A Decomposition Approach for a New Test-Scenario in Complex Problem Solving

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Journal

of

Computational

Science

jo u r n al hom ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / j o c s

A

decomposition

approach

for

a

new

test-scenario

in

complex

problem

solving

Michael

Engelhart

_{, Joachim}

_Funke

_,

_Sebastian

_Sager

a_{InterdisciplinaryCenterforScientificComputing(IWR),HeidelbergUniversity,ImNeuenheimerFeld368,69120Heidelberg,Germany}

b_{DepartmentofPsychology,HeidelbergUniversity,Hauptstr.47–51,69117Heidelberg,Germany}

c_{InstituteofMathematicalOptimization,Otto-von-GuerickeUniversitätMagdeburg,Universitätsplatz2,39106Magdeburg,Germany}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received22January2012

Receivedinrevisedform9May2012 Accepted11June2012

Availableonlinexxx

MSC:

90C30 90C11 90C90 90C59

Keywords:

Mixed-integernonlinearprogramming Complexproblemsolving

Decompositionapproach

a

b

s

t

r

a

c

t

Overthelastyears,psychologicalresearchhasincreasinglyusedcomputer-supportedtests,especiallyin theanalysisofcomplexhumandecisionmakingandproblemsolving.Theapproachistouse computer-basedtestscenariosandtoevaluatetheperformanceofparticipantsandcorrelateittocertainattributes, suchastheparticipant’scapacitytoregulateemotions.However,twoimportantquestionscanonlybe answeredwiththehelpofmodernoptimizationmethodology.Thefirstoneconsidersananalysisofthe exactsituationsanddecisionsthatledtoabadorgoodoverallperformanceoftestpersons.Thesecond importantquestionconcernsperformance,asthechoicesmadebyhumanscanonlybecomparedtoone another,butnottotheoptimalsolution,asitisunknowningeneral.

Additionally,thesetest-scenarioshaveusuallybeendefinedonatrial-and-errorbasis,untilcertain characteristicsbecameapparent.Themorecomplexmodelsbecome,themorelikelyitisthatunforeseen andunwantedcharacteristicsemergeinstudies.Toovercomethisimportantproblem,weproposetouse mathematicaloptimizationmethodologynotonlyasananalysisandtrainingtool,butalsointhedesign stageofthecomplexproblemscenario.

Wepresentanoveltestscenario,theIWRTailorshop,withfunctionalrelationsandmodelparameters thathavebeenformulatedbasedonoptimizationresults.Wealsopresentatailoreddecomposition approachtosolvetheresultingmixed-integernonlinearprogramswithnonconvexrelaxationsandshow somepromisingresultsofthisapproach.

©2012ElsevierB.V.Allrightsreserved.

1. Introduction

Modernlifeimposesdailydecisionmaking,oftenwith impor-tantconsequences.Illustrativeexamplesare,e.g.,politicianswho decideonactionstoovercomeafinancialcrisis,medicaldoctors whodecideoncomplementarychemotherapydrugdelivery strate-gies,orentrepreneurswhodecideonlong-termpricingstrategies fortheproductstheyoffer.

Theprocessofhumandecisionmakinginsuchtasksisthe sub-jectofresearchinthefieldofcomplexproblemsolving(CPS).CPS isdefinedasahigh-ordercognitiveprocess.Inresearch,the per-formanceofparticipantsinclearlydefinedmicroworlds(ortasks)is investigated.Theparticipant’sperformanceisevaluatedand cor-relatedtocertainattributes,suchastheparticipant’scapacityto regulateemotions.

∗_{Correspondingauthor.}

E-mailaddress:[email protected](M.Engelhart).

URL:http://www.mathopt.de(M.Engelhart).

Onemicroworldthatcomprisesavarietyofpropertiessuchas dynamics,complexityandinterdependence,discretechoices,lack oftransparency,andpolytelyinaneconomicalframingisthe Tai-lorshop.Participantshavetomakeeconomicdecisionstomaximize theoverall balanceofa small company,specializedin the pro-ductionandsalesofshirts.TheTailorshopissometimesreferred toasthe“Drosophila”forCPSresearchers[1]andthusa promi-nentexampleforacomputer-basedmicroworld.Ithasbeenused inalargenumberofstudies,e.g.,[2–7].Comprehensivereviewson studieswithTailorshophavebeenpublished,e.g.,[1,8–10].

Thecalculationofindicatorfunctionstomeasureperformance ofCPSparticipantsisbynomeanstrivial.Tomeasureperformance within the Tailorshop microworld, different indicator functions havebeenproposedintheliterature,see[11]forarecentreview. In[12,13]thequestionhowtogetareliableperformance indica-torfortheTailorshopmicroworldhasbeenaddressed.Becauseall previouslyusedindicatorshaveunknownreliabilityandvalidity, decisionsarecomparedtomathematicallyoptimalsolutions.For thefirsttimeacomplexmicroworldsuchasTailorshophasbeen describedintermsofamathematicalmodel.

s.t. xk+1=G(xk,uk,p,), k=ns...N−1

uk,i∈˝i, k=ns...N−1

i=1...n˝

0≤H(xk,uk,p), k=ns...N

xns =xnps

(1)

fordifferentstarttimes0≤ns<Noftheoptimizationandwhere

F,G,andHarenonlinearfunctionals,isarandomvariable,and

˝iare,possiblydiscrete,feasiblesets.Statevariablesaredenoted

byxk,scenarioparametersbyp,anddecisionstobetakenbythe

participantsattimekbyuk.Wedefine

(xp,up)=(xp0,... ,x p

N,u

p 0,...,u

p

N−1) (2)

tobethevectorofdecisionsandstatevariablesforallmonthsofa participant.Certainentriesxpnsenter(1)asfixedinitialvalues. Par-ticipantindependentinitialvaluesxp0=x0arefixedandpartofthe CPSmicroworlddefinition.Themodelisdynamicwithadiscrete timek=0...N,andNthenumberofturns.

Basedon(1),anoptimizationcanbeperformedforeveryturnns oftheparticipant’sdata,startingwithexactlythesameconditions

xpns astheparticipant.Theresultcanbeusedindifferentwaysto copewithquestionslikehowtomeasureperformanceincomplex environmentsinanobjectivewayandhowtodeterminedecisions whichwerecriticalfortheoverallperformanceofaparticipant. Thistechniqueisdescribedindetailin[13].

Thus,theassumptionthatthe“fruitflyofcomplexproblem solv-ing”isnotmathematicallyaccessiblehasbeendisproven.However, solving(1)toprovenglobaloptimalityisalreadyachallengingtask. Thenovelmethodologicalapproachhasalsobeencombinedwith experimentalstudies[6,7,13].

Sofar,allCPSmicroworldshavebeendevelopedinapurely dis-ciplinarytrial-and-errorapproach.Toourknowledge,asystematic developmentofCPSmicroworldsbasedonamathematicalmodel, sensitivityanalysis,andeventuallyoptimizationmethodstochoose parametersthatleadtoawantedbehaviorofthecomplexsystem forallpossibletrajectorieshasnotyetbeenapplied.Asanexample fortheneedtodothis,themathematicalmodelingoftheTailorshop microworldin[13]ledtothediscoveryofaprioriunwantedand unrealisticwinningstrategies(e.g.,thevansbug).

Therefore,inthisarticlewepresentanewmicro-worldbasedon theTailorshop,forwhichoptimizationmethodshavebeen consid-eredalreadythroughoutthemodelingphase,theIWRTailorshop.To overcomethedifficultiesofcomputinggloballyoptimalsolutions forthistest-scenario, whichstill yieldsnonconvexoptimization problems,wedevelopedadecompositionapproachtailoredtothe IWRTailorshop.

Mathematicalmodelreductiontechniquesarequitecommonin otherdomains,seee.g.,[14–16]foranoverview.Thebasicideaof ournewapproachtosolveproblem(1)consistsofadecomposition oftheMINLPintoamasterandseveralsmallersubproblems.This worksiftheobjectivefunctionisseparable.Theideaisrelatedto Lagrangianrelaxation,oneofthemostusedrelaxationstrategiesfor MILPs.Itsfirstapplicationwastheone-treerelaxationofthe travel-ingsalesmanprobleminthefamousHeld-Karpalgorithmin[17,18]. Thetraditionalapplicationfieldsarevariantsoftheknapsack prob-lemlike,e.g.,facilitylocationandcapacityplanning[19],general assignment,networkflowandtheunitcommitmentproblem[20].

themaximum.Hencethementionedtechniquescannotbeapplied inastraightforwardway.

Thearticleisorganizedasfollows.InSection2,theIWR Tailor-shopisintroduced.Thenthetailoreddecompositionapproachis explainedinSection3.Weshowsomepromisingnumericalresults ofthedecompositionappliedtotheIWRTailorshopinSection4and concludewithanoutlookinSection5.

2. TheIWRTailorshop-model

Based on the experience with the original Tailorshop -microworlddescribedin[13] withmodelingoddities,bugs,and otherundesirableproperties,wedecidedtocontinueourworkwith amathematicalmodeldevelopedfromscratch.

We systematically build a new microworld with desirable (mathematical)propertiesbasedontheeconomicalframingof Tai-lorshop.Theseeffortsleadtothenewtest-scenarioIWRTailorshop. Aschematicrepresentationofthisnewmicroworldcanbefoundin Fig.1.Table1listsallstatesandcontrolstheIWRTailorshopcontains togetherwithcorrespondingunits.

ComparedtotheTailorshop,thevarietyofvariableshasbeen shiftedtowardsamoreabstractlevel.Forexample,theparticipants havenolongerthetasktobuyorsellmachines,butinsteadhaveto takecareofthenumberofproductionsitesxPS_{oftheircompany.The}

ratherconcretevariablevanshasbeenreplacedbymoreabstract distributionsitesxDS_{,andsoon.WechosetosetupIWRTailorshopon}

suchanabstractlevel,becausethisyieldsamorerealisticposition ofadecisionmakerfortheparticipants.Forthemajorityof com-panies,itseemsunlikelythattheonewhodecidesonthenumber ofemployees,theshirtprice,andtheamountofmoneyspentfor advertisingisthesamewhohastoensurethatenoughrawmaterial isboughttoproducetheshirts.

Themathematical representation of theIWR Tailorshop con-sistsofthefollowingsetofequationsfork=ns...N,whichwill beexplainedbelow.Remember,thatxkdenotestatevariables,uk

denotecontrolvariables(decisionvariables)andparefixed param-eters.

xEM

k+1=xEMk −udEMk +uDEMk (3a)

xPS_k+1=xPSk −udPSk +uDPSk (3b)

xDS

k+1=xDSk −udDSk +uDDSk (3c)

xDE

k+1 =pDE,

0_·exp(−pDE,1_·uSP k )

·log(pDE,2_·uAD

k +1)·(xREk +pDE,3)

(3d)

xRE

k+1 =pRE, 0_·xRE

k +pRE,

1_log((pRE,2_·uAD k

+pRE,3_·uSP k ·(x

SQ k )

2

+pRE,4_·uWA k )+1)

(3e)

xPR

k+1 =pPR, 0_·xPS

k+1

·log

pPR,1_·xEM k+1

xPS

k+1+xDSk+1+pPR,2

+1

Fig.1. SchematicrepresentationoftheIWRTailorshopmicroworld.Arrowsshowdependencies,thesymbols(+and−)showproportionalandreciprocalinfluencesrespectively. Diamondsindicatetheinfluenceofparticipants’decisions.

xSA

k+1 =min{pSA, 0_·xDS

k+1

·log( p

SA,1_·xEM k+1

xPS

k+1+xkDS+1+pSA,2

+1);

xSH_k +xPR_k+1;pSA,3·xDEk+1}

(3g)

xSH

k+1=xSHk −xkSA+1+xPRk+1 (3h)

xSQ_k+1=pSQ,0_·xMO

k +pSQ,1·xkMQ+pSQ,

2_·uRQ

k (3i)

xMQ_k+1 =x

MQ

k ·pMQ,0·exp

−pMQ,1 x

PR k

xPS k +pMQ,2

+pMQ,3_·log(uMA

k ·pMQ,4+1)

(3j)

xMO

k+1 =(1−pMO, 0_)·xMO

k +pMO,

0

·log (pMO,1_·uDEM k +pMO,

2_·uDPS k

+pMO,3_·uDDS k +pMO,

4_·uWA k

+pMO,5_·xRE

k +pMO,6)

·exp (−(pMO,7_·udEM k +pMO,

8_·udPS k

+pMO,9_·udDS

k )+pMO,10)·pMO,11

(3k)

xCA

k+1 =pCA, 0_·(xCA

k +(xSAk+1·uSPk )+(udPSk ·pCA,

1₎

+(udDS k ·pCA,

2_)−(xEM k+1·uWAk )

−(xPR k+1·u

RQ

k ·pCA,3)−(xPSk ·pCA,4)

−(x_kDS·pCA,5_)−uMA k −uADk

−(xSH k+1·pCA,

6_)−(uDPS_·pCA,7₎

−(uDDS_·pCA,8₎₎

(3l)

Apartoftheseequations,(3a)and(3b),consistofasimplelinear transitionfrommonthktomonthk+1.Theamountofsitescreated

Table1

StatesandcontrolswithcorrespondingunitsintheIWRTailorshop.M.U.meansmonetaryunits.

States Variable Unit Controls Variable Unit

Employees xEM _{Person(s) Shirtprice u}SP _M.U./shirt

Productionsites xPS _{Site(s) Advertising u}AD _M.U.

Distributionsites xDS _{Site(s) Wages u}WA _M.U./person

Shirtsinstock xSH _{Shirt(s) Maintenance u}MA _M.U.

Production xPR _{Shirt(s) Resourcesquality u}RQ _–

Sales xSA _{Shirt(s) Recruit/dismissemployees u}dEM_/uDEM _Person(s)

Demand xDE _{Shirt(s) Create/closeproductionsite u}dPS_/uDPS _Site(s)

Reputation xRE _{– Create/closedistributionsite u}dDS_/uDDS _site(s)

Shirtsquality xSQ _–

Machinequality xMQ _–

Motivationofemployees xMO _–

ThedemandEq.(3d)ismorecomplicatedandcontains three factors.First,thereisanexponentialdecreasewiththeshirtprice, followedbyalogarithm,whichdampstheinfluenceofadvertising. Finally,thesetermsaremultipliedbythereputationandacertain offset.Demandherereferstothedemandatthissinglecompany, notonthewholemarket.

InEq.(3e),determiningthereputation,thereisamemoryterm consistingofafractionofthecurrentreputation.Additionally,there isalogarithmtodampentheeffectsofadvertising,levelofwages, andtheproductvalue– aproductofshirtpriceandshirtqualityto thepoweroftwo.

Theproduction Eq. (3f)consistsof a log-term, which damps theefficiencyofworkerspersite.Theassumptionis,thatallthe employeesare distributed equallyover thesumof distribution andproductionsites.Themoreemployeespersitethereare,the lessproductivityisyieldedbyonemoreemployee,e.g.,because ofthelimitationofspaceormachines.Thistermismultipliedby thenumberofproductionsitesincompensationofthe denomina-torinthelogarithm.ThesalesEq.(3g)isanalogtotheproduction equation,butwithadistributionsitesfactorinsteadofproduction sites.Additionally,salesarelimitedbythenumberofshirts avail-able,i.e.,thesumofshirtsinstockandshirtsproduced,andbythe demand.Thisleadstothemin-expressionwiththreecomponents. Note,however,thatthisexpressioncaneasilybetransformedinto inequalitiesbyintroducingaslack-variable,whichislimitedbyall componentsoftheminimum.Thisworks,becausethesalesonly haveapositiveeffectintheobjectivefunction.

Machinequality, seeEq.(3j),decreaseswiththeload, repre-sentedby shirtsproduced per production site. Maintenance,on theotherhand,increasesmachinequality,dampedbyalogarithm again.

ThemotivationEq.(3k)isaconvexcombinationofoldandnew motivationlevels.Thelevelisdeterminedbyalogarithmcontaining positiveeffects(recruitingemployees,creatingproductionand dis-tributionsites,wages,andreputation)andanegativeexponential, wherenegativefactorsenter(dismissalofemployeesandclosing productionanddistributionsites).

ThelastEq.(3l),thecapital,isacompositionofallexpensesand incomesgivenimplicitlybytheotherequations:revenuepershirt, revenueperproductionanddistributionsitesold(closed),wages peremployee,productioncostsdependingontheresourcequality, fixedcostsforproductionanddistributionsites,maintenanceand advertisingexpenses,storagecosts,andpurchasepricefor produc-tionanddistributionsites.Thecapitalissubjecttoacertaininterest ratepCA,0_.

IWRTailorshopcontainsinequalities.Thereisamaximum stor-agecapacityforshirtsperdistributionsite,

xSH

k ≤pSH,0·xDSk (4)

Recruitmentdependsonaccesstodifferentjobmarketsyieldedby thenumberofsitesandislimited,

uDEM_k ≤pDEM,0·x_kPS+pDEM,1·x_kDS (5) Theoverallnumberofsitesislimited,

xPS

k +xDSk ≤ptS (6)

udEM

k ≤pdEM (8a)

uDPS_k ≤pDPS (8b)

uDDS_k ≤pDDS (8c)

udDS_k ≤pdDS (8d) Furthermore,someofthecontrolshavetobeinteger,

uDEMk ,udEMk ,uDPSk ,ukdPS,uDDSk ,udDSk ∈Z+0 (9) andresourcequalitymustbechosenfromafiniteset:

uRQ_k ∈{pRQ,1_,...,pRQ,nRQ_{} (10)} ComparedtoEq.(1),theseequationsandinequalitiestogether withthereformulationofthesalesequationformthefunctionsG andH.FortheobjectivefunctionF,onecouldeasilythinkof dif-ferentoptions,e.g.,aweightedcombinationofmaximizingprofit, reputation,andsomeotherfactors.Wedecidedtousetheprofit attheend ofthediscretetime-scalein thisarticlefor thesake ofcomparabilitytotheoriginalTailorshop.Hence,wesuggestthe followingobjective:

max

x,u,p x CA

N (11)

Ofcourse,thesetofparametershasasignificantinfluenceon themodelbehavior.Onecoulddefinitelydedicateawholearticle onhowtodetermineanappropriateparametersetforamicroworld likeIWRTailorshop,dependingontheaims–seealsoSection5for futureworkregardingthisissue.Forthisarticle,however,weset upaparametersetmanuallysuchthat themodel fulfillsa cer-taindesiredbehavior.Thechosenparametersalsoyieldamodel behaviorthatmakessensefortheoptimization,i.e.thereare feasi-blesolutionsandtheoptimizationproblemisnotunbounded.The parametervaluesarelistedinTables2and3.

AllthesecomponentsbuildtheIWRTailorshop,which–from amathematicalpointofview–isamixed-integernonlinear pro-gramwithnonconvexrelaxation,i.e.ifthepossiblydiscrete˝iinthe

dMIOCP(1)arereplacedbysomecontinuous ˆ˝i⊇˝i,thisyields

anonconvexnonlinearprogram.Theimplementationofthisnew modelfeaturesaweb-basedinterfaceandusesthewidelyspread AMPLinterface[24],whichallows,e.g.,theuseofavarietyof pow-erfuloptimizationalgorithms.

Table2

ParametersetforstatesusedwithIWRTailorshopinthisarticle.M.U.means mone-taryunits.

Parameter Value

pSH,0 _{2000shirts/site}

pDE,0 _600.0shirts

pDE,1 _2×10−2_shirts/M.U.

pDE,2 _2×10−2_1/M.U.

pDE,3 _0.5

pRE,0 _0.5

pRE,1 _1.0

pRE,2 _2.5×10−5_1/M.U.

pRE,3 ₁₀−4_shirts/M.U.

pRE,4 _6×10−5_persons/M.U.

pPR,0 _{99.9shirts/sites}

pPR,1 _{2.0sites/persons}

pPR,2 ₁₀−6_sites

pSA,0 _{99.9shirts/sites}

pSA,1 _{2.0sites/persons}

pSA,2 ₁₀−6_sites

pSA,3 _1.0

pSQ,0 _0.2

pSQ,1 _0.3

pSQ,2 _0.5

pMQ,0 _0.8

pMQ,1 _0.6×10−2_sites/shirts

pMQ,2 ₁₀−6_sites

pMQ,3 _0.13

pMQ,4 _0.2M.U.−1

pMO,0 _0.5

pMO,1 _4×10−2_persons−1

pMO,2 _0.5sites−1

pMO,3 _0.25sites−1

pMO,4 _2.0×10−4_persons/M.U.

pMO,5 _0.3

pMO,6 _1.0

pMO,7 _0.7persons−1

pMO,8 _2.5sites−1

pMO,9 _2.0sites−1

pMO,10 _1.0

pMO,11 _0.5

pCA,0 _1.03

pCA,1 _{5000M.U./site}

pCA,2 _{3500M.U./site}

pCA,3 _{5.0M.U./shirt}

pCA,4 _{1000M.U./site}

pCA,5 _700M.U./site

pCA,6 _{1.5M.U./shirt}

pCA,7 _{10,000M.U./site}

pCA,8 _{7000M.U./site}

3. Atailoreddecompositionapproach

Nowthatwehaveasystematicallybuiltmicroworldwith desir-ableproperties,wecouldstartdoingstudieswithitandevaluating participants’performancebasedonoptimalsolutionsasexplained

Table3

ParametersetforcontrolsusedwithIWRTailorshopinthisarticle.

Parameter Value

nRQ 4

pRQ,1 _0.25

pRQ,2 _0.5

pRQ,3 _0.75

pRQ,4 _1.0

pDEM,0 _{5persons/site}

pDEM,1 _{10persons/site}

pdEM _10persons

pDPS _1site

pdPS _1site

pDDS _2sites

pdDS _1site

ptS _6sites

max f(x) master problem

min c1(x)

decoupled problems

min c2(x)

costs costs

input

variables variablesinput

Fig.2.Schematicrepresentationofthetailoreddecompositionapproach.

aboveand in [13].The computationof anindicatorfunctionas describedin[13],however,canonlybeclaimedreasonablytobe objective,ifwecanfindguaranteedgloballyoptimalsolutions.But –asalreadymentionedabove–theIWRTailorshopyieldsa non-convexproblem.Thispropertyisunavoidableaslongasweare interested in turn-based scenarios with nonlinear model equa-tions. Hence,itis difficulttocomputeglobalsolutionsfor such test-scenarios.

Andindeed,thecomputationtimeswithCouenne0.4onaIntel Corei7machinewith12GBRAMlookbad:forN=1ittakeslessthan 1s,forN=2already3s,andforN=3byfarmorethan10min(see alsoTable6).ForhighervaluesofN,wecannothopeforasolution atallbeforethemachinerunsoutofmemory.

The idea of the decomposition approach is now, to exploit thestructureoftheproblem–especiallytheseparabilityofthe objectivefunction,see(11)–tocreatearelaxationoftheoriginal problemwherepartsoftheproblemarereplacedbyfreevariables (freewithinsomesimplebounds),forwhichcostsarecomputed indecoupledprograms,which containthecomplexity fromthe originalprogram.Aschematicrepresentationofthis decomposi-tioncanbefoundinFig.2.Thedecouplingofcertainpartsofthe originalproblemobviouslymakestheremainingmasterproblem smallerandthereforeeasiertohandle.Suchadecompositionisnot unique.Wechoseonewithfewoverlappingvariables.Aschematic representationoftheresultingmasterproblemisshowninFig.3.

The costs computation via the decoupledproblems is done offlineonadiscretizedgrid.Thedecoupledproblemsyield them-selvesanoptimizationproblemofthetype

min Costs

s.t. Achievedesired valueoffreevariable (asinmasterproblem)

Theoptimalsolutionsonthegridpointscanbeusedtofitsome model,whichunderestimatesthecosts,detailscanbefoundbelow. Thiscostmodelisnowpluggedintotheobjectivefunctionofthe masterproblemrepresentingcostsforthenewlyintroducedfree variables.Wethencancomputeagloballyoptimalsolutionforthe reducedmasterproblem.Iftherelaxationisvalid,thisyieldsus avalidupperboundfortheoriginalproblem. Thisupperbound determinedbythedecompositioncanthenbeusedasanindicator, howfaralocalsolutionfortheoriginalproblemisawayatthemost fromaglobalone.

Fig.3. IWRTailorshopreducedmasterproblemwithdependenciesandproportional/reciprocalinfluences.Diamondsindicatefreevariables.

Themasterprobleminourdecompositionconsistsofthe follow-ingequations,whichformarelaxationoftheoriginalproblem(2) byunderestimatingnegativeandoverestimatingpositiveeffects:

xDE

k+1 =pDE,

0_·exp(−pDE,1_·uSP k )

·log(pDE,2_·uAD

k +1)·(xREk +pDE,

3₎ (12a)

xRE

k+1 =pRE,0·xREk +pRE,1log(pRE,2·uADk

+pRE,3_·uSP k ·(u

SQ k )

2

+pRE,4_·uWA k +1)

(12b)

xSA

k+1 =min{pSA,0·usitesk+1

·log

pSA,1_·uEM k+1

usites k+1+pSA,2

+1

;

xSH

k +uPRk+1;pSA, 3_·xDE

k+1}

(12c)

xSH

k+1=xSHk −xkSA+1+uPRk+1 (12d)

xCA

k+1 =pCA,0·(xCAk +(xSAk+1·uSPk )−uADk

−uEM

k+1·uWAk −(xSHk+1·pCA,6)

−f1(usites_k ;uPR_k ,u_kEM)−f2(uSQ_k ;uPR_k ))

(12e)

uSP k ∈[lb

SP

,ubSP] (12f)

uSQ_k ∈[lbSQ,ubSQ] (12g)

uPR_k ∈[lbPR,ubPR] (12h)

uWA k ∈[lb

WA

,ubWA] (12i)

usites k ∈[lb

sites

,ubsites]∩Z+

0 (12j)

uAD_k ∈[lbAD,ubAD] (12k)

uEM k ∈[lb

EM

,ubEM]∩Z+

0 (12l)

Here,thefunctionsf1andf2returnthecoststobedeterminedin thedecoupledproblems.Wechoosetheobjectiveagainas max

x,u,p x CA

N. (13)

Thefirstdecoupledprogram,whichdeterminesthecostsfora givenshirtquality,is

min uRQ_k ·uPR

k+1·pPR,cost+uMAk−1 (14a) s.t. u

xMQ_k =pMQ,3·log(pMQ,4·uMA_k−1+1) (14c)

uRQ_k ∈{pRQ,1_,...,pRQ,nRQ_{} (14d)}

uMA_k−1∈[lb

MA

,ubMA] (14e)

Here,thevariableswithahatareconsideredtobegiven,e.g., fromthefreevariablesinthemasterproblem.Inthefollowing,we calltheminputvariablesinthiscontext.Thesecondsubproblem determinesthecostsforagiventotalnumberofsitesandconsists ofthefollowingequations.

min uDS_k+1·pCA, 5₊

uPS_k+1·pCA,

4 _(15a)

s.t. usites

k+1=uPSk+1+uDSk+1 (15b)

uPR

k+1=pPR,0·log(uPSk+1·

pPR,1_·uEM k+1

uPS

k+1+uDSk+1+pPR,2

+1)

(15c)

uDS_k+1∈[lb

DS

,ubDS]∩Z+

0 (15d)

uPS_k+1∈[lb

PS

,ubPS]∩Z+0 (15e)

We evaluate thesedecoupled programs ona grid,i.e., on a discretizationofthefeasibleintervalforeach inputvariable.For

usites_k ∈[2,16],e.g.,wecouldchoosethegrid2,4,8,10,12,14,16. Withmorethanonediscretizedvariable,thisleadsto multidimen-sionalgrids.Foreachgridpoint,wecomputeanoptimalsolution forthecorrespondingdecoupledprogram.Withthesolutionsfor allgridpoints,wecanfite.g.,aquadraticmodel,like

f(uSQ_k ;uPRk )=a0+a1·uPR_k +a2·uSQ_k +a3·uPR_k ·uSQ_k +a4·(uPR_k )2

Table4

Initialvaluesusedforcomputationswithoriginalfullproblemanddecomposition.

Originalmodel Decomposition

xEM

0 =10 uEM0 =10

xPS 0 =1

xDS

0 =1 usites0 =2

xSH

0 =67 x0SH=67

xPR

0 =200 uPR0 =200

xSA

0 =200 x0SA=200

xDE

0 =700 x0DE=700

xRE

0 =0.79 x0RE=0.79

xSQ0 =0.75 u

SQ 0 =0.75

xMQ0 =0.81 –

xMO

0 =0.73 –

xCA

0 =175,000 x0CA=175,000

Ofcourse,wecouldaswellusealinearoracubicmodelor some-thingcompletelydifferent.Thefitcanthenbedonebysolvinga simpleleastsquaresproblem,withXbeingthesetofgridpoints andh(x)afunction,whichreturnstheoptimalobjectivevaluefor eachgridpointx∈X:

min

a,x

x∈X

f(x)−h(x)2

2 (17a)

s.t. f(x)≤h(x)

∀

Especiallywhenconsideringtheintegralityconditions, equal-ityconstraintsareunlikely tobefulfilled exactly.Thereforethe followingreformulationisintroducedforeachequalityconstraint.

uk=... −→ u

Here,shouldbechosenreasonablysmall,suchthatthedecoupled programisfeasibleforalmostallofthegridpoints.

4. Numericalresults

Wepresentfirstresultsofourdecompositionapproachfrom Section3fortheIWRTailorshop.Allcomputationshavebeendone on an Intel Core i7 machine with12GB RAM running Ubuntu 11.10 (64-bit) withthe COIN-OR solvers Ipopt3.10,Bonmin 1.5, and Couenne 0.4.Ipopt 3.10 is a local solverfor nonlinear pro-grams[25],whichimplementsaninteriorpointmethod.Itisnot abletotreatintegerconstraintsandhasonlybeenusedfor refer-ence.Bonmin1.5isa solverforgeneralmixed-integernonlinear programsincludingseveralalgorithms[26].Forthecomputations in this article,B-BB,an NLP-basedbranch-and-boundalgorithm, hasbeenused.Incontrasttothesetwosolvers,Couenne0.4isa globalsolverusingaspatialbranch-and-boundalgorithminorder tofindglobaloptimaformixed-integernonlinearprogramswith nonconvexrelaxations[27].Theparametersetsused areshown inTables2and3.Initialvaluesandsimpleboundsonstatesand controlsusedinallcomputationscanbefoundinTables4and5.

Forthedecomposition,inafirststepthecostfunctionsf1and

f2forthenewfreevariablesuSQ_k andusites_k havebeencomputed. Thereforethesubproblems(3)and (3)havebeensolvedonthe grids

uSQ_k ∈{0.25,0.26,0.27,...,0.74,0.75}, (19a)

uPR

k ∈{100,200,300,...,900,1000}, (19b)

respectively

usites

k ∈{2,3,4,5,6}, (20a)

Table5

Simpleboundsusedforcomputationswithoriginalfullproblemanddecomposition.

Originalmodel Decomposition

uSP

k ∈[35,55] uSPk ∈[35,55]

uAD

k ∈[1000,2000] uADk ∈[1000,2000]

uWA

k ∈[1000,1500] uWAk ∈[1000,1500]

uMA

k ∈[0,5000] uMAk ∈[0,5000]

xEM

k ∈[8,16] uEMk ∈[8,16]

xPS

k,xkDS∈[1,6] usitesk ∈[2,6]

xPR

k ∈[0,1000] uPRk ∈[0,1000]

xSQ

k ∈[0.25,0.75] u

SQ

k ∈[0.25,0.75]

xSH

k ,xDEk ,xREk,xSAk ≥0 xSHk ,xDEk ,xREk,xSAk ≥0

xMO k ,x

MQ

k ≥0 –

uEM_k ∈{8,9,10,...,15,16}, (20b)

uPR

k ∈{100,200,300,...,900,1000}. (20c)

Bysolvingthecorrespondingproblemsoftype(3)withthisdata, wereceivedthefollowingunderestimatorsforthecosts:

f1(usites_k ;uEM_k ,uPR_k )=21.6754−944.6455·u_ksites+1.4968·uPR_k

−28.9341·uEM_k +0.1338·usites_k ·uPR_k −3.3626·usites_k ·uEM_k

−0.0586·uPRk ·ukEM−1.3478·(usitesk )

2

+1.8831·(uEMk )

2

(21a)

f2(uSQ_k ;uPR_k )=−898.0761+0.1991·uPR_k+1+4726.3749·u

SQ k+1

−8.5390·uPR_k+1·u

SQ

k+1+0.0004·(uPRk+1) 2

−5501.7182·(uSQ_k+1) 2

(21b) Theproblemsforallgridpointsofonesubproblemcouldbesolved inlessthan1minincludingthefitofthequadraticmodel.Aplotof theresultingcostfunctionfortheuSQ_{-subproblemcanbefoundin}

Fig.4.However,itwasnecessarytousetheglobalsolverCouenne 0.4atleastinthissubproblem,aswegotdifferentsolutionswith Ipopt3.10forarelaxedversionofthissubproblemwhichobviously arenotgloballyoptimalasonecanobservefromthecomparison tothesolutionsofCouenne0.4inFig.5.Fortheusites_-subproblema

plotofthecostfunctionisnotpossibleduetoitsdimensions. Whencomparingsolutionsandobjectivefunctionvalues,three effects need to be distinguished: integrality, local vs. global

0 200

400 600

800 1000

0.2 0.4 0.6 0.8 1 0 2000 4000 6000 8000 10000

uSQ

uPR

Φ2(uSQ,uPR)

Fig.4.Costvalues˚2 (bluedots)forsolutionsbyCouenne0.4forthe

decou-pledproblemforuSQ_withpRQ,nRQ_=2onthegriduSQ

k ∈{0.25,0.26,...,0.75},uPRk ∈

0 200

400 600

800 1000

0.2 0.4 0.6 0.8 1 0 2000

uSQ _uPR

Couenne

0 200

400 600

800 1000

0.2 0.4 0.6 0.8 1 0 2000

uSQ

uPR

Ipopt

Fig.5.Costvalues˚2(bluedots)forsolutionsbyCouenne0.4andIpopt3.10forthedecoupledproblemforuSQwithpRQ,nRQ=2andrelaxeduRQonthegriduSQ_k ∈

{0.25,0.26,...,0.75},uPR

k ∈{100,200,...,1000}togetherwiththeunderestimatingcostfunction(coloredsurface).FromthedifferencesbetweenCouenne0.4(global

solver)andIpopt3.10(localsolver)onecandetermine,thatitisnecessaryheretouseaglobalsolverevenforthedecoupledproblem.(Forinterpretationofthereferences tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthearticle.)

solutions,andfullversusoverestimatingreducedmodel.We inves-tigatedtwoscenarios.First,thevariablesusites

k respectivelyuPSk and

uDS

k havebeenfixedtotheirlowerbounds2respectively1.The

resultsarelistedinTable7.Here,Ipopt3.10andBonmin1.5found thesamesolutionsfortheoriginalproblem,whichisduetothe factthatthesolutionsdeterminedbyIpopt3.10arealreadyinteger. Thus,thereisnodifferencebetweenthesesolvers.Inthisspecial case,Couenne0.4alsofindsthesamesolutionsfortheoriginal prob-leminanacceptabletime(<1min).Thissettingallowsustofocus exclusivelyonthethirdeffect,thegapbetweenourreducedand thefullmodel.ThegapdeterminedbyCouenne0.4inbothcases reachesfrom4.0%to16.3%.

Fortunately,thisspecialcasewithfixedsitesissomethinglikea worstcase.Thegapismainlyduetoareductioninsales,whichin turnrelatestothedifferencesbetweenEqs.(3g)and(12c).Fixing thenumberofsitesonthelowerboundsresultsinanactivefirst termintheminimumexpressions.Thisisalsotheexpressionthat suffersmost,becausethenewvariableusites_k isinthiscasetwiceas largeasthecorrectexpressionxDS

k intheoriginalmodel.

Ifwe let usites

k free withintheir simple bounds as shown in

Table5,thegapsbetweenlocalsolutiontothefullmodelandglobal solutiontothereducedmodelalternatefrom4.0%to8.1%.Notethat thegaprelatingtoIpopt3.10isonlyforinformation,sinceIpopt3.10 cannothandleintegerconstraintsandthussolvesarelaxedversion oftheproblem.Oneobservesthatthegapfirstincreases,butthen decreases,seemingtoconvergetosomec>0.Thisbehaviorcanbe

Table6

ComparisonofcomputationtimesbetweenIpopt3.10,Bonmin1.5,andCouenne0.4

fortheoriginalproblem,aswellasCouenne0.4forthedecomposition.

N Originalmodel Decomposition

Ipopt Bonmin Couenne Couenne

1 1s <1s <1s <1s

2 1s 4s 3s 1s

3 <1s 45s >10min 2s

4 <1s 537s >10min 3s

5 <1s >10min >10min 5s

6 <1s >10min >10min 10s

7 1s >10min >10min 17s

8 <1s >10min >10min 27s

9 <1s >10min >10min 52s

10 1s >10min >10min 88s

explainedbythefactthatthementionedeffectleadstoanincrease incost(duetostorageofnot-soldshirts)thatisaboutlinearinthe numberofturns.Thepossiblewinningsmakinguseofafreechoice ofusites

k outperformstheseadditionalcostsifthetimescaleforthe

optimizationislongenough.Thus,thegapfirstincreasesandthan againdecreases.

Inthisscenario,Couenne0.4isnotableanymoretofinda solu-tion for the original problemin less than 10min for N≥3. All computationtimescanbefoundinTable6.Obviously,the decom-positioncanbesolvedfasterbyordersofmagnitude.EvenforN=10, ittakeslessthan2minwithCouenne0.4,whileBonmin1.5evenis notabletocomputealocalsolutionfortheoriginalprobleminless than10minforN≥5(seeTables7and8).

Summingup,wecouldestimatethegapbetweenreducedand fullmodeltobeintherangeofafewpercent.Weidentifiedthe mostimportantsourceof gapstobein thedifferencebetween Eqs.(3g)and(12c).Forlongertimehorizonsandmorefreedomof variablechoice,however,ourapproximationbecomesbetterand better.Thecomputationalgainsaredramaticandallowtocalculate globalsolutionsevenonthefulllengthofthetimehorizon.

5. Summaryandoutlook

Wepresentedanewmicroworldforcomplexproblemsolving, theIWRTailorshop.Thisturn-basedtest-scenarioyieldsa

mixed-Table7

Solutionsusingthefullproblemwithfixednumberofsitescomparedtothe decom-positionapproach.NotethatthesolutionsbyIpopt3.10arealreadyinteger,sothat thereisnodifferencebetweenBonmin1.5andIpopt3.10.

N Originalmodel Decomposition Gapin%

Ipopt Bonmin Couenne

1 180995.1 180995.1 188495.0 4.0%

2 187170.0 187170.0 198599.3 5.8%

3 193530.2 193530.2 209006.8 7.4%

4 200081.2 200081.2 219726.5 8.9%

5 206828.8 206828.8 230767.7 10.4%

6 213778.7 213778.7 242140.2 11.7%

7 220937.2 220937.2 253853.9 13.0%

8 228310.4 228310.4 265919.0 14.1%

9 235904.8 235904.8 278346.0 15.2%

Table8

Solutionsusingthefullproblemcomparedtothedecompositionapproach.For solu-tionswitha‘*’,Bonmin1.5didnotfindanoptimalsolutionwithin10min.However, thegapbetweenlowerandupperboundwasinallcasessignificantlybelow1%.

N Originalmodel Decomposition

Ipopt Gapin% Bonmin Gapin% Couenne

1 181835.6 3.5% 180995.1 4.0% 188495.0

2 189161.4 4.8% 187170.0 5.8% 198599.3

3 196180.0 6.1% 193530.2 7.4% 209006.8

4 204760.9 6.8% 201860.5 8.1% 219726.5

5 215097.9 6.8% 212332.9* 8.0% 230767.7

6 226408.7 6.5% 223118.0* 7.9% 242140.2

7 239011.7 5.8% 236196.6* 7.0% 253853.9

8 252536.7 5.0% 250100.3* 6.0% 265919.0

9 266817.6 4.1% 264399.8* 5.0% 278346.0

10 281619.2 3.3% 279119.3* 4.1% 291145.9

integernonlinearprogramwithnonconvexrelaxationandconsists of functional relations based on optimizationresults. Withthe IWR Tailorshop we intend tostart a new era beyond trial-and-errorinthedefinitionofmicroworldsforanalyzinghumandecision making.

Tobeabletosolvetheresultingproblemswithinreasonable times,weproposedatailoreddecompositionapproach,wherethe problemisdividedintoamasterproblemandseveralsubproblems. Thisdecompositionisbuiltsuchthatityieldsavalidupperbound forthecorrespondingglobalsolutionoftheoriginalproblemand thuscanbeusedasanindicatorforthequalityoflocalsolutionsof theoriginalproblem.

We finallypresented promising numerical resultsusing this decompositionapproach,whichindicatedahighpotential.Inafirst (worst-caselike)scenariowithfixedvariables, thegapbetween decompositionandoriginalproblemwasbetween4.0%and16.3%, whiletheoriginalproblemcouldalsobesolvedtoglobal optimal-ity.Inasecondscenario,italternatedbetween4.0%and8.0%.For thisscenario,onlywiththedecompositionitwaspossibletoget agloballyoptimalsolutionformorethan2turns.The computa-tiontimesforthedecompositionarebelow2minevenfor10turns withCouenne0.4,whilethelocalsolverBonmin1.5couldnotfinda localsolutionfortheoriginalproblemwithin10minformorethan 4turns.Infuturework,itcouldbeinterestingtocomparethese resultstoaLagrangianrelaxationtypeapproach.

Theparametersetusedforthecomputationsinthisarticlehas beensetupmanuallytoachieveamoreorlessreasonablemodel behavior.Here westill seehighpotentialforimprovement. For example,onecouldusederivative-freeoptimizationmethodsto optimizetheparametervaluessuchthattwo(orevenmore) previ-ouslydefinedstrategies(e.g.,ahighandalowpricestrategy)yielda similarobjectivevalue.Bythat,participantscouldfollowdifferent strategiesandstillperformquitewell.

Animportantstepinfutureworkwillbetocollectdatawith participants,whichwillthenbeusedtocomputeoptimalsolutions fortheIWRTailorshopstartinginstatesderivedbytheparticipants –aswellfortheoriginalproblemasforthedecomposition.This willyieldanindicatorfunctionwithguaranteedgapstotheglobal solutionfortheoriginalproblem.

Ifwefinallysucceedtocomputeoptimalsolutionsfastenough, wecantakethisapproachevenonestepfurther:bycomputingthe performanceindicatoronline,i.e.,whileparticipantsaresolvingthe IWRTailorshop,wecangiveanimmediatefeedbackbasedon opti-malsolutions.Itwillbesubjectoffutureresearchhowthisfeedback canbeusedtoimprovelearningofcomplexproblemsolving com-petences.Answerstothisquestioncanbeusedtodesignprograms totrainfuturedecisionmakers.

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