices. Distributors bid to buy a range
of quantities at a range of prices. To clear the market, the
market manager solves a linear program in which the decision
variables are the gas to accept from each supplier, the flow
of gas on each pipe segment, and how much gas to provide
to each distributor. As with electricity markets, after solution,
the primal variables prescribe the optimal flows, and the dual
variables provide the market clearing prices.
Conclusion
Smart markets are a hot area in operations research. I’ve left
out a lot of applications! Often, the models are not particularly
complicated, nor are their solution. Rather, the difficult bits are
the market rules, defining what is actually traded, and making
sure that the prices give the right incentives to participants.
The process of determining the initial rights can be politically
charged, and sometimes causes the whole project to fail.
Airplane take-off and landing slots
When implemented, however, society can enjoy enormous
The term smart market appears to have been first used gains from much better allocation of complex resources.
by Rassenti, Smith, and Bulfin (1982), who proposed a
combinatorial auction for airplane take-off and landing slots. Acknowledgement
(Asking a jumbo jet pilot low on fuel how much he’s willing to
pay to land probably wouldn’t work too well!) The idea is that Thanks to E. Grant Read who tutored me in smart markets.
airlines would bid cj for a package, which means at a minimum
a take-off at one airport and a landing at another airport. But References
a package can be more complicated than just a take-off and
I pinched some of this essay from the Wikipedia page on Smart
landing pair. An airline would want to route its planes out of its
Market, which I initially authored, http://en.wikipedia.org/wiki/
home hub and eventually back to its home hub. Constraints
Smart_market.
included capacity at each airport, and routing logic for each
airline.
• Alvey T., Goodwin D., Xingwang M., Streiffert D. and Sun
D. (1998), A security-constrained bid-clearing system for the
Is this a one-sided or two-sided auction? It would be one-sided
NZ wholesale electricity market, IEEE Trans. Power Systems,
if the government decided that it owned all the rights, like radio
13(2), 340-346.
spectrum, and the airlines had to pay for every package. But
• Figliozzi, Andres M., Mahmassani, H. S., and Jaillet, P.
the airlines would think that was too risky, and would want to
(2003). Framework for study of carrier strategies in auctionbuy contracts where they bought continued rights for a long
based transportation marketplace, Transportation Research
time, say 5 years, or permanently. It could then be two-sided,
Record: Journal of the Transportation Research Board, 1854(because an airline could sell its right to another airline. See
1), 162-170.
also Jaikumar (1980).
• Brewer, P. J., and Plott, C. R. (2002). A decentralized,
smart market solution to a class of back-haul transportation
Heaps more.
problems: Concept and experimental test beds. Interfaces,
The Chilean government uses a smart market to choose 13-36.
caterers for school meal programs (Epstein et al 2002). Their • Caplice, C., and Sheffi, Y. (2003). Optimization based
market rules included strict requirements for participating, procurement for transportation services. Journal of Business
Logistics, 24(2), 109-128.
which raised the professionalism and quality of the bidders.
• Chakravorti, B., W.W. Sharkey, Y. Spiegel and S. Wilkie
The University of Chicago’s Booth School of Business uses (1995), Auctioning the Airwaves: The Contest for Broadband
a smart market for course registration (Graves, Sankaran & PCS Spectrum, J. Econ. & Mgt Strategy, 4(2), 267-343.
18
ORbit 19
tutorial
• Epstein, Rafael, Lysette Henriquez, Jaime Catalán, Gabriel
Y. Weintraub, Cristián Martinez, A Combinatorial Auction
Improves School Meals in Chile, Interfaces, 32(6), Nov-Dec
2002, pp. 1-14.
• Graves, R.L., J. Sankaran, and L. Schrage (1993), An
Auction Method for Course Registration, Interfaces, 23(5).
• Groves, T. (1973), Incentives in Teams, Econometrica, pp.
617-631.
• Hogan W.W, Read E.G and Ring B.J. (1996), Using
Mathematical Programming for Electricity Spot Pricing, Int’l
Trans. in Operations Research, 3, (3-4), 243-253.
• Jackson, B.L. and J.M. Brown (1980), Using LP for Crude
Oil Sales at Elk Hills: A Case Study, Interfaces, 10(3) June,
pp. 65-70.
• Jaikumar, R. (1980), Mathematical Programming
Approaches to the Design of Auctions: A Study of the Auction
of Landing Rights at Congested Airports, Economic Forum,
11, pp. 29-47.
• Ledyard, J. O., Olson, M., Porter, D., Swanson, J. A., and
Torma, D. P. (2002). The first use of a combined-value auction
for transportation services, Interfaces, 4-12.
• McCabe, Kevin, Stephen Rassenti, and Vernon Smith
(1990), Auction Design for Composite Goods: The Natural
Gas Industry, J. of Economic Behavior and Organization,
Sep, 127-149.
• McCabe, Kevin, Stephen Rassenti, and Vernon Smith
(1991), Smart computer-assisted markets, Science, v254,
534-538.
• Moore, E.W., J.M.Warmke, and L.R. Gorban (1991), The
Indispensable Role of Management Science in Centralizing
Freight Operations at Reynolds Metals Company, Interfaces,
21(1), Jan.-Feb., pp.107-129.
• Murphy, J. J., Dinar, A., Howitt, R., Rassenti, S. J., and Smith,
V. L. (2000). The Design of ‘Smart’ Water Market Institutions
Using Laboratory Experiments, Env & Resource Economics,
17(4), 375-394.
• O’Neill, R.P., P.M. Sotkiewicz, B.F. Hobbs, M.H. Rothkopf,
W.R. Stewart (2005), Effective market-clearing prices in
markets with non-convexities, European J of Operational
Research, 164 269-285.
• Pekec, Aleksandar and Michael H. Rothkopf (2003),
Combinatorial Auction Design, Management Science, 49(11),
Nov 2003, pp. 1485-1503.
• Raffensperger, J. F., Milke, M. W., and Read, E. G. (2009). A
deterministic smart market model for groundwater, Operations
Research, 57(6), 1333-1346.
• Rassenti, S.J., V.L. Smith, and R.L. Bulfin (1982), A Combinatorial Auction Mechanism for Airport time Slot Allocation,
Bell Journal of Economics, 13(2), pp. 402-417.
• Rothkopf, M.H., A. Pekec, and R.M. Harstad (1998), Computationally Manageable Combinational Auctions, Management
Science, 44(8) Aug, pp. 1131-1147.
• Sheffi, Y. (2004). Combinatorial auctions in the procurement
of transportation services. Interfaces, 245-252.
• Song, J., and Regan, A. (2003). Combinatorial auctions for
transportation service procurement: The carrier perspective,
Transportation Research Record: J of the Transportation
Research Board, 1833(-1), 40-46.
Dr. John F. Raffensperger
is currently Senior Lecturer
in Management Science,
Dept. of Management, University of Canterbury, Christchurch, New Zealand.
He obtained his PhD from
the University of Chicago’s
Graduate School of Business. This article was written while he was at Denmark Technical University
on a IRSES/OptALI faculty
exchange, funded by in part by the EU and the NZ Ministry of Research, Science and Technology.
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ORbit 19
19
nyhed
Af Joachim Dahl
Applications of Optimisation 2012
Det årlige DORS arrangement Applications of Optimisation blev i år afholdt
den 30. maj på Axelborg i København.
Arrangementet var det fjerde af sin art,
hvor det første blev afviklet i 2008, og
formålet med arrangementerne er primært at skabe kontakt mellem dankske
operationsanalytikere gennem en temadag med foredrag fra både industrien og
forskningsinstitutioner. Arrangementet i
år var det hidtil største med 59 registrerede deltagere, hvilket overgik arrangørernes forventninger.
Som et nyt tiltag var de faglige indlæg i
år opdelt i to særskilte temaer; et tema
omkring anvendelsen af optimering i den
finansielle sektor, og et tema omkring
anvendt operationsanalyse og logistik.
Finansiel optimering
investeringsfima. Hans præsentation
gav et både spændende og humoristisk
Finansiel optimering som disciplin er indblik i de praktiske værktøjer en mateområde, hvor der traditionelt anvendes matisk investeringsrådgiver bruger i sit
både optimering, statistisk og operati- daglige virke.
onsanalyse (og mere generelt anvendt
matematik) i stort omfang, hvilket Anvendt operationsanalyse
motiverede planlægningen af en praktisk orienteret foredragsrække, men Arrangement indeholdt også tre indlæg
foredrag omkring både teori og praktisk om mere traditionel operationsanalyse.
anvendelse.
Wim Vandevelde fra Procter & Gamble
Raphael Hauser fra Oxford universitet holdt et spændende foredag "Distriholdt et introducerende foredrag med bution Network Optimization" omkring
titlen "Convex Optimisation in Finance den globale planlægning af fabrikatiand Quantitative Trading", der både onsbeliggenheder og forsyningsruter,
introducerede grundlæggende begreber der foretages i Procter & Gamble med
samt skitserede mere forskingsorien- 5 til 10 års intervaller. Denne optimering
terede ideer. Fra et praktisk synspunkt foretages ved hjælp af firmaets egne
var en af hans vigtigste pointer at den specialudviklede algoritmer, hvilket er
grundlæggende teori bag Markowitz nødvendigt for at håndtere de store prooptimering (af aktieporteføljer) produ- blemdimensioner.
cerer gode afkast i praksis under vise
forbehold, hvilket ellers ofte anfægtes.
Dernæst præsenterede Kourosh M.
Rasmussen med tilknytning til både
danmarks tekniske universitet (DTU)
og firmaet FinE Analytics ApS et webbaseret produkt til portefølje optimering
udviklet med øje for selvstændige private investorer. Produktet har forskellige
risikoprofiler, og fremadrettet arbejder
firmaet på at udvikle heuristikker til at
detektere og forudsige generelle trends
på aktiemarkedet, så investorerne kan
konsolidere sine investeringer i nedgangsperioder.
Figur 1: Raphael Hauser fra Oxford
universitet forelæser om financiel optimering.
20
ORbit 19
Endelig afholdt Thomas Schmelzer et
indlæg med titlen "Convex programming
in quantitative hedgefund". Thomas
Schmelzer har tidligere arbejdet for
forskellige større engelske investerings- Figur 2: Jürgen Kohl fra ILOG Optimization,
fonde, og har for nyligt stiftet et mindre IBM Software Group.
nyhed
Figur 3: Networking ved DORS arrangementet.
Dernæst holdt Jürgen Kohl fra ILOG/
IBM et indlæg "getting the price right
and other applications of business optimisation", hvor han diskuterede forskellige konsulentopgaver han har løst hos
ILOG/IBM.
Networking
En vigtig del af arrangement er networking mellem deltagelerne; for flere
deltagere er DORS arrangementet den
eneste tilknytning til fagområdet og
tidligere bekendte og kollegaer. Defor
Afslutningsvist gav Rune Møller Jensen, vil vi fra arrangørernes side tilstræbe
IT universitet et foredrag "Stowing the at der fremover afsættes mere tid til
right containers on container vessels". networking.
Det gav et spændende indblik i de
problemstillinger og udfordringe besvarelser der er ved pakning af containerskibe. Han illustreredede hvordan en Fremtidige arrangementer
optimal pakning af containerskibe er et
vanskeligt problem, der har potentialle Applications of optimisation er måske
vores forenings vigtigste tilbagevenfor enorme besparelser.
dende begivenhed, og vi forventer at
Overrækkelse af DORS afholde arrangement regelmæssigt hver
prisen
sommer. De umiddelbare tilbagemeldinger vi har modtaget fra deltagerne
Som noget nyt blev vinderen DORS er at et opdelt tema med både klassisk
prisen for bedste speciale indenfor operationsanalyse samt en introduktion
operationsanalyse udnævnt ved arran- til et måske mindre kendt fagområde er
gementet. Vinderen af prisen for bedste blevet vel modtaget. Til planlægningen
speciale 2010-2011 var Jakob Dirksen, af fremtidige arrangementer modtager
med specialet "Disruption Management DORS bestyrelsen meget gerne forslag
in Liner Shipping". Læs mere om spe- til temaer eller foredragsholdere.
cialet på side 9.
AOO 2012 blev arrangeret af Joachim Dahl og Mikkel Sigurd.
Joachim har en Ph.D. grad i signalbehandling fra Aalborg universitet.
Han arbejder for MOSEK ApS med
udvikling af algoritmer til konveks
optimering.
Mikkel har en Ph.D. grad i operationsanalyse fra danmarks tekniske
universitet. Han er ansvarlig for operationsanalyse hos Mærsk Line A/S.
Begge er medlem af DORS bestyrelsen.
ORbit 19
21
undervisning
By Marcel Turkensteen
A sustainable course?
STEP 1: WAREHOUSE LOCATION PROBLEM
This note is about the experiences with an elective course
that I am giving as part of the master program in Logistics
& Supply Chain Management at Aarhus University, called
Sustainable Supply Chain Management, or, to reduce the
energy use in writing this article, SSCM.
ASSUMING THAT:
- EACH POTENTIAL SITE HAS THE SAME FIXED COSTS FOR LOCATING A FACILITY AT
- THE FACILITIES BEING SITED DO NOT HAVE CAPACITIES ON THE DEMAND THAT THEY CAN SERVE, IT IS AN “UNCAPACITATED”
- ONE KNOWS, A PRIORI, HOW MANY FACILITIES SHOULD BE OPENED.
P-MEDIAN MODEL
WITH NEIGHBOURHOOD SEARCH IMPROVEMENT
CASE 1: ONE WAREHOUSE COVERING ALL DEMAND NODES
THE C
But hey, why do we worry about sustainability, about reducing
CO2 emissions, and about saving polar bears? Global warming was a hot issue only a few years ago. Not anymore. After
the economical devastations of the financial crisis and now
the euro crisis, sustainability is something we cannot afford.
Our planet has to be saved at some other point of time.
This line of thinking misses a very important point. The word
’sustainable’ suggests a long term horizon. In recent management theory (for example Hart (1997) and more recently,
Porter (2011)), the purpose of an organization is to add value
to society in the long run; only then it continue to prosper.
Value is not added with destructive practices, such as pollution. In this vision, sustainability and reduction of environmental impact is not a luxury. It should be a key component of an
organization’s long term decision making.
So what does this imply for logistics, or for supply chain
management? An intuitive logistics approach would be to
minimize the CO2 emissions from transportation or, a bit more
advanced, include them in a model with multiple objectives,
where there is some theoretical cost α to each unit emission.
I think this approach is too simple and narrow.
Firstly, the way of modeling suggests that reduction of the environmental impact is a sacrifice. A manager can reduce emissions, but thereby reduces profits as well. Unless of course,
one motivates it from ‘marketing’: if we cut our emissions
from transport by X%, then customers will buy our products.
However, this works if your entire organization and your entire
product can be considered green, not just the transportation.
Secondly, it implies that transportation is the source of pollution and so transportation should be tackled. However, it may
not be smart to view transportation in isolation from the other
components of the logistics process such as purchasing,
warehousing and waste management. One of many examples
where reducing transport can thus deteriorate CO2 emissions
is the following. It can very well be that the collecting of valua-
22
ORbit 19
CASE 2: FOUR WAREHOUSES COVER
WAR
FRED
WAREHOUSE LOCATED IN: AARHUS
TOTAL DEMAND-WEIGHTED DISTANCE: 803,770
TOTA
TONNE KILOMETERS
TONN
AVERAGE DISTANCE: 149.6 KM
AVER
APPROX. CO2 EMISSIONS: 474.7 T
APPR
TOTAL DEMAND WEIGHTED DISTANCE AND THE CO2 EMISSIONS DECREASE BY 8
CO2 SAVINGS: 400 T
STEP 2: WAREHOUSE ENERGY EFFICIENCY AND SUSTAINABLE
DESIGN
THE ENERGY CONSUMPTION OF A WAREHOUSE CONSISTS OF 3 MAIN SOURCES:
SUSTAINING THE TARGETED TEMPERATURE,
WAREHOUSE LIGHTING,
MECHANICAL HANDLING EQUIPMENT.
THE ENERGY CONSUMPTION THROUGH THESE THREE ACTIVITIES CAN BE LOWERED
BY CAREFUL DESIGN AND HOUSEKEEPING MEASURES AND USING THE MOST EFFICIENT
TECHNOLOGIES AND PRODUCTS.
INCORPARATING GREEN ERENGY CAN B
FOR THE WAREHOUSE ENERGY MIX:
BIOMASS (WOOD CHIP O
-
SOLAR PHOTOVOLTAICS
-
RECOVERED PROCESS W
REFRIGERATION PLANTS
-
RECOVERED KINETIC ENE
AIR, GROUND OR WATER
USING ALL THE NEW TECHNOLOGIES AND RENEWABLE ENERGY SOURCES ON SITE CAN LOWER CO2 EMISSIONS BY 39%
TRADITIONAL, NON-SUSTAINABLE SOLUTIONS.
HOWEVER THE MOST EFFICIENT SOLUTIONS CAN BE APPLIED ONLY TO BIG WAREHOUSES, THEREFORE THE CO2 EMISSIONS BURDEN W
CASE(ASSUMING THAT IT HAS 60,000 SQ M) THAN IN THE FOUR, SMALLER WAREHOUSES CASE(10-15,000 SQ M EACH):
2,854 t
4,680 t
1% CO2 emissions
reduction acuired through
sustaianble, energy
efficient warehouse design
Ap
kil
dem
STEP 3: TRADE-OFF BETWEEN LOCATION OF
AND THE WAREHOUSING ENERGY EFF
ble materials after disposal leads to significant reductions of
CO2 emissions, since the materials do not have to be produced from scratch again. Another example is the first poster,
by Jonas Rasmussen (from AU Herning), which presents the
assignment of one of our students. The decision is how many
warehouses to place. Adding more warehouses can reduce
distances and hence emissions, but there are savings from
having larger warehouses, both in environmental impacts and
costs, as well.
undervisning
”
HEURISTIC
RING THE DEMAND NODES LOCATED
CLOSEST TO THE FACILITIES
REHOUSES LOCATED IN: AARHUS,
DERIKSHAVN, KOLDING, COPENHAGEN
AL DEMAND-WEIGHTED DISTANCE: 124,470
NE KILOMETERS
RAGE DISTANCE: 23.2 KM
ROX. CO2 EMISSIONS: 73.5 T
85%.
BE DONE THROUGH USING RENEWABLE SOURCES
OR OTHER WASTE), WIND, SOLAR THERMAL,
S,
WASTE ENERGY, SUCH AS HEAT FROM
S OR AIR COMPRESSORS,
ERGY,
R THERMAL-EXCHANGE UNITS.
AND ENERGY USE BY
40% VERSUS
WILL BE LESS IN THE ONE, BIG WAREHOUSE
pprox. 30,898 tonne
lometers reduction in
mand-weighted distance
acuired by location
optimalization
F WAREHOUSES
FICIENCY
Thirdly, the measurement of the environmental impact of a product is actually quite a
challenge and should be performed properly
before informed decisions can be taken.
This is the field of Life Cycle Analysis (or
Assessment, abbreviated to LCA). Here,
one measures the environmental impact of a
product from some starting point, ideally the
extraction of raw materials, to some ending
point, which can be the customer, the usage
or the disposal. Many exam papers in SSCM
contain basic LCA for some competing alternatives (e.g. stone paper versus wood-based
paper, or electronic distribution of media files
versus physical distribution).
Finally, green or sustainable logistics usually means that the activities themselves
are carried out in a green way. Alternatively,
one could view green logistics as the design of logistics networks to enable activities that are good for society and the
environment, such as the replacement of fossil fuels and the
collection of waste. As another example, the second poster,
by Stefan Maagaard Nielsen (from AU) discusses the problem
of setting up battery stations for electric cars in Århus (after
arguing how the use of these vehicles is environmentally
better than regular cars).
The course SSCM does not provide a standard, off-the-shelf,
one-size-fits-all approach, but it provides a set of tools, some
of them mathematical, some of them conceptual. The students work under supervision with their individually chosen
projects. As the field of research is multi-disciplinary, it is very
interesting to see what students from other fields, such as
management, marketing, and innovation can contribute. This
is challenging, but definitely rewarding.
Marcel Turkensteen
(the Netherlands, 1979) is a
lecturer at CORAL (Center
of Operations Research
And Logistics) at the Aarhus
University. His research
interests include routing and
network problems and in
particular the influence of
the environment on logistics
decisions and vice versa.
ORbit 19
23
artikel
Af Martin Joborn
Optimering för ökad punktlighet ocn minskad
energiförbrukning
Tågtrafikledningssystemet CATO (Computer Aided Train Operation) har nyligen
inköpts av både LKAB och Arlanda
Express, och installeras för närvarande
hos båda företagen.Totalt görs 48
lokinstallationer. Tester har visat mycket
lovande resultat: CATO förbättrar punktligheten, minskar energiförbrukningen
med upp till 25% och ökar den totala
effektiviteten. Kärnan i systemet är en
optimeringsmodul, som beräknar optimal hastighet i varje ögonblick.
optimering och GSM-R, järnvägens CATO-TRAIN beräknar kontinuerligt
digitala kommunikationssystem, gjort den optimala hastighetsprofilen så att
fullskaliga implementeringar möjliga. tåget når sin målpunkt i rätt tid. Den
optimerade hastighetsprofilen skall
CATO består av två delar, CATO-TCC alltid följa den givna tidtabellen, men
- modulen på trafikledningscentralen tar samtidigt hänsyn till andra aspek- och CATO-TRAIN -modulen som ter, som att minimera energiförbrukfinns ombord varje lok. Dessa enheter ningen, minska slitage på fordon och
kommunicerar via GSM-R digital radio. bana,
lokprestanda,
förarergonomi
CATO-TCC är kopplat till Trafikverkets och passagerarkomfort. Den optimala
tågledningssystem och får kontinuerligt hastighetsprofilen visas i förarens
uppdaterad information om t.ex. tågdata CATO-gränssnitt och är rådgivande.
och aktuell tidtabell. CATO-TCC tolkar Optimeringen sker i realtid och hastigCATO är utvecklat av järnvägskon- informationen och skickar instruktioner hetsprofilen uppdateras kontinuerligt så
sult- och mjukvaruföretaget Transrail till tågen uttryckt som målpunkter, dvs de att råden till lokförare alltid är aktuella.
Sweden AB i FoU-projekt, samfinan- positioner och hastigheter som skall nås Förargränssnittet är ergonomiskt och
sierade av Trafikverket, LKAB och vid givna tidpunkter. Detta gör det möjligt utformat i samarbete med avdelningen
Transrail.
att köra tåg enligt det optimala operativa för människa-dator-interaktion vid Uppscenariot beaktande hela trafiksituatio- sala Universitet. HastighetsrekommenTransrail formulerade de konceptuella nen. Inte bara målpunkter, men också dationerna är enkla att följa – utbildning
konturerna av CATO för många år den aktuella linjeprofilen med topografi, av förare har visat sig vara mycket lätt.
sedan, redan innan den nödvändiga hastighetsbegränsningar och hastig- Kommunikationen med driftledningstekniken fanns till hands. Idag har hetsnedsättningar, hämtas till tåget.
centralen ger trafikinformation i realtid
utvecklingen av datorer, algoritmer för
om situationen på en linje, snarare än
bara den statiska
tidtabellen.
Detta
gör att tågen kan
köras i enlighet
med den faktiska
trafiksituationen och
optimeringsberäkningarna
baseras
alltså på den operativa, uppdaterade,
tidtabellen.
Fjärrtågklarerarna
på
driftledningen
får
dessutom kontinuerlig feedback från
CATO så att de vet
att de planer som
görs är möjliga och
Figur 1: LKAB:s IORE-lok, nu med energioptimerande hastighetsförslag och samordnad möteshantering styrd från
på väg att uppfyllas.
driftledningscentralen. Foto: LKAB.
24
ORbit 19
artikel
är en fråga om att alltid använda den
för stunden tillgängliga körtiden på ett
optimalt sätt, utan att riskera försening.
Även i den fasta tidtabellen finns det ett
visst tids-slack, dvs för att uppnå robusthet är körtider i tidtabellen lite långsammare än om tågen skulle kört på absolut
kortast möjliga tid.
Figur 2: Lokförarens gränssnitt till CATO
en bland annat hastighetsförslag och
information om aktuell tidtabell.
CATO gör det möjligt för trafikledningen
att kontrollera tågrörelserna vilket gör att
både lokförare och fjärrtågklarerare kan
lita på att tåget kommer fram till målpunkten i tid och med önskad hastighet.
Onödiga inbromsningar och stopp kan
undvikas.
Den tillgängliga körtiden på en bansträcka kan ändras från dag till dag
beroende på aktuell trafiksituation. Det
gör att den optimala hastighetsprofilen
också varierar mellan olika tåg och olika
dagar. Den optimala hastighetsprofilen
kan i själva verket vara mycket
olik den hastighetsprofil som
normalt väljs av föraren.
Marknadsutsikterna för system som
CATO är mycket goda. Det finns en ständig press på järnvägarna för att minska
driftskostnaderna och energiåtgången,
samtidigt som robustheten i järnvägsnätet måste förbättras. Ökad järnvägstrafik
och behovet av att använda infrastrukturen effektivt i kombination med stigande
energikostnader och miljömedvetenhet
CATO
minskar
driftskostnaderna, kommer att driva efterfrågan på system
förbättrar punktligheten och ökar tra- som CATO.
fikkapaciteten i infrastrukturen. Möjligheten att hantera störningar i trafiken
förbättras kraftigt. Lokföraren är alltid
informerad om den aktuella trafikplanen
och trafikledaren vet att tågen kommer
Martin Joborn
att köras enligt den gällande planen.
Minskade driftskostnader är inte bara en
har disputerat i
fråga om minimerad energiförbrukning,
optimeringslära
utan också t.ex. förbättrad punktlighet,
vid Linköpings
minskat slitage på bromsar och bana,
Universitet inom
bättre utnyttjande av rullande materiel
transportoptimeoch personal, etc. Dessa är alla stora
ring. Han arbefördelar för järnvägssystemet i allmäntar på Transrail
het, för operatörer, infrastrukturförvaltare
Sweden AB som senior konsult och
och kunder. Kostsamma investeringar i
optimeringsexpert. Han har även
infrastruktur kan undvikas eller minskas
eget litet företag som utvecklar
genom användning av CATO. CATO
programvara för schemaläggning
klarar en mycket hög grad av komplexiav sportevenemang. Martin är
tet vad avser typ av dragfordon, rullmotmedlem i SOAF:s styrelse sedan
stånd, linjeprofil, väderförhållanden etc.
2003 och var ordförande 2008vilka alla kan hanteras av systemets
2011.
optimeringsalgoritmer.
Storleken på energibesparingen
som uppnås med CATO beror
på ett antal faktorer, t.ex. den
vertikala profilen på linjen, typ
och vikt av tåget och tillgänglig
körtid. Beräkningar och tester
visar att CATO kan minska energiförbrukningen med så mycket
som 20-25% även om tågen
körs med en högre än normal
medelhastighet. Optimeringen Figur 3: Arlanda Express med CATO är både punktligt och energioptimalt. Foto: Patric Johansson
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25
artikel
By Karina H. Kjeldson, Berit D. Brouer, and Christian E.M. Plum
ESI 2012
EURO Summer Institute on Maritime Logistics
June 3-15 2012, Bremen, Germany
EURO organiserede i perioden 3. Juni – 15. juni, 2012 et
såkaldt EURO Summer Institute (ESI) om Maritime Logistics
i Bremen, Tyskland. EURO financierede deltagelsen og
opholdet og DORS financierede transporten til og fra Bremen.
Antallet af deltagere var begrænset til 17 personer, som
skulle udvælges af EURO blandt alle indstillede kandidater
fra de nationale OR-selskaber i EURO. DORS opstillede 3
kandidater (Karina Hjortshøj Kjeldsen, Berit Dangaard Brouer
og Christian Edinger Munk Plum) og til stor glæde blev alle
3 kandidater fra DORS udvalgt! Virkelig flot, og noget DORS
kan være stolt af, og der skal lyde et stort TILLYKKE til
Karina, Berit og Christian. Læs om deres tur her.
The summer institutes
In October 2011 DORS nominated three
applicants for participation in the EURO
Summer Institute on Maritime Logistics
(ESI2012). The applications were sent
to EURO and the final selection of
participants was made by a Scientific
Committee on the basis of the submitted
papers. From all over the world 23 early
stage researchers were accepted for
ESI2012 representing universities from
Belgium, Denmark, France, Germany,
Italy, the Netherlands, Norway, Switzerland, Turkey, Ukraine, The United Kingdom as well as Australia, Brazil, India,
New Zealand and the United States. In
January 2012 we were informed that all
Danish applicants had been accepted.
The series of EURO Summer and Winter
Institutes (ESWIs) was launched in 1984
at the initiative of J.P. Brans. The idea is
to gather a group of early stage researchers within a particular field of operations research, by arranging a two week
conference for them to present and
discuss their research with each other
and a selected group of invited senior
experts in the field. EJOR has a special
issue for the ESWIs for the research
papers presented at the institute. The
2012 Summer Institute focused on Maritime Logistics and was jointly organized
by the Universities of Bremen and HalleWittenberg and was held in Bremen
The EURO summer institute 2012 was
from June 3rd till June 15th 2012.
sponsored by EURO, the Gesellschaft
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for Operations Research (GOR) and
the organizing universities. Participation
and accommodation was free for all participants. Additionally, the travel costs
were covered by DORS and we are
very grateful to all sponsors for giving
us the opportunity to participate in the
ESI 2012.
Two weeks on maritime
logistics
The Summer Institute started on Sunday
the 3rd of June with a welcome reception at the Bremen youth hostel, which
was the base camp of the ESI2012. The
reception was an opportunity for the 23
participants to meet each other and the
artikel
On Monday afternoon a city tour of
Bremen with native guide Prof. Dr. Christian Bierwirth was organized. We were
introduced to the City Hall of Bremen
and the history of the Hanseatic cities,
which gave a good introduction to the
strength of maritime trade in this city.
Wednesday we visited the impressive semi-automatic container terminal
Hamburg Altenwerder. The visit was
guided by Prof. Stefan Voß, University
of Hamburg, who has been involved in
the design of the terminal, which makes
extensive use of decision support and
automation of the terminal handling
operations. Afterwards, Prof. Stefan Voß
gave a lecture on recent developments
Figure 1: From the left: Prof. Dr. Stefan Voß (University of Hamburg), prof. Dr. Christian
in seaport terminal management at the
Bierwirth (Martin-Luther-Universität Halle-Wittenberg), Dr. Frank meisel (Martin LutherUniversity of Hamburg. The day ended
Universität Halle-Wittenberg) and Prof. Kjetil Fagerholt (NTNU), at Bremenports.
with an unforgettable, but rainy city tour
two main organizers Prof. Dr. Christian rations. Six participants presented their
of Hamburg.
Bierwirth (Halle-Wittenberg) and Dr. work within the subject supplemented by
Frank Meisel (Halle-Wittenberg).
a lecture by Prof. Dr. Christian Bierwirth
Thursday and Friday the focus was on
on Seaside operations planning in conHinterland Transportation. Five particiThe scientific programme was based tainer terminals and a discussion round
pants presented their research within
at the Bremen University Guest House on seaside port operations moderated
this field followed by a lecture and a
and started Monday June 4th. During by Dr. Frank Meisel, who both have
discussion round by Prof. Rommert
the course each participant was given extensive experience within the area.
90 minutes for presenting a research
paper followed by
a group discussion
with the aim of preparing the research
paper for a publication. The scientific
programme
was
divided into five
overall
subjects:
Seaside port operations, Hinterland
Tr a n s p o r t a t i o n ,
Yard Management,
Maritime Ship Routing and Maritime
Network
Design.
The first two days
we focused on
seaside port ope- Figure 2: The ESI2012 participants at Bremen city hall.
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artikel
Dekker, Erasmus University Rotterdam.
The presentations took place at the
Bremen University Campus. We were
invited on a campus tour, which included
a visit to the fascinating drop tower at
the Center of Applied Space Technology and Microgravity. The campus tour
made a final stop at a biergarten close
to Campus.
The first subject of the second week was
yard management. There were five presentations by participants and a lecture
by Prof. Iris Vis, University of Groningen,
on Equipment selection and yard operations in container terminals. The session
on yard management was concluded
with a discussion round moderated by
Prof. Iris Vis assigning us into groups to
discuss new research directions within
the field. On Tuesday Prof. Hans-Dietrich Haasis, University of Bremen gave a
short presentation and introduced Prof.
Mai Sha, Shanghai Maritime University,
who talked about intermodal networks
in China. Tuesday in the afternoon the
session on Maritime Ship Routing and
Scheduling was introduced with two
participant presentations.
Wednesday we visited the city of Bremerhaven. The morning was hosted by
Bremen-ports and included a presentation of the company. The presentation
helped clarify the differences between
a port company and a terminal operator.
The morning ended with a very interesting presentation on maritime ship
scheduling and inventory routing given
by Prof. Kjetil Fagerholt, Norwegian University of Science and Technology. After
lunch at the National Maritime Museum
we visited the automobile terminal
Bremerhaven. The visit started with
two presentations where the first gave
a general presentation of the terminal
and the second one focused on the
terminal’s involvement in the wind mill
industry in the Northern Sea, which was
very impressive. After the presentations
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we took a tour of the automobile terminal
itself and it is safe to say that several of
the participants were dreaming of beautiful cars on their way back to Bremen.
Thursday we continued the focus on
maritime routing and inventory routing
with two participant presentations and
a discussion round moderated by Prof.
Kjetil Fagerholt. The discussion round
focused on best practice in scientific
writing and publishing, of great value
to all of us. The last day of the institute,
Friday, focused on maritime network
design and included three presentations
by participants. The evening was set
aside for the farewell barbecue, which
was a great success.
Thank you for a great event
Before going to the ESI2012 we all felt
that two weeks was a very long time to
set aside for the summer school, but
on the last day of the summer institute,
when we were saying our goodbyes, it
was obvious that we had made strong
connections during our two weeks
together. We had had the opportunity
to discuss our research and life in the
academic world with peer researchers
and we had many laughs together. After
the ESI2012 we are still in contact with
some of the participants for personal
and professional reasons. The ESI2012
was truly well organized with a great
location, inspiring guest lectures, relevant excursions and great social events.
We have gained insight into related
research areas within maritime logistics
and have larger insight into the maturity
of the various research fields. We have
also expanded our network within maritime operations research, which is very
valuable in this relatively small research
field. We would like to thank the organizers, sponsors, guest lecturers and
participants of the ESI2012 for a great
event.
Karina Hjortshøj Kjeldsen
holds a Ph.D.
from Aarhus
University.
Karina previously worked
at CORAL at
Aarhus University. Her research interest is optimization in the liner shipping industry.
Berit D.
Brouer
is a Ph.D.
student at
the Technical
University
of Denmark
(DTU). The
Ph.D. thesis concerns liner shipping
network design and is part of the
ENERPLAN project.
Christian E.M.
Plum
is a Industrial
Ph.D. student
at the Technical
University of
Denmark (DTU)
and Network
Expert at Maersk Line - Network
Strategy. The Ph.D. thesis concerns
liner shipping network design.
artikel
By Steen Nielsen and Erland H. Nielsen
Strategic Balanced Scorecard Simulation
The purpose of this article is to show how a System Dynamics
Modelling approach can be integrated into the Balanced
Scorecard (BSC) for a case company with special focus on
the handling of causality in a dynamic perspective. The case
company’s BSC model includes five perspectives and a
number of financial and non-financial measures. The overall
idea of BSC is to make the strategy operational, as proposed
by Kaplan and Norton (1992; 1996; 2007) and to use the
strategy for simulation. Our results indicate that a company
may gain great learning insight from such simulation studies.
The whole article will be published in a coming issue of
Journal of Business and Systems Research.
Introduction
Modelling is a principal tool for studying complex systems that
may be used for prediction, for analysis or for prescription
The Balanced Scorecard (BSC) emerged in the 1990s almost (Simon, 1990). In a BSC perspective predictions concern difat the same time as two other ideas, Economic Value Added ferent strategies, their related costs, other KPIs and perspec(EVA) and Activity-Based Costing (ABC). Implementation and tives, and finally the effect on the company’s profitability.
utilization of these practices or models could get quite confusing, particularly since each is being passionately advocated To take full advantage of the link of measures and KPIs, meaas the solution for reducing expenses or improving organiza- sures and measurement should be integrated into a single
tional performance. However, each can be implemented inde- management system called a Strategic Management System
pendently, but organizations will derive the greatest benefit by as shown in figure 1.
integrating several of them (Kaplan, 2006). Several surveys
specifically investigate the tendency towards combining elements of different accounting practices have emerged during
the last 10 years.
Original, BSC translates the company’s mission and strategy
into objectives and key performance measures. It recommends
and offers horizontal and vertical views between financial and
non-financial measures, between external and internal measures, between the short and the long term, and between hard
and soft values. BSC should also implies a balance between
lead and lagging indicators (Kaplan & Norton, 1996).
Translating the Vision
Clarifying the vision
Gaining consensus
Feedback and
Learning
Communication and
Linking
Communicating and
education
Setting goals
Linking rewards to
Performance
measures
Balanced
Scorecard
Articulating the
shared vision
Supplying strategic
feedback
Facilitating strategy
review and learning
Business Planning
Setting targets
Aligning strategic
Initiatives
Allocating resources
Establishing
milestones
Our approach shows how different metrics and indicators can
provide the basis for further development of BSC into a strateFigure 1: The strategic management system (Kaplan & Norton,
gic quantitative model.
1996, 2007).
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when managers question their underlying assumptions and reflect on whether
the theory under which they were operating remains consistent (Argyris and
Schön, 1978).
In contrast to Kaplan and Norton (1996)
and Lynch and Cross (1991), the final
choice for the company was to bring
forward the education and training for
the employees and the product innovation and development perspectives. The
main outline and ideas of the BSC from
the case company are shown in figure 2.
The figure includes the classical perspectives and the design of a BSC,
where each perspective is formed by a
Figure 2: Perceived main relations within the balanced scorecard model.
number of KPI’s. Figure 3 illustrates the
Figure 1 puts focus on the strategic aspects of BSC. Innocompany’s design of the main informavation companies must use BSC as a strategic management tion flow and shows how these flows interact and that the
system to manage their strategies in the long run (Kaplan and perspectives interact with each other e.g., by feedbacks.
Norton, 1996). Specifically interesting for the system dynamics approach used in this paper is the feedback and the The vision of the company is a lean and learning strategy as
double-loop learning process. Double-loop learning occurs described by Womack and Jones (2003). The closed loop
Figure 3: The BSC converted into a SDM structure (SD-diagram).
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artikel
causal reasoning described above fits directly into the system
dynamics modelling tradition as it is used within VENSIM™,
which by the way is the software used for this project.
VENSIM™ has all the nice essential features for developing a
complete system dynamics model, e.g., exploration, analysis
and optimization of simulation models (Eberlein & Peterson,
1992). However, one of the basic problems is the problem of
causality in general and in system dynamics specifically.
in relation to this (Barlas, 1996). Below we have demonstrate
two simulation scenarios by the use of system dynamics and
the quantitative BSC model.
Simulation scenario 1
The model is formulated as a monthly model over a five-year
period, and the base-run is set for a company with a ’Production Lead Time’ of six months. This is quite a lengthy production time span and though the model is a ’pull’ based logistics
The system dynamic model approach
structure, the production lead time clearly disqualifies it from
The separate parts above are linked to form a complete being lean at the outset. The outcome is shown in figure 4.
dynamic model. The dynamic and integrated construct makes
it possible to do simulation and strategic learning and to do From the ’Customer Perspective’ we see that ’Customer
real-time research. Strategic learning consists of gathering Loyalty’ is constant over the entire period, and although the
feedback, testing the hypotheses on which the strategy was variable ’Recommendations’ oscillates, its average value
based and making the necessary adjustments. This is exactly does not decrease over time.
what the scorecard should give managers ( Kaplan and
Norton, 2007). If, for instance, a company’s employees have
improved the performance in relation to some of the drivers
(reduced lead times or increased capacity for example), then
a failure to achieve the result that was to be expected based
on the model (higher EVA for example) mostly likely signals
that the model is not fully correct and adequately comprehensive yet. The complete BSC model in the SDM layout is
shown in figure 3.
From the ’Process & Supply perspective’/’Lean perspective’
we note that the company is facing a growing market. The
’Demand’ increases from an initial value of 2,000 to slightly
above 5,000 at the year five. We further notice that all activity
variables are increasing on average with moderate oscillations and no signs of alarm. However, a slight warning turns up
when the ’Average Unit Throughput Cost’ is considered – it
seems to show a non-decreasing path, which is not a good
sign.
However, it is important to keep focus on the purpose of the
model and to continuously perform the necessary validation The ’Financial Perspective’, however, intensifies these worries
even further. Though the
’Profit’ seems to increase
over the five-year period,
the EVA and RoCE measures seem to indicate
that business deteriorates over the five-year
period. Combined with
the message given by
’Average Unit Throughput Cost’, the company is
facing a difficult or even
bad situation. Action
is definitely required in
order to improve the
general business of the
company.
Figure 4: Output and KPIs from the cockpit model.
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artikel
shows a persistent positive economic tendency
– in our scenario setup
this amounts to approximately
two
years.
Especially the drop in
the RoCE measure after
six months might be
very worrying in order
to maintain confidence
in the decision to lower
the ’Production Lead
Time’. One could easily
imagine the discussions
that might arise between
logistics people and
controlling people in the
company, especially if
they do not have a full
BSC ’cockpit’ at their
disposal.
Figure 5: Output and KPIs from the cockpit model.
Simulation scenario 2
This scenario is characterized by a reduction of the ’Production Lead Time’ from six months to three months and an
increase in the number of staff trainings from zero to five
training days per month. Though the nature of the analysis
has not – so far – been very general, some characteristically
features do, however, become evident. The results are shown
in figure 5.
Conclusions
We have used the BSC theory as the basic research frame
for developing and testing a holistic model with illustrations
from different lean scenarios on the financial perspective. A
major issue of senior executives is ‘How do I make sure that
the organization is executing our strategy?’ The research idea
was also motivated by the growing pressure on companies to
introduce more holistic decision models for decision-making
Increasing the ’Number of Trainings’ clearly has a significant
productivity effect as should be expected. From the ’Employees Perspective’ plot we note that the ’Staff’ curve is now
lower than the ’Staff Capacity’ curve owing to the increase in
productivity. However, it can also be noted that the general
need for ’Staff Capacity’ has decreased in this scenario, the
most probable reason being the decrease in the ’Production
Lead Time’. It is a well-known fact that a decrease in the ’Production Lead Time’ has a very positive effect in pull systems
as it increases the performance in all logistics aspects. However, with regard to the financial measures, this well-behaving
aspect seems to tell a somewhat different story. ’Profit’ is not
the best measure to tell us much about the general healthiness of the business. In our scenarios it increases simply
because the market is increasing, whereas both the ’EVA’
and the ’RoCE’ measures did show definitely warning signals
when the ’Production Lead Time’ is at its high value. Lowering
the ’Production Lead Time’ significantly turns ’EVA’ into a positive financial signal, whereas the ’RoCE’ only after some time Figure 6: The simulation framework.
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artikel
(Otley 1999; Kaplan and Norton, 2007). The following figure
6 puts emphasize on the strategic feedback learning process,
where system dynamics simulation models play an important
role. The company can, therefore, use the model in their
active planning for determining when to change or revise the
strategy.
Design, forthcoming, in International Journal of Business and
Systems Research.
• Otley, D. (1999) ‘Performance Management: A Framework
for Management Control Systems Research’, Management
Accounting Research, Vol. 10 (4), pp. 363-382.
• Simon, H. A. (1991) ‘Bounded Rationality and Organizational
Learning’, Organization Science, Vol. 2 No. 1, pp. 125-134.
As pointed out by Kaplan and Norton (1996, 2007), it is vital • Sterman, J. D. (2000) Business Dynamics. System Thinking
that the company is able to evaluate different strategies on and Modelling for a Complex World, McGraw-Hill Higher Eduthe financial result before, during, and after they are executed. cation, Boston USA.
Our model gives this opportunity to get what Kaplan and
Norton (2007) call ‘strategic learning’.
As pointed out by Sterman (2000) and Simon (1990), the use
of SDM may be seen as a natural development for complex
situations. Converting and testing more factual and complicated models are important. An advantage for such a model
approach is that a company can make risk-free experiments
by changing some parts of the model, a strategy or a variable.
It may also be the basis for further discussion of the financial
benefits for both a short and a long run perspective.
References
• Argyris, C. and Schön, D. A. (1978) ‘Organizational Learning: A Theory of Action Perspective’, Addison Wesley
Company, USA.
• Barlas, Y. (1996) ‘Formal Aspects of Model Validity and
Validation in System Dynamics’, System Dynamics Review,
Vol. 12, No. 3 Fall, pp. 183-210.
• Eberlein, R. K., Peterson, D. W. (1992) ‘Understanding
Models with Vensim’, European Journal of Operational
Research, Vol. 59, 216-219.
• Kaplan, R. S. (2006) ‘The Competitive Advance of Management Accounting’, Journal of Management Accounting
Research, Vol. 18, pp. 127-135.
• Kaplan, R. S. and Norton, D. P. (1992) The Balanced Scorecard – Measures that Drive Performance, Harvard Business
Review, No. 1, pp. 71-79.
• Kaplan, R. S. and Norton, D. P. (1996) The Balanced Scorecard – Translating Strategy into Action, Harvard Business
School Press, Boston, USA.
• Kaplan, R. S. and Norton, D. P. (2007) ‘Using the Balanced
Scorecard as a Strategic Management System’, Harvard
Business Review, July-August, pp. 150-161.
• Lynch, R. and Cross, K. E. (1991) Measure Up! Yardstick
for Continuous Improvement, Cambridge, Mass., Basil Blackwell.
• Nielsen, S., and Nielsen, E.H. (2012). Transcribing the
Balanced Scorecard into System Dynamics: From Idea to
Erland Hejn Nielsen
is currently Associate Professor
at at Department of Economics
and Business, Business and
Social Science, Aarhus University,
Denmark.
He received his
MSc (Econ/Math) degree from
University of Aarhus, Denmark. He
is currently a member of Cluster for
Operations Research and Logistics (CORAL), situated
at the Department of Economics and Business, Aarhus
University, Denmark.
Steen Nielsen
is currently Associate Professor
at Department of Economics and
Business, Business and Social
Science,
Aarhus
University,
Denmark. He received his Master’s
degree in Accounting from Aarhus
School of Business, and his PhD
in Cost & Management Accounting
from the Stockholm School of Economics. He is currently a
member of Cluster for Operations Research and Logistics
(CORAL), situated at the Department of Economics and
Business, Aarhus University, Denmark.
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33
artikel
Af Anders Skajaa og Gitte Øskov Skajaa
Om optimal tildeling af KBU-forløb
Vi viser i denne artikel hvorfor Sundhedsstyrelsens
nuværende lodtrækningsmetode til fordeling af KBUforløb (tidligere kaldet turnus) er fundamentalt defekt og
i matematisk forstand suboptimal. Ved hjælp af simple
eksempler anskueliggøres problemet og vi foreslår som
erstatning en optimal metode, som bygger på kendte idéer fra
matematisk optimering. Vi demonstrerer, at en kandidat med
denne optimale tildelingsmetode i gennemsnit kan forvente
at blive tildelt et KBU-forløb der ligger to-tre gange højere
på kandidatens prioriteringsliste. Endelig diskuteres hvordan
denne metode kan implementeres i praksis og hvordan man
kan opnå yderligere forbedringer med advancerede moderne
matematiske metoder.
Introduktion
Efter færdiggørrelsen af kandidatgraden i medicin fra et af
landets universiteter, skal de danske læger, som næste skridt
i deres læge-uddannelse, gennemføre den såkaldte kliniske
basisuddannelse (KBU), der i 2008 erstattede turnus-uddannelsen. Et KBU-forløb består af to halvårlige stillinger, begge
i den samme region i landet. Uanset hvorfra den nye læge
er uddannet, besættes disse stillinger alene ved lodtrækning.
Denne proces forløber ved, at landets i alt N kandidater ved
lodtrækning tildeles tal mellem 1 og N . Fra et givet tidspunkt
har kandidaten med nummer 1 så ti minutter til frit at vælge
blandt de N tilgængelige forløb over hele landet. Derefter
vælger nummer 2 blandt de resterende N − 1 forløb osv. For
tiden har vi ca. N = 400 .
Denne seance er ofte til stor debat og frustration blandt
medicinstuderende fordi der er stor forskel på populariteten
af forløb. Variationen i popularitet skyldes dels den geografiske placering såvel som de stillinger det enkelte KBH-forløb
indeholder. Tildelingsmetoden resulterer ofte i, at personer
der bor, har læst, har venner og eventuelt familie i én region,
bliver tvunget til en helt anden region af landet. Dette er naturligvis til stor skade for denne person og dennes familie og
frygten for at dette sker, har gjort tildelingsmetoden upopulær.
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Men hvordan kan KBU-forløbene tildeles på en mere rimelig
og hensigtsmæssig måde?
I denne artikel vises hvorfor den nuværende procedure fra
et matematisk synspunkt er ugunstig og beviseligt suboptimal. Desuden gives et forslag til hvordan man med allerede
kendte matematiske metoder kan fordele KBU-forløbene så
flere studerende bliver tilfredse - og lige så vigtigt: færre bliver
utilfredse.
Et illustrerende eksempel
Lad os benævne de N nye kandidater som K1 , K 2 ,  , K N .
Tilsvarende findes N KBU-forløb F1 , F2 ,  , FN .
I det følgende illustrerende eksempel antager vi N = 4 . Vi
skal altså fordele K1 , K 2 , K 3 og K 4 på de 4 forløb F1 , F2 , F3
og F4 . Hver kandidat prioriterer nogle forløb fremfor andre, og
vi antager, at alle kandidaterne kan nedskrive en prioriteret
liste med alle forløbene. For eksempel således:
F1 F2 F3 F4
K1 1 4 2 3
K2 1 2 4 3
K3 2 1 4 3
K4 2 3 1 4
artikel
Tabellen forstås sådan, at hver række viser en kandidats prio- Disse to sammensætninger er tydeligt bedre, hvilket kvaliteteriterede rækkefølge af forløbene.
smålet Q også viser.
Sundhedsstyrelsens procedure er at trække lod. Hver kandidat tildeles et nummer fra 1 til 4. Lad os sige at lodtrækningen
faldt ud sådan at K 2 vælger først, efterfulgt at K1, K 4 og til
sidst K 3. Hvis alle kandidater vælger dét tilgængelige forløb,
der står højest i deres prioriteringsliste, vil K 2 først vælge F1
. Dernæst vælges F3 af K1, K 4 vælger F2 og til sidst er K 2
tvunget til at vælge F4. Således:
Lad Qopt være den bedst mulige (eller optimale) værdi af Q
for en given tabel. Den værdi af Q, der i gennemsnit opnås
med lodtrækning kalder vi Q lod .
Det er klart, at det er nemt at bestemme sådan en optimal
sammensætning når N = 4. Vi kan bare betragte tabellen et
øjeblik og bestemme den. Men prøv nu at gøre det samme i
denne tabel:
K1
K2
K3
K4
K5
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10
1 5 7 3 4 6 8 9 10
2
8
9 2 5 3 10
1 7 6 4
8
4
4 10
1 9
5
8
6
7
2
2
3
6
4
5
5
9
4
8
1
2
3
7
1
4
8
1
2
2 10
7 1
9
8
7
5
8
3
3
9
3
7
2 10
9
1
3 7
6 10
1
5
9
3
6 8
K 6 2 10
5
K 7 10
K8 4 5
6
K 9 10
9
7
6
7 10
9 1
3 2
6
4
K10
4
5
2. Summen af tallene i en søjle kan fortolkes som et mål for
den generelle popularitet af et forløb. Jo mindre sum, jo mere I tabellen ovenfor er
skueligt problem.
populært.
N
Vores problem har følgende egenskaber:
1. I hver række af tabellen optræder tallene 1, 2, 3, 4 præcis
én gang hver.
6
8
blot lig 10 og det er allerede et uover-
3. En bestemt sammensætning af kandidater og forløb svarer Man vil opdage, at et valg foretaget tidligt for en bestemt
i tabellen til at lave én cirkel i hver række sådan at der også kandidat har konsekvenser for senere kandidater og netop
dette gør problemet kompliceret. For eksempel: En kandikun er én cirkel i hver søjle.
dat, der egentlig er ligeglad om han får forløb F1, F2 eller F3
4. Gennemsnittet af prioriteterne for de realiserede parringer , vælger bare ét af dem, lad os sige F2. Den næste kandidat
kan fortolkes som et mål for kvaliteten af den samlede sam- ville meget hellere have F2 end F1 og F3 , men er nu tvunget
mensætning af kandidater og forløb.
til at vælge F1 eller F3. Han vælger F . Den næste kandidat
1
ville allerhelst have F1 , men skal nu vælge F3. Altså bliver to
Kvaliteten af en sammensætning
ud af tre kandidater utilfredse, og dét på trods af, at alle kunne
have været tilfredse.
Vi kan definere et kvalitetsmål Q som summen af de
udvalgte prioriteter divideret med N . For en bestemt sam- Dette eksempel blotlægger den helt fundamentale defekt ved
mensætning er Q altså gennemsnittet af de realiserede lodtrækningsmetoden. I lodtrækningsmetoden gennemløbes
prioriteringer. For vores specifikke lodtrækning har vi Q en tabel som dén ovenfor række for række (kandidat for kan= (1 + 2 + 3 + 3)/4 = 2.25 . Men se nu følgende to sam- didat). Men i hvert skridt bruges kun den information, som er
mensætninger, lavet over samme tabel:
indeholdt i dén ene række. Dette tildelingsproblem er imidlertid af en sådan karaktér, at det kun kan løses optimalt, hvis
man i løsningsproceduren simultant bruger al den information,
som tabellen indeholder.
Den Matematiske Løsning
Når N = 400 er det umuligt for mennesker at løse dette problem optimalt indenfor overskuelig tid. Men fra matematikken
ORbit 19
35
artikel
findes systematiske procedurer (algoritmer) til at bestemme
en optimal sammensætning for en given tabel. Vi behøver
ikke forstå detaljerne i, hvordan en sådan virker, men bare
vide, at den bestemmer en optimal sammensætning.
inkluderer ophold
i intern medicin
- vil fordelingen
af populariteter af
forløb være af en
Hvis kandidaternes prioritering af forløbene er uniformt type som dén der
fordelt, er det matematisk bevist1, at Q lod ≈ ln ( N ). Det er er vist i figur 1.
også bevist, at 1.51 ≤ Q opt ≤ 2 . Altså: Hvis populariteten af
forløbene var uniformt fordelt og man bruger Sundhedsty- Hvis vi genererelsens lodtrækningsmetode, kan man ca. forvente at få sin rer en fordeling
sjette prioritet opfyldt (da ln (400) ≈ 6 ). Med den optimale som denne og Figure 1: Mest realitisk fordeling af
sammensætning i stedet, kan hver kandidat ca. forvente at få derpå
udfører populariteten af forløb.
sin anden prioritet opfyldt. Selvom populariteten af alle forløb vores
ekspebestemt ikke er lige stor, fortæller disse resultater noget om riment
som
beskrevet
ovenfor
finder
vi
ca.
egenskaberne ved de to tildelingsmetoder - altså om hvor Q lod = 9.98 og Q opt = 4.23, og derfor nu R ≈ 0.42. Alså
gode de er i forhold til hinanden.
en markant bedre sammensætning ved brug af vores optimeringsalgoritme.
Når populariteten af forløbene er fordelt mere skævt, kan vi
opnå indsigt i hvor meget bedre den optimale tildeling er end Dette resultat kan belyses lidt bedre: Hvis man tæller antallet
lodtrækningsmetoden ved at udføre numeriske eksperimenter af kandidater der får deres første prioritet opfyldt, antallet der
i en computer.
får deres anden prioritet opfyldt, deres tredje, deres fjerde og
så videre, og derefter plotter disse antal med prioriteten ud
Numeriske eksperimenter
ad x -aksen, får man et billede som det, der er vist i figur 2.
Denne figur er dannet ved at generere en fordelingstype som
I dette afsnit undersøges via eksperimenter hvor meget bedre den realitiske i figur 1, udregne sammensætninger for begge
sammensætningen af kandidater og forløb kunne blive, hvis metoder og så optælle som beskrevet ovenfor. Alt dette er
man bestemt den optimale sammensætning ved at bruge gjort 1000 gange og figur 2 viser gennemsnitsfordelingen over
en optimeringsalgoritme. For at udføre disse eksperimenter, disse 1000 forsøg.
følger vi denne procedure:
Vi ser tydeligt hvordan lodtrækningsmetoden er til ugunst for
langt flere kandidater. Begge metoder har en klump sandsynlighedsmasse koncentreret til venstre. Men lodtrækningsmetodens klump er bredere – og den fortsætter langt ud i de
kedelige numre. Man kan altså med den optimale metode
i langt højere grad forhindre, at kandidater ufrivilligt bliver
2. Algoritmen bruges så til at finde den optimale sammensættvunget ud i forløb, de ikke ønsker og/eller er langt fra deres
ning af kandidater og forløb.
hjem.
1. For hver af de N = 400 kandidater genereres en tilfældig prioriteringsliste - dog sådan at populariteten af forløb er
fordelt på en bestemt måde (se nedenfor). Vi danner altså en
tabel som dem vist ovenfor.
3. Nu genereres tilfældigt 100 lodtrækninger og det bestemKan det så bruges i virkeligheden?
mes hvilken sammensætning hver af disse giver. Hver kandidat vælger blot sit højest placerede forløb som stadig er Ja. Men det er åbenlyst, at metoden skal tilpasses virkelighetilgængeligt (som i virkeligheden). Vi beregner endelig det den en smule. Man kunne fx indvende følgende:
gennemsnitlige Q for disse 100 tildelinger, resultatet kaldes
Q lod.
1. Kan man virkelig få alle cand.med’ere til at prioritere alle
400 forløb? Principielt ja, men i praksis er det nok svært. Hel4. Endelig kan vi så udregne R = Q opt //Q lod. Hvis fx R = 0.50 digvis er det ikke nødvendigt. Der findes metoder, hvor alle
var vores metode ca. dobbelt så god som lodtrækningsmeto- ikke behøver prioritere alle forløb, når bare alle forløb er på
den.
nogens prioriteringsliste. Men det er klart, at jo bedre og jo
mere information man har (~ jo længere prioritetslisterne er),
Da visse områder og forløb er specielt eftertragede blandt
jo bedre bliver den optimale sammensætning.
kandidaterne - særligt de store byer og specielt forløb der
36
ORbit 19
artikel
udvælgelsen endnu bedre hvis man stiller kandidaterne
spørgsmål, der afslører endnu mere information: Eksempel: Alle kandidater prioriterer de enkelte komponenter af
et kbu -forløb såsom typen afdeling, speciale, område,
startdato og gør dette uafgængigt af hinanden. Desuden
vægtes disse faktorer. Man kan så bruge denne information i en slags interpolations-metode. Dette kan gøres på
ret sofistikeret vis ved at løse det såkaldte Collaborative
Filtering problem2. Det er med lignende metoder at store
digitaliserede virksomheder (fx Amazon, Netflix, Facebook etc.) laver intelligente anbefalinger af produkter til
forbrugere baseret på deres købe– og adfærdshistorie.
Afsluttende bemærkninger
Figure 2:Fordeling af realiserede prioriteter for de to metoder.
Bemærk logaritmisk skala på den vertikale akse.
2. Er Q det bedste mål for kvaliteten af en bestemt sammensætning? Ikke nødvendigvis. Men man kan nemt bruge
et andet kvalitetsmål ved fx at lade kandidaterne vægte deres
prioriterede forløb med andre tal end 1,2, , N . Fx kunne
man beslutte, at hver kandidat skulle fordele et antal »point«
ud over forløbene – flere forløb måtte gerne tildeles samme
antal point, men der skulle være en øvre grænse for hvor
mange point et forløb kunne tildeles.
3. Bliver dette ikke meget tidskrævende for kandidaterne og
Sundhedsstyrelsen? Under det nuværende lodtrækningssystem tager den samlede parring af kandidater og forløb
ca. seks arbejdsdage to gange om året. For hver kandidat
kulminerer det med ti minutter bag en computerskærm. Under
det foreslåede system skal hver kandidat i løbet af en fast
periode udarbejde en liste med prioriteter og evt svare på
nogle spørgsmål. Vores gæt er, at kandidaterne er villige til
at lægge denne smule arbejde, for derimod at få en meget
højere sandsynlighed for at få et tilfredsstillende forløb.
Hvis rigtig mange kandidater alle foretrækker meget få
forløb, bliver det sværere at stille mange tilfreds – også
selvom man bruger denne optimale metode. Men
bemærk at det bliver ligeså meget sværere for lodtrækningsmetoden. Altså vokser både Q opt og Q lod .
Det vigtige at forstå for den enkelte kandidat er, at man før
lodtrækningen er kendt har større sandsynlighed for at få
et forløb højt på sin liste ved den optimale metode end ved
Sundhedsstyrelsens lodtrækningsmetode. Sandsynligheden
for at ligge øverst i lodtrækningen er jo forsvindende lille.
Dette er specielt tydeligt illustreret i figur 2.
Den foreslåede metode fjerner selvfølgelig ikke upopulære
KBU-forløb. I stedet gives de til de kandidater, der bedst
accepterer dem. Hvis ingen accepterer dem, svarende til, at
alle prioriterer dem nederst, bliver Q opt højst lig Q lod, men
aldrig højere. Altså kan den foreslåede metode ikke levere en
sammensætning af kandidater og forløb der er ringere end
lodtrækningsmetoden, kun bedre. Og i reglen, meget bedre.
Noter
1. Se fx Rainer E. Burkard, Mauro Dell’Amico, Silvano Martello:
4. Får man ikke bare en computer der sender folk ad pom- Assignment Problems. SIAM 2009.
mern til? Husk nu: Det har man allerede! Forskellen er, at 2. Se fx http://en.wikipedia.org/wiki/Collaborative_filtering
denne nye metode, imodsætning til den nuværende, udnytter langt mere af den tilgængelige information til at træffe en
intelligent beslutning. Se specielt figur 2: Sandsynligheden
for at blive tilfreds og specielt for ikke at blive utilfreds vokser
Anders Skajaa er PhD-studerende i anvendt matematik
markant.
på Danmarks Tekniske Universitet.
5. Kan det egentlig ikke gøres endnu bedre? Jo, sagtens. Den
i denne artikel skitserede metode er næsten den simpleste
og mest naive tilgang til denne type problem. Man kan gøre
Gitte Ø­skov Skajaa er nyuddannet læge fra
Københavns Universitet.
ORbit 19
37
DORS pris
By Sune Binzer
Metaheuristics for High School Planning
DORS prisen for 2008/2009 gik til Sune Binzer og Sune
Høj Kjeldsen fra DTU Management for deres speciale
»Metaheuristics for High School Planning«. Redaktionen
ønsker de to kandidater og deres vejleder Thomas Stidsen
tillykke og er glade for her at kunne bringe et sammendrag
af specialet.
The planning jobs of a Danish high
school are often many and complex;
hiring teachers, planning parent-teacher
consultations, choosing elective courses, scheduling the time table. Of these,
timetabling has the largest complexity,
requires the most resources and has a
high impact on the quality of the school’s
primary product – the classes. The focus
of our master thesis was to use metaheuristics to solve two important parts
of the timetabling - elective course planning and the assignment of classes into
timeslots, i.e. the actual timetabling. The
thesis was carried out in cooperation
with the company Macom, which is the
provider of the most popular high school
administrative system, Lectio, and the
produced algorithms were included in
Lectio’s ’Class & subject’ module (DK:
Time-fag modul).
Why metaheuristics?
When we started investigating the
timetabling problem, a fellow computer
science student said: Do not touch this
subject, it is impossible to create software for timetabling that can take into
account all special demands from tea-
chers and principals – many have tried
and many have failed. It did also turn out
to be a challenge just to formulate an
adequate model for Danish high school
timetabling.
We learned from interviews with Macom
and high schools that the most popular
planning tools on the market are GAS,
which has a tool for assigning elective
courses, and TPlan, which assists
the timetabling. These software tools
seemed to work using direct heuristics
developed over many years in cooperation with the high schools. However, the
most common method was still to do the
elective course packing and the timetabling manually, spending several weeks
during the summer period. There was
definitely room for improvement.
The focus of our project was practical
and we were not interested in doing
a completely theoretical study of timetabling of a general model that would
have no use in practice. There were a
few key factors to consider for practical
appliance:
• The high school administrators are
already using the Lectio web administrative system, and would be able to
experiment with this without making a
major new purchase.
38
ORbit 19
DORS pris
The packing of students into classes, and
the production of the
timetable are the problems best aided by
automation, since the
rest is very dependent
on human evaluation.
In theory the packing,
teacher
allocation
and timetabling could
be gathered into one
model, but that would
increase
complexity
• The high schools are competing finaneven
further,
and
cially, and will be interested in results would not be practical.
that can reduce their costs significantly
even if it means neglecting a special The Packing Problem
requirement from a teacher.
The Packing Problem is to
distribute students into a
number of classes according
to their course requests,
maximizing the number of
fulfilled
course
requests.
These classes must be distributed into
a number of blocks in such a way that
any student may only attend one class
in each block. These classes are subject
The problem and its com- to a maximum and a minimum size in
plexity
regards to the number of students in the
class. When distributing the classes into
The time table is a result of a year-long
blocks, there might also be a maximum
chain of planning events:
number of classes of certain courses
Still we had to consider the special
demands for our choice of solution
method. The solution method should
be able to adapt to model changes. It
would also be an advantage if it could
run without an expensive IP-solver. That
is why we chose to investigate the use of
metaheuristics.
• Which elective courses to offer
that a block can accommodate, due to
room restrictions.
This way the blocks can be viewed as
finished chunks of a time plan for the
Timetabling Problem. Thus, the classes
of a block can be assigned one or more
positions in the timetable without the
concern of conflicts.
We will not present our mathematical
model here, as it has 8 types of constraints. See [HSP].
The following graph shows a simplified
version of a sample problem instance:
We have 5 courses, the vertices, and 7
student requests, the edges and wish to
cut the graph into two blocks with a minimum number of edges in the cut.
This graph captures the general idea of
the packing problem. However, it is a
simplified version and ignores the ability
to have doublet classes of the same
course, and that a student can have
more than two requests.
This graph problem is actually the Max
2-Cut problem, which is NP-hard [16]. As
hinted in the simplified graph represen-
• Which elective courses to establish
given the student requests
• How to pack the courses into blocks
of classes
• How to allocate teachers to classes
• How to allocate classes in a timetable
ORbit 19
39
DORS pris
tation of the packing problem, it is pos- resource or room cannot be allocated to
sible to polynomially reduce the the Max the same position.
K-Cut problem to the Packing Problem
• Order constraints. Some classes must
and prove it is also NP-hard [HSP].
followed by another class, or have a
Timetabling
class in parallel.
The following is a general definition of
timetabling:
The problem of allocating a set of events
or tasks, each requiring a set of resources, in time and space, subject to a
number of constraints.
This definition captures the essence
of timetabling, but of course there are
numerous variants of the problem with
additional constraints and types of
objective functions. We have encountered several variants of timetabling in
the literature. Most of them differ by the
organization of the institutions; School,
University, Hospital etc. For educational
timetabling the most common problems
are University Timetabling, Exam Timetabling and School Timetabling.
The Danish high school timetabling is
somewhat unique, lying somewhere
between School and University Course
Timetabling.
In our model, we assume a given set
of time slot positions, p, rooms, o, and
classes, c, with predefined resources,
r, which are a combination of teachers,
t, and students, s. Although rooms,
students and students could all be considered resources, they have different
constraints and must be segregated.
This gives an idea of the nature and
complexity of the problem. Unfortuna• Class-spread constraints. Some clas- tely, the problem is even more complex
ses must be on different days.
as stated earlier, and we have to include
the Order and Spread constraints and
• Soft constraints \ Objective function:
the many soft constraints, to have any
chance of applying solutions to this pro­•
The
cost
of
having
class blem in practice.
c in position p and room o.
Students and teachers should not wait Solution methods
between classes.
• Allocation and exclusion constraints.
Classes must be allocated into exactly
A coloring of the graph can solve the
one room and position.
edge constraints, and in fact, in some
research, graph coloring algorithms
• Edge constraints. Classes that share a
have been tweaked to solve timetabling
ORbit 19
The Packing Problem
•­ Teachers should work for a maximum
number of days in the time table.
In our thesis we briefly look at decomposition schemes and algorithms for
­• Teachers should work a maximum similar problems, such as the Max K-Cut
number of classes each day.
and The Chromatic Number of a graph.
They do not solve the complete Packing
­• and more teacher related constraints. Problem, would be hard to tweak to
change in requirements, and more difThe edge constraints are named such, ficult to apply in practice since it would
because they may be represented as a require an IP-solver. However, the use of
similar graph problem, given the classes these could be further researched.
as the vertices, and the class resource
conflicts as the edges. The following We have investigated two metaheufigure shows a sample graph in which ristics: GRASP (Greedy Randomized
the class vertices are coloured such that Adaptive Search Procedure) and Tabu
all vertices of the same colour may be in Search.
the same position.
We divide our constraints into the following groups:
40
constraints, e.g. to include room allocation. We are able to use the graph coloring problem to show that our timetabling
problem is NP-hard [HSP].
The GRASP metaheuristic is likely to
provide good solutions for this problem,
since experiments showed that a greedy
algorithm gave good initial results. The
greedy construction of a solution:
1. Create elective course classes up
to the maximum number of classes by
greedily choosing the course subject
that has the largest average class quotient.
2. Distribute student groups (students
that have chosen same subjects) to
classes minimizing conflicts between
classes.
DORS pris
3. Assign classes to blocks, selecting use some of the same techniques, such
the largest class that does not create as the diversification strategies for Tabu
any conflicts, or create a new block if Search.
possible.
We chose to design a Tabu Search and
The randomized part of the algorithm Genetic Algorithm for our problem. A
is achieved by selecting an element at Greedy algorithm was also implemented
random from the list of candidates (aka for reference.
Restricted Candidate List), respectively
courses and classes in step 1 and 3. Of Our Genetic Algorithm represents a
course the selection is biased towards solution chromosome as a (position,
the better solutions, using a value based room)-gene with alleles (values) for
scheme [3] for step 1 and a cardinality each class and we experimented with
based scheme [3] for step 1. After the different selection, crossover and mutagreedy, randomized construction a local tion operators [1].
search method ensures local optimality
of the solution.
The Tabu Search searches the solution space by doing moves we call
Student Choice Exchanges which
is a combination of adding a
student to a class - fulfilling a
student request - and if this
generates infeasibility moving
a class to another block, possibly removing students from a
class. A limited sized Tabu list
of such moves is kept to avoid
repeating or reversing moves.
Both heuristics are of course more
detailed than stated here, but this
shows the general ideas.
Timetabling
The use of metaheuristics for timetabling
have been investigated in other timetabling research, e.g. Simulated Annealing
for School Timetabling [11], Tabu Search
for Course and School timetabling [5]
and Genetic Algorithms on School Timetabling [6], [7] and Course Timetabling
[2]. Most of the articles report good
results. We cannot directly use their
work for our problem since their formulations are too different from our problem,
but it suggests that those metaheuristics
may provide good solutions. We can still
function.
A large number of moves are possible:
adding, dropping and swapping classes
to and from positions and rooms. We
chose a combination of these, which
we call ActivityMove, which would
move all classes that had related Order
constraints, i.e. had to be executed in
some order, to another room or position.
This move maintains feasibility for the
relaxed problem and is able to reach
any solution in the search space. The
neighbourhood is large (and thus the
running time high), but after introducing
a room allocation heuristic as part of the
move, it was possible to keep it reasonable. The Tabu in the search is to move
the activity away from the new position.
Also, to ensure proper movement
in the search space, long-term
memory was introduced; three
diversification schemes are part
of the algorithm, which will not
be covered in detail here.
Results
The Packing Problem
The GRASP and Tabu Search algorithms were implemented in C#. The
implementations were unit tested and
tuned to find the best parameter settings
for respectively the size of the restricted
For Tabu
Search,
candidate list and size of the tabu list
a search space with only feasible Alloon real data sets from high schools
cation and Exclusion constrains were
extracted from the Lectio database.
chosen, i.e. classes were allocated,
but we had to relax the Edge and Order
The performance was tested on another
constraints, allowing resource overlaps.
set of real data. The results are shown
As it turns out, it is very difficult just to
in the following table, where b is the
find a feasible solution for the entire set
of hard constraints,
and thus we had
to do the relaxation, including any
violation of those
constraints as a
part of the objective
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number of blocks, μ the mean solution values and gap the An immediate observation from the results is that the only
gap to a simple dual bound. The average running times were feasible timetable found is with Tabu Search for the Bornholm
all within 70 seconds.
data set. It is not necessarily a bad result that such few feasible solutions are found. The data was sampled from timetabMost schools do their Packing with four to five blocks, using les which contained hundreds of violations. We had to tweak
the minimum number of classes, and produce solutions with the data so that the current timetables were feasible and thus
a gap less than a few per mille to the dual bound. A solution at least one feasible timetable existed for the problem. This
to a problem instance with b = 4 where all student requests does not mean that infeasibility should be tolerated, but that in
are granted, is widely regarded as a very good result. From practice, a human scheduler has more liberty to relax some of
the table above, such solutions are found for Høje Taastrup, the hard constraint. E.g. assuming that the 2 violations of hard
Aabenraa and Langkær. For Stenløse and Birkerød for four constraints in the Allerød data were two room clashes, which
blocks, the gap is about one percent which could very well be is effectively just one room clash. Then one of the classes that
acceptable in practice. This gap is still small and could be due were allocated to that room could in practice probably use a
to a gap from the true optimal solution value to the found dual room normally not be suited for that class, although the use
bound, since it is not a tight bound for all instances.
of that particular room would be considered ‘infeasible’ according to the problem definition.
The number of blocks is important in connection with the
Timetabling Problem. It can be directly translated into a To summarize, Tabu Search seems to provide viable timetabnumber of positions in the timetable. If fewer blocks are used, les for the given problems, and the other algorithms do not.
then the probability of finding a feasible solution to the Timetabling Problem is increased.
We will take a closer look at the values for two of the major
soft constraints. The following table shows the average waiTimetabling
ting hour per teacher, SWt, and student, SWs, for the real timetables, μup,, and for a sample Tabu Search solution, Tabu μ.
The Tabu Search and Genetic Algorithm were implemented in
C# and tested. The Tabu Search implementation was tuned
to find the size of the tabu list and the activation start of the
diversifications. The Genetic Algorithm was tuned to find a
good setting of the many parameters of a genetic algorithm,
including which operators performed best.
The following table shows the mean, µ, the deviation, σ, the
number of violations of the relaxed hard constraints, such as As an example, in the given timetable of Bornholm High
student conflicts and day conflicts between classes, for the School, a teacher waits in average for 0.65 modules and a
student waits in average 0.12 modules. For the same school,
Greedy Algorithm, Tabu Search and Genetic Algorithm.
Tabu Search produces a timetable with an average of 0.13
waiting modules for teachers and 0.28 waiting modules for
students in average. We used a penalty of 10 for a student
waiting modules and 1 for teacher. Clearly, these weights of
these soft constraints would be different for both Bornholm
It is no surprise that the best results were found for Bornholm. and Allerød, and it is likely to be different for every school.
It has few classes and a low density, i.e. few conflicts bet- The point is that the results are not that far from the given
ween classes compared to both Allerød and Birkerød. If we timetables and the setting of the weights can be adjusted for
compare the results of the algorithms, Tabu Search clearly every school.
outperforms both the Genetic and Greedy Algorithm. If we
include the deviation of the results, Tabu Search is still better Conclusion
in the worst cases. Only for Bornholm the results are even
Our experiments showed that the solution methods impleclose. We never did expect the greedy algorithm to provide
mented for the Packing Problem - Tabu Search and GRASP
good results.
- provided comparable results. For an average school, any of
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these methods should produce useful results. For the most
difficult problem instances the Tabu Search performed slightly
better than the GRASP and the running time was generally
better. For these reasons we recommend the Tabu Search
algorithm. All the experiments were performed on real data,
and for almost every data set an optimal solution was found,
or a solution with a gap of less than 0.1 percent to the given
bound was found.
Their article [3] shows that in some real instances of the Packing Problem for Danish High Schools, their BnP algorithm
found optimal solutions within an hour that improved solutions
found by our Tabu Search heuristic. Any improvement could
result in a large economical gain for the high schools. However, it also proved, that in most cases our heuristic found optimal solutions. Of course the BnP requires a MIP solver which
Lection does not yet have.
The solution methods implemented for the Timetabling Problem - Tabu Search and Genetic Algorithm - provided results
which were quite far apart. The Tabu Search clearly outperformed the Genetic Algorithm in respect to both running time
and solution value. Due to the popular research of genetic
algorithms for timetabling, we had higher hopes for the GA,
but it did not manage to provide reasonable results. Genetic
algorithms are likely better suited for problems with an easier
representation of a feasible solution.
In a recent interview with S. Kristansen and M. Sørensen
I learned that in recent years more and more Danish high
schools started using the planning tools of the Lectio ‘Timefag’ module and most have been happy with the results. In
March 2012 Simon and Mathias implemented new ALNS
heuristics (Adaptive Large Neighbourhood Search) in Lectio
for both packing and timetabling.
The performance of the Tabu Search matched our expectations to a solution method for this problem. From the schedulers we have interviewed, we have learned that finding
a feasible timetable is a long iterative process. The Tabu
Search algorithm was able to find solutions that were either
feasible or very close to being feasible for all the tested data
sets. Although the tests were performed on real data, the data
was not sufficiently well-formed to label it as completely realistic. Also, a theoretically optimal solution is not necessarily
the best solution in practice. Thus, when using the solution
methods as tools, a number of iterations of adjusting the problem and running the problem solver are inevitable. The best
practical solutions are found this way. The running times of
the implemented solutions should allow for a number of such
iterations.
References are numbered as in the reference list of the thesis where
References
applicable.
[HSP]: Our thesis. S.P. Binzer and S.H. Kjeldsen, 2008, Metaheuristics for High School Planning.
[1] E.K. Burke and G. Kendal, 2005, Search Methodologies, Springer
[2] H. Fang, 1994, Genetic Algorithms in Timetabling and Scheduling.
[3] L.F. Reis and E. Oliviera, A Language for Specifying Complete
Timetabling Problems.
[4] S. Kristiansen, M.Sørensen and T.R. Stidsen, Elective Course
Planning, European Journal of Research.
[5] G. Santos et al., An efficient Tabu Search Heuristic for the School
Timetabling Problem.
[6] G. Beligiannis et. al, 2006, Applying evolutionary computation to
the school timetabling problem: The Greek Case, ScienceDirect.
As expected, the use of the algorithms in the timetabling
tools in Lectio required tweaking and was used primarily as a
method to establish a starting solution and do local searchs in
an iterative manner.
[7] C. Fernandez and M. Santos, 2003, A Non-standard Genetic Algorithm Approach to Solve Constrained School Timetabling Problems.
[11] P. Dige and C. Lund, 1991, Skemalægning ved simuleret udglødning, IMSOR.
Futher Research
Sune Binzer
In 2010 DTU Management students Simon Kristiansen
and Matias Sørensen researched the performance of exact
solution methods on the Packing Problem [4], using Branchand-Price w. Dantzig-Wolfe decomposition. Following that,
they started a PhD in conjunction with Macom to research
both problems further, using both exact solution methods and
meta-heuristics.
er konsulent hos Logis A/S der
specialiserer sig i optimering og
software til logistik og indkøb.
Logis leverer bl.a.planlægningsog disponeringssystemer til akut
og planlagt ambulancekørsel
i flere regioner i Danmark og
udlandet.
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