Optimization Models for Service and Delivery Planning

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Optimization Models for Service and Delivery Planning
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Chen,Shuang
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
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Industrial and Systems Engineering
Committee Chair:
Geunes, Joseph P
Committee Members:
Smith, Jonathan
Guan, Yongpei
Paul, Anand A
Sahni, Sartaj

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Subjects / Keywords:
allocation -- benders -- bilinear -- chain -- decomposition -- heuristic -- integer -- modeling -- nonlinear -- optimization -- prepack -- programming -- resource -- stochastic -- supply
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Industrial and Systems Engineering thesis, Ph.D.
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Abstract:
This dissertation considers the development of efficient algorithms and mixed-integer programming models in the context of several specific real world applications, including service and retail operations. We present models of three specific systems in the service and delivery planning context. We first consider a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers' demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. Our contribution includes providing an exact solution method for this mixed integer nonlinear optimization problem and presenting a family of efficient heuristic approaches based on applied probability. For these service supply chain operations, we demonstrate opportunities for strong financial returns and customer satisfaction by focusing on operational excellence. We then move to a new variant of the class of multi-item lot sizing problems where the planner must order in integer quantities of "case packs". A distribution case pack contains an assortment of varying quantities of different stock keeping units (SKUs) packed together in a single box or pallet, with a goal of reducing handling requirements in the distribution chain. While the inclusion of multiple items in one case pack reduces the number of touches individual items experience in the distribution chain, it can severely increase the complexity involved in retailer procurement planning. This complexity arises because different case packs (and the items within them) share certain (fixed) order costs, and the composition of the case pack requires ordering in defined combinations of multiple SKUs. Thus it is necessary to balance the tradeoff between the handling cost advantages and the reduced ordering flexibility and consequently increased probability of overstock. Since little past research in the literature has addressed this practice, which is widely accepted in industry, our contributions include successfully modeling and solving two retailer problems within this domain: (1) Procurement planning when a supplier of multiple items requires purchasing pre-defined case packs; (2) Integrated case pack configuration and procurement planning problem. The first problem focuses on the short-range decision to determine case pack orders given certain case pack configurations. As for the second long-range decision problem, we provide a model for jointly determining the retailer's optimal case-pack configuration and order timing decisions for a given demand stream over a finite horizon.
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In the series University of Florida Digital Collections.
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Includes vita.
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by Shuang Chen.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Geunes, Joseph P.
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First,Iwouldliketothankmyadvisor,Dr.JoeGeunes,forallhisguidanceandcontinuoussupport.Hepaidattentiontoeverystepofthisdissertation,allowedmethespacetothinkcreatively,motivatedmetopursuequalityresearch,helpedmeimprovemywritingandencouragedmetofollowmyinterests.Ifeelsofortunatetohavehimasmyadvisor.Thankstohisthoughtfulnessandunderstanding,IhaveledahappyandfulllinglifeduringmyPhDprogram.IwouldliketoacknowledgeDr.SartajSahni,Dr.YongpeiGuan,Dr.ColeSmithandDr.AnandPaulforparticipatinginmydissertationcommitteeandtheirinsightfulcomments.SpecialthanksgotoDr.ColeSmith,forhispassionatworkimpactsmeprofessionally,andtoDr.YongpeiGuan,Dr.AnandPaul,Dr.StanUryasevandmyadvisorDr.JoeGeunesfortheirinvaluableadviceduringmyjobsearchprocess.I'mindebtedtomyfriendsattheISEDepartment,inmychurchandinGainesville.Thankyouforyourfriendshipthroughoutthesefouryears.Iremaingratefulforknowingthatyouarealwaystheretolendalisteningearandtotakeawaymyblues.Moreover,Ithankmyparentsandmygrandparents.Ioweagreatdealtothemforalwaysbelievinginmeandfortheirsupportateverystepofmylife.Last,butnotleast,Ithankmybelovedhusband,JunjieLiu.Heismybestfriendandmybiggestemotionalsupport.Hemakesmyhomesweet.Ithankhimforhisunwaveringlove,careandsupport. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2OPTIMALALLOCATIONOFSTOCKLEVELSANDSTOCHASTICCUSTOMERDEMANDSTOACAPACITATEDRESOURCE .................. 17 2.1Motivation .................................... 17 2.2ProblemDenitionandFormulation ..................... 22 2.3GeneralizedBendersDecompositionfor(P) ................. 25 2.4HeuristicSolutionApproachfor(P) ...................... 29 2.5ComputationalResults ............................. 34 2.6ConcludingRemarks .............................. 43 3ALGORITHMSFORMULTI-ITEMPROCUREMENTPLANNINGWITHCASEPACKS ........................................ 45 3.1Motivation .................................... 45 3.2RelatedLiterature ............................... 49 3.3ProblemDenitionandFormulation ..................... 52 3.4SolutionMethodsforSpecialCases ..................... 57 3.4.1SingleCasePack ............................ 57 3.4.2TwoCasePacks ............................ 61 3.4.3NCasePacks .............................. 64 3.5StrengtheningtheFormulation ........................ 64 3.6HeuristicSolutionApproach .......................... 66 3.7AddressingDemandUncertainty ....................... 71 3.8ComputationalResults ............................. 74 3.8.1HeuristicperformanceandcomparisonwithCPLEX ........ 75 3.8.2Sensitivityanalysisofparameters ................... 77 3.8.3Costofrequiringacasepack ..................... 79 3.9ConcludingRemarks .............................. 82 4INTEGRATEDCASE-PACKCONFIGURATIONANDPROCUREMENT .... 84 4.1Motivation .................................... 84 4.2RelatedLiterature ............................... 86 5

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..................... 90 4.4SolutionMethods ................................ 94 4.4.1Reformulationandlinearizationmethod ............... 95 4.4.2Aniterativeheuristic .......................... 96 4.4.3Aniterativeheuristicwithalternativecase-packsizes ........ 101 4.4.4Geometricprogrammingbasediterativeheuristic .......... 102 4.5ComputationalResults ............................. 106 4.6ConcludingRemarks .............................. 113 5CONCLUSIONS ................................... 115 APPENDIX ASOLVINGTHEMULTI-ITEMNEWSVENDORSUBPROBLEM ......... 117 BOBTAININGTHESUBGRADIENTOFv(.)ATXk 119 REFERENCES ....................................... 121 BIOGRAPHICALSKETCH ................................ 128 6

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Table page 2-1Parameterdistributionsusedincomputationaltests. ............... 35 2-2LevelsusedforanalysisoftheimpactofcapacitylevelVandcustomerrevenuevalue^j. ........................................ 36 2-3Computationaltestresultsforaveragerunningtime(second)andperformance 38 2-4Computationaltestresultsforvariouslevelsofcapacityandrevenue. ...... 39 2-5Computationaltestresultsforaveragerunningtime(second)andperformanceoftheheuristicapproach .............................. 42 3-1ExampleofcompositionoftwocasepackswithfourSKUs. ........... 47 3-2Parameterdistributionsusedincomputationaltests. ............... 75 3-3Thenumberoftimewindowsandcutsusedwithinrelax-and-xheuristics. ... 76 3-4Computationaltestresultsforaveragerunningtime(seconds)andoptimalitygapforsolutionapproaches. ............................ 77 3-5Numberoftimes(outoften)eachsolutionmethodattainedthelowest-costsolution. ........................................ 78 3-6Effectsofthetime-windowlength,,onsolutionvalueforthemedium-sizeproblemset. ..................................... 79 3-7Effectsofthelengthofxinginterval,,onsolutionvalueforthemedium-sizeproblemset. ..................................... 79 4-1Parameterdistributionsusedincomputationaltests. ............... 107 4-2Computationaltestresultsforaveragerunningtime(seconds)andoptimalitygap .......................................... 109 4-3Theiterationnumberforiterativeandintegratedheuristics ........... 110 7

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Figure page 2-1Numberofiterationsrequiredforsolvingproblemswithdifferentparameterlevels. ......................................... 41 2-2GBDconvergenceillustrationwhenm=3,n=10,Visatlevel1,and^jisatlevel2. ......................................... 41 3-1Singlecasepackproblemnetwork ......................... 60 3-2Layerednetworkfor(P2),theproblemwithtwocasepacks. ........... 63 3-3Effectsoftimewindowparametersonsolutionvalue.(T=30,N=5,M=6) .... 80 4-1Objectivevaluevsiterationsfortheintegratedapproach(2ndinstanceM=6,T=20) ....................................... 110 4-2ObjectivevaluevsRvaluefortheiterativeandintegratedheuristics(5thinstanceM=6,T=20) ............................ 111 4-3ObjectivevaluevsRvaluefortheIter+R Enum(5thinstanceM=6,T=20) 112 8

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57 ],Deanetal.[ 27 ],Merzifonluogluetal.[ 66 ],andAgralandGeunes[ 2 ],KosuchandLisser[ 58 ]),witheachcustomer's(uncertain)demandquantitycorrespondingtothetypicalitemsizeintheseknapsackproblems.However,ourproblemgeneralizesthisproblemclass,asitinvolvestheadditionaldimensionoflimiteditemavailabilityasaresultoftheitem-to-vehiclestockleveldecisions.Thatis,thevehicle'scapacityconstrainstheallocationofitemstocklevelstothevehicle,andtheitemstocklevelswithinthevehicleconstraintheabilitytomeetcustomerrequirementsduringsalesvisits.Asinseveralofthesecitedworksonstochasticknapsackproblems,wewillfocusonthecasewhereitemsizes(orcustomerpartsrequirements)areindependentbutnotnecessarilyidenticallydistributednormalrandomvariables(thenormaldistributionparametersweusearesuchthattheprobabilityofnegativedemandisnegligible). 18

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49 ]discussedtheconstrainedmulti-itemnewsvendormodelindetail.Anotherclassofclosely-relatedproblemstotheoneweconsiderisknownastheclassofrepairkitproblems(e.g.,Smithetal.[ 88 ],TeunterandHaneveld[ 92 ][ 93 ]andGormanandAhire[ 44 ]).Mostoftheserepairkitproblemsareaimedatminimizingexpectedholdingcostsundergivenservicelevelrequirements.Incontrast,weconsiderthemaximizationofexpectedprotunderanadditionalvehiclecapacityconstraint,usingashortage(penalty)costintheobjective(which,inturn,canbeusedtoensuredesiredservicelevels).ThemostcloselyrelatedrepairkitproblemthatconsidersmultipletypesofitemsandvehiclecapacityistheoneproposedbyGormanandAhire[ 44 ],whodevelopedasingle-passgreedyheuristicapproachforthemultiplevehicleplanningproblem.Withinthesingle-periodcontextweconsider,demandallocationandvehiclestockingdecisionsmustbedeterminedpriortoactualcustomerdemandrealizations.Hence,wefaceajointdemandassignmentandstocklevelproblemwithuncertaincustomerdemands.Ourgoalistosimultaneouslysetstocklevelsforthemultipleitems(parts)withinthevehicleanddeterminecustomerdemandassignmentsinordertomaximizeexpectedprot,equaltotheexpectedrevenuefromcustomervisitslessexpectedshortageandvariablecosts.Tothebestofourknowledge,noexistingworkconsidersanexactsolutionforcombineddemandallocationandvehiclestockingproblemsformultipleitemsunderuncertaindemand,asweaddressinthischapter.Thus,ourcontributionsincludeprovidinganewmodelforthisproblemclass,aswellasexactandheuristicsolutionprocedures,whichwedemonstratetobeextremelyeffectivewhencomparedwithastate-of-the-artmixedintegernonlinearoptimizationsolver. 19

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57 ],Merzifonluogluetal.[ 66 ],AgralandGeunes[ 2 ],KosuchandLisser[ 58 ]).Ifthereisnosharedresource(vehicle)capacityconstraint,wewillhaveastochasticmultidimensionalknapsackproblem(MKP)(Kellereretal.[ 56 ],VasquezandVimont[ 98 ],andAkcayetal.[ 5 ]).TheMKPisawell-knownNP-hardproblem,whichimpliestheNP-hardnessofourproblemundergeneraldemanddistributionsaswell.Further,ifweconsiderthemultiplevehicleversionofourproblemwithoneitemtypeandnosharedresourcecapacity,theproblemtakestheformofthejointfacilityassignmentandcapacityacquisitionproblem(Taaffeetal.[ 91 ]),wheretheoptimalstocklevelforthepartcorrespondstothefacilitycapacity.Similarly,ifthecustomerdemandassignmentsaregiven,weonlyneedtoconsiderthestocklevelsformultipleitemsonthevehicle,whichreducestoasetofconstrainedmulti-itemnewsvendorproblems.Withinthecontextofmake-to-stockqueues,Benjaafaretal.[ 15 ]appeartobethersttoconsiderthejointdemandallocationandinventorycontrolproblem.Theyconsideredthelong-runfractionofdemandforeachproducti(inasetofproducts)allocatedtoeachfacility(fromasetoffacilities,analogoustothevehiclesinthecontextwediscussedabove).Theyconsideredtheminimumlong-runexpectedcostperunittimeundercontinuousassignmentvariables(thedemandallocationproblem,orDAP),andthedemandpartitioningproblem(DPP)withbinaryassignmentvariables.TheyusedconvexoptimizationalgorithmstosolvetheDAPandabranchandboundalgorithmfortheDPP.FedergruenandZipkin[ 33 ]developedacombinedroutingandinventoryallocationmodelforasingleitemunderstochasticcustomerdemands(Federgruenetal.[ 30 ],generalizedthistoaccountforperishableitemswhenfreshandoldstockoftheitemisavailableandout-of-datecostsmayexist).Intheirmodel,customerinventory 20

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39 ],canbeusedtoefcientlysolvenonlinearprogrammingproblemswithcomplicatingvariablesbytemporarilyxingthesevariablesanddealingwiththeremainingproblem,whichisgenerallymucheasiertosolve.ToextendtheapplicabilityoftheGBDapproach,GeromelandBelloni[ 40 ]studiedthedifferentiabilityofasetofrelatedperturbationfunctionsandproposedamethodtohandleproblemsofamoregeneralclassthanGeoffrion[ 39 ]considered.WewillemploythesegeneralizedmethodsofGeromelandBelloni[ 40 ]togenerateBenderscuts.Manypracticalproblems,suchasmulti-commoditynetworkows,quadraticassignmentproblems,andcombinedlocation-inventoryproblemshavebeenefcientlysolvedusingGBDtechniques.Inourapproach,werstxthecomplicatingintegervariables;thatis,foragivenfeasibledemandallocation,wesolvetheremainingmulti-itemconstrainednewsvendorsubproblemusingLagrangianrelaxation(HadleyandWhitin[ 49 ]).Wethenusethesolutionofthissubproblemtoaddthecorrespondingsupportfunction(Benderscut)totheso-calledmasterproblemandsolvearelaxedmasterproblem.Werepeatthisprocessbyusingthedemandallocationdecisionfromthesolutionoftherelaxedmaster 21

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2.2 wedeneandformulateourstochasticdemandallocationandinventorystock-levelproblem.WethenanalyzeimportantpropertiesofoptimalsolutionsforourproblemanddevelopageneralizedBendersdecompositionapproachinSection 2.3 .Section 2.4 presentsaheuristicsolutionapproachforsolvinglargesizeprobleminstancesthatmayprecludetheuseofexactmethods.InSection 2.5 wediscusstheresultsofacomputationalstudy,usedtovalidateoursolutionmethodsandcomparethemwiththeresultsofthreebenchmarkcommercialsolvers,GAMS/LINDOGlobal,GAMS/SBBandGAMS/CoinBonmin.Finally,concludingremarksareprovidedinSection 2.6 22

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item(part)index,i=1,...,m. customerindex,j=1,...,n. expectedrevenuegainedbyservingcustomerj,i.e.,byallocatingcustomerjtothevehicle. unitshortagecostincurredfornotsatisfyingaunitofdemandforitemi. unitsalvagevalueincurredforanunuseditemi. unitvariablecostforcarryingitemiinthevehicle,where^ei>^ci>gi. unitsizeofitemi. vehiclecapacity. randomvariabledenotingthedemandforitemibycustomerj,withmeanandstandarddeviationijandij,respectively.DecisionVariables binarydecisionvariable,equalto1ifcustomerjisassignedtothevehicle,0otherwise.Thevector(column)X=[x1,,xn]Tcharacterizestheassignmentofcustomerstothevehicle. nonnegativedecisionvariableequaltothenumberofunitsofitemicarriedinthevehicle.ThevectorY=[y1,,ym]Tdeterminestheassignmentofitemstothevehicle. 23

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where[x]+=maxfx,0g,ei=^eigi,ci=^cigi,andj=^jPmi=1giijdenotethenetpenaltycost,variablecost,andrevenue,respectively(^ei>^ci>giimpliesthatbotheiandciarenonnegativeforalli).Observethat,foranycustomerj,ifthecoefcientofthethirdtermissuchthatj=^jPmi=1giij0,itisstraightforwardtoshowthatwecansetxj=0withoutlossofoptimality.Wethereforeassumewithoutlossofgeneralitythatj>0forallj.Wecannowformulateourstochasticresourceallocationproblemwitharesourcecapacityconstraintas:(P)minP(Y,X)subjectto:mXi=1siyiV,xj2f0,1g,j=1,...,n,yi0,i=1,...,m.Theconstraintensuresthatthevehicle'scapacityisnotviolated.Observethatwepermitthestocklevelstotakecontinuousvalues.Foragivenvectorofcustomer 24

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49 ]). (MINV)minPmi=1ciyi+eiE(Diyi)+subjectto:Pmi=1siyiV, 49 ]).Moreover,foragivenvectorofitemstocklevelsY,wehavea0-1integerprogrammingproblemwithoutexplicitconstraints.WethusconsiderageneralizedBendersdecompositionapproachforsolvingthisresourceallocationproblem.Werststreamlinethenotationforproblem(P)bydening:H(X,Y)=mXi=1eiE"nXj=1dijxjyi!+#;G1(Y)=mXi=1ciyi;Q(X)=nXj=1jxj;G2(Y)=mXi=1siyiV.Usingthisnotation,problem(P)canbeformulatedas: 25

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39 ]).Deningthedualvariablew0correspondingtotheconstraintin(SP),wecanwritetheLagrangiandualasv(X)=maxw0minY0[H(X,Y)+G1(Y)+wG2(Y)].

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2 )isajointlyconvexfunctiononR+nR+m.Proof.Toshowthat( 2 )isajointlyconvexfunction,weonlyneedtoshowthat(Pnj=1dijxjyi)+isajointlyconvexfunctionbylimitingourattentiontoagivenrealizationofthedemanddij,becausetheothertermsin( 2 )arealllinear,andtheexpectationofaconvexfunctionisaconvexfunction.Notethat(Pnj=1dijxjyi)+=max(0,Pnj=1dijxjyi)isamaximumoftwojointlyconvexfunctionsandis,hence,jointlyconvex.Then,bythefactthatthesummationofjointlyconvexfunctionsisjointlyconvex,theproofiscomplete. @XH(X,Yk)+G1(Yk)+wkG2(Yk)X=Xk.

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53 ],p.525),wecanshowthatv(X)in( 2 )isconvexinx,sinceH(X,Y)andG1(Y)areconvexinbothxandy.Then,becausev(X)isconvex,itiswellknownthatk(X)servesasalinearsupportfunctionatXk.DetailsregardingthecorrectnessofthissupportfunctioncanbefoundinGeromelandBelloni[ 40 ]usingTheorem3andformulas(31)(34).Weomitthedetailsforthesakeofbrevity. Thedetailsofhowwecomputek(X)areprovidedinAppendixB.WecannowrepresenttheRelaxedMasterProblem(assumingKlinearsupportfunctionshavebeengenerated)as:(RMP)minQ(X)subjectto:k(X),k=1,2,,K,X2f0,1g,2R.Herek(X)isaBenderscutforgivenvaluesofXkandwk.Observethatthe(RMP)isamixed0-1linearprogramwithasinglecontinuousvariable.Whilesuchproblemsarenotgenerallyconsideredeasy(theyare,ofcourse,NP-Hard),theyaremuchmoretractablethanouroriginalnonlinearmixed0-1program,andproblemsofreasonablesizecanbesolvedinacceptabletimeusingCPLEX.ThegeneralizedBendersdecompositionapproach(andvariantsthereof)canbeshowntobeconvergentwhenXisanitediscreteset.Wenextformalizeouralgorithmasfollows:Step1:ChooseaninitialvectorX0thatensuresafeasiblesolutionforthesubproblemandselectanoptimalitytolerance.Solvesubproblem(SP)atX0,whichisamulti-itemconstrainednewsvendorproblem,obtainingY0andacorrespondingoptimaldualsolutionw0(AppendixAdiscussesthesolutionapproachfortheMINVproblem).SettheupperboundUB=v(X0)Q(X0)andlet( 28

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2.1 andtheresultsinAppendixB.Step2:Solvetherelaxedmasterproblemwithallpreviouslygeneratedcuts.Let(X,)denoteanoptimalsolutionto(RMP)andsetthelowerboundLB=Q(X).If(UBLB)<,stop;otherwise,gotoStep3.Step3:SolvethesubproblematX=X,denotingYastheoptimalsolutionvectorandv(X)astheoptimalsolutionvalue.Ifv(X)Q(X)
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Pj2ijxjL(z(i)).Substitutingtheseexpressionsforyi(X)andthelossfunctionintoP(Y,X)inEquation( 2 ),withaslightabuseofnotation,theexpectedcostcanbewritteninthefollowingform: Xj2ijxj.(2)LetKi=ciz(i)+eiL(z(i))representtheithcoefcientvalueofthesquareroottermsanddenoterj=j+Piciijasthenegativeoftheexpectednetrevenueforservingcustomerj.Wethenneedtosolvethefollowingprobleminthecaseofnormallydistributeddemands:(RP)minXjrjxj+XiKis Xj2ijxjsubjectto:xj2f0,1g,j=1,...,n.Notethatwecan,withoutlossofoptimality,setxj=0foranyjsuchthatrj0,i.e.,anyjsuchthatcijijj(whichimpliesthattheexpectedcustomercostoutweighsthecustomer'sassociatedrevenue).ThisrelaxedproblemRP,obtainedbydroppingthecapacityconstraintandexpressingYintermsofXhassomespecialproperties:

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41 ]. 84 ]implythatthatP(X)isasubmodularfunction. AsdiscussedinShenetal.[ 84 ],Grotscheletal.[ 48 ]showedthatasubmodularfunctioncanbeminimizedinpolynomialtime.StronglypolynomialtimealgorithmsareavailableforthisproblemclassasaresultofworkbyIwataetal.[ 54 ]andSchrijver[ 80 ].Weareinterestedinexploitingtheparticularspecialstructureofourprobleminordertoobtainafastheuristicapproachforproblem(P).Werstdiscussasolutionmethodforaspecialcaseofourproblem.Thisspecialcaseariseswheneitherij=j,forallj(theequal-item-variancecase)orwhenonlyonetypeofitemm=1is 31

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Xj2jxjsubjectto:xj2f0,1g,j=1,...,n.WecansolvethisspecialcaseusingasimplesortingschemeasinTaaffeetal.[ 91 ].Werstsortcustomersinnonincreasingorderoftheratioofexpectednetrevenuetotheuncertaintyinthatcustomer'sdemand.Thisresultsinindexingcustomerssuchthat 84 ]. Forthegeneralcasewithunequalitemvariances,wecannotshowthatasortingschemeasin( 2 )isdirectlyavailable.Wecan,however,utilizetheinsightfrom( 2 )toarriveataheuristicrankingschemeforcustomers.Sincewewouldliketocapturethetradeoffbetweenrevenuesandvariance-relatedcost,wereplacethedenominatorwithaweighted-averagevarianceacrossitemsforeachcustomer.Thatis,weuseaweighted-averagevariancevalue,wherecustomerj'sproductivarianceisweightedbyafactorij.Wethendene 2.4 (thisfollowsbecauseanyoneofthen!possibleorderedvectorsofcustomerindicescanbeobtainedthroughanappropriatechoiceofijvalues).Oneoptionistosetij=ij,inwhichcase 32

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4-1 summarizesthecommondatausedinourcomputationalstudy.Foreachprobleminstance,thedemand,revenueandallcostdataweregeneratedfromuniformdistributions.WeletU(`,u)denotethecontinuousuniformdistributionwithlowerbound`andupperboundu.TobenchmarktheperformanceofouralgorithmagainstLINDOGlobal,SBB,andCoinBonmin,wetestedoursolutionmethodfor16problemsetsandcomputedtherunningtimeandoptimalityperformance(these16problemsizesusevehiclecapacitylevel3andcustomerrevenuelevel2showninTable 2-2 ).Eachoftheseproblemsetsischaracterizedbyauniquecombinationofthenumberofcustomersandthenumberofitems,(n,m),whereweconsideredn2f5,10,30,50gandm2f3,10,20,50g.Foreachcombinationof(n,m)values,wetested10randomlygeneratedproblem 34

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2-2 )inordertostudytheimpactofdifferentvaluesofthesetwoimportantparameters.Tothisend,wewillconsideranadditional16problemsetscorrespondingtoeachpairofdistributionsusedforthesetwoparameters,withthenumberofcustomersxedat10andthenumberofitemsxedat3.Wetested10randomlygeneratedprobleminstancesforeachoftheseproblemsets,whichconstitutesanother160testcaseswithn=10andm=3.Forthislattersetof160testproblems,whichwereusedtoevaluatetheimpactsofcapacityandrevenuelevels,weusedourgeneralizedBendersdecompositionapproachtoobtainoptimalsolutions(althoughwedidnotcomparetheresultsofthesecaseswiththoseusinganyoftheGAMSsolvers).Wethereforetestedatotalof320probleminstances,ofwhich160casesweresolvedusingbothouralgorithmandthethreedifferentGAMSsolvers,andanadditional160testproblemsweresolvedusingouralgorithmonly. Table2-1. Parameterdistributionsusedincomputationaltests. UnitShortageCost,^eiUnitSalvageValue,giUnitHandlingCost,^ciCapacity,VU(2.5,3.6)U(0.1,0.5)U(1,1.5)U(5mn,80mn)ExpectedDemand,ijStandardDeviation,ijUnititemsize,siFixedRevenue,^jU(50,100)U(14,50)U(0.5,1)U(100m,220m)

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LevelsusedforanalysisoftheimpactofcapacitylevelVandcustomerrevenuevalue^j. CapacitydistributionU(5mn,10mn)U(10mn,20mn)U(20mn,40mn)U(40mn,80mn)RevenuedistributionU(100m,140m)U(140m,180m)U(180m,220m)U(220m,260m) 2 36

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2-3 .Inthetable,GBDrepresentsourgeneralizedBendersdecompositionapproach.Thetableshowsthatourapproachisgenerallyatleast30timesfasterthanGAMS/LINDOGlobal,and2timesfasterthanGAMS/SBBandGAMS/CoinBonmin.Alloftheproblemswetestsweresolvedwithin17secondsusingouralgorithm.However,GAMS/LINDOGlobaltakesaround30secondsforsmallersizeproblems,whilethemajorityofproblemsweresolvedin2to10minutes,withthelargestsizeproblemrequiringnearly1hour.Toprovideabenchmarkfortheperformanceofouralgorithm,weusethePerformanceRatio(PR)asanindex,whichcorrespondstotheaveragesolutionvalueasapercentageoftheoptimalsolutionvalue.Ouralgorithm,SBBandCoinBonmincanfoundanoptimalsolutionforallproblems,whileLINDOGlobalworkswellonlyforsmallersizeproblem(forlargersizeproblems,LINDOGlobaldoesnotperformquiteaswell).Muchofthetimeitstopswithonlylocallyoptimalsolutionsasaresultoftheiterationlimit,whichaccountsforsimplexiterations,barrieriterations,nonlineariterationsandboxiterations.Fortheproblemwith(m,n)=(50,50),for10randomlygeneratedinstances,threeofthecasescouldnotbesolvedwithinthetimelimitof5,000seconds,andinonecaseitincorrectlyconcludedthattheproblemwasinfeasible.TheaveragePRwasaslowas62.14%forGAMS/LINDOGlobal.Basedontheabovecomparison,wefoundthatouralgorithmsignicantlyoutperformedGAMSMINLPsolversLINDOGlobal,SBBandCoinBonminacrossthe160randomlygeneratedprobleminstancesundervehiclecapacitylevel3andcustomerrevenuelevel2.ParameterAnalysis 2-4 ,weshowhowthedifferentlevelsoftwoimportantparameters(capacityandrevenues)affectedtheresultsforthecaseof(m,n)=(3,10).Theaveragevalue(acrossthe10randomlygeneratedinstances)ofexpectedprotisshownincolumn6ofthetable.Therstcolumncorrespondstothefourlevelsoftheresourcecapacity 37

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Computationaltestresultsforaveragerunningtime(second)andperformance 3521.20100.00%1.424.150.531030.20100.00%2.166.490.753086.78100.00%3.769.660.8150365.25100.00%6.0917.741.3510528.89100.00%2.475.851.461046.75100.00%2.656.571.7930394.7599.91%7.0518.473.5550697.2599.96%9.1320.944.3220530.90100.00%2.985.972.211074.75100.00%3.489.363.2030440.2599.63%9.1219.776.6750797.2999.68%16.3921.387.11505166.0099.97%3.796.343.5710314.4487.16%5.309.899.1030547.3398.95%23.4931.1716.70502916.8862.14%45.4530.7314.43 2-1 ,againforthecaseof(m,n)=(3,10). 38

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Computationaltestresultsforvariouslevelsofcapacityandrevenue. 111.089.701.5091.2120.746.902.00296.3730.484.703.30605.6540.242.106.801159.08210.877.802.80205.7620.887.803.50607.3930.716.754.881081.9740.323.007.881835.30311.5813.505.13386.2821.038.505.881041.9030.736.506.631647.4640.201.569.782981.94410.857.508.50515.3820.191.389.881764.5930.211.759.882763.6940.191.309.903907.31 2-1 ,eachbarinthechartcorrespondstoacustomer-specicrevenuelevel,andtheresultsaregroupedbyvehiclecapacitylevel.Theverticalaxisshowstheaveragenumberofiterationsrequired.Wecanseethatforeachvehiclecapacitylevel,theheightsofthefourbarsinarowaredecreasingastherevenuelevelincreasesfromlevel1to4(recallthattheaveragecustomerrevenueincreasesinthelevelnumber).Whenallcustomerrevenuesincrease(allelsebeingequal),wearemorelikelytoassignahighernumberofcustomerstoavehicle,andtherewillberelativelyfewerattractivechoicesfordemandallocation,thusresultinginfeweriterations.Attheextreme,forexample,whenallcustomershaveveryhighrevenuesrelativetocosts,anoptimalsolutionwouldassignallcustomerstothevehicleandsolvethecorrespondingmulti-itemconstrainednewsvendorproblem.Asthegureshows,wecannotestablishadenitepatternasafunctionofthecapacitylevelforagivenrevenuelevel(thevariationintheaveragenumberofiterationsisreasonablysmall,andcanatleastpartiallybeattributedtorandomvariationamongprobleminstances).Aswenotedpreviously,themorecomputationallydifcultproblemsoccurwhentherevenuesarecloselymatchedtotheoverowcosts.Asthecapacitylevelincreases(inparticularatcapacitylevel4),the 39

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2-4 ,undereachcapacitylevel,thenumberofcustomersservedandthevalueoftheexpectedprotbothincreaseasthecustomerrevenuelevelsincrease;similarly,foreachrevenuelevel,whenthevehiclecapacityincreases,thenumberofcustomersservedincreasesandtheexpectedprotincreases.Theseeffectsareallquiteintuitiveandareinaccordancewithwhatwewouldexpectinreal-worldpractice.Wenotethatmostoftheproblemsweresolvedwithanactivevehiclecapacityconstraint;inthelastofthecapacitysettings(Vatlevel4),thevehiclecapacityconstraintwasredundantforabouthalfoftheinstances,whichisresponsibleforthedecreaseinthenumberofrequirediterationsatthiscapacitylevel.InordertoillustratetheconvergenceofourgeneralizedBendersdecompositionapproachforourproblem,weillustrateaninstancesolvedwithinveiterationswithm=3,n=10,Vatlevel1and^jatlevel2,showninFigure 2-2 .Wecanseethattheupperbound(UB)convergesquicklyatthebeginning,whichwefoundwasthecaseingeneral.Sothenumberiterationsrequiredtoimprovethelowerbound(LB)isanimportantfactorwithrespecttorunningtime.Tosummarize,ournumericalresultsshowedthatourgeneralizedBendersdecompositionalgorithmcansolvethemulti-item,multi-customerresourceallocationproblemmuchfasterthanwell-knownbenchmarkcommercialnonlinearsolvers(LINDOGlobal,SBBandCoinBonmin),andtypicallysolvedtheproblemswetestedwithin17seconds.Theoptimalityperformancecomparisonandparameteranalysisprovidedfurtherevidenceoftheefciencyandeffectivenessofouralgorithm.HeuristicPerformance 40

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Numberofiterationsrequiredforsolvingproblemswithdifferentparameterlevels. Figure2-2. GBDconvergenceillustrationwhenm=3,n=10,Visatlevel1,and^jisatlevel2. 41

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2-5 .Forcomparisonpurposes,weusethesame160testinstancesshowninTable 2-3 .Fortheseheuristics,wepresenttheresultingoptimalitygapinTable 2-3 ,equaltooneminusthePerformanceRatio(PR). Table2-5. Computationaltestresultsforaveragerunningtime(second)andperformanceoftheheuristicapproach 350.091.81%0.091.81%0.141.82%0.141.81%100.154.19%0.153.60%0.193.11%0.182.10%300.285.50%0.285.30%0.305.12%0.305.15%500.296.00%0.306.00%0.353.78%0.323.96%1050.143.20%0.143.20%0.163.20%0.172.25%100.202.75%0.203.80%0.222.65%0.222.31%300.432.98%0.423.20%0.463.36%0.473.62%500.582.59%0.592.40%0.623.34%0.623.00%2050.201.66%0.201.66%0.221.27%0.221.65%100.281.33%0.281.20%0.311.01%0.311.01%300.523.24%0.513.15%0.563.50%0.553.21%500.911.75%0.902.36%1.091.68%1.122.39%5050.401.22%0.401.22%0.430.32%0.431.22%100.551.17%0.561.07%0.610.86%0.601.07%301.201.25%1.181.12%1.401.20%1.381.20%502.201.12%2.221.18%2.341.07%2.361.04% 2-5 ,weobservethatthecomputationalrequirementsaresubstantiallylowerforourheuristic,resultingincomputingtimesaround2001000timesfasterthanLINDOGlobaland416timesfasterthanourBendersdecompositionmethod.Theonlytimeconsumingelementoftheheuristicisthecomputationoftheexpectedvaluetermintheobjectivefunctionv(X)formultipleassignmentvectorsXk,k2f1,2,,ng,whileintheBendersdecompositionalgorithm,eachiterationrequiresthecomputationofanexpectedvaluetermandthesolutionofalinearintegerprogrammingproblem.TherequiredcomputationtimeforHCandHDCisalittlelongerthanforHSandHDasaresultoftheneedtocomputethecostparameterKi.Theoptimalitygaponaverageis2.61%forHS,2.64%forHD,2.33%forHC,and2.31%forHDC.However, 42

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74 ])todeterminehowastore'stotalorderquantityshouldbedistributedamongdifferentsizes.Otherrmsusethepreviousyear'ssaleshistoryofdifferentsizesatthechainleveltomakesizedecisionsatthestorelevel(FriendandWalker[ 37 ];Oracle[ 69 ]).However,thetruedemandproleofasetofsizes(e.g.,21%Small,34%Medium,28%Large,17%Extra-Large)canvarywithmerchandisetype,storelocation,andmerchandiseattributeslikevendor,season, 45

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51 ]).Retailchainssometimesreceivebulkpackagesforeachsizeandthenbreak,sort,andrepackagethesetomeetthedemandsofindividualstores.However,manyretailchainsprefertoreceivecasepacks(alsocalledpre-packs)fromvendorsandthenusecombinationsofthesecasepackstosatisfystoredemands.ThetermcasepackreferstoasetofindividualStockKeepingUnits(SKUs)ofmerchandisepackagedintobiggercasesforeasierhandlinginthesupplychain.Casepacksmaybeusedinnumerousindustriesincludinghome/ofceproducts,personalhygiene/beautyproducts,andfoodproducts.Forexample,bulkcasepacksoffoodsnacksmaycontainamixofindividualbagsofdifferentavoredpotatochips,cornchips,andpretzels.Similarly,adepartmentstorewhocarriesmakeupmayreceivecasepackscontainingvariouscolorsoflipstickorothermakeupproduct.Anotherexamplemayariseatthemeparks(e.g.,WaltDisneyWorld),whocarrysouveniritemsassociatedwithmultiplechildren'scharactersandreceivedeliveriesofcasepackscontainingamixofdifferentcharacter-relateditems.Thecasepackservesasthelowestlevelofpackaginghierarchy,andisdesignedtoowthroughfromthevendortotheretailstore.HandlingoftheselargercasepacksratherthanindividualSKUsprovestobecheaperandfasteratalltouchpointsinthevendor-to-retail-storesupplychain.Breakingbulktypicallyinvolvesagreatdealofworkatadistributioncenterandrequiresassociatedinfrastructureandpersonnel,whiledistributioncenterscanprocesscasepacksquicklyandatminimalcost,oftenthroughtheuseofcross-dockingtotransferthemtoindividualstores.Atypicalcasepackforapparelmightcontain12,18,or24pieces,thoughpacksof6mayalsobeusedforreplenishmentduringtheseason.Eachindividualcasepacktypicallycontainsmanydifferentsizes.Forexample,awholesalermaysellboys'solidpiquepoloshirtsincasepacks,eachhaving12piecesofonecolor,withassortedsizes.Formisses'sizes 46

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78 ]).AnexampleofthecontentsoftwocasepacksforcasualshirtsisshowninTable 3-1 .Supply-demandmismatchesatthestorelevelresultfromdecisionprocessesinvolving(Oracle[ 69 ];Parpiaetal.[ 71 ]):(a)thesizedemandproleusedtomakepurchasesatdifferentstores,(b)thecontentsofcasepacks,and(c)theallocationofcasepacksandsizestoindividualstoresfromthedistributioncenter.Inthispaper,wefocuson(c)andassumethatthecompositionofcasepackshasbeenpre-determined. Table3-1. ExampleofcompositionoftwocasepackswithfourSKUs. CasePack12442CasePack20660 47

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78 ]).Sinceavendors'casepacksaresoldtomanycustomers(whilecustomizedcasepacksarepreparedspecicallyforaparticularchain)itisunderstandablethatapremiummaybechargedforcustomizingcasepacks.However,largeretailchainscanusetheirbargainingpowerwithvendorstokeepthispremiumlow.Retailerscanthususeourmodeltoevaluatetheimpactofproposedcustomizedcasepacksonpotentialsavingswhencomparedtoavendor'sexistingcasepacks.Inthispaper,wewillstudytheuseofvendorcasepacksandprocurementplanningformultiplecasepackstomeetstoreleveldemand.Whilethecostsassociatedwithagivensetofproposedcasepacksmaybeaccomplishedusingourmodel,thedesignproblemforcustomizedcasepackswillserveasafutureresearchdirection.Theremainderofthispaperisorganizedasfollows.WediscussrelatedliteratureinSection 3.2 .Section 3.3 denesandformulatesthecasepackprocurementplanningproblem.InSection 3.4 ,westudyspecialcasescontainingoneandtwocasepacks, 48

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3.5 ,andprovideasetofheuristicapproachesinSection 3.6 .WethenprovideadiscussionofpotentialapproachesfordealingwithdemanduncertaintyinSection 3.7 .InSection 3.8 ,wediscusstheresultsofdetailedcomputationalresultsoftheimplementationofourheuristicapproach.Inparticular,wecomparetheresultsforavarietyofprobleminstancesofdifferentsizeswithsolutionviathecommercialsolverCPLEX,andtheninvestigatethesensitivityoftheresultstochangesinseveralparametersusedintheheuristicapproach.Finally,concludingremarksareprovidedinSection 3.9 3.3 .)Theuncapacitated,single-itemdynamiclotsizingproblemhasreceivedasignicantamountofattentionintheliteraturesinceitwasrstintroducedin1958byWagnerandWhitin[ 100 ].TheirdynamicprogrammingalgorithmrunsinO(T2)undergeneralcostparameters,whereTistheplanninghorizonlength.Subsequently,FedergruenandTzur[ 32 ],Wagelmansetal.[ 99 ]andAggarwalandPark[ 1 ]concurrentlyimprovedtheworst-casecomplexityforobtaininganoptimalsolutiontoO(TlogT)undergeneralcostparameterassumptions,andtoO(T)forproblemswithstationarycosts. 49

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90 ]consideredarelatedproblemarisinginsheet/metalmanufacturing,die-casting,andchemicalprocessing,suchthatforeachproductionsetup,productsmustbeproducedinaxedproportiontooneanother,andeachproductcanbeapartofmultipleproductfamilies.Theyviewthemodelasageneralizationofthejointreplenishmentproblem,discussthecomplexityoftheproblemforspecialcases,andsolveamodelwithcontinuousproductionquantities,applicabletochemicalprocessing.TheyuseaLagrangianrelaxationapproachforsolvingtheproblem,whileBhatiaandPalekar[ 17 ]proposeavariableredenitionapproachfortheproblem.Inaddition,theequivalentofasinglecase-packmodelistransformedtothesingle-productuncapacitatedlotsizingprobleminStowersandPalekar[ 90 ]).Aswelaterdiscuss,themodelweconsiderdiffersfromtheirsinthetwo(or,moregenerally,multiple)case-packcase,aswerequirethatdifferentcasepacksincurashared(orjoint)ordercostinanyperiod(whereastheirmodelrequiresincurringaproductionsetupcostforeachproductfamily).Formulti-itemproblemswithniteproductioncapacities,efcientalgorithmsarenotavailable,assuchproblemshavebeenshowntobeNP-hard(Florianetal.[ 35 ]).Asaresult,variousheuristicapproacheshavebeendevelopedforthisclassofproblems.Polyhedralmethodshavealsoledtoimprovedmodelformulationsinordertoprovidebetterlowerboundsonoptimalsolutionvalues.Baranyetal.[ 12 ],PochetandWolsey[ 76 ],Loparicetal.[ 63 ],andAtamturkandMunoz[ 9 ]strengthenedtheoriginalformulationbyaddingvalidinequalities.KrarupandBilde[ 59 ],BelvauxandWolsey[ 14 ],andWolsey[ 101 ]addednewvariablesand/orreformulatedtheproblemsinordertostrengthentheirformulations.Forheuristicapproachesongeneralizationsofthisproblemclassinvolvingmultiplelevels(ormachines)andmultipleechelons(orfacilities),see,forexample,BahlandRitzman[ 10 ],Billingtonetal.[ 18 ],BlackburnandMillen[ 19 ],RollandKarni[ 79 ],ParveenandHaque[ 72 ],andAkartunaliandMiller[ 4 ]. 50

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29 ]developedaclassofprogressiveintervalheuristicsforacapacitateddynamiclotsizingproblemwithjointsetupcoststhatareincurredonceanorderisplacedinagivenperiod.Theyalsoextendedthemodeltoallowforitem-dependentsetupcostsinadditiontothejointsetupcosts.Anilyetal.[ 7 ]alsoconsideredamulti-itemlotsizingproblemwithso-calledjointsetupcosts.However,thissetupcostisincurredforeachbatchofproduction,i.e.,batch-dependentsetupcosts.Anothervariantofdynamiclotsizingthatcontainssimilaritiestothecasepackplanningproblemweconsideristheclassofbatchorderingproblems,inwhichitemsmustbeproducedororderedinmultiplesofacertainbatchsizeorcapacityQ.ThisrequirementisapplicableinsituationswhentheproductisproducedinabatchprocesswithaxedcapacityQ,orproductionisperformedinordertollcontainersortruckloads,eachofsize(capacity)Q.Thesebatch-orderingproblemsusuallyconsiderasingleitem,andhavebeenstudiedbyVander[ 96 ],ChandandSethi[ 23 ],andLietal.[ 62 ],amongothers.Anilyetal.[ 7 ]providedapolynomial-timealgorithmforamulti-itemlot-sizingprobleminwhichabatchmayincludeanymixofitems.Thedifferencebetweencasepackplanningandbatchorderingproblemsisthatabatchdoesnothaveaspecicassortmentcomposition,buthasagiventotalcapacityorbatchoutputsize,whilecasepackassortmentcompositionsarepre-determined.Thus,casepackplanningproblemshaveareduceddegreeofexibility,asboththebatchsizeandthecompositionofthebatch(numberofunitsofeachSKU)arexed.Tothebestofourknowledge,theconceptofretailcasepackplanninghasreceivedverylittleattentionintheoperationsplanningliterature,althoughitisawidelyacceptedpracticeinindustry.WehavefoundonlyoneworkingpaperbyFreimeretal.[ 36 ]thatconsiderscasepackorderingproblemsinadifferentsetting.Theyassumestochasticandstationarydemands,whileminimizingexpectedholdingandbackordercostsoveraninnitehorizon.Theirmodelconsideredonlyonetypeofcasepackwithasingle 51

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3.7 wewilldiscussmethodstohandlesituationsinwhichdemandsareuncertain.)Wealsoassumethatitemswithincasepackscannotserveassubstitutesforoneanotheratthedemandlevel(asaresultof,forexample,differentsizes).Weassumetheunitcostforeachitemisknownandalsotime-varying,andtheunitclearancepriceisincurredonlyattheendoftheplanninghorizon.Eventhoughexcessinventorymaybemarkeddowngraduallyduringtheseason,duringpre-seasonplanning,areasonableassumptionisthatattheendoftheseason,eachunsolditemissoldataclearance(salvage)pricethatislessthanitscost,andthatallitemscanbedisposedofatthisprice.Forseasonalorfashionitemsthisassumptionisvalidand,ideally,wewouldhavenoinventoryattheendoftheseasonsothatnoitemissoldatclearanceprice.(Forbasicitemssoldthroughouttheyear,e.g.,men'swhitedressshirts,aplannermightwanttosettheendinginventoryinthenalperiodtoapositivevaluecorrespondingtoadesiredsafetystocklevel.)WealsoconsiderapurchasingcostcnmtforSKUmthatdependsontheindividualpackn,whichpermitsmodelingpricesbasedoneconomiesofscaleandreducedhandlingrequirementsassociatedwithdifferentcase-packcompositions.Ageneralizationofourmodelwouldpermitvariationsinhowtheitemswithincasepacksmaybeallocatedtostores.Forexample,adistributioncentermaypermitopeningcertaincasepacksandsendingindividualSKUstostoresatsomepremium.Insuch 53

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whereMntisalargenumberthatwecansettoMnt=maxm=1,...,MnlPT=tdm 3 ) 54

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3 )ensuresthatweincurthesetupcostinanyperiodinwhichpositiveprocurementoccurs.Constraintset( 3 )requiresthatcasepackorderquantitiestakenonnegativeintegervalues,whilethenalconstraintsets( 3 )and( 3 )enforcenonnegativityoftheinventoryvariablesandbinaryvaluesforthesetupvariables,respectively.Unlikestandarddynamiclotsizingmodels,wemustrequirethatthenumberofcasepacksreceived(xnt)takesintegervalues,sincewecannotguaranteethatthisconditionwillholdinthelinearprogramming(LP)relaxation,evenwhenwerequirethesetupvariablestotakeintegervalues.Wenextconsideraproblemreformulationthatpermitsreducingthenumberofvariablesrequiredintheformulation.Byrearrangingtermsintheaboveformulation,weknowthat:Imt=tXk=1NXn=1Emnxnkdmk!,m=1,...,M,t=1,...,T.Substitutingtheinventoryvariablesoutoftheformulation,droppingtheconstanttermPMm=1smPTk=1dmk,anddening~Cmnt=(cnmtsm+PTk=thmkEmn,weobtain: subjectto:tX=1NXn=1EmnxntX=1dm,m=1,...,M,t=1,...,T, Notethattheobjectivefunctionof(P)isatranslationoftheobjectiveof(P0),wherewehaveomittedtheconstantPMm=1PTt=1hmtPtk=1dmkfrom( 3 ).Wenextcharacterizethecomplexityofproblem(P). 55

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StowersandPalekar[ 90 ]consideredthecaseinwhichsetupcostsarecase-packdependent.Proposition1showsthattheproblemisstronglyNP-Hardwhenmultiplecasepacksshareajointsetupcostinanyperiod.Becauseofthisnegativecomplexityresult,wewillrstconsiderpracticalspecialcasesthatpermitefcientsolution.Wethengeneralizetheresultsobtainedforthesespecialcasestoprovideapseudopolynomial-timesolutionmethodforthegeneralproblemwithaxednumberofcasepacks.Aswehavenoted,weassumethatcasepackcompositionsarepre-dened.Thus,theproblemwestudyfallswithinahierarchyofdecisionsaftercasepackcompositiondecisionshavebeenmade.AsProposition 3.1 indicates,theproblemwithpre-denedcasepackcompositionsisstronglyNP-Hard.We,therefore,beginbyaddressingthisdifcultproblemclass,andleavetheintegratedcasepackcompositionandprocurementplanningproblemasapromisingareaoffutureresearch. 56

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3.4.1SingleCasePackWerstconsiderthecaseinwhichthesupplieroffersonlyasinglecasepack,whichwerefertoasproblem(P1).Supposethataretailstoreplanstomeetdemandsforsmall-,medium-,large-,andextra-large-sizesofaproductoveratimehorizon,saythecoming13weeks.Thestoremustdeterminehowmanycasepackstoorderatthebeginningofeachperiod.Thismodelisanextensionofthebasicdynamiclot-sizingmodeltoaccommodatecasepacks.ThebasicWagner-Whitinmodelcontainsonlyoneitem,whereasinourmodel,thereareMdifferentsizes.Furthermore,wecanprocureitemsonlybyorderinganintegernumberofcasepackscontainingtheitems.Forexample,inTable 3-1 ,inordertoobtainfoursmallshirts,wemustordertwounitsofCasePack1.StowersandPalekar[ 90 ]showedthatthesingleproductfamilyversionoftheproblemtheyconsiderisequivalenttoasingle-itemlotsizingproblem,aftersomepre-processingtoaccountformultipleproducts.Becauseoursinglecasepackproblemisidenticaltothesingleproductfamilyversiontheyconsider,thesameholdsforproblem(P1).Wenextuseproblem(P1)toillustrateashortestpathsolutionapproach,whichwelatergeneralizetohandlemultiplecasepacks.WenoterstthattheWagner-Whitinzero-inventory-orderingpropertyneednotdirectlyapplywhenmultipleitemsmustbepurchasedincasepacks.Forexample,ifanitemcanonlybeorderedinmultiplesofveandthedemandineachperiodis4units,itisnotnecessarythatinventorymustequalzerobeforeaneworderarrivesinanoptimalsolution.Weshowbelowthatinanoptimalsolution,theinventoryofallSKUsatthebeginningofeveryorderperiod(priortoreceivingareplenishment)canbedetermined.Forthisnewcasepackprocurementplanningproblem,thefollowingmodiedminimumfeasibleinventoryordering(MFIO)propertyapplies.Inthissubsection,EmandcmtcorrespondtoEm1andc1mt,respectively,asweomitthecase-packindexforsimplicity. 57

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TheMFIOProperty( 1 )enablesustodevelopadynamicprogramming(DP)solutionapproachforproblem(P1).InthisDP,timeperiodsserveasstagesandend-of-periodinventorylevelscorrespondtostates(notethattheend-of-periodt1inventory 58

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1 impliesthatthestatevector(ofbeginninginventorylevelsbeforeordering)isindependentofthepathtakenbeforearrivingatastage.Thus,theendinginventoryineachperiodimmediatelypriortoorderplacementcanbecalculatedusingtheordersizesrequiredtomeetthedemandthroughtheendoftheperiod,i.e.,Imt=Emmax~m=1,...,MnlPti=1d~mi 1 doesnotimplythatthebeginninginventory(beforereceivingreplenishment)inanorderperiodmustbeinsufcienttosatisfydemandintheperiodforatleastoneitem.Toillustratethiscounterintuitiveproperty,supposeacasepackcontainsasingleSKUinmultiplesoffour,andthatthedemandinperiodoneequalssix.SupposealsothatthedemandfortheSKUinperiodtwoequalsone,andthatpositivedemandsoccurinperiods3,...,T.Iftheholdingcostinperiodoneissufcientlyhigh,andthexedordercostinperiodthreeissufcientlylarge,itmaybeoptimaltoplaceanorderinperiodtwo,despitethefactthatsufcientinventoryexistsattheendofperiodonetocoverdemandinperiodtwo.Observethatifweknowwhenorderswillbeplaced,theorderquantityineachperiodandtheinventoryattheendofeachperiodcanbedetermined.Theonlyremainingdecisioniswhentoplaceorders.Thus,wecansolvetheDPmodelinamannersimilartothestandarddynamiclot-sizingproblemortheequipmentreplacementproblem.Inparticular,wecanformulatethesinglecasepackproblem 59

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3 ],forabriefdescriptionoftheshortest-pathmodelfordynamiclotsizingwithxedorderingcosts).WethereforecreateashortestpathnetworkcontainingT+1nodesshownasFigure 3-1 .Anarcconnectsnodettoeachhighernumberednodej=t+1,...,T+1.Thus,thearcfromnodettojimpliesthatconsecutiveordersoccurinperiodstandj,andthecostofthisarcincludestheminimumcostsincurredwhilesatisfyingdemandsinperiodstthroughj1.ThisarcthereforehasacostequaltoKt+PMm=1cmtXt,j1Em+Ht,j1,whereHt,j1denotestheinventoryholdingcostfromperiodttoperiodt+j1,i.e.,Ht,j1=MXm=1j1X=thm"Xt,j1Em+Im(t1)Xi=tdmi#.Theexpressionsinbracketsdenotereceipts,startinginventory,andcumulativedemand,respectively.Theshortestpathfromnode1toT+1correspondstoanoptimalsolutionforthesinglecasepackversionof(P). Figure3-1. Singlecasepackproblemnetwork Clearly,thenumberofarcsisboundedbyO(T2),andthetimerequiredtocomputeanarc'scostisboundedbyO(M(jt))O(MT).Thus,theconstructionofthisnetworkisboundedbyO(MT3).UsingDijkstra'salgorithm(e.g.,Ahujaetal.[ 3 ])tosolvetheshortestpathproblemthenrequiresO(T2)time.Ourdiscussionillustrateshowthesinglecasepackproblemdiffersfromtheclassicsingleitemdynamiclotsizingproblem.Inparticular,wehaveaknowninventorydeterminedbytheMFIOpropertypriortoreceivinganewreplenishmentratherthan 60

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3-2 ).DeneV1(V2)asthesetofitemscontainedinCP1(CP2).LetN1(t)denotetheminimumnumberofCP1casesrequiredtocoverdemandforitemsinthesetV1fromperiod1toperiodt,i.e.,N1(t)=maxm2V1nlPti=1dmi 61

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1 ). E~m2')t1Xi=1dmi.Proof.TheprooffollowsthesamelogicastheproofofProperty 1 ,usingtheconditionthatjunitsofCP1havebeenorderedpriortoperiodtandourassumptionthatthecnmtvaluesarenon-increasingintforeachmandn(wethusomitthedetails). Thefollowingtwopropositionsareimpliedbytheabovepropertyandourdescriptionoftheshortestpathnetworkstructure. 62

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Layerednetworkfor(P2),theproblemwithtwocasepacks. Allotherregulararccostscanbecomputedusing:A(t,j),(s,k)=Kt+MXm=10BBB@c1mtEm1(kj)+c2mtEm2maxm2V28>>><>>>:2666666s1Pi=tdmiIjm,t1Em1(kj) 63

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12 ],whoshowthattheseinequalitiesarefacetsforthesingle-itemuncapacitatedlotsizingproblemthataresufcienttodescribetheconvexhullofsolutionsforthisproblem).Forproblem(P),wedeneassociated(l,s)cpinequalitiesforcasepackproblemsasfollows.Givenadividingperiodl2f1,...,T1g,anonemptysubsetSbelongingtotheperiodsetfromperiod1throughthedividingperiodl,Sf1,...,lganditscomplementS,;6=S=f1,...,lgnS,wedeneacasepackbased(l,s)cpinequalityas: wheredmrt=Pt=rdm. 3 )arevalidfor(P).Proof.Forafeasiblesolution(X,Y),ifyi=0foralli2S,thenxni=0foralli2SbecausexntMnyt,soXi2SNXn=1Emnxni+Xi2Sdmilyi=Xi2SNXn=1Emnxni+0=lXi=1NXn=1Emnxnidm1l,8mwherethelastinequalityfollowsfromconstraintset( 3 )in(P).Otherwise,letk=minfi2S:yi=1g.Thenxni=0foralli2S,i
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77 ].Wenextconsideranadditionalsetofinequalities.LettingEmaxm=maxn=1,...,NfEmng,thenwehaveEmaxmtX=1NXn=1xntX=1NXn=1EmnxntX=1dm,m=1,...,M,t=1,...,T,wherethesecondinequalityabovecorrespondstoconstraintset( 3 ).Theaboveinequality,alongwiththeintegralityofthexntvariables,impliesthatthefollowinginequalitiesarevalidfor(P):tX=1NXn=1xnPt=1dm Addinginequalities( 3 )and( 3 )helpsinimprovingthequalityoftheLPrelaxation-basedlowerbound.Inordertoimprovethesolutionupperbound(UB),wenextturntoaheuristicsolutionapproachforprovidinggoodfeasiblesolutions,and,hence,upperbounds. 3.8 showthattheresultingsolution 66

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77 ].Therstproductionplanningtooltouserelax-and-xwasproposedbyBelvauxandWolsey[ 13 ].Undertheirapproach,thetimewindowsarepredenedinlengthwithoutanyoverlap.Stadtler[ 89 ]extendedthebasicideaofsuccessivetimewindowstoincludeoverlapbetweenconsecutivewindowsforhigherqualityresults.Aftersolvingthesubproblemforeachtimewindow,onlythevariablesassociatedwiththetimeperiodsthatdonotoverlapwiththenexttimewindowarexed.ThedisadvantageofStadtler'sheuristicliesinitsmorecompleximplementation.TherecentpaperbyAkartunaliandMiller[ 4 ]modiedpriorrelax-and-xapproachesbyusingoverlappingtimewindowsand 67

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4 ],ourheuristicsdonotapplyaninitializationphasetoprovideaninitialsolution,andweonlybegingeneratingupperboundsduringthesecondtimewindow,inordertoreducetherequiredrunningtime.Onekeyparameterthataffectsourheuristicperformanceisthelengthofthetimewindow,denotedby.Whenusingoverlappingtimewindows,thelengthoftherstpartofthewindowthatdoesnotoverlapwiththefollowingtimewindowiscalledthexinginterval,sincetheintegervariableswithinthistimeintervalwillbexedoncethesubproblemissolved.Thelengthofthexingintervalisdenotedby.Wereporttheresultsoftestsontheeffectsoftheseparametersinthenextsection. 70

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3.4.1 ,sincetheoptimalsolutionpropertiesstillhold.Moreover,wecanadjustorderquantitiesbeforeorderinginanorderperiodbasedonobserveddemands(aswearemoreconcernedwiththeamountontheshelfafterreceivinganorderthanwearewithadheringtotheoriginalorderquantityprescribedbythemodelatthebeginningoftheplanninghorizon).WecanalternativelymodifytheSilver-Meal(1973)heuristic(aswellasothersimilarheuristicapproaches),whichcanenablequickersolutionaswellasanabilitytoadjustsuccessiveorderperiodsandquantitiesinresponsetoobserveddemands.Usingthisapproach,wewouldbeginwithperiod1andcomputetheexpectedcostperperiodassociatedwithstockingmaxm=1,...,MF1D(m,t)() 73

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Afteraddingtheaboveconstraintstotheproblemformulation(P0),wecanthenobtainanequivalentformulationwithnonnegativeinventoryvariablesbysubstitutingI0mt=ImtLmtthroughouttheformulation,andrequiringI0mt0forallmandt.Inordertoensureequivalence,wethendeneemt=LmtLm,t1andaddemttotheforecasteddemandvaluedmtforallvaluesofmandt(observethattheemtvaluesmaybenegative,unlessLmtLm,t1forallvaluesofmandt;thisimpliesthatthemodieddemandvaluesmaybenegativeaswell,althoughforthemajorityofpracticalsettingsitislikelysafetoassumethatdmtemt).Clearlythisservesasaheuristicapproachtosettingsafetystocklevels(evenincasesinwhichatargetservicelevelisspecied),asthelowerbound(Lmt)levelsmustbesetpriortohavingknowledgeofthetimebetweenreplenishmentorders,andtheresultingmodelsetsallorderquantitiesinadvanceoftheplanninghorizon. 4-1 illustratesthegeneralparametersettingsweusedinourcomputationalstudy.Exceptfortherstrow,whichprovidesspecicvaluesofthetimehorizonT,thenumberofcasepacksN,andthenumberofitemsM,allotherdatashowninthe 74

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Table3-2. Parameterdistributionsusedincomputationaltests. Period,TCasepack,NSKU,M10,30,503,5,7,103,6,10Unitclearanceloss,cmsmUnitholdingcost,hmFixedorderingcost,KU[0.1,0.5]U[1,1.5]U[10,25]Demand,dmtCompositionofcasepacks,EmnU[40,100]U[3,20] 75

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3-3 indicatesthenumberofwindowsprocessedandthenumberof(l,s)cpcutsandtheadditionalsetofinequalities,denotedasExtraCuts,addedtotheproblemformulation.Generallyspeaking,the(l,s)cpinequalitieshelptoimprovethelowerboundsobtainedattherootnodeduringtheapplicationofCPLEX'sbranchandboundmethod.Thelargertheproblem'ssize,themorecutsareaddedtotheinitialproblemformulation.FortheExtraCuts,althoughthenumberofaddedsuchcutsissmaller,theystillplayaroletowardgettingbetterlowerbounds.Additionally,problemswithalongerplanninghorizonrequireprocessingagreaternumberoftime-windowsubproblems. Table3-3. Thenumberoftimewindowsandcutsusedwithinrelax-and-xheuristics. 10338271.1568540.37108900.1303328876.856281740.17102829005033481473.5564829407104849001010484900.1 Tables 3-4 and 3-5 summarizetheaveragerunningtime,optimalitygap,andsolutioninformationresultingfromtheapplicationofourheuristicsandCPLEX(assuminga600stimelimitfortheheuristicsandCPLEX).OurheuristicapproachrunsfasterwhencomparedtoCPLEX.Formostcases,itstopswithin1minuteandgeneratesagoodsolutionwithasmalleroptimalitygap.CPLEX,ontheotherhand,frequentlyreachedthetimelimitof600s,andstoppedwithasuboptimalsolution,evenformedium-sizeproblems.Forthelargestproblemset,thesolutionsobtainedbyourheuristicshavegapsassmallashalfthatofCPLEX,obtainedinsubstantiallylesstime.Table 3-5 reportsthenumberoftimesthatthesolutionmethodcorrespondingtothecolumnprovidedthebestsolutionvalues,among10randomlygeneratedinstances 76

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Computationaltestresultsforaveragerunningtime(seconds)andoptimalitygapforsolutionapproaches. 10335.330.111.290.111.300.110.740.121.100.1256293.020.112.100.132.610.131.730.142.180.13710600.000.568.170.479.310.477.130.488.310.483033460.000.165.980.138.630.134.080.136.680.1356540.070.3516.620.2520.820.2511.590.2617.240.25710600.000.5369.220.3398.350.3340.730.3790.160.345033540.250.1919.970.1624.370.1610.970.1617.790.1656600.000.3144.780.2158.930.2132.800.2142.60.21710600.000.51127.640.28169.740.25106.420.26152.420.251010600.000.52126.100.26181.380.25115.300.26168.600.25 3-6 andFigure 3-3 )underdifferenttime-windowlengths(i.e.,=2,3,4,5)forourmedium-sizeproblemsetwithT=30, 77

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Numberoftimes(outoften)eachsolutionmethodattainedthelowest-costsolution. Dataset#BestsolutionsTNMCPLEXA1B1A2B2 10331099995610666671038767303337666561768671004808503334646561252671004838101001928 3-7 andFigure 3-3 ).Itindicatesthat=1isthebestchoiceforthismedium-sizeproblemset.Asthevalueincreases,therunningtimedecreasesforeachheuristic,becausemorevariablesbecomexedandtheMIPsubproblemsareeasiertosolve;the 78

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Effectsofthetime-windowlength,,onsolutionvalueforthemedium-sizeproblemset. 13889883890303889883890143889883890303889883889982275067274969274998274998275067274969274998275029326569026557926561926562826566326569626564826576142598292597902597752597752598292597902597752597755282839282842282839282839282839282842282843282968625967625964925965025964525967625964925965025965972970582970002970092970232970582970002970022970438279668279662279555279552279671279662279666279556926181426184526171026171026182426183626175226176110309396309315309208309217309396309315309208309227Avg.time(s):11.1416.6230.0756.077.1711.5920.8934.84Avg.win#:2928272629282726Avg.solu:288002.0287968.1287935.1287940.1288001.1287978.9287953.0287977.7 Table3-7. Effectsofthelengthofxinginterval,,onsolutionvalueforthemedium-sizeproblemset. 13890303891003890193890303890083890192274969275047274983274969275152274983326557926568926571026569626583826566542597902598442597892656962658382656655282842282861282853282842282979282853625964925974925969025964925975925969072970002971242971862970002972072971868279662279693279665279662279745279665926184526199126175926183626205326175910309315309373309391309315309458309391Avg.value:287968.1288047.1288004.5287978.9288115.0288000.0Avg.time(s):16.629.867.1911.596.624.42Avg.win#:281410281410 79

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Effectsoftimewindowparametersonsolutionvalue.(T=30,N=5,M=6) packshasonthebuyer'scostperformance.Inordertoestimatethiscost,wewouldliketobeabletodetermineanoptimalcase-packcompositionforthebuyer.Findinganoptimalcase-packdesignandprocurementplan,however,requiressolvinganextremelychallengingnonlinearoptimizationproblemwithabilinearobjectiveandasetofbilinearconstraints(inwhichboththeobjectiveandconstraintset( 3 )offormulation(P0)wouldnowcontainaproductofEmnandxntvariables).Forexample,forevensmallproblemsizes(withN=3ormorecasepacks),ourcomputationalexperience 80

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24 ]).Thus,weareonlyabletoobtainanoptimalcase-packcompositionforrelativelysmallproblemsizesusingGAMS/BARON.Inordertogainsomeinitialinsightonthisproblem,weconsideredasmallsetoftestproblemsinvolvingM=2itemsandT=8planningperiods.ClearlyifM=2,thenweneedonlyconsiderproblemssuchthatthenumberofcasepacks,N,isequaltoeitheroneortwo(withN=1wemustnecessarilyincludebothSKUsinasinglecasepack,whilethecaseofN=2permitseachitemtohaveitsowncasepack).Thus,thissettingenablesustoconsidertheretailer'sincrementalcostthatresultsfromrequiringacasepack.UsingthecostanddemandgenerationparametersinTable 3-2 ,wegeneratedabaselinesetof12probleminstancesinordertoassessthevalueofpermittingN=2casepacks(relativetothecostofrequiringN=1casepack).Forthisbaselinesetofinstances,theaveragepercentagesavingsofpermittingN=2casepackswasequalto3.72%.Thus,theholdingcostpenaltyforrequiringonecasepackisnontrivial,andthesecostsmustbebalancedagainstthecostsavingsthatcanbegainedinwarehousingoperationsfromusingthecasepack.Inadditiontoconsideringthebaselinesetof12probleminstances,wealsoconsideredthreeadditionaltestsetsthatalteredtheoriginal12instancesby(a)doublingthevariablepurchaseandholdingcostsofoneoftheitems;(b)doublingthedemandvaluesofoneoftheitems;and(c)multipliedeachxedordercostbyten(eachofthesechangeswasmadewithrespecttotheoriginalbaselineproblems,i.e.,thesechangeswerenotappliedsuccessivelytothedata).Ourgoalinrunningtheseadditional36testproblemswastodeterminehoweachofthesefactorsinuencestherelativecostofrequiringacasepack.Thesechangesresultedincostpenaltiesof(a)3.48%,(b)2.58%,and(c)1.6%,foreachoftherespectivecases.Ourresultsshowedthatthesepercentagedecreases(withrespect 81

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1. Howmanycasepackpatternsshouldbeusedinagivensystem? 2. WhichSKUsandwhatSKUquantitiesshouldbecontainedineachcasepack? 3. Howshouldinventoryreplenishmentbeplannedtoensurethataretailermeetsdemandwithaslittleleftoverinventoryaspossible,withagoalofminimizingthetotalprocurement-relatedcost?Designinglargercase-packcongurationscontainingagreaternumberofSKUsimplieshandlingcostadvantages,butalsoreducesorderingexibilityandconsequentlyincreasestheprobabilityofoverstockingand/ortheneedforbreakingbulkattheDC.Thus,itisnecessarytobalancethesetradeoffsinordertotakefulladvantageofthereducedDChandlingcostsassociatedwithcasepacks.Case-packdesignsarenormallydeterminedbytheretailerinconsultationwithavendor(ChettriandSharma[ 26 ]).Inpractice,theresearchquestionsaboveareusuallyaddressedsequentially(byrstdeterminingcase-packcongurationsandthenmakingreplenishmentdecisions),whichleadstosub-optimaldecisionmaking.Clearly,thereplenishmentproblemisheavilyinuenced(andconstrained)bythenumberandcompositionofthedifferentcasepacksavailable.Thismotivatesourmodelingandsolutionapproachesforsolvingintegratedjointcase-packcongurationandprocurementplanningdecisionproblems.Inthischapter,weassumethatindividualitemsmustbeassignedtoalimitednumberofcasepacks(denedbyanassortmentofitemsincludedwithinalargercase 85

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4.2 .Section 4.3 describesandformulatesthejointcase-packdesignandprocurementproblem,whichisanintegernonlinear(nonconvex)optimizationproblem.WerstproposeanexactlinearizationmethodforsmallprobleminstancesusingareformulationinSection 4.4.1 .Then,inSection 4.4.2 ,wedescribeabasiciterativeheuristicapproachthatalternativelyxescase-packcompositionandprocurementdecisionvariablesandsolvestheremainingrestrictedproblemoptimally.InSection 4.4.3 ,wethenmodifytheiterativeheuristictoimproveperformancebychangingtheinitialboundoncase-packcapacity.Tocapturethebenetsofintegrateddecisionmaking,Section 4.4.4 furtherstudiesanotheriterativeapproachthatalterativelyconsidersageometricprogrammingproblemandadynamiclot-sizingproblem,whileoptimizingthecase-packcompositionandorderquantitiessimultaneously.Insection 4.5 wediscussthedetailsofanimplementationofourapproaches.Finally,concludingremarksarepresentedinSection 4.6 29 ]developedaclass 86

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7 ]alsoconsideredamulti-itemproblemwithjointsetupcosts.However,inthiscase,thesetupcostisincurredforeachbatchofproduction,whereeachbatch(ortruckload)hasaconstantcapacityconsistingofanymixoftheitems.Afterassumingthestoragecostsobeycertaindominanceconditions,Anilyetal.[ 7 ]reformulatethemodelbyintroducingsurrogateproductsandaddinginequalities.Theythendescribetheconvexhullofsolutionsandproposealinearprogrammingapproach.Ingeneral,dynamiclot-sizingproblemswithcapacityconstraintsarenotoriouslydifcultproblems.Whenonlyonebatchisallowedineachperiodandcapacitylimitsaretime-varying,eventhesingle-itemcaseisNP-Hard(Florianetal.[ 35 ]).Noefcientsolutionmethodsareknownforsuchmulti-itemproblems,withtheexceptionofAnilyandTzur's[ 8 ]dynamicprogrammingalgorithmforthecaseinvolvingconstantcapacitiesandcostparameterswithdeterministicdemands,whichtheyshowispolynomiallysolvablewhenthenumberofitemsisxed(butisexponentialotherwise).Thelot-sizingliteratureconsidershardcapacities(e.g.,Florianetal.[ 35 ])aswellassoftcapacitiesintheformofbatchquantities(e.g.,PochetandWolsey[ 75 ]).Inthelattercase,nQunitsofcapacitymaybemadeavailableforanypositiveintegervaluen(whereQdenotesthebatchcapacity)atacostofFn,whereFdenotesthexedbatchcapacitycost.Case-packprocurementproblemsdifferfromproblemswithsoftcapacitiesbecausethelatterclassofproblemsdoesnotrequireaxednumberofeachproductineachbatch,butallowsthebatchsizeQtocontainanymixofitems,whilecase-packcompositionsarexed.Thus,case-packprocurementproblemshaveareduceddegreeofexibility,asboththebatchsizeandthecompositionofthebatch(numberofunitsofeachSKU)arexed.StowersandPalekar[ 90 ]andBhatiaandPalekar[ 17 ]consideredasimilarclassoflot-sizingproblemswithso-calledstrongset-upinteractions,wheremultipleproductscanbeproducedonlyinxed 87

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90 ]showedthattheproblemisNP-hardandpresentedaLagrangianrelaxationapproachforsolvingthelinearprogramming(LP)relaxation,whileBhatiaandPalekar[ 17 ]usedavariableredenitionapproach.Chenetal.[ 25 ]proposedapolynomial-timeshortestpathalgorithmforthespecialcaseinvolvingasinglecasepack,andapseudo-polynomialtimealgorithmforthegeneralmultiple-case-packproblemwhenthenumberofcasepacksisxed.Forlargerproblems,theyappliedvalidinequalitiestostrengthentheproblem'sLPrelaxation,andproposedheuristicsforsolvingthemixedintegerproblem.Ineachofthesecases,theassortmentandnumberofitemsinacasepack(ortheproportionsinwhichdifferentitemsmustbeproduced)areexogenoustothemodel.Incontrast,weconsidertheprobleminwhichthedecisionvariablesincludenotonlytheorderquantities(oftheindividualcasepacks)asaddressedinaforementionedliterature,butthecomposition(ordesign)ofeachcasepackaswell.Ifwerestrictourproblemtoasingleperiodsetting,thenthisspecialcaseissimilartothecuttingstockandmodulardesignproblems.ThecuttingstockproblemisawellstudiedclassofintegerlinearprogrammingproblemsrstintroducedbyGilmoreandGomory[ 42 ].Afewpastworkshaveconsideredcombinedcuttingstockandlot-sizingproblems.NonasandThorstenson([ 67 ],[ 68 ])consideredthecombinedproblemunderstaticanddeterministicconditions.Intheabsenceofjointsetupcostsandintegerlotsizes,theysuggestedseveralspecializedheuristics.GramaniandFranca[ 46 ]consideredadynamiclot-sizingproblemwheretheintegralityconstraintsonlotsizeswererelaxed.Theysolvedthisproblemheuristically,usingananalogousshortestpathalgorithminwhicheacharccorrespondstoacuttingstockproblemparameterizedbythecuttingpatternvariables.Thisapproachdidnotapplyanylimitonthenumberofplate(case-pack)types,andassumedthatpatternchangesarecostless.Asaresult,alargenumberofdifferent 88

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47 ]thendevelopedaslightlydifferentmodelthatignoredsetupcostsandrelaxedintegralityrestrictionsonlotsizes,andsolvedthisproblemusingacolumngenerationtechnique.However,thismodelalsoappliednorestrictionsonthenumberofpatterns,andthusthenumberofdifferentpatternsinthesolutionisusuallyverylarge.Toreducethenumberofcuttingpatternchanges,heuristicmethodshavebeenstudiedforcuttingstockproblems,includingasequentialheuristicprocedurewherethepatternchangecostisincorporatedintheobjective(Haessler,[ 50 ],LefrancoisandGascon[ 61 ]),aswellasapatterncombinationheuristic(Johnston[ 55 ],Goulimis[ 45 ]).Forpatternminimizationproblemsinwhichtheobjectiveisonlytominimizethenumberofdifferentcuttingpatterns,exactalgorithmsexist(e.g.,Vanderbeck[ 97 ],Umetanietal.[ 94 ]).Inadditiontotheselinearformulationsofcuttingstockproblemsthatparameterizeoncuttingpatternvariables,somepastworksformulatethisproblemasanon-convex,integer,non-lineartrim-lossproblem,andthenlinearizeandsolvetheproblemusingexistinglinearizationmethods(HarjunkoskiandWesterlund[ 52 ]),anapproachthatisconsistentwiththelinearizationapproachwelaterdiscuss.Ifthereisnophysicalrestrictiononthesheetsizeinthecuttingstockproblem,theresultingproblembecomesanonlinearModularDesignProblem(MDP).Thisisasingle-periodproblemrstproposedbyEvans[ 28 ],whereboththemodulesizeanditscomposition,intermsofparts,mustbedetermined.MostpapersintheliteratureonMDPstargetthesinglemoduleandcontinuousversionsoftheproblem.Passy[ 73 ],Smeers[ 87 ],andShaftelandThompson[ 82 ]successivelydevelopedandrenedsimilarefcientsimplex-likealgorithmsbyexploitingthespecialgeometricprogrammingstructure,orbyusingtheKarush-Kuhn-Tucker(KKT)conditions.OnlythepaperbyShaftel[ 81 ]investigatedtheprobleminwhichanintegersolutionisrequired,usinganenumerationmethodwhentheproblemissmall.Al-Khayyal[ 6 ]modeledtheMDPas 89

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16 ]presenteddualityresultsandprovidedtwoapproaches(generalizedBendersdecompositionandtheseparableprogrammingapproach)forsolvingtheMDP.Althoughillustrativeexampleswereprovidedfortheseapproaches,nocomputationalresultsforproblemsofpracticalsizewerepresented.Silverman[ 86 ],ShaftelandThompson[ 83 ],andAl-Lhayyal[ 6 ]havestudiedthemultiplemoduledesignproblem,althoughtheapproachestheysuggestedareonlyguaranteedtoleadtogoodlocalsolutionsfortheproblem.Intheproblemweconsider,wenotonlyneedtodesignthecase-packcomposition,asintheMDPproblem,butwemustalsosolveanembeddedmulti-period,multi-itemdynamiclot-sizingproblemforcase-packreplenishment.Forpracticalpurposes,anupperlimitonthenumberofcase-packtypesusedovertimeshouldbeenforced,sincemostretailersdonotwishtohandlealargenumberofcasepacks,asthisimposestoohighadegreeofcomplexityonthedistributionsystemandreducesthebenetsofusingcasepacks.Tothebestofourknowledge,thecombineddynamiccase-packprocurementanddesignproblemwithpatternlimitationshasnotbeenaddressedpreviouslyintheliterature.Ourcontributionsincludeanalyzingtheproblem'sspecialstructureandprovidingefcientlinearandnonlinearsolutionmethodsforsolvingthisclassofproblems. 90

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whereMtisalargenumberthatwecansettoMt=maxm=1,...,MnPTj=tdmtoinperiodt,withoutlossofoptimality.Theobjectivefunctionexpresses,respectively,thetheinventorycostsforallSKUs,thepurchasingcostforallcasepacks,andtheordersetupcostsoverallperiods.Therstconstraintset( 4 )ensuresinventorybalanceforeachitem,whilethecase-packcapacitylimitisgivenby( 4 ).Theconstraintset( 4 )ensuresthatweincurthesetupcostinanyperiodinwhichpositiveprocurementoccurs,whilethenalconstraintsets( 4 ),( 4 )and( 4 )enforcenonnegativityoftheinventoryvariables,nonnegativeintegervaluesforthecase-packorderquantitiesxntandcompositionvariablesEmn,andbinaryvaluesforthesetupvariables,respectively.Inthecase-packcapacityconstraints( 4 ),theparameterRprovidesanupperboundonthelargestfeasiblecase-packsize.Inpractice,thecase-packsizemayalsobeadecisionvariableitself.Thus,thequantityRmayserveasanupperlimit,andanoptimalsolutionmayresultcontainingfewerthanRitems.Thatis,ifanoptimalsolutionresultsinwhichconstraint( 4 )isnottightforsomecase-packn,wecandesignacasepackwhosesizeisdeterminedbytheoptimalvalueoftheleft-handsideofthisconstraint.Aswelaterdiscuss,theresultsofouriterativeheuristicmethodsmaybestronglyinuencedbythevalueofRusedintheproblemformulation,andourheuristicapproachwillthereforeparameterizeonRinordertoimprovesolutionquality. 92

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subjectto:tX=1Xn=1EmnxnDmt,m=1,...,M,t=1,...,T, Wemightalsoconsiderformulatingproblem(P)byenumeratingallpossiblecase-packpatternsNandchoosingsomesubsetofthese,usingasetpartitioningformulation.Inthisapproach,thesetofEmnvaluesplaytheroleofasetofparametersofdimensionMNinsteadofasetofdecisionvariablesofdimensionM.However,limitingthenumberofcasepackswouldthenrequiredeninganewbinaryvariableyn,forn=1,...,N,indicatingwhetherpatternnisselected.WewouldthenneedtoaddtheconstraintPNn=1yntoanappropriateset-partitioningformulationof 93

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70 ]).Thefollowingobservationfurtheraddressesitscomplexity. 4.4.1 wediscussareformulationandlinearizationapproachthatpermitssolvingtheproblemusingacommercialmixedintegerlinearprogramming(MILP)solver,suchasCPLEX.Section 4.4.2 thenconsidersaniterativeheuristicapproachthatalternatesbetweenxedvaluesofthecase-packdesignvariablesandproductionplanningvariables. 94

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4.4.3 modiesthisheuristicbyapplyingitusingeachlevelofpossiblecase-packsize.Wethenpresentanintegratedgeometricprogrammingbasedapproachforthesingle-case-packspecialcaseinSection 4.4.4 6 ]).Inourproblem,since0mnk1,0xntMt,wecanaddthefollowingfoursetsofinequalitiestotheoriginalproblemtoeliminatethenonlineartermEmnxntinboththeobjectivefunctionandtheconstraints,whilemaintainingequivalencewithouroriginalformulation. 95

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64 ]).Thisapproachbeginswithacase-packdesignanddeterminesanoptimalprocurementplanforthiscase-packcomposition.Then,wedetermineanoptimalcase-packdesignforthegivenprocurementplan.Foreaseofexposition,wepresenttheapproachindetailforthesingle-case-packspecialcase.Attheendofthesectionwethendiscusstheimplicationsforthemultiplecase-packproblem.Asweshowbelow,ouralgorithmquicklyconvergeswithinatmosttwoiterations,duetocertainspecialpropertiesinthe 96

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4.4.3 wefurthermodifytheiterativeheuristicinanattempttoimproveuponlocallyoptimalsolutions.Inthesingle-case-packproblem(=1,andthustheindexnisomitted),ifthecase-packdesignvariables(Em,m=1,...,M)arexedtovaluesEm,m=1,...,M,theremainingproblemcanbeformulatedasadynamiclot-sizingproblemwithcasepacks,denotedasPE,asfollows: subjectto:tX=1xmaxmDmt WecansolvePEasashortestpathprobleminordertodeterminetheoptimalorderperiodsandcase-packorderquantities(Chenetal.[ 25 ]).Ontheotherhand,ifwexthextandztvariablestoxedvaluesxt,zt,t=1,...,T,andletCm=PTt=1~cmtxt,m=1,...,M,andwt=Pt=1x,t=1,...,T,theresultingproblemPx,zbecomes: subjectto:wtEmDmt,m=1,...,M,t=1,...,T, Wecansolvethisproblembyinspection,becausefeasibilityoftherstsetofconstraintsandtheintegralityrestrictionsrequiresEmmaxt=1,...,TdDmt=wte.Because 97

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28 ]showedthataninnitenumberofoptimalsolutionsexistforsuchproblemswhenthevariableintegralityrestrictionsarerelaxed,sinceforanyoptimalsolution(E,x),(E=,x)isalsoanoptimalsolutionforany>0.Goldberg[ 43 ]observedthatforcontinuousmoduledesignproblemswithsideconstraints,anoptimalsolutionexistsinwhichatleastoneofthesideconstraintsholdsatequality.Inoursingle-periodproblem,thereisonlyonesideconstraint(PMm=1EmR),anditiseasytoshowthatanoptimalsolutionexistswhentheintegralityrestrictionsarerelaxedsuchthatthisinequalityistight.Inaddition,inanyoptimalsolutiontothecontinuousversionofthisproblem,eachofthedemandconstraintswillbetight,andthusanoptimalsolutionexiststhatsatises assumingpositivedemandvalues.TheuniquesolutiontothissystemisthengivenbyEm=RDm=PMm=1Dm,m=1,...,M.Thus,anoptimalsolutiontothecontinuousrelaxationofthesingle-periodproblemllsthecapacityRwithSKUsinproportiontoeachSKU'spercentageofthetotaldemandintheperiod.Thissuggestsapotential 98

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4 )andsettingDmtotheaveragedemandperperiodforSKUm(andapplyingaroundingheuristicinordertosatisfythecase-packcapacityconstraint).Seti=0. 99

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wherethesecondinequalityholdssinceE1mDmt=w0tforallt=1,...,T.Then,Proposition 4.3 impliesthat:E1m=maxtj(t+1)2Q0Dmt=w0tmaxtj(t+1)2Q0Dmt=w1tmaxt=1,...,TDmt=w1t=E2m.CombiningthiswiththeresultofProposition 4.1 impliesE2=E1. 100

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25 ],Akcayetal.[ 5 ]).Forthemultiplecase-packprocurementplanningsubproblemforagivensetofcase-packcongurations(PE),apseudopolynomialtimeapproachexistsforanyxedvalueof,inadditiontoheuristicsolutionmethods(Chenetal.[ 25 ]).Theassociatedcase-packdesignsubproblem(Px,z)foragivensetofprocurementdecisionsx,zwillbe:minXnXmCmnEmnsubjectto:XnwntEmnDmt,m=1,...,M,t=1,...,T,XmEmnR,n=1,...,,m=1,...,M,Emn2Z+,n=1,...,,m=1,...,M,whereCmn=PTt=1~cmtxntandwnt=Pt=1xn.ThisproblemcanbeshowntobestronglyNP-Hardbyareductionfromthemulti-dimensionalknapsackproblem(Akcayetal.[ 5 ]). 4 )stronglyinuencesthenalsolutionquality.ThevalueofRusedin( 4 )is,therefore,acriticaldeterminantofsolutionquality.Recallthatthecase-packsizemayinpracticebeadecisionvariableitself,andweuseRasanupperlimitoncase-packcapacity.ByconsideringvariousstartingvaluesofR,wearethereforeabletoevaluatenumerouslocallyoptimalsolutionsusingtheiterativeheuristicapproach.Whiletheiterativeapproachimprovesthesolutionsateachiteration,theseimprovementsonlyoccurovertwoiterations.Therefore,whilewedomakesomeprogressindeterminingabettercase-packsize(fromtheupperlimitRtoPmEm,which 101

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4.5 102

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21 ],weknowthatageometricprogram 103

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AccordingtoBoydandVandenberghe[ 22 ],standardinterior-pointalgorithmscansolveaGPwith1,000variablesand10,000constraintsinunderaminuteonasmalldesktopcomputer.Withrecentlydevelopedsolutionmethods,wecansolveeven 104

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4.4.2 ,exceptthatthevalueofRusedintheroundingschemechangesatsuccessiveiterations(inordertocapturethebenetofevaluatingvariousvaluesofRaspreviouslydiscussedinsection 4.4.3 ).Thatis,ateachiteration,weupdateRtobeequaltothesummationofEmvalues(overallm)obtainedinthepreviousiteration,andusethisvalueofRfortheroundingschemediscussedinSection 4.4.2 .GivenanintegerfeasiblesolutionE,wethensetwQi=maxm=1,...,MlDm(Qi+11) 4.4.2 .Thisprovidesanewvalueofthevectorz,andthusanewdenitionofthesetQ.Wethusapplyaniterativeapproachasdescribedintheprevioussection,wherethegeometricprogramisusedtodeterminethebestcase-packcompositionforaxedsetQ.Thisiterativeapproachisapplieduntilthezvariablesdonotchange,oruntilreachinganiterationnumberlimit,whichwesetto5inourcomputationaltests.Theadvantageofthisiterativeapproachovertheonedescribedintheprevioussectionliesinthefactthatthegeometricprogram(PQ)permitsintegratedoptimizationofthecase-packcompositionandthecase-packorderquantities,eventhoughtheorderperiodsarexed. 105

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4.4.2 .Inthissection,wedescribedageometricprogrammingbasedheuristicasitappliestothesingle-case-packdesignandprocurementplanningproblem.Generalizingthisapproachtothemultiplecase-packsettingintroducesseveraldifculties.Inparticular,thegeometricprogrammingapproachrequiresassumingthatallvariablesmusttakepositivevalues.However,inthemultiplecase-packsetting,wecannotgenerallyrequirethateverycasepackcontainsatleastoneunitofeachSKU,orthateveryordercontainsapositiveorderquantityforeachcasepack.Forthespecialcasewithstationarycosts,themultiple-case-packversionofsubproblem(PQ)willcorrespondtoamultiplemoduledesignproblem.AlthoughSilverman[ 86 ],ShaftelandThompson[ 83 ]andAl-Khayyal[ 6 ]havedevelopedsomeheuristicschemesforsuchproblems,theseapproachesareoftendifculttoimplement,andoptimalsolutionsarenotguaranteed. 106

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4-1 summarizesthecommondatausedinourcomputationalstudy.Foreachprobleminstance,thedataweregeneratedfromuniformdistributions.WeletU[l,u]denotethecontinuousuniformdistributionwithlowerboundlandupperboundu.ThecapacityRhereindicatestheupperboundforanallowablecase-packsize,whichissettotwentytimesthenumberofitems,i.e.,R=20M.Thedatawerechosensothattrivialcaseswouldbeavoided(e.g.,problemswithonlyonereplenishmentatoptimalityorproblemsinwhichareplenishmentoccursineveryperiod). Table4-1. Parameterdistributionsusedincomputationaltests. Sizedata(integer)andcostdata(continuous) Period,TSKU,MCapacity,RDemand,dmt5,12,20,303,6,1020MU[50m,100+50m]Purchasing,cmHolding,hmFixedOrdering,OtU[10,60]U[0.05,1.00]U[100,180] 4-2 .WecomparedtheaveragerunningtimeandoptimalitygapofourapproacheswiththoseobtainedbyGAMS/BARON.Wesetthetimelimitforeachtestto5000seconds.Intherstrow,CPLEXcorrespondstoourlinearizationmethod,whichusesCPLEXtosolvethetransformedMILPmodel,whichisanexactmethod.TheIterativeandIntegratedcolumnscorrespondtoourbasicandgeometricprogrammingbasediterativeheuristicintroducedinSections 4.4.2 and 4.4.4 ,respectively.ThelastboldcolumnisourmodiedIterative+Modapproach.Thegapisthedifferencebetweenthevalueofthesolutionobtainedbyeachapproachandthelowerbound(LB)achievedbyCPLEX,takenasapercentageoftheLBvalue.FromTable 4-2 ,weseethatalthoughBARONisacomputationalsystemforsolving 107

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4-2 underthecolumnheadingsIterative,IntegratedandIterative+Mod.Asthetableshows,theiterativeapproachisextremelyquick,withrunningtimesoflessthan0.01secondsevenforthelargeproblemsizes.However,theintegratedapproachprovidesmuchlowergaps,onaverage,attheexpenseoflongercomputingtimes.Itsrunningtimeisalsoacceptable,withsolutiontimeswithinoneminuteforsmallandmediumproblems,and2-20minutesforthelargerinstances.Weobservethatourheuristicsquicklyprovidebettersolutionsthantheexactapproachesforthelargersizeproblems(with 108

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Computationaltestresultsforaveragerunningtime(seconds)andoptimalitygap DataSetBARONCPLEXIterativeIntegratedIter+ModMTTime%GapTime%GapTime%GapTime%GapTime%Gap 35206.270.0050.440.00<0.011.123.010.91<0.0010.101250000.4040470.26<0.011.563.640.53<0.0010.18612500020.9650001.08<0.011.6817.031.250.0080.952050003211.0750001.49<0.014.0071.261.850.0121.22101250001300.6850006.46<0.016.38198.182.420.0181.712050003610.8550006.47<0.013.24146.653.360.0302.143050003701.42500022.16<0.017.961259.952.220.0401.85 4-3 providesthenumberofiterationsusedbyeachheuristicapproach.RecallthattheIterativeheuristicterminateswithintwoiterations.TheIntegratedheuristic,ontheotherhand,usuallyrequires2-10iterations,dependingonthesizeanddifcultyofjumpingoutofthelocaloptimumsassociatedwitheachproblem.Whiletheshortest-pathbasedIterativeheuristicguaranteesfastsolution,thelimitednumberofiterationsrestrictsitsperformance.Onthecontrary,byalternatingbetweenthegeometricprogramPQandtheshortest-pathsolutionasdiscussedinSection 4.4.4 ,theIntegratedapproachenablesmovingawayfromalocalminimum,althoughthegeometricprogrammingsubproblemrequiresagreateramountoftime.Figure 4-1 illustratestheprogressoftheIntegratedapproachforaninstancewithM=6andT=20.Thisprobleminstancerequired10iterationsbeforetermination.Asthegureshows,atthenaliteration,boththereddot(correspondingtotheobjectivevalueforthesubproblem(PQ)withxedzvariables)andthebluedot(theobjectivevalueforthedynamicprogramwithxedEvariables)reachthesamelevel,meaning 109

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Theiterationnumberforiterativeandintegratedheuristics 351.62.6121.43.66121.84.6201.27.810121.86201.64.2301.810.2 thatnoimprovementwasseeninsuccessiveiterations.Atpreviousiterations,thesolutioncorrespondingtothebluedotalwaysslightlyimprovesuponitsimmediateprecedingsolutionindicatedbythereddot(bydecreasingtheobjectivevalue).Atsubsequentiterations,thereddotfollowingeachbluedoteitherjumpsoutofthepreviouslocalarea(asinIterations4,7,and10),orfurtherimprovesupontheobjective(asinIterations2,3,5,6,8,and9).Ifthereisnoimprovementoverthebestsolutionfoundafter5iterations,thealgorithmterminates. Figure4-1. Objectivevaluevsiterationsfortheintegratedapproach(2ndinstanceM=6,T=20) 110

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ObjectivevaluevsRvaluefortheiterativeandintegratedheuristics(5thinstanceM=6,T=20) BecausetheIterativeheuristicrunssoquickly,ourmodiedIterative+Modapproachcantakeadvantageofthisbyusingtheiterativeheuristicasasubroutine.Theresultingperformanceisquitegood,asshowninTable 4-2 .Thisapproachofenumeratingdifferentinitialcase-packcapacityvaluesisveryefcient,withallproblemssolvedwithin0.04seconds,andagapthatisthesmallest,exceptforthesmallestset,whichcanbesolvedtooptimalityusingthelinearizationmethodinCPLEX.WehavetestedthecasewhenM=100,T=100,whichcanprovideareasonablygoodsolutiononlyinaround24seconds.Sothepotentialofthisheuristicissignicant.Todemonstratehowthevalueofthecase-packcapacityRinuencestheheuristicperformance,werstimplementthebasicandgeometricprogrammingbasediterativeheuristicwithseveraltestvaluesofRbetween4Mand20Matintervalsof4M.Thatis,wetestedthecase-packcapacityRatvaluesof4M,8M,12M,16M,and20M.TheresultsareshowninFigure 4-2 .WefoundthatusingdifferentvaluesofRcanstronglyaffectheuristicperformance.AtvaluesofR=20M,16M,and4M,theIntegratedheuristicgeneratedabettersolutionthanthatobtainedbytheIterativealgorithm,whichislikelyaresultoftheabilityoftheformertoperformmoreiterationsthanthe 111

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4-3 illustratestheenumerationprocessoftheIterative+Modheuristic.Bychangingthecase-packcapacitylimitRsuccessivelyfrom20Mto2M,theobjectivevalueobtainedthroughthebasiciterativeheuristicateachiterationalsochanges,withthenalsolutiontakenasthesmallestamongthese.AsshowninFigure 4-3 ,thesolutionforthisprobleminstanceturnsouttobeworseiftheinitialcase-packcapacityissettoavaluesmallerthan8M.Thus,duringimplementation,itisimportanttochooseareasonablerangeofRvaluestoenumerateinordertoavoidunnecessarycomputations.InourtestingoftheIterative+Modmethod,wesuccessivelysetRequaltoeachintegervaluefrom20Mto8M. Figure4-3. ObjectivevaluevsRvaluefortheIter+R Enum(5thinstanceM=6,T=20) Insummary,wecomparedourapproacheswiththebenchmarknonlinearsolverGAMS/BARON.TheresultsshowthatourproposedmethodsarefarsuperiortoBARONfortherangeofproblemswetested.Ourexactlinearizationmethodcanprovidebetter 112

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49 ]).WeintroduceaLagrangemultiplierw0andformtheLagrangianfunctionLg(Xk,Y)(notethatintermsofourpriornotation,Lg(Xk,Y)isequivalenttotheLagrangianrelaxationobjectiveofv(Xk)in( 2 )):Lg(Xk,Y)=mXi=1eiE"nXj=1dijxkjyi!+#+mXi=1ciyi+wmXi=1siyiV!Then,foragivenw,usingthenecessaryandsufcientrst-orderconditions,theyivaluesaredeterminedbysolvingthesetofequations@Lg(Xk,Y)

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2.1 ,(Xk)=@ @XH(X,Yk)+G1(Yk)+ukG2(Yk)X=Xk=@ @XH(X,Yk)X=Xk.UndertheassumptionthatdijN(ij,2ij),weletDi=Pnj=1dijxj;thusE(Di)=Pnj=1ijxj=~Ui(X),Var(Di)=Pnj=12ijx2j=2i(X),andDiN(~Ui(X),2i(X)).WerepresentH(X,Yk)usingthefollowingequation:H(X,Yk)=mXi=1eiE"nXj=1dijxjyki!+#=mXi=1eii(X) (~Ui(X)yki) @XfH(X,Yk)gX=Xk=mXi=1ei"1 @X(i(X)exp((yki~Ui(X))2 @X8<:(~Ui(X)yki)Z1yki~Ui(X) 119

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@xj(i(X)exp((yki~Ui(X))2 @xj8<:(~Ui(X)yki)Z1yki~Ui(X) @xjfH(X,Yk)gX=Xk=mXi=1ei1 @xjfH(X,Yk)gX=Xk=0=mXi=1eiij.(B) 120

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ShuangChenwasborninBeijing,Chinain1984.ShegraduatedfromthehighschoolattachedtoTsinghuaUniversityin2003.SheearnedaBachelorofScienceinIndustrialEngineeringfromBeihangUniversity(BUAA)inJuly2007.Afterherundergraduatestudies,sheenrolledinthePhDprogramintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.ShereceivedherPh.D.inIndustrialandSystemsEngineeringfromtheUniversityofFloridainthesummerof2011.Followinggraduation,shewilljoinBankofAmericaasanOperationsResearchAnalyst. 128