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Models for Optimal Utilization of Production Resources under Demand Selection Flexibility


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MODELSFOROPTIMALUTILIZATIONOFPRODUCTIONRESOURCES UNDERDEMANDSELECTIONFLEXIBILITY By KEVINMICHAELTAAFFE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2004

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Copyright2004 by KevinMichaelTaae

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Idedicatethisworktomyfamilyandtomyfuturestudents.

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ACKNOWLEDGMENTS Iwouldliketothankeveryonewhohasprovidedwordsofsuppor tand encouragementduringthepastfouryears.Mostimportantly,It hankmywife, Mary,forprovidingunwaveringsupportformypursuitofthisd ream.Shechose tosacriceherowndesiresandneedstoassistmebycaringforourch ildrenand maintainingherbusinesswhileIcompletedmydegree.Idonotk nowmanypeople whocoulddothatevenonce ::: andshedoesitallthetime.Furthermore,shehas alwaysbeenmybiggestemotionalsupport.Afterbeingawayfrom schoolforso longandworkinginindustryformanyyears,Ifounditdicultto resumewhere Ihadlefto15yearsagowithadvancedmathandtheoreticalr esearch.Mary remindedmethatitwouldbehardattimes,butshealwaysmanage dtocalmme downandgetmebackontrack.Simplysaid,Icouldnothaveacco mplishedthis goalwithouther.Sheistheloveofmylife,andIwillalwaysl oveherfromthe bottomofmyheart. Lookingback,Icouldnothaveaskedforabetterpersontobemyt hesis advisorthanJoeGeunes.Hehasbeenagreatrolemodelformyfutu recareerin academia,andIthankhimforalloftheexperienceswehavesha red.Heallowed methespacetothinkcreatively,buthewasalwaystherewhenI neededhelpor guidance.Ourfamiliesbecameverycloseovertheyears,andIh opewecontinueto staycloseformanyyearstocome. Therearemanypeoplewhohavetouchedmylifeinaspecialwaysi nceI arrivedinGainesville.Fromourneighborswhobecamelikefa mily,tomyentire churchfamily,andallofthefriendsIhavemetalongtheway, IcanhonestlysayI haveneverfeltsuchwarmthonsomanydierentlevels.Everyoneo fthesepeople iv

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hashadanimpactonwhoIamtoday,andIcansaythattheyallhav eservedas dailyremindersastowhatistrulyimportantinlife.Itwill beasadfarewellwhen weleaveGainesville,butIhavedevelopedmanyrelationships thatwillnevergo away.ForthatIameternallygrateful. v

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TABLEOFCONTENTS pageACKNOWLEDGMENTS.............................ivLISTOFTABLES.................................ixLISTOFFIGURES................................xABSTRACT....................................xiCHAPTER 1INTRODUCTION..............................12INTEGRATEDORDERSELECTIONAND REQUIREMENTSPLANNING.....................52.1Introduction..............................52.2OrderSelectionProblemDenitionandFormulation........92.2.1TheUncapacitatedOrderSelectionProblem........112.2.2SolutionPropertiesforUOSP................122.3OSPModels-LimitedProductionCapacity............152.3.1OSPSolutionMethods....................182.3.2StrengtheningtheOSPFormulation.............192.3.3HeuristicSolutionApproachesforOSP...........232.3.3.1LagrangianRelaxationBasedHeuristic........242.3.3.2GreatestUnitProtHeuristic.............262.3.3.3LinearProgrammingRoundingHeuristic.......282.4ScopeandResultsofComputationalTests.............292.4.1ComputationalTestSetup..................292.4.2ResultsfortheOSPandtheOSP-NDC...........332.4.3ResultsfortheOSP-AND...................372.5Conclusions..............................412.6Appendix...............................423PRICING,PRODUCTIONPLANNING,AND ORDERSELECTIONFLEXIBILITY..................453.1Introduction..............................453.2RequirementsPlanningwithPricing.................493.2.1ShortestPathApproachfortheUncapacitatedRPP....523.2.2Dual-ascentMethodfortheUncapacitatedRPP......55 vi

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3.2.3PolynomialSolvabilityofMoreGeneralModels .......65 3.2.3.1Multipleprice-demandcurves .............65 3.2.3.2Piecewise-linearconcaveproductioncosts ......67 3.2.4ProductionCapacities .....................68 3.3PricingandOrderSelectionInterpretations .............73 3.4Conclusions ..............................78 3.5Appendix ...............................78 4SELECTINGMARKETSUNDERDEMANDUNCERTAINTY .....85 4.1Introduction ..............................85 4.2TheSelectiveNewsvendorProblem .................92 4.2.1ProblemFormulationandSolutionApproach ........92 4.2.2ManagerialInsightsfortheSNP ...............97 4.3SNPandtheRoleofAdvertising ..................103 4.3.1SelectiveNewsvendorwithMarketingEort ........104 4.3.2IndependentDemandVariance ................105 4.3.2.1ConcaveDemand ....................106 4.3.2.2S-curvedDemand ....................108 4.3.3DependentDemandVariance .................109 4.3.4MarketingInsights .......................111 4.4OperatingwithLimitedMarketingResources ............114 4.4.1FormulationoftheLimitedResourcesProblem .......114 4.4.2SolutionApproachtotheLimitedResourcesProblem ...116 4.4.3SubproblemSolutionandB&BImplementation .......119 4.5ComputationalResults ........................127 4.5.1SNPValue:MinimumMarketRequirement .........127 4.5.2SNPValue:ProtImprovement ...............129 4.5.3SolvingtheLimitedResourcesProblem ...........132 4.6OtherConsiderations .........................135 4.6.1TheInniteHorizonPlanningProblem ...........135 4.6.2LimitedMarketingEortunderaFixedContract .....138 4.7Conclusions ..............................140 5AIRPORTCAPACITYLIMITATIONS{ SELECTINGFLIGHTSFORGROUNDHOLDING ..........142 5.1Introduction ..............................142 5.2StaticStochasticGroundHoldingProblem .............145 5.2.1ProblemDenitionandFormulation .............145 5.2.2SolutionProperties ......................147 5.3MotivationforStochasticProgrammingApproach .........148 5.3.1ArrivalDemandandRunwayCapacityData ........148 5.3.2KeyStochasticProgrammingMeasurements ........151 5.4RiskAversionMeasures ........................155 5.4.1ConditionalValueatRisk(CVaR)Model ..........155 vii

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5.4.2ModelComparison ......................157 5.4.3AlternateRiskAversionModels ...............159 5.5Conclusions ..............................162 6CONCLUDINGREMARKS .........................165 REFERENCES ...................................167 BIOGRAPHICALSKETCH ............................174 viii

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LISTOFTABLES Table page 2{1Counterexampleillustratingdecreasingcumulativedema ndsatisfaction. 15 2{2Classicationofmodelspecialcasesandrestrictions. ..........16 2{3ProblemsizecomparsionforcapacitatedversionsoftheOSP .....17 2{4Probabilitydistributionsusedforgeneratingproblemin stances. ....30 2{5OSP-NDCandOSPproblemoptimalitygapmeasures. .........35 2{6OSP-NDCandOSPsolutiontimecomparison. .............36 2{7OSPandOSP-NDCheuristicsolutionperformancemeasures. .....37 2{8OSP-ANDoptimalitygapmeasures. ...................40 2{9OSP-ANDsolutiontimecomparison. ...................40 2{10OSP-ANDheuristicsolutionperformancemeasures. ..........41 4{1ResultsforSNPwithlimitedresources{CaseII. ............134 4{2ResultsforSNPwithlimitedresources{CaseI. ............135 5{1EVPIandVSSstatistics(Minimizetotalexpecteddelaycostmo del). .154 5{2Overallmodelcomparisons. ........................158 5{3Performancecomparisonofalternateriskmodels. ............162 ix

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LISTOFFIGURES Figure page 2{1FixedchargenetworkowrepresentationofUOSP. ..........11 2{2ShortestpathnetworkstructureforUOSP. ...............13 3{1Pricinginterpretationsbasedontotalrevenueanddeman d. ......74 4{1Optimalmarketingeortforconcaveexpecteddemandfunc tions. ..107 4{2OptimalmarketingeortforS-curveddemandresponsefunct ions. ..109 4{3ApproximationoftheS-curveddemandresponsefunction. ......110 4{4Minimummarketrequirementbasedonindividualcostparam eters. .128 4{5ProtimprovementusingSNPbasedontotalmarketsavailable ...131 4{6ProtimprovementusingSNPbasedondemandvariance. ......132 5{1Aircraftarrivaldemandatthecapacitatedairport. ...........148 5{2Weather-inducedarrivalcapacityscenarios. ...............150 5{3Totaldelayoutputforarrivaldemandlevel1. .............160 x

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MODELSFOROPTIMALUTILIZATIONOFPRODUCTIONRESOURCES UNDERDEMANDSELECTIONFLEXIBILITY By KevinMichaelTaae August2004 Chair:JosephGeunesMajorDepartment:IndustrialandSystemsEngineering Optimaldemandselectionappliestocontextsinwhichanorga nizationhas somediscretionindecidingthesetofdemandsitwilluseitsresou rcestosatisfy. Insuchcases,thedecisionmakerwishestodeterminethesetofdown stream demandsthatprovidesthebestmatchforitsresourcecapabili ties.Thissteps awayfromtraditionalstreamsofresearchthatignoretheselec tiondecisionand assumealldemandsourcesmustbesatised.Wefocusondevelopingne wmodels andsolutionmethodsforproblemsthatintegratedemandselec tionwiththe planningandutilizationofproductionresources,forbothun limitedandlimited productioncapacities.Capacitylimitsoftenrestricttheto talamountofdemand thatanorganizationcansatisfy.Whentotaldemandforresourc esexceedscapacity limits,selectingtheoptimalsubsetofdemandsourcesisachalle ngingoptimization problem.Evenincontextswherecapacitylimitsaretypical lynotaconstraining factor,theproblemremainsdicultduetoeconomiesofscalei nproductionand theattractivenessandtimingofindividualdemands.Givenase tofheterogeneous downstreamdemandsources,whichmaybedeterministicorstochast icinnature, alongwithnonlinearcapacityusagecostsinvolume,wepropose modelsthat xi

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provideoptimaldemandsourceselectionsthatachievemaximu mprotability. Inthisdissertation,wespecicallyaddressdemandselectionexi bilityforthe applicationsareasofgeneralproductionandinventorypla nningproblemsand airportgroundholdingproblems. xii

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CHAPTER1 INTRODUCTION Thisdissertationfocusesbroadlyonmodelsforoptimaldemand selection. Suchmodelsapplytocontextsinwhichanorganizationhassom ediscretionin decidingthesetofdemandsitwilluseitsresourcestosatisfy.In suchcases,the decisionmakerwishestodeterminethesetofdownstreamdemands thatprovides thebestmatchforitsresourcecapabilities.Capacitylimitso ftenrestrictthe totalamountofdemandthatanorganizationcansatisfy.Whent otaldemand forresourcesexceedscapacitylimits,thedecisionmakermustd eterminethe bestwayinwhichtoallocateitslimitedresources.Givenasetof heterogeneous downstreamdemandsources,alongwithnonlinearcapacityusage costsinvolume, itisnotatrivialproblemtoselectthesubsetofdemandsourcest hatwillprovide themaximumprottotherm.Evenincontextswherecapacityli mitsare typicallynotaconstrainingfactor,economiesofscaleinpro duction,combined withtime-varyingcustomerdemandpatternsfromcustomerswh ohavedierent reservationprices,maketheproblemofchoosingthebestsetofde mandstosatisfy achallengingtask.Wefocusondevelopingnewmodelsandsolut ionmethodsfor problemsthatfallinthisgeneralclassofdemandselectionpr oblems. Weexploretwoapplicationscontextswithintheclassofdema ndselection problems: Generalproductionandinventoryplanningproblems,and Airportoperationsgroundholdingproblems. Weexamineseveraltypesofproductionandinventoryplannin gproblemsin Chapters 2 3 ,and 4 .InChapter 2 ,weconsideruncapacitatedandcapacitated versionsofthesingle-stage,multi-periodproductionplanni ngproblemfora 1

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2 producerwhocanselectanynumberoforders,ordemandsources,f romatotalset ofpotentialdemands.Theproblemhasanitehorizon,andthep roducerhasthe discretiontochoosewhentoproduceandhowmuchdemandtosatisf yinorder tomaximizeprot.Wedenethisnewclassofproductionplannin gproblems with orderselectionexibility andprovideoptimization-basedmodelingand solutionmethodsfortheseproblems.Weprovideapolynomial-t imealgorithmfor solvingtheuncapacitatedversionoftheproblem,andwepropo sestrongproblem formulationsandheuristicsolutionalgorithmsforseveralca pacitatedversions. InChapter 3 ,weextendourdiscussionofthissingle-stageproductionplanni ng problemtoaddresstheimportanceofpricing.Firmsthatmanu factureandsell productswithprice-elasticdemandfacethechallengeofdet erminingprices,and thereforedemandvolumes,thatprovidemaximumprottotherm .Nonlinearities indemandasafunctionofpriceandinproductioncostsasafun ctionofdemand volumescreatecomplexitiesindeterminingpricingstrateg iesthatmaximizecontributiontoprotafterproduction.Now,insteadofdirectlyse lectingthedesired demandquantitiestosatisfy,asshowninChapter 2 ,wepresentaproductionplanningmodelthatimplicitlydecides,throughpricingdecision s,thedemandlevels thermshouldsatisfyinordertomaximizecontributiontoprot. Wepresenttwo polynomial-timesolutionapproachesfortheseproblemswhen productioncapacities areeectivelyunlimited,andshowthattheseapproachesapply acrossarangeof applicablerevenueandcostfunctions.Wealsodescribeapolyno mial-timesolution approachundertime-invariantniteproductioncapacities andpiecewise-linearand concaverevenuefunctionsintheamountofdemandsatised. Thesechapterstogetherillustratetheimportanceofintegra teddemandand productionplanningdecisionsbyenablingaproducertoleve rageproduction economiesofscaletothegreatestextentpossiblethroughmatch ingtheright amountofdemandtoproductioncapabilities.

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3 InChapter 4 ,weintroduceastochasticversionofthedemandselection probleminasingle-periodsetting,aproblemthatwerefertoa stheselective newsvendorproblem.Inthisproblem,asellerfacesalongproc urementleadtime fromanexternalsupplier,andmustsimultaneouslydecidethema rketsinwhich itwillsellitsproductalongwiththeprocurementquantityf romtheexternal supplier.Foreachselectedmarket,theselleralsodeterminest heamountof marketingeortitwillexertinthemarket,andthismarketin geortinuences thedistributionofdemandinthemarket(e.g.,increasedmark etingeortimplies higherexpecteddemandinthemarketandalsoimpactstheunce rtaintyofthe market'sdemand).Thegoalistochoosethemarkets,advertisin glevels,andoverall procurementquantitythatmaximizestheseller'sexpectedp rotintheselling season. First,wepresentsolutionapproachesformarketselectiondeci sionsinwhich themarketinglevelsarexedorpre-denedbythermorsupplier .Wethen extendthemarketselectionapproachtoallowthermtodeterm inethebest levelofadvertisingtoapplyineachmarketselected.Weillust ratethisapproach forboththeunlimitedandlimitedresourcescases,andweevalu atemultiple functionalformsforthemannerinwhichmarketdemandlevel srespondtomarket advertising. Weconcludebypresentingtheairportoperationsgroundhold ingproblem inChapter 5 .Thisprobleminvolvesdeterminingwhichightsdestinedfor a givenairportshouldbedispatchedunderuncertaintyinfutur eweather.Inthis context,ightsdestinedforanairportconstitutethe(future )demandforarrival capacity,whiletheuncertaintyinfutureweatherleadstou ncertainanddynamic capacitylevelsforreceivingightsatthedestinationpoint .Acceptingdemands atagivendestinationcanbeverycostlyiftheresultingcapaci tyatightarrival timeislow(duetobadweather).Conversely,denyingdemands byholdingights

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4 attheiroriginationpointscanalsobequitecostly,particul arlyiftheresulting capacityatthescheduledightarrivaltimeishigh(i.e.,whe npreviouslypredicted badweatherdoesnotmaterializeatarrivaltime).Thegroun dholdingproblem introducedinChapter 5 addressesthiscriticalissueofoptimalightarrival selectiondecisionsunderuncertainty. Thegroundholdingproblemprovidesanexcellentillustrati onofthebenets ofusingastochasticoveradeterministicapproachinmathemati calprogramming. Wesummarizethesebenetswithinthechapter.Thegroundholdi ngproblem isalsoaninterestingproblemtostudyduetothenumberofdiere ntpotential decisionmakersinuencingthechoiceofightdemandstogroun dhold.Sincethe FederalAviationAdministration,localairportauthorities,a ndindividualairlines allhaveconictingoperationalgoals,weaddressnewriskaversi onmodelsthat allowmultipledecisionmakerstoachieveacceptableperfor manceatthesametime. Asshowninthisdissertation,thedemandselectionproblemcanap pearin manyforms,andweprovideathoroughdiscussionofthemainfocus areaswithin demandselection,suchasdeterministicdemandvs.stochasticdem and,unlimited resourcesvs.limitedresources,andxedpricingvs.variablepri cing.These chaptersprovideasolidfoundationforfutureresearchindem andsourceselection, andwealsomakeseveralsuggestionsforresearchdirectionsinth etheconcluding remarks.

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CHAPTER2 INTEGRATEDORDERSELECTIONAND REQUIREMENTSPLANNING 2.1 Introduction Firmsthatproducemade-to-ordergoodsoftenmakecritical orderacceptance decisionspriortoplanningproductionfortheorderstheyul timatelyaccept.These decisionsrequiretherm'srepresentatives(typicallysales/m arketingpersonnel inconsultationwithmanufacturingmanagement)todetermin ewhichamongall customerordersthermwillsatisfy.Incertaincontexts,suchast hoseinvolving highlycustomizedgoods,thecustomerworkscloselywithsalesre presentativesto deneanorder'srequirementsand,basedontheserequirements, thestatusofthe productionsystem,andthepriorityoftheorder,thermquotes aleadtimefor orderfulllment,whichisthenacceptedorrejectedbythecu stomer(seeYano [ 85 ]).Inothercompetitivesettings,thecustomer'sneedsaremor erigidandthe customer'sordermustbefullledataprecisefuturetime.Thema nufacturercan eithercommittofulllingtheorderatthetimerequestedbyth ecustomer,or declinetheorderbasedonseveralfactors,includingthemanuf acturer'scapacity tomeettheorderandtheeconomicattractivenessoftheorder .These\order acceptanceanddenial"decisionsaretypicallymadepriorto establishingfuture productionplansandaremostoftenmadebasedonthecollectiv ejudgmentof sales,marketing,andmanufacturingpersonnel,withouttheai dofthetypesof mathematicaldecisionmodelstypicallyusedintheproductio nplanningdecision process. Whenthemanufacturingorganizationishighlycapacitycon strainedand customershavermdeliverydaterequirements,itisoftennece ssarytosatisfya 5

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6 subsetofcustomerordersandtodenyanadditionalsetofpotenti allyprotable orders.Insomecontexts,themanufacturercanchoosetoemploya rationing schemeinanattempttosatisfysomefractionofeachcustomer'sde mand(see Lee,Padmanabhan,andWhang[ 46 ]).Inothersettings,sucharationingstrategy cannotbeimplemented;i.e.,itmaynotbedesirableorpossible tosubstituteitems orderedbyonecustomerinordertosatisfyanothercustomer'sde mand.Thus,it maybenecessaryforthermtodenycertaincustomerorders(orpa rtsoforders) sothatthemanufacturercanmeetthecustomer-requesteddueda tesfortheorders itaccepts.Incontextswherecapacitylimitsarenon-bindin g,itisalsonotalways clearthatcommittingtoaparticularcustomerorderisinthe bestinterestofthe rm,eveniftheunitpricethecustomerwillpayexceedsthevar iableproduction cost.Thisisevidentinenvironmentswithsignicantxedprodu ctioncosts. Regardlessofwhethertheoperationisconstrainedorunconstr ainedby productioncapacity,assessingtheprotabilityofanorderiniso lation,priorto productionplanning,leadstomyopicdecisionrulesthatfai ltoconsiderthebest setofactionsfromanoverallprotabilitystandpoint.Thepro tabilityofan order,whengaugedsolelybytherevenuesgeneratedbytheord erandperceived customerpriorities,neglectstheimpactsofimportantopera tionscostfactors,such astheopportunitycostofmanufacturingcapacityconsumedby theorder,aswell aseconomiesofscaleinproduction.Decisionsonthecollecti vesetofordersthe organizationshouldacceptcanbeacriticaldeterminantoft herm'sprotability. SinceWagnerandWhitin's[ 83 ]seminalpaperaddressedthebasiceconomic lot-sizingproblem(ELSP),numerousextensionsandgenerali zationsofthisbasic problemhavefollowed,includingextensionstoincorporate backlogging(Zangwill [ 86 ]),serialsystemstructures(Love[ 51 ]),andmultistageassemblyandgeneral multistagestructures(Afentakis,Gavish,andKarmarkar[ 2 ],andAfentakis andGavish[ 1 ]).Intensiveresearchonthecapacitatedversionofthedynami c

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7 requirementsplanningproblembeganinthe1970's(seeFlori anandKlein[ 28 ], Baker,Dixon,Magazine,andSilver[ 5 ],andFlorian,Lenstra,andRinnooyKan [ 29 ]),andhasreceivedincreasedattentionrecentlyasaresulto ftheapplicationof strongvalidinequalitiesthatenablefastersolutionofthesed icultproblems(e.g., Barany,VanRoy,andWolsey[ 10 ],Pochet[ 61 ],andLeung,Magnanti,andVachani [ 47 ]).LeeandNahmias[ 45 ],Shapiro[ 70 ],andBaker[ 4 ]provideexcellentoverall analysesofthegeneralizationsandsolutionapproachesford ynamicrequirements planningproblems,includingvariousheuristicapproachest hathaveproven eectiveforthecapacitatedversionoftheproblem. Withafewnotableexceptionsthatwelaterdiscuss,thispastrese archon dynamicrequirementsplanningproblemsnearlyalwaysassume sthatdemands arepre-speciedbytimeperiodandthatalldemandsmustbecomp letelylled atthetimetheyoccur(oraftertheyoccurinmodelsthatperm itbacklogging). Incontrast,weconsiderarequirementsplanningmodelthatim plicitlydetermines thebestdemandlevelstosatisfyinordertomaximizecontribut iontoprot. Whiletheuncapacitatedversionissolvableinpolynomialtim e,aswelaterdiscuss, thecapacitatedversionisNP-Hardandthereforerequirescusto mizedheuristic solutionapproaches.WeproposestrongLPformulationsoftheca pacitatedversion, whichoftenallowssolvinggeneralcapacitatedinstancesvia branch-and-boundin reasonablecomputingtime.Forthoseproblemsthatcannotbeso lvedviabranchand-boundinreasonabletime,weprovideasetofthreeeective heuristicsolution methods.Computationaltestresultsindicatethattheproposed solutionmethods forthegeneralcapacitatedversionoftheproblemareveryee ctive,producing solutionswithin0.67%ofoptimality,onaverage,forabroad setof3,240randomly generatedprobleminstances. Loparic,Pochet,andWolsey[ 50 ]recentlyconsideredarelatedproblemin whichaproducerwishestomaximizenetprotfromsalesofasingl eitemand

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8 doesnothavetosatisfyalloutstandingdemandineveryperiod. Theirmodel assumesthatonlyonedemandsourceexistsineveryperiod,andth attherevenue fromthisdemandsourceisproportionaltothevolumeofdeman dsatised.The \orderselection"interpretationofthemodelwepresent,ont heotherhand,allows thermtoconsideranynumberoforders(ordemandsources)ineac hperiod, eachwithauniqueassociatedperunitrevenue(i.e.,weallowf orcustomerswith dierentreservationprices).Inthisrespect,theirmodelrepr esentsasingle-order specialcaseofoneofthemodelswepropose.Morerecently,Lee, Cetinkaya,and Wagelmans[ 43 ]consideredcontextsinwhichdemandscanbemeteitherearli er (throughearlyproductionanddelivery)orlater(throughb acklogging)than speciedwithoutpenalty,providedthatdemandissatisedwithi ncertain demand timewindows fortheuncapacitated,single-stagelotsizingproblem.Their model stillassumesultimately,however,thatallpre-specieddemand smustbelled duringtheplanninghorizon. Theremainderofthischapterisorganizedasfollows.Sectio n 2.2 presents aformaldenitionandmixedintegerprogrammingformulatio nofthegeneral productionplanningproblemwithorderselectionexibility .Wethenpresenta solutionapproachfortheuncapacitatedversionoftheproble mthatgeneralizes theWagner-Whitin[ 83 ]shortestpathsolutionmethodforsingle-stagedynamic requirementsplanningproblems.InSection 2.3 weconsidervariousmixedinteger programmingformulationsofthecapacityconstrainedprobl em,alongwiththe advantagesanddisadvantagesofeachformulationstrategy.W ealsoprovideseveral heuristicsolutionapproachesforeachofthecapacitatedpro bleminstances.Section 2.4 thenprovidesasummaryofasetofcomputationaltestsusedtogau gethe eectivenessoftheformulationstrategiesandheuristicsoluti onmethodsdescribed inSection 2.3

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9 2.2 OrderSelectionProblemDenitionandFormulation Consideraproducerwhomanufacturesagoodtomeetasetofoutst anding ordersoveranitenumberoftimeperiods, T .Producingthegoodinanytime period t requiresaproductionsetupatacost S t andeachunitcostsanadditional p t tomanufacture.Welet M ( t )denotethesetofallordersthatrequestdeliveryin period t (weassumezerodeliveryleadtimeforeaseofexposition;themod eleasily extendstoaconstantdeliveryleadtimewithoutlossofgenera lity),andlet m denoteanindexfororders.Themanufacturerhasacapacityto produce C t unitsin period t t =1 ;:::;T .Weassumethatthatthereisnoplannedbacklogging 1 (i.e., noshortagesarepermitted)andthatitemscanbeheldininven toryatacostof h t perunitremainingattheendofperiod t .Let d mt denotethequantityofthe goodrequestedbyorder m forperiod t delivery,forwhichthecustomerwillpay r mt perunit,andsupposetheproducerisfreetochooseanyquantity between zeroand d mt insatisfyingorder m inperiod t (i.e.,rationingispossible,andthe customerwilltakeasmuchofthegoodasthesuppliercanprovid e,upto d mt ). Theproducerthushastheexibilitytodecidewhichordersit willchoosetosatisfy ineachperiodandthequantityofdemanditwillsatisfyforeac horder.Ifthe producerndsitunprotabletosatisfyacertainorderinaperio d,itcanchoose torejecttheorderatthebeginningoftheplanninghorizon. Themanufacturer incursaxedshippingcostfordeliveringorder m inperiod t equalto F mt (any variableshippingcostcanbesubtractedfromtherevenueterm, r mt ,withoutlossof generality).Theproducer,therefore,wishestomaximizene tprotovera T -period horizon,denedasthetotalrevenuefromorderssatisedminust otalproduction 1 Extendingourmodelsandsolutionapproachestoallowbacklo ggingataper unitperperiodbackloggingcostisfairlystraightforward.W ehavechosentoomit thedetailsofthisextension.

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10 (setup+variable),holding,anddeliverycostsincurredover thehorizon.To formulatethisproblemwedenethefollowingdecisionvariab les: x t =Numberofunitsproducedinperiod t y t = 8><>: 1 ; ifwesetupforproductioninperiod t; 0 ; otherwise, I t =Producer'sinventoryremainingattheendofperiod t v mt =Proportionoforder m satisedinperiod t z mt = 8><>: 1 ; ifwesatisfyanypositivefractionoforder m inperiod t; 0 ; otherwise. WeformulatetheCapacitatedOrderSelectionProblem(OSP) asfollows. [OSP] maximize: T P t =1 P m 2 M ( t ) ( r mt d mt v mt ¡ F mt z mt ) ¡ S t y t ¡ p t x t ¡ h t I t (2.1) subjectto: I t ¡ 1 + x t = P m 2 M ( t ) d mt v mt + I t t =1 ;:::;T; (2.2) 0 x t C t y t t =1 ;:::;T; (2.3) 0 v mt z mt t =1 ;:::;T;m 2 M ( t ) ; (2.4) I 0 =0 ;I t 0 t =1 ;:::;T; (2.5) y t ;z mt 2f 0 ; 1 g t =1 ;:::;T;m 2 M ( t ) : (2.6) Theobjectivefunction( 2.1 )maximizesnetprot,denedastotalrevenueless xedshippingandtotalproductionandinventoryholdingcosts. Constraintset ( 2.2 )representsinventorybalanceconstraints,whileconstraintse t( 2.3 )ensures thatnoproductionoccursinperiod t ifwedonotperformaproductionsetupin theperiod.Ifasetupoccursinperiod t ,theproductionquantityisconstrainedby theproductioncapacity, C t .Constraintset( 2.4 )encodesourassumptionregarding theproducer'sabilitytosatisfyanyproportionoforder m uptotheamount d mt

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11 while( 2.5 )and( 2.6 )providenonnegativityandintegralityrestrictionsonvar iables. Observethatwecanforceanyorderselection( z mt )variabletooneifqualitative and/orstrategicconcerns(e.g.,marketsharegoals)requiresa tisfyinganorder regardlessofitsprotability. 2.2.1 TheUncapacitatedOrderSelectionProblem Ifasetupoccursinperiod t ,theproductionquantityisunconstrained bysetting C t equaltoalargenumber.Alternatively,wecanset C t equalto P T = t P m 2 M ( ) d m withoutlossofgenerality,sincethisisthemaximumamountof demandthatcouldbesatisedbyperiod t production(intheabsenceofbacklogging).Wedenotetheresultinguncapacitatedorderselection problemas[UOSP]. Problem[UOSP]canberepresentedasaxed-chargenetworkowpr oblem asillustratedbytheexampleinFigure 2{1 ,where T =4and M ( t )=2for t =1, ..., T .Thenetworkcontainsthreetypesofarcs: productionarcs inventoryarcs Demand, d mt Inventory holding arcs Flow cost = h t Order selection arcs Production Source Supply = D Dummy Source Supply = D Dummy Sink Demand = D 1 T mt tmMt Dd Cost tttt Sypx Cost mtmtmtmtmt Fzrdv Production flow arcs Capacity = C t Figure2{1:FixedchargenetworkowrepresentationofUOSP. and orderselectionarcs .Thedummysourcenodeimpliesthatitisnotnecessary tosatisfyalldemand|thedummysourcecansupplytheentiredema ndoverthe horizonifnecessary.Wealsoaddadummysinknodethatcanreceiv eowfrom boththeproductionsourceandthedummysource.Flowonaprodu ctionarc impliesthatasetupoccursinthatperiod,whileowonanorder selectionarc

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12 impliesthatwesatisfyatleastsomeofthatorderinthecorrespon dingperiod. Sincetheowcostoneacharcisconcaveorlinear(andhencealso concave),the objectivefunction( 2.1 )isconvexand[UOSP]maximizesaconvexfunctionovera polyhedronforagiven y;z .Thisimpliesthatanoptimalextremepointsolution existsfor[UOSP].Sincetheproblemisanetworkowproblem,th isimplies thatanoptimal spanningtree solutionexists(seeAhujaetal.[ 3 ]),inwhichthe subgraphinducedbythearcswithpositiveowinasolutionforms aspanning tree.Weexploitthisspanningtreepropertytoderivecertai npropertiesofoptimal solutionsto[UOSP].Notethat[UOSP]generalizestheWagner-Wh itinsingle-stage requirementsplanningproblemunderdynamicdemand,whoseso lutionapproach wewillextendtosolveourorderselectionproblem. 2.2.2 SolutionPropertiesandShortestPathApproachforUOSP Theexistenceofanoptimalspanningtreesolutionfor[UOSP]imp liesthe followingproperty: All-or-nothingordersatisfactionproperty: Giventhechoicetosatisfy anyquantityofdemandlessthanorequalto d mt fororder m inperiod t ,an optimalsolutionexistswitheither v mt equalto0or1forall m and t ;i.e.,foreach order-periodcombination( m t )theproducereitherprovides d mt unitsornoneat all. WenextconsiderhowtoextendtheWagner-Whitindynamicprog ramming solutionmethodforsolvingUOSP.Theirdynamicprogrammingsol utionmethod canbeequivalentlyposedasashortestpathproblemonagraphco ntaining T +1nodes(seeFigure 2{2 ).Notethatthismethodreliesontheexistenceofan optimal Zero-InventoryOrdering (ZIO)policyinwhichasetuponlyoccursin period t ifweholdnoinventoryattheendofperiod t {1(thevalidityofthis propertycanalsobeshowntoholdfor[UOSP]asaresultofthespann ingtree propertyoftheequivalentxedchargenetworkrepresentatio nof[UOSP]).Since

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13 theWagner-Whitinapproachminimizestotalcost,eacharc( t;t 0 )inthegraph isassignedacost, c ( t;t 0 ),where c ( t;t 0 )equalsthesetupcostinperiod t plusthe variableproductionandholdingcostsincurredforsatisfying alldemandinperiods t;t +1 ;:::;t 0 ¡ 1usingonlythesetupinperiod t .Thisapproachensuresthata pathexistsintheshortestpathnetworkforeveryfeasiblecombi nationofsetups andthatthecostofapathcorrespondstotheminimumcostincurr edinusingthe setupstosatisfyalldemand. 1 2 3 4 5 c (1,5) c (1,4) c (1,3) c (1,2) c (2,3) c (3,4) c (4,5) c (2,5) c (2,4) c (3,5) Figure2{2:ShortestpathnetworkstructureforUOSP. SincetheZIOpropertyalsoholdsfortheuncapacitatedorder selection problem,wecansolvetheUOSPproblemusingashortestpathgraphc ontaining thesamestructureasthatusedforsolvingtheWagner-Whitinpro blem.Thearc lengthcalculationforUOSP,however,requiresanewapproac h.Theorderselection problemseekstomaximizenetprotandsoweinterpretarclengt hsintermsof netcontributiontoprotandseekthelongestpathinthegraph. Themethod usedforarclengthcalculationproceedsasfollows.Weinterp retthelengthofarc ( t;t 0 )asthemaximumprotpossiblefromsatisfyingordersinperiods t;:::;t 0 ¡ 1 assumingthattheonlysetupavailabletosatisfydemandinthesepe riodsmust occurinperiod t ,ifatall.Supposewechoosetoperformthesetupinperiod t and incuritscorrespondingcost, S t .Toosetthecostofthissetupwewillsatisfythe demandfororder m inperiod t ifandonlyif ( r mt ¡ p t ) d mt F mt ;(2.7)

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14 i.e.,ifthenetrevenuegeneratedfromorder m isatleastasgreatasthexed deliverycostfororder m inperiod t .Similarly,forallperiods suchthat t<< t 0 ,wewillsatisfyorder m inperiod ifandonlyif ( r m ¡ p t ¡ X ¡ 1 k = t h k ) d m F m ;(2.8) i.e.,ifthenetrevenuefromorder m inperiod ,lessanyholdingcostsincurred fromperiod t toperiod ,exceedsthexeddeliverycostfororder m inperiod Let O t ( t )denotethesetofordersinperiod t suchthat r mt p t ,andlet O t ( ) denotethesetofordersinperiod suchthat( 2.8 )holdsfor >t .Thenthe maximumprotpossibleifwedoasetupinperiod t andusethatsetuptosatisfy demandsinperiods t;:::;t 0 ¡ 1,whichwedenotebyMP S ( t;t 0 ),isgivenby MP S ( t;t 0 )= P t 0 ¡ 1 = t P m 2 O t ( ) (( r m ¡ p t ) d m ¡ F m ) ¡ P t 0 ¡ 2 = t h P t 0 ¡ 1 k = +1 P m 2 O t ( ) d mk ¡ S t : (2.9) IfMP S ( t;t 0 )isgreaterthan0wesetthelengthofarc( t;t 0 )equaltoMP S ( t;t 0 ); otherwisewesetthelengthofarc( t;t 0 )equaltozeroandassumenosetupoccurs inperiod t iftheoptimalsolution(thelongestpathinthegraph)traverse sarc ( t;t 0 ).Afterndingthelongestpathinthegraphwecandeterminewhi chordersto satisfyineachperiodbycheckingtheelementsofthesets O t ( t )and O t ( )forall arcs( t;t 0 )containedinthelongestpath. Letting m =max t =1 ;:::;T fj M ( t ) jg ,thetotalcomputationaleortofarccost calculationsisboundedby O ( mT 2 )andtheshortestpathcalculationisnoworse than O ( T 2 ),sotheworstcasecomplexityofthisalgorithmisboundedby O ( mT 2 ). Recentworkontheuncapacitatedlotsizingproblem(e.g.,Fe dergruenandTzur [ 24 ]andWagelmans,vanHoesel,andKolen[ 82 ])hasreducedthecomplexityof thisproblemfromthe O ( T 2 )boundto O ( T log T )(oreven O ( T )incertainspecial cases).Theseapproaches,however,relyonanimportantpropert ythatholdsfor

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15 theELSP,andthispropertystatesthatthecumulativedemand satisedaswe increasethenumberofperiodsinaprobleminstanceisnondecr easing.Thatis,for theELSP,thetotaldemandsatisedinatwo-periodproblemisat leastasgreat asthatsatisedinaone-periodproblem(wheredemandinperiod 1isthesame inbothprobleminstances).Interestingly,wecanshowthatthis property doesnot hold ingeneralfortheUOSPproblem.Assumingthattheholdingcostine very periodequalszero,weintroducethefollowingdataforathr ee-periodproblem: Table2{1:Counterexampleillustratingdecreasingcumulati vedemandsatisfaction. Period SetupCost ProductionCost Demand UnitRevenue 1 $50 $1.50 20 $1.80 1 $50 $1.25 20 $4.00 1 $1,000 $1.20 10 $10.00 Considertheperiod1problemalone.Ifwesetupandsatisfyall20 unitsof demand,therevenueequals$36,whilethesetupplusvariable productioncost equals$80.Thuswesatisfyzerounitsofdemandintheperiod1p roblem.Inthe period1+period2problem,anoptimalsolutionsatises20units inperiods1and 2,usingthesetupinperiod1,foratotalof40unitsofdemandsat ised.Finally, fortheproblemcontainingperiods1,2,and3,itisoptimalt osetupinperiod2 andsatisfythe30unitsofdemandinperiods2and3.Thisexampl eillustrates whywecannotapplymethodspreviouslydevelopedtoreduceth ecomplexityof ELSPto O ( T log T )inaneorttoreducethecomplexityofouralgorithmto,say, O ( mT log T ),sincecumulativedemandsatisedisnotnecessarilynondecreasi ng fortheUOSPproblem(notethatitispossibletoprovideasimilar example underwhichcumulativedemandsatisedfromperiod t to T isnotnecessarily nondecreasingaswemovebackwardsintime,or,as t decreases). 2.3 OSPModels-LimitedProductionCapacity Wenowturnourattentiontothecapacitatedversionofourmod el.We investigatenotonlytheOSPmodelasformulatedabove,butal socertainspecial

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16 casesandrestrictionsofthismodelthatareofbothpractical andtheoretical interest.Inparticular,weconsiderthespecialcaseinwhichno xeddelivery chargesexist(i.e.,thecaseinwhichallxeddeliverycharge( F mt )parameters equalzero).WedenotethisversionofthemodelastheOSP-NDC. Wealsoexplore contextsinwhichcustomersdonotpermitpartialdemandsatisf action.Thisis arestrictedversionoftheOSPinwhichthecontinuous v mt variablesmustequal thebinarydelivery-chargeforcing( z mt )variablevalues,andcanthereforebe substitutedoutoftheformulation;letOSP-ANDdenotethisversi onofthemodel (whereANDimpliesall-or-nothingdemandsatisfaction).Observ ethatforthe OSP-ANDmodelwecanintroduceanewrevenueparameter R mt r mt d mt ,where thetotalrevenuefromorder m inperiod t mustnowequal R mt z mt .Table 2{2 denesournotationwithrespecttothedierentvariantsofthe OSPproblem. Table2{2:Classicationofmodelspecialcasesandrestrictions. FixedDelivery PartialOrder Model ChargesExist SatisfactionAllowed OSP Y Y OSP-NDC N Y OSP-AND U N Y=Yes;N=No.U:Modelandsolutionapproachesunaectedbythisassumptio n. Wedistinguishbetweenthesemodelvariantsnotonlybecausethe ybroaden themodel'sapplicabilitytodierentcontexts,butalsobecau setheycansubstantiallyaectthemodel'sformulationsizeandcomplexity, aswenextbriey discuss.Let M § = P Tt =1 j M ( t ) j denotethetotalnumberofcustomerordersover the T -periodhorizon,where j M ( t ) j isthecardinalityoftheset M ( t ).Notethat formulation[OSP]contains M § + T binaryvariablesand M § +2 T constraints,not includingthebinaryandnonnegativityconstraints.TheOSPNDCmodel,onthe otherhand,inwhich F mt =0forallorder-period( m;t )combinations,allowsusto replaceeach z mt variableontheright-hand-sideofconstraintset( 2.4 )witha1,and eliminatethesevariablesfromtheformulation.TheOSP-NDCm odelcontainsonly

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17 T binaryvariablesandthereforerequires M § fewerbinaryvariablesthan[OSP], asignicantreductioninproblemsizeandcomplexity.IntheOS P-ANDmodel, customersdonotallowpartialdemandsatisfaction,andsowereq uire v mt = z mt forallorder-period( m;t )combinations;wecanthereforeeliminatethecontinuous v mt variablesfromtheformulation.WhiletheOSP-AND,liketheOS P,contains M § + T binaryvariables,itrequires M § fewertotalvariablesthan[OSP]asaresult ofeliminatingthe v mt variables.Table 2{3 summarizesthesizeofeachofthese variantsoftheOSPwithrespecttothenumberofconstraints,bi naryvariables, andtotalvariables. Table2{3:Problemsizecomparsionforcapacitatedversionsof theOSP. OSP OSP-NDC OSP-AND NumberofConstraints a M § +2 T M § +2 T 2 T NumberofBinaryVariables M § + T T M § + T NumberofTotalVariables 2 M § +3 T M § +3 T M § +3 T a Binaryrestrictionandnonnegativityconstraintsarenoti ncluded. Basedontheinformationinthistable,wewouldexpecttheOSP andOSPANDtobesubstantiallymorediculttosolvethantheOSP-NDC.Aswewi ll showinSection 2.4 ,theOSP-ANDactuallyrequiresthegreatestamountof computationtimeonaverage,whiletheOSP-NDCrequiresthel east. NotethattheOSP-ANDisindierenttowhetherxeddeliverycharg esexist, sincewecansimplyreducethenetrevenueparameter, R mt r mt d mt ,bythexed delivery-chargevalue F mt ,withoutlossofgenerality.IntheOSP-ANDthen,the netrevenuereceivedfromanorderequals R mt z mt ,andwethusinterpretthe z mt variablesasbinary\orderselection"variables.Incontrast, intheOSP,thepurpose ofthebinary z mt variablesistoforceustoincurthexeddeliverychargeifwe satisfyanyfractionoforder m inperiod t .Inthismodelwethereforeinterpretthe z mt variablesasxeddelivery-chargeforcingvariables,sinceth eirobjectivefunction coecientsarexeddeliverycosttermsratherthannetrevenue terms,asinthe OSP-AND.NotealsothatsinceboththeOSP-NDCandtheOSP-ANDrequire

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18 onlyonesetoforderselectionvariables(thecontinuous v mt variablesfortheOSPNDCandthebinary z mt variablesfortheOSP-AND),theirlinearprogramming relaxationformulationswillbeidentical(sincerelaxingt hebinary z mt variables isequivalenttosetting z mt = v mt ).TheOSPlinearprogrammingrelaxation formulation,ontheotherhand,explicitlyrequiresbothth e v mt and z mt variables, resultinginalargerLPrelaxationformulationthanthatfor theOSP-NDCand theOSP-AND.Thesedistinctionswillplayanimportantroleinin terpretingthe dierenceinourabilitytoobtainstrongupperboundsontheop timalsolution valuefortheOSPandtheOSP-ANDinSection 2.4.3 .Wenextdiscusssolution methodsfortheOSPandtheproblemvariantswehavepresented 2.3.1 OSPSolutionMethods TosolvetheOSP,wemustdecidewhichorderstoselectand,among the selectedorders,howmuchoftheorderwewillsatisfywhileobeyi ngcapacity limits.WecanshowthatthisproblemisNP-Hardthroughareducti onfromthe capacitatedlot-sizingproblemasfollows.Ifweconsiderthesp ecialcaseofthe OSPinwhich P jt =1 C t P jt =1 P m 2 M ( t ) d mt for j =1,..., T (whichimpliesthat satisfyingallordersisfeasible)andmin t =1 ;:::;T;m 2 M ( t ) f r mt g max t =1 ;:::;T f S t g +max t =1 ;:::;T f p t g + P T ¡ 1 t =1 h t (whichimpliesthatitisprotabletosatisfyallordersinever yperiod), thentotalrevenueisxedandtheproblemisequivalenttoaca pacitatedlot-sizing problem,whichisanNP-Hardoptimizationproblem(seeFlorian andKlein[ 28 ]). GiventhattheOSPisNP-Hard,wewouldliketondanecientmetho dfor obtaininggoodsolutionsforthisproblem.Asourcomputation altestresultsin Section 2.4 latershow,wewereabletondoptimalsolutionsusingbranch-an dboundformanyofourrandomlygeneratedtestinstances.Whilet hisindicates thatthemajorityofprobleminstancesweconsideredwerenott erriblydicultto solve,therewerestillmanyinstancesinwhichanoptimalsoluti oncouldnotbe foundinreasonablecomputingtime.Basedonourcomputationa ltestexperience

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19 ineectivelysolvingprobleminstancesviabranch-and-bound usingtheCPLEX 6.6solver,wefocusonstrongLPrelaxationsfortheOSPthatpr ovidequality upperboundsonoptimalnetprotquickly,andoftenenablesol utionviabranchand-boundinacceptablecomputingtime.Forthoseproblemst hatcannotbe solvedviabranch-and-bound,weemployseveralcustomizedheu risticmethods, whichwediscussinSection 2.3.3 .Beforewediscusstheheuristicsusedtoobtain lowerboundsfortheOSP,werstpresentourreformulationstrat egy,whichhelps tosubstantiallyimprovetheupperboundprovidedbythelinea rprogramming relaxationoftheOSP. 2.3.2 StrengtheningtheOSPFormulation Thissectionpresentsanapproachforprovidinggoodupperbou ndsonthe optimalnetprotfortheOSP.Inparticular,wedescribetwoLP relaxationsfor theOSP,bothofwhichdierfromtheLPrelaxationobtainedby simplyrelaxing thebinaryrestrictionsof[OSP](constraintset( 2.6 ))inSection 2.2 .Wewillrefer tothissimpleLPrelaxationof[OSP]asOSP-LP,todistinguisht hisrelaxation fromthetwoLPrelaxationapproachesweprovideinthissecti on. ThetwoLPrelaxationformulationswenextconsiderarebasedo nareformulationstrategydevelopedfortheUOSP.InChapter 3 ,wewillpresenta\tight" formulationofasimilarproblemtotheUOSP,forwhichweshowth attheoptimal LPrelaxationsolutionvalueequalstheoptimal(mixedinteg er)UOSPsolution value.Wediscussthisreformulationstrategyingreaterdetai lbyrstproviding atightlinearprogrammingrelaxationfortheUOSP.Werstnote thatforthe UOSP,anoptimalsolutionexistssuchthatweneversatisfypartofa norder;i.e., v mt equalseither0or1.Thuswecansubstitutethe v mt variablesoutof[OSP]by setting v mt = z mt forall t and m 2 M ( t ). Nextobservethatsince I t = P tj =1 x j ¡ P tj =1 P m 2 M ( j ) d mj z mj ,wecaneliminate theinventoryvariablesfrom[OSP]viasubstitution.Afterint roducinganew

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20 variableproductionandholdingcostparameter, c t ,where c t p t + P Tj = t h j ,the objectivefunctionoftheUOSPcanberewrittenas maximize T X j =1 X m 2 M ( j ) R mj z mj + T X t =1 h t 0@ T X j =1 X m 2 M ( j ) d mj z mj 1A ¡ T X t =1 ( S t y t + c t x t )(2.10) Wenextdene mt asanadjustedrevenueparameterfororder m inperiod t ,where mt = P Tj = t h j + R mt .Ourreformulationprocedurerequirescapturing theexactamountofproductionineachperiodallocatedtoev eryorder.Wethus dene x mtj asthenumberofunitsproducedinperiod t usedtosatisfyorder m in period j ,for j t ,andreplaceeach x t with P Tj = t P m 2 M ( j ) x mtj .Thefollowing formulationprovidesthe\strong"linearprogrammingrelax ationoftheUOSP. [UOSP 0 ] maximize: T P j =1 P m 2 M ( j ) mj d mj z mj ¡ T P t =1 S t y t + c t T P j = t P m 2 M ( j ) x mtj (2.11) subjectto: j P t =1 x mtj ¡ d mj z mj =0 j =1 ;:::;T;m 2 M ( j ) ; (2.12) P m 2 M ( j ) d mj y t ¡ P m 2 M ( j ) x mtj 0 t =1 ;:::;T;j = t;:::;T; (2.13) ¡ z mj ¡ 1 j =1 ;:::;T;m 2 M ( j ) ; (2.14) y t ;x mtj ;z mj 0 t =1 ;:::;T; j = t;:::;T;m 2 M ( j ) : (2.15) Notethatsinceapositivecostexistsforsetups,wecanshowthatthec onstraint y t 1isunnecessaryintheaboverelaxation,andsoweomitthisconst raintfrom therelaxationformulation.Itisstraightforwardtoshowtha t[UOSP 0 ]withthe additionalrequirementsthatall z mj and y t arebinaryvariablesisequivalenttoour [OSP]whenproductioncapacitiesareinnite.(Wewillalsosho winChapter 3 that

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21 bydisaggregatingthesetupforcingconstraints( 2.13 ),theresultingformulationhas zerointegralitygapthroughadualsolutionapproach.) ToobtaintheLPrelaxationfortheOSP(whenproductioncapa citiesare nite),weaddnitecapacityconstraintsto[UOSP 0 ]byforcingthesumof x mtj overall j t andall m 2 M ( j )tobelessthantheproductioncapacity C t in period t .Thatis,wecanaddthefollowingconstraintsetto[UOSP]toobta inan equivalentLPrelaxationfortheOSP: T X j = t X m 2 M ( j ) x mtj C t t =1 ;:::;T: (2.16) NotethatthisLPrelaxationapproachisvalidforallthreeva riantsoftheOSP:the generalOSP,theOSP-NDC,andtheOSP-AND.Observethattheabove constraint canbestrengthenedbymultiplyingtheright-hand-sidebythe setupforcing variable y t .Toseehowthisstrengthenstheformulation,notethatconstra intset ( 2.13 )in[UOSP 0 ]impliesthat T X j = t X m 2 M ( j ) x mtj T X j = t 0@ X m 2 M ( j ) d mj 1A y t t =1 ;:::;T: Tostreamlineournotation,wedenethefollowing.Let X tT = P Tj = t P m 2 M ( j ) x mtj and D tT = P Tj = t P m 2 M ( j ) d mj for t =1 ;:::;T denoteaggregatedproduction variablesandorderamounts,respectively.Constraintset( 2.16 )canberewrittenas X tT C t t =1 ;:::;T; andtheaggregateddemandforcingconstraints( 2.13 )cannowbewrittenas X tT D tT ¢ y t .Ifwedonotmultiplytheright-hand-sideofcapacityconstra intset ( 2.16 )bytheforcingvariable y t ,theformulationallowssolutions,forexample,such that X tT = C t forsome t ,while X tT equalsonlyafractionof D tT .Insuchacase, theforcingvariable y t takesthefractionalvalue X tT D tT ,andweonlyabsorbafraction

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22 ofthesetupcostinperiod t .Multiplyingtheright-hand-sideof( 2.16 )by y t ,on theotherhand,wouldforce y t = X tT C t =1insuchacase,leadingtoanimproved upperboundontheoptimalsolutionvalue.Wecanthereforestr engthentheLP relaxationsolutionthatresultsfromaddingconstraintset( 2.16 )byinsteadusing thefollowingcapacityforcingconstraints. X tT min f C t ;D tT g¢ y t t =1 ;:::;T: (2.17) Notethatinthecapacitatedcasewenowexplicitlyrequirestat ingthe y t 1 constraintsintheLPrelaxation,sinceitmayotherwisebeprot abletoviolateproductioncapacityinordertosatisfyadditionalorders.Werefe rtotheresultingLP relaxationwiththeseaggregatedsetupforcingconstraintsas the[ASF]formulation, whichweformulateasfollows.[ASF] maximize: T P j =1 P m 2 M ( j ) mj d mj z mj ¡ T P t =1 S t y t + c t T P j = t P m 2 M ( j ) x mtj subjectto:Constraints( 2.12 { 2.15 2.17 ) y t 1 t =1 ;:::;T: (2.18) WecanfurtherstrengthentheLPrelaxationformulationbydi saggregating thedemandforcingconstraints( 2.13 )(seeErlenkotter[ 23 ],whousesthisstrategy fortheuncapacitatedfacilitylocationproblem).Thiswil lforce y t tobeatleast asgreatasthemaximumvalueof x mtj d mj forall j = t;:::;T and m 2 M ( j ).The resultingDisaggregatedSetupForcing(DASF)LPrelaxationis formulatedas follows.[DASF] maximize: T P j =1 P m 2 M ( j ) mj d mj z mj ¡ T P t =1 S t y t + c t T P j = t P m 2 M ( j ) x mtj

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23 subjectto:Constraints( 2.12 2.14 2.15 2.17 2.18 ) x mtj d mj y t t =1 ;:::;T; (2.19) j = t;:::;T; m 2 M ( j ) : EachoftheLPrelaxationswehavedescribedprovidessomevalu einsolving thecapacitatedversionsoftheOSP.BoththeOSP-LPandASFrel axationscan besolvedveryquickly,andtheyfrequentlyyieldhighqualit ysolutions.TheDASF relaxationfurtherimprovestheupperboundontheoptimalso lutionvalue.But astheproblemsizegrows(i.e.,thenumberofordersperperio dorthenumber oftimeperiodsincreases),[DASF]becomesintractable,evenv iastandardlinear programmingsolvers.Wepresentresultsforeachoftheserelaxat ionapproachesin Section 2.4 .Beforedoingthis,however,wenextdiscussmethodsfordeterm ining goodfeasiblesolutions,andthereforelowerbounds,fortheOSP viaseveral customizedheuristicsolutionprocedures. 2.3.3 HeuristicSolutionApproachesforOSP Whilethemethodsdiscussedintheprevioussubsectionoftenprov idestrong upperboundsontheoptimalsolutionvaluefortheOSP(andits variants),we cannotguaranteetheabilitytosolvethisprobleminreasonab lecomputingtime usingbranch-and-boundduetothecomplexityoftheproblem. Wenextdiscuss threeheuristicsolutionapproachesthatallowustoquicklyg eneratefeasible solutionsforOSP.AsourresultsinSection 2.4 report,usingacompositesolution procedurethatselectsthebestsolutionamongthosegeneratedb ythethree heuristicsolutionapproachesprovidedfeasiblesolutionswit hobjectivefunction values,onaverage,within0.67%oftheoptimalsolutionvalue .Wedescribeour threeheuristicsolutionapproachesinthefollowingthreesub sections.

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24 2.3.3.1 LagrangianRelaxationBasedHeuristic Lagrangianrelaxation(Georion[ 32 ])isoftenusedformixedintegerprogrammingproblemstoobtainstrongerupperbounds(formaximizat ionproblems)than providedbytheLPrelaxation.AswediscussedinSection 2.3.2 ,ourstrengthened linearprogrammingformulationstypicallyprovideverygo odupperboundson theoptimalsolutionvalueoftheOSP.Moreover,aswelaterdi scuss,ourchoiceof relaxationresultsinaLagrangiansubproblemforwhichwecan ndanoptimal extremepointsolutionequivalenttothesolutionfoundforou rLPrelaxation.This impliesthattheupperboundprovidedbyourLagrangianrela xationschemewill notprovidebetterboundsthanourLPrelaxation.Ourpurpose forimplementing aLagrangianrelaxationheuristic,therefore,isstrictlyto obtaingoodfeasiblesolutionsusingaLagrangian-basedheuristic.Becauseofthisweo mitcertaindetails oftheLagrangianrelaxationalgorithmandimplementation ,anddescribeonlythe essentialelementsofthegeneralrelaxationschemeandhowweo btainaheuristic solutionateachiterationoftheLagrangianalgorithm. UnderourLagrangianrelaxationscheme,weadd(redundant)co nstraints oftheform x t My t ;t =1 ;:::;T to[OSP](where M issomelargenumber), eliminatetheforcingvariable y t fromtheright-handsideofthecapacity/setup forcingconstraints( 2.3 ),andthenrelaxtheresultingmodiedcapacityconstraint (withoutthe y t multiplierontheright-handside)ineachperiod.TheLagran gian relaxationsubproblemisthensimplyanuncapacitatedOSP(or UOSP)problem. AlthoughtheLagrangianmultipliersintroducethepossibilit yofnegativeunit productioncostsintheLagrangiansubproblem,weretainthec onvexityofthe objectivefunction,andallpropertiesnecessaryforsolvingt heUOSPproblemvia ashortestpathnetworkapproachstillhold(seeSection 2.2.2 ).Wecantherefore solvetheLagrangiansubproblemsinpolynomialtime.Becausew ehaveatight formulationoftheUOSP(aswewillproveinChapter 3 ),thisimpliesthatthe

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25 Lagrangiansolutionwillnotprovidebetterupperboundstha ntheLPrelaxation. Wedo,however,usethesolutionoftheLagrangiansubproblemat eachiteration ofasubgradientoptimizationalgorithm(seeFisher[ 26 ])asastartingpointfor heuristicallygeneratingafeasiblesolution,whichservesasa candidatelowerbound ontheoptimalsolutionvalueforOSP. Observethatthesubproblemsolutionfromthisrelaxationwill satisfyall constraintsoftheOSPexceptfortherelaxedcapacityconstra ints( 2.3 ).We thereforecallafeasiblesolutiongenerator(FSG)ateachstep ofthesubgradient algorithm,whichcantakeanystartingcapacity-infeasible( butotherwisefeasible) solutionandgenerateacapacity-feasiblesolution.(Wealsouse thisFSGinour otherheuristicsolutionschemes,aswelaterdescribe.)TheFSGw orksinthree mainphases.PhaseIrstconsidersperformingadditionalproduct ionsetups (beyondthoseprescribedbythestartingsolution)totrytoacco mmodatethe desiredproductionlevelsandorderselectiondecisionsprovi dedinthestarting solution,whileobeyingproductioncapacitylimits.Thatis,w econsidershifting productionfromperiodsinwhichcapacitiesareviolatedto periodsinwhichno setupwasoriginallyplannedinthestartingsolution.Itispossi ble,however,that westillviolatecapacitylimitsafterPhaseI,sincewedonotel iminateanyorder selectiondecisionsinPhaseI. InPhaseII,afterdeterminingwhichperiodswillhavesetupsi nPhaseI,we considerthosesetupperiodsinwhichproductionstillexceedsc apacityand,for eachsuchsetupperiod,indextheorderssatisedfromproduction inthesetup periodinnondecreasingorderofcontributiontoprot.Forea chperiodwith violatedcapacity,inincreasingprotabilityindexorder,w eshiftorderstoan earlierproductionsetupperiod,iftheorderremainsprotab leandsuchanearlier productionsetupperiodexistswithenoughcapacitytoaccomm odatetheorder. Otherwiseweeliminatetheorderfromconsideration.Ifremov ingtheorderfrom

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26 thesetupperiodwillleaveexcesscapacityinthesetupperiodu nderconsideration, weconsidershiftingonlypartoftheordertoapriorproductio nperiod;wealso considereliminatingonlypartoftheorderwhencustomersdon otrequireall-ornothingordersatisfaction.Thisprocessiscontinuedforeach setupperiodinwhich productioncapacityisviolateduntiltotalproductionint heperiodsatisesthe productioncapacitylimit.Followingthissecondphaseofthe algorithm,wewill havegeneratedacapacity-feasiblesolution.Inthethirdand nalphase,wescan allproductionperiodsforavailablecapacityandassignaddi tionalprotableorders thathavenotyetbeenselectedtoanyexcesscapacityifpossible .TheChapter AppendixinSection 2.6 containsadetaileddescriptionoftheFSGalgorithm. 2.3.3.2 GreatestUnitProtHeuristic Ournextheuristicsolutionprocedureismotivatedbyanappro achtakenin severalwell-knownheuristicsolutionapproachesfortheELSP .Inparticular,we useasimilar\myopic"approachtothoseusedintheSilver-Meal[ 72 ]andLeast UnitCost(seeNahmias[ 56 ])heuristics.Theseheuristicsproceedbyconsidering aninitialsetupperiod,andthendeterminingthenumberofco nsecutiveperiod demands(beginningwiththeinitialsetupperiod)thatprodu cethelowestcostper period(Silver-Meal)orperunit(LeastUnitCost)whenallocat edtoproductionin thesetupperiod.Thenextperiodconsideredforasetupistheon eimmediately followingthelastdemandperiodassignedtothepriorsetup;the heuristicsproceed untilalldemandhasbeenallocatedtosomesetupperiod.Ourap proachdiers fromtheseapproachesinthefollowingrespects.Sinceweareco ncernedwiththe protfromorders,wetakeagreatestprotratherthanalowestcost approach. Wealsoallowforacceptingorrejectingvariousorders,which impliesthatweneed onlyconsiderthoseordersthatareprotablewhenassigningorde rstoaproduction period.Moreover,wecanchoosenottoperformasetupifnoselec tionoforders

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27 producesapositiveprotwhenallocatedtothesetupperiod.Fi nally,weapply our\greatestunitprot"heuristicinacapacitatedsetting,wh ereasamodication oftheSilver-MealandLeastUnitCostheuristicsisrequiredfor applicationtothe capacitatedlot-sizingproblem. Ourbasicapproachbeginsbyconsideringasetupinperiod t (where t initially equals1)andcomputingthemaximumprotperunitofdemandsat isedinperiod t usingonlythesetupinperiod t .Notethat,givenasetupinperiod t ,wecansort ordersinperiods t;:::;T innonincreasingorderofcontributiontoprotbased solelyonthevariablecostsincurredwhenassigningtheorderto thesetupinperiod t (fortheOSPwhenxeddeliverychargesexistwemustalsosubtract thiscost fromeachorder'scontributiontoprot).Ordersarethenall ocatedtothesetup innonincreasingorderofcontributiontoprotuntileithert hesetupcapacityis exhaustedornoadditionalattractiveordersexist.Aftercomp utingthemaximum protperunitofdemandsatisedinperiod t usingonlythesetupinperiod t ,we thencomputethemaximumprotperunitsatisedinperiods t;:::;t + j using onlythesetupinperiod t ,for j =1 ;:::;j 0 ,whereperiod j 0 istherstperiodinthe sequencesuchthatthemaximumprotperunitinperiods t;:::;t + j 0 isgreater thanorequaltothemaximumprotperunitinperiods t;:::;t + j 0 +1.The capacity-feasiblesetofordersthatleadstothegreatestprot perunitinperiods t;:::;j 0 usingthesetupinperiod t isthenassignedtoproductioninperiod t assumingthemaximumprotperunitispositive.Ifthemaximumpr otperunit foranygivensetupperioddoesnotexceedzero,however,wedo notassignany orderstothesetupandthuseliminatethesetup. Sinceweconsideracapacity-constrainedproblem,wecaneith erconsider period j 0 +1(asisdoneintheSilver-MealandLeastUnitCostheuristics)or period t +1asthenextpossiblesetupperiodfollowingperiod t .Weuseboth approachesandretainthesolutionthatproduceshighernetp rot.Notethatifwe

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28 considerperiod t +1asthenextpotentialsetupperiodfollowingperiod t ,wemust keeptrackofthoseordersinperiods t +1andhigherthatarealreadyassignedto period t (andprior)production,sincethesewillnotbeavailablefora ssignmentto period t +1production.Finally,afterapplyingthisgreatestunitpr otheuristic, weapplyPhaseIIIoftheFSGalgorithm(seetheChapterAppendix inSection 2.6 )totheresultingsolution,inaneorttofurtherimprovethehe uristicsolution valuebylookingforopportunitiestoeectivelyuseanyunused setupcapacity. 2.3.3.3 LinearProgrammingRoundingHeuristic OurthirdheuristicsolutionapproachusestheLPrelaxationsol utionasa startingpointforalinearprogrammingroundingheuristic.W efocusonrounding thesetup( y t )andorderselection( z mt )variablesthatarefractionalintheLP relaxationsolution(roundingtheorderselectionvariables isnot,however,relevant fortheOSP-NDCproblem,sincethe z mt variablesdonotexistinthisspecialcase). Werstconsiderthesolutionthatresultsbysettingall(non-zero )fractional y t and z mt variablesfromtheLPrelaxationsolutiontoone.Wethenappl ythesecond andthirdphasesofourFSGalgorithmtoensureacapacityfeasib lesolution,and tosearchforunselectedorderstoallocatetoexcessproductio ncapacityinperiods wherethesetupvariablewasroundedtoone. Wealsouseanalternativeversionofthisprocedure,wherewero undupthe setupvariableswithvaluesgreaterthanorequalto0.5inthe LPrelaxationsolution,androunddownthosewithvalueslessthan0.5.Againwesu bsequently applyPhasesIIandIIIoftheFSGalgorithmtogenerateagoodc apacity-feasible solution(ifthemaximumsetupvariablevaluetakesavaluebet ween0and0.5,we rounduponlythesetupvariablewiththemaximumfractionalv ariablevalueand applyPhasesIIandIIIoftheFSGalgorithm).Finally,basedon ourdiscussionin

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29 Section 2.3.2 ,wehaveachoiceofthreedierentformulationsforgenerati ngLPrelaxationstartingsolutionsfortheroundingprocedure:form ulation[OSP](Section 2.2 ),[ASF](Section 2.3.2 ),or[DASF](Section 2.3.2 ).Asourcomputationalresults laterdiscuss,startingwiththeLPrelaxationsolutionfromthe[ DASF]formulation providessolutionsthatare,onaverage,farsuperiortothosep rovidedusingthe otherLPrelaxationsolutions.However,thesizeofthisLPrelax ationalsofar exceedsthesizeofourotherLPrelaxationformulations,maki ngthisformulation impracticalasproblemsizesbecomelarge.Weusetheresulting LPrelaxation solutionundereachoftheseformulationsandapplytheLProun dingheuristicto allthreeoftheseinitialsolutionsforeachprobleminstance, retainingthesolution thatprovidesthehighestnetprot. 2.4 ScopeandResultsofComputationalTests Thissectiondiscussesabroadsetofcomputationaltestsintended toevaluate ourupperboundingandheuristicsolutionapproaches.Ourresul tsfocuson gaugingboththeabilityofthedierentLPrelaxationspresen tedinSection 2.3.2 toprovidetightupperboundsonoptimalprot,andtheperfor manceofthe heuristicproceduresdiscussedinSection 2.3.3 inprovidinggoodfeasiblesolutions. Section 2.4.1 nextdiscussesthescopeofourcomputationaltests,whileSection s 2.4.2 and 2.4.3 reportresultsfortheOSP,OSP-NDC,andOSP-ANDversionsof theproblem. 2.4.1 ComputationalTestSetup Thissectionpresentstheapproachweusedtocreateatotalof3, 240randomly generatedprobleminstancesforcomputationaltesting,whic hconsistof1,080 problemsforeachoftheOSP,OSP-NDC,andOSP-ANDversionsofthe problem. Withineachproblemversion(OSP,OSP-NDC,andOSP-AND),weusedt hree dierentsettingsforthenumberofordersperperiod,equalto 25,50,and200.In ordertocreateabroadsetoftestinstances,weconsideredarange ofsetupcost

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30 values,productioncapacitylimits,andperunitorderrevenu es. 2 Table 2{4 providesthesetofdistributionsusedforrandomlygeneratingthe separametervalues inourtestcases.Thetotalnumberofcombinationsofparameter distribution settingsshowninTable 2{4 equals36,andforeachuniquechoiceofparameter distributionsettingswegenerated10randomprobleminstance s.Thisproduced atotalof360probleminstancesforeachofthethreevaluesof thenumberof ordersperperiod(25,50,and200),whichequals1,080probl eminstancesforeach problemversion.Asthedistributionsusedtogenerateproducti oncapacitiesin Table2{4:Probabilitydistributionsusedforgeneratingpro bleminstances. Numberof Distributionsused a Parameter Distribution forParameter Settings Generation Setupcost(varies 3 U[350,650] fromperiod-to-period) U[1750,3250] U[3500,6500] Perunitperperiodholdingcost b 2 0 : 15 £ p= 50 0 : 25 £ p= 50 Productioncapacityina 3 U[ d= 3 ¡ : 05 d d= 3+ : 05 d ] period(variesfrom U[ d= 2 ¡ : 1 d d= 2+ : 1 d ] period-to-period) c U[ d ¡ : 15 d d + : 15 d ] Perunitorderrevenue(varies 2 U[28,32] fromorder-to-order) U[38,42] a U[a,b]denotesauniformdistributionontheinterval[a,b] b p denotesthevariableproductioncost.Weassume50workingw eeksinoneyear. c d denotestheexpectedper-periodtotaldemand,whichequalst hemeanofthe distributionofordersizesmultipliedbythenumberoforde rsperperiod. Table 2{4 indicate,wemaintainaconstantratioofaverageproduction capacityper periodtoaveragetotaldemandperperiod.Thatis,wemaintai nthesameaverage ordersize(averageof d mt values)acrosseachofthesetestcases,buttheaverage capacityperperiodforthe200-orderproblemsetsisfourtim esthatofthe50-order 2 Thesethreeparametersappearedtobethemostcriticalonesto varywidely inordertodeterminehowrobustoursolutionmethodsweretopr oblemparameter variation.

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31 problemsetsandeighttimesthatofthe25-orderproblems.Bec ausethetotal numberofavailableordersperperiodtendstostronglyaectt herelativequalityof oursolutions(aswelaterdiscuss),wereportperformancemeasur esacrossalltest casesandalsoindividuallywithinthe25,50,and200orderpro blemsets. Inordertolimitthescopeofourcomputationalteststoamanag eable size,wechosetolimitthevariationofcertainparametersacr ossallofthetest instances.Theperunitproductioncostfollowedadistribution ofU[20,30]forall testinstances(whereU[ a;b ]denotesaUniformdistributionontheinterval[ a;b ]), andallprobleminstancesuseda16-periodplanninghorizon.W ealsousedan ordersizedistributionofU[10,70]foralltestproblems(i.e., the d mt valuesfollowa uniformdistributionon[10,70]).FortheOSP,thedistributi onusedforgenerating xeddeliverychargeswasU[100,600]. 3 Byincludingawiderangeoflevelsof productioncapacity,setupcost,andordervolumes,wetestedase tofproblems whichwouldfairlyrepresentavarietyofactualproductionsc enarios. Observethatthetwochoicesfordistributionsusedtogenerate perunit orderrevenuesuserelativelynarrowranges.Giventhatthedi stributionusedto generatevariableproductioncostisU[20,30],therstofthesep erunitrevenue distributions,U[28,32],producesprobleminstancesinwhicht hecontributionto prot(aftersubtractingvariableproductioncost)isquitesma ll|leadingtofewer attractiveordersafterconsideringsetupandholdingcosts.Th eseconddistribution, U[38,42],providesamoreprotablesetoforders.Wechosetokeep theseranges verynarrowbecauseourpreliminarytestresultsshowedthatati ghterrange,which 3 Weperformedcomputationaltestswithsmallerper-orderdeli verycharges,but theresultswerenearlyequivalenttothosepresentedfortheOS P-NDCinTable 2.4.2 ,sincetheprotabilityoftheordersremainedessentiallyunch anged.Aswe increasedtheaveragedeliverychargeperorder,moreorders becameunprotable, creatingprobleminstancesthatwerequitedierentfromtheO SP-NDCcase.

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32 implieslessperunitrevenuedierentiationamongorders,pro ducesmoredicult probleminstances.Thoseprobleminstanceswithagreaterrange ofperunit revenuevaluesamongorderstendedtobesolvedinCPLEXviabr anch-and-bound muchmorequicklythanthosewithtightranges,andwewishedtoe nsurethatour computationaltestsreectedmoredicultprobleminstances. Atighterrangeofunitrevenuesproducesmoredicultproble minstances duetotheabilitytosimply`swap'orderswithidenticalunitr evenuesinthe branch-and-boundalgorithm,leadingtoalternativeoptim alsolutionsatnodesin thebranch-and-boundtree.Forexample,ifanorder m inperiod t issatisedat thecurrentnodeinthebranch-and-boundtree,andsomeother order m 0 isnot satised,but r mt = r m 0 t and d mt = d m 0 t ,thenasolutionwhichsimplyswapsorders m and m 0 hasthesameobjectivefunctionastherstsolution,andnoimpro vement intheboundoccursasaresultofthisswap.So,wefoundthatwhe ntheproblem instancehaslessdierentiationamongorders,thebranch-andboundalgorithmcan takesubstantiallylonger,leadingtomoredicultprobleminst ances.Barnhartet al.[ 7 ]andBalakrishnanandGeunes[ 6 ]observedsimilarswappingphenomenain branch-and-boundformachineschedulingandsteelproductio nplanningproblems, respectively. Alllinearandmixedintegerprogramming(MIP)formulations weresolved usingtheCPLEX6.6solveronanRS/6000machinewithtwoPowerP C(300MHz) CPUsand2GBofRAM.Wewillrefertothebestsolutionprovidedbyt heCPLEX branch-and-boundalgorithmasthe MIPsolution .Theremainingsubsections summarizeourresults.Section 2.4.2 reportstheresultsofourcomputational experimentsfortheOSP-NDCandtheOSP,andSection 2.4.3 presentsthendings fortheOSP-AND(all-or-nothingordersatisfaction)problem.F ortheOSPANDprobleminstancesdiscussedinSection 2.4.3 ,weassumethattherevenue parametersprovidedrepresentrevenuesinexcessofxeddeliv erycharges(sincewe

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33 alwayssatisfyallornoneofthedemandfortheOSP-AND,thisiswi thoutlossof generality). 2.4.2 ResultsfortheOSPandtheOSP-NDC RecallthattheOSPassumesthatwehavetheexibilitytosatisfya ny proportionofanorderinanyperiod,aslongaswedonotexcee dtheproduction capacityintheperiod.Becauseofthis,whennoxeddeliverych argesexist,the onlybinaryvariablesintheOSP-NDCcorrespondtothe T binarysetupvariables, andsolvingtheseprobleminstancestooptimalityusingCPLEX'sM IPsolverdid notprovetobeverydicult.Thesameisnotnecessarilytrueofth eOSP-AND, aswelaterdiscussinSection 2.4.3 .Surprisingly,theOSP(whichincludesabinary xeddelivery-chargeforcing( z mt )variableforeachorder-periodcombination)was notsubstantiallycomputationallychallengingeither.Allof theOSP-NDCandall buttwooftheOSPinstancesweresolvedoptimallyusingbranchand-boundwithin theallottedbranch-and-boundtimelimitofonehour.Event houghweareableto solvetheOSPandOSP-NDCprobleminstancesusingCPLEXwithrela tiveease, westillreporttheupperboundsprovidedbythedierentLPrel axationsforthese problemsinthissection.Thisallowsustogaininsightregard ingthestrengthof theserelaxationsasproblemparameterschange,withknowle dgeoftheoptimal mixedintegerprogramming(MIP)solutionvaluesasabenchma rk. Table 2.4.2 presentsoptimalitygapmeasuresbasedonthesolutionvalues resultingfromtheLP(OSP-LP)relaxationupperbound,theagg regatedsetup forcing(ASF)relaxationupperbound,andthedisaggregatedse tupforcing(DASF) relaxationupperboundfortheOSP-NDCandOSPprobleminstanc es.Thelast rowofthetableshowsthepercentageofprobleminstancesforw hichCPLEXwas abletondanoptimalsolutionviabranch-and-bound.AsTable 2.4.2 shows,for theOSP-NDC,allthreerelaxationsprovidegoodupperbounds ontheoptimal solutionvalue,consistentlyproducinggapsoflessthan0.25%, onaverage.As

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34 expected,the[ASF]formulationprovidesbetterboundsthan thesimpleOSP-LP relaxation,andthe[DASF]formulationprovidesthetightest bounds.Wenote thatasthenumberofpotentialordersandtheper-periodpro ductioncapacities increase,therelativeperformanceoftherelaxationsimpro ves,andtheoptimality gapdecreases.Sinceanoptimalsolutionexistssuchthatatmoston eorderper periodwillbepartiallysatisedunderanyrelaxation,asthep roblemsizegrows, wefulllagreaterproportionofordersintheirentirety.So theimpactofour choiceofwhichordertopartiallysatisfydiminisheswithlarg erproblemsizes. Notealso,however,thatasmallportionofthisimprovementisa ttributabletothe increasedoptimalsolutionvaluesinthe50-and200-ordercase s. FortheOSP,wehavenon-zeroxeddeliverycostsandcannotthe refore eliminatethebinary z mt variablesfromformulation[OSP].Inaddition,sinceformulation[OSP]includesthecontinuous v mt variables,ithasthehighestnumberof variablesofanyofthecapacitatedversionsweconsider.This doesnotnecessarily, however,makeitthemostdicultproblemclassforsolutionviaC PLEX,asa latercomparisonoftheresultsfortheOSPandOSP-ANDindicates. TheupperboundoptimalitygapresultsreportedinTable 2.4.2 forthe OSParesignicantlylargerthanthosefortheOSP-NDC. 4 Thisisbecausethis formulationpermitssettingfractionalvaluesofthexeddel ivery-chargeforcing ( z mt )variables,andthereforedoesnotnecessarilychargetheenti rexeddelivery costwhenmeetingafractionofsomeorder'sdemand.Forthispr oblemsetthe [DASF]formulationprovidessubstantialvalueinobtainingstr ongupperboundson theoptimalnetprotalthough,asshowninTable 2{6 ,thesizeofthisformulation 4 Forthetwoproblemsthatcouldnotbesolvedtooptimalityvia branch-andboundusingCPLEXduetomemorylimitations,theMIPsolutionva lueusedto computetheupperboundoptimalitygapisthevalueofthebest solutionfoundby CPLEX.

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35 makessolutionviaCPLEXsubstantiallymoretimeconsumingasth enumberof ordersperperiodgrowsto200. Table2{5:OSP-NDCandOSPproblemoptimalitygapmeasures. OSP-NDC OSP %Gap OrdersperPeriod Overall OrdersperPeriod Overall (fromMIP) 25 50 200 Average 25 50 200 Average OSP-LP a 0.24% 0.14% 0.05% 0.14% 9.26% 6.09% 0.57% 5.31% ASF b 0.18 0.12 0.04 0.11 9.21 6.07 0.56 5.28 DASF c 0.11 0.07 0.03 0.07 1.58 0.35 0.10 0.68 %Opt d 100 100 100 100 100 99.7 99.7 99.8 Note:Entriesineach\ordersperperiod"classrepresentan averageamong360testinstances. a (OSP-LP{MIP)/MIP £ 100%. b (ASF{MIP)/MIP £ 100%. c (DASF{MIP)/MIP £ 100%. d %ofproblemsforwhichCPLEXbranch-and-boundfoundanoptima lsolution. Table 2{6 summarizesthesolutiontimesforsolvingtheOSP-NDCandthe OSP.TheMIPsolutiontimesreecttheaveragetimerequiredto ndanoptimal solutionforthoseproblemsthatweresolvedtooptimalityinCP LEX(thetwo problemsthatCPLEXcouldnotsolvetooptimalityarenotincl udedintheMIP solutiontimestatistics).WeusedtheOSP-LPformulationastheb aseformulation forsolvingallmixedintegerprograms.Thetablealsoreportst hetimesrequired tosolvetheLPrelaxationsforeachofourLPformulations(OS P-LP,ASF,and DASF).Wenotethatthe[ASF]and[DASF]LPrelaxationsoftentak elonger tosolvethanthemixedintegerproblemitself.The[DASF]formu lation,despite providingthebestupperboundsonsolutionvalue,quicklybec omeslessattractive astheproblemsizegrowsbecauseofthesizeofthisLPformulati on.Nonetheless, therelaxationsprovideextremelytightboundsontheoptim alsolutionasshown inthetable.Aswelatershow,however,solvingtheproblemtoop timalityin CPLEXisnotalwaysaviableapproachfortherestrictedOSP-AND discussedin thefollowingsection.

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36 Table 2{6 revealsthattheMIPsolutiontimesfortheOSPwerealsomuch greaterthanfortheOSP-NDC.Thisisduetotheneedtosimultan eouslytrack thebinary( z mt )andcontinuous( v mt )variablesfortheOSPwithnon-zeroxed deliverycosts.Asexpected,theaverageandmaximumsolutionti mesforeach relaxationincreasedwiththenumberofordersperperiod.Asw enotedpreviously, thepercentageoptimalitygaps,however,substantiallydecre aseasweincreasethe numberofordersperperiod. Table2{6:OSP-NDCandOSPsolutiontimecomparison. OSP-NDC OSP OrdersperPeriod OrdersperPeriod TimeMeasure(CPUseconds) 25 50 200 25 50 200 AverageMIPSolutionTime 0.1 0.1 0.2 3.3 19.1 129.4 MaximumMIPSolutionTime 0.1 0.1 0.3 44.8 541.3 3417.2 AverageOSP-LPSolutionTime 0.1 0.1 0.3 0.1 0.1 0.3 MaximumOSP-LPSolutionTime 0.1 0.2 0.5 0.1 0.1 0.5 AverageASFSolutionTime 0.5 1.5 14.0 0.4 1.0 8.3 MaximumASFSolutionTime 0.7 2.2 25.2 0.6 1.6 15.4 AverageDASFSolutionTime 5.3 27.3 727.2 3.3 15.7 333.8 MaximumDASFSolutionTime 18.4 64.3 1686.7 12.1 47.1 1251.9 Note1:Entriesrepresentaverage/maximumamong360testin stances. Note2:LPrelaxationsolutiontimesincludetimeconsumeda pplyingtheLP roundingheuristictotheresultingLPsolution,whichwasn egligible. Wenextpresenttheresultsofapplyingourheuristicsolutionap proachesto obtaingoodsolutionsfortheOSPandOSP-NDC.Weemploythethr eeheuristic solutionmethodsdiscussedinSection 2.3.3 ,denotingtheLagrangian-based heuristicasLAGR,thegreatestunitprotheuristicasGUP,andth eLProunding heuristicasLPR.Table 2{7 providestheaveragepercentagedeviationfromthe bestupperbound(asapercentageofthebestupperbound)forea chheuristic solutionmethod.Notethatsincewefoundanoptimalsolutionfor allbuttwoof theOSPandOSP-NDCprobleminstances,theupperboundusedincom putingthe heuristicsolutiongapsisnearlyalwaystheoptimalmixedint egersolutionvalue. ThelastrowinTable 2{7 showstheresultinglowerboundgapfromourcomposite

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37 solutionprocedure,whichselectsthebestsolutionamongallof theheuristic methodsapplied.Theaveragelowerboundpercentagegapisw ithin0.06%of optimalityfortheOSP-NDC,whilethatfortheOSPis1.69%,in dicatingthat overall,ourheuristicsolutionmethodsarequiteeective.Ast hetableindicates, theheuristicsperformmuchbetterintheabsenceofxeddelive rycosts.Forthe Lagrangian-basedandLProundingheuristics,wecanattribute thisinparttothe dicultyinobtaininggoodrelaxationupperboundsfortheOS Pascomparedto theOSP-NDC.Observethatastheupperbounddecreases(i.e.,as thenumberof ordersperperiodincreases),theseheuristicstendtoimprovesu bstantially.The GUPheuristic,ontheotherhand,appearstohavedicultyident ifyingagood combinationofsetupperiodsinthepresenceofxeddeliverych arges.Although itappears,basedonaverageperformance,thattheLPRheuristi cdominates theLAGRandGUPheuristics,thelastrowofthetablerevealsthat thisisnot universallytrue.Eachofourheuristicapproachesprovidedt hebestsolutionvalue forsomenontrivialsubsetoftheproblemstested. Table2{7:OSPandOSP-NDCheuristicsolutionperformancemeasu res. OSP-NDC OSP %Gap OrdersperPeriod Overall OrdersperPeriod Overall (fromUB) 25 50 200 Average 25 50 200 Average LAGRv.UB a 1.34% 0.58% 0.32% 0.75% 6.35% 4.07% 2.16% 4.19% GUPv.UB b 1.00 0.69 0.44 0.71 7.27 6.91 5.39 6.52 LPRv.UB c 0.25 0.15 0.05 0.15 8.32 5.31 0.96 4.86 BestLB d 0.10 0.07 0.02 0.06 3.08 1.55 0.44 1.69 Note:Entriesineach\ordersperperiod"classrepresentan averageamong360testinstances. a (LAGR{UB)/UB £ 100%. b (GUP{UB)/UB £ 100%. c (LPR{UB)/UB £ 100%. d Usesthebestheuristicsolutionvalueforeachprobleminst ance. 2.4.3 ResultsfortheOSP-AND WenextprovideourresultsfortheOSP-ANDwhere,ifwechoosetoa cceptan order,wemustsatisfytheentireorder(i.e.,nopartialordersa tisfactionisallowed).

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38 FindingtheoptimalsolutiontotheOSP-ANDcanbemuchmorechal lengingthan fortheOSP,sincewenowfaceamoredicultcombinatorial\pac king"problem (i.e.,determiningthesetofordersthatwillbeproducedine achperiodissimilarto amultipleknapsackproblem). Table 2{8 providesupperboundoptimalitygapmeasuresbasedonthesolut ion valuesresultingfromourdierentLPrelaxationformulation s,alongwiththe percentageofprobleminstancesthatweresolvedoptimallyvi atheCPLEXbranchand-boundalgorithm.Observethattheupperboundoptimalit ygapmeasuresare quitesmallandonlyslightlylargerthanthoseobservedfortheO SP-NDC.The reasonforthisisthattheLPrelaxationformulationsareide nticalinbothcases (asdiscussedinSection 2.2 ),andtheoptimalLPrelaxationsolutionviolatesthe all-or-nothingrequirementforatmostoneorderperperiod. Thus,eveninthe OSP-NDCcase,almostallordersthatareselectedarefullysatised intheLP relaxationsolution.Incontrasttothe[OSP]formulation,th ebinary z mt variables intheOSP-ANDmodelnowrepresent\orderselection"variablesr atherthanxed delivery-chargeforcingvariables.Thatis,sincewenetanyxe ddeliverycharge outofthenetrevenueparameters R mt ,andthetotalrevenueforanorderina periodnowequals R mt z mt inthisformulation,wehavestrongpreferencefor z mt variablevaluesthatareeitherclosetooneorzero.Inthe[OS P]formulation,on theotherhand,the z mt variablesaremultipliedbythexeddelivery-chargeterms ( F mt )intheobjectivefunction,leadingtoastrongpreferencefo rlowvaluesofthe z mt variablesand,therefore,aweakerupperboundonoptimalne tprot.Note alsothatasthenumberofpossibleordersincreases(fromthe25ordercasetothe 200-ordercase),theinuenceofthesinglepartiallysatisedord erineachperiod ontheobjectivefunctionvaluediminishes,leadingtoareduc edoptimalitygapas thenumberofordersperperiodincreases.AsthelastrowofTable 2{8 indicates, wewerestillquitesuccessfulinsolvingtheseprobleminstancesto optimalityin

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39 CPLEX.Thetimerequiredtodoso,however,wassubstantiallygre aterthanthat foreithertheOSPorOSP-NDC,becauseofthecomplexitiesintr oducedbythe all-or-nothingordersatisfactionrequirement. Table 2{9 summarizestheresultingsolutiontimeperformancefortheOSP AND.Wenoteherethatourrelaxationsolutiontimesarequitere asonable, especiallyascomparedtotheMIPsolutiontimes,indicatingth atqualityupper boundscanbefoundveryquickly.Again,theMIPsolutiontimes reectthe averagetimerequiredtondanoptimalsolutionforthoseprobl emsthatwere solvedtooptimalityinCPLEX(thoseproblemswhichCPLEXcoul dnotsolveto optimalityarenotincludedintheMIPsolutiontimestatistics) .Thetabledoesnot reportthetimerequiredtosolveourdierentLPrelaxationfo rmulations,sincethe OSP-ANDLPrelaxationisidenticaltotheOSP-NDCLPrelaxation ,andthese timesarethereforeshowninTable 2{6 UnlikeourpreviouscomputationalresultsfortheOSPandtheO SP-NDC,we foundseveralprobleminstancesoftheOSP-ANDinwhichanoptima lsolutionwas notfoundeitherduetoreachingthetimelimitofonehourorb ecauseofmemory limitations.Fortheprobleminstanceswewereabletosolveopt imally,theMIP solutiontimeswerefarlongerthanthosefortheOSPproblem.T hisisdueto theincreasedcomplexityresultingfromtheembedded\packin gproblem"inthe OSP-ANDproblem.Interestingly,however,incontrasttoourpre viousresultsfor theOSP,theaverageandmaximumMIPsolutiontimesfortheOSP -ANDwere smaller forthe200-orderperperiodproblemsetthanforthe25and50orderper periodproblemsets.Thereasonforthisappearstobebecauseoft henearlynonexistentintegralitygapsoftheseprobleminstances,whereast hesegapsincrease whenthenumberofordersperperiodissmaller.

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40 Table2{8:OSP-ANDoptimalitygapmeasures. OrdersperPeriod GapMeasurement 25 50 200 OverallAverage OSP-LPvs.MIPSolution a 0.34% 0.20% 0.06% 0.20% ASFvs.MIPSolution b 0.28 0.18 0.05 0.17 DASFvs.MIPSolution c 0.21 0.10 0.03 0.11 %Optimal d 96.7 94.2 100 97 Note:Entrieswithineach\ordersperperiod"classreprese ntaverage among360testinstances. a (OSP-LP{MIP)/MIP £ 100%. b (ASF{MIP)/MIP £ 100%. c (DASF{MIP)/MIP £ 100%. d %ofproblemsforwhichCPLEXbranch-and-boundfoundanoptima lsolution. Table2{9:OSP-ANDsolutiontimecomparison. OrdersperPeriod TimeMeasure(CPUseconds) 25 50 200 AverageMIPSolutionTime 42.0 67.9 21.9 MaximumMIPSolutionTime 1970.1 1791.8 1078.8 Note:Entriesrepresentaverage/maximumamong360testins tances. Table 2{10 showsthatonceagainourcompositeheuristicprocedureperfor med extremelywellontheproblemswetested.Thepercentagedevi ationfromoptimalityinoursolutionsisveryclosetothatoftheOSP-NDC,andmuch betterthan thatoftheOSP,withanoverallaverageperformancewithin0 .25%ofoptimality.Wenote,however,thatthebestheuristicsolutionperform anceforboththe OSP-NDCandtheOSP-ANDoccurredusingtheLProundingheuristica pplied totheDASFLPrelaxationsolution.AsTable 2{6 showed,solvingtheDASFLP relaxationcanbequitetimeconsumingasthenumberoforders perperiodgrows, duetothesizeofthisformulation.Wenote,however,thatfor theOSP-NDCand OSP-AND,applyingtheLProundingheuristictotheASFLPrelaxat ionsolution producedresultsveryclosetothoseachievedusingtheDASFLPrel axationsolutioninmuchlesscomputingtime.Amongallofthe3,240OSP,OSP -NDC,and OSP-ANDproblemstests,thebestheuristicsolutionvaluewaswithi n0.67%of optimalityonaverage,indicatingthatoverall,theheurist icsolutionapproaches

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41 wepresentedprovideanextremelyeectivemethodforsolvingt heOSPandits variants. Table2{10:OSP-ANDheuristicsolutionperformancemeasures. OSP-AND OrdersperPeriod Overall GapMeasurement 25 50 200 Average LAGRvs.UB a 3.95% 3.92% 0.33% 2.73% GUPvs.UB b 1.85 0.83 0.46 1.04 LPRvs.UB c 0.80 0.31 0.12 0.41 BestLB d 0.49 0.19 0.06 0.25 Note:Entrieswithineach\ordersperperiod"classreprese nt averageamong360testinstances a (LAGR{UB)/UB £ 100%. b (GUP{UB)/UB £ 100%. c (LPR{UB)/UB £ 100%. d Usesthebestheuristicsolutionvalueforeachprobleminst ance. 2.5 Conclusions Whenaproducerhasdiscretiontoacceptordenyproductionor ders,determiningthebestsetoforderstoacceptbasedonbothrevenueandprod uction/delivery costimplicationscanbequitechallenging.Forsituationswh ennoproduction capacitiesexist,weshowhowtheorderselectionproblemcanbe solvedusing asimilarapproachtotheWagner-Whitin[ 83 ]dynamicprogrammingalgorithm employedfortheELSP.Whenfacingproductioncapacities,sev eralvariationsof theproblememerge,andweformulatedandpresentedsolutiona pproachestothese aswell. Weconsideredvariantsoftheproblembothwithandwithoutxe ddelivery charges,aswellascontextsthatpermittheproducertosatisfy anychosenfraction ofanyorderquantity,thusallowingtheproducertorationi tscapacity.We providedthreelinearprogrammingrelaxationsthatproduc estrongupperbound valuesontheoptimalnetprotfromintegratedorderselectio nandproduction planningdecisions.Wealsoprovidedasetofthreeeectiveheuri sticsolution methodsfortheOSP.Computationaltestsperformedonabroad setofrandomly

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42 generatedproblemsdemonstratedtheeectivenessofourheuri sticmethodsand upperboundingprocedures.Probleminstancesinwhichthepro ducerhasthe exibilitytodetermineanyfractionofeachorderitwillsupp ly,andnoxed deliverychargesexist,wereeasilysolvedusingtheMIPsolverin CPLEX.When xeddeliverychargesarepresent,however,theproblembecom esmoredicult, particularlyasthenumberofavailableordersincreases.Opt imalsolutionswere stillobtained,however,fornearlyalltestinstanceswithino nehourofcomputing timewhenpartialordersatisfactionwasallowed.Whenthepro ducermusttake anall-or-nothingapproach,satisfyingtheentireamountofe achorderitchooses tosatisfy,theproblembecomessubstantiallymorechallenging ,andtheheuristic solutionswepresentedbecomeamorepracticalapproachforsol vingsuchproblems. Weexpandourdiscussionofdemand(ororder)selectionexibilit yina productionplanningcontextoverthenexttwochapters.Spec ically,wewill introducepricingasadecisionvariableintherequirements planningproblem inChapter 3 .Thenwewillconsidertherolethatdemanduncertaintyplays in demandsourceselectiondecisionsinChapter 4 2.6 Appendix DescriptionofFeasibleSolutionGenerator(FSG)AlgorithmforOSP ThisappendixdescribestheFeasibleSolutionGenerator(F SG)algorithm,which takesasinputasolutionthatisfeasibleforallOSPproblem constraintsexceptthe productioncapacityconstraints,andproducesacapacity-f easiblesolution.Notethat wepresenttheFSGalgorithmasitappliestotheOSP,andthat certainstraightforward modicationsmustbemadefortheOSP-ANDversionoftheproble m. PhaseI:Assessattractivenessofadditionalsetups0) Let j denoteaperiodindex,let p ( j )bethemostrecentproductionperiodprior toandincludingperiod j ,andlet s ( j )bethenextsetupafterperiod j .Ifno productionperiodexistspriortoandincluding j ,set p ( j )=0.Set j = T and

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43 s ( j )= T +1andlet X j denotethetotalplannedproduction(inthecurrent, possiblycapacity-infeasiblesolution)forperiod j 1) Determinethemostrecentsetup p ( j )asdescribedinStep0.If p ( j )=0,gotoPhase II.If X p ( j ) C p ( j ) ,set s ( p ( j ) ¡ 1)= p ( j )and j = p ( j ) ¡ 1andrepeatStep1(note thatwemaintain s ( j )= j +1).Otherwise,continue. 2) Comparethedesiredproductioninperiod p ( j ), X p ( j ) ,withactualcapacitiesover thenext s ( j ) ¡ p ( j )periods.If X p ( j ) > P s ( j ) ¡ 1 t = p ( j ) C t ,andthesumoftherevenues forallselectedordersforperiod j exceedthesetupcostinperiod j ,thenadd aproductionsetupinperiod j andtransferallselectedordersinperiod j to thenewproductionperiod j .Otherwisedonotaddthesetupinperiod j .Set s ( p ( j ) ¡ 1)= p ( j ) ;j = p ( j ) ¡ 1,andreturntoStep1. PhaseII:Transfer/removeleastprotableproductionorders0) Let d m;p ( j ) ;j denotetheamountofdemandfromorder m inperiod j tobesatised byproductioninperiod p ( j )inthecurrent(possiblycapacity-infeasible)plan. Whenreadingintheproblemdata,allprotableorderandprod uctionperiod combinationsweredetermined.Basedonthesolution,wemai ntainalistof allordersthatweresatised,andthislistiskeptin nondecreasing orderof per-unitprotability.Per-unitprotabilityisdenedasfollo ws:¦ m;p ( j ) ;j = r mj ¡ p p ( j ) ¡ P j ¡ 1 t = p ( j ) h t ¡ F mj d mj .Wewillusethislisttodeterminetheleastdesirable productionorderstomaintain. 1) Ifnoperiodshaveplannedproductionthatexceedscapacity ,gotoPhaseIII.While therearestillperiodsinwhichproductionexceedscapacit y,ndthenextleast protableorderperiodcombination,( m ¤ ;p ( j ¤ ) ;j ¤ ),inthelist. 2) If X p ( j ¤ ) >C p ( j ¤ ) ,considershiftingorremovinganamountequalto d ¤ = min f d m ¤ ;p ( j ¤ ) ;j ¤ ;X p ( j ¤ ) ¡ C p ( j ¤ ) g fromproductioninperiod p ( j ¤ )(otherwise, returntoStep1).Ifanearlierproductionperiod


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44 inperiod ;i.e., d m ¤ ;;j ¤ =min( d ¤ ;C ¡ X ).Otherwise,reducetheamountof productioninperiod p ( j ¤ )by d ¤ andset d m ¤ ;p ( j ¤ ) ;j ¤ = d m ¤ ;p ( j ¤ ) ;j ¤ ¡ d ¤ 3) Updateallplannedproductionlevelsandorderassignments andupdatethenumber ofperiodsinwhichproductionexceedscapacity.ReturntoS tep1. PhaseIII:Attempttoincreaseproductioninunder-utilized periods 0) Createanewlistforeachperiodofallprotableordersnot fullled.Eachlistis indexedin nonincreasing orderofper-unitprotability,asdenedearlier.Let j denotetherstproductionperiod. 1) If j = T +1,STOPwithafeasiblesolution.Otherwise,continue. 2) If C p ( j ) >X p ( j ) ,excesscapacityexistsinperiod p ( j ).Choosethenextmost protableorderfromperiod j ,andlet m ¤ denotetheorderindexforthisorder. Let d m ¤ ;p ( j ) ;j =min d m ¤ ;j ;C p ( j ) ¡ X p ( j ) ,andassignanadditional d m ¤ ;p ( j ) ;j to productioninperiod p ( j ). 3) Ifthereisremainingcapacityandadditionalprotableorde rsexistforperiod j ,the repeatStep2.Otherwise,set j = j +1andreturntoStep1.

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CHAPTER3 PRICING,PRODUCTIONPLANNING,AND ORDERSELECTIONFLEXIBILITY 3.1 Introduction Firmsthatproducemade-to-ordergoodsoftenmakepricingd ecisionsprior toplanningtheproductionrequiredtosatisfydemands.Thesede cisionsrequire therm'srepresentatives(oftensales/marketingpersonnelinc onsultationwith manufacturingmanagement)todetermineprices,whichimply certaindemand volumesthermwillneedtosatisfy.Suchpricingdecisionsaret ypicallymade priortoestablishingfutureproductionplansandareinmanyc asesmadebasedon thecollectivejudgmentofsalesandmarketingpersonnel.Thi sresultsindecisions thatdonotaccountfortheinteractionbetweenpricingdeci sionsandproduction requirements,andhowthesefactorsaectoverallprotability .Lee[ 44 ]recently notedthatoneofthecommonpitfallsofsupplychainmanageme ntpracticeoccurs whenthosewhoinuencedemandwithintherm(e.g.,marketing, sales)donot properlyaccountforoperationscostsindemandplanning,wh ilesupplychainmanagersfailtorecognizethatdemandisnotcompletelydeterm inedexogenously.He arguesthatintegratingsupplyanddemand-basedmanagemento ersgreatopportunityforfuturevaluecreationandservesas\thenextcompe titivebattlegroundin the21stcentury." Sinceproductionenvironmentsofteninvolvesignicantxedp roductioncosts, justifyingthesexedcostsrequiresademandlevelatwhichreve nuesexceednot onlyvariablecosts,butthexedcostsincurredinproductionas well.Decisions onthedemandvolumetheorganizationmustsatisfy,andtheimpl iedrevenues andcosts,canbeacriticaldeterminantoftherm'sprotabilit y.Pastoperations 45

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46 modelingliteraturehasnotfullyaddressedintegratedprici ngandproduction planningdecisionsinmake-to-ordersystemswiththetypesofn onlinearproduction coststructuresoftenfoundinpracticeasaresultofproductio neconomiesof scale.Weoermodelingandsolutionapproachesforintegratin gthesedecisionsin single-stagesystems. Mostoftherequirementsplanningliteraturefocusesonprodu ctionrequirementsbasedonpre-specieddemands,withnoadjustmentsforpric eexibility. Inthischapter,weintroducearequirementsplanningmodel thatimplicitlydeterminesthebestdemandlevelstosatisfyinordertomaximizecont ributiontoprot whendemandisadecreasingfunctionofprice.Inotherwords,t hermwillselect thedemandleveltosatisfybysettingasinglepricefortheprodu ct. Wemakeseveralcontributionstotheliteraturethroughourm odeland solutionapproachesintroducedinthischapter.First,ourco mbinedpricingand productionplanningmodelpermitsmultipleprice-demandc urvesineachperiod, whicheectivelyrepresentsthepossibilityofoeringdierentp ricesindierent markets,whereeachmarkethasauniqueresponsetomarketprice .Moreover, thismodelgeneralizesthe orderselection approachpresentedinChapter 2 ,where armfacedasetofcustomerorders,fromwhichitselectedthemostp rotable subset.Inthe orderselection context,wecanuseourrequirementsplanning withpricingmodelandapplyauniquepricetoeachorder,rat herthanasingle priceforalldemands.Oursolutionapproachalsoaccommodates moregeneral productioncostfunctionsthanpreviouslyconsideredinthere quirementsplanning andpricingliterature,alongwithexplicitconsiderationo fbothgeneralconcaveand piecewise-linearconcaverevenuefunctions. Givenxedpluslinearproductioncostsandpiecewise-linearc oncaverevenue functions,wealsoprovidea`tight'linearprogrammingformu lationofourmodel, usingadual-basedsolutionapproachtoshowthatthisformulati onhaszeroduality

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47 gap.Thisresult,andtheformulationsdiscoveredwhiledevel opingtheapproach, playedakeyroleinformulatingtherelaxationsusedinsolvin gthecapacitated OSPmodelsinChapter 2 ,whereproductioncapacitiesvariedovertime.Ournal majorcontributionalsoaddressesacapacitatedversionofthem odel.Assuming time-invariantproductioncapacitylimitsandpiecewise-l inearconcaverevenue functionsinthetotaldemandsatised,weshowthatthisproblem canbesolvedin polynomialtime. Giventherecentemphasisondierentialpricinganddemandma nagement inmanufacturing(e.g.,Lee[ 44 ],ChopraandMeindl[ 22 ]),thesemodelsand associatedsolutionapproacheshavethepotentialforbroadap plicationinpractice. AnalyticsOperationsEngineering,Inc.,anoperationsstrat egyandexecution consultingrm,recentlycitedapplicationcontextsinthespe cialtypapersand timberindustriesinwhichintegratedpricingandproductio nplanningmodelssuch astheoneswediscusscanaddsubstantialvalueinpractice(form oredetailson theseapplications,pleaseseeBurman[ 17 ]). Thomas[ 74 ]providedananalysisandsolutionalgorithmforarelatedint egratedpricingandproductionplanningdecisionmodel.Hismo delgeneralizedthe Wagner&Whitin[ 83 ]modelbycharacterizingdemandineachofasetofdiscrete timeperiodsasadownward-slopedfunctionofthepriceineac hperiod,thustreatingeachperiod'spriceasadecisionvariable.Themodelprop osedbyThomas[ 74 ] setsonlyasinglepriceforalldemandsinanygivenperiod,whe reasourmodel permitsdierentialpricingindierentmarkets.Moreover,we demonstratethat a`tight'linearprogrammingformulationexistsforthispro blemunderpiecewiselinearconcaverevenuefunctions.Wealsoextendtheanalysist oaccountformore generalproductioncostfunctionsineachperiod. Additionalcontributionstotheintegratedpricingandprod uctionplanning problemincludetheworkofKunreutherandSchrage[ 41 ]andGilbert[ 33 ],who

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48 consideredtheproblemwhenasinglepricemustbeusedovertheen tirehorizon. KunreutherandSchrage[ 41 ]providedboundsontheoptimalsolutionvalue undertimevaryingproductioncostassumptions,whileGilbert[ 33 ]assumedtimeinvariantproductionsetupandholdingcostsandprovidedane xactpolynomialtimealgorithm.RecallthepaperbyLoparic,Pochet,andWol sey[ 50 ]thatwe introducedinChapter 2 .Theyconsideredaprobleminwhichaproducerwishes tomaximizenetprotfromsalesofasingleitemanddoesnothave tosatisfyall outstandingdemandineveryperiod.Theirmodelcontainsnop ricingdecisions, eectivelyassumingthatonlyonedemandsourceexistsineverype riod,andthat therevenuefromasingledemandsourceisproportionaltothev olumeofdemand satised.Incontrast,weallowrevenuetobeageneralconcaveno ndecreasing functionoftheamountofdemandsatised,whichisconsistentwit hadownwardslopeddemandcurveasafunctionofprice.AlsodiscussedinChapte r 2 was thepaperbyLee,Cetinkaya,andWagelmans[ 43 ],inwhichtheyintroducea productionplanningmodelwith demandtimewindows .Whiletheirmodelassumes thatallpre-specieddemandsmustbelledduringtheplanningh orizon,our approachimplicitlydeterminesdemandlevelsthroughpric ing. BhattacharjeeandRamesh[ 13 ]consideredthepricingproblemforperishable goodsusingaverygeneralfunctiontocharacterizedemandas afunctionofprice. Theyalsoassumedupperandlowerboundsonprices,characterize dstructural propertiesoftheoptimalprotfunction,anddevelopedheur isticmethodsfor solvingtheresultingproblems.Biller,Chan,Simchi-Levi,an dSwann[ 14 ]analyzed amodelsimilartooursunderstrictlylinearproductioncosts( i.e.,noxedsetup costs,andassumingtime-varyingproductioncapacitylimits),w hichtheysolved ecientlyusingagreedyalgorithm.Whileourdiscussionofthere levantliterature hasfocusedondeterministicapproachesforintegratedprici ngandproduction planningproblems,someadditionalworkondynamicpricingex iststhataddresses

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49 stochasticdemandenvironments;forpastworkonintegratedpri cingandproduction/inventoryplanninginastochasticdemandsetting,please seeThomas[ 75 ], GallegoandvanRyzin[ 30 ],andChan,Simchi-Levi,andSwann[ 21 ]. Theremainderofthischapterisorganizedasfollows.Sectio n 3.2 presents aformaldenitionandmixedintegerprogrammingformulatio nofthegeneral requirementsplanningproblemwithpricing.Inthissection weprovideoursolution approachesforthisproblem,therstofwhichextendstheWagn er-Whitin[ 83 ] shortestpathsolutionmethod(discussedinChapter 2 )tocontextswithgeneral concaverevenuefunctionsandxed-chargeproductioncosts.Assu mingpiecewiselinearconcaverevenuefunctions,wethenprovideadual-base dpolynomialtimealgorithmforsolvingtheuncapacitatedproblem.Thisd ual-basedsolution approachallowsustoshowthattheproblemreformulationinS ection 3.2.2 has alinearprogrammingrelaxationwhoseoptimalvalueequalst hatoftheoptimal mixedintegersolution;i.e.,theproblemformulationis\ti ght".Wealsoexplore thegeneralityofoursolutionapproacheswithrespecttodier entfunctionalforms fortheproductioncostfunctionsandundermultiplemarketp rice-demandcurves inanygivenperiod.Inadditiontopresentingsolutionapproa chestoseveral uncapacitatedversionsoftheproblem,weprovideananalysis oftheequal-capacity versionofthemodelunderpiecewise-linearconcaverevenuef unctions.Section 3.3 discussesdierentpricinginterpretationsfromourmodels,and illustrateshow ourpricingmodelcanbecastasanequivalent\orderselection "problem,thus broadeningitspotentialforapplicationinpractice. 3.2 RequirementsPlanningwithPricing Consideraproducerwhomanufacturesagoodtomeetdemandove ranite numberoftimeperiods, T .Theproductioncostfunctioninperiod t isdenoted g t ( ¢ ),andisanondecreasingconcavefunctionoftheamountprodu cedinperiod t ,whichwedenoteby x t .Similarly,therevenuefunctioninperiod t isdenoted

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50 by R t ( ¢ ),andisanondecreasingconcavefunctionofthe totaldemandsatised in period t ,whichwedenoteby D t ,with R t (0)=0forall t =1 ;:::;T .Weassume that D t ,thetotaldemandsatisedinanyperiod t ,isthesumofthedemands satisedfromsome M t distinctmarkets.Ineachmarketweemployastandard assumptionofaone-to-onecorrespondencebetweenpriceandma rketdemand volumeinanyperiod,wheremarketdemandisadownward-slopi ngfunctionof price(seeGilbert[ 33 ]),andeachmarket'srevenueisanondecreasingconcave functionofdemandsatisedinthemarket.Givenatotaldemandv alueof D t in period t wesolveanoptimizationsubproblemtodetermineapricevalue inperiod t ineverymarket m (equivalently, D t = P M t m =1 d mt ( mt )where mt isthepricein market m inperiod t and d mt ( ¢ )isthetotaldemandinmarket m inperiod t as afunctionofprice).Section 3.2.3.1 discusseshowtodeterminethepriceineach marketinperiod t givenademandvolumeof D t ;fornowitissucienttosimply considerthedecisionvariablesforthetotaldemandineachpe riod(i.e.,the D t variables). Inventorycostsarechargedagainstendinginventory,where h t denotestheunit holdingcostinperiod t and I t isadecisionvariablefortheend-of-periodinventory inperiod t .Letting C denotetheproductioncapacitylimit(whichdoesnotdepend ontime),weformulatethe requirementsplanningwithpricing (RPP)problemas follows.[RPP] maximize P Tt =1 ( R t ( D t ) ¡ ( g t ( x t )+ h t I t )) subjectto: D t + I t = x t + I t ¡ 1 t =1 ;:::;T; (3.1) x t Ct =1 ;:::;T; (3.2) x t ;I t ;D t 0 t =1 ;:::;T: (3.3)

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51 Theobjectivefunctionmaximizesnetprotafterproduction andholding costs;constraintset( 3.1 )ensuresinventorybalanceinallperiodsandconstraint set( 3.2 )enforcesproductioncapacitylimits.ThegeneralRPPprobl emdened abovemaximizesthedierencebetweenconcavefunctionsand is,therefore,in generaladicultglobaloptimizationproblem(seeHorstandTuy [ 37 ]).By providingcertainsomewhatmildrestrictionsonthefunction alformsoftherevenue andproductioncostfunctions, R t ( D t )and g t ( x t ),wearriveatafamilyofspecial casesoftheRPPproblem,severalofwhichhavebroadapplicabi lityinpractice. Consistentwiththevastmajorityofpastproductionplanningli terature, exceptwherespecicallynoted,wehenceforthassumethatprodu ctioncosts containaxed-chargestructure;i.e.,axedcostof S t isincurredwhenperforming aproductionsetupinanyperiod t ,whilethevariablecostperunitinperiod t equals p t (welaterdiscussinSection 3.2.3.2 thenecessaryextensionstohandle productioncoststhatcontainamoregeneralpiecewise-linea rnondecreasing concavecoststructure).Underxedpluslinearproductioncosts, unlimited productioncapacity,andasinglepriceoeredtoallmarketsi neachperiodwehave themodelrstanalyzedbyThomas[ 74 ],whoproposedadynamicprogramming recursionforsolvingtheproblem.Thealgorithmissimilartot heWagner-Whitin [ 83 ]algorithmfortheELSP,andreliesonsimilarkeystructuralp ropertiesofthe problem.Thesepropertiesincludethezeroinventoryorderi ng(ZIO)property(if inventoryisheldattheendofperiod t ¡ 1thenwedonotperformasetupin period t ).Thefollowingsectiondescribesanequivalentshortestpatha lgorithm (refertoSection 2.2.2 foracompletediscussion)forthisproblem,alongwithan explicitcharacterizationofthesolutionapproachunderco ncaverevenuefunctions. Whiletheshortestpathmethodwepresentisgenerallyequivale nttothedynamic programmingmethodproposedbyThomas[ 74 ]whenproductioncostscontaina xed-chargestructure,wedepartfromthisworkinthefollowi ngrespects:

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52 (i)weprovideanexactsolutionapproachforcontextsinwhic htotalrevenueis concaveandnondecreasingintheamountofdemandsatised; (ii)weshowthattheshortestpathmethodgeneralizestocaseswi thmultiple demandsources,eachwithauniqueconcaverevenuecurve;and (iii)weshowhowtogeneralizetheshortestpathapproachtopro videanexact procedureforthecaseofpiecewise-linearandconcaveproduc tioncosts. Thus,thefollowingsectionlaysthefoundationforsubsequentg eneralizationsof oursolutionmethodologytobroadercontexts. 3.2.1 ShortestPathApproachfortheUncapacitatedRPP Retainingourassumptionofaxed-chargeproductioncoststruct ureand assumingtherevenuefunction R t ( D t )ineveryperiod t isageneralnondecreasing concavefunctionof D t with R t (0)=0,wenowupdatetheWagner-Whitin [ 83 ]shortestpathapproach(introducedinChapter 2 )fortheuncapacitated RPPproblem.Notethatundertheseassumptions,foranyxedchoice ofthe demandvector( D 1 ;D 2 ;:::;D T ),theresultingproblemisasimpleELSP.Now, wecandecomposethe T -periodRPPproblemsintoasetofsmallercontiguous intervalsubproblems,usingtheshortestpathgraphstructurepre viouslyshown inFigure 2{2 .Toillustratethecomputationofarclength c ( t;t 0 ),whereasetup isperformedinperiod t andthenextsetupoccursinperiod t 0 >t ,wesolvethe period t;:::;t 0 ¡ 1subproblemofmaximizingnetprotintheseperiods.This period t;:::;t 0 ¡ 1subproblemcanbestatedas maximize: t 0 ¡ 1 P = t R ( D ) ¡ h P t 0 ¡ 1 j = +1 D j ¡ p t P t 0 ¡ 1 j = t D j (3.4) subjectto: D 0 = t:::;t 0 ¡ 1 : (3.5) Thisdecisionproblemseparatesbyperiod,andsincewearemaxi mizingasetof nondecreasingconcavefunctions,wearriveatthefollowingc haracterizationofthe optimalamountofdemandtosatisfyinperiod ,givenamostrecentsetupin

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53 period t .Fornotationalconveniencewelet v t p t + P ¡ 1 j = t h j denotethecostper unitofdemandsatisedinperiod usingasetupinperiod t Theorem1 FortheuncapacitatedRPP,givenaproductionsetupinperiod t only, ifademandquantity D existssuchthat v t isinthesetofsubgradientsof R ( ¢ ) at D ,then D isanoptimaldemandquantityforthesubproblemgivenby( 3.4 )and ( 3.5 ). AproofofTheorem 1 canbefoundinAppendixAofSection 3.5 .Notethatif R ( ¢ )iseverywheredierentiablewithlim D !1 R 0 ( D ) v t lim D # 0 R 0 ( D ), thentheoptimaldemandquantityasstatedinthetheoremcanb edeterminedby nding D suchthat R 0 ( D )= v t Givenany t t 0 ¡ 1,ifa D > 0existsthatsatisestheconditionof Theorem 1 ,thentheoptimalvalueof D forthesubproblem,whichwedenoteby D ¤ ( t ),equalsthisdemandvalue.Otherwise,assuminganite(non-ne gative)value of v t ,wemusthaveeither D ¤ ( t )=0(ifallsubgradientsatall D > 0arelessthan v t )or D ¤ ( t )= 1 (ifasubgradientexistsforeach D > 0thatisgreaterthan v t ). Thenthemaximumpossibleprotinperiods t;:::;t 0 ¡ 1(assumingtheonlysetup withintheseperiodsoccursinperiod t ,whichwedenoteby¦( t;t 0 ))isgivenby ¦( t;t 0 )= t 0 ¡ 1 P = t R ( D ¤ ( t )) ¡ h t 0 ¡ 1 P j = +1 D ¤ j ( t ) ¡ p t t 0 ¡ 1 P j = t D ¤ j ( t ) ¡ S t ; (3.6) andthearclengthforarc( t;t 0 )isthereforegivenby c ( t;t 0 )=max f 0 ; ¦( t;t 0 ) g : (3.7) Withappropriatepreprocessingandrecursivecomputationsof the¦( t;t 0 )values, wecandetermineall¦( t;t 0 )valuesin O ( T 2 )time.Asdiscussedpreviously,the longestpathonanacyclicnetworkcanbefoundin O ( T 2 )timeintheworstcase (seeLawler[ 42 ]).Therefore,theoverallsolutioneortisnoworsethan O ( T 2 ).

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54 Wenextconsideraparticularspecialcaseoftheconcaverevenu efunctions, whichwewilluseformoredetailedanalysisinsubsequentsection s.Suppose thattherevenuefunctionineachperiodcanberepresentedas anondecreasing piecewise-linearconcavefunctionofdemand.Weassumethatth erevenuefunction inperiod t has J t +1consecutive(contiguous)linearsegments.Therst J t ofthese segmentshaveinterval width values d 1 t ;d 2 t ;:::;d J t t ,andwelet r jt denotetheslope (perunitrevenue)withinthe j th linearsegment;the( J t +1) st segmenthasslope zero(i.e.,themaximumpossibletotalrevenueisnitewithval ue P J t j =1 r jt d jt for t =1 ;:::;T ).Thisimpliesthatwecanstateourrevenuefunctionsasfoll ows: R t ( D t )= 8>>>>>><>>>>>>: k ¡ 1 P j =1 r jt d jt + r kt D t ¡ k ¡ 1 P j =1 d jt for k ¡ 1 P j =1 d jt D t < k P j =1 d jt ; k =1 ;:::;J t ; J t P j =1 r jt d jt for J t P j =1 d jt D t : (3.8) where r 1 t >r 2 t > ¢¢¢ >r J t t > 0.Theorem 1 impliesthatanoptimalsolution existssuchthatthetotaldemandsatisedineachperiod t occursatoneof thebreakpointvalues;i.e.,at P kj =1 d jt forsome k betweenoneand J t (note thatanoptimaldemandvaluecannotexistinthe( J t +1) st intervalifcostsare positive,whichweassumethroughout,sincecostswillincreasean drevenuesremain constant).Denotesuchavalueof D by D ¤ ( t ).Then, c ( t;t 0 )=max 0 ; t 0 ¡ 1 X = t ( R t ( D ¤ ( t )) ¡ v t D ¤ ( t )) ¡ S t : Thetimeneededtocomputethesevaluesis O ( T 2 )multipliedbythetimerequired tond D ¤ ( t )andevaluate R t ( D ¤ ( t ))forall t; .Notethatifthefunctions R t ( ¢ ) arepiecewise-linearandconcavewithatmost J max segments,theslopesateach breakpointandtheresulting R t ( D ¤ ( t ))computationscanbeperformedin O ( J max ) time,foratotalarc`cost'calculationtimeof O ( J max T 2 ).Sincetheacycliclongest

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55 pathproblemrequires O ( T 2 )operations,ourtotalsolutiontimeisnoworsethan O ( J max T 2 ). 3.2.2 Dual-ascentMethodfortheUncapacitatedRPP Whentherevenuefunctionsarepiecewise-linearandconcave ineveryperiod, andproductioncostscontainaxedplusvariablecoststructure ,wecanalsouse adual-basedalgorithmtosolvetheuncapacitatedRPP,whichw enextdescribe. ThisapproachrequiresrstreformulatingtheRPP.Aswelatersh ow,thisnew formulationis\tight";i.e.,itslinearprogrammingrelax ationobjectivefunction valueequalstheoptimalobjectivefunctionvalueofRPP.We beginbyprovidingan explicitbaseformulationoftheuncapacitatedRPPunderpie cewise-linearconcave revenuefunctionsandxedpluslinearproductioncosts,usingm uchofthenotation alreadydenedintheprevioussections.Wedeneasetofbinaryva riables z jt for t =1 ;:::;T and j =1 ;:::;J t ,suchthat z jt =1if D t P jk =1 d kt (i.e.,when thetotaldemandsatisedinperiod t occursatthe j th orhigherbreakpointofthe piecewise-linearconcaverevenuecurve);otherwise z jt =0when D t P j ¡ 1 k =1 d kt Bythedenitionofthe z jt variablesandthefactthatanoptimalsolution existswheretotaldemandfallsatanintervalbreakpointine achperiod,we thereforehavethatthetotaldemandsatisedinperiod t equals D t = P J t j =1 d jt z jt andthecorrespondingtotalrevenueequals P J t j =1 r jt d jt z jt .Wenextdeneanewset ofbinarysetupvariables, y t ,for t =1 ;:::;T; where y t =1ifweperformasetup inperiod t ,and y t =0otherwise.WecanthusformulatetheuncapacitatedRPP withpiecewise-linearconcaverevenuefunctions,whichwere fertoastheRPP PLC asfollows.

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56 [RPP PLC ] maximize: P Tt =1 P J t j =1 r jt d jt z jt ¡ S t y t ¡ p t x t ¡ h t I t subjectto: I t ¡ 1 + x t = P J t j =1 d jt z jt + I t t =1 ;:::;T; (3.9) 0 x t P T = t P J j =1 d j y t t =1 ;:::;T; (3.10) I 0 =0 ;I t 0 ;t =1 ;:::;T; (3.11) 0 z jt 1 t =1 ;:::;T; j =1 ;:::;J t ; (3.12) y t 2f 0 ; 1 g t =1 ;:::;T: (3.13) Intheabove[RPP PLC ]formulation,theobjectivefunctionprovidesthenetreve nue aftersubtractingproductionandholdingcosts.Constraintset( 3.9 )ensures inventorybalance,whilethesetupforcingconstraints( 3.10 )enforcesetting y t equal tooneifanyproductionoccursinperiod t .Notethatthecoecientof y t inthese constraintsequalsthetotaldemandfromperiod t through T ,therebyeectively leavingtheproblemuncapacitated.Constraints( 3.11 )through( 3.13 )encode ourvariablerestrictions.Sinceanoptimalsolutionexistsfor theuncapacitated versionoftheproblemsuchthatthedemandsatisedinanyperiodo ccursatone ofthebreakpointvaluesoftheperiod'srevenuefunction,[ RPP PLC ]providesthe sameoptimalsolutionvalueastheformulationobtainedbyexp licitlyimposing thebinaryrestrictiononthe z jt variables.Weformulatetheproblemwiththe relaxedbinaryrestrictions,however,forlaterextensiontot heequal-capacitycase inSection 3.2.4 Notethatwehavenotimposedanyspecicconstraintsontherelati onship between z jt variablescorrespondingtothesamerevenuefunctioninagive n period t .Thefollowingpropertyallowsustoconsidereachoftheinte rvalsofthe piecewise-linearconcaverevenuefunctionindependentlyf romoneanotherinour

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57 mixedintegerprogrammingformulation(thatis,weneednoti ntroduceanyexplicit constraintsinourformulationthatspecifythestrictorderin gofthepiecewise-linear segmentsoftherevenuefunctions).Property1 ContiguityProperty :Forthe[RPP PLC ]problemdenedabove,if anoptimalsolutionexistssuchthat z j ¡ 1 ;t =0 ,then z kt =0 for k = j;:::;J t inany optimalsolution.Proof :Supposethatanoptimalsolutionexistswithobjectivefuncti onvalue Z ¤ with z kt =0and z lt =1forsome l>k ,andletperiod s t denotethesetup periodinwhichtheproductionoccurredthatsatiseddemandin period t .Since z lt =1,wemusthavethat r lt p s + P t ¡ 1 = s h ;otherwiseasolutionexistssuch that z lt =0withobjectivefunctionvaluegreaterthan Z ¤ ,whichcontradicts theoptimalityofthesolutionwith z lt =1.Since r kt >r lt wemustalsohave r kt >p s + P t ¡ 1 = s h ,andasolutionexistswith z kt =1andanobjectivefunction valuegreaterthan Z ¤ ,acontradictionoftheoptimalityofthesolutionwith z kt =0 and z lt =1,whichimpliesthatif z kt =0inanoptimalsolution z lt mustequalzero for l = k +1 ;:::;J t inanyoptimalsolution(i.e.,thecontiguityproperty). } Wecanalsousetheargumentsinthecontiguitypropertyprooft oshow thatif z kt =1inanoptimalsolution,thenwemustalsohave z jt =1for j =1 ;:::;k ¡ 1.Thecontiguitypropertythusensuresthatthequantities P J t j =1 d jt z jt and P J t j =1 r jt d jt z jt correctlyprovidethetotaldemandsatisedandthe totalrevenueinperiod t ,withouttheneedtointroduceanyexplicitdependencies amongthe z jt variablesinourmixedintegerprogrammingformulation. Whilethe[RPP PLC ]formulationcorrectlycapturestheRPP PLC problem wehavedened,itslinearprogrammingrelaxationvaluedoes notnecessarily equaltheoptimalvalueoftheRPP PLC ;i.e.,itsintegralitygapisnotnecessarily zero.Wenextderiveanequivalentproblemformulationforw hichtheintegrality gapisindeedequaltozero.Weshowthisbydevelopingadual-a scentalgorithm

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58 forthedualofthisformulationthatprovidesanoptimaldua lsolutionwhose complementaryprimalsolutionisfeasibleforalloftheinteg errestrictionsofthe [RPP PLC ]formulation.Wenotethatthisapproachgeneralizesarela tedapproach fortheELSPdevelopedbyWagelmans,vanHoesel,andKolen[ 82 ].Alternative approachesalsoincludeextendingtheprooftechniquesfort hereformulatedELSP showninNemhauserandWolsey[ 58 ],Barany,VanRoy,andWolsey[ 10 ],and Barany,Edmonds,andWolsey[ 9 ]. Startingwiththe[RPP PLC ]formulation,wecanequivalentlystatethe objectivefunctionas: minimize: P Tt =1 ( S t y t + p t x t + h t I t ) ¡ P Tt =1 P J t j =1 r jt z jt (3.14) Since I t = P t =1 x ¡ P t =1 P J j =1 d j z j ,wecaneliminatetheinventoryvariables fromtheformulationviasubstitution.Wenextintroduceanew costparameter, c t where c t p t + P T = t h .TheobjectivefunctionoftheRPP PLC cannowbewritten as: minimize: T P t =1 ( S t y t + c t x t ) ¡ T P t =1 h t t P =1 J P j =1 d j z j ¡ T P =1 J P j =1 r j d j z j (3.15) Wenextdene jt asamodiedrevenueparameterforlinearsegment j inperiod t where jt = P T = t h + r jt .Thedevelopmentofourdual-ascentprocedurerequires capturingtheexactamountofproductionineachperiodthat correspondstothe amountofdemandsatisedwithineachlinearsegmentofthepiece wise-linear revenuefunctioninthecurrentandallfutureperiods.Wethu sdene x jt asthe numberofunitsproducedinperiod t correspondingtodemandsatisfactionwithin linearsegment j inperiod ,for t ,andreplaceeach x t with P T = t P J j =1 x jt WenextprovideareformulationoftheLPrelaxationoftheRP P PLC ,whichwe denoteby[RPP 0PLC ],thatlendsitselfnicelytoourdual-basedapproach.

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59 [RPP 0PLC ] minimize: T X t =1 S t y t + c t T X = t J X j =1 x jt ¡ T X t =1 J t X j =1 jt d jt z jt subjectto: P t =1 x jt ¡ d j z j =0 =1 ;:::;T;j =1 ;:::;J ; (3.16) d j y t ¡ x jt 0 t =1 ;:::;T; = t;:::;T;j =1 ;:::;J ; (3.17) ¡ z j ¡ 1 =1 ;:::;T;j =1 ;:::;J ; (3.18) y t ;x jt ;z jt 0 t =1 ;:::;T; = t;:::;T;j =1 ;:::;J : (3.19) Recallthatweintroducedaverysimilarformulation([UOSP 0 ])inChapter 2 forthe purposeofdevelopingheuristicsolutionapproachestotheOSP problem.Inthis section,wedisaggregatethesetupforcingconstraints( 2.13 )from[UOSP 0 ]toarrive attheaboveformulation[RPP 0PLC ]. Notethatif z jt =1inasolutionwesaythatthe demandcorrespondingto segment j inperiod t issatised inthecorrespondingsolution.Thismannerof describingthesolutionwillfacilitateaclearerdescription ofourformulationand thedualalgorithmandsolutionthatlaterfollow.Constraint s( 3.16 )ensurethat ifthedemandinsegment j inperiod issatised,thenaproductionamount equaltothisdemandmustoccurinsomeperiodlessthanorequalt o .Ifany productionoccursinperiod t ,constraintset( 3.17 )forces y t =1,thusallowing productioninperiod t forsegment j demandinperiod toequalanyamount upto d j ;otherwise,if y t =0,noproductioncanbeallocatedtoperiod t Constraints( 3.18 )and( 3.19 )representthe(relaxed)variablerestrictions.Note thatsinceapositivecostexistsforsetups,wecanshowthattheconst raint y t 1isunnecessaryintheaboverelaxation,andsoweomitthisconst raintfrom

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60 therelaxationformulation.Itisstraightforwardtoshowtha tthe[RPP 0PLC ] formulationwiththeadditionalrequirementthatall y t arebinaryvariablesis equivalenttoouroriginalRPP PLC Toformulatethedualof[RPP 0PLC ],let j w jt ,and j denotedualmultipliersassociatedwithconstraints( 3.16 ),( 3.17 ),and( 3.18 ),respectively.Takingthe dualof[RPP 0PLC ],wearriveatthefollowingdualformulation[DP]: [DP] maximize: P T =1 P J j =1 ¡ j subjectto: T P = t J P j =1 d j w jt S t t =1 ;:::;T; (3.20) j ¡ w jt c t t =1 ;:::;T; = t;:::;T;j =1 ;:::;J (3.21) ¡ d j j ¡ j ¡ j d j =1 ;:::;T;j =1 ;:::;J ; (3.22) j ;w jt 0; j unrestricted t =1 ;:::;T; = t;:::;T;j =1 ;:::;J : (3.23) Inspectionofformulation[DP]indicatesthatwecanset w jt equaltothemaximumbetween0and j ¡ c t withoutlossofoptimality;similarly,anoptimal solutionexistswith ¡ j equaltotheminimumbetween0and d j ( j ¡ j ).The aboveformulationcan,therefore,bere-writteninamoreco mpactformas: [CDP] maximize: T P =1 J P j =1 min(0 ;d j ( j ¡ j )) subjectto: T P = t J P j =1 d j f max(0 ; j ¡ c t ) g S t t =1 ;:::;T: (3.24) Wenotesomeimportantpropertiesofthe[CDP]formulation.F irst,wehaveno incentivetosetany j variablevalueinexcessof j ,sinceanyincreaseabovethis valuedoesnotaecttheobjectivefunctionvalue.Second,we caninitiallyseteach

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61 j =min t =1 ;:::; f c t g forall =1,..., T and j =1 ;:::;J ,withoututilizinganyof the\capacity", S t ,ineachconstraint.Wecanalsoeliminateanysegment-period combination( j; )suchthatmin t =1 ;:::; f c t g j ,sinceanydemandsatisedwithin suchasegmentwillneverprovideapositivecontributiontopro t.Indescribing oursolutionapproach,wewillrefertotheconstraintforperi od t in[CDP]asthe t th constraint(orconstraint t )oftheformulation.Ourapproachforsolving[CDP] istouseadual-ascentprocedurethatincreasesthedualvariab lesinincreasing timeindexorder.Thatis,weincreasethevaluesofthe j 1 variablesbeforewe increaseany jt valuesfor t> 1.Wethenfocusonincreasingthe j 2 variables, andsoon.Webeginbysimultaneouslyincreasingthevalueofall j 1 variables. Ifforsomesegment l inperiod1, l 1 reachesavalueof l 1 beforeconstraint1 becomestight,wesaythatthissegment\dropsout"inperiod1a ndwedonot furtherincreasethevalueof l 1 (i.e., l 1 isxedat l 1 inthesolution).Wethen continuetoincreaseallother j 1 valuesuntilconstraint1becomestight.Let J 0 t denotethesetofallsegmentsthatdropoutinperiod t ,andlet J 1 t denotetheset ofallsegmentsthatdonotdropoutinperiod t .Wedene ¤1 asthevalueof j 1 forallsegmentsthatdonotdropoutinperiod1,where ¤1 = c 1 + S 1 ¡ P j 2 J 0 1 d j 1 max(0 ; j 1 ¡ c 1 ) P j 2 J 1 1 d j 1 : Notethatatthispoint,afterdetermining ¤1 ,therstconstraintof[CDP]istight (assuming J 1 1 6 = ; ;welaterdiscussthenecessarymodicationsif J 1 1 = ; ).We nextfocusonincreasingthe j 2 variablevalues.Whenweincreasethevaluesofthe j 2 variables,thesevariablescanbeblockedfromincreasebyeith erdroppingout (i.e.,when l 2 = l 2 forsomesegment l ),bytighteningconstraint2,orbyhitting thevalue c 1 (observethatno j 2 valuecanbegreaterthan c 1 sinceconstraint1is alreadytight,andsuchavaluewould,therefore,violatecon straint1).Letting ¤2

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62 equalthevalueof j 2 forall j 2 J 1 2 ,wehave ¤2 =min ( c 1 ; c 2 + S 2 ¡ P j 2 J 0 2 d j 2 max(0 ; j 2 ¡ c 2 ) P j 2 J 1 2 d j 2 ) : Applyingthissameapproachinperiod3produces ¤3 =min 8>>>>><>>>>>: c 1 ; c 2 + S 2 ¡ P j 2 J 0 2 d j 2 max(0 ; j 2 ¡ c 2 ) ¡ P j 2 J 1 2 d j 2 max ( ¤2 ¡ c 2 ; 0 ) ¡ P j 2 J 0 3 d j 3 max(0 ; j 3 ¡ c 2 ) P j 2 J 1 3 d j 3 ; c 3 + S 3 ¡ P j 2 J 0 3 d j 3 max(0 ; j 3 ¡ c 3 ) P j 2 J 1 3 d j 3 9>>>>>=>>>>>; ; oringeneral,forperiod : ¤ =min i 8><>: c i + S i ¡ P t = i P j 2 J 0 t d jt max(0 ; jt ¡ c i ) ¡ ¡ 1 P t = i P j 2 J 1 t d jt max( ¤t ¡ c i ; 0) P j 2 J 1 j d j 9>=>; : (3.25) Ournaldualsolutiontakestheform: j = 8><>: j ;j 2 J 0 ; ¤ ;j 2 J 1 ; for =1 ;:::;T; and j =1 ;:::;J : Notethatitispossiblethattheset J 1 isemptyforsome afterapplyingthe algorithm,sinceallordersinperiod maydropoutbeforehittinganyofthe constraints.Insuchcases ¤ requiresnodenition.Wecansummarizethisdualascentsolutionapproachasfollows:CDPDual-AscentSolutionAlgorithm0. Deleteanysegment-periodcombination( j )suchthatmin t =1 ;:::; f c t g j 1. Set j =min t =1 ;:::; f c t g forall =1,..., T and j =1 ;:::;J .Setiteration counter k =1. 2. Let J 0 k = J 1 k = f;g .Simultaneouslyincreaseall jk for j =1 ;:::;J k fromtheinitialvalueofmin t =1 ;:::;k f c t g .If,whileincreasingthe jk values, some lk = lk beforethe jk valuesareblockedfromincreasebyany

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63 constraint,x lk at lk ,insertsegment l into J 0 k ,andcontinuetosimultaneouslyincrease jk forall j= 2 J 0 k untilsomeconstraint ( k ) k blocksthe jk valuesfromfurtherincrease.Whenconstraint ( k ) k blocksthe jk valuesfromfurtherincreasethen,forallsegments j= 2 J 0 k insert j into J 1 k andset jk usingequation( 3.25 );i.e.,set jk = ¤k = c ( k ) + S ( k ) ¡ k P t = ( k ) P j 2 J 0 t d jt max ( 0 ; jt ¡ c ( k ) ) ¡ k ¡ 1 P t = ( k ) P j 2 J 1 t d jt max ( ¤t ¡ c ( k ) ; 0 ) P j 2 J 1 k d jk .(Ifall j =1 ;:::;J k enter J 0 k beforesomeconstraintbecomestight,then ¤k requiresnodenition.) 3. Set k = k +1.If k = T ,stopwithdualfeasiblesolution.Otherwise,repeat Step2. Notethatineachperiod k wemustcheckthevalueof jk foreachsegment j =1 ;:::;J k anddeterminewhetherthisvalueof jk willtightenorviolateany oftheconstraints1,..., k .Sinceweneedtoapplythiscomparisonfor k =1,..., T ,wecanboundthecomplexityofthisdual-ascentalgorithmby O ( J max T 2 ),the sameasthatoftheshortest-pathalgorithmintheprevioussecti on.Wenextshow thatthedual-ascentsolutionprocedureoutlinedabovenoton lysolves[CDP],but alsoleadstoaprimalcomplementarysolutioninwhichallofth ebinaryrestrictions informulation[RPP PLC ]aresatised;i.e.,thedual-ascentproceduresolvesthe RPP PLC .Beforeshowingthis,werstneedthefollowinglemma. Lemma1 Foranypairofpositiveintegers and l suchthat + l T and ¤ and ¤ + l aredenedasinthedual-ascentalgorithm,wenecessarilyhave ¤ ¤ + l Proof: Let k< besuchthat ¤ = c k + S k ¡ P t = k P j 2 J 0 t d jl max(0 ; jt ¡ c k ) ¡ P ¡ 1 t = k P j 2 J 1 t d jl max( ¤t ¡ c k ; 0) P J 2 J 1 d j ; fromwhichwecanconcludethat ¤ c k (sincethenumeratorontherighthand sideistheslackofconstraint k ,whichmustbenonnegative,sincewemaintaindual

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64 feasibilityatalltimes).Nextconsider ¤ + l : ¤ + l c k + S k ¡ P + l t = k P j 2 J 0 t d jl max(0 ; jt ¡ c k ) ¡ P + l ¡ 1 t = k P j 2 J 1 t d jt max( ¤t ¡ c k ; 0) P j 2 J 1 + l d j; + l Since ¤ tightensconstraint k ,thequantityinthenumeratorabovemustbezero andwethereforehave ¤ + l c k ¤ forall =1 ;:::;T and + l T ,since waschosenarbitrarily. } Lemma 1 isrequiredforprovingthefollowingresult,theproofofwhi chcanbe foundinAppendixBofSection 3.5 Theorem2 Thedual-ascentalgorithmpresentedabovesolves[CDP].Mor eover, thecomplementaryprimalsolutiontothedualsolutionprodu cedbythealgorithm satisestheintegralityrestrictionsoftheRPP PLC andthereforeprovidesan optimalsolutionfortheRPP PLC Theorem 2 impliesthatformulation[RPP PLC ]istight,andwecaneasilynd thesolutionvaluefortheRPP PLC usingalinearprogrammingsolver.Thealgorithmswehavedeveloped,however,havebetterworstcasecomp lexity( O ( J max T 2 )) thansolutionvialinearprogramming.Toprovidesomeinsighto nthestructureof theprimalsolution,giventhedualsolutions,wecanshowthatth etightconstraints inthedualsolutioncorrespondtoperiodsinwhichwesetupinth ecomplementary primalsolution.Further,if jk = ¤k ,thenthedemandinsegment j inperiod k issatisedusingthesetupcorrespondingtotheconstraintthatblo cked ¤k from furtherincrease(i.e.,period ( k )fromStep2ofthedual-ascentalgorithm). AswasshowninChapter 2 ,wecannotreducethisboundto O ( T log T ),as FedergruenandTzur[ 24 ]andWagelmans,vanHoesel,andKolen[ 82 ]doforthe ELSP,sincewecannotensurethatcumulativedemandsatisedaswe increasethe numberofperiodsinaprobleminstanceisnondecreasing.SeeS ection 2.2.2 fora presentationofacounterexample.

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65 3.2.3 PolynomialSolvabilityforOtherProductionCostsandPrice -DemandCurves Tothispointwehavemadetwosetsofkeyassumptionsthathavefa cilitated providingpolynomial-timesolutionmethodsfortheuncapac itatedRPP.Therst oftheseassumptionsreliesontheproductioncostfunctiontaki ngaxed-charge structureineachperiod,whilethesecondassumesthatasinglepr ice-demand curveexistsineachperiod.Wenextexplorethedegreetowhic hwecanrelaxthese assumptions,whileretainingourabilitytoapplythepolynomi al-timesolution methodswehavepresented.Firstweconsidercontextsinwhichm ultiplepricedemandresponsecurvesexistineachperiod;thiswouldcorrespo ndtocontextsin whichtheproducerhasmultipleavailablemarketsinwhicht osellitsoutput,with eachmarkethavingauniqueresponsetoprice.Wethenconsidert heimpactsofa piecewise-linearconcaveproductioncoststructure(whichma yincludeaxedsetup cost)ineachperiod. 3.2.3.1 Multipleprice-demandcurves InthissectionweshowthatanyuncapacitatedRPPwithmultipl edemand curvesinaperiodcanbereformulatedasanRPPwithonlyasing ledemandcurve perperiod.Wewillshowthatthisholdsforgeneralconcavere venuefunctionsand forpiecewise-linearconcavefunctionsinparticular.This impliesthatthepiecewiselinearconcavityoftherevenuefunctionsispreservedunder thetransformation fromamultipledemandcurveperperiodproblemtoasingledem andcurveper periodproblem.Supposewenowhave M t distinctrevenuefunctionsinperiod t eachcorrespondingtoadistinctrevenue source ,andthat D mt isnowthedecision variablefortheamountofdemandwesatisfyforrevenuesource m inperiod t ; R mt ( D mt )istherevenuefunctionassociatedwithrevenuesource m inperiod t (arevenuesourcemaybeanindividualmarketorcustomer).Wec anrewritethe

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66 uncapacitatedRPPas maximize P Tt =1 P M t m =1 R mt ( D mt ) ¡ P Tt =1 ( g t ( x t )+ h t I t ) subjectto: P M t m =1 D mt = D t t =1 ;:::;T; D t + I t = x t + I t ¡ 1 t =1 ;:::;T; x t ;I t ;D t 0 t =1 ;:::;T; D mt 0 m =1 ;:::;M t ;t =1 ;:::;T: Nowobservethat,foragivenchoiceof D t ,wewillchoosethedemandquantities foreachrevenuesourcethatyieldthemaximumprot.Sotheunc apacitatedRPP isequivalentto maximize P Tt =1 ~ R t ( D t ) ¡ P Tt =1 ( g t ( x t )+ h t I t ) subjectto: D t + I t = x t + I t ¡ 1 t =1 ;:::;T; x t ;I t ;D t 0 t =1 ;:::;T: wherethe aggregaterevenuefunction forperiod t ~ R t ( D t ),isdenedthroughthe followingsubproblem(SP)as[SP] ~ R t ( D t ) max ( M t X m =1 R mt ( D mt ): M t X m =1 D mt = D t ; D mt 0 ;m =1 ;:::;M t ) : Thefunction ~ R t ( D t )isconcave(seeRockafellar[ 66 ]Theorem5.4),andclearly ~ R t (0)=0.Itnowalsoeasilyfollowsthatif R mt ( ¢ )ispiecewise-linearandconcave (and R mt (0)=0)forall m and t ~ R t ( ¢ )ispiecewise-linearandconcaveforall t (and ~ R t (0)=0).Thiscanbeshownbyorderingtheslopesofallsegmentsi nagiven periodindecreasingorder,andnotingthatthefunction ~ R t ( D t )will\use"these segmentsinnondecreasingindexorder(ornonincreasingvalue order).

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67 Observethatifthe R mt ( ¢ )functionsarealleverywheredierentiable,thenthe demandvaluesselectedforeachrevenuesourceinagivenperio d t asaresultof solvingsubproblem[SP]willbesuchthat R 0 1 t ( D 1 t )= R 0 2 t ( D 2 t )= ¢¢¢ = R 0 M t t ( D M t t ). Inotherwords,attheoptimaldemandlevel,themarginalreve nueforeachrevenue sourcewillbeequal.Thus,iftherevenuesourcesaredistinctbu thaveidentical revenuefunctions,wewillofcoursechargethesamepricetoeve ryrevenuesource. 3.2.3.2 Piecewise-linearconcaveproductioncosts Wenextconsiderthecaseinwhichtheproductioncostfunctioni neach periodispiecewise-linearconcaveandnondecreasinginthep roductionvolume intheperiod.Notethatanynondecreasingpiecewise-linearco ncavefunctioncan beviewedastheminimumofanumberofxed-chargefunctions.T herefore,if theproductionfunctionsarepiecewise-linearandconcavew ithanitenumber ofsegments,wecanviewthisasachoicebetweenanitenumberof alternative productionmodes.Itiseasytoseethat,inanyperiod,wewillof courseonlyusea singleproductionmodewithoutlossofoptimality. Wecanwritesuchaproductioncostfunctioninthefollowingfo rm: g t ( x )= 8><>: 0if x =0 ; min k =1 ;:::;` t f S kt + p kt x g if x> 0 ; where k denotesanindexfordierentproduction\modes".Givenaseque nce ofperiods t;:::;t 0 ¡ 1andpositiveproductioninperiod t ,wenowessentially alsoneedtochoosewhichofthe ` t costfunctions(orproduction modes )touse. Givenaproductionsetupinperiod t only,theunitproductionplusholdingcost associatedwithperiod ( = t;:::;t 0 ¡ 1)underproductionmode k equals v kt p kt + P ¡ 1 s = t h s .Aswithourpreviousanalysisanddevelopmentoftheshortest pathalgorithm(seeTheorem 1 ),theoptimalquantityofdemandsatisedinperiod

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68 underproduction mode k usingasetupinperiod t ,whichwedenoteby D ¤ k ( t ), isthenequaltoanyvalueof D suchthat v kt isinthesetofsubgradientsof R ( D ) at D .Let ¦ k ( t;t 0 )= t 0 ¡ 1 X = t ( R t ( D ¤ k ( t )) ¡ v kt D ¤ k ( t )) ¡ S kt and c ( t;t 0 )=max 0 ; max k =1 ;:::;` t ¦ k ( t;t 0 ) : Thevalueof¦ k ( t;t 0 )providesthemaximumprotpossibleinperiods t;:::;t 0 ¡ 1 underproductionmode k assumingwesatisfydemandamountsof D ¤ k ( t )for = t;:::;t 0 ¡ 1.Asaresult, c ( t;t 0 ),asbefore,providesthemaximumpossible protinperiods t;:::;t 0 ¡ 1assumingtheonlysetupthatcansatisfydemand intheseperiodsmustoccurinperiod t (ifatall).Wecanthereforeusethesame shortestpathgraphstructureasbefore(showninFigure 2{2 )withthesemodied arclengthcomputationstodetermineanoptimalsolution.Not ethatduetothe concavityoftheproductioncostfunction,automatically,t heproductionquantity correspondingtothebestproductionmode k liesinthecorrectsegment;i.e.,the productioncostshavebeencomputedcorrectly.Thetimerequ iredtondallarc protsis O ( LT 2 )multipliedbythetimerequiredtond D ¤ k ( t )forsome k;t; where L =max t =1 ;:::;T ` t isthemaximumnumberoflinearsegmentsforanyof the T piecewise-linearconcaveproductioncostfunctions.Asthisan alysisshows, thecaseofpiecewise-linearconcaveproductioncostfunction scanbehandledina straightforwardmanner,evenundergeneralconcaverevenue functions,withouta substantialincreaseinproblemcomplexity. 3.2.4 ProductionCapacities ThissectionconsidersacapacitatedversionoftheRPP PLC whereproduction capacitiesareequalinallperiods.InChapter 2 ,weshowedthatRPP PLC with time-varyingniteproductioncapacitiesisNP-Hardbydemonst ratingthatit generalizesthecapacitatedlotsizingproblem(CLSP).Thesp ecialcaseofthe

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69 CLSPwhereproductioncapacitiesareequalineveryperiod, however,canbe solvedinpolynomialtime(seeFlorianandKlein[ 28 ])withacomplexityof O ( T 4 ). Becauseofthis,wenextinvestigatewhethertheequal-capacit yversionofthe RPP PLC containsasimilarspecialstructurethatwemightexploittosol vethis probleminpolynomialtime. Thepolynomialsolvabilityoftheequal-capacityCLSPrelie soncharacterizing so-called regenerationintervals (FlorianandKlein[ 28 ]).Aregenerationintervalis characterizedbyapairofperiods, and 0 (with < 0 )suchthat I = I 0 =0, and I +1 I +2 ,..., I 0 ¡ 1 > 0inanoptimalsolution.Anoptimalsolutiontherefore consistsofasequenceofregenerationintervals(includingth epossibilityofasingle regenerationinterval(0 ;T )).A capacityconstrainedsequence betweenperiods +1 and 0 isoneinwhich x t =0or C forallperiodsbetween(andincluding) +1and 0 exceptforatmostone.Fortheequal-capacityCLSP,anoptima lsolutionexists consistingofacapacityconstrainedsequencewithineachregen erationinterval(see FlorianandKlein[ 28 ]).GivenanychoiceofdemandsineveryperiodfortheequalcapacityRPP PLC problem,theresultingproblemisanequal-capacityCLSP;th us, anoptimalsolutionexistsfortheequal-capacityRPP PLC problemthatconsists ofcapacityconstrainedproductionsequenceswithineachofa setofconsecutive regenerationintervals. Let D 0 = P 0 t = +1 d t denotethetotaldemandsatisedbetweenperiods +1and 0 ,where d t isthedemandsatisedinperiod t .If( ; 0 )comprises aregenerationinterval,weknowthattotalproductioninpe riods +1 ;:::; 0 mustequal D 0 (since I = I 0 =0and D 0 isthedemandsatisedinperiods +1 ;:::; 0 ).Sinceatmostoneperiodcontainsproductionatavalueothe rthan 0or C inacapacity-constrainedsequence,wemusthave D 0 = kC + ,where k issomenonnegativeinteger,and istheamountproducedintheperiodinwhich wedonotproduceat0or C (assuming D 0 isnotevenlydivisibleby C ,inwhich

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70 case equalszero).So,given D 0 ,ineachoftheperiods +1 ;:::; 0 ,weeither produce0, ,or C ,withaproductionamountof inonlyoneoftheperiods.We caneasilydetermineboth k and given D 0 and C ;i.e., = D 0 (mod C ),and k = b D 0 =C c .Wethenconstructashortest-pathgraphthatcontainsapathfo r everyfeasiblecapacity-constrainedproductionsequencebet weenperiods +1 and 0 .Solvingthisshortest-pathproblemprovidestheminimumcost capacity constrainedsequenceforevery( ; 0 )pair(with 0 > ).Givenavalueof D 0 for everypossible( ; 0 )pair,wecanusethis O ( T 4 )CLSPsolutionapproachtosolve theequal-capacityRPP PLC .Thechallengethenliesindeterminingappropriate D 0 valuesforeachpossible( ; 0 )pair.Toaddressthisissue,wenextshowthat thecandidatesetof D 0 valuesforeach( ; 0 )paircanbelimitedtoamanageable numberofchoices.NotethatLoparic[ 49 ]providesasimilaranalysisforalot-sizing modelinwhichtotalrevenueislinearintheamountofdemand satised. 1 Consideraregenerationinterval( ; 0 ),andrecallthatbydenitionwemust have I =0 ;I j > 0for j = +1 ;:::; 0 ¡ 1 ; and I 0 =0.Theadjustedrevenue parameterthatweintroducedinSection 3.2 (i.e., jt = r jt + P Ts = t h s for
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71 optimalsolutionexistssuchthat it 0 > 0 ,thenanoptimalsolutionalsoexistswith jt = d jt Proof :Considertheregenerationinterval( ; 0 )andconsidersome jt it 0 with 0,and jt 0for t = +1 ;:::; 0 ¡ 1,wecanincrease jt bysome > 0anddecrease it 0 by withoutchangingtheamountproducedineachoftheperiods +1 ;:::; 0 Inparticular,if t t 0 wecanset =min f d jt ¡ jt ; it 0 ;min f I t ;:::;I t 0 ¡ 1 gg .The resultingchangeinobjectivefunctionvalueequals( r jt ¡ ( r it 0 ¡ P t 0 ¡ 1 s = t h s )) = ( jt ¡ it 0 ) 0,andweeitherhave jt = d jt it 0 =0,or I s =0forsome s = t;:::;t 0 ¡ 1(inthelatercase, t and t 0 nolongerbelongtothesameregeneration interval).Similarly,if t 0
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72 AppendixCinSection 3.5 containsaproofofLemma 3 .Lemmas 2 and 3 taken togetherimplythatalimitednumberofcandidateoptimalsol utionsmustbe consideredforeachpossibleregenerationinterval(notethat thenumberofpossible regenerationintervalsisboundedby O ( T 2 )).Letting J max denotethemaximum numberoflinearsegmentsoftherevenuefunctionsamongallp eriods(i.e., J max = max s =1 ;:::;T f J s g ),Lemmas 2 and 3 leadtothefollowingtheorem: Theorem3 Theequal-capacityRPP PLC problemcanbesolvedin O ( J max T 7 ) time. Proof :Considerapotentialregenerationinterval( ; 0 )containing n periods,and let J ( ; 0 )denotethetotalnumberoflinearsegmentsinperiods +1 ;:::; 0 .For potentialregenerationinterval( ; 0 )wesort J ( ; 0 )valuesof jt .Letthisindex sequenceofsortedvaluesbedenotedby ¡ 1 ; 2 ;:::; J ( ; 0 ) ¢ (i.e., 1 2 ::: J ( ; 0 ) ),whereeachindex i identiesauniquesegment-periodpairwithinthe regenerationinterval.Forpotentialregenerationinterv al( ; 0 ),notethatLemma 2 impliesthatif i takesavaluestrictlybetween0and d i wemusthave i + k =0 for k =1 ;:::;J ( ; 0 ) ¡ i Lemma 3 impliesthatwithineachpotentialregenerationinterval( ; 0 )of length n weneedtoconsidertwotypesofsolutions.Thersttypeofsolution producesaquantityofzeroor C ineachofthe n periods.Forthistypeofsolution wewillhaveatmostone i
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73 Thesecondtypeofsolutionwemustconsidersetseach i equaltozeroor d i forall i =1 ;:::;J ( ; 0 )andproducesatavaluestrictlybetweenzeroand C inat mostoneperiodintheregenerationinterval.Thechoiceofth eindex i suchthat i ¡ k = d j i ¡ k for k =0 ;:::;i ¡ 1and i + k =0for k =1 ;:::;J ( ; 0 ) ¡ i uniquely determinesthenumberofperiodsinwhichproductionatfull capacityisrequired, andthevalueofproductionrequiredinthesingleperiodsucht hat x t
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74 orequivalently mt = d ¡ 1 mt ( D t ),where mt denotesthepriceoeredtomarket m inperiod t ,and d ¡ 1 mt ( D t )isdeterminedbysolvinganoptimizationsubproblem, asdiscussedinSection 3.2.3.1 .Givenatotaldemandsatisedof D t inperiod t ,wealsoassumedthatatotalrevenueof R t ( D t )isrealized,where R t ( D t )isa nondecreasingconcavefunctionofdemand D t Giventhisrelationshipbetweendemandandrevenue,wecanin terpret theactualpricespaidfortheunitssoldinatleasttwoways,dep endingonthe model'sintendedapplicationcontext.WeuseFigure 3{1 toillustratetwosuch interpretations.Figures 3{1 (a)and 3{1 (b)showidenticalpiecewise-linearrevenue curveswiththreesegmentsandsegmentslopes r 1 >r 2 >r 3 .Inbothcases,the totalrevenueachievedatthedemandlevel D 0 equals R ( D 0 )= r 1 d 1 + r 2 d 2 + r 3 d 3 .In Figure 3{1 (a)weassumethatamarketexistswithatotalof d 1 unitsofdemand, eachofwhichiswillingtopayanamountof r 1 perunitofthegood,whileasecond marketwithatotalof d 2 unitsofdemandprovidesarevenueof r 2 perunit,and athirdmarketcontains d 3 unitsofdemandwitharevenueof r 3 perunit.Inthis case,thepricepaidforunitsofdemandfallingwithinasegmen tcorrespondsto theslopeofthesegment.Thisinterpretationmightapplywhen dierentmarket segments(e.g.,geographicalsegments)actuallypaydierentp rices,andeachofthe d j valuescorrespondstoagivenmarket m 'stotalavailabledemandvalue, d mt ,in period t ;i.e., d j representsamarket\size"inFigure 3{1 (a). D R ( D ) r 1 r 2 r 3 d 1 d 2 d 3 Slope = R ( D ')/ D ( = single price value for all demands) D D R ( D ) r 1 ( = price value 1) r 2 ( = price value 2) r 3 ( = price value 3) d 1 d 2 d 3 D (a) (b) Figure3{1:Pricinginterpretationsbasedontotalrevenuea nddemand.

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75 InFigure 3{1 (b),ontheotherhand,weassumethatwehaveonlyasinglemarketavailable,andallsatiseddemandsprovidethesamep er-unitrevenue(price),whichatademandlevelof D 0 isgivenby ( D 0 )= R ( D 0 ) =D 0 = P 3j =1 r j d j = P 3j =1 d j .Thisinterpretationimpliesthatatotalof D 0 demandsexist thatarewillingtopay ( D 0 ),whichequalstheslopeofthelineconnectingtheoriginto R ( D 0 ).Thisinterpretationappliestocasesinwhichthesupplierm ustcharge asinglepricetoallcustomersinthemarket.Ineithercase,the modelsarecompletelythesame,butthepricinginterpretationsandthecon textstowhichthese interpretationsapplyarequitedierent.Notethatwhenther evenuefunctionis characterizedbyadierentiableconcavefunction,theonly practicalinterpretation isoneinwhichthepricepaidforeachunitequalstheslopeoft helineconnecting theoriginto R ( D 0 ),whichis R ( D 0 ) =D 0 ( R 0 ( D 0 )ofcourseindicatesthe marginal total revenueat D 0 ). GivenourinterpretationofFigure 3{1 (a),wecannowviewtheindividual segmentsofthepiecewise-linearrevenuecurveinadierentli ght.Thatis,each linearsegmentmaynotonlycorrespondtoseparateunitsofdema ndfroman individualmarket,butmightalternativelybeassociatedwit hanaggregateorder fromanindividualcustomer,whereindividualcustomersarew illingtopay dierentunitpricesfortheitem(or,alternatively,dieren tcustomershavea dierentunitcostassociatedwithfulllingtheirorders).Given thisinterpretation, theRPP PLC modelcanbeutilizedinabroadersetofcontexts,wheretherm doesnotsetprices,butcanselectfromanumberofcustomerorder s,eachofwhich oersaparticularnetrevenueperunitordered. Recallthat,inthe\orderacceptanceanddenial"environme ntintroduced inChapter 2 ,rmscaneithercommittofulllinganorderordeclinetheord er basedonseveralfactors,includingthecapacitytomeettheord erandtheeconomic attractivenessoftheorder.TheRPP PLC modelcanalsobeappliedwithinsuch

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76 contexts.Inthisorderselectionsetting,wenowassumethataseto fordersfor thesupplier'sgoodexistsineachofthe T periodsoftheplanninghorizon,and redene J t asthenumberofordersthatrequestfulllmentinperiod t .Theindex j nowcorrespondstoindividualorderindices,andwelet r jt denotetherevenue perunitprovidedbyorder j inperiod t ,while d jt istheorderquantityassociated withorder j inperiod t .Weindexallorderswithinaperiodinnonincreasingorder ofunitrevenues(i.e., r 1 t r 2 t ¢¢¢ r J t t ).Weredenethebinary z jt variables previouslyusedinthe[RPP PLC ]formulationasfollows: z jt =1ifweacceptorder j inperiod t ,and z jt =0otherwise.Thesevariablescannowbeinterpretedas orderselectionvariables.Theremainingproductionquantit y( x t ),productionsetup ( y t ),andinventory( I t )variablesinthe[RPP PLC ]formulationretaintheiroriginal denition. Sincetheformulationiscompletelyunchangedexceptforou rinterpretation ofthemeaningofcertainparametersanddecisionvariables,w ecanusethesame shortestpathanddual-ascentmethodswepresentedtosolvethis orderselection problem.Intheuncapacitatedproductionsetting,recallth atanoptimalsolution existsfortheRPP PLC problemsuchthattheamountofdemandselectedineach periodfallsatoneofthebreakpointsoftherevenuefunctio n.Undertheorder selectioninterpretation,thisimpliesthatanoptimalsolut ionexistsinwhichevery orderwillbeeitherfullyaccepted(andfullledinitsentir ety),orwillbedeclined. Wenextbrieydiscusstheimplicationsofnitecapacitylimits withinthe orderselectioncontext,againrestrictingourdiscussiontothe equal-capacity case.Sincethe[RPP PLC ]formulationservedasourstartingpointfortheanalysis oftheequal-capacitycase,andtheuncapacitatedorderselec tionproblemis formulatedexactlythesameasthe[RPP PLC ]formulation,wecanessentiallyfollow thediscussioninSection 3.2.4 withourneworderselectioninterpretation.This approachassumes,however,thatcustomerswillpermitpartialo rdersatisfaction;

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77 i.e.,fororder j inperiod t wearefreetosatisfyanyamountoftheorderquantity betweenzeroand d jt .Forcontextsinwhichsuchpartialordersatisfactionis allowed,wecanthereforeapplythesameapproachdiscussedinSe ction 3.2.4 tosolvetheequal-capacityversionoftheorderselectionprob lem.If,however, customersdonotpermitpartialordersatisfaction,theproble misNP-Hard. Todemonstratethedicultyoftheproblemwhenpartialordersa tisfactionis notallowed,wenextbrieyconsiderthesingle-periodspecialc aseofthisproblem, where T =1.Notethatwenowexplicitlyrequirethebinaryrestriction sonthe z j 1 variablesforthisproblem.Forthissingle-periodspecialca sewecanwritethe inventorybalanceconstraintsas x 1 = P J 1 j =1 d j 1 z j 1 ,whichsimplyimpliesthatthe productionintheonlyperiodmustequalthedemandwechooseto satisfy.Given thatwehaveonlyasingle-periodproblem,wewilleitherperf ormasetupornot.If wedonotperformasetup,thentheobjectivefunctionequalsz ero.Ifwedosetup, thenweneedtosolvethefollowingproblemtodeterminetheop timalsolution: maximize: J 1 P j =1 ( R j 1 ¡ p 1 d j 1 ) z j 1 subjectto: J 1 P j =1 d j 1 z j 1 C; z j 1 2f 0 ; 1 g ;j =1 ;:::;J 1 : Theaboveproblemisaknapsackprobleminitsmostgeneralform (sincethe R j 1 and d j 1 parameterscantakearbitrarynonnegativevalues),indicat ingthat theall-or-nothingordersatisfactionversionofthecapacita tedproblemproblem isNP-Hard,eveninthesingle-periodspecialcase(althoughthesi ngle-period versionisnotstronglyNP-Hard).Thisproblemisthereforeclea rlyNP-Hardfor themultiple-periodcasewithorwithoutequalcapacitiesin allperiods.

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78 3.4 Conclusions Allocatingappropriateamountsofresourcestoanticipatedd emandsources hasbeenawell-researchedprobleminrevenuemanagement,al thoughthework hasprimarilyfocusedonserviceindustryapplications(e.g., airlineandhospitality industryapplications;see,forexample,vanRyzinandMcGill[ 79 ]).Aswehave discussed,anincreasingamountofattentionisbeingplacedonr evenuemanagement,throughpricingmodels,inmanufacturingcontexts.Wec ontributetothis eortbyprovidingmodelsandecientsolutionmethodsforagen eralsetofpricing problemsinmanufacturingsettingswherexedsetupcostscompr iseasubstantial partofoperationscosts.Inadditiontopricingapplications, weshowedthatour modelingapproachalsoappliestoorderselectionproblems,th efocusofChapter 2 inwhichasuppliermustchoosefromasetofoutstandingorderstom aximizeits contributiontoprotafterproductioncosts.Aswehaveshown,ou rmodelsand methodsalsoallowforecientlysolvingproblemsinwhichtime -invariantnite productioncapacitiesexist. Mostrevenuemanagementliteratureaddressesanticipateddem andthatis stochasticinnature,whichiswhyselectingthebestutilizatio nofresourcesto achievemaximumprotissuchadicultproblem.Inthenextchap ter,wealso considertheeectsofstochasticdemandonourdemandselectiond ecisions. 3.5 Appendix AppendixATheorem 1 FortheuncapacitatedRPP,givenaproductionsetupinperiod t only, ifademandquantity D existssuchthat v t isinthesetofsubgradientsof R ( ¢ ) at D ,then D isanoptimaldemandquantityforthesubproblemgivenby( 3.4 )and ( 3.5 ). Proof: Givenaperiod t ,andassumingasetupinperiod t only,weneed tochoosethedemandquantity d inperiod t suchthatthetotalrevenueat d in

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79 period t minusthetotalcostincurredinsatisfyingthequantity d inperiod t is maximized.Bythedenitionof v t ,thetotalcost(excludingthesetupcost,which hasalreadybeenincurred)forsatisfying d unitsinperiod t usingproductionin period t equals v t d .Weneedtothereforesolvethefollowingproblemtodetermin e theoptimaldemandvaluetosatisfyinperiod : maximize: R ( d ) ¡ v t d subjectto: d 0 : Consideravalue D suchthat v t isinthesetofsubgradientsof R ( d )at d = D .Thisimpliesbythedenitionofasubgradientofaconcavefun ctionthat R ( d ) R ( D )+ v t ( d ¡ D )forall d 0(thedomainof R ( ¢ )).Thisimpliesthat R ( d ) ¡ v t d R ( D ) ¡ v t D forall d 0,whichimpliestheresult. } AppendixBTheorem 2 Thedual-ascentalgorithmpresentedinSection 3.2.2 solves[CDP]. Moreover,thecomplementaryprimalsolutiontothedualsoluti onproducedbythe algorithmsatisestheintegralityrestrictionsofformulatio n[RPP PLC ]andtherefore providesanoptimalsolutionfortheRPP PLC Proof: Let F ( )denotetheoptimalvalueofaproblemconsistingofperiods1, ..., .Aswehavedemonstratedthroughourshortest-pathapproach,th efollowing recursiverelationshipholdsfortheRPP PLC : F ( )=min i F ( i ¡ 1)+min S i + c i d i ¡ i ; where d i = P t = i P j 2 J ¤ ( t;i ) d jt , i = P t = i P j 2 J ¤ ( t;i ) jt d jt ,and J ¤ ( t;i )= f j : jt c i g Inourdualproblem,theonlyvariablescontributingvaluet otheobjective functionarethosecontainedinthesets, J 1 for =1,..., T .Inotherwords, letting Z T D denotetheobjectivefunctionvalueofourdualsolutionfora T -period

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80 problem,wehave Z T D = T X t =1 X j 2 J 1 t min(0 ;d jt ( ¤t ¡ jt ))= T X t =1 X j 2 J 1 t d jt ( ¤t ¡ jt ) ; since jt ¤t forall j 2 J 1 t bydenition.Toshowtheoptimalityofourdualascentprocedure,weneedtoshowthat F ( )= X t =1 X j 2 J 1 t d jt ( ¤t ¡ jt ) ; wherewehave F ( ) P t =1 P j 2 J 1 t d jt ( ¤t ¡ jt )forallfeasible ¤t byweakduality. Werstshowthat F (1)= P j 2 J 1 1 d j 1 ( ¤1 ¡ j 1 )and F (2)= 2 P t =1 P j 2 J 1 t d jt ( ¤t ¡ jt ) directly,andthenuseinductiontoshowthegeneralresult.For =1,theresult isstraightforward,sincethenalobjectivefunctionafterim plementingthedual procedureisequalto P j 2 J 1 1 d j 1 ( ¤1 ¡ j 1 ).If J 1 1 isempty,thentheobjectivefunction equalszero,whichimplieswedonosetupandsatisfynodemand.O therwise, Z 1 D = X j 2 J 1 1 d j 1 ( ¤1 ¡ j 1 ) = X j 2 J 1 1 d j 1 264 c 1 + S 1 ¡ P j 2 J 0 1 d j 1 max(0 ; j 1 ¡ c 1 ) P j 2 J 1 1 d j 1 375 ¡ X j 2 J 1 1 j 1 d j 1 = S 1 + X j 2 J 1 1 d j 1 ( c 1 ¡ j 1 )+ X j 2 J 0 1 d j 1 max(0 ; j 1 ¡ c 1 ) = S 1 + X j 2 J 1 1 d j 1 ( c 1 ¡ j 1 )+ X j 2 J 0 1 d j 1 ( c 1 ¡ j 1 ) = S 1 + X j 2 J 1 1 [ J 0 1 d j 1 ( c 1 ¡ j 1 ) ; where J 0 1 isthesetofall j 2 J 0 1 suchthat j 1 c 1 .Wenowhaveconstructeda dualfeasiblesolutionwithanobjectivefunctionvalueequal tothatofaprimal feasiblesolutionthatsetsupinperiod1andsatisesalldemandfo rsegments j suchthat j 2 J 1 1 [ J 0 1 ,implyingthatthissolutionisoptimalfortheprimalproble m.

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81 Wenextconsiderthecaseof =2.Inthiscasewehave Z 2 D = 2 P t =1 P j 2 J 1 t d jt ( ¤t ¡ jt ). If J 1 1 isempty,thenwehaveasingle-periodproblem(forperiod2)a ndwecanrefer totheproofaboveforthecaseof =1.Supposethenthatneither J 1 1 nor J 1 2 is empty.Intheprocessofapplyingourdual-ascentalgorithm,w eencounteroneof thetwocasesbelow:CaseI: ¤2 = c 1 .Thisimpliesthatconstraint2doesnotbecometightandfurt her increasesin ¤2 areblockedbytherstconstraint.Inthiscasethedualobjectiv e equals Z 2 D = 2 X t =1 X j 2 J 1 t d jt ( ¤t ¡ jt )= S 1 + X j 2 J 1 1 [ J 0 1 d j 1 ( c 1 ¡ j 1 )+ X j 2 J 1 2 d j 2 ( c 1 ¡ j 2 ) ; whichequalstheprimalobjectivefunctionvalueofaprimal feasiblesolutionthat setsupinperiod1onlyandusesthissetuptosatisfysegments j inperiod1such that j 2 J 1 1 [ J 0 1 andinperiod2suchthat j 2 J 1 2 CaseII: ¤2 = c 2 + S 2 ¡ P j 2 J 0 2 d j 2 max(0 ; j 2 ¡ c 2 ) P j 2 J 1 2 d j 2 .Thisimpliesthatconstraint2becomes tightbefore ¤2 reaches c 1 andfurtherincreasesin ¤2 areblockedbythesecond constraint.Inthiscasethedualobjectiveequals Z 2 D = 2 X t =1 X j 2 J 1 t d jt ( ¤t ¡ jt )= S 1 + S 2 + X j 2 J 1 1 [ J 0 1 d j 1 ( c 1 ¡ j 1 )+ X j 2 J 1 2 [ J 0 2 d j 2 ( c 2 ¡ j 2 ) ; where J 0 2 isthesetofall j 2 J 0 2 suchthat j 2 c 2 .Thisvalueof Z 2 D isequaltothe primalobjectivefunctionvalueofaprimalfeasiblesolution thatsetsupinperiods 1and2andsatisesdemandinallsegments j inperiod1suchthat j 2 J 1 1 [ J 0 1 usingthesetupinperiod1andsatisesdemandinallsegments j inperiod2such that j 2 J 1 2 [ J 0 2 usingthesetupinperiod2.Wehavesofarshownthat Z D = F ( ) for =1and2.Wenextuseinductiontoshowthatthisholdsforall > 2. Assumethereisatleastoneattractivesegmentinsomeperiod ;i.e., O 1 existsfor some 2f 1,..., T g (otherwisetheoptimaldualsolutionvalueequalszeroandno

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82 demandissatised).Forsome k wemusthave ¤ = c k + S k ¡ P t = k P j 2 J 0 t d jl max(0 ; jt ¡ c k ) ¡ ¡ 1 P t = k P j 2 J 1 t d jl max( ¤t ¡ c k ; 0) P j 2 J 1 d j : The -periodobjectivefunctionthenbecomes Z D = X t =1 X j 2 J 1 t d jt ( ¤t ¡ jt ) = k ¡ 1 X t =1 X j 2 J 1 t d jt ( ¤t ¡ jt )+ ¡ 1 X t = k X j 2 J 1 t d jt ( ¤t ¡ jt )+ X j 2 J 1 d j ( ¤ ¡ j ) Substitutingfor ¤ ,thelastexpressioncanbewrittenas: X j 2 J 1 d j ( ¤ ¡ j ) = P j 2 J 1 d j c k + S k ¡ P t = k P j 2 J 0 t d jl max(0 ; jt ¡ c k ) ¡ ¡ 1 P t = k P j 2 J 1 t d jl max( ¤t ¡ c k ; 0) ¡ P j 2 J 1 j d j Returningtothe -periodobjectivefunction,andusingourinductionhypothe sis, wenowhave Z D = F ( k ¡ 1)+ X j 2 J 1 d j c k + S k + ¡ 1 X t = k X j 2 J 1 t d jt ( ¤t ¡ jt ¡ max( ¤t ¡ c k ; 0)) ¡ X t = k X j 2 J 0 t d jt max(0 ; jt ¡ c k ) ¡ X j 2 J 1 j d j : Sincefor t< ,if ¤t isdened(i.e., J 1 t 6 = f;g ),wehave ¤t ¤ c k (fromLemma 1 ),theabovecanberewrittenas Z D = F ( k ¡ 1)+ X j 2 J 1 d j ( c k ¡ j )+ S k + ¡ 1 X t = k X j 2 J 1 t d jt ( c k ¡ jt ) ¡ X t = k X j 2 J 0 t d jt max(0 ; jt ¡ c k ) = F ( k ¡ 1)+ S k + X t = k X j 2 J 1 t d jt ( c k ¡ jt )+ X t = k X j 2 J 0 t d jt ( c k ¡ jt ) ;

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83 where J 0 t = f j 2 J 0 t : jt c k g Fromourpreviousdenitions,wecannowsimplify Z D to Z D = F ( k ¡ 1)+ S k + c k d k ¡ k ; whichcorrespondstotheobjectivefunctionvalueofaprimal feasiblesolution, implyingthat Z D F ( ).Butbyweakdualitywehave Z D F ( ),andso wemusthave Z D = F ( ),theoptimalsolutionvalueoftheprimal.Moreover, thecomplementaryprimalsolutionisalsofeasibleforthebina ryrestrictionsof [RPP PLC ]. } AppendixCLemma 3 AnoptimalsolutionexistsforRPP PLC containingconsecutiveregenerationintervals ( ; 0 ) wheretheproductionsub-planinperiods +1 ;:::; 0 isofone ofthefollowingtypes: (i)Weproduce0or C ineveryproductionperiodintheinterval +1 ;:::; 0 with atmostone 0 < jt
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84 t .Sinceinventoryineachperiodintheregenerationinterva lispositive,afeasible solutionexistsfortheregenerationintervalthatusesthesame setupperiodsand reduces jt byoneunit,alongwithinventoryinperiods s;:::;t ¡ 1,andproduction inperiod s .Sincethissolutiondoesnotimproveoverouroptimalsolutio n(and giventhelinearityofcosts),thisimpliesthatatleastasgood asolutionexiststhat increases jt byoneunit,alongwithinventoryinperiods s;:::;t ¡ 1andproduction inperiod s .Repeatingthisargumentuntileither x s = C or jt = d jt impliesthe resultofthelemma.Similarly,ifperiod t isbeforeperiod s ,afeasiblesolution existsfortheregenerationintervalthatusesthesamesetupper iods,increases jt byoneunit,reducesinventoryinperiods t;:::;s ¡ 1byoneunit,andincreases x s byoneunit.Sincethissolutiondoesnotimproveoverouropti malsolution,this impliesthatatleastasgoodasolutionexiststhatreduces jt byoneunit,increases inventoryinperiods t;:::;s ¡ 1byoneunit,andreduces x s byoneunit.Repeating thisargumentuntileither jt =0or x s =0provestheresult. }

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CHAPTER4 SELECTINGMARKETSUNDERDEMANDUNCERTAINTY 4.1 Introduction Thusfar,ourapproachtoselectingthebestdemandsources(ord ers,markets, etc.)tosatisfyreliedondeterministicinformationconcerni ngthesizeandtiming ofeachdemand.Whileweusuallyhavesomedataforplanningpur poses,typically viascheduledordersordemandforecasts,theexactamountsare ofteninaccurate. Therefore,itisextremelyimportantforarmmakingsuchprod uctordering (ormanufacturing)decisionstoaccountforthestochasticnat ureofdemand.As demandbecomeslesspredictable,ourselectiondecisionswill surelybeinuenced. Westudythemarketselectionproblemwithdemanduncertainty inorderto developarobustmodelingapproachthataddressessuchtypesofd emand. Theclassicnewsvendorproblemhasbeenstudiedextensivelyinre search literaturedueinlargeparttoitsindustryapplications.The retailandairline industrieshaveshownthatoperatingwithaperishablegood(e. g.,seasonable fashionitems,airlineseatsorights)requirestheattentionof asingleselling seasonmodel,whichisaddressedthroughthenewsvendormodel.In asimilar vein,manufacturingrmsareproducingitemswithever-decr easingproductlives, inaneorttostaycompetitivewiththelatestoeringofotherrm s.Thisis especiallytrueinthetechnologysectorwhere,bythetimearm startstorealize demandduringthesellingseason,itisoftentoolatetoplacease condorderwitha supplierduetolongleadtimes.Inotherwords,thermmustlivewi thitsprevious orderquantitydecisionandnowpossiblypayapremiumforexped itingadditional producttocaptureanyadditionalunforeseendemand. 85

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86 Nomatterhowmucheortisspentontryingtoreduceproductandp rocess leadtimes,certainindustrieswilllikelyexistwhereobtaini ngmaterialsormore productatareasonableunitcostwillrequireasubstantialamou ntoftime. Evenifthermoperatesinaso-calledQuickResponse(QR)modewi thits suppliers,theleadtimesmaystillbelongrelativetothesellin gseason(seeIyer andBergen[ 39 ]foradiscussionofQRintheapparelindustry).Thisleadsusto studyquestionsconcerningintegratedorderquantityandmar ketselectiondecisions underuncertaindemand. Weconsiderarmthatoersaproductforasinglesellingseason.The rmusesanoverseasor\longleadtime"supplierastheprimarysou rceforits product,andthustheorderquantitymustbedecidedfarinadv anceofactual sales.Thermhastheexibilitytoselectwhichmarketdemandsour cestosatisfy, whereeachdemandsourceisarandomvariable.Intheclassicnew svendormodel, thepreferredorderquantityisdependentonthedistributio noftotaldemand. However,inourcontext,thedemanddistributionisdependent onthemarketsthe rmselects.Thus,themarketselectiondecisionmustbemadepriort oordering fromtherm'ssuppliersothatanappropriateordercanberecei vedintimefor thesellingseason.Inadditiontoeachmarket'sdemanddistribu tionbeingrandom, weassumethatthisdistributioncanbeinuencedbythelevelofa dvertisingeort usedwithineachmarket.Byexpendingmoreeortinamarket,th ermcan increasethedemandforitsproduct.Weaddressappropriatead vertisingresponse functionswhichmeasuremarketingeectivenessbasedonthelev elofadvertising spending(seeVakratsas,Feinberg,Bass,andKalyanaram[ 78 ]).Wealsoexamine theeectofbudgetaryconstraints.Themarketingbudgetcould preventtherm fromcapturingadditionalexpectedmarketdemand,and,thu s,additionalprots, regardlessoftherm'sorderingorproductioncapacity.

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87 Asproductlifecyclescontinuetodecreaseandassessingdemandri skfor marketentrybecomesincreasingcritical,manycompaniesnd themselvesfaced withsimilarissuesthatweaddresshere.Claritas,amarketresear chandstrategic planningcompany,hascitedseveralclients,includingEddie Bauer,thatwanted betterknowledgeoftheircustomersinordertominimizedema ndrisk.Claritashas hadmanysuccessstoriesinidentifyingprotablecustomers,assessin gpotential markets,andrankingopportunities.Recently,Fisher,Raman, andMcClelland [ 27 ]studiedhow32leadingretailers,allofwhichoershort-lifecycleproducts (somewithasinglesellingseason)withunpredictabledemand,to determine howeectivelyeachcompanyusedavailabledatasourcestounde rstandtheir customers.Inthepresentmarketplace,theseretailersaresayin gthattheymust makebetteruseofdemandinformationiftheywanttomakeprot ablemarket selections.Finally,CarrandLovejoy[ 20 ]alsodiscussthisproblem'smotivation fromaninversenewsvendorpoint-of-view.Theyciteaclientr mmakingindustrial products,andthisrmdesiresamarketingstrategythatselectsa ppropriate demandsormarketstoenterwhileworkingwithinaxedproduc tionlevel. IncontrasttotheapproachdevelopedbyCarrandLovejoy[ 20 ],wedonot assumeapredeterminedcapacitylimitthatthermmustobeywhen selecting markets.Rather,ourmodeljointlydeterminesthecapacitya cquiredandthe marketsselectedinordertomaximizearm'sprot.Moreover,t hermcan inuencemarketdemandsthroughjudicioususeofadvertisingr esources.The resultingmodelsleadtointerestingnewnonlinearandintege roptimization problems,forwhichwedeveloptailoredsolutionmethods.These modelsalsoallow ustodevelopinsightsregardingtheparametersandtradeost hatareinuential inintegratedmarketselectionandcapacityacquisitiondeci sionsforitemswitha singlesellingseason.Thus,thisworkprovidesnewcontribution stotheoperations

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88 modelingandmanagementliteratureaswellastheliteratur eonoperations researchmethodologies. Manyresearchershavecontributedtothewiderangeoflitera turethatexists onstochasticinventorycontrol,forwhichPorteus[ 62 ]providesaniceoverview. Particularlyrelevanttoourworkistheliteraturethatfoc usesonthenewsvendor problem.InadditiontotheworkbyPorteus,reviewsbyTsay,Nah mias,and Agrawal[ 76 ]andCachon[ 18 ]providemorerecentresearchdirectionsconcerning supplychaincontractsandcompetitiveinventorymanagemen tinthecontextofa single-period\newsvendor"setting. Additionalliteratureconsidersthemulti-itemnewsvendorpr oblemaswell, forwhichwecandrawsomesimilaritiestoour\multiple-marke t"setting.Inour problemwehaveonecostforproducingasingleproduct,andthe individualmarket deliverycosts,salesandadvertisingcosts,andrevenuesprovide dierentiation amongmarkets.Eachmarkethasacertainamountofrandomdema nd,andweattempttosatisfythedemandfromthemarketsweselecttomaximiz eoverallprot. Inthetypicalmultiple-productsetting,eachproducthasun iqueproduction,salvage,orderingandperhapsdistributioncosts.However,therei snodierentiation betweenthedemandsourcesforaparticularitem.MoonandSil ver[ 55 ]present heuristicapproachesforsolvingthemulti-itemnewsvendorpr oblemwithabudget constraint. Otherresearchershaveinvestigatedhowproductioncapacity canbeadjusted withintheframeworkofanewsvendor-typeproblem.FineandF reund[ 25 ]consider cost-exibilitytradeosininvestinginproduct-exiblemanuf acturingcapacity. Theyformulatethecapacityinvestmentdecisionasatwo-stage stochasticprogram, whereallfutureproductionandinventorydecisionsareroll edintoonefutureperiod.Thisisnotablydierentfromourapproachinthattheyc onsiderproduction capacityconstrainedproblemsasopposedtobudgetconstraine dproblems.They

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89 improveprotabilitybyincreasingcapacityandthenallocat ingitappropriately, whileweimproveprotabilitybyworkingwithintheadvertisi ngbudget,selecting specicmarketstoserve,andexpendingadvertisingeortinthese marketsto achievetheappropriatelevelofexpecteddemand. Yieldmanagement,whichhasbeenanactiveeldofresearchfora long time,isalsocloselyrelatedtoourstudyofmarketselectiondec isions.Givena setofrandomdemands(ordemanddistributions)andsellingpric eswithineach demandsegment,armwilluseyieldmanagementtechniquestode terminethe amountofproducttooerateachsellingpricesoastomaximizeo verallprot. Someofthemostnotableyield(orrevenue)managementresearc hfocuseson pricingandseatallocationdecisionswithintheairlineindu stry.Belobaba[ 12 ]and Williamson[ 84 ]providenicereviewsofthistopic.Otherexcellentpapers onthe eectsofpricingandyieldmanagementincludeGallegoandva nRyzin[ 30 ],[ 31 ], Biller,Chan,Simchi-LeviandSwann[ 14 ],PetruzziandDada[ 59 ],andMonahan, Petruzzi,andZhao[ 54 ].ThemorerecentworkbyMonahanetal.[ 54 ]focuses ontheparallelbetweenthedynamicpricingproblemandthed ynamicinventory problem.Theyoerareinterpretationofthedynamicpricing problemasapricesettingnewsvendorproblemwithrecourse,whichleadstoinsigh tsintotheactions andprotsofaprice-settingnewsvendor. Additionaldemandormarketselectiondecisionscanbefoundin thegametheoryliteratureconcerningmarketentry,asdiscussedinRhim,Ho ,andKarmarkar [ 63 ].Whenfacedwithcompetitionforanymarketdemandsegment, armmust makeselectionsconcerningproductionsites,capacitiesandq uantities.Theyfocus onamulti-rmapproach,whereeachcompetitor'sdecisionswi llbeaectedby thetimingofentryoftheothercompetitors,andeachcompeti tor'slevelofentry. Ourcurrentresearchdoesnotconsidermulti-rmdecisions,andt hiscouldbean interestingextensiontothesingle-rmdecisionswepresentinth isdissertation.

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90 Ourworkisbasedlargelyontherelationshipbetweenexpected revenue anddemanduncertainty,whichisdirectlycorrelatedwitht heareaofportfolio optimization.TheseminalworkbyMarkowitz[ 53 ]over50yearsago,followed bycountlessarticlesinthisstreamofresearch,introducedth econceptofmeanvarianceoptimization.Themean-varianceapproachattemp tstoachieveadesired rateofreturnwhileminimizingtheriskinvolvedwithobtain ingthatreturn.We determineanoptimalsetofmarketsbasedontheirexpectedrev enues(orreturns) andtheassociateddemanduncertainties(returnrisk).Ourmod elingapproach diersinthatwedonotplaceaminimumlevelonexpectedprot, nordowefocus onriskminimizationwhilemeetingadesiredprotlevel.Foram orerecentreview ofportfoliooptimizationandriskaversion,seeBrealyandMye rs[ 16 ]. Someofthemoreclosely-relatedstochasticworkondemandando rder selectionarePetruzziandMonahan[ 60 ],CarrandDuenyas[ 19 ],andCarrand Lovejoy[ 20 ].PetruzziandMonahan[ 60 ]addressselectingbetweentwosourcesof demands,theprimarymarketandthesecondary(oroutletstore) market,forwhich tosupplyproduct.Whilethesedemandsmightoccursimultaneou sly,thermmust decidethepreferredtimetomovetheproducttotheoutletsto remarket.Carr andDuenyas[ 19 ]considerasequentialproductionsystemthatreceivesdemand forbothmake-to-stockandmake-to-orderproducts.Acontrac tualobligation existstoproducethemake-to-stockdemand,andthermcansuppl ementits productionbyacceptingadditionalmake-to-orderdemandso urces.Theyapproach thisproblemusingqueueingtheory,inaneorttoprovideanop timaladmissionof make-to-orderdemandandoverallsequencingofproductionj obs.BeyondtheCarr andDuenyas[ 19 ]paperonjointadmissioncontrolandproductiondecisions,the mostrelevantworkisfoundinHa[ 35 ],whichpresentsaqueueingtheoryapproach tostockrationingacrossseveraldemandclassesforasingle-item ,make-to-stock productionsystem.

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91 CarrandLovejoy[ 20 ]examineaninversenewsvendorproblem,whichoptimallychoosesademanddistributionbasedonsomepre-denedorde rquantityor capacitysetbyasupplier.Theyselectademanddistributionfro masetoffeasible demandportfolios,whichmayincludeseveralcustomerclasses.To createademand portfolio,theyselectthesecustomerclasses,eachofwhichhasra ndomdemand thatfollowsanormaldemanddistribution,anddeterminethe amountofdemand tosatisfywithineachclasswhilenotexceedingthepre-denedc apacity.Because theyconsideraninversenewsvendor,thereisnodecisiontomake insettingthe orderquantity.Furthermore,theyassumethatallcustomercla sseswithineach portfoliohavealreadybeenrankedbysomeexogenouscriteri a,anddemandisallocatedinsuchawaythathigherpriorityclassesarelledcompl etelybeforelower priorityclassesareconsidered.Ourdecisionprocessisdierent forseveralkeyreasons.Wesimultaneouslyselectthecustomerdemandstosatisfyandde terminean appropriateorderquantitytorequestfromthesupplier,maki ngtheorderquantity adecisionvariable.Sincetherankingofalldemandsmaychan gebasedonthe availablefundsformarketing,wecannotprovideanapriori rankingofdemands, butallowthemodeltoimplicitlydeterminethemostattracti vesetofmarkets.We alsorequirethatallunmetdemandfromthesesourceswillbeexp edited,ensuring thatthedemandofall\selected"customersisfullled. First,wedeneandformulateourselectivenewsvendorproblemi nSection 4.2 .Givenasetofdemanddistributionsandnobudgetingorcapaci tyconstraints inplace,weprovideademandselectionalgorithmthatdeterm inesthemarkets topenetratetomaximizeprot.Severalmanagerialinsightsa realsoprovided.In Section 4.3 ,weevaluatetheeectthatmarketingplaysinincreasingthee xpected valueandvarianceofdemandwithinanyindividualdemandsou rce.Weaddress severalfunctionalformsoftheadvertisingresponsefunction, assuminganunlimited advertisingbudget.Wethenpresentthegeneralselectivenewsv endorproblemwith

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92 limitedmarketresourcesinSection 4.4 ,aswellasasolutionapproachbasedonthe problem'sKKTconditions.InSection 4.5 ,werstprovidecomputationalresults thatillustratethebenetsofusingthebasicselectivenewsvendo rproblem.Then, weexaminethesolutiontothelimitedresourcesproblemgiven severalformsof marketingeort.Afterpresentingtheseresults,wenallyoersomei nsightintoa fewothermodelingconsiderationspresentedinSection 4.6 4.2 TheSelectiveNewsvendorProblem 4.2.1 ProblemFormulationandSolutionApproach Assumewehaveasetof n potentialmarketsthatasuppliercanserve.Denote r i astheperunitrevenueoftheiteminmarket i ,where i =1 ;:::;n .Let D i denotetherandomvariablefordemandfrommarket i ,where D i haspdfandcdf f i ( D i )and F i ( D i )withmean i andvariance 2 i .Wealsoassumethatmarket demandsarestatisticallyindependent.Thermmustdecidefari nadvanceofthe sellingseasonboththeactualmarketsitwillserve(andthusinw hichmarketsit willapplymarketingeortpriortothesellingseason 1 )andthetotalquantity Q itwillprocurefromtheoverseassupplierataperunitcostof c .Thexedcostof enteringmarket i is S i Weassumewithoutlossofgeneralitythat r i isnetofanymarket-specicunit variablecostduetoproductionordeliveryoftheitem.Wecan thenassume,again withoutlossofgenerality,that r i >c ,otherwiseitwouldbeunprotabletoenter market i ,andwecouldimmediatelyeliminatethemarketfromconsider ation.We assumethat,givenasetofselectedmarkets,thermmustultimately satisfyall 1 Notethatweinitiallyassumethatthermwillapplyaxedamounto fselling eortineachmarketselected,andthatthisamountofsellingeo rtdeterminesthe market'sdemanddistribution.Thatis,initially,marketing eortisnotadecision variable.Thesexedamountsofmarketingeortarealsoindepen dentacrossall markets.

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93 realizeddemandinthesemarkets,andthatahigh-costdomesticsu pplierexists fromwhichthermcanexpediteunitsofthegood(afterobservi ngdemand)at aperunitcostof e ,where e>c .Anyunsolditemsremainingattheendofthe sellingseasonwillbesoldatasalvagevalueof v perunit,where c>v .Therm wishestomaximizenetprotfromitsmarketselectiondecisionf orthesingle sellingseason. Inadditiontotheorderquantitydecisionvariable Q ,thermmustdecide (beforeplacingtheorderfor Q units)themarketsitwillsatisfy.Let y i =1ifthe rmdecidestosatisfydemandinmarket i ,and0otherwise.Givenabinaryvector ofmarketselectionvariables y ,let D y = P ni =1 D i y i denotethetotaldemandof theselectedmarkets,anddenoteitspdfby f y anditscdfby F y .Itiseasytosee thatthetotalselecteddemandhasmean E ( D y )= P ni =1 i y i = y andvariance Var( D y )= P ni =1 2 i y i = 2 y .Wecanthenexpresstherm'sexpectedprotasa function G ( Q;y )oftheorderquantity Q andthebinaryvectorofmarketselection variables y : G ( Q;y )= X ni =1 ( r i i ¡ S i ) y i ¡ cQ + v Z Q 0 ( Q ¡ x ) f y ( x )d x ¡ e Z 1 Q ( x ¡ Q ) f y ( x )d x: Foragivenvector y ,theprotfunction G ( Q;y )isconcave,andmaximizingthe protisequivalenttominimizingthecostintheassociatednewsv endorproblem. Thisthenyieldsanoptimalorderquantityasafunctionof y ,say Q ¤y ,givenby F y ( Q ¤y )= e ¡ c e ¡ v : Assumingthat F y ( Q ¤y )isinvertible,wehave Q ¤y = F ¡ 1 y ( ) ; (4.1) where = e ¡ c e ¡ v

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94 Tofacilitateourlateranalysisoftheoptimalmarketselecti ondecisions,we dene¤ y ( Q )asthe lossfunction foragivenorderquantity Q andmarketselection vector y ;i.e.,¤ y ( Q )= R 1 Q ( x ¡ Q ) f y ( x )d x .Usingthisnotation,therm'sexpected protcanbewrittenas G ( Q ¤y ;y )= X ni =1 ( r i i ¡ S i ) y i ¡ ( c ¡ v ) Q ¤y ¡ v y ¡ ( e ¡ v )¤ y ( Q ¤y ) : Theformofthelossfunction¤ y ( Q ¤y )dependsonthedistributionof D y ,and ingeneral,canbequitediculttocharacterize.Ifeachmark et'sdemandis normallydistributed,wecaneasilycharacterizethedistribu tionof D y (itisalso normallydistributed),andwecanalsoemploythe standardnormallossfunction L ( z )= R 1 z ( u ¡ z ) ( u ) du (where ( u )isthep.d.f.ofthestandardnormal distribution,withc.d.f.( u ))tosimplifyouranalysis.Thatis,if D y isnormally distributed,thenwecanwrite¤ y ( Q ¤y )= y L ( z ( )),where z ( )= Q ¤y ¡ y y = ¡ 1 ( ) isthestandardnormalvariatevalueassociatedwiththefracti le .Moreover, assumingnormallydistributeddemand,wecanrewriteouroptim alorderquantity equation( 4.1 )as Q ¤y = X ni =1 i y i + z ( ) r X ni =1 2 i y i : (4.2) Usingequation( 4.2 )andtheidentity¤ y ( Q )= y L ( z ),undernormallydistributed demand,wecanwritetherm'sexpectedprotasG ( Q ¤y ;y )= X ni =1 [( r i ¡ c ) i ¡ S i ] y i ¡f ( c ¡ v ) z ( )+( e ¡ v ) L ( z ( )) g r X ni =1 2 i y i : Notethatgiventhecostparameters c v ,and e ,thecoecientofthesquare roottermisanonnegativeconstant.Letting K ( c;v;e )equalthiscoecient(i.e., K ( c;v;e )= f ( c ¡ v ) z ( )+( e ¡ v ) L ( z ( )) g ),wecanrewritetheexpectedprot

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95 equationas G ( Q ¤y ;y )= X ni =1 (( r i ¡ c ) i ¡ S i ) y i ¡ K ( c;v;e ) r X ni =1 2 i y i : Nowdene r i =( r i ¡ c ) i ¡ S i asthetotalexpectednetrevenuefromservingmarket i withtheoverseassupplier,afterincludingmarketentrycosts. Tomaximizethe rm'sexpectedprot,wemustsolvethefollowingselectivenewsve ndorproblem (SNP):[SNP] maximize P ni =1 r i y i ¡ K ( c;v;e ) p P ni =1 2 i y i (4.3) subjectto: y i 2f 0 ; 1 g i =1 ;:::;n: (4.4) BasedonasimilarapproachrstdenedinShen,Coullard,andDaski n[ 71 ],we cansolvethisproblemusingasimplesortingschemeandaselection algorithmthat wenextdescribe.Werstsortmarketsin nonincreasingorder oftheratioofthe coecientof y i intherst(linear)termintheobjectivefunction,tothecoec ient of y i inthesecond(squareroot)termintheobjectivefunction.Thi sresultsin indexingthemarketssuchthat r 1 2 1 r 2 2 2 ¢¢¢ r n 2 n : Notethatthenumeratoroftheratio r i = 2 i istheexpectednetrevenue,whilethe denominatorreectstheuncertaintyinmarketdemand.Wewil lthereforereferto thisratiogenericallyastheexpectedrevenuetouncertain tyratioforamarket i Thefollowingpropertyallowsustoapplyanecientmarketsel ectionalgorithm. Property2 DecreasingExpectedRevenuetoUncertainty(DERU) RatioProperty :Afterindexingmarketsindecreasingorderofexpectednet revenuetouncertainty,anoptimalsolutionto[SNP]existssu chthatifweselect customer l ,wealsoselectcustomers 1 ; 2 ;:::;l ¡ 1

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96 ThisDecreasingExpectedRevenuetoUncertainty(DERU)Ratiop roperty indicatesthatthereare n candidatesolutionsfromwhichwecanndanoptimal solution.Therefore,thedominantcomputationaleortinvol vessortingmarkets accordingtothisratio,whichcanbedonein O ( n log n )time.Oncethemarkets aresorted,startingwithmarket1,wecandeterminetherm'sex pectedprot byaddingonemarketatatimetoourcandidatesolution.Let z ( i )representthe expectedprotwhenmarkets1 ;:::;i areselected.When z ( i +1) >z ( i ),update theoptimalsolutiontoselectmarkets1through i .Continuethisprocedureuntil i = n .Itispossiblefortotalprottodecreaseafteraddingsomemarke t i ,but thentoincreaseandachieveamaximumtotalprotvaluebasedon selectingall marketsuptoandincluding i +1.Thus,wemustenumerateallcandidatesolutions providedbytheDERURatiopropertytondtheoptimalsolution Thisratioorderingisintuitivelyappealing,asahigherne trevenuemakesa marketmoreattractive,whileincreasesinthemarket'sunce rtaintyleadstoaless attractivemarket.Itisimportanttopointoutthatamarket willnotnecessarily beattractiveeventhough,byassumption,eachmarket'snetre venueispositive. Also,whentwomarketshavethesameratio,theyareequallyattr active.Ifwehave markets j and k suchthat R j = 2 j = R k = 2 k = ,thesemarketscanbetreatedasa single\aggregatemarket"withdemand D j + k = D j + D k ,sincetheresultingratio ofthisnewmarket j + k equals( R j + R k ) = ( 2 j + 2 k )= .Wecanthereforeusethe term\decreasing"inplaceof\nonincreasing"inourdescripti onofthisproperty withoutambiguity. ByexaminingEquation( 4.3 ),weseethattherm'sexpectedprotislimited bythedemandaccuracyestimatewithineachmarket i .Byreducingthisvariance estimate(throughimprovingforecasts,reducingsupplierlead times,orother measures)ineachmarket,wecouldincreasetheexpectedprotan dpossibly includeagreaternumberofmarketsintheoptimalsolution.No te,however,that

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97 thexedcost S i mightincreasetoincludesomemarketresearcheorttoimprove onexistingforecastingtechniques.Or,therm'ssuppliersmayw anttosharein theincreasedprotstoosettheircostsofoeringshorterleadtim es.Insuch cases,thermwouldneedtoweighthecostofimprovedforecastsag ainstthe benetoflowerdemanduncertainty. 4.2.2 ManagerialInsightsfortheSNP Section 4.2.1 providesanicesolutionapproachfortheSNP,givenanyset ofmarketsandeachmarket'sper-unitrevenue(sellingprice ¡ productioncost), expecteddemand,andstandarddeviationofdemand.Inthissec tion,weprovide insightsandobservationstoassistasupplymanagerindeterminin gtheinuence thateachofthesefactorshasontheacceptanceorrejectiono faparticularmarket. Wecanalsoshowhowamarket'sprotabilitywillchangebasedont hesetof marketsknowntoexistintheoptimalsolution.Furthermore,w ecanusethis informationtoexaminethesensitivityofaparticularmarket tochangesinselling price,expecteddemand,orothermarketparameters.Weassumet hatallmarkets notpre-selectedforentryhavebeensortedaccordingtotheDE RURatioproperty. Thus,anycandidatesolutioncontainingmarket k +1mustalsocontainmarkets 1 ;:::;k First,weassumethatnomarketshavebeenselected.Inorderfora rmto protfromenteringanyonemarket k ,wemusthave G ( Q ¤y k ;y k )= r k ¡ K ( c;v;e ) q 2 k 0 where y k representsthesolutionvectorinwhich y k =1and y i =0forall i 6 = k .Let r 0 k betheminimumnetrevenuerequiredtoachieveaprotinthissi nglemarket k Then, r 0 k = K ( c;v;e ) k : (4.5)

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98 Thesinglemarket k willbeattractivesolongastheunitsellingpricecanbeset suchthat r k r 0 k .Since r k =( r k ¡ c ) k ¡ S k ,wecanstatethissellingpriceas r 0 k = K ( c;v;e ) k k + S k k + c = K ( c;v;e )CV( k )+ S k k + c; whereCV( k )representsthecoecientofvariationformarket k .Theabove equationdirectlyshowstheeectthatvariabilityhasonther equiredsellingprice. Nowsupposewearegivenasetofmarkets1 ;:::;k knowntobeinanoptimal solution.Wewouldincludemarket k +1ifitsincrementalnetrevenueexceedsits incrementaluncertaintycost.Sincethereisnolimitationo ntheamountofproduct suppliedfromtheoverseassupplier,wewouldincreasetheoptim alquantity Q ¤ to Q ¤ = X k +1 i =1 i + z ( ) r X k +1 i =1 2 i : Thisimpliesthatthechangeintotalcostwillbe K ( c;v;e ) r X k +1 i =1 2 i ¡ r X ki =1 2 i : Let k = P ki =1 2 i representthetotalvarianceofallselectedmarkets1 ;:::;k .Then theexpectedincrementaluncertaintycost( EIUC )ofincludingmarket k +1is EUIC k +1 = K ( c;v;e ) h q k + 2 k +1 ¡ p k i = K ( c;v;e ) k +1 r k + 2 k +1 2 k +1 ¡ q k 2 k +1 = r 0 k +1 q 1+ k 2 k +1 ¡ q k 2 k +1 (4.6) Notethatas k increases,andlikewise, k increases,theincrementaluncertainty costisdominatedbytheuncertaintyofmarket k ,or k .Wecannowprovidea necessaryconditionforincludingornotincludingmarket k +1inanoptimal solution.Property3 IfanoptimalsolutionfortheSNPcontainsonlymarkets 1 ; 2 ;:::;k where k
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99 optimalSNPsolutionexiststhatselectsmarkets 1 ;:::;k ,and r k +1 satises r k +1 r 0 k +1 q 1+ k 2 k +1 ¡ q k 2 k +1 ,thenanoptimalsolutionexiststhatselectsmarkets 1 ;:::;k +1 Noticethatwehavegeneralizedthesingle-marketconditiont hatdetermineswhetherselectingmarket k byitselfisworthwhile.Nowwearetestingif weshouldacceptornotacceptanadditionalmarketintoanexi stingoptimal solution.Wecancalculatetheincrementalcostofaddingmark et k +1tobe r 0 k +1 q 1+ k 2 k +1 ¡ q k 2 k +1 .Iftheincrementalnetrevenuefrommarket k +1, or r k +1 ,isgreaterthanthisincrementalcost,thenitwouldbeprota bletoenter market k +1aswell.Ifthereareadditionalmarketsbeyond k +1,say k +2 ;:::;n andtheaboveconditionisnotsatised,thenwecannotsaywhethe rmarket k +1 willultimatelybeincludedintheoptimalsolution.However, if k +1istheonly neworadditionalmarket,thenwecanuseProperty2asasucient condition forselectingornotselectingthemarket.Nowlet'sassumethat k +1istheonly additionalmarket,anditisnotprotabletoinclude.Then,i fadditionalmarkets alsobecomeavailabletopenetrate,weshouldconsiderallmark etsnotyetselected (whichincludes k +1)todetermineanupdatedoptimalselection.Thisapproach leadsdirectlytothedevelopmentofsucientconditionsforse lectingornotselectingagroupofadditionalmarkets,whichwillbediscussedshortly .Itisalsoworth notingthefollowingspecialcase.Considerascenarioinwhicht hemarketentry cost S k +1 iseithernegligibleornonincreasingbasedonthemarketinde x.This impliesthattherequiredper-unitnetrevenue, r k +1 ¡ c ,toincludemarket k +1will naturallydecreaseasadditionalmarketsareincludedinthe solutionandthevalue of k increases. Wemayalsobeabletosetthepriceorinuencetheamountofdeman din market k +1suchthatthereexistssomemarginalprotandtheconditionfo r selectingmarket k +1issatised.Thisallowsasupplymanagergreaterexibility

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100 whenconsideringexpandingonacurrentsetofmarketsbeingser ved.Ifarm hascontractedwith k marketsandhasanoptiontoaddthe k +1 st market,this marginalprotcheckcanbeusedasabenchmarkforexpandingit soperation. Wealsoobservethatasarmservesmoremarkets,thevariabilityt hatexists intheadditionalmarketbecomeslessimportant;i.e.,theva riabilityfromthe marketsalreadybeingservedwillprovideanexistingbuerinsa fetystockthatcan accommodatethevariabilitybroughtinbytheadditionalma rket. Aspreviouslystated,Property 3 providesanecessaryconditionforselecting ornotselectingamarket.Thenextproperty,basedontheDERUR atioproperty, providesasucientconditionfornotselectinganyadditional marketsbeyondsome currentbestselectionof1 ;:::;k markets. Property4 If P k + j i = k +1 r i
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101 protachievedbyaddingmarkets k +1 ;:::;k + j forsome j = k +1 ;:::;n thatis greaterthantheincrementalprotachievedbyincludingmar kets k +1 ;:::;k + l forall l =1 ;:::;j ¡ 1. Thermmayalsoliketosetasinglepriceacrossallnewmarketsent ered. Thisisastraightforwardadjustmenttotheconditionstatedin Property 5 .Let r k +1 ;k + j representthesinglepriceformarkets k +1 ;:::;k + j ;i.e., r k +1 = ::: = r k + j = r k +1 ;k + j .Also,byassigning k +1 ;k + j = q P ji =1 2 k + i ,( S= ) k +1 ;k + j = P ji =1 S k + i P ji =1 k + i andCV( k +1 ;k + j )= p k +1 ;k + j P ji =1 k + i ,wecanexpresstheminimumpriceacrossallnew selectedmarketsasr k +1 ;k + j = K ( c;v;e )CV( k +1 ;k + j ) s 1+ k k +1 ;k + j ¡ s k k +1 ;k + j +( S= ) k +1 ;k + j + c: Wenoteagainthatreducingthecoecientofvariationwillre ducetheminimum sellingpricerequired.Onecanseethatbyincreasingtheexpec teddemandin anyofthenewmarkets(withoutincreasingtherespectivemark et'sdemand variance),wecanreducethecoecientofvariation.Ontheot herhand,the pricewouldhavetobesethigherasthevariabilityofthenewm arketsincreases. Infact,asthedemandvariabilitywithinmarkets k +1 ;:::;k + j increases suchthat k +1 ;k + j >> k ,welosethebenetofhavingalreadyenteredinto markets1 ;:::;k .As k +1 ;k + j growslarge,wecanapproximatetheminimum requiredsellingpricetooeracrossalladditionalmarkets k +1 ;:::;k + j as K ( c;v;e )CV( k +1 ;k + j )+( S= ) k +1 ;k + j + c ,whichactuallyrepresentsthecostof only enteringmarkets k +1 ;:::;k + j Ourlastpropertyintroducedinthissectionaddresseshowarmca naccommodateemergingmarkets.Initially,armmaybefacedwithan n -marketselection problem.AftersortingbasedontheDERURatioproperty,wecand etermine whichofthese n marketstoacceptandwhichnottoaccept.Astimepasses,and

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102 additionalmarketsemerge,thermmustre-evaluatetheovera llselectiondecision.Notonlyshouldthenewmarketormarketsbeconsidered,but allmarkets originallyrejectedmayalsobereconsideredforselection.Assum ingwehavean emergingmarket m ,thenwecanuseProperty 5 todecideiftheinclusionofmarket m makessomeadditionalsetofmarketsattractive.Wewillinclu deadditional marketsonlyifthermachievesanincrementalprotfordoing so.Regardless,it shouldbeclearthatthenewoptimalsolutionwillcontainatle asttheoriginalset ofmarkets.Thefollowingpropertyaddressesthispoint.Property6 Wearegivenanoptimalsolutionthatselectsmarkets 1 ;:::;k ,and asinglenewmarket m emerges.Forthenew ( n +1) -marketselectionproblem, ifmarket m isnotchosen,thentheoptimalsolutionisthesameforbotht he ( n +1) -marketand n -marketproblems. First,assumethattheoptimalsolutionincludesmarkets1 ;:::;k .Sincethe original n marketsaresortedbytheDERUproperty,weshouldplacemarket m in itscorrectDERUratioorderposition.Let m representtheindexedpositionwithin theordering,whichnowcontains n +1markets.Ifmarket m hasahigherDERU ratiothanmarket k ,thenmarket m willbeimmediatelyaddedtothesolution. Moreover,wecanuseProperty 5 todetermineifsomepreviouslyunprotable marketsshouldnowbeincludedintheoptimalsolution.Ifmark et m hasalower DERUratiothanmarket k ,thenwemustevaluatetheinclusionofmarkets k +1 ;:::;m ,andpossiblymoremarkets,intothesolutionbasedonProperty 5 (i.e., set j = m;:::;n +1 ¡ k inProperty 5 ,andtestthecondition).Iftheconditionis metforsome j ,thenmarkets k +1 ;:::;j willbeaddedtotheoptimalselection. Otherwise,theoriginalsolutionremainsunchanged. Itisclearthatarmwouldliketheexibilityofeithersetting thepriceor inuencingtheproductdemandtoensuremarket k +1isprotable.Property 3 allowsustocheckprotabilitybasedonthismarket-specicinf ormation.

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103 Let'sassumethatweholdthesellingpriceconstant.Wecanthend erivethe minimumrequiredexpecteddemandinorderforthermtobreak evenbysetting r k +1 = K ( c;v;e ) q k + 2 k +1 ¡ p k ,giventhisparticularsellingprice.Letting mink +1 representtheminimumrequiredexpecteddemand,andassuming k +1 canbe estimatedandremainsconstantfordierentlevelsofexpected demand,wehave mink +1 = K ( c;v;e ) r k +1 ¡ c q k + 2 k +1 ¡ p k + S k +1 r k +1 ¡ c : (4.7) Thiscouldprovetobequitevaluabletotherm.Forexample,c onsiderthatthe rmwouldliketoentermarket j ,whichhasanexpecteddemandof j ,basedon axedorpre-determinedlevelofmarketing.But,inordertoa ddmarket j andbe protable,thermmusthaveaminimumexpecteddemandof minj > j .Through additionalsalesandadvertising,wecouldincreasetheexpect eddemanduptothe desiredlevel, minj .Ofcourse,thiscomesatacost,andtheadditionalmarketing expensetermwouldalsoneedtobeconsideredinequation( 4.7 ).Weaddress thisinSection 4.3 byallowingsalesandadvertisingeortineachmarkettobea decisionvariable;i.e.,themarketingeortwillnolongerbe xedforeachmarket. 4.3 SNPandtheRoleofAdvertising Intheprevioussection,weconsideredaprobleminwhichthede mandwithin anymarket i followsadistributionwithaknownmean i andstandarddeviation i .Both i and i implicitlyassumedthatthesalesandadvertisingeorts werexedforallmarkets.Inthissection,wegeneralizethemod eltoalloweach market'sdemanddistributiontobeafunctionofmarketingeo rtexpended,which impliesthat i and i arenotnecessarilyxedvalues.Moreover,weexamine contextsinwhichdemandishighlydependentonadvertising, andamarketisnot protablewithoutsomelevelofadvertising.Thisimpliesthee xpecteddemandin market i withnoadvertisingeort(whichwedenoteby i )providesinsucientnet revenuetocoverthexedmarketcost;i.e., S i > ( r i ¡ c ) i

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104 Asmentionedearlier,theexpectedmarketdemandresultingfr omagiven levelofadvertisingisusuallydenedthroughanadvertisingre sponsefunction. Vakratsasetal.[ 78 ],Lilienetal.[ 48 ],MahajanandMuller[ 52 ],andJohansson [ 40 ]discussmanyformsthattheadvertisingresponsefunctioncanta ke.Common functionalformsusedfortheadvertisingresponsefunctionare concave,linear,or S-shaped.Formostindustriesorproducts,oneoftheseresponsefun ctionswill approximatethebehaviorofdemandincreaseswithrespecttot headvertising level.Inrecentyears,researchershavefocusedmoreonS-curv eddemandfunctions, whicharebelievedtobemorebroadlyapplicableinindustry. Incontrast,Simon [ 73 ]showsthatcertainproprietarybrandsactuallybehaveinan asymmetric fashiontoadvertising;i.e.,demandpeaksimmediatelyafter theadvertising increase,butthelong-termdemandlevelismuchlowerthanth einitialpeak.Since wefocusonasingletimeperiod,thistime-basedeectdoesnota pplyinourcase. 4.3.1 SelectiveNewsvendorwithMarketingEort Inordertoformulateamoregeneralmodelthatincludessales andadvertising, weintroducesomeadditionalnotation.Let a i denotethenumberofunits(e.g., hours,days,employees)ofmarketingeortexpendedinmarket i ,andlet t i > 0 betheper-unitcostofthiseort.Withaslightabuseofnotation ,let i ( a i ) and 2 i ( a i )denotethemeanandvarianceofexpecteddemandinmarket i asa functionofmarketingeort a i .Weassumethatthefunction i ( a i )isnonnegative, nondecreasing,continuous,andbounded.Wealsoassumethatsomem arketing level b i existsformarket i ,suchthat a i >b i providesnoadditionalexpected demand.Inparticular,let i ( a )= i forall a b i and i ( a ) < i forall a
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105 eort.Thereisalsoamarketingeort w i beyondwhichdemandvarianceinmarket i iseectivelyconstant,atavalueof 2 i Wenextformulateamodelforprotmaximizationinthepresenc eofmarket choiceandmarketingeortexibility,ortheso-calledselecti venewsvendorproblem withmarketingeort(SNP-M).[SNP-M] maximize P ni =1 (( r i ¡ c ) i ( a i ) ¡ t i a i ¡ S i ) y i ¡ K p P ni =1 2 i ( a i ) y i subjectto: a i 0 i =1 ;:::;n y i 2f 0 ; 1 g i =1 ;:::;n: Notethat[SNP-M]isanonlinear,integeroptimizationproble m,which initiallyappearstobequitediculttosolve.Werstexaminesev eralformsof theadvertisingresponsefunctionwheredemandvarianceisind ependentofthe marketingeort.Wethenpresentaselectivenewsvendormodelin whichexpected demandanddemandvariancebothdependonthemarketingeort 4.3.2 CaseI:DemandVarianceIndependentofMarketingEort Inthissection,weanalyzethecasewheredemandvarianceisin dependent ofmarketingeort(i.e., 2 i ( a i )= 2 i ).Recallthatinthebasic(SNP),thetotal expectednetrevenuefromservingmarket i wasdenedas r i =( r i ¡ c ) i ¡ S i Similarly,wenowdenemarket i 'sexpectednetrevenueasafunctionofthe marketingeort a i spentinmarket i .Werepresentthisas r i ( a i )=( r i ¡ c ) i ( a i ) ¡ t i a i ¡ S i : Nownotethatthisoptimizationproblemisequivalentto maximize P ni =1 (max a i 0 r i ( a i )) y i ¡ K ( c;v;e ) p P ni =1 2 i y i subjectto: y i 2f 0 ; 1 g i =1 ;:::;n;

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106 whichmeansthattheoptimumlevelofmarketingeorttoexert ineachmarketis independent ofthemarketselectiondecision(where,ofcourse,marketinge ortsare onlyexertedinselectedmarkets).Thatis,wemay,foreachmark et i =1 ;:::;n ndtheoptimalmarketingeort^ a i inmarket i thatwillbeexerted if market i is selectedbysolvingtheoptimizationproblem maximize a i 0 ( r i ¡ c ) i ( a i ) ¡ t i a i : (4.8) Anoptimallevelofmarketingeort^ a i canbefoundamongallvalues0 a i b i forwhichtherstorderconditionsaresatised;i.e., t i = ( r i ¡ c ) 2 @ i ( a i ),where @ i ( a i )denotesthesetofsubgradientsat a i (seeBazaraa,Sherali,andShetty[ 11 ] foradiscussiononnecessaryconditionsforoptimality). Assumingwecanndan^ a i thatsolves( 4.8 )forall i ,wethenxthemarketinglevelineachmarketatthisoptimumvalue,whichredu cestheselected newsvendorproblemto(SNP^ D): [SNP^ D] maximize P ni =1 r i (^ a i ) y i ¡ K ( c;v;e ) p P ni =1 2 i y i subjectto: y i 2f 0 ; 1 g i =1 ;:::;n: Whenformulatedinthisway,weimmediatelyndtheoptimalsol utiontothis problembyusingtheDERUpropertywiththerankingratiodene dby r i (^ a i ) = 2 i where r i (^ a i )replaces r i fromtheoriginalratiopresentedinSection 4.2.1 4.3.2.1 ConcaveDemand If,inadditiontotheassumptionsmentionedabove,thedemand function i ( a i )isaconcavefunctionof a i ,then^ a i caneasilybefound.Inparticular,wemay ecientlyndtheoptimalvalueof a i usingbinarysearchontheinterval a i 2 [0 ;b i ]. SeeFigure 4{1 fortwoexamplesofconcaveexpecteddemand.Noticethatifwe

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107 assume ( a i )isdierentiableeverywhere,asinFigure 4{1 (a),thenwesimplynd avalue^ a i forwhich 0i (^ a i )= t i ( r i ¡ c ) (ifitexists),otherwise^ a i =0.If,inaddition, ( a i )isstrictlyconcave,thentheoptimallevelofmarketingeor t^ a i isunique. a ˆ i a t i /R i () i a a s i j+1 ,1 ij a ˆ i a ,1 ij a t i /R i s i j (a) (b) i b i b () i a i i Figure4{1:Optimalmarketingeortforconcaveexpecteddem andfunctions. Wenowpresentthespecialcaseinwhich ( a i )isnotdierentiableeverywhere.Werstconsiderthatexpecteddemandinmarket i increasesasa piecewise-linearfunctionwithdecreasingslopes s i 1 >s i 1 > ¢¢¢ >s i;J i >s iJ i +1 =0 (wherethereare J i +1consecutivesegments)andcorrespondingbreakpoints a i = a i 0 >>><>>>>: i + P k ¡ 1 j =1 s ij ( a ij ¡ a i;j ¡ 1 )+ s ik ( a ¡ a i;k ¡ 1 )for a i;k ¡ 1 a
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108 Wenextconsiderthecaseinwhichexpecteddemandinmarket i increases linearlywithslope s i > 0formarketingeortintheinterval[0 ;b i ]: i ( a )= 8><>: i + s i a for0 a b i ; i + s i b i for a>b i : Infact,thisisjustaspecialcaseofpiecewise-linearconcaved emand,inwhich thereisonlyonesloperepresentingtherateofincreaseindema nd.Recallingthe optimizationproblemstatedin( 4.8 ),^ a i willbeofthefollowingform: ^ a i = 8><>: 0for( r i ¡ c ) s i ¡ t i 0 ; b i for( r i ¡ c ) s i ¡ t i > 0 : Akeyresultofthisspecialcaseisthattheoptimalmarketingeo rtwillalways resideateithertheminimumlevelormaximumlevelofadverti singeortallowed (i.e.,^ a i 2f 0 ;b i g forthiscase).Wewillexploitasimilarpropertyforother functionsinsubsequentsectionsofthepaper. 4.3.2.2 S-curvedDemand Supposethattheexpecteddemandasafunctionofmarketeortf ollowsanScurve.Suchacurvecanberepresentedbyacontinuousfunctio nthatisconvexand nondecreasinguptosomemarketingeortlevelandconcaveandn ondecreasing beyondthatlevel.Thatis,thefunction i isgivenby i ( a i )= 8><>: (1)i ( a i )for0 a i i (2)i ( a i )for a i i where (1)i ( i )= (2)i ( i )and (2)i ( a i )= i for a i b i .AnexampleofanS-curveis giveninFigure 4{2 Weareinterestedinndinganoptimalmarketingeortlevel^ a i forthis demandfunction.Tothisend,weexaminethetwodierentcomp onents (1)i and

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109 a 1 ˆ i a 2 ˆ i a i b () i a i t i / ( r i – c ) t i / ( r i – c ) i Figure4{2:OptimalmarketingeortforS-curveddemandrespo nsefunctions. (2)i independently.For k =1 ; 2,let^ a ik denoteavalueof a i atwhich t i = ( r i ¡ c ) 2 @ ( k ) i ( a i ).Then,notethat (i)^ a i 1 and^ a i 2 correspondtoalocalminimumandalocalmaximum,respectivel y, ofsubproblem( 4.8 )forndingtheoptimalmarketeort,unless^ a i 1 =^ a i 2 = i ; (ii)subproblem( 4.8 )hasalocalmaximumat a i =0unless^ a i 1 =0(inwhichcase observation(i)applies); (iii)subproblem( 4.8 )hasalocalminimumat a i = b i unless^ a i 2 = b i (inwhich caseobservation(i)applies). Combiningobservations(i){(iii),wecannowimmediatelyco ncludethatthatthe onlycandidatesfor^ a i thatweneedtoconsiderare^ a i =0and^ a i =^ a i 2 4.3.3 CaseII:DemandVarianceDependentonMarketingEort Uptothispoint,wehaveassumedthatthestandarddeviationofde mand isindependentofanymarketingeortexerted.Inthissection ,wegeneralizethe eectofmarketingondemandbyallowingamarket'sdistributi on(bothmean andvariance)ofdemandtobeafunctionofthemarketingeort .Toaddressthis case,wewilladoptanapproximationoftheS-shapedcurve,abr oadlyapplicable advertisingresponsefunctionforexpecteddemand. Assume i ( a i )isaconvexincreasingfunctionfor0 a i b i ,andlet i (0)= i and i ( a )= i forall a b i .ThisdescribestheS-curvefunctionforexpected

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110 demandshowninFigure 4{3 below.Inaddition,assumethedemandvariancein market i isaconcaveandnondecreasingfunction, 2 i ( a i ),for0 a i w i ,where 2 i (0)= 2i and 2 i ( a )= 2 i forall a w i a 2 ˆ iii ab () i a i Figure4{3:ApproximationoftheS-curveddemandresponsefunc tion. Thefollowingtheoremshowsthat,undertheseassumptions,weonl yneedto considertwodistinctadvertisinglevelsineachmarketwhensol ving[SNP-M],our selectivenewsvendorproblemwithmarketing.Theorem4 Theoptimalmarketingeortinmarket i iseither ^ a i =0 or ^ a i = b i ( i =1 ;:::;n ). Proof: Fixthemarketingeortlevelsinallmarketsexceptone,aswe llasthe marketselectionvariables.Withoutlossofgenerality,wemay lettheunrestricted marketbemarket1.Furthermore,let a i = a i for i =2 ;:::;n and y i = y i for i =1 ;:::;m .Sincetheexpectedprotisindependentof a 1 if y 1 =0,weonlyneed toconsiderthecase y 1 =1.Finally,deneforconvenience V = n X i =2 2 i ( a i ) y i : Thenletting G 1 ( a 1 )denotetheexpectedprotasafunctionofmarketingeortin market1alone,andignoringconstantterms: G 1 ( a 1 )=( r 1 ¡ c ) 1 ( a 1 ) ¡ t 1 a 1 ¡ K ( c;v;e ) q 2 1 ( a 1 )+ V:

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111 Thesquarerootfunctionisaconcaveincreasingfunction,and wealsoknowthat 2 i ( ¢ )isconcave.Itthenfollowsthattheentiresquareroottermi sconcavefrom Theorem5.1ofRockafellar[ 66 ],whichstatesthatanincreasingconcavefunction ofaconcavefunctionisitselfconcave.Fromthisresult,weca nnowsaythat G 1 is convex. Thisimpliesthattheoptimummarketingeortintheinterval 0 a 1 b 1 is atoneofthetwobounds: a 1 =0or a 1 = b 1 .Furthermore,for a 1 >b 1 weknowthat G 1 ( a 1 )=( r 1 ¡ c ) 1 ( b 1 ) ¡ t 1 a 1 ¡ K ( c;v;e ) q 2 1 ( b 1 )+ V isdecreasing,whichmeansthatwedonotneedtoconsidermarke tingeortlevels inexcessof b 1 .Thisprovesthedesiredresult. } Notethatif y i =1,wewillset a i = b i ,since S i > ( r i ¡ c ) i (0)=( r i ¡ c ) i If y i =0,wecanstillreplace a i with b i intheformulationwithoutaectingthe objectivefunctionvalue.Therefore,weset a i = b i for i =1 ;:::;n .Thisleads tothefollowingformulationoftheselectivenewsvendorprob lemwheredemand varianceisdependentonmarketingeort(SNP-DV):[SNP-DV] maximize P ni =1 r i ( b i ) y i ¡ K ( c;v;e ) p P ni =1 2 i ( b i ) y i subjectto: y i 2f 0 ; 1 g i =1 ;:::;n; where r i ( b i )=( r i ¡ c ) i ( b i ) ¡ t i b i ¡ S i Tosolvefortheoptimalselectionofmarkets,wecanusethesameDER U propertyintroducedinSection 4.2.1 ,wheretheratioforeachmarket i isnow denedby r i ( b i ) = 2 i ( b i ). 4.3.4 MarketingInsights InSection 4.2.2 ,wepresentedseveralpropertiestoaidtherminmaking marketselectiondecisions.Thesepropertiescanbeupdatedtoi ncludetheeects

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112 ofadvertisinginastraightforwardmanner,basedonasuitabler edenitionof thevalueof R 0 k .Wealsoprovidedexpressionsfortheminimumsellingpriceand minimumexpecteddemandrequirementsinordertoachieveap rot.Here,we expandthisdiscussiontoincludeminimummarketingeortrequi rements. Sincemarketingeortisadecisionvariable,thermcanincrea setheexpected demandand,thus,changetheexpectednetrevenueinamarket, atacostequal totheamountofmarketingeortexpended.Whereaswepreviou slydenedthe minimumrequiredsellingpriceinordertoachieveaprotinma rket k +1,we willnowaddresstheminimumrequirementintermsofmarketin geort.There certainlyaresituationswhenthermwouldprefernottochang etheunitselling price, r k +1 .Byxingthisamount,wecanstateaconditionforselectingorn ot selectingmarket k +1as ( r k +1 ¡ c ) k +1 (^ a k +1 ) ¡ t k +1 ^ a k +1 ¡ S k +1 K ( c;v;e ) q k + 2 k +1 ¡ p k : Itwouldbedesirabletoisolatethemarketingeorttodetermin etheminimum eortrequiredtosatisfytheabovecondition.Weaddressthecase whereexpected demandisalinearlyincreasingfunctionoftheexpendedmark etingeort. Denote a mink +1 tobetherequiredmarketingeorttoselectmarket k +1.Then, inthelinearlyincreasingcase,wecandenethisminimummarke tingeortas a mink +1 = K ( c;v;e )( q k + 2 k +1 ¡ p k )+ S k +1 ¡ k +1 ( r k +1 ¡ c ) s k +1 ¡ t k +1 : If a mink +1 >b k +1 ,addingsolelymarket k +1willnotbeprotable.Otherwise, market k +1canbeselectedwithamarketingeortof a mink +1 b k +1 .Ofcourse, anyadditionalmarketingupto b k +1 willonlyprovideadditionalprot,sothe rmwouldsimplychoose b k +1 astheappropriatemarketinglevel.However,if marketingresourcesareconstrained,choosingthespecicamoun tofeorttouse beyond a mink +1 isnotsoclear,unlessmarket k +1istheonlyadditionalmarketunder

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113 consideration.Wewillexaminethelimitedmarketingresourc esprobleminthe followingsection. Nowconsiderascenariowhereboththeadvertisinglevelandthese llingprice areexible;i.e., a l and r l arebothdecisionvariables.Wewouldthenneedtond anappropriatesettingforbothvariablesthatequalstheunc ertaintycost.First, restateequation( 4.5 )as ( r l ¡ c ) l ( a l ) ¡ t l a l ¡ S l = K ( c;v;e ) l ; where l ( a l )isdenedasanyadvertisingresponsefunctionwhoseexpectedde mandisdependentonthelevelofmarketingexerted,and a l isthemarketingeort usedinmarket l .(Notethatdemandvarianceisindependentofthemarketing eortinthissection.) Heuristic:DeterminingValidPrice/AdvertisingSettings : Therecommended approachtosolvingthisproblemisasfollows.Beginbysetting r l suchthatavery lowprotmarginisobtained;i.e., r l ¡ c isverysmall.Next,searchforafeasible solutiontotheequationwheretheonlydecisionvariableis a l .Callthesevalues r 1 l and a 1l .Increasethesellingpricebysomeunitamount.Withthenewsett ing of r 2 l ,solvefor a 2l .Continueuntilsucientdatapointshavebeencollectedorno advertisingisrequiredtomeettheminimumnetrevenue(i.e. a 0l =0).Wenow havearangeofvalidpriceandadvertisingsettingsfromwhich thermcould operate. Wecanassignanymarketingeortwhenresourcesareunlimited,so any valuefor a l canbecalculated.However,ifwendasolutionsuchthat a l >b l thisdoesnotnecessarilymeanthatthemarketwouldnotbechose n.Themarket couldbemademoreattractivebyincreasing r l ,which,inturn,woulddecreasethe requirementforalargevalueof a l .Thismeansthatthisprocesswillnotresultin

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114 anoptimalpriceandadvertisinglevelsetting.Instead,itoer sthermarangeof valuesfromwhichtoconsideroperatingwiththismarket. 4.4 OperatingwithLimitedMarketingResources InSection 4.3 ,wepresentedseveralsolutionapproachestotheSNPdepending onthemarketingeort'sinuenceonexpectednetrevenue.Ine achcase,we assumedthattherewereunlimitedmarketingresourcesavailab le.Sincesuppliers andproducerstypicallyoperatewithinanannualsalesandad vertisingbudget, thereismostlikelyanupperlimitontheeortthatcanbeexpen ded.However, thermmaynotbeabletospendthisdesiredamountofmarketinge ortwhen facedwithlimitedresources.Moreover,theDERUpropertynol ongernecessarily holdsunderthebudgetconstraint.Inthissection,wewillpre sentthelimited resourcesproblemanddiscussmethodsforobtainingtheoptima lsolution. 4.4.1 FormulationoftheLimitedResourcesProblem Wewillexaminethelimitedresourcesproblemusingtherelati onshipbetween marketingeortanddemanddistributionspreviouslyintroduc edforCasesIandII inSection 4.3 .WerstdetailthesolutionapproachusingCaseII,themoregener al casethatallowsforbothexpecteddemandanddemandvariance tobefunctionsof marketingeort.AsimilarapproachcanalsobeappliedtotheCa seIproblem,in whichdemandvarianceisxed. InCaseII,wearegiventhat i ( a i )isaconvexandnondecreasingfunctionfor 0 a i b i ,andlet i (0)= i and i ( a )= i forall a b i ;i.e.,theexpected demandfunctionfollowstheS-curveapproximationfunctio nshowninFigure 4{3 Inaddition,denotethedemandvarianceinmarket i asageneralconcaveand nondecreasingfunction 2 i ( a i ),for0 a i w i ,where 2 i (0)= 2i and 2 i ( a )= 2 i forall a w i .Furthermore,recallfromSection 4.3 thatmarketentrywithoutany advertisingisassumedtobeunprotable(i.e., S i > ( r i ¡ c ) i formarket i ).Now,

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115 inthissection,weimposeacapacityconstraintontheamountof marketingeort expended. Byisolatingthexedportionofexpecteddemandanddemandunc ertainty fromthecomponentsthatdependonmarketeort,weredeneeac hfunctionas i ( a i )= i +~ i ( a i ) ; i 0 ; ~ i (0)=0 ; 2 i ( a i )= 2i +~ 2 i ( a i ) ; 2i 0 ; ~ 2 i (0)=0 ; where~ i ( a i )= i ( a i ) ¡ i and~ 2 i ( a i )= 2 i ( a i ) ¡ i for a i > 0.Theformulation oftheselectivenewsvendorwithlimitedmarketingresources( LM)iscloselylinked to[SNP-M],exceptthatthermnowhasamaximumof B unitsofmarketing resourcesavailable.Inaddition,weintroducethenotation S 0 i = S i ¡ ( r i ¡ c ) i > 0, andformulation[LM]becomes[LM] maximize P ni =1 [( r i ¡ c )~ i ( a i ) ¡ t i a i ¡ S 0 i ] y i ¡ K ( c;v;e ) p P ni =1 [ 2i +~ 2 i ( a i )] y i subjectto: P ni =1 a i B; 0 a i b i i =1 ;:::;n; (4.9) y i 2f 0 ; 1 g i =1 ;:::;n: InSection 4.3 ,wepresentedseveralcasesinwhichwecouldxthemarketing eortvariables(i.e., a i 's),reducingtheproblemintoaformsuchthattheoptimal selectionofmarketscanbefoundusingtheDERUproperty.Thesa meapproach cannotworkhereduetothemarketingbudgetconstraint,since wecannot necessarilyseteach a i toavaluethatachievesmaximumnetrevenueinmarket i .Thediscussionthatfollowswillillustrateanappropriatesolu tionapproach formarketentrydecisionswithbudgetaryconsiderations.Tho ughtherangeof

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116 potentialvaluesfor a i isimplicitlydenedwithinthedemandfunction i ( a i ),we includeconstraintset( 4.9 )for a i ,whosepurposewillbeclearonceweintroduce oursolutionstrategyfortheproblem. 4.4.2 SolutionApproachtotheLimitedResourcesProblem Wewillimplementabranch-and-bound(B&B)proceduretosolv eourproblem.Branchingwillbedonebyxingmarketselectionvariable s y i toappropriate values.Let I 0 denotethesetofmarketsthatarenotselected(i.e., y i =0for i 2 I 0 )and I 1 denotethesetofmarketsthatareselected(i.e., y i =1for i 2 I 1 ). Theremainingmarketsarein I 2 .AnodeintheB&Btreecanthusbeviewedas characterizedbysets I 1 and I 2 ,andthecorrespondingsubproblemof[LM],say [LM( I 1 ;I 2 )],is [LM( I 1 ;I 2 )] maximize P i 2 I 1 [( r i ¡ c )~ i ( a i ) ¡ t i a i ¡ S 0 i ]+ P i 2 I 2 [( r i ¡ c )~ i ( a i ) ¡ t i a i ¡ S 0 i ] y i ¡ K ( c;v;e ) q P i 2 I 1 [ 2i +~ 2 i ( a i )]+ P i 2 I 2 [ 2i +~ 2 i ( a i )] y i subjectto: P i 2 I 1 [ I 2 a i B; 0 a i b i i 2 I 1 [ I 2 ; (4.10) y i 2f 0 ; 1 g i 2 I 2 : Atagivennode,wewillndanupperboundontheoptimalvalueo fthe subprobleminthatnodebysolvingarelaxationofthisproblem asdescribedbelow. First,observethatwemaywriteconstraintset( 4.10 )for i 2 I 2 as 0 a i b i y i i 2 I 2 : ( 4.10 ')Thisnowenforcesthat a i =0whenever y i =0.Next,notethatformarkets i 2 I 1 wecanintroduceanarticialcontinuousvariable z i thatmeasuresthe

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117 fractionofthemaximummarketingeortthatisexertedintha tmarket;i.e.,we makethesubstitution a i = b i z i for i 2 I 1 .Wecanthenrewrite[LM( I 1 ;I 2 )]as maximize P i 2 I 1 [( r i ¡ c )~ i ( b i z i ) ¡ t i b i z i ¡ S 0 i ]+ P i 2 I 2 [( r i ¡ c )~ i ( a i ) ¡ t i a i ¡ S 0 i y i ] ¡ K ( c;v;e ) q P i 2 I 1 [ 2i +~ 2 i ( b i z i )]+ P i 2 I 2 [ 2i y i +~ 2 i ( a i )] subjectto: P i 2 I 1 b i z i + P i 2 I 2 a i B; 0 a i b i y i i 2 I 2 ; (4.11) 0 z i 1 i 2 I 1 ; y i 2f 0 ; 1 g i 2 I 2 : (4.12) Notethatthisproblemis equivalent to[LM( I 1 ;I 2 )].Weobtainanupper boundtotheoptimalvalueofthisproblembyrelaxingconstra intset( 4.12 )and usingthefollowingtheorem.Theorem5 Thereexistsanoptimalsolutiontothelinearrelaxationof [LM( I 1 ;I 2 )] suchthattheupperboundingconstraintsetin( 4.11 )willbetight. Proof: Observethatifwereducethevalueof y i byanamount i > 0suchthat a i = b i y i ,wedonotviolateanyconstraint.Moreover,thisimpliesach angein objectivefunctionvalueequalto i S 0 i ¡ K ( c;v;e ) h p C ¡ i 2i ¡ p C i > 0,where C = P ni =1 [ 2i y i +~ 2 i ( a i )], K ( c;v;e )isknowntobenonnegative,and S 0 i > 0. Therefore,wehaveincreasedtheobjectivefunctionvalue,w hichimpliesthat a i = b i y i foranoptimalsolution. } Sowecan,withoutthelossofoptimality,assumethat a i = b i y i inthefollowing relaxedproblem:maximize P i 2 I 1 [( r i ¡ c )~ i ( b i z i ) ¡ t i b i z i ¡ S 0 i ]+ P i 2 I 2 [( r i ¡ c )~ i ( b i y i ) ¡ t i b i y i ¡ S 0 i y i ] ¡ K ( c;v;e ) q P i 2 I 1 [ 2i +~ 2 i ( b i z i )]+ P i 2 I 2 [ 2i y i +~ 2 i ( b i y i )]

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118 subjectto: P i 2 I 1 b i z i + P i 2 I 2 b i y i B; 0 z i 1 i 2 I 1 ; 0 y i 1 i 2 I 2 : Finally,notethatbytheconvexityofthefunctions~ i andtheconcavityofthe functions~ 2 i wecanfurtherrelaxthisproblembynotingthat~ i ( b i y i ) ~ i ( b i ) y i and~ 2 i ( b i y i ) ~ 2 i ( b i ) y i .Therelaxationtooursubproblem,[R-LM( I 1 ;I 2 )],isstated asfollows: maximize P i 2 I 1 [(( r i ¡ c )~ i ( b i ) ¡ t i b i ) z i ¡ S 0 i ]+ P i 2 I 2 [( r i ¡ c )~ i ( b i ) ¡ t i b i ¡ S 0 i ] y i ¡ K ( c;v;e ) q P i 2 I 1 [ 2i +~ 2 i ( b i ) z i ]+ P i 2 I 2 [ 2i +~ 2 i ( b i )] y i subjectto: P i 2 I 1 b i z i + P i 2 I 2 b i y i B; 0 z i 1 i 2 I 1 ; 0 y i 1 i 2 I 2 : Bysubstituting y 0 i = z i for i 2 I 1 and y 0 i = y i for i 2 I 2 ,weobtainamore compactformulationoftherelaxationofthesubproblem:[R-LM( I 1 ;I 2 )] maximize P i 2 I 1 ¡ S 0 i + P i 2 I 1 [ I 2 R i y 0 i ¡ K ( c;v;e ) q P i 2 I 1 2i + P i 2 I 1 [ I 2 C i y 0 i subjectto: P i 2 I 1 [ I 2 b i y 0 i B; 0 y 0 i 1 i 2 I 1 [ I 2 : where R i = 8><>: ( r i ¡ c )~ i ( b i ) ¡ t i b i i 2 I 1 ( r i ¡ c )~ i ( b i ) ¡ t i b i ¡ S 0 i i 2 I 2 ,and

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119 C i = 8><>: ~ 2 i ( b i ) i 2 I 1 2 i ( b i ) i 2 I 2 Theaboveformservesasourupperboundingproblematanode.W edescribe thesolutionapproachtothisproblem,aswellastheB&Bimple mentation,inthe nextsection. 4.4.3 SubproblemSolutionandB&BImplementation Problem[R-LM( I 1 ;I 2 )]canbewrittenasaspecialcaseofamoregeneral problemdiscussedinRomeijn,Geunes,andTaae[ 69 ].Theirstrategyutilizesthe KKToptimalityconditionsandapreferentialorderingofsel ectionvariablesto ndanoptimalsolution.Infact,wecanusethissolutionapproac htosolve[RLM( I 1 ;I 2 )]inpolynomialtime,basedonthestructureofourproblem.We develop thesolutionapproachfor[R-LM( I 1 ;I 2 )]asfollows. Introducingthenonnegativedualvariables ; i ; i ,wepresenttheKKT conditionsfor[R-LM( I 1 ;I 2 )]asfollows: R i ¡ K ( c;v;e ) C i 2 q P nj =1 C j y j ¡ b i ¡ i + i =0 i =1 ;:::;n; (4.13) ( X ni =1 b i y i ¡ B )=0 ; (4.14) i (1 ¡ y i )=0 i =1 ;:::;n; (4.15) ¡ i y i =0 i =1 ;:::;n (4.16) X ni =1 b i y i ¡ B 0 ; 0 y i 1 i =1 ;:::;n: Notethatfor[R-LM( I 1 ;I 2 )],theKKTconditionsarenecessarybutnotsucient foroptimality.Thus,ourapproachproceedsbyenumeratinga llcandidateKKT points.ToconstructcandidateKKTpoints.Werstassumethatwehav esome candidatevalueoftheKKTmultiplier ;wewilllaterdiscusshowtodetermine appropriatecandidate values.Dening i = i ¡ i ,wecanrewriteKKT

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120 condition( 4.13 )as i = 24 ( R i ¡ b i ) C i ¡ K ( c;v;e ) 2 q P nj =1 C j y j 35 C i i =1 ;:::;n: (4.17) ObservingtheKKTconditions,if i 0weset i = i and i =0;otherwiseset i = i and i =0.Intermsoftheprimalsolution,wehave i > 0 ) i > 0 ) y i =1 i =0 ) i = i =0 ) 0 y i 1 i < 0 ) i > 0 ) y i =0 : AsindicatedinEquation( 4.17 ),weneedtoknowthevaluesofthe y variables inordertodetermineeach i givenanappropriatevalueof .Itactuallyturnsout, however,thatwedonotneedtoknowthespecicvaluesofthe i variablesinorder toevaluateprimalsolutionscorrespondingtocandidateKKTso lutions.Toshow this,notethatthesecondtermintheequationfor i isthesameforall i ,andthe valueoftheratio R i ¡ b i C i (4.18) completelydeterminesthesignof i foreachmarket 2 .Ifwerankmarketsin nonincreasingorderof( 4.18 ),wecanbecertainthatifsomemarket k has k 0 thenforallmarkets1,..., k {1, i 0.Similarly,ifsomemarket l has l < 0, then i < 0forall i>l .Then,foranyKKTpoint,wemusthavesome k 1 suchthat i > 0for i k 1 andsome k 2 suchthat i =0for k 1 k 2 ,where0 k 1 k 2 n .Wethereforeneedtoevaluatealimitednumberof possible k 1 and k 2 valuesforanygivenvalueof ,whereweseteach y i accordingto ( 4.18 ). 2 ThisistheratioobtainedbyapplyingLagrangianrelaxatio nto[R-LM( I 1 ;I 2 )] whenwerelaxthebudgetconstraintanduseLagrangianmultipl ier

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121 Tofurtherreducethevaluestoconsiderfor k 1 and k 2 (andthecorresponding solutionsinthe y variables),wenotethat,basedonaresultshowninHuanget al.[ 38 ],anoptimalsolutionexistsfor[R-LM( I 1 ;I 2 )]suchthatatmostone y i variableisfractional.Moreover,afractionalvalueof y i canonlyoccurwhenthe budgetconstraintistight.Asaresultwehavethat,afteranapp ropriatesecondary rankingscheme(whichwelaterdiscuss),asingleindex k existssuchthat y i =1for i =1 ;:::;k ¡ 1, y k 2 [0 ; 1],and y i =0for i = k +1 ;:::;n ,where k 1 +1 k k 2 Whenthebudgetconstraintistight, y k isoftheform0 y k 1.Whenthe budgetconstraintisnottight,thenusingtheKKTconditions,w emusthave =0,whichprovidesonechoiceoforderingaccordingtothera tio( 4.18 ).Wenext considertwotypesofKKTsolutions,thosewith =0andthosewith > 0. TypeI: =0 If =0,usingratio( 4.18 )werankmarketsinnondecreasingorderof R i =C i ThisratioorderingcorrespondstoourDERUratiopropertydi scussedpreviously intheabsenceofabudgetconstraint,andthisratioalsodeterm inesthesignsof the i variables.Nowwemustensurenotonlythatacandidatesolutionob eys theDERUproperty,butalsothatthebudgetconstraintissatised .Assume allmarketsaresortedinDERUorder.GivenaDERUsolutionthat iscapacity feasiblecontaininguptomarket k (i.e., P ki =1 b i B ),thisimplies y i =1 i =1 ;:::;k ¡ 1 ; y k =min n B ¡ P k ¡ 1 i =1 b i b k ; 1 o ; y i =0 i = k +1 ;:::;n:

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122 Iftheassociatedsolutionresultsin i > 0for i =1 ;:::;k ¡ 1 ; i < 0for i = k +1 ;:::;n; k 0for y k =1 ; k =0for0 0)willhavethesameeectontheobjective functionvalue.However,eachincreaseof "=C i consumes "b i =C i oftheadvertising budget.Therefore,weshouldadoptasecondaryorderingscheme (whentiesare presentintheprimaryordering)thatselectsmarketsbasedoni ncreasingorderof b i =C i .Ifthereisatieinboththeprimaryandsecondaryordering,t hisimpliesthat

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123 wehavecompletelyidenticalmarkets.Thesemarketscouldbet reatedasonelarger marketsuchthattherestillisonlyonefractional y i TypeII: > 0 HereweobservefromtheKKTconditionsthatwhen > 0,wemusthavea tightbudgetconstraint(i.e., P ki =1 b i y i = B ).Givensomevalueof ,wecanagain sortmarketsinnonincreasingorderoftheratio( 4.18 )andusethesecondaryrank orderingpreviouslydescribedtobreakanytiesintheratio.S incewemusthavea tightbudgetconstraintandbasedoncondition( 4.18 )(andthefactthattheratio ( 4.18 )alongwiththeappropriatechoiceoftheindex k determinesthesignsofthe i variables)weincludetherst k marketsinthesolutionsuchthat P ki =1 b i B and P k ¡ 1 i =1 b i
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124 zerooccursatthepointwheretheirratiosareequal;i.e.,w hen R i ¡ b i C i = R j ¡ b j C j : Thisimpliesthatthecriticalvalueof ,whichwedenoteby ij ,isgivenby ij = C i R j ¡ C j R i C i b j ¡ C j b i : Inotherwords,if R i =C i >R j =C j ,thenwehavethat( R i ¡ b i ) =C i > ( R j ¡ b j ) =C j forall < ij and( R j ¡ b j ) =C j > ( R i ¡ b i ) =C i forall > ij .Therefore therankorderingofratioscanonlychangeat( n ( n ¡ 1)) = 2possiblediscrete valuesof ij .Let p indexthesecriticalbreakpointsinincreasingorder,andlet p denotethemidpointoftheintervalbetween p ¡ 1 and p .Thisisequivalentto p =( p ¡ p ¡ 1 ) = 2for p =1 ;:::; ( n ( n ¡ 1)) = 2. Weconstructtheindexorderingforeachofthesevaluesof p usinganonincreasingorderingoftheratio( R i ¡ p b i ) =C i ,againbreakinganytiesusingthe secondaryorderingdescribedforTypeIsolutions.Wethenconstr uctthesolution inthe y variablesaccordingto( 4.20 ),where k issuchthat P ki =1 b i B and P k ¡ 1 i =1 b i 0.ThecomputationaleortrequiredtoevaluateallTypeIand Type IIsolutionsandndtheoptimalmarketselectionis O ( n 3 ).

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125 Asstatedpreviously,thissolutionprocedurendsanoptimalsolu tiontothe relaxedsubprobleminwhichatmostonemarkethasafractional valuefor y 0 i Thisimpliesthatif y 0 i isfractionalforsomemarket i 2 I 1 ,thenthecorresponding marketeort a i islessthan b i .Infact,sincethebranchingstrategydoesnot guaranteethatallmarketsin I 1 willactuallybeservedintheoptimalsolutionto theoriginalproblem[LM],itmaybeoptimalinthesubproblem [R-LM( I 1 ;I 2 )]to performnomarketingforsome i 2 I 1 WearenowpreparedtointroducetheB&Bprocedureforsolving problem [LM].Assumethatwearegiventhemarketscontainedinsets I 0 I 1 ,and I 2 .At therootnodeofthetree,wethensolvetherelaxedproblem[RLM( ; ;I )],where I representsthesetofallpotentialmarkets.Basedonsomebranchi ngstrategy, eitherbranchingonthefractionalmarketselectionvariabl eorfollowingapredeterminedbranchingorder,wexonemarketselectionvariab leto0or1andsolve anewrelaxedproblem,wheretheonemarketselectionvariabl ehasbeenplaced eitherinset I 0 or I 1 .Aswemovefurtherdownthetree,additionalmarketsare addedtosets I 0 and I 1 (andsubsequentlyremovedfromset I 2 ),andwesolvea problemoftheform[R-LM( I 1 ;I 2 )]ateverynode.Sinceatmostone y 0 i willbe fractionalinanysubproblemsolution,wecanquicklyconstruc tafeasibleinteger solutionsimplybyroundingthefractional y 0 i tozero.Thisheuristiccanprovidea quickmethodfortighteningthelowerbound.Ofcourse,ifata nynode y 0 i 2f 0 ; 1 g for i 2 I 2 ,wecanfathomandcheckifthiscurrentintegersolutionisan improved lowerbound.WeformallypresenttheB&Bprocedurebelow,and weprovide computationalresultsusingthisB&Bschemeinthefollowingsec tion. Wealsonotethatourmodelingapproachcanhandleproblemswh eretherm hascontractualobligationsincertainmarketsorapriorik nowledgeofunprotable markets.Insuchsituations,sets I 0 and I 1 wouldnotbeemptyattherootnodeof thetree.

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126 Branch-and-BoundSolutionto[LM]0) Assumewearegivensets I 0 I 1 ,and I 2 .Setthelowerbound( LB )to0,and settheupperbound( UB )toinnity.SolvetheLPrelaxationproblem [R-LM( I 1 ;I 2 )]associatedwiththesesetassignments.Denote Z ¤ asthe optimalsolutionvalue.Let ¤ representthevalueforwhichwedetermined theoptimalrankingofmarkets.DeneLISTasthelistofallmark etsin I 2 rankedaccordingtotheratio R i ¡ ¤ b i C i 1) If( z;y )isintegral,STOPwiththeoptimalsolution.Themarketinge ort associatedforeach z i =1andeach y i =1is b i .Otherwise,set UB = Z ¤ and continue. 2) (BranchingonNewVariable)DenotetherstmarketinLISTas k .Update I 1 = I 1 [f k g and I 2 = I 2 nf k g 3) If k 2 I 1 and y 0 k =1(or k 2 I 0 and y 0 k =0)inthelastsubproblem,retain thesolution( z;y ),solutionvalue Z ¤ ,ratioranking,and ¤ fromthelast subproblem,andgotoStep4.Otherwise,solveanewsubproblemof theform [R-LM( I 1 ;I 2 )].UpdateLISTtoincludeallmarketsin I 2 rankedaccordingto theratio R i ¡ ¤ b i C i ,wheretheoptimalsolutiontothissubproblemwasfound usingamarketrankingbasedon ¤ .Recordthenewsolutionas( z;y ),witha solutionvalueof Z ¤ 4) If( z;y )isfeasibleandnotintegral,and Z ¤ >LB ,gotoStep2.If( z;y )is feasibleandintegral,and Z ¤ >LB ,set LB = Z ¤ .Continuetofathoming step. 5) (FathomingStep)Forthecurrentnodewithbranchingmarket variable k ,check whetherthisparent'sotherchildnodehasalreadybeenenum erated.Ifso, thenremove k fromtheappropriateset( I 0 or I 1 ),let I 2 = I 2 [f k g ,update k totheparentnode'sbranchingvariableandrepeatStep5.Ot herwise, continue.

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127 6) (BranchingonSameVariable)If k 2 I 1 ,nowset y 0 k =0, I 1 = I 1 nf k g ,and I 0 = I 0 [f k g .Otherwise,update I 1 = I 1 [f k g ,and I 0 = I 0 nf k g .Returnto Step3. 4.5 ComputationalResults WenowdiscusscomputationaltestsforseveralvariantsoftheSNP model. First,weaddresstheimportanceofthebasicSNP,introducedinS ection 4.2 ,for whichwearegivenxedlevelsofmarketingtoapplyineachmar ket.Then,we evaluateoursolutionapproachforsolvingtheSNPwithlimited resources,where marketingeortisadecisionvariable. 4.5.1 SNPValue:MinimumMarketRequirement Atthisstage,itisimportanttoquantifythevaluetoarmwhen usingthe selectivenewsvendorapproach.Givenasetofpotentialcustome rsormarkets, andforecastestimatesforexpecteddemandineachmarket,wesh ouldbeableto discernwhetherusingthebaseSNPmodelwithxedadvertisingwill provideany protimprovement.Wehavepreviouslydiscussedthebenetofrisko runcertainty poolingprovidedbytheselectionofadditionalmarkets.Toga infurtherinsight intotherolethatthenumberofmarketsplays,consideranoper ationthatis unprotablewithasmallmarketsetbutbecomesprotablewithth eadditionof newmarkets.Thisimpliesthatthereisaminimumnumberofmar ketsrequired beforeachievingaprot.Thequestionthenbecomes,\Whichpar ametershavethe mostinuenceontheminimumnumberofrequiredmarkets?" Considerasetof n identicalmarkets;i.e.,theexpecteddemand( ),demand variance( 2 ),unitrevenue( r ),andentrycost( S )isthesameforeachmarket. Assumingallmarketswillbeentered(sincetheyareidentical), theresulting expectedprotequationis G ( Q ¤ )= n ( r ¡ c ) ¡ nS ¡ K ( c;v;e ) p n: (4.21)

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128 Inordertoachieveprotability,werequire G ( Q ¤ ) 0.Solvingfor n inequation ( 4.21 ),wehave n ¤ K ( c;v;e ) ( r ¡ c ) ¡ S 2 ; (4.22) where n ¤ istheminimumnumberofmarketsrequiredtoobtainaprot.We can useequation( 4.22 )todrawseveralconclusionsabouttheinuencethateach parameterhasontheminimummarketrequirements.Ifthermis facedwitha lowprotmargin( r ¡ c ),alargecoecientofvariation( CV = = ),orahigh marketentrycost( S ),theyshouldexpectahigherminimummarketrequirement. Recallthat K ( c;v;e )= f ( c ¡ v ) z ( )+( e ¡ v ) L ( z ( )) g .Todeterminetheeect that K ( c;v;e )hasontheminimummarketrequirement,weproceedasfollow s. Holdingallotherparametersconstantin( 4.22 ),wecanplotthevalueof n ¤ as c v; or e isincreased.Figure 4{4 illustratestheeectthateachparameter( c v e ,and themarketentrycost S )hasontheminimummarketrequirement. Unit Salvage Value ( v )Min # Markets Unit Expediting Cost ( e )Min # Markets Unit Production Cost ( c )Min # Markets Market Entry Cost ( S )Min # Markets Unit Production Cost Unit Revenue 5 x (Unit Revenue) Expected Net Revenue Unit Revenue (b) (a) (d) (c) Figure4{4:Minimummarketrequirementbasedonindividualc ostparameters.

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129 ConsiderFigure 4{4 (a),whichdepictsunitsalvagevalueagainsttheminimum numberofmarketsrequired.Astheunitsalvagevalueapproach estheunitproductioncost,theminimummarketrequirementdecreases.Thisimpl iesthatifmostor alloftheproductioncostcanbesalvagedonunuseditems,therisk ofenteringa marketismuchlower,andtheresultingminimummarketrequir ementisalsolower. InFigure 4{4 (b),weobservethattheminimummarketrequirementincreases linearlywithincreasesintheunitexpeditingcost.Aswouldbe expected,higher expeditingcostswillresultinalargerminimummarketrequir ement.Figure 4{4 (c) comparestheunitproductioncostagainsttheminimumnumbero frequiredmarkets.Astheunitcostapproachestheunitrevenuesetting,there quirednumberof marketstoobtainanexpectedprotincreasesexponentially. Thisresultreiterates thepreviousconclusionthatitemswithlowprotmarginswill haveasignicant impactontheminimummarketrequirement.Similarly,Figur e 4{4 (d)further supportsthisargumentbyillustratingthattheminimummarke trequirementalso increasesexponentiallyasmarketentrycostsincrease. Sowhatwillhappenwhentherearenotenoughcandidatemarke tsinwhich armcanoperate?First,wemustrememberthatthemarketsinthe previous examplewereassumedtobeidentical,andthisisnotlikelytoo ccurinanactual operation.Thismeansthatthermcanexpectsomedierentiati onbetween markets.Ifwecanidentifyandselectonlythosemarketsthatwi lladdbenetto therm'soperation,thenwemayachieveprotabilityevenwit houthavingenough candidatemarkets.WemaynotneedtoincludeALLmarketsinthe plan,andthis iswheretheselectivenewsvendorapproachbecomesimportant 4.5.2 SNPValue:ProtImprovement Wereturntoourdiscussionofthetypicalmarketselectionprobl eminwhich eachmarketcontainsuniqueper-unitrevenues,marketentry cost,expected demand,andvariancedata.Thusfar,wehaveshownthatspecicm arketdata,

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130 therm'scostparameters,alongwiththetotalnumberofpossible marketsinthe operation,willeachinuencethemarketselectionandtheove rallexpectedprot. InordertodemonstratethevalueoftheSNPapproach,wepresent acomparison oftheoptimalorderquantitydecisionsbasedonthefollowing twomethodsof operation: Method1:Selectingasubsetofmarketstoenter(SelectiveNewsv endor Approach) Method2:Selectingtheorderquantityusingallmarketsinwh ichunit revenueexceedsunitcost(MaximumMarketShareApproach) Afterallcostsandrevenueshavebeendetermined,wecancalcu latetherm's protforeachmodelingapproach.Anyprotimprovementresulti ngwiththeSNP approachwillthenberecorded. Weusethefollowingtestdataforthecomparison.Everymarketh asunit revenueintherangeU[$200,$240],whiletheunitproduction costissetat$200. Expecteddemandanddemandvarianceforeachmarketaredistr ibutedaccording toU[500,1000]unitsandU[50000,100000],respectively.Thex edcostformarket entryaredrawnfromU[$2500,$7500].Finally,thesalvageval ueis$50perunit, andweusethreesettingsforexpeditingcost:$350,$425,and$5 00perunit, respectively.Evenatthehighestexpeditingcostof$500perun it,westillonly obtainacriticalfractileof =( e ¡ c ) = ( e ¡ v )=0 : 67.Thisimpliesthatthermis willingtoacceptproductexpeditingone-thirdofthetime. Notethatitispossible fortheexpectednetrevenueinmarket i tobenegative(i.e.,( r i ¡ c ) i ¡ S i < 0). Allsuchmarketswillberemovedfromconsideration. Figure 4{5 presentsthepercentimprovementinprotwhenimplementing theSNPapproachoverthemaximummarketshareapproach,basedo nthetotal numberofpotentialmarkets.Dataareshownforeachsettingoft heexpediting cost.Thereisanoticeableprotimprovementwhenthermhas20 orfewer

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131 candidatemarketstoconsider,whichsuggeststhatmodelingth isoperationwith aselectivenewsvendorapproachisquiteappealing.Forhighe rlevelsofexpediting costs,theimprovementinexpectedprotwhenusingtheSNPapproa chiseven moredramatic.Asthenumberofcandidatemarketsapproaches 50,theprot gainedfromimplementingthisapproachbecomesminimal,wh ichisanillustration oftheeectofuncertaintyorriskpooling.Onecanclearlyseet hebenetofhaving alargesetofcandidatemarketsfromwhichtochoose. % Profit Improvement vs. # Markets0% 10% 20% 30% 40% 50% 01020304050 # Markets Expediting Cost = 350 Expediting Cost = 425 Expediting Cost = 500 Figure4{5:ProtimprovementusingSNPbasedontotalmarketsav ailable. Figure 4{6 presentsthepercentimprovementinprotobtainedwiththeSNP approach,basedonvariouslevelsofdemandvariance.Forthi scomparison,we usedasingleexpeditingcostof$500perunit.Wetestedthefollo wingrangesfor demandvariancewithinamarket:U[100,2000],U[1000,10000] ,U[2500,25000], U[5000,50000],U[5000,75000],andU[5000,100000]. Whenmarketdemandisquitepredictable,thereisofcourseve rylittlebenet ofusingaselectivenewsvendorapproach.Littleornodemandun certaintyimplies thatthereisnotmuchdierenceinthecandidatemarkets,andse lectingall marketsbecomesthemostprotableapproach.Moreover,whend emandvariance withineachmarketissmall,theminimummarketrequirementi salsoverylow.

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132 % Profit Improvement vs. Variance Level0% 10% 20% 30% 40% 50% 01000020000300004000050000 Variance Level 10 Markets 20 Markets U[1000,10000] U[2500,25000] U[5000,50000] U[5000,75000] U[5000,100000] Figure4{6:ProtimprovementusingSNPbasedondemandvariance Thisisillustratedbythenearlyidenticalresultspresentedf orthe10-and20marketcasesatlowdemandvariancesettings.Asweincreasetheav eragedemand variancebeyondalevelof25,000unitspermarket,however, theimprovement inprotissignicant.Thedemandvarianceincreaseandthesmall ercandidate marketsetbothcontributetotheprotimprovementshownonthe graph. Basedontheexamplesofpresentedinthissection,wecanconclu dethat incertaincontexts,rmsmayhavetheopportunitytoachievesu bstantialprot improvementsbyusingaselectivenewsvendorapproach. 4.5.3 SolvingtheLimitedResourcesProblem Inthissection,weexaminetheeectivenessoftheB&Bapproach insolving theselectivenewsvendorproblemwithlimitedresources.Wealso performcomputationalteststhatshowtheintegralitygapthatresultsfromso lvingarelaxation oftheoriginalproblem[LM].Considertherelaxationsubprob lem[R-LM( I 1 ;I 2 )]at therootnode,whereweassumethatwearegiventhemarketscont ainedinsets I 0 I 1 ,and I 2 .Wewillrefertothisrelaxationproblemas[R-LM( I 1 ;I 2 )].Recall thatwehavepresentedsolutionapproachesto[LM]fordemandf unctionsthat followeitherCaseIorCaseIIassumptions.CaseIIassumesexpectedd emand

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133 followsanS-shapedapproximationthatisconvexandnondecr easing,anddemand variancefollowsaconcavenondecreasingfunction.First,we presentcomputational resultsforthespecialcaseofCaseII,whenbothexpecteddemand anddemand varianceincreaselinearlywithmarketingeort.Wethenprov ideananalysisof CaseI,wheredemandvarianceisassumedxed.Again,weusealinear lyincreasingexpecteddemandfunctioninplaceofthemoregeneralcon vexnondecreasing function. Weorganizethecomputationaltestsasfollows.Weconsiderthr eequantities forthesizeofthepotentialmarketpool:10,20,and50.Withi neachmarketpool scenario,wesettheadvertisingbudgetateachofthefollowing levels:25%,50%, 75%,100%,and200%ofthetotalexpecteddemandacrossallava ilablemarkets. AspreviouslyintroducedinSection 4.5.2 ,unitrevenueandentrycostforeach marketweredrawnfromU[$200,$240]andU[$2500,$7500],respe ctively,aswellas assumingtheproductioncost,expeditingcost,andsalvagevalue sare$200,$500, and$50,respectively.Totalexpecteddemanddependsonthef ollowingmarketing eortparameters.Thedemandperadvertisingunitisdistribute daccordingto U[10,20],theunitadvertisingcostisdistributedaccordingto U[$30,$50],whilethe marketinglevelbeyondwhichnoadditionaldemandcanbegen eratedisdrawn fromU[75,125].Forthecomputationaltests,wegenerated500r andomproblem instancesforeachmarketpoolsizeandeachcase(IandII),fora totalof3000 probleminstances. Table 4{1 comparesthesolutionqualityofproblem[R-LM( I 1 ;I 2 )]toproblem [LM]forCaseII,inwhichexpecteddemandanddemandvariance increaselinearly withthelevelofmarketingeort.Onlyforanadvertisingbudg etoflessthan 50%oftotalexpecteddemandisthereasignicantintegrality gapbetweenthe relaxedandmixedintegerformulations.Asthemarketpoolinc reasestoasizeof 50potentialmarkets,thesolutionprovidedby[R-LM( I 1 ;I 2 )]iswithin0.31%of

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134 Table4{1:ResultsforSNPwithlimitedresources{CaseII. AdvertisingBudget:%ofExpectedDemand Scenario/Measurement 25% 50% 75% 100% 200% 10Markets [R-LM( I 1 ;I 2 )]{[LM]%Gap 6.02% 1.21% 0.31% 0.02% 0.00% [LM]SolutionTime 0.00sec. 0.00sec. 0.00sec. 0.00sec. 0.00sec. Avg#NodesUsed 10 11 7 2 1 20Markets [R-LM( I 1 ;I 2 )]{[LM]%Gap 1.11% 0.59% 0.13% 0.00% 0.00% [LM]SolutionTime 0.01sec. 0.02sec. 0.01sec. 0.00sec. 0.00sec. Avg#NodesUsed 17 26 17 1 1 50Markets [R-LM( I 1 ;I 2 )]{[LM]%Gap 0.31% 0.20% 0.06% 0.00% 0.00% [LM]SolutionTime 0.67sec. 1.17sec. 0.86sec. 0.03sec. 0.03sec. Avg#NodesUsed 47 76 59 1 1 theexact[LM]solution.Itisalsoworthpointingoutthatforp roblemswith50 orfewermarkets,solving[LM]tooptimalitytypicallyrequir eslessthan1.0CPU second.WealsoincludethenumberofsubproblemssolvedintheB& Btreetond theoptimal[LM]solution. Table 4{2 comparesthesolutionqualityofproblem[R-LM( I 1 ;I 2 )]toproblem [LM]forCaseI,inwhichexpecteddemandincreaseslinearlywi ththelevelof marketingeortanddemandvarianceremainsconstantatanyad vertisinglevel. Noticethattheintegralitygapforinstanceswhentheadverti singbudgetisless than50%ofexpecteddemandissignicantlyhigherthanthesame resultsfor CaseII.Recallthat,for[R-LM( I 1 ;I 2 )],weinvokealinearityassumptionforall demandfunction'sresponsetoadvertising.Thisimpliesthat, forCaseIandII, [R-LM( I 1 ;I 2 )]attherootnodewillhavethesamevalue.(Oncesubproblem[R LM( I 1 ;I 2 )]issolvedatothernodesinthetree,solutionsmaybedierent between thetwocases.)However,underCaseI,theentireuncertaintycost isspentassoon asthemarketisentered.So,the[LM]solutionprovidedunder CaseIwillalways belessthanorequaltothe[LM]solutionunderCaseIIforthesame problemdata, andthisresultsinalargergap.Thisconclusionisalsosupporte dbythefactthat

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135 Table4{2:ResultsforSNPwithlimitedresources{CaseI. AdvertisingBudget:%ofExpectedDemand Scenario/Measurement 25% 50% 75% 100% 200% 10Markets [R-LM( I 1 ;I 2 )]{[LM]%Gap 17.21% 3.03% 0.48% 0.02% 0.00% [LM]SolutionTime 0.00sec. 0.00sec. 0.00sec. 0.00sec. 0.00sec. Avg#NodesUsed 13 12 8 1 1 20Markets [R-LM( I 1 ;I 2 )]{[LM]%Gap 2.42% 1.15% 0.18% 0.00% 0.00% [LM]SolutionTime 0.00sec. 0.00sec. 0.00sec. 0.00sec. 0.00sec. Avg#NodesUsed 28 33 15 1 1 50Markets [R-LM( I 1 ;I 2 )]{[LM]%Gap 0.64% 0.30% 0.08% 0.00% 0.00% [LM]SolutionTime 1.70sec. 1.81sec. 0.92sec. 0.03sec. 0.03sec. Avg#NodesUsed 83 102 60 1 1 solving[LM]requiresadditionalnodesintheB&Btree.Yet,o naverage,thetime requiredtoobtainthe[LM]solutionisstilllessthan2CPUsecon ds. Asprobleminstancesincreasebeyond50markets,thesolutiontim eforthe mixedintegerformulationmaybecomesubstantial.Sinceweob servethat[RLM( I 1 ;I 2 )]attherootnodeprovidesaverytightboundon[LM]forthese large marketpoolproblems,wecouldsimplysolve[R-LM( I 1 ;I 2 )]androundtoobtainan integersolution.Thisheuristicapproachislikelytoyieldh ighqualitysolutions. 4.6 OtherConsiderations 4.6.1 ExtensiontotheInniteHorizonPlanningProblem Whilemanyoftheapplicationsinthenewsvendorliteraturei nvolveproblems withasingleplanningperiodandsellingcycle,therearemany situationswhenthis cyclewillrepeatitselfinfuturesellingseasons.Therefore,it wouldbedesirableto examinehowtoselectmarketsandsettheorderquantityforeac hsellingseason. First,weassumethateachmarkethasperiodicindependent,stat ionary,and identicallydistributed(iid)demandoveraninnitehorizon .Anysuppliershortages incurredwithinaperiodarenowback-orderedatacostof b perunit.Givena multi-periodproblem,thereisnolongeraneedtosalvageexc essproduct.Instead,

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136 wemaintaintheproductininventoryatacostof h perunitineachperiod.We alsoassumethattheproductdoesnotchangeandcanbesoldinfutu reperiods. Thermmustalsohaveanunconstrainedsupplyandeitherhavenegl igibleorder costsperperiodorhavescheduleddeliveriesineveryperiod. Intheinnitehorizonproblem,wewanttomaximizethelong-r unexpected protperperiod.Assumingfullback-orderingofdemand,wecane xtendthe SNPmodeltoaninnitehorizoninasimilarmannershowninNahmias [ 56 ].To determinethelong-runexpectedprot,westartbyassumingan N -periodproblem. Inordertocomputethe N -periodprot,wemustknowthequantitiessoldineach period.Let D i t representtherealizeddemandfrommarket i inperiod t .Then, let D y t representthetotalquantitydemandedacrossallmarketsforp eriod t (i.e., D y t = D 1 t + D 2 t + ¢¢¢ + D I t ).Intherstperiod,thermwouldorder Q .Infact,since eachmarkethasiiddemandperperiod,thermwouldmaintaina norderof Q for everyperiod.Sinceanydemandnotmetisback-ordered,this impliesthattherm willalwaysplaceanorderequivalenttolastperiod'stotald emand.Thisisshown below: Unitssoldinperiod1=min( Q;D y 1 ) Unitssoldinperiod2=max( D y 1 ¡ Q; 0)+min( Q;D y 2 ) Unitssoldinperiod3=max( D y 2 ¡ Q; 0)+min( Q;D y 3 ) ¢¢¢ Unitssoldinperiod N =max( D y N ¡ 1 ¡ Q; 0)+min( Q;D y N ) :

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137 Wecanthenstatethe N -periodexpectedprotas G ( Q;y )= P i 2 I (( r i ¡ c ) E ( D i 1 + D i 2 + ¢¢¢ + D i N ¡ 1 ) ¡ S i ) y i + rE [min( Q; P i 2 I D i N y i )] ¡ cQ ¡ N h h R Q 0 ( Q ¡ x ) f y ( x )d x + b R 1 Q ( x ¡ Q ) f y ( x )d x i = P i 2 I (( r i ¡ c )( N ¡ 1) i ¡ S i ) y i + rE [min( Q; P i 2 I D i N y i )] ¡ cQ ¡ N h h R Q 0 ( Q ¡ x ) f y ( x )d x + b R 1 Q ( x ¡ Q ) f y ( x )d x i ; where r representstheperunitaveragerevenueofthequantitysoldin period N Thiswouldbearelativelydicultvaluetocalculate,butitw illnotbenecessary. Dividingby N andletting N !1 ,wecanrepresentthelong-runexpectedprot perperiodas G ( Q;y )= P i 2 I ( r i ¡ c ) i y i ¡ h R Q 0 ( Q ¡ x ) f y ( x )d x ¡ b R 1 Q ( x ¡ Q ) f y ( x )d x = P i 2 I ( r i ¡ c ) i y i ¡ hQ + h y ¡ ( b + h ) R 1 Q ( x ¡ Q ) f y ( x )d x: Foragivenvector y ,thisexpectedprotequationisconvexin Q ,andtherstorderconditionimpliesanoptimalorderquantity, Q ¤y ,satisfying F y ( Q ¤y )= b b + h ; wherewenowhave = b b + h .Foragivenvector y ,andtheresultinglossfunctionof ¤ y ( Q )= R 1 Q ( x ¡ Q ) f y ( x )d x ,werewritetherm'slong-runexpectedprotas: G ( Q ¤y ;y )= X ni =1 [( r i ¡ c ) i ¡ S i ] y i ¡ hQ ¤y + h y ¡ ( b + h )¤ y ( Q ¤y ) : Assumingnormallydistributeddemand,weusethesameoptimalorde r quantityequation( 4.2 )andidentity¤ y ( Q ¤y )= y L ( z )asbefore.Letting z ( )= z ( b b + h )and r i =( r i ¡ c ) i ,wecanwritetherm'slong-runexpectedprotas: G ( Q ¤y ;y )= X ni =1 r i y i ¡f hz ( )+( b + h ) L ( z ( )) g r X ni =1 2 i y i :

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138 Replacing K ( c;v;e )fromtheoriginalmodelwith K ( b;h )= f hz ( )+( b + h ) L ( z ( )) g weformulatetheselectivenewsvendorproblemoveraninniteh orizon(SNP-I): [SNP-I] maximize P ni =1 r i y i ¡ K ( b;h ) p P ni =1 2 i y i subjectto: y i 2f 0 ; 1 g i =1 ;:::;n: Noticethatthisformulationhasthesamestructureasthesingle periodversion oftheproblem,withthefollowingminorexceptions.First,th emarketentrycost isnotpresentintheformulation.Thisisduetothefactthata costincurredonly intherstperiodwillnotbesignicantinthelongrun.Second,t he K ( c;v;e ) constanthasbeenreplacedwith K ( b;h ),whichmeansthattheproductioncost nolongerinuencesuncertainty-relatedcosts.Aswithformula tion[SNP],wecan applytheDERURatiopropertyto[SNP-I]anddeterminetheopt imalselectionof markets,wheretheratioforeachmarket i isdenedby r i = 2 i 4.6.2 LimitedMarketingEortunderaFixedContract Instead,let'sassumewearegivenapre-denedselectionofmarke ts;i.e.,the rmisoperatingunderaxedcontractthatstatesitwillserveth isgivensetof markets.Thisassumption,infact,allowsustoformulateaprob lemforwhichwe willoerastraightforwardsolutionapproach.SimilartoSect ion 4.2.1 ,denea binaryvectorofmarketselectionvariables y ,andlet I 1 denotethesetofmarkets suchthat y i =1.Wecanwritetherm'sexpectedprotequationas G ( Q ¤y ;y )= X i 2 I 1 ( r i i ( a i ) ¡ t i a i ¡ S 0 i ) ¡ ( c ¡ v ) Q ¤y ¡ v y ¡ ( e ¡ v )¤ y ( Q ¤y ) : Again,assumingnormallydistributeddemandandamarketingeor t a i thatonly aectsthemeanvalueofdemandinmarket i ,wecanrestatetheoptimalorder

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139 quantityas Q ¤y = X i 2 I 1 i ( a i )+ z ( ) r X i 2 I 1 2 i : where z ( )= ¡ 1 ( )isthestandardnormalvariatevalueassociatedwiththe fractile .ThroughthesamesimplicationsusedinSection 4.2.1 ,therm's expectedprotbecomes G ( Q ¤y ;y )= X i 2 I 1 r i ( a i ) ¡ K ( c;v;e ) r X i 2 I 1 2 i : (4.23) where r i ( a i )=( r i ¡ c ) i ( a i ) ¡ t i a i ¡ S 0 i .Noticethattheuncertaintyterminthe aboveequationissimplyaxedcost,regardlessoftheamountofm arketingeort expendedinanymarket.Excludingthistermfromtheoptimiz ationproblem,and assumingamaximumavailablemarketingeortof B ,theselectivenewsvendor withlimitedresources(SNP-LR)canbeformulatedas:[SNP-LR] maximize P i 2 I 1 r i ( a i ) subjectto: P i 2 I 1 t i a i Bi 2 I 1 ; 0 a i b i i 2 I 1 : Thisproblemcanbeclassiedasanonlinearknapsackproblem,an dwewill presentsolutionapproachesbasedonthevariousformsthatthe expecteddemand functioncantake.Usingthespecialcasesoflinearlyincreasing expecteddemand andpiecewiselinearlyincreasingexpecteddemandshowninCase IofSection 4.3.2 wenowdevelopsolutionapproachesunderaxedcontract. First,let'sconsiderthecaseofhavingpiecewise-linearnonde creasingconcave expecteddemandasafunctionofmarketingeort.InSection 4.3.2 ,wedetermined thattheoptimalmarketingeortinmarket i willbeequaltoavalue^ a i suchthat t i = ( r i ¡ c ) 2 @ i ( a i )at^ a i .Thisisthevaluewhere( r i ¡ c ) s ij t i ( r i ¡ c ) s i;j +1 formarket i .Assumethatthereare M i lineardemandsegmentsinmarket i

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140 where( r i ¡ c ) s ij t i forsome j .Wewouldliketochoosethemostprotable segmentsfromthe P i 2 I 1 M i protablesegmentsacrossallmarkets.Wecanplace thesesegmentsinnondecreasingorderof( r i ¡ c ) s ij values,andsimplyassign themarketingeorttothecorrespondingprotablemarketsegme ntuntilthe marketingeortlimitof B isreachedorexceeded.Let k i representthenumber ofsegmentsfrommarket i includedintheoptimalsolution,andlet a k i denote themaximummarketinglevelassociatedwithsegment k i .Finally,let j represent themarketforwhichthelimitof B wasreached.Then,theoptimalmarketing eortinmarket i 2 I 1 n j willbe^ a i = a k i ,andwecandeterminetheresulting expecteddemandinmarket i basedonthepiecewise-linearfunctiondenedin Section 4.3.2 .Sincesegment k j frommarket j isthelastsegmentassigned,then theoptimalmarketingeortinmarket j isrepresentedby^ a j = B ¡ P i 2 I 1 n j a k i Thisassignmentprocedurerequires O ( n P i 2 I 1 M i log P i 2 I 1 M i ).Thereasonthat thisapproachwillworkisthatwehavecommittedup-frontto enteringallofthe marketsincludedin[SNP-LR],andtheuncertaintycostdepict edinequation( 4.23 ) isaconstant.Therefore,theonlydecisionistodeterminehow muchprotto obtainfromeachmarketthathasbeenentered. Actually,thelinearlyincreasingcaseisnowjustaspecialcaseof thepiecewiselinearcase,whereeachmarketonlyhasonelineardemandsegme nt(i.e., M i =1 forall i ).Then,theabovealgorithmwillalsosolvethelinearlyincre asingcase optimallyaswell. 4.7 Conclusions Tofullyaddressdemandselectionmodeling,onemustsurelyconsi derthe eectthatdemanduncertaintyhasonourselectiondecisions.Ev enwhentherm decidesaprioritoapplycertainxedmarketinglevelsineac hmarket,wearefaced withsolvinganintegerproblemwithanonlinearprotmaximiz ationobjective.

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141 AswehaveshownforthebasicSNP,ourmodelingapproachutilizes the problemstructuretoprovideaclosed-formsolutionbasedonthe rankingofeach market'sattractiveness,whichwecalltheDERU(DecreasingEx pectedRevenue toUncertainty)ratio.Oncewehaveanoptimalranking,weoer severalwhat-if scenariosforthermtoconsider.Eachoftheseinsightscanplaya roleindeciding whethertoexpanditspresenceinnewmarkets,increasethemark etingeort,or oersomeformofpricediscounting. Wealsoconsidertherolethatsalesandadvertisingplaysindete rminingthe demanddistributionobservedwithineachmarket.Whenthermd ecidesapriori toapplycertainxedmarketinglevelsineachmarket,wearef acedwithsolving anintegerproblemwithanonlinearprotmaximizationobjec tive.Andwehave shownthroughtheDERUratiopropertythatthisproblemissurp risinglyeasyto solve.Beyondthisxed-advertisingbasicSNP,weevaluatedafai rlygeneralsetof advertisingresponsefunctions,eachofwhichhasuniqueproper tiesfortheoptimal selectionofmarketsandthecorrespondingadvertisinglevels inthesemarkets. Forthecasesinwhichtherewasanunlimitedmarketingbudget ,wecandeterminetheoptimalmarketingleveltoexpendineachmarket,an dweshowhowthe problemsimpliestothebasicSNP.Combiningtheeectsofalimit edmarketing budgetwithaprotobjectivebasedonexpectedrevenueswithd emanduncertainty,wearethenfacedwithanonlinearknapsackproblemwi thanonseparable objectivefunction,whichisaverydicultproblemtosolvein general.Forthe limitedresourcescase,weprovideabranch-and-boundproced uretoobtainthe optimalsolution.However,wecanactuallysolvethenonlinear knapsackrelaxation subproblemsinpolynomialtimeateachnode.Infact,forprob lemswith50markets,wehavealsoshownthattheintegralitygapprovidedbythe originalproblem relaxationislessthan1%fortherandomlygeneratedproblem instanceswetested.

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CHAPTER5 AIRPORTCAPACITYLIMITATIONS{ SELECTINGFLIGHTSFORGROUNDHOLDING 5.1 Introduction Wenowshiftourattentionfromtheproductionandorderingsyst emsdescribedinChapters 2 4 toademandselectionproblemarisingintheairtransportationindustry.Throughoutthisdissertation,wedevelop solutionapproachesto decisionproblemsthatrequireselectingtheappropriatedem andsourcestosatisfy, basedonavailableresources.Inthischapter,weaddresshowtose lectaircraftfor arrivaltoasingleairportexperiencingbadweather,throug htheimplementationof agroundholdingplan.Wealsodemonstratetheadvantagesofusi ngastochastic programmingapproachtoaddressweather-relateduncertain ties. Overthepast20years,businessandleisureairtravelhaveconsiste ntly increasedinpopularity.Withmoreandmorepassengerswanting totravel,airlines andairportshavecontinuedtoexpandtomeetpassengers'needs. Andnow,many airportsareatornearcapacitywithfewoptionsforexpansio n(seeU.S.House SubcommitteeonAviation[ 77 ]).Asmoreairportsapproachtheircapacity,the airtravelindustryiswitnessinghigheraveragedelays.Whileso medelaysresult fromanairline'soperations(groundservicing,ightcrews,l atebaggage,etc.),a majorityoftheseveredelaysareweatherrelated.Duringbad weather,theFederal AviationAdministration(FAA)imposesrestrictionsonthenumbero faircraftan airportcanacceptinanhour.Intechnicalterms,theairport willbeinstructedto operateunderoneofthreeightrulepolicies:VFR(VisualFlight Rules),IFR1 (InstrumentFlightRules1),orIFR2(InstrumentFlightRules 2{morerestrictive thanIFR1).AnairportoperatesunderVFRduringgoodweathero rnormal 142

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143 conditions.Astheweatherconditionsdeteriorate,theFAAmay restrictairport capacitybyrequiringanairporttooperateunderIFR1orIFR 2.Inthemost extremecases,anairportwilltemporarilycloseuntilthepoor weatherconditions subside. Asaresultoftheserules,theairportandairlinesmustdecidewha ttodowith alloftheaircraftwantingtoarriveatanairportexperienc ingbadweather.The aircraftcanbeallowedtotakeoandapproachtheairport,r esultinginsomeair delayswhileightcontrollerssequencethesearrivingaircra ft.Alternatively,the aircraftcanbeheldattheiroriginatingstations,incurring whatiscalledaground holdingdelay.Findingthedesiredbalancebetweengroundde laysandairdelays undersevereweatherconditionsthatachievesthelowestcosti sthefocusofthis paper. Theairportacceptancerate(AAR)playsanimportantroleinde termining groundholdingpoliciesatairportsacrossthenation.Since allairportshave nitecapacity,thereisacontinuingeorttomaximizecapaci tyutilization,while avoidingunwantedgroundandairdelays,whichimpactfuelco sts,crewand passengerdisruptions,andotherintangiblecosts.Weuseastochasti cprogramming approach,whichtakesintoconsiderationtherandomnatureo ftheeventsthat putagroundholdingplanintoplace.Astheseplanscannotpred ictthefuture, theremaybeunnecessarygroundholdsatoriginatingorupline stations,resulting inunusedcapacityattheairportinquestioniftheprojectedc apacityreductions (weather-relatedornot)donotoccur. Researchhasbeenconductedonthegroundholdingproblemfor singleand multipleairports,andbothstaticanddynamicversionsofthep roblemexist.The staticversionassumesthatthecapacityscenariosaredenedonce atthebeginning ofthegroundholdingperiodunderevaluation.Ouranalysisf ocusesonthestatic, stochasticsingleairportversionoftheproblem.Foradditiona lbackgroundonthe

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144 static,single-airportproblem,seeBall,etal.[ 8 ],Grignon[ 34 ],HomanandBall [ 36 ],RichettaandOdoni[ 64 ],andRifkin[ 65 ].Whileourcontributionstothestatic modelingapproachoerseveralnewinsights,werecognizethepo tentiallimitation incorrectlyrepresentingthereal-timechangesinairporto perationsduetoweather uncertainties.Incontrast,adynamicversionhasbeenstudiedi nwhichcapacity reductionscenariosareupdatedasthedayprogresses.Inotherw ords,aswemove fromoneperiodtothenext,thecapacityestimatesforallrem ainingperiodscanbe updatedtoreectmorerecentweatherforecasts.Thus,therecou ldbeatotalof k uniquesetsofcapacityscenariosfortimeperiod k .Forresearchonthedynamicor multipleairportproblems,pleaseseeVranas,Bertsimas,andOdon i[ 80 ],[ 81 ],and NavazioandRomanin-Jacur[ 57 ],ofwhichonlyVranasetal.[ 80 ]considerscapacity tobestochasticinnature. InSection 5.2 ,wepresenttheRifkin[ 65 ]stochasticformulationoftheground holdingproblem,alongwithsomesolutionproperties.Weadopt thismodel formulationanddevelopnewndings,whicharepresentedinSec tions 5.3 and 5.4 First,Section 5.3 illustratesthebenetofstudyingastochasticasopposedtoa deterministicgroundholdingproblem.Ajusticationfortheu seofastochastic modelispresentedthroughaseriesofcomputationalexperime ntsperformed withvariousinputdata.Benchmarkmeasures,suchasthevalueo fthestochastic solutionandexpectedvalueofperfectinformation(seeBirge andLouveaux[ 15 ]), areusedforthisjustication.InSection 5.4 ,weconsidertheeectofintroducing risk-averseconstraintsinthemodel.Inotherwords,byrestrict ingthesizeofthe worst-casedelays,howistheoverallexpecteddelayaected?Th isisnotthesame asamaximumdelaymodel,whichwouldplaceastrictupperboun donworst-case delays.Finally,wesummarizeourndingsanddiscussdirections forfuturework.

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145 5.2 StaticStochasticGroundHoldingProblem 5.2.1 ProblemDenitionandFormulation Wedeneapotentialgroundholdingperiodtoconsistofanitenu mber, T ,of15-minuteincrementsorperiods : Supposethearrivalschedulecontains F ightsduringthetimehorizonunderevaluation.Furtherassum ethatwecan groupindividualightarrivalsintouniquetimeperiods,whe rethesequencingof ightswithinthetimeperiodisnotimportant.Wecanthenden otethenumberof arrivalsinitiallyscheduledtoarriveinperiod t as D t Whiletheairportmayhaveanominalarrivalcapacityof X aircraftperperiod, theestimatesbasedonthepoorweatherconditionswillproduc e Q possiblecapacityscenarioswithinanyinterval.Foreachcapacityscenario q ,thereisaprobability p q ofthatscenarioactuallyoccurring.Foreachtimeperiodand scenario,let M qt bethearrivalcapacityforscenario q duringperiod t: Let c g denotetheunitcostof incurringagrounddelayinperiod t .Assumethatgrounddelaysdonotincrease incostbeyondthersttimeperiodforanyaircraft.Similarly, deneanairdelay cost, c a ,astheunitcostofincurringanairdelayforoneperiod.Wewi llusethese parameterstoexaminemodelperformancefordierentground /airdelayratiosin Section 5.3 .Wenextdenethefollowingdecisionvariables: A t =Numberofaircraftallowedtodepartfromanuplinestationan d arrive\intotheairspace"ofthecapacitatedairportduring period t: W qt =Numberofaircraftexperiencingairdelayduringperiodtun der scenario q: G j =Numberofaircraftincurringgrounddelaysinperiod j: Thisisthe dierencebetweentheactualnumberofarrivals( P jt =1 D t )through period j andthetotalnumberofexpectedarrivals( P jt =1 A t ) throughperiod j:

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146 Asstatedpreviously,wefocusonthestatic,stochasticsingleairp ortversionof theproblem.BasedontheproblemrstpresentedinRichettaandO doni[ 64 ]and laterrevisedinRifkin[ 65 ],theformulationisasfollows: [SSGHP] StaticStochasticGroundHoldingProblem(GeneralCase) minimize: c g T P t =1 G t + c a Q P q =1 T P t =1 p q W qt (5.1) subjectto: ScheduleArrivalTimes: j P t =1 A t j P t =1 D t j =1 ;:::;T; T +1 P t =1 A t = T +1 P t =1 D t (5.2) ArrivalPeriodCapacities: A t + W q;t ¡ 1 ¡ W qt M qt t =1 ;:::;T; q =1 ;:::;Q; (5.3) InitialPeriodAirDelays: W q 0 =0 q =1 ;:::;Q; (5.4) GroundDelays: G j + j P t =1 A t = j P t =1 D t j =1 ;:::;T; (5.5) Integrality: A t 2 Z + ;W qt 2 Z + ;G t 2 Z + t =1 ;:::;T; q =1 ;:::;Q: (5.6) Theobjectivefunctionminimizestotalexpecteddelaycost, accountingfor bothgroundandairdelays.Constraintset( 5.2 )showsthatnoaircraftwillarrive earlierthanitsplannedarrivalperiod.Theequalityconstr aintthatincludesa summationtoperiod T +1requiresthatweaccountforallaircraftthatcould notlandbyperiod t .Therefore,anyaircraftnotlandingby T willlandinperiod T +1.Whenexamininganentireplanningday,thisisfairlyrea listicsinceeven thebusiestairportswillreducetheiroperationsto10{20%of capacitylateinthe evening.Whenweonlywanttoconsideraportionofaplanningd ay,thenwe needtorealizethattheremaynotbeenoughcapacityinperio d T +1tolandall aircraft.Sosomeadditionaldelaywillbepresent.Constraint set( 5.3 )requires that,foragiventimeperiod,allen-routeaircraft,includ ingthoseon-timeand

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147 thosealreadyexperiencingairdelay,willeitherlandorexp erienceanairdelay untilthenexttimeperiod.Constraintset( 5.4 )assumesthattherearenoaircraft currentlyexperiencingdelaysandwaitingtoland,priorto thebeginningofthe groundholdingpolicy.Constraintset( 5.5 )assignsgrounddelaystothoseaircraft notlandingintheiroriginallydesiredtimeperiods.Thus,one canseethataircraft canbeassignedgrounddelay,airdelay,orboth,whenairportc apacityisrestricted. 5.2.2 SolutionProperties Ithasbeenshown(RichettaandOdoni[ 64 ],Rifkin[ 65 ])thattheconstraint matrixof[SSGHP]istotallyunimodular,anditfollowsthatt helinearprogrammingrelaxationof[SSGHP]isguaranteedtoyieldanintegerso lution.Thus, constraintset( 5.6 )canbereplacedwiththenonnegativityconstraintset: Nonnegativity: A t ;W qt ;G t 0 t =1 ;:::;T;q =1 ;:::;Q: (5.7) Thispropertywillnotholdwhenweintroducetheriskaversion measuresin Section 5.4 ,sowecannotremovetheintegralityconstraintsfromallofou rmodels inthisanalysis. Whiletheoriginalproblemitselfisnotverylarge,itisalwa ysdesirableto reducetheproblemtoalinearprogram.Thenumberofinteger decisionvariables is O ( T + T ¤ Q + T )= O ( QT ),where Q isthenumberofscenariosand T isthe numberof15-minuteperiods.Aswillbeshown,theexperimentsp resentedinthis reportarebasedona22-scenario,24-periodproblem,whichim pliestheevaluation coversasix-hourtimeframe.Thistranslatesto576integerva riables,whichisvery manageable.However,asmoreaccuratedataonweatherpatter nsareavailable,we wouldmostlikelyhavehistoricaldataonmanymoreairportcap acityscenarios. Theresultingmodelwouldincreaseinsizeveryquickly,andthe benetofnot requiringtheintegerrestrictionnowbecomesmoreimportan t.

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148 5.3 MotivationforStochasticProgrammingApproach 5.3.1 ArrivalDemandandRunwayCapacityData Whenagroundholdingpolicyisbeingconsidered,theexpecte darrivalstothe airportwillbeaected.Soitisimportanttoknowthearrival stream.Weconsider asix-hourtimeframe.Basedonactualdatafromamajorairport ,anestimated arrivaldemandin15-minuteincrementswasobtained.Figur e 5{1 presentsthis arrivaldemanddata.Eachchartshowstypicalarrivalpatter nsforanairportwith Arrival Demand Test 2 0 5 10 15 20 25 30147 101316192215-min window Arrival Demand Test 3 0 5 10 15 20 25 30147 101316192215-min window Arrival Demand Test 4 0 5 10 15 20 25 30147 101316192215-min window Arrival Demand Test 1 0 5 10 15 20 25 30147 101316192215-min window Figure5{1:Aircraftarrivaldemandatthecapacitatedairpo rt. huboperationsintheU.S.Thisisrepresentedinthecyclicald emandforarrivals throughouttheperiodunderconsideration.Typically,arri valswilldominatethe tracpatternofanairportforapproximatelyonehour,follo wedbyanhourof tracdominatedbydepartures.Thelengthofthesecycleswilld ependonthe

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149 airportandairlinesoperatingattheairport.Test1andTest2 usethesame underlyingarrivalpattern,withTest2having25%morearriv alsperperiod.Atthe beginningofthesix-hourperiod,thereisalowarrivaldeman d,indicatingthatthe groundholdingpolicyisbeingputintoplaceduringadepart urepeak.Test3and 4alsousethesamedistributionasTests1and2,witha\timeshift"t orecognize thatagroundholdingpolicyisjustaslikelytobeginduringa peakarrivalow. EventhoughtheFAAmayimposeonlyoneofthreepolices(VFR,IFR 1, IFR2),theactualweatherandightsequencingwillfurtherae ctanairport's capacity.So,althoughtheremayonlybethreeocialAARsunder agivenrunway conguration,manymorecapacitycaseswillbeseen.Considerrst thepossibility thatnoinclementweathermaterializes.Wedenotethiscaseas CapacityScenario 0,orCP0.Wethenincludereducedcapacityscenariosinsetsof three.Foreach capacity-reducedset,therstscenariorepresentsaparticular weather-induced capacityrestriction.Thesecondandthirdscenariosinthesetr educethecapacity ineachperiodbyanadditional15%and30%,respectively.Unde rthesescenarios, therewillexistnoperiodsinwhichthenominalorVFRcapacity isrealized. Figure 5{2 presentsseveral\badweather"scenariosandtheireectonreal ized runwaycapacity.Weonlyshowtherstscenariofromeachsetofthr eecreated. Insomeextremecases,thebadweathermayappeartwicewithinon esixhourperiod,andthisisconsideredinCP16{CP21.Theprobabi litiesassociated withtheseseverelyaectedcapacityscenariosarerelativelysm all.Wecreated sevencapacity-reducedsets,foratotalof22capacityscenario s(includingthefull capacityscenario,CP0)inourstochasticproblem.Asecondsetof \badweather" scenarios(notpresented)wasalsousedinthecomputationalexp eriments. Withallofthearrivalcapacityscenarios,wehaveassignedreaso nableprobabilities.This,ofcourse,iswheremuchofthedicultyofusingst ochasticprogrammingisseen.Airtraccontrollers,FAA,andairlinepersonnel donothave

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150 CP1 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP0 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP19 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP4 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP7 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP16 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP10 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window CP13 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315-min window Figure5{2:Weather-inducedarrivalcapacityscenarios. (NOTE:Onlytherstscenarioineachsetofthreeisshown.)

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151 historicaldatatoprovidethemwithsuchcapacityscenariosan dprobabilities. Untilthisinformationbecomesmorereadilyavailable,wemu stmakesomeassumptionsabouthowtoobtainsuchdata.Sincetherewouldbesom edebateas totheappropriateprobabilitiestoassigntoeachscenario,we testthreesetsof probabilities. Recallthat[SSGHP]uses c g (or c a )todenotetheunitcostofincurringa ground(orair)delayforoneperiod, t .Weevaluatethreereasonableestimatesfor therelativecostofincurringdelaysonthegroundorintheai r.Sincemostofthe costisrelatedtofuel,airdelayswillusuallybemuchhigher. Buttheremaybe othernegativeimpactsofholdinganaircraftbeforeittake so.Keeping c g =2,we createthreetestcasesfor c a =3,5,and10.Thesetestcasesarealsousedbasedon priorexperimentsconductedanddiscussedinBall,etal.[ 8 ],RichettaandOdoni [ 64 ],andRifkin[ 65 ]. Inall,theexperimentsincludefourarrivaldemandproles,t wosetsof capacityscenarios,threesetsofcapacityprobabilities,andt hreeground/airdelay ratios,foratotalof72testproblems. 5.3.2 ExpectedValueofPerfectInformation(EVPI)andValueofSto chasticSolution(VSS) Twokeymeasurestogaugethevalueofstochasticprogrammingar ethe expectedvalueofperfectinformation(EVPI)andthevalueof thestochastic solution(VSS).EVPImeasuresthemaximumamountadecisionmaker wouldbe readytopayinreturnforcompleteandaccurateinformation aboutthefuture. Usingperfectinformationwouldenablethedecisionmakertode viseasuperior groundholdingpolicybasedonknowingwhatweatherconditio nstoexpect.It isnothardtoseethatobtainingperfectinformationisnotli kely.Butwecan quantifyitsvaluetoseetheimportanceofhavingaccuratewe atherforecasts.VSS measuresthecostofignoringuncertaintyinmakingaplanning decision.First, thedeterministicproblem(i.e.,theproblemthatreplacesa llrandomvariables

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152 bytheirexpectedvalues),issolved.Pluggingthissolutionba ckintotheoriginal probabilisticmodel,wenowndthe\expectedvalue"solutionc ost. Thisvalueiscomparedtothevalueobtainedbysolvingthestoc hasticprogram,resultingintheVSS.ApplyingthistotheGHP,weobtainasol utionto thedeterministicproblem,whichprovidesasetofrecommende dgroundholdsper periodbasedontheexpectedreductionofarrivalcapacitype rperiod.Wethen re-solvethestochasticproblemusingtheserecommendedgroundh oldsasxed amounts.Thedierencebetweentheoriginalstochasticsolution andthesolutionusingthispre-denedgroundholdingpolicyistheVSS.Bot htheEVPIand VSSmeasuresaretypicallypresentedintermsofeitherunitcost orpercent.(We havechosentoshowEVPIandVSSaspercentagevalues.)Foramoreth orough explanation,refertoBirgeandLouveaux[ 15 ]. Werstintroducethefourproblemsthatweresolvedincalculat ingEVPI andVSS.The\DeterministicSolution"usesanexpectedarrival capacityper period, M t ,basedontheprobabilityofeachweatherscenariooccurring. Denoting M t = P Qq =1 p q M qt ,wecanrewrite[SSGHP]withoutanyscenariosand,thus, withoutanyuncertainty.Wepresentthefollowingformulati on: [DGHP] DeterministicGroundHoldingProblem minimize: c g T P t =1 G t + c a T P t =1 W t (5.8) subjectto: Constraints( 5.2 ),( 5.5 ), ArrivalPeriodCapacities: A t + W t ¡ 1 ¡ W t M t t =1 ;:::;T; (5.9) InitialPeriodAirDelays: W 0 =0 ; (5.10) Nonnegativity: A t ;W t ;G t 0 t =1 ;:::;T; (5.11)

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153 The\PerfectInformationSolution"assumesthatweknow,inad vance,the arrivalcapacityperperiod.Sincewehave Q possiblecapacityscenarios,wesolve Q individualproblems,setting M t = M qt foreachscenario q .Using[DGHP],we determineaminimumcostsolution, S q ,foreachscenario.Then,wecalculatethe \PerfectInformationSolution"(PIS)bytakingtheweighte daverageofthesolution values,orPIS= P Qq =1 p q S q The\StochasticSolution,"ourrecommendedapproach,repre sentstheresults ofsolving[SSGHP].Finally,tocalculatethe\ExpectedValue Solution,"wewill use[SSGHP].However,werstsetthegrounddelayvariables, G t ,andtheactual departurevariables, A t ,tothevaluesobtainedwiththe\DeterministicSolution." Whenwesolvethisversionof[SSGHP],weareactuallysupplyinga xedground holdingplanandobservingtheadditionalairdelaysthatresu ltfromnottakingthe randomnessofeachweatherscenario, q ,intoaccountexplicitly. Runswereperformedacrossallofthecombinationsofdemandp roles, capacityproles,probabilitysets,andground/airdelayratio s.Inordertoarriveat somesummarystatistics,thetwoarrivalcapacitiesandthreesets ofprobabilities weregroupedtogether.Thus,eachsummarytestcaseisanaverage ofsixruns. Denoteeachrun'sground/airdelayratioasG2A#,whereG2rep resentsaunitcost of2forincurringgrounddelayandA#representsaunitcostof#f orincurringair delay.Eachsummarytestcaseisthenuniquebasedonitsarrivald emandprole anditsground/airdelayratio(G2A#).Table 5{1 summarizestheresultsover thesegroupsoftestcases. Boththedeterministicandperfectinformationsolutionsdon otchangewithin aparticulararrivaldemandprole.Thisindicatesthatalld elaysarebeingtaken asgroundholds.Sincethegrounddelaycostislessthantheaird elaycostin eachtestcase,themodelwillalwaysassigngrounddelaysrst,reg ardlessofthe magnitudeofthecapacityreduction.Sincedeterministicin formationisnotusually

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154 Table5{1:EVPIandVSSstatistics(Minimizetotalexpecteddela ycostmodel). Perfect Expected Summary Deterministic Information Stochastic Value TestCase Solution Solution Solution Solution EVPI VSS ArrDem1 G2A3 199 562 821 862 46% 4.9% G2A5 199 562 1246 1304 122% 4.6% G2A10 199 562 1874 2409 235% 28.3% ArrDem2 G2A3 1034 1459 2033 2154 39% 6.0% G2A5 1034 1459 2728 2901 87% 6.4% G2A10 1034 1459 3484 4767 140% 36.8% ArrDem3 G2A3 130 602 897 930 49% 3.6% G2A5 130 602 1387 1464 131% 5.4% G2A10 130 602 2094 2798 250% 33.3% ArrDem4 G2A3 1035 1497 2145 2340 43% 9.1% G2A5 1035 1497 3007 3210 101% 6.8% G2A10 1035 1497 3906 5384 162% 37.8% Note:Delaycostsrepresentunitcosts,notmonetaryamounts. available,introducinguncertaintythroughstochasticprog rammingresultsin solutionswithmuchhighertotaldelaycosts.Arrivaldemandpro les2and4both increasetheamountoftracarrivingtothecapacitatedairpo rt.Thisisclearly shownthroughthelargeincreaseindelays,eveninthedetermin isticcase. FortheG2A3cases,thevalueofthestochasticsolution(VSS)isatle ast 3.6%,whichcanbequiteimportantgiventhemagnitudeinthe costperdelay unit.And,aswemovetotheG2A10cases,VSSisgreaterthan28%.For example, intheG2A10caseunderArrivalDemandProle4,theexpectedvalu esolution givesavalueof5384,andtheexpectedsavingsbyusingastochast icsolution wouldbe1478.Thisindicatesthat,ifairdelaysareexpecte dtobemorethanve timesascostlyasgrounddelays,thenevaluatingthegroundho ldingpolicyusing stochasticprogrammingisessential.Similarly,withEVPIvalue srangingfrom40% to250%,itisquiteevidentthatobtaininghigherqualitywe atherforecastswould

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155 beverybenecial.TheEVPImeasurecanbeusedasajusticationfor investingin improvedweatherforecastingtechniques. 5.4 RiskAversionMeasures 5.4.1 ConditionalValueatRisk(CVaR)Model ThesolutiontotheSSGHPmodelisdeterminedbyminimizingtot alexpected delaycostovertheentiresetofscenariospresented.However,th erestillmaybe instanceswhere,incertainweatherscenarios,thedelayincur redasaresultofa particulargroundholdingstrategyismuchlongerthanthede layincurredunder anyotherscenario.Inthissituation,wemaywanttondsolution sthatattempt tominimizethespreadofdelaysacrossallscenarios,ortominim izetheextentto whichextremelypooroutcomesexist.Thiscanbedonethrough theadditionof riskaversionmeasures.Suchmeasuresallowustoplacearelative importanceon otherfactorsoftheproblembesidestotaldelay.TheValue-a t-Risk(VaR)measure hasbeenextensivelystudiedinthenancialliterature.Morer ecently,researchers havediscoveredthatanothermeasure,ConditionalValue-atRisk(CVaR),proves veryusefulinidentifyingthemostcriticalorextremedelays fromthedistribution ofpotentialoutcomes,andinreducingtheimpactthattheseou tcomeshaveon theoverallobjectivefunction.Foramoredetaileddescript ionofCVaRandsome applications,seeRockafellarandUryasev[ 67 ],[ 68 ]. CVaRcanbeintroducedinmorethanoneformfortheGHP,depend ingon theconcernsoftheairlines,theairtraccontrollers,andthe FAA.Wecandene anewobjectivethatfocusesontheriskmeasure,orwecanaddthe riskmeasure intheformofriskaversionconstraints(seeSection 5.4.3 foralternateCVaR models).Inthissection,wepresentanewformulationthatatte mptstominimize theexpectedvalueofapercentileoftheworst-casedelays;i.e .,weplacetheCVaR measureintheobjectivefunction.

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156 InordertosetuptheCVaRmodel,additionalvariablesandpar ametersare required.Let representthesignicancelevelforthetotaldelaycostdistribu tion acrossallscenarios,andlet beadecisionvariabledenotingtheValue-at-Risk forthemodelbasedonthe -percentileofdelaycosts.Inotherwords,in %of thescenarios,theoutcomewillnotexceed .Then,CVaRisaweightedmeasure of andthedelaysexceeding ,whichareknowntobetheworst-casedelays. Next,weintroduce q torepresentthe\tail"delayforscenario q .Wedene \tail"delayastheamountbywhichtotaldelaycostinascenari oexceeds whichcanberepresentedmathematicallyas q =MAX D q ¡ ; 0 ,where D q = c g P Tt =1 G t + c a P Tt =1 W qt .Theriskaversionproblemisnowformulated. [GHP{CVaR] GroundHoldingProblem(ConditionalValue-at-Risk) minimize: +(1 ¡ ) ¡ 1 Q P q =1 p q q (5.12) subjectto: ScheduleArrivalTimes: j P t =1 A t j P t =1 D t j =1 ;:::;T; T +1 P t =1 A t = T +1 P t =1 D t ; ArrivalPeriodCapacities: A t + W q;t ¡ 1 ¡ W qt M qt t =1 ;:::;T;q =1 ;:::;Q; InitialPeriodAirDelays: W q 0 =0 q =1 ;:::;Q; GroundDelays: G j + j P t =1 A t = j P t =1 D t j =1 ;:::;T; Worst-CaseTailDelays: q c g T P t =1 G t + c a T P t =1 W qt ¡ q =1 ;:::;Q; (5.13) Nonnegativity: ; q 0 q =1 ;:::;Q; (5.14) Integrality: A t 2 Z + ;W qt 2 Z + ;G t 2 Z + t =1 ;:::;T;q =1 ;:::;Q: Thismodelwillactuallyhaveanobjectivefunctionvalueeq ualto -CVaR.In ordertocomparethissolutiontothesolutionprovidedby[SSG HP],wemuststill

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157 calculatetotalexpecteddelaycost.Totaldelaycost,aswell asmaximumscenario delaycost,canbedeterminedafter[GHP{CVaR]issolved. Operatingunderthisholdingpolicy,wecanaddresstheriskin volvedwith incurringextremelylonggroundandairdelays.Thismaysacri cegoodperformanceunderanothercapacityscenariosincethebadweatheris notrealizedunder allscenarios.Inotherwords,acapacityscenariothatwouldresu ltinlittleorno delaymaynowexperienceagreaterdelaybasedontheholdingp olicy'sattempt toreducethedelayunderamoreseverelyconstrainedcapacity case.Sothese dierenceswouldneedtobedealtwithonacase-by-casebasis,and wepresent somealternativemodelstoaccommodatethegoalsofdierentd ecisionmakersin Section 5.4.3 5.4.2 MinimizeTotalDelayCostModelvs.MinimizeConditionalValu e-at-RiskModel TheCVaRmodelrequirestheadditionalinputofasignicancel evel,and = 0.9ischosenfortheanalysis.Table2presentsacomparisonofthe delaystatistics fortheMinimizeTotalDelayCostModel(SSGHP)andtheMinimiz eConditional VaRmodel(GHP{CVaR). ResultsofthemodelcomparisonsshowthatintheCVaRmodel,tot alexpecteddelaywillbeincreasedinordertoreducetheworst-case delaysacrossall testcases,whichsupportsourexplanationdescribingtheuseofri skaversion.By examiningtheresultsmoreclosely,wenotesomeinterestingndi ngs. Observethedierenceinvaluesfor -VaRand -CVaRwhennoriskis modeled.Thisillustratestheimportanceofconsideringthea verageofworst-case delaycostswhenyouchoosetomodelrisk.VaRtendstooverlookt hedierences indelaysbeyondthecriticalvalue,anditmaynotbeabletor educetheworst-case delaycostsaseectively.Whenminimizing -CVaRinthesecondmodel,notice thatthe -VaRand -CVaRvaluesaremuchcloser.

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158 Table5{2:Overallmodelcomparisons. MinimizeTotalExpectedDelay MinimizeConditionalVaR Summary Total Total TestCase DelayCost -VaR -CVaR DelayCost -VaR -CVaR ArrDem1 G2A3 821 1661 2628 1767 1875 2087 G2A5 1246 2337 3943 2059 2222 2471 G2A10 1874 3075 4856 2757 2855 2934 ArrDem2 G2A3 2033 3667 4777 3426 3505 3717 G2A5 2728 4450 6299 3743 3852 4101 G2A10 3484 4809 6755 4411 4485 4564 ArrDem3 G2A3 897 2109 3078 1981 2139 2378 G2A5 1387 3087 4703 2371 2531 2749 G2A10 2094 3209 5624 2960 3166 3225 ArrDem4 G2A3 2145 4315 5520 3878 3953 4192 G2A5 3007 5544 7552 4263 4345 4563 G2A10 3906 5092 7624 4869 4980 5039 Note:Delaycostsrepresentunitcosts,notmonetaryamounts. Also,thepercentageincreaseintotalexpecteddelaycostbetwe enthetwo modelsismoredrasticforsmallerairdelaycosts.Butastheaird elaycost rises,thetotaldelaycostincurredwhenminimizing -CVaRisnotseverely aected.ConsiderthefollowingresultsobservedfortheArrival Demand4prole. TheG2A3caseexperiencesanincreaseinaveragedelaycostof80% ,whilethe G2A10caseexperiencesonlya25%increase.Themagnitudeofair delaycosts willsignicantlyimpacttheeectivenessofusingriskconstraint s.Recallfrom Table 5{1 theexamplethatwaspreviouslydescribed.IntheG2A10casethe expectedvaluesolutiongivesavalueof5384,andtheexpecte dsavingswithoutrisk constraintswouldbe1478.Now,byminimizingthe10%worst-case delays(using theGHP{CVaRmodel),theexpectedsavingsreducesto515.Butw ealsohave reducedtheworst-casedelaysfrom7624to5039. SincetheCVaRanalysisuptothispointonlyconsidersusing =0.9,itis worthwhiletoshowhowCVaRcanshapethedistributionofoutcom esatanother

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159 signicancelevel.Consider =0.75,whichimpliesthatthe25%-largestdelay costsareminimized.Figure 5{3 showsthedistributionofdelaysacrossallscenarios foraparticulartestcase(ArrDem1,G2A5,witharrivalcapacity 1andprobability set1).WepresentthedelaydistributionswithandwithoutCVaR ,andatboth levelsofsignicancefortheCVaRmodel.Noticehowthedelaysb yscenariovary greatlywhenminimizingtotalexpecteddelay.Infact,ther earethreescenarios whichwouldresultindelaycostsexceeding4500.Byminimizin g -CVaR,these longdelayswouldnotoccur.Thetrade-owiththismodeling approachisthat therearenoscenarioswithonlyminimaldelays.Withfurthera djustmentstothe valueof ,thedecisionmakershavesomecontroloverhowtheresultingde lay outcomeswouldlook.Itthendependsonwhatthedecisionmake rsarewillingto accept.Thisisoneoftheunderlyingpowersofintroducingr iskaversionsuchas minimizing -CVaR. 5.4.3 AlternateRiskAversionModels Dependingontheinputfromeachgroupinvolvedinconstructi ngaground holdingpolicy,therewillbeconictingdesirestoreducetot alexpecteddelayand toreducetheworst-caseoutcomes.Forthesepurposes,wecanactua llychoose amongseveralriskaversionmodels.Ifyoursoledesireweretoredu cethetotal expecteddelaycost,youwouldnotrequiretheuseofriskaversio n.Butifyouwant toreducemaximumdelaycosts,youmightusetheGHP{CVaRmodel.F orother cases,whichwillaccountformostcollaborativeeorts,somecomb inationofthese modelswillbechosen. Forthisreasoning,weintroduceamoregeneralformulationo ftheground holdingproblem.Now,weconsiderthattheobjectivemaybetom inimizetotal expecteddelay,tominimizeworst-casedelay( -CVaR),ortominimizesome combinationofthesemeasures.WepresenttherstalternateGHPfor mulationas:

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160 Minimize Total Expected Delay Cost Total Exp. Delay Cost = 1114 0 2 4 6 8 10 12 14250 750 1250 1750 2250 2750 3250 3750 4250 4750 M oreDelay Minimize Conditional VaR (alpha = 0.90) Total Exp. Delay Cost = 2063 0 2 4 6 8 10 12 142 5 0 7 5 0 1 2 50 1 7 50 2 2 50 2 7 50 3 2 50 3 7 50 4 2 50 4 7 50 M o r eDelay Minimize Conditional VaR (alpha = 0.75) Total Exp. Delay Cost = 1609 0 2 4 6 8 10 12 14250 750 1250 1750 2250 2750 3250 3750 4250 4750 MoreDelay Figure5{3:Totaldelayoutputforarrivaldemandlevel1.

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161 [GHP{CVaR1] AlternateGHPRiskFormulation1 minimize: w 1 c g T P t =1 G t + c a Q P q =1 T P t =1 p q W q;t + w 2 +(1 ¡ ) ¡ 1 Q P q =1 p q q (5.15) subjectto: Constraints(2 ¡ 6 ; 13 ¡ 14) : Byincludingweights w 1 and w 2 ,thedecisionmakercanhavecompletecontrolover theimportanceofeachobjectivemeasure.Notethatfor w 1 =0and w 2 =any constant,wehavethespecialcaseof[GHP{CVaR].Likewise,for w 1 =anyconstant and w 2 =0,wehavethespecialcaseof[SSGHP]. Weintroduceasecondalternateformulationthatimposesarest rictionon allowablelosses.Weusetheoriginalobjectivefunctionfrom[S SGHP],minimizing theexpectedtotaldelaycost,whilesatisfyingaconstraintreq uiringthepercentile ofworst-casedelaystobenomorethansomeparameter, v [GHP{CVaR2] AlternateGHPRiskFormulation2 minimize: c g T P t =1 G t + c a Q P q =1 T P t =1 p q W qt subjectto: Constraints(2 ¡ 6 ; 13 ¡ 14) ; Worst-CaseDelayBound: +(1 ¡ ) ¡ 1 Q P q =1 p q q v: (5.16) Byplacinganupperboundonalossfunction,asin( 5.16 ),itapproachesa maximumlossconstraint.Butsomescenarioscanactuallyexceed thisparameter value,aslongastheweightedaverageoflosseswithintheperc entileremainsbelow v .InTable 5{3 ,weprovideanillustrationoftheeectofusingeachmodeltoset thegroundholdingpolicy. NoticethatSSGHPprovidesthelowestexpectedtotaldelaycost, basedon consideringthelikelihoodofeachweatherscenarioactually occurring.Ontheother

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162 Table5{3:Performancecomparisonofalternateriskmodels. TotalExpected Maximum Model DelayCost -VaR -CVaR DelayCost SSGHP 3336 4890 7354 8570 GHP{CVaR 4325 4446 4521 5076 GHP{CVaR1( w 1 = w 2 ) 3746 4136 4703 5696 GHP{CVaR2( v =5000) 3566 4068 5000 5898 GHP{CVaR2( v =6000) 3383 4332 6000 6882 Note:Resultsarebasedon(ArrDem2,G2A10,ArrivalCapacity1,Pro babilitySet1). hand,GHP{CVaRproducesthebestvalueof -CVaRandthelowestmaximum delaycostinanyscenario.Thetradeoisthatthemorelikelysc enarioswillnow encounterincreasedgroundholdings.Combiningthesetwoobje ctiveswithGHP{ CVaR1,wegainasubstantialamountofthebenetoftheprevious twomodels, withtotalexpecteddelaycostat3746andmaximumscenariodel aycostat5696. Andwecanevenne-tuneourobjectivefurtherthroughtheuseof theCVaR constraint.As -CVaRisincreased,weapproachouroriginalSSGHPmodel. Inadditiontotheaboveriskmodels,Rifkin[ 65 ]brieypresentstheMaximumAirDelayModel(MADM).MADMcanbethoughtofasamaximum-lo ss constraintforanyscenario,andifsuchanumberexists,thiscoul dbeaddedto anyoftheaboveformulations.WhatMADMfailstoaddressisthec ontinuing eorttominimizetotaldelays.Amax-lossconstraintcanbeadde dtoanyofthe formulationspresentedinthispaper,allowingtheuseraddit ionalinsightintoa particularairport'sgroundholdingpolicies.Aswiththepar ameter v ,settingthe maximumlosstootightmaypreventthemodelfromndingafeasib lesolution. Thereisnooneanswerwhendecidingwhichproblemformulatio ntouse.Each willshapetheresultingtotaldelayindierentways,andthusit isdependenton thegroupsmakingthedecisionsindeterminingtheamountofa cceptabledelay. 5.5 Conclusions Aswehaveshowninthischapter,theairportgroundholdingpro blemis essentiallyademandselectionprobleminwhichwedesiretheopt imalallocation

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163 ofightarrivalrequeststotheairport'srunwaycapacity,th elimitedresource.By selectingwhentoholdaircraftattheiruplinestations,wecan reducethequantity ofcostlyairdelayswhilemaintainingsomedesiredlevelofserv ice. Wehavealsoshownthatmodelingthegroundholdingproblemasa stochastic problemismostcertainlybenecial.Evenundercaseswhendela ycostsarelow anduniform,thevalueofthestochasticsolutionissignicant.Ad ditionally, introducingriskaversionallowsadecisionmakertooerseveral potentialoutcomes basedonvariousworst-casedelayscenarios.TheFAA,airportautho rities,and airlinesallworkwithinsomepre-denedperformancemeasures, andprovidinga modelthatallowsconstraintstobeadjustedtomeetsuchperfor mancemeasuresis veryimportant. Thereareseveralissuesthatwerenotaddressedandareareasforf uture research.Bymodelingtheproblemattheindividualightdeta il,wemaybeable togainmoreaccuracyindeterminingthetruecapacityandre alizedarrivalows intoanairport.Additionalresearchisstillrequiredtodeter minewhetherthe addedbenetsofuniqueightinformationmerittheundertaki ngofworkingwith amorecomplexmodel.Oncethemodelisattheightlevel,arri valsequencing, banking,andotherarrival/departuredisruptionscanbemod eled.Wewouldalso liketoincorporatespecicightdurationstomoreaccurately representthetrac fortheairportinquestion. Aswebrieymentioned,thenatureofastaticGHPmodeldoesnotal lowfor systemupdatestoairportcapacity,whichinreality,arevery likelytooccur.We handlethisbysimplyre-solvingthestaticmodelastimeprogre sses.Alternatively, wewouldlikeourrstgroundholdingdecisiontotakeintoaccou ntthepossible changestofutureweatherconditions,anddevelopingadynam icmodelwould providethisadditionallevelofforecastingdetail.

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164 Consideringtheoriginatingstationsoftheaircraftcouldal sobeworthwhile. Amultipleairportmodelwouldbeabletoprovidemorerealist icinformationon thedecisionsandactionsofeachindividualdepartingaircr aftenroutetothe capacitatedairportunderstudy.However,thesemodelswillgr owinsizequickly, evenundertheassumptionthatairportcapacitiesaredetermi nistic,asinNavazio andRomanin-Jacur[ 57 ]. Finally,CollaborativeDecisionMaking(CDM),describedinB all,etal.[ 7 ], hasbeenanareaoffocusrecently.Itallowsairlinestobein volvedinthedecisions onwhichaircraftwillbedelayedduringagroundholdingpla n.Thisislikelyto achieveareductioninoverallcoststoindividualairlinesb yallowing\morecritical" aircrafttotakeoattheirscheduleddeparturetimesandnot incurgroundholding delays.CDMmaybemorediculttomodel,butitisimportanttoi ncludethis fundamentalinteractiveapproachinordertorepresentorsim ulatetheactual environment.

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CHAPTER6 CONCLUDINGREMARKS Whenadecisionmakerhasdiscretiontoacceptordenydemandsou rces, especiallywhenfacinglimitedresources,determiningthebest setofdemandsto selectbasedontheresultingrevenueandproduction/delivery costscanbequite challenging.Inordertoadequatelyaddressthischallenge, westudiedawide varietyofdemandselectionproblems.Theproduction-relate dproblemsincluded uncapacitatedandcapacitatedversionsoftheorderselectio nproblem,demand selectionwithpricingasadecisionvariable,andalsostochasti cdemandselection problemsthatallowthedecisionmakertoinuencedemandthro ughmarketing eort.Wethentiedtheseeortsinwiththeapplicationsareaof airportoperations. Thedemandselectionproblem,especiallyinthemanufacturin gsetting,hasgone relativelyunnoticedintheliteratureuntilrecently.Weh aveprovidedathorough discussionofafamilyofmodelsthatexistinthisarea. Themodelswepresentserveasastartingpointforfutureresearc honmore generalmodels.Werstaddressfutureresearchareasspecictothe multi-period problemspresentedinChapters 2 and 3 .Supposethatinsteadofpickingand choosingindividualordersbyperiod,theproducermustsatisfy agivencustomer's ordersin every periodiftheproducersatisesthatcustomer'sdemandinany singleperiod.Inotherwords,acustomercannotbeservedonlywh enitisdesirable fortheproducer,sincethiswouldresultinpoorcustomerservic e.Wemightalso poseaslightlymoregeneralversionofthisproblem,whichrequ iresservinga customerinsomecontiguoussetoftimeperiods,ifwesatisfyanypo rtionofthe customer'sdemand.Thiswouldcorrespondtocontextsinwhich theproduceris freetobeginservingamarketatanytimeandcanlaterstopservi ngthemarketat 165

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166 anytimeintheplanninghorizon;however,fullservicetothe marketmustcontinue betweenthestartandendperiodchosen. Futureresearchmightalsoconsidervaryingdegreesofproduce rexibility (wherecertainminimumorderfulllmentrequirementsmustbe met)ormodelingmorecomplexmarketinteractionsandpriceasadecisio nvariable(where marketdemandisafunctionoftheproducer'spriceand/orth epriceoeredby competitors).Wemightalsoconsiderasituationinwhichthepro ducercanacquire additionalcapacity(eitherintermsofproductioncapacit yormarketingbudget) atsomecostinordertoaccommodatemoreordersthancurrentca pacitylevels allow.Thesegeneralizationsoftheorderselectionmodelsma yfurtherincreasenet protfromintegratedorderselection,capacityplanning,an dproductionplanning decisions. Someofthemajorareasforfutureresearchintheairportoper ationsground holdingcontextwillfocusonmodelingtheindividualdeman dsources(orights) atanerlevelofdetail.Aswasshownforthedemandselectionpro blemsin Chapters 2 { 4 ,theuniquecharacteristicsofeachdemandsourcecansignican tly inuencethedesirabilityoffulllingthedemandsource.Thesam ewillholdtrue fordeterminingwhichaircraftaremostsuitableforgroundho lds.Wealsowould liketointroducecapacityupdatestothegroundholdingpro blem,whichimplies thatourrstgroundholdingdecisionwouldtakeintoaccountth epossiblechanges tofutureweatherconditions.However,asweincludetheseaddi tionaldetails,our demandselectionproblembecomesamulti-period,dynamicsto chasticdecision problem,whichcouldprovetobeverydiculttosolve.And,fort hisreason,it alsoremainsaveryinterestingtopicforfutureresearch.

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BIOGRAPHICALSKETCH IwasbornonApril27,1966,inBerwyn,Illinois.Aftermaintain ingasteady interestinmathematicsthroughoutmiddleschoolandhighscho ol,Iattended theUniversityofIllinoisatUrbana-Champaign,whereIreceiv edaB.S.and M.S.inindustrialengineeringin1988and1990.Ibeganmypro fessionalcareer atAmericanAirlinesDecisionTechnologies(AADT)inFortWorth, Texas.I consultedonavarietyofprojectsconcerningairandpassenger transportation systems,usingsimulationandothermodelingtoolstoaidinmakin gproject recommendations.In1994,thetechnologygroupswerereorga nizedunderthename Sabre,acompanyindependentfromAmericanAirlines.Inadditi ontoAmerican, ourclientsincludedairportauthorities,airportboards,go vernmentalagencies, packagedeliverycompanies,hotels,carrentalagencies,aswe llasotherairlines.I spenteightyearswithAADTandSabre,servinginrolesfromconsul tanttosenior projectmanager. Insearchofdeeperprofessionalreward,Itookstepsinanewdire ctionin1998, workingasanInstructorintheDepartmentofIndustrialEngin eeringatNorthern IllinoisUniversity.Iworkedtherefortwoyearsandgainedin valuableexperience, teachingseveraldierentundergraduatecourses.RealizingIst illhadonenalstep totake,IchosetoleaveNIUin2000andpursueaPh.D.inindustria landsystems engineeringattheUniversityofFlorida.IwasawardedaSteph enC.O'Connell PresidentialFellowshipforthedurationofthedoctoralprog ram.Duringmy graduatestudies,Iacquiredanewareaofexpertiseinproducti onandinventory controlandsupplychainmanagement,andthesespecialization snowcomplement myexistingexperienceandinterestintransportationandlogi stics. 174


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MODELS FOR OPTIMAL UTILIZATION OF PRODUCTION RESOURCES
UNDER DEMAND SELECTION FLEXIBILITY
















By

KEVIN MICHAEL TAAFFE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004

































Copyright 2004

by

Kevin Michael Taaffe















I dedicate this work to my family and to my future students.















ACKNOWLEDGMENTS

I would like to thank everyone who has provided words of support and

encouragement during the past four years. Most importantly, I thank my wife,

Mary, for providing unwavering support for my pursuit of this dream. She chose

to sacrifice her own desires and needs to assist me by caring for our children and

maintaining her business while I completed my degree. I do not know irn r: people

who could do that even once ... and she does it all the time. Furthermore, she has

ah--,i-x been my 'i-.-. -1 emotional support. After being away from school for so

long and working in industry for many years, I found it difficult to resume where

I had left off 15 years ago with advanced math and theoretical research. Mary

reminded me that it would be hard at times, but she ah--,i-i managed to calm me

down and get me back on track. Simply said, I could not have accomplished this

goal without her. She is the love of my life, and I will ah--,i-x love her from the

bottom of my heart.

Looking back, I could not have asked for a better person to be my thesis

advisor than Joe Geunes. He has been a great role model for my future career in

academia, and I thank him for all of the experiences we have shared. He allowed

me the space to think creatively, but he was ah--,i-i there when I needed help or

guidance. Our families became very close over the years, and I hope we continue to

stay close for many years to come.

There are many people who have touched my life in a special way since I

arrived in Gainesville. From our neighbors who became like family, to my entire

church family, and all of the friends I have met along the way, I can honestly -,- I

have never felt such warmth on so many different levels. Every one of these people









has had an impact on who I am 'itv, and I can iv that they all have served as

daily reminders as to what is truly important in life. It will be a sad farewell when

we leave Gainesville, but I have developed many relationships that will never go

away. For that I am eternally grateful.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ...................... ......... ix

LIST OF FIGURES ................................ x

ABSTRACT ...................... ............. xi

CHAPTER

1 INTRODUCTION .................... ....... 1

2 INTEGRATED ORDER SELECTION AND
REQUIREMENTS PLANNING ........ ......... .... 5

2.1 Introduction .............................. 5
2.2 Order Selection Problem Definition and Formulation ....... 9
2.2.1 The Uncapacitated Order Selection Problem ........ 11
2.2.2 Solution Properties for UOSP ............ . 12
2.3 OSP Models Limited Production Capacity . . 15
2.3.1 OSP Solution Methods ............. .. .. 18
2.3.2 Strengthening the OSP Formulation . . 19
2.3.3 Heuristic Solution Approaches for OSP . . ... 23
2.3.3.1 Lagrangian Relaxation Based Heuristic . ... 24
2.3.3.2 Greatest Unit Profit Heuristic . . ..... 26
2.3.3.3 Linear Programming Rounding Heuristic ...... 28
2.4 Scope and Results of Computational Tests . . 29
2.4.1 Computational Test Setup ..... . . 29
2.4.2 Results for the OSP and the OSP-NDC . .... 33
2.4.3 Results for the OSP-AND .................. .. 37
2.5 Conclusions .................. ........... .. 41
2.6 Appendix .................. ............ .. 42

3 PRICING, PRODUCTION PLANNING, AND
ORDER SELECTION FLEXIBILITY ................. .. 45

3.1 Introduction ........... . . . ...... 45
3.2 Requirements Planning with Pricing . . . .... 49
3.2.1 Shortest Path Approach for the Uncapacitated RPP . 52
3.2.2 Dual-ascent Method for the Uncapacitated RPP . 55









3.2.3 Polynomial Solvability of More General Models ...... 65
3.2.3.1 Multiple price-demand curves . . 65
3.2.3.2 Piecewise-linear concave production costs . 67
3.2.4 Production Capacities .................. .. 68
3.3 Pricing and Order Selection Interpretations . . 73
3.4 Conclusions .................. ........... .. 78
3.5 Appendix .................. ............ .. 78

4 SELECTING MARKETS UNDER DEMAND UNCERTAINTY ..... 85

4.1 Introduction .............. . . ...... 85
4.2 The Selective N. v.- '--. dor Problem ................ .. 92
4.2.1 Problem Formulation and Solution Approach . ... 92
4.2.2 Managerial Insights for the SNP .............. .. 97
4.3 SNP and the Role of Advertising . . . 103
4.3.1 Selective N. v.- i. idor with Marketing Effort ....... 104
4.3.2 Independent Demand Variance . . 105
4.3.2.1 Concave Demand ................... ... 106
4.3.2.2 S-curved Demand ........... ... 108
4.3.3 Dependent Demand Variance . . 109
4.3.4 Marketing Insights ................ .... .. 111
4.4 Operating with Limited Marketing Resources . . ... 114
4.4.1 Formulation of the Limited Resources Problem ...... 114
4.4.2 Solution Approach to the Limited Resources Problem .. 116
4.4.3 Subproblem Solution and B&B Implementation ....... 119
4.5 Computational Results .................. ...... 127
4.5.1 SNP Value: Minimum Market Requirement ........ 127
4.5.2 SNP Value: Profit Improvement . . .... 129
4.5.3 Solving the Limited Resources Problem . .... 132
4.6 Other Considerations .................. ... .. 135
4.6.1 The Infinite Horizon Planning Problem . .... 135
4.6.2 Limited Marketing Effort under a Fixed Contract ..... 138
4.7 Conclusions ............... .......... 140

5 AIRPORT CAPACITY LIMITATIONS
SELECTING FLIGHTS FOR GROUND HOLDING . .... 142

5.1 Introduction ........... .. ... ..... ..... 142
5.2 Static Stochastic Ground Holding Problem . . ... 145
5.2.1 Problem Definition and Formulation . . ... 145
5.2.2 Solution Properties ................ 147
5.3 Motivation for Stochastic Programming Approach ........ 148
5.3.1 Arrival Demand and Runway Capacity Data ....... 148
5.3.2 Key Stochastic Programming Measurements . ... 151
5.4 Risk Aversion Measures . . . . .... 155
5.4.1 Conditional Value at Risk (CVaR) Model . .... 155









5.4.2 Model Comparison .................. ... 157
5.4.3 Alternate Risk Aversion Models . . 159
5.5 Conclusions .. .. ... .. .. .. .. ... .. .. .. ..... 162

6 CONCLUDING REMARKS .................. ..... .. 165

REFERENCES .................. .............. .. .. 167

BIOGRAPHICAL SKETCH ........... . ........ 174















LIST OF TABLES
Table page

2-1 Counterexample illustrating decreasing cumulative demand satisfaction. 15

2-2 Classification of model special cases and restrictions. . .... 16

2-3 Problem size comparsion for capacitated versions of the OSP. ..... ..17

2-4 Probability distributions used for generating problem instances . 30

2-5 OSP-NDC and OSP problem optimality gap measures . .... 35

2-6 OSP-NDC and OSP solution time comparison. ........... ..36

2-7 OSP and OSP-NDC heuristic solution performance measures. ...... 37

2-8 OSP-AND optimality gap measures. ................. 40

2-9 OSP-AND solution time comparison. .................. 40

2-10 OSP-AND heuristic solution performance measures. . .... 41

4-1 Results for SNP with limited resources -Case II. . ..... 134

4-2 Results for SNP with limited resources -Case I. . ..... 135

5-1 EVPI and VSS statistics (\!iiiiiii..e total expected d, li cost model).. 154

5-2 Overall model comparisons. ............... .... 158

5-3 Performance comparison of alternate risk models . ...... 162















LIST OF FIGURES
Figure page

2-1 Fixed charge network flow representation of UOSP. . .... 11

2-2 Shortest path network structure for UOSP. .............. ..13

3-1 Pricing interpretations based on total revenue and demand. . 74

4-1 Optimal marketing effort for concave expected demand functions. .. 107

4-2 Optimal marketing effort for S-curved demand response functions... 109

4-3 Approximation of the S-curved demand response function. ..... ..110

4-4 Minimum market requirement based on individual cost parameters. 128

4-5 Profit improvement using SNP based on total markets available. 131

4-6 Profit improvement using SNP based on demand variance. ..... ..132

5-1 Aircraft arrival demand at the capacitated airport . . .... 148

5-2 Weather-induced arrival capacity scenarios. .............. ..150

5-3 Total delay output for arrival demand level 1. ........... ..160















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MODELS FOR OPTIMAL UTILIZATION OF PRODUCTION RESOURCES
UNDER DEMAND SELECTION FLEXIBILITY

By

Kevin Michael Taaffe

August 2004

C(!i r: Joseph Geunes
Major Department: Industrial and Systems Engineering

Optimal demand selection applies to contexts in which an organization has

some discretion in deciding the set of demands it will use its resources to satisfy.

In such cases, the decision maker wishes to determine the set of downstream

demands that provides the best match for its resource capabilities. This steps

away from traditional streams of research that ignore the selection decision and

assume all demand sources must be satisfied. We focus on developing new models

and solution methods for problems that integrate demand selection with the

planning and utilization of production resources, for both unlimited and limited

production capacities. Capacity limits often restrict the total amount of demand

that an organization can satisfy. When total demand for resources exceeds capacity

limits, selecting the optimal subset of demand sources is a challenging optimization

problem. Even in contexts where capacity limits are typically not a constraining

factor, the problem remains difficult due to economies of scale in production and

the attractiveness and timing of individual demands. Given a set of heterogeneous

downstream demand sources, which may be deterministic or stochastic in nature,

along with nonlinear capacity usage costs in volume, we propose models that









provide optimal demand source selections that achieve maximum profitability.

In this dissertation, we specifically address demand selection flexibility for the

applications areas of general production and inventory planning problems and

airport ground holding problems.















CHAPTER 1
INTRODUCTION

This dissertation focuses broadly on models for optimal demand selection.

Such models apply to contexts in which an organization has some discretion in

deciding the set of demands it will use its resources to satisfy. In such cases, the

decision maker wishes to determine the set of downstream demands that provides

the best match for its resource capabilities. Capacity limits often restrict the

total amount of demand that an organization can satisfy. When total demand

for resources exceeds capacity limits, the decision maker must determine the

best way in which to allocate its limited resources. Given a set of heterogeneous

downstream demand sources, along with nonlinear capacity usage costs in volume,

it is not a trivial problem to select the subset of demand sources that will provide

the maximum profit to the firm. Even in contexts where capacity limits are

typically not a constraining factor, economies of scale in production, combined

with time-varying customer demand patterns from customers who have different

reservation prices, make the problem of choosing the best set of demands to satisfy

a challenging task. We focus on developing new models and solution methods for

problems that fall in this general class of demand selection problems.

We explore two applications contexts within the class of demand selection

problems:

General production and inventory planning problems, and

Airport operations ground holding problems.

We examine several types of production and inventory planning problems in

C'! lpters 2, 3, and 4. In C'! lpter 2, we consider uncapacitated and capacitated

versions of the single-stage, multi-period production planning problem for a









producer who can select any number of orders, or demand sources, from a total set

of potential demands. The problem has a finite horizon, and the producer has the

discretion to choose when to produce and how much demand to satisfy in order

to maximize profit. We define this new class of production planning problems

with order selection ft i, .:,.:. and provide optimization-based modeling and

solution methods for these problems. We provide a polynomial-time algorithm for

solving the uncapacitated version of the problem, and we propose strong problem

formulations and heuristic solution algorithms for several capacitated versions.

In ('!i lpter 3, we extend our discussion of this single-stage production planning

problem to address the importance of pricing. Firms that manufacture and sell

products with price-elastic demand face the challenge of determining prices, and

therefore demand volumes, that provide maximum profit to the firm. Nonlinearities

in demand as a function of price and in production costs as a function of demand

volumes create complexities in determining pricing strategies that maximize con-

tribution to profit after production. Now, instead of directly selecting the desired

demand quantities to satisfy, as shown in ('!i lpter 2, we present a production plan-

ning model that implicitly decides, through pricing decisions, the demand levels

the firm should satisfy in order to maximize contribution to profit. We present two

polynomial-time solution approaches for these problems when production capacities

are effectively unlimited, and show that these approaches apply across a range of

applicable revenue and cost functions. We also describe a polynomial-time solution

approach under time-invariant finite production capacities and piecewise-linear and

concave revenue functions in the amount of demand satisfied.

These chapters together illustrate the importance of integrated demand and

production planning decisions by enabling a producer to leverage production

economies of scale to the greatest extent possible through matching the right

amount of demand to production capabilities.









In C'!i pter 4, we introduce a stochastic version of the demand selection

problem in a single-period setting, a problem that we refer to as the selective

newsvendor problem. In this problem, a seller faces a long procurement lead time

from an external supplier, and must simultaneously decide the markets in which

it will sell its product along with the procurement quantity from the external

supplier. For each selected market, the seller also determines the amount of

marketing effort it will exert in the market, and this marketing effort influences

the distribution of demand in the market (e.g., increased marketing effort implies

higher expected demand in the market and also impacts the uncertainty of the

market's demand). The goal is to choose the markets, advertising levels, and overall

procurement quantity that maximizes the seller's expected profit in the selling

season.

First, we present solution approaches for market selection decisions in which

the marketing levels are fixed or pre-defined by the firm or supplier. We then

extend the market selection approach to allow the firm to determine the best

level of advertising to apply in each market selected. We illustrate this approach

for both the unlimited and limited resources cases, and we evaluate multiple

functional forms for the manner in which market demand levels respond to market

advertising.

We conclude by presenting the airport operations ground holding problem

in C'! lpter 5. This problem involves determining which flights destined for a

given airport should be dispatched under uncertainty in future weather. In this

context, flights destined for an airport constitute the (future) demand for arrival

capacity, while the uncertainty in future weather leads to uncertain and dynamic

capacity levels for receiving flights at the destination point. Accepting demands

at a given destination can be very costly if the resulting capacity at flight arrival

time is low (due to bad weather). Conversely, denying demands by holding flights









at their origination points can also be quite costly, particularly if the resulting

capacity at the scheduled flight arrival time is high (i.e., when previously predicted

bad weather does not materialize at arrival time). The ground holding problem

introduced in Chapter 5 addresses this critical issue of optimal flight arrival

selection decisions under uncertainty.

The ground holding problem provides an excellent illustration of the benefits

of using a stochastic over a deterministic approach in mathematical programming.

We summarize these benefits within the chapter. The ground holding problem

is also an interesting problem to study due to the number of different potential

decision makers influencing the choice of flight demands to ground hold. Since the

Federal Aviation Administration, local airport authorities, and individual airlines

all have conflicting operational goals, we address new risk aversion models that

allow multiple decision makers to achieve acceptable performance at the same time.

As shown in this dissertation, the demand selection problem can appear in

many forms, and we provide a thorough discussion of the main focus areas within

demand selection, such as deterministic demand vs. stochastic demand, unlimited

resources vs. limited resources, and fixed pricing vs. variable pricing. These

chapters provide a solid foundation for future research in demand source selection,

and we also make several sl::.----, i..i for research directions in the the concluding

remarks.















CHAPTER 2
INTEGRATED ORDER SELECTION AND
REQUIREMENTS PLANNING

2.1 Introduction

Firms that produce made-to-order goods often make critical order acceptance

decisions prior to planning production for the orders they ultimately accept. These

decisions require the firm's representatives (typically sales/marketing personnel

in consultation with manufacturing management) to determine which among all

customer orders the firm will satisfy. In certain contexts, such as those involving

highly customized goods, the customer works closely with sales representatives to

define an order's requirements and, based on these requirements, the status of the

production system, and the priority of the order, the firm quotes a lead time for

order fulfillment, which is then accepted or rejected by the customer (see Yano

[85]). In other competitive settings, the customer's needs are more rigid and the

customer's order must be fulfilled at a precise future time. The manufacturer can

either commit to fulfilling the order at the time requested by the customer, or

decline the order based on several factors, including the manufacturer's capacity

to meet the order and the economic attractiveness of the order. These "order

acceptance and denial" decisions are typically made prior to establishing future

production plans and are most often made based on the collective judgment of

sales, marketing, and manufacturing personnel, without the aid of the types of

mathematical decision models typically used in the production planning decision

process.

When the manufacturing organization is highly capacity constrained and

customers have firm delivery date requirements, it is often necessary to satisfy a









subset of customer orders and to deny an additional set of potentially profitable

orders. In some contexts, the manufacturer can choose to employ a rationing

scheme in an attempt to satisfy some fraction of each customer's demand (see

Lee, Padmanabhan, and Whang [46]). In other settings, such a rationing strategy

cannot be implemented; i.e., it may not be desirable or possible to substitute items

ordered by one customer in order to satisfy another customer's demand. Thus, it

may be necessary for the firm to deny certain customer orders (or parts of orders)

so that the manufacturer can meet the customer-requested due dates for the orders

it accepts. In contexts where capacity limits are non-binding, it is also not ahv--,-

clear that committing to a particular customer order is in the best interest of the

firm, even if the unit price the customer will p ,i exceeds the variable production

cost. This is evident in environments with significant fixed production costs.

Regardless of whether the operation is constrained or unconstrained by

production capacity, assessing the profitability of an order in isolation, prior to

production planning, leads to myopic decision rules that fail to consider the best

set of actions from an overall profitability standpoint. The profitability of an

order, when gauged solely by the revenues generated by the order and perceived

customer priorities, neglects the impacts of important operations cost factors, such

as the opportunity cost of manufacturing capacity consumed by the order, as well

as economies of scale in production. Decisions on the collective set of orders the

organization should accept can be a critical determinant of the firm's profitability.

Since Wagner and Whitin's [83] seminal paper addressed the basic economic

lot-sizing problem (ELSP), numerous extensions and generalizations of this basic

problem have followed, including extensions to incorporate backlogging (Zangwill

[86]), serial system structures (Love [51]), and multistage assembly and general

multistage structures (Afentakis, Gavish, and Karmarkar [2], and Afentakis

and Gavish [1]). Intensive research on the capacitated version of the dynamic









requirements planning problem began in the 1970's (see Florian and Klein [28],

Baker, Dixon, Magazine, and Silver [5], and Florian, Lenstra, and Rinnooy Kan

[29]), and has received increased attention recently as a result of the application of

strong valid inequalities that enable faster solution of these difficult problems (e.g.,

B 1I il, Van Roy, and Wolsey [10], Pochet [61], and Leung, Magnanti, and Vachani

[47]). Lee and N lii,;i [45], Shapiro [70], and Baker [4] provide excellent overall

analyses of the generalizations and solution approaches for dynamic requirements

planning problems, including various heuristic approaches that have proven

effective for the capacitated version of the problem.

With a few notable exceptions that we later discuss, this past research on

dynamic requirements planning problems nearly ahv-- -i assumes that demands

are pre-specified by time period and that all demands must be completely filled

at the time they occur (or after they occur in models that permit backlogging).

In contrast, we consider a requirements planning model that implicitly determines

the best demand levels to satisfy in order to maximize contribution to profit.

While the uncapacitated version is solvable in polynomial time, as we later discuss,

the capacitated version is NP-Hard and therefore requires customized heuristic

solution approaches. We propose strong LP formulations of the capacitated version,

which often allows solving general capacitated instances via branch-and-bound in

reasonable computing time. For those problems that cannot be solved via branch-

and-bound in reasonable time, we provide a set of three effective heuristic solution

methods. Computational test results indicate that the proposed solution methods

for the general capacitated version of the problem are very effective, producing

solutions within 0.1.7'. of optimality, on average, for a broad set of 3,240 randomly

generated problem instances.

Loparic, Pochet, and Wolsey [50] recently considered a related problem in

which a producer wishes to maximize net profit from sales of a single item and









does not have to satisfy all outstanding demand in every period. Their model

assumes that only one demand source exists in every period, and that the revenue

from this demand source is proportional to the volume of demand satisfied. The

"order selection" interpretation of the model we present, on the other hand, allows

the firm to consider any number of orders (or demand sources) in each period,

each with a unique associated per unit revenue (i.e., we allow for customers with

different reservation prices). In this respect, their model represents a single-order

special case of one of the models we propose. More recently, Lee, Qetinkaya, and

Wagelmans [43] considered contexts in which demands can be met either earlier

(through early production and delivery) or later (through backlogging) than

specified without penalty, provided that demand is satisfied within certain demand

time windows for the uncapacitated, single-stage lot sizing problem. Their model

still assumes ultimately, however, that all pre-specified demands must be filled

during the planning horizon.

The remainder of this chapter is organized as follows. Section 2.2 presents

a formal definition and mixed integer programming formulation of the general

production planning problem with order selection flexibility. We then present a

solution approach for the uncapacitated version of the problem that generalizes

the Wagner-Whitin [83] shortest path solution method for single-stage dynamic

requirements planning problems. In Section 2.3 we consider various mixed integer

programming formulations of the capacity constrained problem, along with the

advantages and disadvantages of each formulation strategy. We also provide several

heuristic solution approaches for each of the capacitated problem instances. Section

2.4 then provides a summary of a set of computational tests used to gauge the

effectiveness of the formulation strategies and heuristic solution methods described

in Section 2.3.









2.2 Order Selection Problem Definition and Formulation

Consider a producer who manufactures a good to meet a set of outstanding

orders over a finite number of time periods, T. Producing the good in any time

period t requires a production setup at a cost St and each unit costs an additional

pt to manufacture. We let M(t) denote the set of all orders that request delivery in

period t (we assume zero delivery lead time for ease of exposition; the model easily

extends to a constant delivery lead time without loss of generality), and let m

denote an index for orders. The manufacturer has a capacity to produce Ct units in

period t, t = 1,... T. We assume that that there is no planned backlogging1 (i.e.,

no shortages are permitted) and that items can be held in inventory at a cost of

ht per unit remaining at the end of period t. Let dt denote the quantity of the

good requested by order m for period t delivery, for which the customer will pI i,

rt per unit, and suppose the producer is free to choose any quantity between

zero and dmt in satisfying order m in period t (i.e., rationing is possible, and the

customer will take as much of the good as the supplier can provide, up to dmt).

The producer thus has the flexibility to decide which orders it will choose to satisfy

in each period and the quantity of demand it will satisfy for each order. If the

producer finds it unprofitable to satisfy a certain order in a period, it can choose

to reject the order at the beginning of the planning horizon. The manufacturer

incurs a fixed shipping cost for delivering order m in period t equal to Fmt (any

variable shipping cost can be subtracted from the revenue term, r,t, without loss of

generality). The producer, therefore, wishes to maximize net profit over a T-period

horizon, defined as the total revenue from orders satisfied minus total production



1 Extending our models and solution approaches to allow backlogging at a per
unit per period backlogging cost is fairly straightforward. We have chosen to omit
the details of this extension.









(setup + variable), holding, and delivery costs incurred over the horizon. To

formulate this problem we define the following decision variables:

xt = Number of units produced in period t,

S1, if we setup for production in period t,
t =
0, otherwise,

It = Producer's inventory remaining at the end of period t,

vt = Proportion of order m satisfied in period t,

{ 1, if we satisfy any positive fraction of order m in period t,
Zmt =
0, otherwise.

We formulate the Capacitated Order Selection Problem (OSP) as follows.

[OSP]

T
maximize: m (rmtdrmtVmt FmtZmt) Styt ptXt htlt (2.1)
t=1 TmeM(t)


subject to: It-_ + xt = dtvt + It t 1,..., T, (2.2)
mEM(t)
0< x < Ctyt t = ,..., T, (2.3)

0 < vmt <_ zmt t =1,..., T, m M(t), (2.4)

o 0,I > 0 t 1,...,T, (2.5)

yt, mt E {0,1} t- 1,...,T, E M(t). (2.6)

The objective function (2.1) maximizes net profit, defined as total revenue less

fixed shipping and total production and inventory holding costs. Constraint set

(2.2) represents inventory balance constraints, while constraint set (2.3) ensures

that no production occurs in period t if we do not perform a production setup in

the period. If a setup occurs in period t, the production quantity is constrained by

the production capacity, Ct. Constraint set (2.4) encodes our assumption regarding

the producer's ability to satisfy any proportion of order m up to the amount d,t,









while (2.5) and (2.6) provide nonnegativity and integrality restrictions on variables.

Observe that we can force any order selection (zxt) variable to one if qualitative

and/or strategic concerns (e.g., market share goals) require satisfying an order

regardless of its profitability.

2.2.1 The Uncapacitated Order Selection Problem

If a setup occurs in period t, the production quantity is unconstrained

by setting Ct equal to a large number. Alternatively, we can set Ct equal to

C ,t ECM() drn. without loss of generality, since this is the maximum amount of
demand that could be satisfied by period t production (in the absence of backlog-

ging). We denote the resulting uncapacitated order selection problem as [UOSP].

Problem [UOSP] can be represented as a fixed-charge network flow problem

as illustrated by the example in Figure 2-1, where T = 4 and M(t) = 2 for t = 1,

... T. The network contains three types of arcs: production arcs, inventory arcs,

Production Source
Supply = D
Production flow arcs Inventory holding arcs
Cost= S =y + px ---, Flow cost = h,
LOt- +^, -... ^ T ''<-'
Capacity = C -C't -
Order selection arcs------- Dummy Sink
Cost = FI rm ,dmtvmt n Demand = D

Demand, dn'--: / ,
''^....-----"
D t=1-imM() mt Dummy Source
Supply = D

Figure 2-1: Fixed charge network flow representation of UOSP.

and order selection arcs. The dummy source node implies that it is not necessary

to satisfy all demand-the dummy source can supply the entire demand over the

horizon if necessary. We also add a dummy sink node that can receive flow from

both the production source and the dummy source. Flow on a production arc

implies that a setup occurs in that period, while flow on an order selection arc









implies that we satisfy at least some of that order in the corresponding period.

Since the flow cost on each arc is concave or linear (and hence also concave), the

objective function (2.1) is convex and [UOSP] maximizes a convex function over a

polyhedron for a given y, z. This implies that an optimal extreme point solution

exists for [UOSP]. Since the problem is a network flow problem, this implies

that an optimal -'i ,.'.' :, tree solution exists (see Am!li et al. [3]), in which the

subgraph induced by the arcs with positive flow in a solution forms a spanning

tree. We exploit this spanning tree property to derive certain properties of optimal

solutions to [UOSP]. Note that [UOSP] generalizes the Wagner-Whitin single-stage

requirements planning problem under dynamic demand, whose solution approach

we will extend to solve our order selection problem.

2.2.2 Solution Properties and Shortest Path Approach for UOSP

The existence of an optimal spanning tree solution for [UOSP] implies the

following property:

All-or-nothing order satisfaction property: Given the choice to satisfy

any quantity of demand less than or equal to dmt for order m in period t, an

optimal solution exists with either vt equal to 0 or 1 for all m and t; i.e., for each

order-period combination (m, t) the producer either provides dmt units or none at

all.

We next consider how to extend the Wagner-Whitin dynamic programming

solution method for solving UOSP. Their dynamic programming solution method

can be equivalently posed as a shortest path problem on a graph containing T

+ 1 nodes (see Figure 2-2). Note that this method relies on the existence of an

optimal Zero-Inventory Ordering (ZIO) policy in which a setup only occurs in

period t if we hold no inventory at the end of period t 1 (the validity of this

property can also be shown to hold for [UOSP] as a result of the spanning tree

property of the equivalent fixed charge network representation of [UOSP]). Since









the Wagner-Whitin approach minimizes total cost, each arc (t, t') in the graph

is assigned a cost, c(t, t'), where c(t, t') equals the setup cost in period t plus the

variable production and holding costs incurred for satisfying all demand in periods

t, t + 1,...., 1 using only the setup in period t. This approach ensures that a

path exists in the shortest path network for every feasible combination of setups

and that the cost of a path corresponds to the minimum cost incurred in using the

setups to satisfy all demand.

c(1,5)


S c(O 2 c(2,3 3 c(3,4 c(4,5 5
2,4) c 3 5
c(2,5)

Figure 2-2: Shortest path network structure for UOSP.

Since the ZIO property also holds for the uncapacitated order selection

problem, we can solve the UOSP problem using a shortest path graph containing

the same structure as that used for solving the Wagner-Whitin problem. The arc

length calculation for UOSP, however, requires a new approach. The order selection

problem seeks to maximize net profit and so we interpret arc lengths in terms of

net contribution to profit and seek the longest path in the graph. The method

used for arc length calculation proceeds as follows. We interpret the length of arc

(t, t') as the maximum profit possible from satisfying orders in periods t,..., t' 1

assuming that the only setup available to satisfy demand in these periods must

occur in period t, if at all. Suppose we choose to perform the setup in period t and

incur its corresponding cost, St. To offset the cost of this setup we will satisfy the

demand for order m in period t if and only if


(rrt pt)dmt > Fmt;


(2.7)









i.e., if the net revenue generated from order m is at least as great as the fixed

delivery cost for order m in period t. Similarly, for all periods r such that t < r <

t', we will satisfy order m in period 7 if and only if


(rmt Pt hk)dmT > Fm,; (2.8)

i.e., if the net revenue from order m in period T, less any holding costs incurred

from period t to period T, exceeds the fixed delivery cost for order m in period r.

Let Ot(t) denote the set of orders in period t such that rmt > Pt, and let Ot(T)

denote the set of orders in period 7 such that (2.8) holds for 7 > t. Then the
maximum profit possible if we do a setup in period t and use that setup to satisfy

demands in periods t,..., t' 1, which we denote by MPs(t, t'), is given by

MPs(t, t') = J ZCt(,,) ((,, -P t) dt F,, )
-2 hT (zY hi Zc+o1(T) dk) s,. (2.9)


If MPs(t, t') is greater than 0 we set the length of arc (t, t') equal to MPs(t, t');

otherwise we set the length of arc (t, t') equal to zero and assume no setup occurs
in period t if the optimal solution (the longest path in the graph) traverses arc

(t, t'). After finding the longest path in the graph we can determine which orders to
satisfy in each period by checking the elements of the sets Ot(t) and Ot(r) for all

arcs (t, t') contained in the longest path.

Letting m = max { M(t) }, the total computational effort of arc cost
t=1,...,T
calculations is bounded by 0 (mT2) and the shortest path calculation is no worse

than O(T2), so the worst case complexity of this algorithm is bounded by O (mT2).

Recent work on the uncapacitated lot sizing problem (e.g., Federgruen and Tzur

[24] and Wagelmans, van Hoesel, and Kolen [82]) has reduced the complexity of

this problem from the O(T2) bound to O(T log T) (or even O(T) in certain special

cases). These approaches, however, rely on an important property that holds for









the ELSP, and this property states that the cumulative demand satisfied as we

increase the number of periods in a problem instance is nondecreasing. That is, for

the ELSP, the total demand satisfied in a two-period problem is at least as great

as that satisfied in a one-period problem (where demand in period 1 is the same

in both problem instances). Interestingly, we can show that this property does not

hold in general for the UOSP problem. Assuming that the holding cost in every

period equals zero, we introduce the following data for a three-period problem:

Table 2-1: Counterexample illustrating decreasing cumulative demand satisfaction.

Period Setup Cost Production Cost Demand Unit Revenue
1 $50 $1.50 20 $1.80
1 $50 $1.25 20 $4.00
1 $1,000 $1.20 10 $10.00

Consider the period 1 problem alone. If we setup and satisfy all 20 units of

demand, the revenue equals ;:, while the setup plus variable production cost

equals 1I. Thus we satisfy zero units of demand in the period 1 problem. In the

period 1 + period 2 problem, an optimal solution satisfies 20 units in periods 1 and

2, using the setup in period 1, for a total of 40 units of demand satisfied. Finally,

for the problem containing periods 1, 2, and 3, it is optimal to setup in period 2

and satisfy the 30 units of demand in periods 2 and 3. This example illustrates

why we cannot apply methods previously developed to reduce the complexity of

ELSP to O(T log T) in an effort to reduce the complexity of our algorithm to, iv,

0 (mT log T), since cumulative demand satisfied is not necessarily nondecreasing

for the UOSP problem (note that it is possible to provide a similar example

under which cumulative demand satisfied from period t to T is not necessarily

nondecreasing as we move backwards in time, or, as t decreases).

2.3 OSP Models Limited Production Capacity

We now turn our attention to the capacitated version of our model. We

investigate not only the OSP model as formulated above, but also certain special









cases and restrictions of this model that are of both practical and theoretical

interest. In particular, we consider the special case in which no fixed delivery

charges exist (i.e., the case in which all fixed delivery charge (Fmt) parameters

equal zero). We denote this version of the model as the OSP-NDC. We also explore

contexts in which customers do not permit partial demand satisfaction. This is

a restricted version of the OSP in which the continuous Vmt variables must equal

the binary delivery-charge forcing (zmt) variable values, and can therefore be

substituted out of the formulation; let OSP-AND denote this version of the model

(where AND implies all-or-nothing demand satisfaction). Observe that for the

OSP-AND model we can introduce a new revenue parameter Rmt rmtdmt, where

the total revenue from order m in period t must now equal Rmtzmt. Table 2-2

defines our notation with respect to the different variants of the OSP problem.
Table 2-2: Classification of model special cases and restrictions.

Fixed Delivery Partial Order
Model Charges Exist Satisfaction Allowed
OSP Y Y
OSP-NDC N Y
OSP-AND U N
Y = Yes; N = No.
U: Model and solution approaches unaffected by this assumption.

We distinguish between these model variants not only because they broaden

the model's applicability to different contexts, but also because they can sub-

stantially affect the model's formulation size and complexity, as we next briefly

discuss. Let My = 1 IM(t)) denote the total number of customer orders over

the T-period horizon, where |M(t)| is the cardinality of the set M(t). Note that

formulation [OSP] contains My + T binary variables and My + 2T constraints, not

including the binary and nonnegativity constraints. The OSP-NDC model, on the

other hand, in which Fmt = for all order-period (m, t) combinations, allows us to

replace each Zmt variable on the right-hand-side of constraint set (2.4) with a 1, and

eliminate these variables from the formulation. The OSP-NDC model contains only









T binary variables and therefore requires Ms fewer binary variables than [OSP],

a significant reduction in problem size and complexity. In the OSP-AND model,

customers do not allow partial demand satisfaction, and so we require Vmt = mt

for all order-period (m, t) combinations; we can therefore eliminate the continuous

Vmt variables from the formulation. While the OSP-AND, like the OSP, contains

Ms + T binary variables, it requires Ms fewer total variables than [OSP] as a result

of eliminating the Vmt variables. Table 2-3 summarizes the size of each of these

variants of the OSP with respect to the number of constraints, binary variables,

and total variables.

Table 2-3: Problem size comparsion for capacitated versions of the OSP.

OSP OSP-NDC OSP-AND
Number of Constraints" Ms + 2T Ms + 2T 2T
Number of Binary Variables Ms + T T Ms + T
Number of Total Variables 2Ms + 3T Ms + 3T Ms + 3T
SBinary restriction and nonnegativity constraints are not included.

Based on the information in this table, we would expect the OSP and OSP-

AND to be substantially more difficult to solve than the OSP-NDC. As we will

show in Section 2.4, the OSP-AND actually requires the greatest amount of

computation time on average, while the OSP-NDC requires the least.

Note that the OSP-AND is indifferent to whether fixed delivery charges exist,

since we can simply reduce the net revenue parameter, Rmt rmtdmt, by the fixed

delivery-charge value Fmt, without loss of generality. In the OSP-AND then, the

net revenue received from an order equals RmtZmt, and we thus interpret the Zmt

variables as binary "order selection" variables. In contrast, in the OSP, the purpose

of the binary zmt variables is to force us to incur the fixed delivery charge if we

satisfy any fraction of order m in period t. In this model we therefore interpret the

Zmt variables as fixed delivery-charge forcing variables, since their objective function

coefficients are fixed delivery cost terms rather than net revenue terms, as in the

OSP-AND. Note also that since both the OSP-NDC and the OSP-AND require









only one set of order selection variables (the continuous vmt variables for the OSP-

NDC and the binary Zmt variables for the OSP-AND), their linear programming

relaxation formulations will be identical (since relaxing the binary Zmt variables

is equivalent to setting Zmt = vmt). The OSP linear programming relaxation

formulation, on the other hand, explicitly requires both the vmt and Zmt variables,

resulting in a larger LP relaxation formulation than that for the OSP-NDC and

the OSP-AND. These distinctions will p1 i, an important role in interpreting the

difference in our ability to obtain strong upper bounds on the optimal solution

value for the OSP and the OSP-AND in Section 2.4.3. We next discuss solution

methods for the OSP and the problem variants we have presented.

2.3.1 OSP Solution Methods

To solve the OSP, we must decide which orders to select and, among the

selected orders, how much of the order we will satisfy while obeying capacity

limits. We can show that this problem is NP-Hard through a reduction from the

capacitated lot-sizing problem as follows. If we consider the special case of the

OSP in which 1>J Ct > E -1 ECmeM(t) drnt for j =1, ..., T (which implies that

satisfying all orders is feasible) and min {rmt} > max {St + max {pt} +
t=1,...,T,mEM(t) t=1,...,T t=1,...,T
:t 1> ht (which implies that it is profitable to satisfy all orders in every period),
then total revenue is fixed and the problem is equivalent to a capacitated lot-sizing

problem, which is an NP-Hard optimization problem (see Florian and Klein [28]).

Given that the OSP is NP-Hard, we would like to find an efficient method for

obtaining good solutions for this problem. As our computational test results in

Section 2.4 later show, we were able to find optimal solutions using branch-and-

bound for many of our randomly generated test instances. While this indicates

that the i1 i i ,i i ly of problem instances we considered were not terribly difficult to

solve, there were still many instances in which an optimal solution could not be

found in reasonable computing time. Based on our computational test experience









in effectively solving problem instances via branch-and-bound using the CPLEX

6.6 solver, we focus on strong LP relaxations for the OSP that provide quality

upper bounds on optimal net profit quickly, and often enable solution via branch-

and-bound in acceptable computing time. For those problems that cannot be

solved via branch-and-bound, we employ several customized heuristic methods,

which we discuss in Section 2.3.3. Before we discuss the heuristics used to obtain

lower bounds for the OSP, we first present our reformulation strategy, which helps

to substantially improve the upper bound provided by the linear programming

relaxation of the OSP.

2.3.2 Strengthening the OSP Formulation

This section presents an approach for providing good upper bounds on the

optimal net profit for the OSP. In particular, we describe two LP relaxations for

the OSP, both of which differ from the LP relaxation obtained by simply relaxing

the binary restrictions of [OSP] (constraint set (2.6)) in Section 2.2. We will refer

to this simple LP relaxation of [OSP] as OSP-LP, to distinguish this relaxation

from the two LP relaxation approaches we provide in this section.

The two LP relaxation formulations we next consider are based on a reformu-

lation strategy developed for the UOSP. In C'! lpter 3, we will present a "- i,!ii

formulation of a similar problem to the UOSP, for which we show that the optimal

LP relaxation solution value equals the optimal (mixed integer) UOSP solution

value. We discuss this reformulation strategy in greater detail by first providing

a tight linear programming relaxation for the UOSP. We first note that for the

UOSP, an optimal solution exists such that we never satisfy part of an order; i.e.,

Vmt equals either 0 or 1. Thus we can substitute the Vmt variables out of [OSP] by

setting Vmt = Zmt for all t and m E M(t).

Next observe that since It = Z 1 Xj Y IYEmMm(j) dmj mj, we can eliminate

the inventory variables from [OSP] via substitution. After introducing a new









variable production and holding cost parameter, ct, where ct pt + Yj=t hj, the

objective function of the UOSP can be rewritten as

ST T T
maximize > RmjZmj + ht dmjzmjj (Styt + ctxt) (2.10)
j=1 mCM(j) t=1 j=1 rmCM(j) t 1

We next define pnt as an adjusted revenue parameter for order m in period

t, where pit = j=t hj + Rt. Our reformulation procedure requires capturing

the exact amount of production in each period allocated to every order. We thus

define xtj as the number of units produced in period t used to satisfy order m in

period j, for j > t, and replace each xt with T7, EamM(j) xmtj. The following

formulation provides the -I 'ng" linear programming relaxation of the UOSP.

[UOSP']


T
maximize: Y Y Pmjdmjzmj
j= 1 mM(j)


(2.11)


subject to:


x Xmtj dmj zmj 0
t=1

]C( >_ > 0
mCM(j) / mEM(j)
-Zmj > -1


t Xmtj, Zmj > 0


j 1,...,T,m m M(j), (2.12)


t= ,... ,T,j = t,..., T,(2.13)

j 1,...,T,m e M(j), (2.14)

t= 1,...,T,

j t,...,T,m E M(j). (2.15)


Note that since a positive cost exists for setups, we can show that the constraint

yt < 1 is unnecessary in the above relaxation, and so we omit this constraint from

the relaxation formulation. It is straightforward to show that [UOSP'] with the

additional requirements that all zmj and yt are binary variables is equivalent to our

[OSP] when production capacities are infinite. (We will also show in C'!i pter 3 that


T ( T j
YE St + c E E Xmtj
t=1 j=tmI=M(j)









by di'l,--_-regating the setup forcing constraints (2.13), the resulting formulation has

zero integrality gap through a dual solution approach.)

To obtain the LP relaxation for the OSP (when production capacities are

finite), we add finite capacity constraints to [UOSP'] by forcing the sum of Xmtj

over all j > t and all m E M(j) to be less than the production capacity Ct in

period t. That is, we can add the following constraint set to [UOSP] to obtain an

equivalent LP relaxation for the OSP:
T
S xTt j=t mCM(j)

Note that this LP relaxation approach is valid for all three variants of the OSP: the

general OSP, the OSP-NDC, and the OSP-AND. Observe that the above constraint

can be strengthened by multiplying the right-hand-side by the setup forcing

variable yt. To see how this strengthens the formulation, note that constraint set

(2.13) in [UOSP'] implies that

T T
S Xmtj <> d(j yt t 1,... ,T.
j t mnEM(j) j t /TGM(j)

To streamline our notation, we define the following. Let XtT j= E ZmM(j) Xrtj

and DT = Y= t (KZmCe(j) dmj) for t = 1,... T denote related production

variables and order amounts, respectively. Constraint set (2.16) can be rewritten as


Xt, <- C t= ,...,T,

and the .. :i regated demand forcing constraints (2.13) can now be written as

XtT < DtT Y.t If we do not multiply the right-hand-side of capacity constraint set
(2.16) by the forcing variable yt, the formulation allows solutions, for example, such

that XtT = Ct for some t, while XtT equals only a fraction of DtT. In such a case,

the forcing variable yt takes the fractional value XT-, and we only absorb a fraction
DLT









of the setup cost in period t. Multiplying the right-hand-side of (2.16) by yt, on

the other hand, would force yt = X = 1 in such a case, leading to an improved

upper bound on the optimal solution value. We can therefore strengthen the LP

relaxation solution that results from adding constraint set (2.16) by instead using

the following capacity forcing constraints.


XtT < min{Ct, DtT} t t =1,..., T. (2.17)


Note that in the capacitated case we now explicitly require stating the yt < 1

constraints in the LP relaxation, since it may otherwise be profitable to violate pro-

duction capacity in order to satisfy additional orders. We refer to the resulting LP

relaxation with these .,-.--regated setup forcing constraints as the [ASF] formulation,

which we formulate as follows.

[ASF]


maximize: Y Y pmrjdrmjzmj cE Stt + Ct E Xmtj
j 1 zCM(j) t=1 j=t rtzaM(j)


subject to: Constraints (2.12 -2.15,2.17)

Yt < I t= ,...,T. (2.18)


We can further strengthen the LP relaxation formulation by di- .- -'egating

the demand forcing constraints (2.13) (see Erlenkotter [23], who uses this strategy

for the uncapacitated facility location problem). This will force yt to be at least

as great as the axium value of m for all = t,...,T and m M(j). The

resulting Disaggregated Setup Forcing (DASF) LP relaxation is formulated as

follows.

[DASF]

T T T
maximize: Y Y pmrjdrjZrj S- E tt + Ct E E Xmtj
j= ImzM(j) t=1 j=t mrzCM(j)









subject to: Constraints (2.12, 2.14, 2.15,2.17, 2.18)

Xmtj < dmjyt t 1,.. T, (2.19)

j t,... ,T,

mn M(j).


Each of the LP relaxations we have described provides some value in solving

the capacitated versions of the OSP. Both the OSP-LP and ASF relaxations can

be solved very quickly, and they frequently yield high quality solutions. The DASF

relaxation further improves the upper bound on the optimal solution value. But

as the problem size grows (i.e., the number of orders per period or the number

of time periods increases), [DASF] becomes intractable, even via standard linear

programming solvers. We present results for each of these relaxation approaches in

Section 2.4. Before doing this, however, we next discuss methods for determining

good feasible solutions, and therefore lower bounds, for the OSP via several

customized heuristic solution procedures.

2.3.3 Heuristic Solution Approaches for OSP

While the methods discussed in the previous subsection often provide strong

upper bounds on the optimal solution value for the OSP (and its variants), we

cannot guarantee the ability to solve this problem in reasonable computing time

using branch-and-bound due to the complexity of the problem. We next discuss

three heuristic solution approaches that allow us to quickly generate feasible

solutions for OSP. As our results in Section 2.4 report, using a composite solution

procedure that selects the best solution among those generated by the three

heuristic solution approaches provided feasible solutions with objective function

values, on average, within 0.1'7'. of the optimal solution value. We describe our

three heuristic solution approaches in the following three subsections.









2.3.3.1 Lagrangian Relaxation Based Heuristic

Lagrangian relaxation (Geoffrion [32]) is often used for mixed integer program-

ming problems to obtain stronger upper bounds (for maximization problems) than

provided by the LP relaxation. As we discussed in Section 2.3.2, our strengthened

linear programming formulations typically provide very good upper bounds on

the optimal solution value of the OSP. Moreover, as we later discuss, our choice of

relaxation results in a Lagrangian subproblem for which we can find an optimal

extreme point solution equivalent to the solution found for our LP relaxation. This

implies that the upper bound provided by our Lagrangian relaxation scheme will

not provide better bounds than our LP relaxation. Our purpose for implementing

a Lagrangian relaxation heuristic, therefore, is strictly to obtain good feasible so-

lutions using a Lagrangian-based heuristic. Because of this we omit certain details

of the Lagrangian relaxation algorithm and implementation, and describe only the

essential elements of the general relaxation scheme and how we obtain a heuristic

solution at each iteration of the Lagrangian algorithm.

Under our Lagrangian relaxation scheme, we add (redundant) constraints

of the form xt < Myt, t = 1,..., T to [OSP] (where M is some large number),

eliminate the forcing variable yt from the right-hand side of the capacity/setup

forcing constraints (2.3), and then relax the resulting modified capacity constraint

(without the yt multiplier on the right-hand side) in each period. The Lagrangian

relaxation subproblem is then simply an uncapacitated OSP (or UOSP) problem.

Although the Lagrangian multipliers introduce the possibility of negative unit

production costs in the Lagrangian subproblem, we retain the convexity of the

objective function, and all properties necessary for solving the UOSP problem via

a shortest path network approach still hold (see Section 2.2.2). We can therefore

solve the Lagrangian subproblems in polynomial time. Because we have a tight

formulation of the UOSP (as we will prove in C'! lpter 3), this implies that the









Lagrangian solution will not provide better upper bounds than the LP relaxation.

We do, however, use the solution of the Lagrangian subproblem at each iteration

of a subgradient optimization algorithm (see Fisher [26]) as a starting point for

heuristically generating a feasible solution, which serves as a candidate lower bound

on the optimal solution value for OSP.

Observe that the subproblem solution from this relaxation will satisfy all

constraints of the OSP except for the relaxed capacity constraints (2.3). We

therefore call a feasible solution generator (FSG) at each step of the subgradient

algorithm, which can take any starting capacity-infeasible (but otherwise feasible)

solution and generate a capacity-feasible solution. (We also use this FSG in our

other heuristic solution schemes, as we later describe.) The FSG works in three

main phases. Phase I first considers performing additional production setups

(beyond those prescribed by the starting solution) to try to accommodate the

desired production levels and order selection decisions provided in the starting

solution, while obeying production capacity limits. That is, we consider shifting

production from periods in which capacities are violated to periods in which no

setup was originally planned in the starting solution. It is possible, however, that

we still violate capacity limits after Phase I, since we do not eliminate any order

selection decisions in Phase I.

In Phase II, after determining which periods will have setups in Phase I, we

consider those setup periods in which production still exceeds capacity and, for

each such setup period, index the orders satisfied from production in the setup

period in nondecreasing order of contribution to profit. For each period with

violated capacity, in increasing profitability index order, we shift orders to an

earlier production setup period, if the order remains profitable and such an earlier

production setup period exists with enough capacity to accommodate the order.

Otherwise we eliminate the order from consideration. If removing the order from









the setup period will leave excess capacity in the setup period under consideration,

we consider shifting only part of the order to a prior production period; we also

consider eliminating only part of the order when customers do not require all-or-

nothing order satisfaction. This process is continued for each setup period in which

production capacity is violated until total production in the period satisfies the

production capacity limit. Following this second phase of the algorithm, we will

have generated a capacity-feasible solution. In the third and final phase, we scan

all production periods for available capacity and assign additional profitable orders

that have not yet been selected to any excess capacity if possible. The C'! lpter

Appendix in Section 2.6 contains a detailed description of the FSG algorithm.


2.3.3.2 Greatest Unit Profit Heuristic

Our next heuristic solution procedure is motivated by an approach taken in

several well-known heuristic solution approaches for the ELSP. In particular, we

use a similar i'y" '1.. approach to those used in the Silver-Meal [72] and Least

Unit Cost (see N ilii ,4 [56]) heuristics. These heuristics proceed by considering

an initial setup period, and then determining the number of consecutive period

demands (beginning with the initial setup period) that produce the lowest cost per

period (Silver-Meal) or per unit (Least Unit Cost) when allocated to production in

the setup period. The next period considered for a setup is the one immediately

following the last demand period assigned to the prior setup; the heuristics proceed

until all demand has been allocated to some setup period. Our approach differs

from these approaches in the following respects. Since we are concerned with the

profit from orders, we take a greatest profit rather than a lowest cost approach.

We also allow for accepting or rejecting various orders, which implies that we need

only consider those orders that are profitable when assigning orders to a production

period. Moreover, we can choose not to perform a setup if no selection of orders









produces a positive profit when allocated to the setup period. Finally, we apply

our ,i. ii. -I unit profit" heuristic in a capacitated setting, whereas a modification

of the Silver-Meal and Least Unit Cost heuristics is required for application to the

capacitated lot-sizing problem.

Our basic approach begins by considering a setup in period t (where t initially

equals 1) and computing the maximum profit per unit of demand satisfied in period

t using only the setup in period t. Note that, given a setup in period t, we can sort

orders in periods t,..., T in nonincreasing order of contribution to profit based

solely on the variable costs incurred when assigning the order to the setup in period

t (for the OSP when fixed delivery charges exist we must also subtract this cost

from each order's contribution to profit). Orders are then allocated to the setup

in nonincreasing order of contribution to profit until either the setup capacity is

exhausted or no additional attractive orders exist. After computing the maximum

profit per unit of demand satisfied in period t using only the setup in period t, we

then compute the maximum profit per unit satisfied in periods t,..., t + j using

only the setup in period t, for j = 1,..., j', where period j' is the first period in the

sequence such that the maximum profit per unit in periods t,... t + j' is greater

than or equal to the maximum profit per unit in periods t,..., t + j' + 1. The

capacity-feasible set of orders that leads to the greatest profit per unit in periods

t,..., j' using the setup in period t is then assigned to production in period t,

assuming the maximum profit per unit is positive. If the maximum profit per unit

for any given setup period does not exceed zero, however, we do not assign any

orders to the setup and thus eliminate the setup.

Since we consider a capacity-constrained problem, we can either consider

period j' + 1 (as is done in the Silver-Meal and Least Unit Cost heuristics) or

period t + 1 as the next possible setup period following period t. We use both

approaches and retain the solution that produces higher net profit. Note that if we









consider period t + 1 as the next potential setup period following period t, we must

keep track of those orders in periods t + 1 and higher that are already assigned to

period t (and prior) production, since these will not be available for assignment to

period t + 1 production. Finally, after applying this greatest unit profit heuristic,

we apply Phase III of the FSG algorithm (see the C'! lpter Appendix in Section

2.6) to the resulting solution, in an effort to further improve the heuristic solution

value by looking for opportunities to effectively use any unused setup capacity.


2.3.3.3 Linear Programming Rounding Heuristic

Our third heuristic solution approach uses the LP relaxation solution as a

starting point for a linear programming rounding heuristic. We focus on rounding

the setup (yt) and order selection (zmt) variables that are fractional in the LP

relaxation solution (rounding the order selection variables is not, however, relevant

for the OSP-NDC problem, since the Zmt variables do not exist in this special case).

We first consider the solution that results by setting all (non-zero) fractional yt and

z,t variables from the LP relaxation solution to one. We then apply the second

and third phases of our FSG algorithm to ensure a capacity feasible solution, and

to search for unselected orders to allocate to excess production capacity in periods

where the setup variable was rounded to one.

We also use an alternative version of this procedure, where we round up the

setup variables with values greater than or equal to 0.5 in the LP relaxation so-

lution, and round down those with values less than 0.5. Again we subsequently

apply Phases II and III of the FSG algorithm to generate a good capacity-feasible

solution (if the maximum setup variable value takes a value between 0 and 0.5, we

round up only the setup variable with the maximum fractional variable value and

apply Phases II and III of the FSG algorithm). Finally, based on our discussion in









Section 2.3.2, we have a choice of three different formulations for generating LP re-

laxation starting solutions for the rounding procedure: formulation [OSP] (Section

2.2), [ASF] (Section 2.3.2), or [DASF] (Section 2.3.2). As our computational results

later discuss, starting with the LP relaxation solution from the [DASF] formulation

provides solutions that are, on average, far superior to those provided using the

other LP relaxation solutions. However, the size of this LP relaxation also far

exceeds the size of our other LP relaxation formulations, making this formulation

impractical as problem sizes become large. We use the resulting LP relaxation

solution under each of these formulations and apply the LP rounding heuristic to

all three of these initial solutions for each problem instance, retaining the solution

that provides the highest net profit.

2.4 Scope and Results of Computational Tests

This section discusses a broad set of computational tests intended to evaluate

our upper bounding and heuristic solution approaches. Our results focus on

gauging both the ability of the different LP relaxations presented in Section 2.3.2

to provide tight upper bounds on optimal profit, and the performance of the

heuristic procedures discussed in Section 2.3.3 in providing good feasible solutions.

Section 2.4.1 next discusses the scope of our computational tests, while Sections

2.4.2 and 2.4.3 report results for the OSP, OSP-NDC, and OSP-AND versions of

the problem.

2.4.1 Computational Test Setup

This section presents the approach we used to create a total of 3,240 randomly

generated problem instances for computational testing, which consist of 1,080

problems for each of the OSP, OSP-NDC, and OSP-AND versions of the problem.

Within each problem version (OSP, OSP-NDC, and OSP-AND), we used three

different settings for the number of orders per period, equal to 25, 50, and 200. In

order to create a broad set of test instances, we considered a range of setup cost









values, production capacity limits, and per unit order revenues.2 Table 2-4 pro-

vides the set of distributions used for randomly generating these parameter values

in our test cases. The total number of combinations of parameter distribution

settings shown in Table 2-4 equals 36, and for each unique choice of parameter

distribution settings we generated 10 random problem instances. This produced

a total of 360 problem instances for each of the three values of the number of

orders per period (25, 50, and 200), which equals 1,080 problem instances for each

problem version. As the distributions used to generate production capacities in

Table 2-4: Probability distributions used for generating problem instances.

Number of Distributions used"
Parameter Distribution for Parameter
Settings Generation
Setup cost (varies 3 U[350,650]
from period-to-period) U[1750,3250]
U[3500,6500]
Per unit per period holding cost 2 0.15 x p/50
0.25 x p/50
Production capacity in a 3 U[d/3 .05d, d/3 + .05d]
period (varies from U[d/2 .d, d/2 + .ld]
period-to-period) U[d .15d, d + .15d]
Per unit order revenue (varies 2 U[28,32]
from order-to-order) U[38,42]
a U[a, b] denotes a uniform distribution on the interval [a, b].
b p denotes the variable production cost. We assume 50 working weeks in one year.
Sd denotes the expected per-period total demand, which equals the mean of the
distribution of order sizes multiplied by the number of orders per period.

Table 2-4 indicate, we maintain a constant ratio of average production capacity per

period to average total demand per period. That is, we maintain the same average

order size (average of dmt values) across each of these test cases, but the average

capacity per period for the 200-order problem sets is four times that of the 50-order



2 These three parameters appeared to be the most critical ones to vary widely
in order to determine how robust our solution methods were to problem parameter
variation.









problem sets and eight times that of the 25-order problems. Because the total

number of available orders per period tends to strongly affect the relative quality of

our solutions (as we later discuss), we report performance measures across all test

cases and also individually within the 25, 50, and 200 order problem sets.

In order to limit the scope of our computational tests to a manageable

size, we chose to limit the variation of certain parameters across all of the test

instances. The per unit production cost followed a distribution of U[20,30] for all

test instances (where U[a, b] denotes a Uniform distribution on the interval [a, b]),

and all problem instances used a 16-period planning horizon. We also used an

order size distribution of U[10,70] for all test problems (i.e., the dt values follow a

uniform distribution on [10,70]). For the OSP, the distribution used for generating

fixed delivery charges was U[100,600].3 By including a wide range of levels of

production capacity, setup cost, and order volumes, we tested a set of problems

which would fairly represent a variety of actual production scenarios.

Observe that the two choices for distributions used to generate per unit

order revenues use relatively narrow ranges. Given that the distribution used to

generate variable production cost is U[20,30], the first of these per unit revenue

distributions, U[28,32], produces problem instances in which the contribution to

profit (after subtracting variable production cost) is quite small-leading to fewer

attractive orders after considering setup and holding costs. The second distribution,

U[38,42], provides a more profitable set of orders. We chose to keep these ranges

very narrow because our preliminary test results showed that a tighter range, which



3 We performed computational tests with smaller per-order delivery charges, but
the results were nearly equivalent to those presented for the OSP-NDC in Table
2.4.2, since the profitability of the orders remained essentially unchanged. As we
increased the average delivery charge per order, more orders became unprofitable,
creating problem instances that were quite different from the OSP-NDC case.









implies less per unit revenue differentiation among orders, produces more difficult

problem instances. Those problem instances with a greater range of per unit

revenue values among orders tended to be solved in CPLEX via branch-and-bound

much more quickly than those with tight ranges, and we wished to ensure that our

computational tests reflected more difficult problem instances.

A tighter range of unit revenues produces more difficult problem instances

due to the ability to simply 'swap' orders with identical unit revenues in the

branch-and-bound algorithm, leading to alternative optimal solutions at nodes in

the branch-and-bound tree. For example, if an order m in period t is satisfied at

the current node in the branch-and-bound tree, and some other order m' is not

satisfied, but rt = rmt and dt = dmt, then a solution which simply swaps orders

m and m' has the same objective function as the first solution, and no improvement

in the bound occurs as a result of this swap. So, we found that when the problem

instance has less differentiation among orders, the branch-and-bound algorithm can

take substantially longer, leading to more difficult problem instances. Barnhart et

al. [7] and Balakrishnan and Geunes [6] observed similar swapping phenomena in

branch-and-bound for machine scheduling and steel production planning problems,

respectively.

All linear and mixed integer programming (VI P) formulations were solved

using the CPLEX 6.6 solver on an RS/6000 machine with two PowerPC (300MHz)

CPUs and 2GB of RAM. We will refer to the best solution provided by the CPLEX

branch-and-bound algorithm as the MIP solution. The remaining subsections

summarize our results. Section 2.4.2 reports the results of our computational

experiments for the OSP-NDC and the OSP, and Section 2.4.3 presents the findings

for the OSP-AND (all-or-nothing order satisfaction) problem. For the OSP-

AND problem instances discussed in Section 2.4.3, we assume that the revenue

parameters provided represent revenues in excess of fixed delivery charges (since we









alv--i- satisfy all or none of the demand for the OSP-AND, this is without loss of

generality).

2.4.2 Results for the OSP and the OSP-NDC

Recall that the OSP assumes that we have the flexibility to satisfy any

proportion of an order in any period, as long as we do not exceed the production

capacity in the period. Because of this, when no fixed delivery charges exist, the

only binary variables in the OSP-NDC correspond to the T binary setup variables,

and solving these problem instances to optimality using CPLEX's MIP solver did

not prove to be very difficult. The same is not necessarily true of the OSP-AND,

as we later discuss in Section 2.4.3. Surprisingly, the OSP (which includes a binary

fixed delivery-charge forcing (zmt) variable for each order-period combination) was

not substantially computationally challenging either. All of the OSP-NDC and all

but two of the OSP instances were solved optimally using branch-and-bound within

the allotted branch-and-bound time limit of one hour. Even though we are able to

solve the OSP and OSP-NDC problem instances using CPLEX with relative ease,

we still report the upper bounds provided by the different LP relaxations for these

problems in this section. This allows us to gain insight regarding the strength of

these relaxations as problem parameters change, with knowledge of the optimal

mixed integer programming (\I P) solution values as a benchmark.

Table 2.4.2 presents optimality gap measures based on the solution values

resulting from the LP (OSP-LP) relaxation upper bound, the .,.-.-regated setup

forcing (ASF) relaxation upper bound, and the di,- _regated setup forcing (DASF)

relaxation upper bound for the OSP-NDC and OSP problem instances. The last

row of the table shows the percentage of problem instances for which CPLEX was

able to find an optimal solution via branch-and-bound. As Table 2.4.2 shows, for

the OSP-NDC, all three relaxations provide good upper bounds on the optimal

solution value, consistently producing gaps of less than 0.25'. on average. As









expected, the [ASF] formulation provides better bounds than the simple OSP-LP

relaxation, and the [DASF] formulation provides the tightest bounds. We note

that as the number of potential orders and the per-period production capacities

increase, the relative performance of the relaxations improves, and the optimality

gap decreases. Since an optimal solution exists such that at most one order per

period will be partially satisfied under any relaxation, as the problem size grows,

we fulfill a greater proportion of orders in their entirety. So the impact of our

choice of which order to partially satisfy diminishes with larger problem sizes.

Note also, however, that a small portion of this improvement is attributable to the

increased optimal solution values in the 50- and 200-order cases.

For the OSP, we have non-zero fixed delivery costs and cannot therefore

eliminate the binary Zmt variables from formulation [OSP]. In addition, since for-

mulation [OSP] includes the continuous Vmt variables, it has the highest number of

variables of any of the capacitated versions we consider. This does not necessarily,

however, make it the most difficult problem class for solution via CPLEX, as a

later comparison of the results for the OSP and OSP-AND indicates.

The upper bound optimality gap results reported in Table 2.4.2 for the

OSP are significantly larger than those for the OSP-NDC.4 This is because this

formulation permits setting fractional values of the fixed delivery-charge forcing

(zmt) variables, and therefore does not necessarily charge the entire fixed delivery
cost when meeting a fraction of some order's demand. For this problem set the

[DASF] formulation provides substantial value in obtaining strong upper bounds on

the optimal net profit although, as shown in Table 2-6, the size of this formulation



4 For the two problems that could not be solved to optimality via branch-and-
bound using CPLEX due to memory limitations, the MIP solution value used to
compute the upper bound optimality gap is the value of the best solution found by
CPLEX.









makes solution via CPLEX substantially more time consuming as the number of

orders per period grows to 200.

Table 2-5: OSP-NDC and OSP problem optimality gap measures.

OSP-NDC OSP
% Gap Orders per Period Overall Orders per Period Overall
(from MIP) 25 50 200 Average 25 50 200 Average
OSP-LPa 0.2!', 0.1 !' 0.05' 0.1 'I, 9.2i.' 6.0',' 0.57' 5.31 ,
ASFb 0.18 0.12 0.04 0.11 9.21 6.07 0.56 5.28
DASFc 0.11 0.07 0.03 0.07 1.58 0.35 0.10 0.68
% Optd 100 100 100 100 100 99.7 99.7 99.8
Note: Entries in each ,.-I. I per period" class represent an
average among 360 test instances.
a (OSP-LP -MIP)/MIP x 100%.
b (ASF MIP)/MIP x 100%.
c(DASF MIP)/MIP x 100%.
d % of problems for which CPLEX branch-and-bound found an optimal solution.

Table 2-6 summarizes the solution times for solving the OSP-NDC and the

OSP. The MIP solution times reflect the average time required to find an optimal

solution for those problems that were solved to optimality in CPLEX (the two

problems that CPLEX could not solve to optimality are not included in the MIP

solution time statistics). We used the OSP-LP formulation as the base formulation

for solving all mixed integer programs. The table also reports the times required

to solve the LP relaxations for each of our LP formulations (OSP-LP, ASF, and

DASF). We note that the [ASF] and [DASF] LP relaxations often take longer

to solve than the mixed integer problem itself. The [DASF] formulation, despite

providing the best upper bounds on solution value, quickly becomes less attractive

as the problem size grows because of the size of this LP formulation. Nonetheless,

the relaxations provide extremely tight bounds on the optimal solution as shown

in the table. As we later show, however, solving the problem to optimality in

CPLEX is not alv--i- a viable approach for the restricted OSP-AND discussed in

the following section.









Table 2-6 reveals that the MIP solution times for the OSP were also much

greater than for the OSP-NDC. This is due to the need to simultaneously track

the binary (zmt) and continuous (vmt) variables for the OSP with non-zero fixed

delivery costs. As expected, the average and maximum solution times for each

relaxation increased with the number of orders per period. As we noted previously,

the percentage optimality gaps, however, substantially decrease as we increase the

number of orders per period.

Table 2-6: OSP-NDC and OSP solution time comparison.

OSP-NDC OSP
Orders per Period Orders per Period
Time Measure (CPU seconds) 25 50 200 25 50 200
Average MIP Solution Time 0.1 0.1 0.2 3.3 19.1 129.4
Maximum MIP Solution Time 0.1 0.1 0.3 44.8 541.3 3417.2
Average OSP-LP Solution Time 0.1 0.1 0.3 0.1 0.1 0.3
Maximum OSP-LP Solution Time 0.1 0.2 0.5 0.1 0.1 0.5
Average ASF Solution Time 0.5 1.5 14.0 0.4 1.0 8.3
Maximum ASF Solution Time 0.7 2.2 25.2 0.6 1.6 15.4
Average DASF Solution Time 5.3 27.3 727.2 3.3 15.7 333.8
Maximum DASF Solution Time 18.4 64.3 1686.7 12.1 47.1 1251.9
Note 1: Entries represent average/maximum among 360 test instances.
Note 2: LP relaxation solution times include time consumed applying the LP
rounding heuristic to the resulting LP solution, which was negligible.

We next present the results of applying our heuristic solution approaches to

obtain good solutions for the OSP and OSP-NDC. We employ the three heuristic

solution methods discussed in Section 2.3.3, denoting the Lagrangian-based

heuristic as LAGR, the greatest unit profit heuristic as GUP, and the LP rounding

heuristic as LPR. Table 2-7 provides the average percentage deviation from the

best upper bound (as a percentage of the best upper bound) for each heuristic

solution method. Note that since we found an optimal solution for all but two of

the OSP and OSP-NDC problem instances, the upper bound used in computing the

heuristic solution gaps is nearly alv-- the optimal mixed integer solution value.

The last row in Table 2-7 shows the resulting lower bound gap from our composite









solution procedure, which selects the best solution among all of the heuristic

methods applied. The average lower bound percentage gap is within 0.0 .' of

optimality for the OSP-NDC, while that for the OSP is 1.-'.I' indicating that

overall, our heuristic solution methods are quite effective. As the table indicates,

the heuristics perform much better in the absence of fixed delivery costs. For the

Lagrangian-based and LP rounding heuristics, we can attribute this in part to the

difficulty in obtaining good relaxation upper bounds for the OSP as compared to

the OSP-NDC. Observe that as the upper bound decreases (i.e., as the number of

orders per period increases), these heuristics tend to improve substantially. The

GUP heuristic, on the other hand, appears to have difficulty identifying a good

combination of setup periods in the presence of fixed delivery charges. Although

it appears, based on average performance, that the LPR heuristic dominates

the LAGR and GUP heuristics, the last row of the table reveals that this is not

universally true. Each of our heuristic approaches provided the best solution value

for some nontrivial subset of the problems tested.

Table 2-7: OSP and OSP-NDC heuristic solution performance measures.

OSP-NDC OSP
% Gap Orders per Period Overall Orders per Period Overall
(from UB) 25 50 200 Average 25 50 200 Average
LAGR v. UB" 1.3 !'. 0..', 0.32'-. 0.7".' 6.35' 4.07.' 2.1.' 4.19
GUP v. UBb 1.00 0.69 0.44 0.71 7.27 6.91 5.39 6.52
LPR v. UBe 0.25 0.15 0.05 0.15 8.32 5.31 0.96 4.86
Best LB 0.10 0.07 0.02 0.06 3.08 1.55 0.44 1.69
Note: Entries in each -.I. i, per period" class represent an
average among 360 test instances.
a (LAGR -UB)/UB x 100%.
b (GUP -UB)/UB x 100%.
(LPR -UB)/UB x 100%.
d Uses the best heuristic solution value for each problem instance.

2.4.3 Results for the OSP-AND

We next provide our results for the OSP-AND where, if we choose to accept an

order, we must satisfy the entire order (i.e., no partial order satisfaction is allowed).









Finding the optimal solution to the OSP-AND can be much more challenging than

for the OSP, since we now face a more difficult combinatorial p1 I1" .!:, problem

(i.e., determining the set of orders that will be produced in each period is similar to

a multiple knapsack problem).

Table 2-8 provides upper bound optimality gap measures based on the solution

values resulting from our different LP relaxation formulations, along with the

percentage of problem instances that were solved optimally via the CPLEX branch-

and-bound algorithm. Observe that the upper bound optimality gap measures are

quite small and only slightly larger than those observed for the OSP-NDC. The

reason for this is that the LP relaxation formulations are identical in both cases

(as discussed in Section 2.2), and the optimal LP relaxation solution violates the

all-or-nothing requirement for at most one order per period. Thus, even in the

OSP-NDC case, almost all orders that are selected are fully satisfied in the LP

relaxation solution. In contrast to the [OSP] formulation, the binary Zmt variables

in the OSP-AND model now represent "order selection" variables rather than fixed

delivery-charge forcing variables. That is, since we net any fixed delivery charge

out of the net revenue parameters Rmt, and the total revenue for an order in a

period now equals Rmtzmt in this formulation, we have strong preference for Zmt

variable values that are either close to one or zero. In the [OSP] formulation, on

the other hand, the Zmt variables are multiplied by the fixed delivery-charge terms

(Fmt) in the objective function, leading to a strong preference for low values of the

Zmt variables and, therefore, a weaker upper bound on optimal net profit. Note

also that as the number of possible orders increases (from the 25-order case to the

200-order case), the influence of the single partially satisfied order in each period

on the objective function value diminishes, leading to a reduced optimality gap as

the number of orders per period increases. As the last row of Table 2-8 indicates,

we were still quite successful in solving these problem instances to optimality in









CPLEX. The time required to do so, however, was substantially greater than that

for either the OSP or OSP-NDC, because of the complexities introduced by the

all-or-nothing order satisfaction requirement.

Table 2-9 summarizes the resulting solution time performance for the OSP-

AND. We note here that our relaxation solution times are quite reasonable,

especially as compared to the MIP solution times, indicating that quality upper

bounds can be found very quickly. Again, the MIP solution times reflect the

average time required to find an optimal solution for those problems that were

solved to optimality in CPLEX (those problems which CPLEX could not solve to

optimality are not included in the MIP solution time statistics). The table does not

report the time required to solve our different LP relaxation formulations, since the

OSP-AND LP relaxation is identical to the OSP-NDC LP relaxation, and these

times are therefore shown in Table 2-6.

Unlike our previous computational results for the OSP and the OSP-NDC, we

found several problem instances of the OSP-AND in which an optimal solution was

not found either due to reaching the time limit of one hour or because of memory

limitations. For the problem instances we were able to solve optimally, the MIP

solution times were far longer than those for the OSP problem. This is due to

the increased complexity resulting from the embedded 'p I.ig pi5 i ii1 in the

OSP-AND problem. Interestingly, however, in contrast to our previous results for

the OSP, the average and maximum MIP solution times for the OSP-AND were

smaller for the 200-order per period problem set than for the 25 and 50-order per

period problem sets. The reason for this appears to be because of the nearly non-

existent integrality gaps of these problem instances, whereas these gaps increase

when the number of orders per period is smaller.









Table 2-8: OSP-AND optimality gap measures.

Orders per Period
Gap Measurement 25 50 200 Overall Average
OSP-LP vs. MIP Solution" 0.3 !' 0.211' 0.0.' 0.21i' ,
ASF vs. MIP Solutionb 0.28 0.18 0.05 0.17
DASF vs. MIP Solution' 0.21 0.10 0.03 0.11
% Optimald 96.7 94.2 100 97
Note: Entries within each ~..I. Is per period" class represent average
among 360 test instances.
a(OSP-LP -MIP)/MIP x 100%.
b (ASF -MIP)/MIP x 100%.
S(DASF MIP)/MIP x 100%.
d % of problems for which CPLEX branch-and-bound found an optimal solution.

Table 2-9: OSP-AND solution time comparison.

Orders per Period
Time Measure (CPU seconds) 25 50 200
Average MIP Solution Time 42.0 67.9 21.9
Maximum MIP Solution Time 1970.1 1791.8 1078.8
Note: Entries represent average/maximum among 360 test instances.

Table 2-10 shows that once again our composite heuristic procedure performed

extremely well on the problems we tested. The percentage deviation from optimal-

ity in our solutions is very close to that of the OSP-NDC, and much better than

that of the OSP, with an overall average performance within 0.25'. of optimal-

ity. We note, however, that the best heuristic solution performance for both the

OSP-NDC and the OSP-AND occurred using the LP rounding heuristic applied

to the DASF LP relaxation solution. As Table 2-6 showed, solving the DASF LP

relaxation can be quite time consuming as the number of orders per period grows,

due to the size of this formulation. We note, however, that for the OSP-NDC and

OSP-AND, applying the LP rounding heuristic to the ASF LP relaxation solution

produced results very close to those achieved using the DASF LP relaxation solu-

tion in much less computing time. Among all of the 3,240 OSP, OSP-NDC, and

OSP-AND problems tests, the best heuristic solution value was within 0.i.7' of

optimality on average, indicating that overall, the heuristic solution approaches









we presented provide an extremely effective method for solving the OSP and its

variants.

Table 2-10: OSP-AND heuristic solution performance measures.

OSP-AND
Orders per Period Overall
Gap Measurement 25 50 200 Average
LAGR vs. UB" 3.95' 3.92' 0. :' 2.7 '
GUP vs. UBb 1.85 0.83 0.46 1.04
LPR vs. UBe 0.80 0.31 0.12 0.41
Best LBd 0.49 0.19 0.06 0.25
Note: Entries within each -.Il. I per period" class represent
average among 360 test instances
a (LAGR -UB)/UB x 100%.
b (GUP -UB)/UB x 100%.
(LPR -UB)/UB x 100%.
d Uses the best heuristic solution value for each problem instance.

2.5 Conclusions

When a producer has discretion to accept or deny production orders, determin-

ing the best set of orders to accept based on both revenue and production/delivery

cost implications can be quite challenging. For situations when no production

capacities exist, we show how the order selection problem can be solved using

a similar approach to the Wagner-Whitin [83] dynamic programming algorithm

employ, ,- for the ELSP. When facing production capacities, several variations of

the problem emerge, and we formulated and presented solution approaches to these

as well.

We considered variants of the problem both with and without fixed delivery

charges, as well as contexts that permit the producer to satisfy any chosen fraction

of any order quantity, thus allowing the producer to ration its capacity. We

provided three linear programming relaxations that produce strong upper bound

values on the optimal net profit from integrated order selection and production

planning decisions. We also provided a set of three effective heuristic solution

methods for the OSP. Computational tests performed on a broad set of randomly









generated problems demonstrated the effectiveness of our heuristic methods and

upper bounding procedures. Problem instances in which the producer has the

flexibility to determine any fraction of each order it will supply, and no fixed

delivery charges exist, were easily solved using the MIP solver in CPLEX. When

fixed delivery charges are present, however, the problem becomes more difficult,

particularly as the number of available orders increases. Optimal solutions were

still obtained, however, for nearly all test instances within one hour of computing

time when partial order satisfaction was allowed. When the producer must take

an all-or-nothing approach, satisfying the entire amount of each order it chooses

to satisfy, the problem becomes substantially more (1 i 1, -ii:- and the heuristic

solutions we presented become a more practical approach for solving such problems.

We expand our discussion of demand (or order) selection flexibility in a

production planning context over the next two chapters. Specifically, we will

introduce pricing as a decision variable in the requirements planning problem

in C'!i lter 3. Then we will consider the role that demand uncertainty pl ,l- in

demand source selection decisions in C'! lpter 4.

2.6 Appendix

Description of Feasible Solution Generator (FSG) Algorithm for OSP

This appendix describes the Feasible Solution Generator (FSG) algorithm, which

takes as input a solution that is feasible for all OSP problem constraints except the

production capacity constraints, and produces a capacity-feasible solution. Note that

we present the FSG algorithm as it applies to the OSP, and that certain straightforward

modifications must be made for the OSP-AND version of the problem.

Phase I: Assess attractiveness of additional setups

0) Let j denote a period index, let p(j) be the most recent production period prior

to and including period j, and let s(j) be the next setup after period j. If no

production period exists prior to and including j, set p(j) = 0. Set j = T and









s(j) = T + 1 and let Xj denote the total planned production (in the current,

possibly capacity-infeasible solution) for period j.

1) Determine the most recent setup p(j) as described in Step 0. If p(j) = 0, go to Phase

II. If Xp(j) < Cp(j), set s(p(j) 1) = p(j) and j = p(j) 1 and repeat Step 1 (note

that we maintain s(j) = j + 1). Otherwise, continue.

2) Compare the desired production in period p(j), Xp(j), with actual capacities over

the next s(j) p(j) periods. If Xp(j) > 3 Ct, and the sum of the revenues

for all selected orders for period j exceed the setup cost in period j, then add

a production setup in period j and transfer all selected orders in period j to

the new production period j. Otherwise do not add the setup in period j. Set

s(p(j) 1) = p(j), j p(j) 1, and return to Step 1.

Phase II: Transfer/remove least profitable production orders

0) Let dm,p(j),j denote the amount of demand from order m in period j to be satisfied

by production in period p(j) in the current (possibly capacity-infeasible) plan.

When reading in the problem data, all profitable order and production period

combinations were determined. Based on the solution, we maintain a list of

all orders that were satisfied, and this list is kept in nondecreasing order of

per-unit profitability. Per-unit profitability is defined as follows: Im,p(j),j

rmj pp(j) EC- ) ht Tj. We will use this list to determine the least desirable

production orders to maintain.

1) If no periods have planned production that exceeds capacity, go to Phase III. While

there are still periods in which production exceeds capacity, find the next least

profitable order period combination, (m*, p(j*), j*), in the list.

2) If Xp(j*) > Cp(j*), consider shifting or removing an amount equal to d*

min{dm*,p(j*),j*, Xp(j*) Cp(j*)} from production in period p(j*) (otherwise,

return to Step 1). If an earlier production period 7 < p(j*) exists such that

X, < C,, then move an amount equal to min (d*, C, X,) to the production










in period 7; i.e., dm*,r,j* = min (d*, C X,). Otherwise, reduce the amount of

production in period p(j*) by d* and set d*,p(j*),j* dm*,p(j*),j* d*.

3) Update all planned production levels and order assignments and update the number

of periods in which production exceeds capacity. Return to Step 1.

Phase III: Attempt to increase production in under-utilized periods

0) Create a new list for each period of all profitable orders not fulfilled. Each list is

indexed in nonincreasing order of per-unit profitability, as defined earlier. Let j

denote the first production period.

1) If j = T + 1, STOP with a feasible solution. Otherwise, continue.

2) If Cp(j) > Xp(j), excess capacity exists in period p(j). C'!, ...-- the next most

profitable order from period j, and let m* denote the order index for this order.

Let dm*,p(j),j min {dm*,j, Cp(j) Xp(j) }, and assign an additional dm*,p(j)j to

production in period p(j).

3) If there is remaining capacity and additional profitable orders exist for period j, the

repeat Step 2. Otherwise, set j = j + 1 and return to Step 1.















CHAPTER 3
PRICING, PRODUCTION PLANNING, AND
ORDER SELECTION FLEXIBILITY

3.1 Introduction

Firms that produce made-to-order goods often make pricing decisions prior

to planning the production required to satisfy demands. These decisions require

the firm's representatives (often sales/marketing personnel in consultation with

manufacturing management) to determine prices, which imply certain demand

volumes the firm will need to satisfy. Such pricing decisions are typically made

prior to establishing future production plans and are in many cases made based on

the collective judgment of sales and marketing personnel. This results in decisions

that do not account for the interaction between pricing decisions and production

requirements, and how these factors affect overall profitability. Lee [44] recently

noted that one of the common pitfalls of supply chain management practice occurs

when those who influence demand within the firm (e.g., marketing, sales) do not

properly account for operations costs in demand 1p1 .lii I. while supply chain man-

agers fail to recognize that demand is not completely determined exogenously. He

argues that integrating supply and demand-based management offers great oppor-

tunity for future value creation and serves as "the next competitive battleground in

the 21st century."

Since production environments often involve significant fixed production costs,

j -1 iVi.-; these fixed costs requires a demand level at which revenues exceed not

only variable costs, but the fixed costs incurred in production as well. Decisions

on the demand volume the organization must satisfy, and the implied revenues

and costs, can be a critical determinant of the firm's profitability. Past operations









modeling literature has not fully addressed integrated pricing and production

planning decisions in make-to-order systems with the types of nonlinear production

cost structures often found in practice as a result of production economies of

scale. We offer modeling and solution approaches for integrating these decisions in

single-stage systems.

Most of the requirements planning literature focuses on production require-

ments based on pre-specified demands, with no adjustments for price flexibility.

In this chapter, we introduce a requirements planning model that implicitly deter-

mines the best demand levels to satisfy in order to maximize contribution to profit

when demand is a decreasing function of price. In other words, the firm will select

the demand level to satisfy by setting a single price for the product.

We make several contributions to the literature through our model and

solution approaches introduced in this chapter. First, our combined pricing and

production planning model permits multiple price-demand curves in each period,

which effectively represents the possibility of offering different prices in different

markets, where each market has a unique response to market price. Moreover,

this model generalizes the order selection approach presented in C'! lpter 2, where

a firm faced a set of customer orders, from which it selected the most profitable

subset. In the order selection context, we can use our requirements planning

with pricing model and apply a unique price to each order, rather than a single

price for all demands. Our solution approach also accommodates more general

production cost functions than previously considered in the requirements planning

and pricing literature, along with explicit consideration of both general concave and

piecewise-linear concave revenue functions.

Given fixed plus linear production costs and piecewise-linear concave revenue

functions, we also provide a 'tight' linear programming formulation of our model,

using a dual-based solution approach to show that this formulation has zero duality









gap. This result, and the formulations discovered while developing the approach,

pl ill I a key role in formulating the relaxations used in solving the capacitated

OSP models in ('! Ilpter 2, where production capacities varied over time. Our final

1 ii, contribution also addresses a capacitated version of the model. Assuming

time-invariant production capacity limits and piecewise-linear concave revenue

functions in the total demand satisfied, we show that this problem can be solved in

polynomial time.

Given the recent emphasis on differential pricing and demand management

in manufacturing (e.g., Lee [44], Chopra and Meindl [22]), these models and

associated solution approaches have the potential for broad application in practice.

Analytics Operations Engineering, Inc., an operations strategy and execution

consulting firm, recently cited application contexts in the specialty papers and

timber industries in which integrated pricing and production planning models such

as the ones we discuss can add substantial value in practice (for more details on

these applications, please see Burman [17]).

Thomas [74] provided an analysis and solution algorithm for a related inte-

grated pricing and production planning decision model. His model generalized the

Wagner & Whitin [83] model by characterizing demand in each of a set of discrete

time periods as a downward-sloped function of the price in each period, thus treat-

ing each period's price as a decision variable. The model proposed by Thomas [74]

sets only a single price for all demands in any given period, whereas our model

permits differential pricing in different markets. Moreover, we demonstrate that

a 'tight' linear programming formulation exists for this problem under piecewise-

linear concave revenue functions. We also extend the analysis to account for more

general production cost functions in each period.

Additional contributions to the integrated pricing and production planning

problem include the work of Kunreuther and Schrage [41] and Gilbert [33], who









considered the problem when a single price must be used over the entire horizon.

Kunreuther and Schrage [41] provided bounds on the optimal solution value

under time varying production cost assumptions, while Gilbert [33] assumed time-

invariant production setup and holding costs and provided an exact polynomial-

time algorithm. Recall the paper by Loparic, Pochet, and Wolsey [50] that we

introduced in Chapter 2. They considered a problem in which a producer wishes

to maximize net profit from sales of a single item and does not have to satisfy all

outstanding demand in every period. Their model contains no pricing decisions,

effectively assuming that only one demand source exists in every period, and that

the revenue from a single demand source is proportional to the volume of demand

satisfied. In contrast, we allow revenue to be a general concave nondecreasing

function of the amount of demand satisfied, which is consistent with a downward-

sloped demand curve as a function of price. Also discussed in C'! lpter 2 was

the paper by Lee, Qetinkaya, and Wagelmans [43], in which they introduce a

production planning model with demand time windows. While their model assumes

that all pre-specified demands must be filled during the planning horizon, our

approach implicitly determines demand levels through pricing.

Bhattacharjee and Ramesh [13] considered the pricing problem for perishable

goods using a very general function to characterize demand as a function of price.

They also assumed upper and lower bounds on prices, characterized structural

properties of the optimal profit function, and developed heuristic methods for

solving the resulting problems. Biller, C'!i i, Simchi-Levi, and Swann [14] analyzed

a model similar to ours under strictly linear production costs (i.e., no fixed setup

costs, and assuming time-varying production capacity limits), which they solved

efficiently using a greedy algorithm. While our discussion of the relevant literature

has focused on deterministic approaches for integrated pricing and production

planning problems, some additional work on dynamic pricing exists that addresses









stochastic demand environments; for past work on integrated pricing and produc-

tion/inventory planning in a stochastic demand setting, please see Thomas [75],

Gallego and van Ryzin [30], and Chan, Simchi-Levi, and Swann [21].

The remainder of this chapter is organized as follows. Section 3.2 presents

a formal definition and mixed integer programming formulation of the general

requirements planning problem with pricing. In this section we provide our solution

approaches for this problem, the first of which extends the Wagner-Whitin [83]

shortest path solution method (discussed in Chapter 2) to contexts with general

concave revenue functions and fixed-charge production costs. Assuming piecewise-

linear concave revenue functions, we then provide a dual-based polynomial-

time algorithm for solving the uncapacitated problem. This dual-based solution

approach allows us to show that the problem reformulation in Section 3.2.2 has

a linear programming relaxation whose optimal value equals that of the optimal

mixed integer solution; i.e., the problem formulation is "tight". We also explore

the generality of our solution approaches with respect to different functional forms

for the production cost functions and under multiple market price-demand curves

in any given period. In addition to presenting solution approaches to several

incapacitated versions of the problem, we provide an analysis of the equal-capacity

version of the model under piecewise-linear concave revenue functions. Section

3.3 discusses different pricing interpretations from our models, and illustrates how

our pricing model can be cast as an equivalent "order selection" problem, thus

broadening its potential for application in practice.

3.2 Requirements Planning with Pricing

Consider a producer who manufactures a good to meet demand over a finite

number of time periods, T. The production cost function in period t is denoted

gt(), and is a nondecreasing concave function of the amount produced in period

t, which we denote by xt. Similarly, the revenue function in period t is denoted









by Rt(-), and is a nondecreasing concave function of the total demand -./I.:- i, in

period t, which we denote by Dt, with Rt(0) = 0 for all t = 1,..., T. We assume

that Dt, the total demand satisfied in any period t, is the sum of the demands

satisfied from some i., distinct markets. In each market we employ a standard

assumption of a one-to-one correspondence between price and market demand

volume in any period, where market demand is a downward-sloping function of

price (see Gilbert [33]), and each market's revenue is a nondecreasing concave

function of demand satisfied in the market. Given a total demand value of Dt in

period t we solve an optimization subproblem to determine a price value in period

t in every market m (equivalently, Dt = 1 dmt(Omt) where Ot is the price in

market m in period t and dt() is the total demand in market m in period t as

a function of price). Section 3.2.3.1 discusses how to determine the price in each

market in period t given a demand volume of Dt; for now it is sufficient to simply

consider the decision variables for the total demand in each period (i.e., the Dt

variables).

Inventory costs are charged against ending inventory, where ht denotes the unit

holding cost in period t and It is a decision variable for the end-of-period inventory

in period t. Letting C denote the production capacity limit (which does not depend

on time), we formulate the requirements 'l."i.".'::, with pricing (RPP) problem as

follows.

[RPP]


maximize zf1 (Rt(D) (t(xt) + hjlt))

subject to: Dt + It = t + It- 1= ,..., T, (3.1)

xt < C t 1,...,T, (3.2)

xt, It,Dt > 0 t =1,...,T. (3.3)









The objective function maximizes net profit after production and holding

costs; constraint set (3.1) ensures inventory balance in all periods and constraint

set (3.2) enforces production capacity limits. The general RPP problem defined

above maximizes the difference between concave functions and is, therefore, in

general a difficult global optimization problem (see Horst and Tuy [37]). By

providing certain somewhat mild restrictions on the functional forms of the revenue

and production cost functions, Rt(Dt) and gt(xt), we arrive at a family of special

cases of the RPP problem, several of which have broad applicability in practice.

Consistent with the vast in i i liy of past production planning literature,

except where specifically noted, we henceforth assume that production costs

contain a fixed-charge structure; i.e., a fixed cost of St is incurred when performing

a production setup in any period t, while the variable cost per unit in period t

equals pt (we later discuss in Section 3.2.3.2 the necessary extensions to handle

production costs that contain a more general piecewise-linear nondecreasing

concave cost structure). Under fixed plus linear production costs, unlimited

production capacity, and a single price offered to all markets in each period we have

the model first analyzed by Thomas [74], who proposed a dynamic programming

recursion for solving the problem. The algorithm is similar to the Wagner-Whitin

[83] algorithm for the ELSP, and relies on similar key structural properties of the

problem. These properties include the zero inventory ordering (ZIO) property (if

inventory is held at the end of period t 1 then we do not perform a setup in

period t). The following section describes an equivalent shortest path algorithm

(refer to Section 2.2.2 for a complete discussion) for this problem, along with an

explicit characterization of the solution approach under concave revenue functions.

While the shortest path method we present is generally equivalent to the dynamic

programming method proposed by Thomas [74] when production costs contain a

fixed-charge structure, we depart from this work in the following respects:









(i) we provide an exact solution approach for contexts in which total revenue is

concave and nondecreasing in the amount of demand satisfied;

(ii) we show that the shortest path method generalizes to cases with multiple

demand sources, each with a unique concave revenue curve; and

(iii) we show how to generalize the shortest path approach to provide an exact

procedure for the case of piecewise-linear and concave production costs.

Thus, the following section lays the foundation for subsequent generalizations of

our solution methodology to broader contexts.

3.2.1 Shortest Path Approach for the Uncapacitated RPP

Retaining our assumption of a fixed-charge production cost structure and

assuming the revenue function Rt(Dt) in every period t is a general nondecreasing

concave function of Dt with Rt(0) = 0, we now update the Wagner-Whitin

[83] shortest path approach (introduced in C'! plter 2) for the uncapacitated

RPP problem. Note that under these assumptions, for any fixed choice of the

demand vector (D1, D2,..., DT), the resulting problem is a simple ELSP. Now,

we can decompose the T-period RPP problems into a set of smaller contiguous

interval subproblems, using the shortest path graph structure previously shown

in Figure 2-2. To illustrate the computation of arc length c(t, t'), where a setup

is performed in period t and the next setup occurs in period t' > t, we solve the

period t,..., t' 1 subproblem of maximizing net profit in these periods. This

period t,..., t' 1 subproblem can be stated as

t'-1- t- t- 1
maximize: R, (D,) hT EY -+ DD) pt E 1 Dj (3.4)

subject to: D, > 0 r t... ,' 1. (3.5)


This decision problem separates by period, and since we are maximizing a set of

nondecreasing concave functions, we arrive at the following characterization of the

optimal amount of demand to satisfy in period r, given a most recent setup in









period t. For notational convenience we let pt+ E- hj denote the cost per

unit of demand satisfied in period 7 using a setup in period t < T.

Theorem 1 For the uncapacitated RPP, given a production setup in period t only,

if a demand ;',;, i./.:/ D, exists such that I,- is in the set of subgradients of R,(.) at

D,, then D, is an optimal demand ,;.r ,'.i/.:/i for the subproblem given by (3.4) and

(3.5).

A proof of Theorem 1 can be found in Appendix A of Section 3.5. Note that if

R,(.) is everywhere differentiable with limD,, R'(D) < i,- < limDio R(D),

then the optimal demand quantity as stated in the theorem can be determined by

finding D, such that R'(D,) = ,-.

Given any t < r < t' 1, if a D, > 0 exists that satisfies the condition of

Theorem 1, then the optimal value of D, for the subproblem, which we denote by

D*(t), equals this demand value. Otherwise, assuming a finite (non-negative) value

of ',-, we must have either D*(t) = 0 (if all subgradients at all D, > 0 are less than

, -) or D*(t) = oo (if a subgradient exists for each D, > 0 that is greater than -).

Then the maximum possible profit in periods t,..., t' 1 (assuming the only setup

within these periods occurs in period t, which we denote by I(t, t')) is given by

t'-1 t7-1 tI-1
n(t, ) R- D(t) -ptYED t) St, (3.6)
T=t j=7+1 j=t

and the arc length for arc (t, t') is therefore given by


c(t, t') max {0, n(t, t')}. (3.7)


With appropriate preprocessing and recursive computations of the H(t, t') values,

we can determine all H(t, t') values in O(T2) time. As discussed previously, the

longest path on an .. l 1,i network can be found in O(T2) time in the worst case

(see Lawler [42]). Therefore, the overall solution effort is no worse than O(T2).









We next consider a particular special case of the concave revenue functions,

which we will use for more detailed analysis in subsequent sections. Suppose

that the revenue function in each period can be represented as a nondecreasing

piecewise-linear concave function of demand. We assume that the revenue function

in period t has Jt + 1 consecutive (contiguous) linear segments. The first Jt of these

segments have interval width values dut, d2t,... dt, and we let rjt denote the slope

(per unit revenue) within the jth linear segment; the (Jr + I)st segment has slope

zero (i.e., the maximum possible total revenue is finite with value J' rTjtdjt for

t = 1,..., T). This implies that we can state our revenue functions as follows:

k- k-1 k-i k
Srjtdjt + rkt Dt djt for Z djt < Dt < C djt,
j=1 j=1 j=1 j=1
R(D) = k= ,..., J, (3.8)
Jt Jt
E rjtdjt for E d4j < D-
j=1 j=1

where rit > r2t > > rit > 0. Theorem 1 implies that an optimal solution

exists such that the total demand satisfied in each period t occurs at one of

the breakpoint values; i.e., at -1i djt for some k between one and Jt (note

that an optimal demand value cannot exist in the (Jt + 1)st interval if costs are

positive, which we assume throughout, since costs will increase and revenues remain

constant). Denote such a value of D, by D*(t). Then,


c(t, t') = max (0, ( (Rt(D(t)) -D(t)) t


The time needed to compute these values is O(T2) multiplied by the time required

to find D*(t) and evaluate Rt(Dn(t)) for all t, r. Note that if the functions Rt()

are piecewise-linear and concave with at most Jmax segments, the slopes at each

breakpoint and the resulting Rt(D*(t)) computations can be performed in O(Jmax)

time, for a total arc 'cost' calculation time of O(JmaxT2). Since the ., i-, i. longest









path problem requires O(T2) operations, our total solution time is no worse than

O(JmaxT2).
3.2.2 Dual-ascent Method for the Uncapacitated RPP

When the revenue functions are piecewise-linear and concave in every period,

and production costs contain a fixed plus variable cost structure, we can also use

a dual-based algorithm to solve the uncapacitated RPP, which we next describe.

This approach requires first reformulating the RPP. As we later show, this new

formulation is "tight"; i.e., its linear programming relaxation objective function

value equals the optimal objective function value of RPP. We begin by providing an

explicit base formulation of the uncapacitated RPP under piecewise-linear concave

revenue functions and fixed plus linear production costs, using much of the notation

already defined in the previous sections. We define a set of binary variables zjt for

t = 1,..., T and j = ,..., Jt, such that zjt 1 if Dt > Y i dtM (i.e., when

the total demand satisfied in period t occurs at the jth or higher breakpoint of the

piecewise-linear concave revenue curve); otherwise zjt = 0 when Dt < CYj: dkt.

By the definition of the zjt variables and the fact that an optimal solution

exists where total demand falls at an interval breakpoint in each period, we

therefore have that the total demand satisfied in period t equals Dt = E 1 djtzjt,

and the corresponding total revenue equals j3 1 rjtdjtzjt. We next define a new set

of binary setup variables, yt, for t = 1,..., T, where yt = 1 if we perform a setup

in period t, and yt = 0 otherwise. We can thus formulate the uncapacitated RPP

with piecewise-linear concave revenue functions, which we refer to as the RPPPLc,

as follows.









[RPPPLC]

maximize: _t (Ejl 1 rjtdjtzjt Stt ptxt -htIt

subject to: It-1 + = j1 djtZjt + It t 1,.. T, (3.9)
0-- (E tEj ldi t- l,...,r, (3.10)

lo = 0, It > 0, t 1,...,T, (3.11)

0 < Zjt < 1 t = 1,...,T,

j 1, J (3.12)

t E {0,1} t =1,...,T. (3.13)

In the above [RPPPLC] formulation, the objective function provides the net revenue

after subtracting production and holding costs. Constraint set (3.9) ensures

inventory balance, while the setup forcing constraints (3.10) enforce setting yt equal

to one if any production occurs in period t. Note that the coefficient of yt in these

constraints equals the total demand from period t through T, thereby effectively

leaving the problem uncapacitated. Constraints (3.11) through (3.13) encode

our variable restrictions. Since an optimal solution exists for the uncapacitated

version of the problem such that the demand satisfied in any period occurs at one

of the breakpoint values of the period's revenue function, [RPPPLC] provides the

same optimal solution value as the formulation obtained by explicitly imposing

the binary restriction on the zjt variables. We formulate the problem with the

relaxed binary restrictions, however, for later extension to the equal-capacity case

in Section 3.2.4.

Note that we have not imposed any specific constraints on the relationship

between zjt variables corresponding to the same revenue function in a given

period t. The following property allows us to consider each of the intervals of the

piecewise-linear concave revenue function independently from one another in our









mixed integer programming formulation (that is, we need not introduce any explicit

constraints in our formulation that specify the strict ordering of the piecewise-linear

segments of the revenue functions).

Property 1 Contiguity Property: For the [RPPpLc] problem I. J7, ., above, if

an optimal solution exists such that zj-,t = 0, then Zkt = 0 for k = j,..., J in ;.1

optimal solution.

Proof: Suppose that an optimal solution exists with objective function value Z*

with Zkt = 0 and zt = 1 for some 1 > k, and let period s < t denote the setup

period in which the production occurred that satisfied demand in period t. Since

zut 1,- we must have that rut > ps + t 1 h,; otherwise a solution exists such

that z t 0 with objective function value greater than Z*, which contradicts

the optimality of the solution with zut 1. Since rkt > rlt we must also have

rkt > Ps + Y :1 h, and a solution exists with kt = 1 and an objective function

value greater than Z*, a contradiction of the optimality of the solution with kt = 0

and z=t 1, which implies that if Zkt = 0 in an optimal solution zlt must equal zero

for 1 k + ,... Jt in any optimal solution (i.e., the contiguity property). )

We can also use the arguments in the contiguity property proof to show

that if Zkt = 1 in an optimal solution, then we must also have zjt = 1 for

j = 1,..., k The contiguity property thus ensures that the quantities

jl djtijt and j-it rjtdjtzjt correctly provide the total demand satisfied and the
total revenue in period t, without the need to introduce any explicit dependencies

among the zjt variables in our mixed integer programming formulation.

While the [RPPPLc] formulation correctly captures the RPPPLC problem

we have defined, its linear programming relaxation value does not necessarily

equal the optimal value of the RPPPLC; i.e., its integrality gap is not necessarily

zero. We next derive an equivalent problem formulation for which the integrality

gap is indeed equal to zero. We show this by developing a dual-ascent algorithm









for the dual of this formulation that provides an optimal dual solution whose

complementary primal solution is feasible for all of the integer restrictions of the

[RPPPLc] formulation. We note that this approach generalizes a related approach

for the ELSP developed by Wagelmans, van Hoesel, and Kolen [82]. Alternative

approaches also include extending the proof techniques for the reformulated ELSP

shown in Nemhauser and Wolsey [58], B ,1 i-il, Van Roy, and Wolsey [10], and

B II ili, Edmonds, and Wolsey [9].

Starting with the [RPPPLc] formulation, we can equivalently state the

objective function as:


minimize: t 1 (Stt + PtXt + httI) T- E 1 Tltzt (3.14)


Since It = t1 X1 t 1 LJ 1 djzj,, we can eliminate the inventory variables

from the formulation via substitution. We next introduce a new cost parameter, ct,

where ct = pt + L=t h". The objective function of the RPPPLC can now be written

as:

T T tJ TJ,
minimize: (Styt + ctxt) E hE E djzj, EE rjrdjzjr (3.15)
t=1 t=1 -T=lj=1 T=lj1=

We next define pjt as a modified revenue parameter for linear segment j in period t,

where pjt -= r=t h, + rjt. The development of our dual-ascent procedure requires

capturing the exact amount of production in each period that corresponds to the

amount of demand satisfied within each linear segment of the piecewise-linear

revenue function in the current and all future periods. We thus define xjt. as the

number of units produced in period t corresponding to demand satisfaction within

linear segment j in period T, for r > t, and replace each xt with L t ~ji 1 xjtr.

We next provide a reformulation of the LP relaxation of the RPPPLC, which we

denote by [RPP'LC], that lends itself nicely to our dual-based approach.









[RPPPLC]
T T Jr \ T Jt
minimize: (Stt + Ct jt )- pjtdjtzjt
t=1 Tt j=1 t=1 j=1

T
subject to: E xjtr dJzJ, =0 1,..., T,j 1,...,Jr, (3.16)
t= 1
djryt xjt, > 0 t = 1, T,

S=t,..., T,j = 1, ..., J, (3.17)

-zJ, > -1 1,...,T,j ,...,J, (3.18)

Yt, xjt, Zjt > 0 t 1,...,T,

7- t,...,T,j 1 ,..., Jr. (3.19)


Recall that we introduced a very similar formulation ([UOSP']) in C'! plter 2 for the

purpose of developing heuristic solution approaches to the OSP problem. In this

section, we di- i::regate the setup forcing constraints (2.13) from [UOSP'] to arrive

at the above formulation [RPPPLCI.

Note that if zjt = 1 in a solution we -- that the demand corresponding to

segment j in period t is .,/.:l/7 ,1 in the corresponding solution. This manner of

describing the solution will facilitate a clearer description of our formulation and

the dual algorithm and solution that later follow. Constraints (3.16) ensure that

if the demand in segment j in period 7 is satisfied, then a production amount

equal to this demand must occur in some period less than or equal to 7. If any

production occurs in period t, constraint set (3.17) forces yt 1, thus allowing

production in period t for segment j demand in period 7 to equal any amount

up to dj,; otherwise, if yt 0, no production can be allocated to period t.

Constraints (3.18) and (3.19) represent the (relaxed) variable restrictions. Note

that since a positive cost exists for setups, we can show that the constraint yt <

1 is unnecessary in the above relaxation, and so we omit this constraint from









the relaxation formulation. It is straightforward to show that the [RPPPLc]

formulation with the additional requirement that all yt are binary variables is

equivalent to our original RPPPLC.

To formulate the dual of [RPP'PLCI, let ijT, Ii,t-, and 7j, denote dual multipli-

ers associated with constraints (3.16), (3.17), and (3.18), respectively. Taking the

dual of [RPP'Lc], we arrive at the following dual formulation [DP]:

[DP]


maximize: Y 1 T j -Tjrj
T J,
subject to: EE ,1,-,,t- < St t= ,...,T, (3.20)
T=tj=
fl-r- ',,- < ct t 1,... ,T,

S=t,... ,T,j = 1,... ,J (3.21)

-dJPJ < -Pjddj, = 1,..., T, = 1,... J,-, (3.22)

7rj, ,'t- > 0 ; pijunrestricted t= 1,...,T,

7r t,...,T,j 1,..., J, (3.23)


Inspection of formulation [DP] indicates that we can set i',t- equal to the max-

imum between 0 and p j ct without loss of optimality; similarly, an optimal

solution exists with -Tj, equal to the minimum between 0 and dj,(pj, pj-). The

above formulation can, therefore, be re-written in a more compact form as:

[CDP]

T J,
maximize: EE min (0, dj T(pjr pjI))
T=1 j=1
T J,
subject to: YE dj, {max (Oj, ct)} < S, t = ,...,T. (3.24)
T =tj =

We note some important properties of the [CDP] formulation. First, we have no

incentive to set any Pij variable value in excess of pj-, since any increase above this

value does not affect the objective function value. Second, we can initially set each









/jT min {ct} for all r = 1, ..., T and j = 1,..., Jr, without utilizing any of
t 1,...yT
the "( S1' 1 '1y St, in each constraint. We can also eliminate any segment-period

combination (j, r) such that min {ct} > pj,, since any demand satisfied within
t 1,...,T
such a segment will never provide a positive contribution to profit. In describing

our solution approach, we will refer to the constraint for period t in [CDP] as the

tth constraint (or constraint t) of the formulation. Our approach for solving [CDP]

is to use a dual-ascent procedure that increases the dual variables in increasing

time index order. That is, we increase the values of the fjl variables before we

increase any ijt values for t > 1. We then focus on increasing the i/.. variables,

and so on. We begin by simultaneously increasing the value of all Pjl variables.

If for some segment I in period 1, p i reaches a value of pll before constraint 1

becomes tight, we -,i that this segment "drops out" in period 1 and we do not

further increase the value of / 1 (i.e., /il is fixed at pl in the solution). We then

continue to increase all other fjl values until constraint 1 becomes tight. Let Jfo

denote the set of all segments that drop out in period t, and let J1 denote the set

of all segments that do not drop out in period t. We define p1 as the value of pjl

for all segments that do not drop out in period 1, where

S1 Ejjo djl max (0, pjl ci)
i Ci + j djl

Note that at this point, after determining p, the first constraint of [CDP] is tight

(assuming J, / 0; we later discuss the necessary modifications if J) = 0). We

next focus on increasing the i/._ variable values. When we increase the values of the

/._. variables, these variables can be blocked from increase by either dropping out

(i.e., when P2 P12 for some segment 1), by tightening constraint 2, or by hitting

the value cl (observe that no i/.. value can be greater than cl since constraint 1 is

already tight, and such a value would, therefore, violate constraint 1). Letting p2










equal the value of i _. for all j E J, we have

SS2 YEjjo .: max (0, p2 C2)
2 min cl; 2 + -jj4-


Applying this same approach in period 3 produces

S2 E d2, i -)- E dj2max (P-c2,0)- E dj31.. -C2)
jeJg jeJ^ jeJ3
cl; c2 + E- dj3
m3 Sin 3- dFE 1.. 2..." -C3)
C3+ -
C3 E dj3
j JC

or in general, for period r:

si- Ed dit .. -ci)-E E djtmax(pQ-ci,0)
t*i jE6 o t=i j(EjI
pI- min c+ }=-J- (3.25)
i

Our final dual solution takes the form:


{ Pjr, J E o r
IjT j for 7- 1 ,..., T, and j= 1,...,J.


Note that it is possible that the set J, is empty for some '- after applying the

algorithm, since all orders in period 7 may drop out before hitting any of the

constraints. In such cases p* requires no definition. We can summarize this dual-

ascent solution approach as follows:

CDP Dual-Ascent Solution Algorithm

0. Delete any segment-period combination (j, r) such that min {ct}> pjr.
t 1,...IT
1. Set pij = min {ct for all 7 1, ..., T and j =1,..., Jr. Set iteration
t 1,...,T
counter k = 1.

2. Let JO = J {0}. Simultaneously increase all Pjk for j = ,..., ,

from the initial value of min {ct}. If, while increasing the Pjk values,
t 1,...,k
some Plk Pl1k before the Pjk values are blocked from increase by any









constraint, fix Plk at Plk, insert segment I into J0, and continue to simul-

taneously increase Pjk for all j 0 J until some constraint 7(k) < k

blocks the pjk values from further increase. When constraint 7(k) < k

blocks the pjk values from further increase then, for all segments j O

insert j into and set i;., using equation (3.25); i.e., set Pjk
k k-1
S (4)- E E djt max 0,pjt-c((k))- E E djt max(pl -c(),0)
t-(2) 2tmxJ t=()jJ (If all
C7(k) + 2 d-k t (If all

j 1,... Jk enter J0 before some constraint becomes tight, then p* re-

quires no definition.)

3. Set k = k + 1. If k = T, stop with dual feasible solution. Otherwise, repeat

Step 2.

Note that in each period k we must check the value of pjk for each segment

j = 1,..., Jk and determine whether this value of Pjk will tighten or violate any

of the constraints 1, ..., k. Since we need to apply this comparison for k= 1, ...,

T, we can bound the complexity of this dual-ascent algorithm by (JmaxT2), the

same as that of the shortest-path algorithm in the previous section. We next show

that the dual-ascent solution procedure outlined above not only solves [CDP], but

also leads to a primal complementary solution in which all of the binary restrictions

in formulation [RPPPLc] are satisfied; i.e., the dual-ascent procedure solves the

RPPPLc. Before showing this, we first need the following lemma.

Lemma 1 For i~'; pair of positive integers 7 and I such that 7 + I < T and p* and

#/+* are /, 1,,, as in the dual-ascent il,>rithm, we '... -.;7lq have /p > # +i.

Proof: Let k < 7 be such that

Sk EtCk E ejo djl max (0, pit Ck) 1 Ej dj max Ck, 0)
PT = Ck + -- t- -

from which we can conclude that p* > Ck (since the numerator on the right hand

side is the slack of constraint k, which must be nonnegative, since we maintain dual









feasibility at all times). Next consider p*+.*:

S k E' Ei o djl max (0, pjt ) -1 Zj i max ( -( C)k, 0)
Pr+I < Ck+ --j-j-+ dj,7-+l
ZJCJI d +

Since p* tightens constraint k, the quantity in the numerator above must be zero

and we therefore have /*+1 < Ck < p* for all 1,..., T and '- + 1 < T, since '-

was chosen arbitrarily. <

Lemma 1 is required for proving the following result, the proof of which can be

found in Appendix B of Section 3.5.

Theorem 2 The dual-ascent ,l.' -.:thhm presented above solves [CDP/. Moreover,

the complementary primal solution to the dual solution produced by the dil.>rithm

l.,/7. 4 the ':,/. ,'il:l;, restrictions of the RPPPLC and therefore provides an

optimal solution for the RPPPLC.

Theorem 2 implies that formulation [RPPPLc] is tight, and we can easily find

the solution value for the RPPPLC using a linear programming solver. The algo-

rithms we have developed, however, have better worst case complexity ( (JmaxT2))

than solution via linear programming. To provide some insight on the structure of

the primal solution, given the dual solutions, we can show that the tight constraints

in the dual solution correspond to periods in which we setup in the complementary

primal solution. Further, if Pjk = p, then the demand in segment j in period k

is satisfied using the setup corresponding to the constraint that blocked p* from

further increase (i.e., period 7(k) from Step 2 of the dual-ascent algorithm).

As was shown in C'! lpter 2, we cannot reduce this bound to O(T log T), as

Federgruen and Tzur [24] and Wagelmans, van Hoesel, and Kolen [82] do for the

ELSP, since we cannot ensure that cumulative demand satisfied as we increase the

number of periods in a problem instance is nondecreasing. See Section 2.2.2 for a

presentation of a counterexample.









3.2.3 Polynomial Solvability for Other Production Costs and Price-Demand Curves

To this point we have made two sets of key assumptions that have facilitated

providing polynomial-time solution methods for the uncapacitated RPP. The first

of these assumptions relies on the production cost function taking a fixed-charge

structure in each period, while the second assumes that a single price-demand

curve exists in each period. We next explore the degree to which we can relax these

assumptions, while retaining our ability to apply the polynomial-time solution

methods we have presented. First we consider contexts in which multiple price-

demand response curves exist in each period; this would correspond to contexts in

which the producer has multiple available markets in which to sell its output, with

each market having a unique response to price. We then consider the impacts of a

piecewise-linear concave production cost structure (which may include a fixed setup

cost) in each period.


3.2.3.1 Multiple price-demand curves

In this section we show that any uncapacitated RPP with multiple demand

curves in a period can be reformulated as an RPP with only a single demand curve

per period. We will show that this holds for general concave revenue functions and

for piecewise-linear concave functions in particular. This implies that the piecewise-

linear concavity of the revenue functions is preserved under the transformation

from a multiple demand curve per period problem to a single demand curve per

period problem. Suppose we now have if, distinct revenue functions in period t,

each corresponding to a distinct revenue source, and that Dt is now the decision

variable for the amount of demand we satisfy for revenue source m in period t;

Rt(Dt) is the revenue function associated with revenue source m in period t

(a revenue source may be an individual market or customer). We can rewrite the









uncapacitated RPP as


maximize 1 mtDm) E (g (xt) + hlt)

subject to: EMt Dmnt D

Dt + It= xt + It-1

xt, It, Dt > 0

Dmt > 0


t= 1,... T,

t 1,... T,

t 1,... ,T ,

m 1,...,311,,t 1,... ,T.


Now observe that, for a given choice of Dt, we will choose the demand quantities

for each revenue source that yield the maximum profit. So the uncapacitated RPP

is equivalent to


maximize 1 Rt (Dt) (t(t) + htt)

subject to: Dt + It = xt + It-1

xt, It, Dt > 0


t 1,...,T ,

t =1,..., T.


where the ag/;, ./ l. revenue function for period t, Rt(Dt), is defined through the

following subproblem (SP) as

[SP]

(Mt Mt
Rt(Dt) max Rmt (D^t): Dmn = Dt; D > 0, m = 1,..., 11, .
m=1 m= 1

The function Rt(Dt) is concave (see Rockafellar [66] Theorem 5.4), and clearly

Rt(0) = 0. It now also easily follows that if Rt(.) is piecewise-linear and concave

(and Rt(0) = 0) for all m and t, Rt(-) is piecewise-linear and concave for all t (and

Rt(0) = 0). This can be shown by ordering the slopes of all segments in a given

period in decreasing order, and noting that the function Rt(Dt) will ii-, these

segments in nondecreasing index order (or nonincreasing value order).









Observe that if the Rmt(') functions are all everywhere differentiable, then the

demand values selected for each revenue source in a given period t as a result of

solving subproblem [SP] will be such that R't(Du) = R't(D2t) .= R (D

In other words, at the optimal demand level, the marginal revenue for each revenue

source will be equal. Thus, if the revenue sources are distinct but have identical

revenue functions, we will of course charge the same price to every revenue source.


3.2.3.2 Piecewise-linear concave production costs

We next consider the case in which the production cost function in each

period is piecewise-linear concave and nondecreasing in the production volume

in the period. Note that any nondecreasing piecewise-linear concave function can

be viewed as the minimum of a number of fixed-charge functions. Therefore, if

the production functions are piecewise-linear and concave with a finite number

of segments, we can view this as a choice between a finite number of alternative

production modes. It is easy to see that, in any period, we will of course only use a

single production mode without loss of optimality.

We can write such a production cost function in the following form:


gt() 0 if x = 0,
mink ,...,t {Skt + Pktx} if x > 0,

where k denotes an index for different production "modes". Given a sequence

of periods t,..., t' 1 and positive production in period t, we now essentially

also need to choose which of the it cost functions (or production modes) to use.

Given a production setup in period t only, the unit production plus holding cost

associated with period T (r = t,..., t' 1) under production mode k equals

, ,- pkt + ~sY hs. As with our previous analysis and development of the shortest

path algorithm (see Theorem 1), the optimal quantity of demand satisfied in period









- under production mode k using a setup in period t, which we denote by D((t),

is then equal to any value of D such that i,,- is in the set of subgradients of R,(D)

at D. Let
tf-1
,tt)= (Rt(,(t)) D, (t)) Skt
T-=t
and

c(t, t') = max 0, max k(t, t1)

The value of IIk(t, t') provides the maximum profit possible in periods t,..., t' 1

under production mode k assuming we satisfy demand amounts of D,(t) for

S= t,... ,t' 1. As a result, c(t,t'), as before, provides the maximum possible

profit in periods t,..., t' 1 assuming the only setup that can satisfy demand

in these periods must occur in period t (if at all). We can therefore use the same

shortest path graph structure as before (shown in Figure 2-2) with these modified

arc length computations to determine an optimal solution. Note that due to the

concavity of the production cost function, automatically, the production quantity

corresponding to the best production mode k lies in the correct segment; i.e., the

production costs have been computed correctly. The time required to find all arc

profits is O(LT2) multiplied by the time required to find D*,(t) for some k,t,r,

where L = maxt 1,...,T t is the maximum number of linear segments for any of

the T piecewise-linear concave production cost functions. As this analysis shows,

the case of piecewise-linear concave production cost functions can be handled in a

straightforward manner, even under general concave revenue functions, without a

substantial increase in problem complexity.

3.2.4 Production Capacities

This section considers a capacitated version of the RPPPLC where production

capacities are equal in all periods. In C'! lpter 2, we showed that RPPPLC with

time-varying finite production capacities is NP-Hard by demonstrating that it

generalizes the capacitated lot sizing problem (CLSP). The special case of the









CLSP where production capacities are equal in every period, however, can be

solved in polynomial time (see Florian and Klein [28]) with a complexity of O(T4).

Because of this, we next investigate whether the equal-capacity version of the

RPPPLC contains a similar special structure that we might exploit to solve this

problem in polynomial time.

The polynomial solvability of the equal-capacity CLSP relies on characterizing

so-called regeneration intervals (Florian and Klein [28]). A regeneration interval is

characterized by a pair of periods, r and r' (with r < T') such that I, = I, = 0,

and +I, +, ..., 1,-1 > 0 in an optimal solution. An optimal solution therefore

consists of a sequence of regeneration intervals (including the possibility of a single

regeneration interval (0, T)). A .' .ri.:/;/ constrained sequence between periods 'r + 1

and r' is one in which xt = 0 or C for all periods between (and including) r + 1 and

r' except for at most one. For the equal-capacity CLSP, an optimal solution exists

consisting of a capacity constrained sequence within each regeneration interval (see

Florian and Klein [28]). Given any choice of demands in every period for the equal-

capacity RPPPLC problem, the resulting problem is an equal-capacity CLSP; thus,

an optimal solution exists for the equal-capacity RPPPLC problem that consists

of capacity constrained production sequences within each of a set of consecutive

regeneration intervals.

Let D,,, = 7+1 dt denote the total demand satisfied between periods

- + 1 and r', where dt is the demand satisfied in period t. If (r, r') comprises

a regeneration interval, we know that total production in periods 'T + 1,..., '

must equal D.,, (since I, = I,, = 0 and D.,, is the demand satisfied in periods

- + 1,..., r'). Since at most one period contains production at a value other than

0 or C in a capacity-constrained sequence, we must have D.,, = kC + c, where k

is some nonnegative integer, and c is the amount produced in the period in which

we do not produce at 0 or C (assuming D.,, is not evenly divisible by C, in which









case c equals zero). So, given D,,', in each of the periods r + 1,..., r', we either

produce 0, c, or C, with a production amount of c in only one of the periods. We

can easily determine both k and c given D,,, and C; i.e., D,,, (mod C), and

k [DTT,/C]. We then construct a shortest-path graph that contains a path for

every feasible capacity-constrained production sequence between periods T + 1

and r'. Solving this shortest-path problem provides the minimum cost capacity

constrained sequence for every (r, T') pair (with r' > T). Given a value of D,,, for

every possible (T, ') pair, we can use this O(T4) CLSP solution approach to solve

the equal-capacity RPPPLC. The challenge then lies in determining appropriate

D,,, values for each possible (r, r') pair. To address this issue, we next show that

the candidate set of D,,, values for each (r, T') pair can be limited to a manageable

number of choices. Note that Loparic [49] provides a similar analysis for a lot-sizing

model in which total revenue is linear in the amount of demand satisfied.1

Consider a regeneration interval (r, -'), and recall that by definition we must

have I, = 0,lI > 0 for j = 7 + 1,..., 1, and I, = 0. The adjusted revenue

parameter that we introduced in Section 3.2 (i.e., pit =jt + s t hs for T < t < r')

will pl i, an important role in the analysis that follows. We also let 6jt denote the

decision variable for the amount of demand within segment j in period t that we

satisfy, and recall that at most djt units of demand exist within segment j in period

t. The following lemma is important in developing a useful solution algorithm.

Lemma 2 Suppose an optimal solution for RPPpLC contains a regeneration

interval (7r,7'), and suppose pjt > pit, with T < t,t' < r'. If an optimal solution

exists with 6jt < djt, an optimal solution also exists with 6t, = 0. E,!;.: ; .. ,:/Il; if an



1 I would like to gratefully acknowledge the insightful comments and direction
provided by Yves Pochet at the International Workshop on Optimization in Sup-
ply ('!I ni Planning in Maastricht, The Netherlands June 2001, that significantly
strengthened the material in this section.









optimal solution exists such that 6it, > 0, then an optimal solution also exists with

6jt = djt.
Proof: Consider the regeneration interval (7, -') and consider some pjt > pit, with

7 < t, t' < T'. Suppose we have an optimal solution with 6it, > 0, and 6jt < djt.

Since It > 0 for t = + 1,..., r' 1, we can increase 6jt by some c > 0 and decrease

6it, by c without changing the amount produced in each of the periods 7- + 1,..., '.

In particular, if t < t' we can set c = min {djt 6jt; it',; min {It,... It-}}. The

resulting change in objective function value equals (rjt (rit' _, hs))7

(Pjt Pit,)c > 0, and we either have 6jt = djt, 6it 0, or Is = 0 for some
s = t,... t'- 1 (in the later case, t and t' no longer belong to the same regeneration

interval). Similarly, if t' < t, we take e = min { ; djt 6jt}, and the resulting

change in objective function value equals ((rit t- 1, hs) Ti')c (pjt pit')e > 0,

with either 6it, = 0 or 6jt = djt.

Lemma 2 ensures that an optimal solution exists such that, within each

regeneration interval (7, 7'), there is at most one period from 7 + 1 to r' in which

demand will not be satisfied at a value equal to one of the breakpoints of the

revenue function. The following additional lemma allows us to further reduce the

number of potential values of D,,, that we must consider for a given regeneration

interval.

Lemma 3 An optimal solution exists for RPPPLC conr/.'I,:',:' consecutive regenera-

tion intervals (7, 7') where the production plan in periods 7 + 1,..., r' is one of the

following types:

(i) We produce 0 or C in every production period in the interval + 1,..., 7' with

at most one 0 < 6jt < djt in the interval; or

(ii) We produce at a value of c, with 0 < c < C, in at most one production period

in the interval 7 + 1,.. 7r' (and all other production levels are either 0 or C

in this interval), with all 6jt values equal to either 0 or djt within the interval.









Appendix C in Section 3.5 contains a proof of Lemma 3. Lemmas 2 and 3 taken

together imply that a limited number of candidate optimal solutions must be

considered for each possible regeneration interval (note that the number of possible

regeneration intervals is bounded by 0 (T2)). Letting Jmax denote the maximum

number of linear segments of the revenue functions among all periods (i.e., Jmax

max, 1,..., r {Js}), Lemmas 2 and 3 lead to the following theorem:

Theorem 3 The equal-, '/'i. /;:1 RPPPLC problem can be solved in 0 (JmaxT7) time.

Proof: Consider a potential regeneration interval (r, T') containing n periods, and

let J (r, -') denote the total number of linear segments in periods + 1,..., r'. For

potential regeneration interval (-r, T') we sort J ('-, T') values of pjt. Let this index

sequence of sorted values be denoted by (7r, -2,..., -Tj(, ')) (i.e., p, > p, > ... >

pr, ), where each index Tn identifies a unique segment-period pair within the
regeneration interval. For potential regeneration interval (-, T'), note that Lemma

2 implies that if 6, takes a value strictly between 0 and d. we must have 6, = 0

for k = 1,..., J (r, T') i.

Lemma 3 implies that within each potential regeneration interval (r, r') of

length n we need to consider two types of solutions. The first type of solution

produces a quantity of zero or C in each of the n periods. For this type of solution

we will have at most one 6, < d. for 1 < ni < J(r, 7'), with 6T, = d, for

k = 1,..., i 1 and 6+k = 0 for k = 1,..., J(, -,') i. The choice of the segment

-ri such that 6. < d. (of which there are J(r, -') + 1 possible choices, including

the choice to produce zero for all periods) uniquely determines the number of

periods in which we must produce at full capacity and, therefore, the values of 6,

for i = 1,..., J(r, -'). This in turn determines fixed demand levels that must be

satisfied in an equal-capacity lot sizing problem for the regeneration interval (r, -'),

which is solvable in 0 (n4) using Florian and Klein's [28] algorithm.









The second type of solution we must consider sets each 6, equal to zero or d,

for all i = 1,... J(r, 7') and produces at a value strictly between zero and C in at

most one period in the regeneration interval. The choice of the index r such that

6,_ = dji- for k 0,..., 1 and 6 + = 0 for k = 1,..., J(r, r') i uniquely
determines the number of periods in which production at full capacity is required,

and the value of production required in the single period such that xt < C. Again,

there are J(r, -') + 1 possible choices, including the choice to produce zero for all

periods. In total we must consider 2J(r, r') + 1 potential values of the demand

vector (6 6,2 ; J, for each regeneration interval of length n, which implies

that the number of potential demand vectors for an interval of length n is bounded

by O(Jmax,). For each of these vector values, we solve an 0 (n4) equal-capacity

lot-sizing problem. So the time required to evaluate a regeneration interval of

length n is 0 (Jmax,5), which is clearly bounded by 0 (JmaxTS). Since we have at

most 0 (T2) potential regeneration intervals, the total effort required is 0 (JmaxT7).

0
Note that, unlike the uncapacitated case, in the capacitated case it is now

possible to choose an optimal demand level in any period that is strictly between

breakpoint values of the revenue function. While this is not a critical distinction

in the pricing setting, we must explicitly consider this factor in the order selection

problem setting discussed in the following Section.

3.3 Pricing and Order Selection Interpretations

Although our discussion has centered on the concept of pricing, up to this

point we have said little about the actual pricing decisions that result from our

models. That is, since our model assumes a one-to-one correspondence between

price and demand within a market in every period, we have worked solely with

demand levels as decision variables. Recall that we assumed that this price-demand

relationship in each period is represented by the function Dt = Y I1 dmt(Omt),










or equivalently Omt = d>(Dt), where Omt denotes the price offered to market m

in period t, and d4 (Dt) is determined by solving an optimization subproblem,

as discussed in Section 3.2.3.1. Given a total demand satisfied of Dt in period

t, we also assumed that a total revenue of Rt(Dt) is realized, where Rt(Dt) is a

nondecreasing concave function of demand Dt.

Given this relationship between demand and revenue, we can interpret

the actual prices paid for the units sold in at least two v--v depending on the

model's intended application context. We use Figure 3-1 to illustrate two such

interpretations. Figures 3-1(a) and 3-1(b) show identical piecewise-linear revenue

curves with three segments and segment slopes ri > r2 > r3. In both cases, the

total revenue achieved at the demand level D' equals R(D') = rid1 + r2d2 + r3d3. In

Figure 3-1(a) we assume that a market exists with a total of dl units of demand,

each of which is willing to p ,iv an amount of r, per unit of the good, while a second

market with a total of d2 units of demand provides a revenue of r2 per unit, and

a third market contains d3 units of demand with a revenue of r3 per unit. In this

case, the price paid for units of demand falling within a segment corresponds to

the slope of the segment. This interpretation might apply when different market

segments (e.g., geographical segments) actually p ,iv different prices, and each of the

dj values corresponds to a given market m's total available demand value, dt, in

period t; i.e., dj represents a market -. in Figure 3-1(a).

(a) (b)
R(D). R(D)


-'- Slope = R(D')ID'
r2 (=price value 2) r (= single price value
S-' for all demands)
I, I
I I A
r (I= price value 1) I i'
YD D
di d2 da D' di d2 d3 D'

Figure 3-1: Pricing interpretations based on total revenue and demand.









In Figure 3-1(b), on the other hand, we assume that we have only a sin-

gle market available, and all satisfied demands provide the same per-unit rev-

enue (price), which at a demand level of D' is given by 0(D') = R(D')/D'

= rjdjZll 1 dj. This interpretation implies that a total of D' demands exist
that are willing to 1p v 0(D'), which equals the slope of the line connecting the ori-

gin to R(D'). This interpretation applies to cases in which the supplier must charge

a single price to all customers in the market. In either case, the models are com-

pletely the same, but the pricing interpretations and the contexts to which these

interpretations apply are quite different. Note that when the revenue function is

characterized by a differentiable concave function, the only practical interpretation

is one in which the price paid for each unit equals the slope of the line connecting

the origin to R(D'), which is R(D')/D' (R'(D') of course indicates the ,,,.,,j:,,i.1

total revenue at D').

Given our interpretation of Figure 3-1(a), we can now view the individual

segments of the piecewise-linear revenue curve in a different light. That is, each

linear segment may not only correspond to separate units of demand from an

individual market, but might alternatively be associated with an ..:-iegate order

from an individual customer, where individual customers are willing to p li

different unit prices for the item (or, alternatively, different customers have a

different unit cost associated with fulfilling their orders). Given this interpretation,

the RPPPLC model can be utilized in a broader set of contexts, where the firm

does not set prices, but can select from a number of customer orders, each of which

offers a particular net revenue per unit ordered.

Recall that, in the "order acceptance and denial" environment introduced

in C'! lpter 2, firms can either commit to fulfilling an order or decline the order

based on several factors, including the capacity to meet the order and the economic

attractiveness of the order. The RPPPLC model can also be applied within such









contexts. In this order selection setting, we now assume that a set of orders for

the supplier's good exists in each of the T periods of the planning horizon, and

redefine Jt as the number of orders that request fulfillment in period t. The index

j now corresponds to individual order indices, and we let rjt denote the revenue

per unit provided by order j in period t, while djt is the order quantity associated

with order j in period t. We index all orders within a period in nonincreasing order

of unit revenues (i.e., rut > r2t > > rJet). We redefine the binary zjt variables

previously used in the [RPPPLc] formulation as follows: zjt = 1 if we accept order

j in period t, and zjt = 0 otherwise. These variables can now be interpreted as

order selection variables. The remaining production quantity (xt), production setup

(yt), and inventory (It) variables in the [RPPPLc] formulation retain their original

definition.

Since the formulation is completely unchanged except for our interpretation

of the meaning of certain parameters and decision variables, we can use the same

shortest path and dual-ascent methods we presented to solve this order selection

problem. In the uncapacitated production setting, recall that an optimal solution

exists for the RPPPLC problem such that the amount of demand selected in each

period falls at one of the breakpoints of the revenue function. Under the order

selection interpretation, this implies that an optimal solution exists in which every

order will be either fully accepted (and fulfilled in its entirety), or will be declined.

We next briefly discuss the implications of finite capacity limits within the

order selection context, again restricting our discussion to the equal-capacity

case. Since the [RPPPLc] formulation served as our starting point for the analysis

of the equal-capacity case, and the uncapacitated order selection problem is

formulated exactly the same as the [RPPPLc] formulation, we can essentially follow

the discussion in Section 3.2.4 with our new order selection interpretation. This

approach assumes, however, that customers will permit partial order satisfaction;










i.e., for order j in period t we are free to satisfy any amount of the order quantity

between zero and djt. For contexts in which such partial order satisfaction is

allowed, we can therefore apply the same approach discussed in Section 3.2.4

to solve the equal-capacity version of the order selection problem. If, however,

customers do not permit partial order satisfaction, the problem is NP-Hard.

To demonstrate the difficulty of the problem when partial order satisfaction is

not allowed, we next briefly consider the single-period special case of this problem,

where T = 1. Note that we now explicitly require the binary restrictions on the

zjl variables for this problem. For this single-period special case we can write the

inventory balance constraints as x1i = 1 dji zjl which simply implies that the

production in the only period must equal the demand we choose to satisfy. Given

that we have only a single-period problem, we will either perform a setup or not. If

we do not perform a setup, then the objective function equals zero. If we do setup,

then we need to solve the following problem to determine the optimal solution:

J1
maximize: > (Rji pidjl) Zjl
j=1
J1
subject to: > djl ji < C,
j=1
zji {0,1}, j 1t,... ,J1.


The above problem is a knapsack problem in its most general form (since the

Rjl and djl parameters can take arbitrary nonnegative values), indicating that

the all-or-nothing order satisfaction version of the capacitated problem problem

is NP-Hard, even in the single-period special case (although the single-period

version is not strongly NP-Hard). This problem is therefore clearly NP-Hard for

the multiple-period case with or without equal capacities in all periods.









3.4 Conclusions

Allocating appropriate amounts of resources to anticipated demand sources

has been a well-researched problem in revenue management, although the work

has primarily focused on service industry applications (e.g., airline and hospitality

industry applications; see, for example, van Ryzin and McGill [79]). As we have

discussed, an increasing amount of attention is being placed on revenue manage-

ment, through pricing models, in manufacturing contexts. We contribute to this

effort by providing models and efficient solution methods for a general set of pricing

problems in manufacturing settings where fixed setup costs comprise a substantial

part of operations costs. In addition to pricing applications, we showed that our

modeling approach also applies to order selection problems, the focus of C'! Ilpter 2,

in which a supplier must choose from a set of outstanding orders to maximize its

contribution to profit after production costs. As we have shown, our models and

methods also allow for efficiently solving problems in which time-invariant finite

production capacities exist.

Most revenue management literature addresses anticipated demand that is

stochastic in nature, which is why selecting the best utilization of resources to

achieve maximum profit is such a difficult problem. In the next chapter, we also

consider the effects of stochastic demand on our demand selection decisions.

3.5 Appendix

Appendix A

Theorem 1 For the uncapacitated RPP, given a production setup in period t only,

if a demand /;,,r'/,:.'/; D, exists such that I,- is in the set of subgradients of R,(-) at

D,, then D, is an optimal demand i;,,u;i.:'I. for the subproblem given by (3.4) and

(3.5).

Proof: Given a period 7 > t, and assuming a setup in period t only, we need

to choose the demand quantity d in period t such that the total revenue at d in









period t minus the total cost incurred in satisfying the quantity d in period t is

maximized. By the definition of ,,-, the total cost (excluding the setup cost, which

has already been incurred) for satisfying d units in period t using production in

period t equals -.1 We need to therefore solve the following problem to determine

the optimal demand value to satisfy in period T:


maximize: R,(d) ,-.

subject to: d > 0.


Consider a value D such that is in the set of subgradients of R,(d) at d

D. This implies by the definition of a subgradient of a concave function that

R,(d) < R,(D) +, ,-(d D) for all d > 0 (the domain of R,(.)). This implies that

R,(d) -.1 < R,(D) i,-D for all d > 0, which implies the result. 0

Appendix B

Theorem 2 The dual-ascent i,'':thhm presented in Section 3.2.2 solves [CDPI.

Moreover, the complementary primal solution to the dual solution produced by the

i,l.,rithm rl. -[7. the ':,., ./,l.:l;/ restrictions of formulation [RPPPLCI and therefore

provides an optimal solution for the RPPPLC.

Proof: Let F(Tr) denote the optimal value of a problem consisting of periods 1,

..., 7. As we have demonstrated through our shortest-path approach, the following

recursive relationship holds for the RPPPLc:


F(r) = min F(i 1) + min Si + cid, Pi}} ,
i
where dsi = E E djt pir E E p jtdjt, and J*(t, i) {j : pjt > ci}.
t=ijCJ*(t,i) t=i jCJ*(t,i)
In our dual problem, the only variables contributing value to the objective

function are those contained in the sets, J1 for 7 = 1, ..., T. In other words,

letting ZDT denote the objective function value of our dual solution for a T-period









problem, we have
T T
ZDT min (0,d( pt)) PEt)
t -1 jcJtl t1 jcJtl

since pjt > pf for all j E J1 by definition. To show the optimality of our dual-

ascent procedure, we need to show that
T

t 1 j3Jil

where we have F(r) > E E djt (p pjt) for all feasible p* by weak duality.
t=1 jcJt
2
We first show that F(1) = djl (p* pjl) and F(2) = E djt (* pjt)
jCJi1 t-1 j1
directly, and then use induction to show the general result. For 7 = 1, the result

is straightforward, since the final objective function after implementing the dual

procedure is equal to Y djl (p* prj). If J, is empty, then the objective function

equals zero, which implies we do no setup and satisfy no demand. Otherwise,




S1 djl max (0, pjl ci)
djl c1 + jCJ
SdjIpjldjl

Sl E (djl (c jl) E djl max (O' jl cl)


Sl E+ djl (cl P- )j) djl (c pjl)
jC i1 jJi
Sl + E dl (ci p),


where J1 is the set of all j E J1o such that pjl > cl. We now have constructed a

dual feasible solution with an objective function value equal to that of a primal

feasible solution that sets up in period 1 and satisfies all demand for segments j

such that j E Jl U JO, implying that this solution is optimal for the primal problem.









2
We next consider the case of 7 = 2. In this case we have ZS = E djt (P pit).
t-1 jGcJ
If J, is empty, then we have a single-period problem (for period 2) and we can refer

to the proof above for the case of r 1. Suppose then that neither J, nor J2 is

empty. In the process of applying our dual-ascent algorithm, we encounter one of

the two cases below:

Case I: p = cl. This implies that constraint 2 does not become tight and further

increases in p are blocked by the first constraint. In this case the dual objective

equals
2
Z% Z djtq< Pit) S+ > djl (ci -pji) + -(cl Pj2),
t= 1 jiJ jcJfuJQ jiJ%

which equals the primal objective function value of a primal feasible solution that

sets up in period 1 only and uses this setup to satisfy segments j in period 1 such

that j E J U Jo and in period 2 such that j J2.
S2- F dj2 l... _.-c2)
7'JO
Case II: p* = c2 + --. This implies that constraint 2 becomes
j Jc
tight before p reaches ci and further increases in p/ are blocked by the second

constraint. In this case the dual objective equals
2
Z2 2d t (li pit) S1 + S2+ Y dil (Cl pjl) + (C2 P2),
t= 1 jcGi jfJ1uJQ j'UJ20

where J2 is the set of all j 2 JO such that pj2 > C2. This value of ZS is equal to the

primal objective function value of a primal feasible solution that sets up in periods

1 and 2 and satisfies demand in all segments j in period 1 such that j E Ji U J1o

using the setup in period 1 and satisfies demand in all segments j in period 2 such

that j J1 U J2 using the setup in period 2. We have so far shown that Z3 = F(-)

for 7 = 1 and 2. We next use induction to show that this holds for all 7 > 2.

Assume there is at least one attractive segment in some period 7r; i.e., O exists for

some r E {1, ..., T} (otherwise the optimal dual solution value equals zero and no










demand is satisfied). For some k < r we must have


P = Ck +


T T-1
Sk E E dj max (0, pt Ck) djl max (p~
t=kjGO t=k ji:


C dj,
j GJ"


The T-period objective function then becomes

T
Z d t (/ 1 pit)
t 1 jeJl


k-1

E d (/ 4
t 1 j~Jl


7-1

P t= j+ t (
t-k jk


Pit) + dj T (/
i G J1


Substituting for p*, the last expression can be written as:


C djr (r
j3^)


Pji)


Y djrCk + Sk
G3J


f dj max(0, pit
t-k jyGo


7-1
Ck) djl max (p
t k jcG~


Ck,O) Y pjdj
icJ


Returning to the T-period objective function, and using our induction hypothesis,

we now have

7-1



T
Zj= F(k -1)+5djCk+Sk+ di t Pjt maxQ -t Ck, 0))



djti max (0, pit Ck) Pj Tdj
t-kc jEo jGJ

Since for t < T, if p, is defined (i.e., J1 / {0}), we have p* > p* > Ck (from Lemma

1), the above can be rewritten as


F(k 1) + Y di, (fk P) +

T-l T
+dit (Ck Pit)
t k~j j t k jG
T
F(k- 1))Sk+ + dt,(ck
t=k jCJt


Sk


dt max (0, it
Jt?


Ck,O)


Pjr)


pit) + d j (Ck
t-k jiCJ?


Pjt) ,









where Jo {j- E jt : pjt > Ck.

From our previous definitions, we can now simplify Zf to


Z1 = F(k 1) + Sk + kdk PkT,

which corresponds to the objective function value of a primal feasible solution,

implying that ZL > F(r). But by weak duality we have ZL < F(r), and so

we must have Zf = F(r), the optimal solution value of the primal. Moreover,

the complementary primal solution is also feasible for the binary restrictions of

[RPPpLc]. 0

Appendix C

Lemma 3 An optimal solution exists for RPPPLC containing consecutive regenera-

tion intervals (7, -') where the production sub-plan in periods 7 + 1,...., 7' is of one

of the following ';/'p

(i) We produce 0 or C in every production period in the interval + 1,..., T' with

at most one 0 < 6jt < djt in the interval; or

(ii) We produce 0 < c < C in at most one production period in the interval

7 + 1,..., 7' (and all other production levels are either 0 or C in this

interval), with all 6jt values equal to either 0 or djt within the interval.

Proof: We have shown that an optimal solution exists containing a sequence

of regeneration intervals, and that at most one ijt value exists in a regeneration

interval with 0 < 6jt < djt (Lemma 2); we also know that a capacity-constrained

production sequence exists. We therefore need to show that given an optimal solu-

tion satisfying these properties with a production quantity x, within a regeneration

interval such that 0 < x, < C, and with a 6jt for some period t in the regeneration

interval such that 0 < 6jt < djt, an optimal solution also exists satisfying the

conditions stated in the lemma. Suppose that we do have such an optimal solution,

and that the production period s occurs prior to or including the demand period









t. Since inventory in each period in the regeneration interval is positive, a feasible

solution exists for the regeneration interval that uses the same setup periods and

reduces 6jt by one unit, along with inventory in periods s,... ,t 1, and production

in period s. Since this solution does not improve over our optimal solution (and

given the linearity of costs), this implies that at least as good a solution exists that

increases 6jt by one unit, along with inventory in periods s,..., t- 1 and production

in period s. Repeating this argument until either x, = C or 6jt = djt implies the

result of the lemma. Similarly, if period t is before period s, a feasible solution

exists for the regeneration interval that uses the same setup periods, increases 6jt

by one unit, reduces inventory in periods t,..., s 1 by one unit, and increases x,

by one unit. Since this solution does not improve over our optimal solution, this

implies that at least as good a solution exists that reduces 6jt by one unit, increases

inventory in periods t,..., s 1 by one unit, and reduces x, by one unit. Repeating

this argument until either 6jt = 0 or x, = 0 proves the result. O















CHAPTER 4
SELECTING MARKETS UNDER DEMAND UNCERTAINTY

4.1 Introduction

Thus far, our approach to selecting the best demand sources (orders, markets,

etc.) to satisfy relied on deterministic information concerning the size and timing

of each demand. While we usually have some data for planning purposes, typically

via scheduled orders or demand forecasts, the exact amounts are often inaccurate.

Therefore, it is extremely important for a firm making such product ordering

(or manufacturing) decisions to account for the stochastic nature of demand. As

demand becomes less predictable, our selection decisions will surely be influenced.

We study the market selection problem with demand uncertainty in order to

develop a robust modeling approach that addresses such types of demand.

The classic newsvendor problem has been studied extensively in research

literature due in large part to its industry applications. The retail and airline

industries have shown that operating with a perishable good (e.g., seasonable

fashion items, airline seats or flights) requires the attention of a single selling

season model, which is addressed through the newsvendor model. In a similar

vein, manufacturing firms are producing items with ever-decreasing product lives,

in an effort to stay competitive with the latest offering of other firms. This is

especially true in the technology sector where, by the time a firm starts to realize

demand during the selling season, it is often too late to place a second order with a

supplier due to long lead times. In other words, the firm must live with its previous

order quantity decision and now possibly 1 iv a premium for expediting additional

product to capture any additional unforeseen demand.









No matter how much effort is spent on trying to reduce product and process

lead times, certain industries will likely exist where obtaining materials or more

product at a reasonable unit cost will require a substantial amount of time.

Even if the firm operates in a so-called Quick Response (QR) mode with its

suppliers, the lead times may still be long relative to the selling season (see Iyer

and Bergen [39] for a discussion of QR in the apparel industry). This leads us to

study questions concerning integrated order quantity and market selection decisions

under uncertain demand.

We consider a firm that offers a product for a single selling season. The

firm uses an overseas or "long lead time" supplier as the primary source for its

product, and thus the order quantity must be decided far in advance of actual

sales. The firm has the flexibility to select which market demand sources to satisfy,

where each demand source is a random variable. In the classic newsvendor model,

the preferred order quantity is dependent on the distribution of total demand.

However, in our context, the demand distribution is dependent on the markets the

firm selects. Thus, the market selection decision must be made prior to ordering

from the firm's supplier so that an appropriate order can be received in time for

the selling season. In addition to each market's demand distribution being random,

we assume that this distribution can be influenced by the level of advertising effort

used within each market. By expending more effort in a market, the firm can

increase the demand for its product. We address appropriate advertising response

functions which measure marketing effectiveness based on the level of advertising

spending (see Vakratsas, Feinberg, Bass, and Kalyanaram [78]). We also examine

the effect of budgetary constraints. The marketing budget could prevent the firm

from capturing additional expected market demand, and, thus, additional profits,

regardless of the firm's ordering or production capacity.









As product life cycles continue to decrease and assessing demand risk for

market entry becomes increasing critical, many companies find themselves faced

with similar issues that we address here. Claritas, a market research and strategic

planning company, has cited several clients, including Eddie Bauer, that wanted

better knowledge of their customers in order to minimize demand risk. Claritas has

had many success stories in identifying profitable customers, assessing potential

markets, and ranking opportunities. Recently, Fisher, Raman, and McClelland

[27] studied how 32 leading retailers, all of which offer short-life-cycle products

(some with a single selling season) with unpredictable demand, to determine

how effectively each .m1 in'' used available data sources to understand their

customers. In the present marketplace, these retailers are -zvIing that they must

make better use of demand information if they want to make profitable market

selections. Finally, Carr and Lovejoy [20] also discuss this problem's motivation

from an inverse newsvendor point-of-view. They cite a client firm making industrial

products, and this firm desires a marketing strategy that selects appropriate

demands or markets to enter while working within a fixed production level.

In contrast to the approach developed by Carr and Lovejoy [20], we do not

assume a predetermined capacity limit that the firm must obey when selecting

markets. Rather, our model jointly determines the capacity acquired and the

markets selected in order to maximize a firm's profit. Moreover, the firm can

influence market demands through judicious use of advertising resources. The

resulting models lead to interesting new nonlinear and integer optimization

problems, for which we develop tailored solution methods. These models also allow

us to develop insights regarding the parameters and tradeoffs that are influential

in integrated market selection and capacity acquisition decisions for items with a

single selling season. Thus, this work provides new contributions to the operations









modeling and management literature as well as the literature on operations

research methodologies.

Many researchers have contributed to the wide range of literature that exists

on stochastic inventory control, for which Porteus [62] provides a nice overview.

Particularly relevant to our work is the literature that focuses on the newsvendor

problem. In addition to the work by Porteus, reviews by T-zv, N il in i i- and

Agrawal [76] and Cachon [18] provide more recent research directions concerning

supply chain contracts and competitive inventory management in the context of a

single-period i,. v--1., i'dor" setting.

Additional literature considers the multi-item newsvendor problem as well,

for which we can draw some similarities to our "multiple-i1 il:l. I setting. In our

problem we have one cost for producing a single product, and the individual market

delivery costs, sales and advertising costs, and revenues provide differentiation

among markets. Each market has a certain amount of random demand, and we at-

tempt to satisfy the demand from the markets we select to maximize overall profit.

In the typical multiple-product setting, each product has unique production, sal-

vage, ordering and perhaps distribution costs. However, there is no differentiation

between the demand sources for a particular item. Moon and Silver [55] present

heuristic approaches for solving the multi-item newsvendor problem with a budget

constraint.

Other researchers have investigated how production capacity can be adjusted

within the framework of a newsvendor-type problem. Fine and Freund [25] consider

cost-flexibility tradeoffs in investing in product-flexible manufacturing capacity.

They formulate the capacity investment decision as a two-stage stochastic program,

where all future production and inventory decisions are rolled into one future pe-

riod. This is notably different from our approach in that they consider production

capacity constrained problems as opposed to budget constrained problems. They