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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:Attempttoincreaseproductioninunderutilized periods 0) Createanewlistforeachperiodofallprotableordersnot fullled.Eachlistis indexedin nonincreasing orderofperunitprotability,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 Firmsthatproducemadetoordergoodsoftenmakepricingd 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 arguesthatintegratingsupplyanddemandbasedmanagemento 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 planningdecisionsinmaketoordersystemswiththetypesofn onlinearproduction coststructuresoftenfoundinpracticeasaresultofproductio neconomiesof scale.Weoermodelingandsolutionapproachesforintegratin gthesedecisionsin singlestagesystems. Mostoftherequirementsplanningliteraturefocusesonprodu ctionrequirementsbasedonprespecieddemands,withnoadjustmentsforpric eexibility. Inthischapter,weintroducearequirementsplanningmodel thatimplicitlydeterminesthebestdemandlevelstosatisfyinordertomaximizecont ributiontoprot whendemandisadecreasingfunctionofprice.Inotherwords,t hermwillselect thedemandleveltosatisfybysettingasinglepricefortheprodu ct. Wemakeseveralcontributionstotheliteraturethroughourm odeland solutionapproachesintroducedinthischapter.First,ourco mbinedpricingand productionplanningmodelpermitsmultiplepricedemandc 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 piecewiselinearconcaverevenuefunctions. Givenxedpluslinearproductioncostsandpiecewiselinearc oncaverevenue functions,wealsoprovidea`tight'linearprogrammingformu lationofourmodel, usingadualbasedsolutionapproachtoshowthatthisformulati onhaszeroduality
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47 gap.Thisresult,andtheformulationsdiscoveredwhiledevel opingtheapproach, playedakeyroleinformulatingtherelaxationsusedinsolvin gthecapacitated OSPmodelsinChapter 2 ,whereproductioncapacitiesvariedovertime.Ournal majorcontributionalsoaddressesacapacitatedversionofthem odel.Assuming timeinvariantproductioncapacitylimitsandpiecewisel 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 timeperiodsasadownwardslopedfunctionofthepriceineac 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 thatallprespecieddemandsmustbelledduringtheplanningh orizon,our approachimplicitlydeterminesdemandlevelsthroughpric ing. BhattacharjeeandRamesh[ 13 ]consideredthepricingproblemforperishable goodsusingaverygeneralfunctiontocharacterizedemandas afunctionofprice. Theyalsoassumedupperandlowerboundsonprices,characterize dstructural propertiesoftheoptimalprotfunction,anddevelopedheur isticmethodsfor solvingtheresultingproblems.Biller,Chan,SimchiLevi,an dSwann[ 14 ]analyzed amodelsimilartooursunderstrictlylinearproductioncosts( i.e.,noxedsetup costs,andassumingtimevaryingproductioncapacitylimits),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,SimchiLevi,andSwann[ 21 ]. Theremainderofthischapterisorganizedasfollows.Sectio n 3.2 presents aformaldenitionandmixedintegerprogrammingformulatio nofthegeneral requirementsplanningproblemwithpricing.Inthissection weprovideoursolution approachesforthisproblem,therstofwhichextendstheWagn erWhitin[ 83 ] shortestpathsolutionmethod(discussedinChapter 2 )tocontextswithgeneral concaverevenuefunctionsandxedchargeproductioncosts.Assu mingpiecewiselinearconcaverevenuefunctions,wethenprovideadualbase dpolynomialtimealgorithmforsolvingtheuncapacitatedproblem.Thisd ualbasedsolution approachallowsustoshowthattheproblemreformulationinS ection 3.2.2 has alinearprogrammingrelaxationwhoseoptimalvalueequalst hatoftheoptimal mixedintegersolution;i.e.,theproblemformulationis\ti ght".Wealsoexplore thegeneralityofoursolutionapproacheswithrespecttodier entfunctionalforms fortheproductioncostfunctionsandundermultiplemarketp ricedemandcurves inanygivenperiod.Inadditiontopresentingsolutionapproa chestoseveral uncapacitatedversionsoftheproblem,weprovideananalysis oftheequalcapacity versionofthemodelunderpiecewiselinearconcaverevenuef 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 assumptionofaonetoonecorrespondencebetweenpriceandma rketdemand volumeinanyperiod,wheremarketdemandisadownwardslopi 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 isadecisionvariablefortheendofperiodinventory 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 containaxedchargestructure;i.e.,axedcostof S t isincurredwhenperforming aproductionsetupinanyperiod t ,whilethevariablecostperunitinperiod t equals p t (welaterdiscussinSection 3.2.3.2 thenecessaryextensionstohandle productioncoststhatcontainamoregeneralpiecewiselinea rnondecreasing concavecoststructure).Underxedpluslinearproductioncosts, unlimited productioncapacity,andasinglepriceoeredtoallmarketsi neachperiodwehave themodelrstanalyzedbyThomas[ 74 ],whoproposedadynamicprogramming recursionforsolvingtheproblem.Thealgorithmissimilartot heWagnerWhitin [ 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 xedchargestructure,wedepartfromthisworkinthefollowi ngrespects:
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52 (i)weprovideanexactsolutionapproachforcontextsinwhic htotalrevenueis concaveandnondecreasingintheamountofdemandsatised; (ii)weshowthattheshortestpathmethodgeneralizestocaseswi thmultiple demandsources,eachwithauniqueconcaverevenuecurve;and (iii)weshowhowtogeneralizetheshortestpathapproachtopro videanexact procedureforthecaseofpiecewiselinearandconcaveproduc tioncosts. Thus,thefollowingsectionlaysthefoundationforsubsequentg eneralizationsof oursolutionmethodologytobroadercontexts. 3.2.1 ShortestPathApproachfortheUncapacitatedRPP Retainingourassumptionofaxedchargeproductioncoststruct ureand assumingtherevenuefunction R t ( D t )ineveryperiod t isageneralnondecreasing concavefunctionof D t with R t (0)=0,wenowupdatetheWagnerWhitin [ 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(nonne 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 piecewiselinearconcavefunctionofdemand.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 ( ¢ ) arepiecewiselinearandconcavewithatmost 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 DualascentMethodfortheUncapacitatedRPP Whentherevenuefunctionsarepiecewiselinearandconcave ineveryperiod, andproductioncostscontainaxedplusvariablecoststructure ,wecanalsouse adualbasedalgorithmtosolvetheuncapacitatedRPP,whichw enextdescribe. ThisapproachrequiresrstreformulatingtheRPP.Aswelatersh ow,thisnew formulationis\tight";i.e.,itslinearprogrammingrelax ationobjectivefunction valueequalstheoptimalobjectivefunctionvalueofRPP.We beginbyprovidingan explicitbaseformulationoftheuncapacitatedRPPunderpie cewiselinearconcave 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 piecewiselinearconcaverevenuecurve);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 withpiecewiselinearconcaverevenuefunctions,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 heequalcapacitycase inSection 3.2.4 Notethatwehavenotimposedanyspecicconstraintsontherelati onship between z jt variablescorrespondingtothesamerevenuefunctioninagive n period t .Thefollowingpropertyallowsustoconsidereachoftheinte rvalsofthe piecewiselinearconcaverevenuefunctionindependentlyf romoneanotherinour
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57 mixedintegerprogrammingformulation(thatis,weneednoti ntroduceanyexplicit constraintsinourformulationthatspecifythestrictorderin gofthepiecewiselinear 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.Weshowthisbydevelopingaduala 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 .Thedevelopmentofourdualascentprocedurerequires capturingtheexactamountofproductionineachperiodthat correspondstothe amountofdemandsatisedwithineachlinearsegmentofthepiece wiselinear 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 ],thatlendsitselfnicelytoourdualbasedapproach.
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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,berewritteninamoreco 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.Wecanalsoeliminateanysegmentperiod 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] istouseadualascentprocedurethatincreasesthedualvariab 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:CDPDualAscentSolutionAlgorithm0. Deleteanysegmentperiodcombination( 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 ,wecanboundthecomplexityofthisdualascentalgorithmby O ( J max T 2 ),the sameasthatoftheshortestpathalgorithmintheprevioussecti on.Wenextshow thatthedualascentsolutionprocedureoutlinedabovenoton lysolves[CDP],but alsoleadstoaprimalcomplementarysolutioninwhichallofth ebinaryrestrictions informulation[RPP PLC ]aresatised;i.e.,thedualascentproceduresolvesthe RPP PLC .Beforeshowingthis,werstneedthefollowinglemma. Lemma1 Foranypairofpositiveintegers and l suchthat + l T and ¤ and ¤ + l aredenedasinthedualascentalgorithm,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 Thedualascentalgorithmpresentedabovesolves[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 )fromStep2ofthedualascentalgorithm). 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 providingpolynomialtimesolutionmethodsfortheuncapac itatedRPP.Therst oftheseassumptionsreliesontheproductioncostfunctiontaki ngaxedcharge structureineachperiod,whilethesecondassumesthatasinglepr icedemand curveexistsineachperiod.Wenextexplorethedegreetowhic hwecanrelaxthese assumptions,whileretainingourabilitytoapplythepolynomi altimesolution methodswehavepresented.Firstweconsidercontextsinwhichm ultiplepricedemandresponsecurvesexistineachperiod;thiswouldcorrespo ndtocontextsin whichtheproducerhasmultipleavailablemarketsinwhicht osellitsoutput,with eachmarkethavingauniqueresponsetoprice.Wethenconsidert heimpactsofa piecewiselinearconcaveproductioncoststructure(whichma yincludeaxedsetup cost)ineachperiod. 3.2.3.1 Multiplepricedemandcurves InthissectionweshowthatanyuncapacitatedRPPwithmultipl edemand curvesinaperiodcanbereformulatedasanRPPwithonlyasing ledemandcurve perperiod.Wewillshowthatthisholdsforgeneralconcavere venuefunctionsand forpiecewiselinearconcavefunctionsinparticular.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 ( ¢ )ispiecewiselinearandconcave (and R mt (0)=0)forall m and t ~ R t ( ¢ )ispiecewiselinearandconcaveforall 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 Piecewiselinearconcaveproductioncosts Wenextconsiderthecaseinwhichtheproductioncostfunctioni neach periodispiecewiselinearconcaveandnondecreasinginthep roductionvolume intheperiod.Notethatanynondecreasingpiecewiselinearco ncavefunctioncan beviewedastheminimumofanumberofxedchargefunctions.T herefore,if theproductionfunctionsarepiecewiselinearandconcavew 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 piecewiselinearconcaveproductioncostfunctions.Asthisan alysisshows, thecaseofpiecewiselinearconcaveproductioncostfunction scanbehandledina straightforwardmanner,evenundergeneralconcaverevenue functions,withouta substantialincreaseinproblemcomplexity. 3.2.4 ProductionCapacities ThissectionconsidersacapacitatedversionoftheRPP PLC whereproduction capacitiesareequalinallperiods.InChapter 2 ,weshowedthatRPP PLC with timevaryingniteproductioncapacitiesisNPHardbydemonst ratingthatit generalizesthecapacitatedlotsizingproblem(CLSP).Thesp ecialcaseofthe
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69 CLSPwhereproductioncapacitiesareequalineveryperiod, however,canbe solvedinpolynomialtime(seeFlorianandKlein[ 28 ])withacomplexityof O ( T 4 ). Becauseofthis,wenextinvestigatewhethertheequalcapacit yversionofthe RPP PLC containsasimilarspecialstructurethatwemightexploittosol vethis probleminpolynomialtime. ThepolynomialsolvabilityoftheequalcapacityCLSPrelie soncharacterizing socalled 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.FortheequalcapacityCLSP,anoptima lsolutionexists consistingofacapacityconstrainedsequencewithineachregen erationinterval(see FlorianandKlein[ 28 ]).GivenanychoiceofdemandsineveryperiodfortheequalcapacityRPP PLC problem,theresultingproblemisanequalcapacityCLSP;th us, anoptimalsolutionexistsfortheequalcapacityRPP 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 inacapacityconstrainedsequence,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 .Wethenconstructashortestpathgraphthatcontainsapathfo r everyfeasiblecapacityconstrainedproductionsequencebet weenperiods +1 and 0 .Solvingthisshortestpathproblemprovidestheminimumcost capacity constrainedsequenceforevery( ; 0 )pair(with 0 > ).Givenavalueof D 0 for everypossible( ; 0 )pair,wecanusethis O ( T 4 )CLSPsolutionapproachtosolve theequalcapacityRPP PLC .Thechallengethenliesindeterminingappropriate D 0 valuesforeachpossible( ; 0 )pair.Toaddressthisissue,wenextshowthat thecandidatesetof D 0 valuesforeach( ; 0 )paircanbelimitedtoamanageable numberofchoices.NotethatLoparic[ 49 ]providesasimilaranalysisforalotsizing 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
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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 TheequalcapacityRPP 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 identiesauniquesegmentperiodpairwithinthe 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)showidenticalpiecewiselinearrevenue 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 erunitrevenue(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 segmentsofthepiecewiselinearrevenuecurveinadierentli 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 shortestpathanddualascentmethodswepresentedtosolvethis orderselection problem.Intheuncapacitatedproductionsetting,recallth atanoptimalsolution existsfortheRPP PLC problemsuchthattheamountofdemandselectedineach periodfallsatoneofthebreakpointsoftherevenuefunctio n.Undertheorder selectioninterpretation,thisimpliesthatanoptimalsolut ionexistsinwhichevery orderwillbeeitherfullyaccepted(andfullledinitsentir ety),orwillbedeclined. Wenextbrieydiscusstheimplicationsofnitecapacitylimits withinthe orderselectioncontext,againrestrictingourdiscussiontothe equalcapacity case.Sincethe[RPP PLC ]formulationservedasourstartingpointfortheanalysis oftheequalcapacitycase,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 tosolvetheequalcapacityversionoftheorderselectionprob lem.If,however, customersdonotpermitpartialordersatisfaction,theproble misNPHard. Todemonstratethedicultyoftheproblemwhenpartialordersa tisfactionis notallowed,wenextbrieyconsiderthesingleperiodspecialc aseofthisproblem, where T =1.Notethatwenowexplicitlyrequirethebinaryrestriction sonthe z j 1 variablesforthisproblem.Forthissingleperiodspecialca sewecanwritethe inventorybalanceconstraintsas x 1 = P J 1 j =1 d j 1 z j 1 ,whichsimplyimpliesthatthe productionintheonlyperiodmustequalthedemandwechooseto satisfy.Given thatwehaveonlyasingleperiodproblem,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 theallornothingordersatisfactionversionofthecapacita tedproblemproblem isNPHard,eveninthesingleperiodspecialcase(althoughthesi ngleperiod versionisnotstronglyNPHard).Thisproblemisthereforeclea rlyNPHardfor themultipleperiodcasewithorwithoutequalcapacitiesin allperiods.
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78 3.4 Conclusions Allocatingappropriateamountsofresourcestoanticipatedd emandsources hasbeenawellresearchedprobleminrevenuemanagement,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 ThedualascentalgorithmpresentedinSection 3.2.2 solves[CDP]. Moreover,thecomplementaryprimalsolutiontothedualsoluti onproducedbythe algorithmsatisestheintegralityrestrictionsofformulatio n[RPP PLC ]andtherefore providesanoptimalsolutionfortheRPP PLC Proof: Let F ( )denotetheoptimalvalueofaproblemconsistingofperiods1, ..., .Aswehavedemonstratedthroughourshortestpathapproach,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,thenwehaveasingleperiodproblem(forperiod2)a ndwecanrefer totheproofaboveforthecaseof =1.Supposethenthatneither J 1 1 nor J 1 2 is empty.Intheprocessofapplyingourdualascentalgorithm,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 ) wheretheproductionsubplaninperiods +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,manufacturingrmsareproducingitemswitheverdecr 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. EvenifthermoperatesinasocalledQuickResponse(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,allofwhichoershortlifecycleproducts (somewithasinglesellingseason)withunpredictabledemand,to determine howeectivelyeachcompanyusedavailabledatasourcestounde rstandtheir customers.Inthepresentmarketplace,theseretailersaresayin gthattheymust makebetteruseofdemandinformationiftheywanttomakeprot ablemarket selections.Finally,CarrandLovejoy[ 20 ]alsodiscussthisproblem'smotivation fromaninversenewsvendorpointofview.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 singleperiod\newsvendor"setting. Additionalliteratureconsidersthemultiitemnewsvendorpr oblemaswell, forwhichwecandrawsomesimilaritiestoour\multiplemarke t"setting.Inour problemwehaveonecostforproducingasingleproduct,andthe individualmarket deliverycosts,salesandadvertisingcosts,andrevenuesprovide dierentiation amongmarkets.Eachmarkethasacertainamountofrandomdema nd,andweattempttosatisfythedemandfromthemarketsweselecttomaximiz eoverallprot. Inthetypicalmultipleproductsetting,eachproducthasun iqueproduction,salvage,orderingandperhapsdistributioncosts.However,therei snodierentiation betweenthedemandsourcesforaparticularitem.MoonandSil ver[ 55 ]present heuristicapproachesforsolvingthemultiitemnewsvendorpr oblemwithabudget constraint. Otherresearchershaveinvestigatedhowproductioncapacity canbeadjusted withintheframeworkofanewsvendortypeproblem.FineandF reund[ 25 ]consider costexibilitytradeosininvestinginproductexiblemanuf acturingcapacity. Theyformulatethecapacityinvestmentdecisionasatwostage 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,SimchiLeviandSwann[ 14 ],PetruzziandDada[ 59 ],andMonahan, Petruzzi,andZhao[ 54 ].ThemorerecentworkbyMonahanetal.[ 54 ]focuses ontheparallelbetweenthedynamicpricingproblemandthed ynamicinventory problem.Theyoerareinterpretationofthedynamicpricing problemasapricesettingnewsvendorproblemwithrecourse,whichleadstoinsigh tsintotheactions andprotsofapricesettingnewsvendor. Additionaldemandormarketselectiondecisionscanbefoundin thegametheoryliteratureconcerningmarketentry,asdiscussedinRhim,Ho ,andKarmarkar [ 63 ].Whenfacedwithcompetitionforanymarketdemandsegment, armmust makeselectionsconcerningproductionsites,capacitiesandq uantities.Theyfocus onamultirmapproach,whereeachcompetitor'sdecisionswi llbeaectedby thetimingofentryoftheothercompetitors,andeachcompeti tor'slevelofentry. Ourcurrentresearchdoesnotconsidermultirmdecisions,andt hiscouldbean interestingextensiontothesinglermdecisionswepresentinth isdissertation.
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90 Ourworkisbasedlargelyontherelationshipbetweenexpected revenue anddemanduncertainty,whichisdirectlycorrelatedwitht heareaofportfolio optimization.TheseminalworkbyMarkowitz[ 53 ]over50yearsago,followed bycountlessarticlesinthisstreamofresearch,introducedth econceptofmeanvarianceoptimization.Themeanvarianceapproachattemp tstoachieveadesired rateofreturnwhileminimizingtheriskinvolvedwithobtain ingthatreturn.We determineanoptimalsetofmarketsbasedontheirexpectedrev enues(orreturns) andtheassociateddemanduncertainties(returnrisk).Ourmod elingapproach diersinthatwedonotplaceaminimumlevelonexpectedprot, nordowefocus onriskminimizationwhilemeetingadesiredprotlevel.Foram orerecentreview ofportfoliooptimizationandriskaversion,seeBrealyandMye rs[ 16 ]. Someofthemorecloselyrelatedstochasticworkondemandando 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 forbothmaketostockandmaketoorderproducts.Acontrac tualobligation existstoproducethemaketostockdemand,andthermcansuppl ementits productionbyacceptingadditionalmaketoorderdemandso urces.Theyapproach thisproblemusingqueueingtheory,inaneorttoprovideanop timaladmissionof maketoorderdemandandoverallsequencingofproductionj obs.BeyondtheCarr andDuenyas[ 19 ]paperonjointadmissioncontrolandproductiondecisions,the mostrelevantworkisfoundinHa[ 35 ],whichpresentsaqueueingtheoryapproach tostockrationingacrossseveraldemandclassesforasingleitem ,maketostock productionsystem.
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91 CarrandLovejoy[ 20 ]examineaninversenewsvendorproblem,whichoptimallychoosesademanddistributionbasedonsomepredenedorde rquantityor capacitysetbyasupplier.Theyselectademanddistributionfro masetoffeasible demandportfolios,whichmayincludeseveralcustomerclasses.To createademand portfolio,theyselectthesecustomerclasses,eachofwhichhasra ndomdemand thatfollowsanormaldemanddistribution,anddeterminethe amountofdemand tosatisfywithineachclasswhilenotexceedingthepredenedc 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 isnetofanymarketspecicunit 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,andthatahighcostdomesticsu 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'sperunitrevenue(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 notpreselectedforentryhavebeensortedaccordingtotheDE 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 Noticethatwehavegeneralizedthesinglemarketconditiont 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 impliesthattherequiredperunitnetrevenue, 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,thermmustreevaluatetheovera 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 allowsustocheckprotabilitybasedonthismarketspecicinf 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 axedorpredeterminedlevelofmarketing.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 Sshaped.Formostindustriesorproducts,oneoftheseresponsefun ctionswill approximatethebehaviorofdemandincreaseswithrespecttot headvertising level.Inrecentyears,researchershavefocusedmoreonScurv eddemandfunctions, whicharebelievedtobemorebroadlyapplicableinindustry. Incontrast,Simon [ 73 ]showsthatcertainproprietarybrandsactuallybehaveinan asymmetric fashiontoadvertising;i.e.,demandpeaksimmediatelyafter theadvertising increase,butthelongtermdemandlevelismuchlowerthanth einitialpeak.Since wefocusonasingletimeperiod,thistimebasedeectdoesnota pplyinourcase. 4.3.1 SelectiveNewsvendorwithMarketingEort Inordertoformulateamoregeneralmodelthatincludessales andadvertising, weintroducesomeadditionalnotation.Let a i denotethenumberofunits(e.g., hours,days,employees)ofmarketingeortexpendedinmarket i ,andlet t i > 0 betheperunitcostofthiseort.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,orthesocalledselecti venewsvendorproblem withmarketingeort(SNPM).[SNPM] 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[SNPM]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 i a t i /R i () i a a s i j+1 ,1 ij a 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 piecewiselinearfunctionwithdecreasingslopes 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,thisisjustaspecialcaseofpiecewiselinearconcaved 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 ScurvedDemand 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 .AnexampleofanScurveis giveninFigure 4{2 Weareinterestedinndinganoptimalmarketingeortlevel^ a i forthis demandfunction.Tothisend,weexaminethetwodierentcomp onents (1)i and
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109 a 1 i a 2 i a i b () i a i t i / ( r i c ) t i / ( r i c ) i Figure4{2:OptimalmarketingeortforScurveddemandrespo 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,wewilladoptanapproximationoftheSshapedcurve,abr oadlyapplicable advertisingresponsefunctionforexpecteddemand. Assume i ( a i )isaconvexincreasingfunctionfor0 a i b i ,andlet i (0)= i and i ( a )= i forall a b i .ThisdescribestheScurvefunctionforexpected
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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 iii ab () i a i Figure4{3:ApproximationoftheScurveddemandresponsefunc tion. Thefollowingtheoremshowsthat,undertheseassumptions,weonl yneedto considertwodistinctadvertisinglevelsineachmarketwhensol ving[SNPM],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(SNPDV):[SNPDV] 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 demandfunctionfollowstheScurveapproximationfunctio 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[SNPM],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 Wewillimplementabranchandbound(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,[RLM( 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:[RLM( 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[RLM( 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[RLM( I 1 ;I 2 )]asfollows. Introducingthenonnegativedualvariables ; i ; i ,wepresenttheKKT conditionsfor[RLM( 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[RLM( 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[RLM( 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[RLM( 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
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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 [RLM( 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[RLM( 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 BranchandBoundSolutionto[LM]0) Assumewearegivensets I 0 I 1 ,and I 2 .Setthelowerbound( LB )to0,and settheupperbound( UB )toinnity.SolvetheLPrelaxationproblem [RLM( 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 [RLM( 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 eachmarketcontainsuniqueperunitrevenues,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 willingtoacceptproductexpeditingonethirdofthetime. 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 orthe10and20marketcasesatlowdemandvariancesettings.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[RLM( I 1 ;I 2 )]at therootnode,whereweassumethatwearegiventhemarketscont ainedinsets I 0 I 1 ,and I 2 .Wewillrefertothisrelaxationproblemas[RLM( I 1 ;I 2 )].Recall thatwehavepresentedsolutionapproachesto[LM]fordemandf unctionsthat followeitherCaseIorCaseIIassumptions.CaseIIassumesexpectedd emand
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133 followsanSshapedapproximationthatisconvexandnondecr 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[RLM( I 1 ;I 2 )]toproblem [LM]forCaseII,inwhichexpecteddemandanddemandvariance increaselinearly withthelevelofmarketingeort.Onlyforanadvertisingbudg etoflessthan 50%oftotalexpecteddemandisthereasignicantintegrality gapbetweenthe relaxedandmixedintegerformulations.Asthemarketpoolinc reasestoasizeof 50potentialmarkets,thesolutionprovidedby[RLM( 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 [RLM( 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 [RLM( 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 [RLM( 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[RLM( I 1 ;I 2 )]toproblem [LM]forCaseI,inwhichexpecteddemandincreaseslinearlywi ththelevelof marketingeortanddemandvarianceremainsconstantatanyad vertisinglevel. Noticethattheintegralitygapforinstanceswhentheadverti singbudgetisless than50%ofexpecteddemandissignicantlyhigherthanthesame resultsfor CaseII.Recallthat,for[RLM( I 1 ;I 2 )],weinvokealinearityassumptionforall demandfunction'sresponsetoadvertising.Thisimpliesthat, forCaseIandII, [RLM( 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 [RLM( 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 [RLM( 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 [RLM( 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[RLM( 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 incurredwithinaperiodarenowbackorderedatacostof b perunit.Givena multiperiodproblem,thereisnolongeraneedtosalvageexc essproduct.Instead,
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136 wemaintaintheproductininventoryatacostof h perunitineachperiod.We alsoassumethattheproductdoesnotchangeandcanbesoldinfutu reperiods. Thermmustalsohaveanunconstrainedsupplyandeitherhavenegl igibleorder costsperperiodorhavescheduleddeliveriesineveryperiod. Intheinnitehorizonproblem,wewanttomaximizethelongr unexpected protperperiod.Assumingfullbackorderingofdemand,wecane xtendthe SNPmodeltoaninnitehorizoninasimilarmannershowninNahmias [ 56 ].To determinethelongrunexpectedprot,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.Sinceanydemandnotmetisbackordered,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 ,wecanrepresentthelongrunexpectedprot 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'slongrunexpectedprotas: 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'slongrunexpectedprotas: 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(SNPI): [SNPI] 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 nolongerinuencesuncertaintyrelatedcosts.Aswithformula tion[SNP],wecan applytheDERURatiopropertyto[SNPI]anddeterminetheopt imalselectionof markets,wheretheratioforeachmarket i isdenedby r i = 2 i 4.6.2 LimitedMarketingEortunderaFixedContract Instead,let'sassumewearegivenapredenedselectionofmarke 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(SNPLR)canbeformulatedas:[SNPLR] 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'sconsiderthecaseofhavingpiecewiselinearnonde 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 basedonthepiecewiselinearfunctiondenedin 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 thisapproachwillworkisthatwehavecommittedupfrontto enteringallofthe marketsincludedin[SNPLR],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 problemstructuretoprovideaclosedformsolutionbasedonthe rankingofeach market'sattractiveness,whichwecalltheDERU(DecreasingEx pectedRevenue toUncertainty)ratio.Oncewehaveanoptimalranking,weoer severalwhatif scenariosforthermtoconsider.Eachoftheseinsightscanplaya roleindeciding whethertoexpanditspresenceinnewmarkets,increasethemark etingeort,or oersomeformofpricediscounting. Wealsoconsidertherolethatsalesandadvertisingplaysindete rminingthe demanddistributionobservedwithineachmarket.Whenthermd ecidesapriori toapplycertainxedmarketinglevelsineachmarket,wearef acedwithsolving anintegerproblemwithanonlinearprotmaximizationobjec tive.Andwehave shownthroughtheDERUratiopropertythatthisproblemissurp risinglyeasyto solve.BeyondthisxedadvertisingbasicSNP,weevaluatedafai rlygeneralsetof advertisingresponsefunctions,eachofwhichhasuniqueproper tiesfortheoptimal selectionofmarketsandthecorrespondingadvertisinglevels inthesemarkets. Forthecasesinwhichtherewasanunlimitedmarketingbudget ,wecandeterminetheoptimalmarketingleveltoexpendineachmarket,an dweshowhowthe problemsimpliestothebasicSNP.Combiningtheeectsofalimit edmarketing budgetwithaprotobjectivebasedonexpectedrevenueswithd emanduncertainty,wearethenfacedwithanonlinearknapsackproblemwi thanonseparable objectivefunction,whichisaverydicultproblemtosolvein general.Forthe limitedresourcescase,weprovideabranchandboundproced 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 programmingapproachtoaddressweatherrelateduncertain 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 (weatherrelatedornot)donotoccur. Researchhasbeenconductedonthegroundholdingproblemfor singleand multipleairports,andbothstaticanddynamicversionsofthep roblemexist.The staticversionassumesthatthecapacityscenariosaredenedonce atthebeginning ofthegroundholdingperiodunderevaluation.Ouranalysisf ocusesonthestatic, stochasticsingleairportversionoftheproblem.Foradditiona lbackgroundonthe
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144 static,singleairportproblem,seeBall,etal.[ 8 ],Grignon[ 34 ],HomanandBall [ 36 ],RichettaandOdoni[ 64 ],andRifkin[ 65 ].Whileourcontributionstothestatic modelingapproachoerseveralnewinsights,werecognizethepo tentiallimitation incorrectlyrepresentingtherealtimechangesinairporto 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 NavazioandRomaninJacur[ 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 riskaverseconstraintsinthemodel.Inotherwords,byrestrict ingthesizeofthe worstcasedelays,howistheoverallexpecteddelayaected?Th isisnotthesame asamaximumdelaymodel,whichwouldplaceastrictupperboun donworstcase delays.Finally,wesummarizeourndingsanddiscussdirections forfuturework.
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145 5.2 StaticStochasticGroundHoldingProblem 5.2.1 ProblemDenitionandFormulation Wedeneapotentialgroundholdingperiodtoconsistofanitenu mber, T ,of15minuteincrementsorperiods : 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,allenrouteaircraft,includ ingthoseontimeand
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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 numberof15minuteperiods.Aswillbeshown,theexperimentsp resentedinthis reportarebasedona22scenario,24periodproblem,whichim pliestheevaluation coversasixhourtimeframe.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 asixhourtimeframe.Basedonactualdatafromamajorairport ,anestimated arrivaldemandin15minuteincrementswasobtained.Figur e 5{1 presentsthis arrivaldemanddata.Eachchartshowstypicalarrivalpatter nsforanairportwith Arrival Demand Test 2 0 5 10 15 20 25 30147 101316192215min window Arrival Demand Test 3 0 5 10 15 20 25 30147 101316192215min window Arrival Demand Test 4 0 5 10 15 20 25 30147 101316192215min window Arrival Demand Test 1 0 5 10 15 20 25 30147 101316192215min 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 beginningofthesixhourperiod,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 capacityreducedset,therstscenariorepresentsaparticular weatherinduced 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 sevencapacityreducedsets,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 1113151719212315min window CP0 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window CP19 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window CP4 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window CP7 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window CP16 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window CP10 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window CP13 Arrival Capacity 0 5 10 15 20 2513579 1113151719212315min window Figure5{2:Weatherinducedarrivalcapacityscenarios. (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 resolvethestochasticproblemusingtheserecommendedgroundh oldsasxed amounts.Thedierencebetweentheoriginalstochasticsolution andthesolutionusingthispredenedgroundholdingpolicyistheVSS.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.TheValuea tRisk(VaR)measure hasbeenextensivelystudiedinthenancialliterature.Morer ecently,researchers havediscoveredthatanothermeasure,ConditionalValueatRisk(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 theexpectedvalueofapercentileoftheworstcasedelays;i.e .,weplacetheCVaR measureintheobjectivefunction.
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156 InordertosetuptheCVaRmodel,additionalvariablesandpar ametersare required.Let representthesignicancelevelforthetotaldelaycostdistribu tion acrossallscenarios,andlet beadecisionvariabledenotingtheValueatRisk forthemodelbasedonthe percentileofdelaycosts.Inotherwords,in %of thescenarios,theoutcomewillnotexceed .Then,CVaRisaweightedmeasure of andthedelaysexceeding ,whichareknowntobetheworstcasedelays. 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(ConditionalValueatRisk) 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; WorstCaseTailDelays: 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 dierenceswouldneedtobedealtwithonacasebycasebasis,and wepresent somealternativemodelstoaccommodatethegoalsofdierentd ecisionmakersin Section 5.4.3 5.4.2 MinimizeTotalDelayCostModelvs.MinimizeConditionalValu eatRiskModel TheCVaRmodelrequirestheadditionalinputofasignicancel evel,and = 0.9ischosenfortheanalysis.Table2presentsacomparisonofthe delaystatistics fortheMinimizeTotalDelayCostModel(SSGHP)andtheMinimiz eConditional VaRmodel(GHP{CVaR). ResultsofthemodelcomparisonsshowthatintheCVaRmodel,tot alexpecteddelaywillbeincreasedinordertoreducetheworstcase delaysacrossall testcases,whichsupportsourexplanationdescribingtheuseofri skaversion.By examiningtheresultsmoreclosely,wenotesomeinterestingndi ngs. Observethedierenceinvaluesfor VaRand CVaRwhennoriskis modeled.Thisillustratestheimportanceofconsideringthea verageofworstcase delaycostswhenyouchoosetomodelrisk.VaRtendstooverlookt hedierences indelaysbeyondthecriticalvalue,anditmaynotbeabletor educetheworstcase 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%worstcase delays(using theGHP{CVaRmodel),theexpectedsavingsreducesto515.Butw ealsohave reducedtheworstcasedelaysfrom7624to5039. 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.Thetradeowiththismodeling approachisthat therearenoscenarioswithonlyminimaldelays.Withfurthera djustmentstothe valueof ,thedecisionmakershavesomecontroloverhowtheresultingde lay outcomeswouldlook.Itthendependsonwhatthedecisionmake rsarewillingto accept.Thisisoneoftheunderlyingpowersofintroducingr iskaversionsuchas minimizing CVaR. 5.4.3 AlternateRiskAversionModels Dependingontheinputfromeachgroupinvolvedinconstructi ngaground holdingpolicy,therewillbeconictingdesirestoreducetot alexpecteddelayand toreducetheworstcaseoutcomes.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,tominimizeworstcasedelay( 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 ofworstcasedelaystobenomorethansomeparameter, 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) ; WorstCaseDelayBound: +(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. Andwecanevennetuneourobjectivefurtherthroughtheuseof theCVaR constraint.As CVaRisincreased,weapproachouroriginalSSGHPmodel. Inadditiontotheaboveriskmodels,Rifkin[ 65 ]brieypresentstheMaximumAirDelayModel(MADM).MADMcanbethoughtofasamaximumlo ss constraintforanyscenario,andifsuchanumberexists,thiscoul dbeaddedto anyoftheaboveformulations.WhatMADMfailstoaddressisthec ontinuing eorttominimizetotaldelays.Amaxlossconstraintcanbeadde 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 basedonvariousworstcasedelayscenarios.TheFAA,airportautho rities,and airlinesallworkwithinsomepredenedperformancemeasures, 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 handlethisbysimplyresolvingthestaticmodelastimeprogre 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 andRomaninJacur[ 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.Theproductionrelate dproblemsincluded uncapacitatedandcapacitatedversionsoftheorderselectio nproblem,demand selectionwithpricingasadecisionvariable,andalsostochasti cdemandselection problemsthatallowthedecisionmakertoinuencedemandthro ughmarketing eort.Wethentiedtheseeortsinwiththeapplicationsareaof airportoperations. Thedemandselectionproblem,especiallyinthemanufacturin gsetting,hasgone relativelyunnoticedintheliteratureuntilrecently.Weh aveprovidedathorough discussionofafamilyofmodelsthatexistinthisarea. Themodelswepresentserveasastartingpointforfutureresearc honmore generalmodels.Werstaddressfutureresearchareasspecictothe multiperiod 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 demandselectionproblembecomesamultiperiod,dynamicsto chasticdecision problem,whichcouldprovetobeverydiculttosolve.And,fort hisreason,it alsoremainsaveryinterestingtopicforfutureresearch.
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BIOGRAPHICALSKETCH IwasbornonApril27,1966,inBerwyn,Illinois.Aftermaintain ingasteady interestinmathematicsthroughoutmiddleschoolandhighscho ol,Iattended theUniversityofIllinoisatUrbanaChampaign,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



Full Text  
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,ix 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,ii 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,ix 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,ii 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 OSPNDC . .... 33 2.4.3 Results for the OSPAND .................. .. 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 Dualascent Method for the Uncapacitated RPP . 55 3.2.3 Polynomial Solvability of More General Models ...... 65 3.2.3.1 Multiple pricedemand curves . . 65 3.2.3.2 Piecewiselinear 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 Scurved 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 21 Counterexample illustrating decreasing cumulative demand satisfaction. 15 22 Classification of model special cases and restrictions. . .... 16 23 Problem size comparsion for capacitated versions of the OSP. ..... ..17 24 Probability distributions used for generating problem instances . 30 25 OSPNDC and OSP problem optimality gap measures . .... 35 26 OSPNDC and OSP solution time comparison. ........... ..36 27 OSP and OSPNDC heuristic solution performance measures. ...... 37 28 OSPAND optimality gap measures. ................. 40 29 OSPAND solution time comparison. .................. 40 210 OSPAND heuristic solution performance measures. . .... 41 41 Results for SNP with limited resources Case II. . ..... 134 42 Results for SNP with limited resources Case I. . ..... 135 51 EVPI and VSS statistics (\!iiiiiii..e total expected d, li cost model).. 154 52 Overall model comparisons. ............... .... 158 53 Performance comparison of alternate risk models . ...... 162 LIST OF FIGURES Figure page 21 Fixed charge network flow representation of UOSP. . .... 11 22 Shortest path network structure for UOSP. .............. ..13 31 Pricing interpretations based on total revenue and demand. . 74 41 Optimal marketing effort for concave expected demand functions. .. 107 42 Optimal marketing effort for Scurved demand response functions... 109 43 Approximation of the Scurved demand response function. ..... ..110 44 Minimum market requirement based on individual cost parameters. 128 45 Profit improvement using SNP based on total markets available. 131 46 Profit improvement using SNP based on demand variance. ..... ..132 51 Aircraft arrival demand at the capacitated airport . . .... 148 52 Weatherinduced arrival capacity scenarios. .............. ..150 53 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 timevarying 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 singlestage, multiperiod 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 optimizationbased modeling and solution methods for these problems. We provide a polynomialtime 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 singlestage production planning problem to address the importance of pricing. Firms that manufacture and sell products with priceelastic 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 polynomialtime 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 polynomialtime solution approach under timeinvariant finite production capacities and piecewiselinear 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 singleperiod 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 predefined 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 madetoorder 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 customerrequested due dates for the orders it accepts. In contexts where capacity limits are nonbinding, 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 lotsizing 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 prespecified 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 NPHard and therefore requires customized heuristic solution approaches. We propose strong LP formulations of the capacitated version, which often allows solving general capacitated instances via branchandbound in reasonable computing time. For those problems that cannot be solved via branch andbound 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 singleorder 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, singlestage lot sizing problem. Their model still assumes ultimately, however, that all prespecified 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 WagnerWhitin [83] shortest path solution method for singlestage 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 Tperiod 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 fixedcharge network flow problem as illustrated by the example in Figure 21, 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=1imM() mt Dummy Source Supply = D Figure 21: Fixed charge network flow representation of UOSP. and order selection arcs. The dummy source node implies that it is not necessary to satisfy all demandthe 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 WagnerWhitin singlestage 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: Allornothing 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 orderperiod combination (m, t) the producer either provides dmt units or none at all. We next consider how to extend the WagnerWhitin 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 22). Note that this method relies on the existence of an optimal ZeroInventory 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 WagnerWhitin 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 22: 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 WagnerWhitin 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 twoperiod problem is at least as great as that satisfied in a oneperiod 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 threeperiod problem: Table 21: 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 OSPNDC. 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 deliverycharge forcing (zmt) variable values, and can therefore be substituted out of the formulation; let OSPAND denote this version of the model (where AND implies allornothing demand satisfaction). Observe that for the OSPAND 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 22 defines our notation with respect to the different variants of the OSP problem. Table 22: Classification of model special cases and restrictions. Fixed Delivery Partial Order Model Charges Exist Satisfaction Allowed OSP Y Y OSPNDC N Y OSPAND 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 Tperiod 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 OSPNDC model, on the other hand, in which Fmt = for all orderperiod (m, t) combinations, allows us to replace each Zmt variable on the righthandside of constraint set (2.4) with a 1, and eliminate these variables from the formulation. The OSPNDC 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 OSPAND model, customers do not allow partial demand satisfaction, and so we require Vmt = mt for all orderperiod (m, t) combinations; we can therefore eliminate the continuous Vmt variables from the formulation. While the OSPAND, 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 23 summarizes the size of each of these variants of the OSP with respect to the number of constraints, binary variables, and total variables. Table 23: Problem size comparsion for capacitated versions of the OSP. OSP OSPNDC OSPAND 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 OSPNDC. As we will show in Section 2.4, the OSPAND actually requires the greatest amount of computation time on average, while the OSPNDC requires the least. Note that the OSPAND is indifferent to whether fixed delivery charges exist, since we can simply reduce the net revenue parameter, Rmt rmtdmt, by the fixed deliverycharge value Fmt, without loss of generality. In the OSPAND 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 deliverycharge forcing variables, since their objective function coefficients are fixed delivery cost terms rather than net revenue terms, as in the OSPAND. Note also that since both the OSPNDC and the OSPAND require only one set of order selection variables (the continuous vmt variables for the OSP NDC and the binary Zmt variables for the OSPAND), 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 OSPNDC and the OSPAND. 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 OSPAND 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 NPHard through a reduction from the capacitated lotsizing 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 lotsizing problem, which is an NPHard optimization problem (see Florian and Klein [28]). Given that the OSP is NPHard, 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 branchand 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 branchandbound 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 andbound in acceptable computing time. For those problems that cannot be solved via branchandbound, 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 OSPLP, 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 Note that this LP relaxation approach is valid for all three variants of the OSP: the general OSP, the OSPNDC, and the OSPAND. Observe that the above constraint can be strengthened by multiplying the righthandside 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 righthandside 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 righthandside 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 OSPLP 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 branchandbound 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 Lagrangianbased 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 righthand side of the capacity/setup forcing constraints (2.3), and then relax the resulting modified capacity constraint (without the yt multiplier on the righthand 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 capacityinfeasible (but otherwise feasible) solution and generate a capacityfeasible 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 allor 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 capacityfeasible 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 wellknown heuristic solution approaches for the ELSP. In particular, we use a similar i'y" '1.. approach to those used in the SilverMeal [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 (SilverMeal) 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 SilverMeal and Least Unit Cost heuristics is required for application to the capacitated lotsizing 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 capacityfeasible 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 capacityconstrained problem, we can either consider period j' + 1 (as is done in the SilverMeal 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 OSPNDC problem, since the Zmt variables do not exist in this special case). We first consider the solution that results by setting all (nonzero) 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 capacityfeasible 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, OSPNDC, and OSPAND 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, OSPNDC, and OSPAND versions of the problem. Within each problem version (OSP, OSPNDC, and OSPAND), 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 24 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 24 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 24: 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 periodtoperiod) 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] periodtoperiod) U[d .15d, d + .15d] Per unit order revenue (varies 2 U[28,32] from ordertoorder) 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 perperiod total demand, which equals the mean of the distribution of order sizes multiplied by the number of orders per period. Table 24 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 200order problem sets is four times that of the 50order 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 25order 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 16period 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 smallleading 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 perorder delivery charges, but the results were nearly equivalent to those presented for the OSPNDC 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 OSPNDC 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 branchandbound 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 branchandbound algorithm, leading to alternative optimal solutions at nodes in the branchandbound tree. For example, if an order m in period t is satisfied at the current node in the branchandbound 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 branchandbound 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 branchandbound 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 branchandbound algorithm as the MIP solution. The remaining subsections summarize our results. Section 2.4.2 reports the results of our computational experiments for the OSPNDC and the OSP, and Section 2.4.3 presents the findings for the OSPAND (allornothing 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 alvi satisfy all or none of the demand for the OSPAND, this is without loss of generality). 2.4.2 Results for the OSP and the OSPNDC 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 OSPNDC 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 OSPAND, as we later discuss in Section 2.4.3. Surprisingly, the OSP (which includes a binary fixed deliverycharge forcing (zmt) variable for each orderperiod combination) was not substantially computationally challenging either. All of the OSPNDC and all but two of the OSP instances were solved optimally using branchandbound within the allotted branchandbound time limit of one hour. Even though we are able to solve the OSP and OSPNDC 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 (OSPLP) relaxation upper bound, the .,..regated setup forcing (ASF) relaxation upper bound, and the di, _regated setup forcing (DASF) relaxation upper bound for the OSPNDC 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 branchandbound. As Table 2.4.2 shows, for the OSPNDC, 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 OSPLP relaxation, and the [DASF] formulation provides the tightest bounds. We note that as the number of potential orders and the perperiod 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 200order cases. For the OSP, we have nonzero 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 OSPAND indicates. The upper bound optimality gap results reported in Table 2.4.2 for the OSP are significantly larger than those for the OSPNDC.4 This is because this formulation permits setting fractional values of the fixed deliverycharge 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 26, the size of this formulation 4 For the two problems that could not be solved to optimality via branchand 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 25: OSPNDC and OSP problem optimality gap measures. OSPNDC OSP % Gap Orders per Period Overall Orders per Period Overall (from MIP) 25 50 200 Average 25 50 200 Average OSPLPa 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 (OSPLP MIP)/MIP x 100%. b (ASF MIP)/MIP x 100%. c(DASF MIP)/MIP x 100%. d % of problems for which CPLEX branchandbound found an optimal solution. Table 26 summarizes the solution times for solving the OSPNDC 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 OSPLP 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 (OSPLP, 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 alvi a viable approach for the restricted OSPAND discussed in the following section. Table 26 reveals that the MIP solution times for the OSP were also much greater than for the OSPNDC. This is due to the need to simultaneously track the binary (zmt) and continuous (vmt) variables for the OSP with nonzero 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 26: OSPNDC and OSP solution time comparison. OSPNDC 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 OSPLP Solution Time 0.1 0.1 0.3 0.1 0.1 0.3 Maximum OSPLP 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 OSPNDC. We employ the three heuristic solution methods discussed in Section 2.3.3, denoting the Lagrangianbased heuristic as LAGR, the greatest unit profit heuristic as GUP, and the LP rounding heuristic as LPR. Table 27 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 OSPNDC 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 27 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 OSPNDC, 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 Lagrangianbased 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 OSPNDC. 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 27: OSP and OSPNDC heuristic solution performance measures. OSPNDC 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 OSPAND We next provide our results for the OSPAND 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 OSPAND 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 28 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 andbound algorithm. Observe that the upper bound optimality gap measures are quite small and only slightly larger than those observed for the OSPNDC. 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 allornothing requirement for at most one order per period. Thus, even in the OSPNDC 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 OSPAND model now represent "order selection" variables rather than fixed deliverycharge 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 deliverycharge 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 25order case to the 200order 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 28 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 OSPNDC, because of the complexities introduced by the allornothing order satisfaction requirement. Table 29 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 OSPAND LP relaxation is identical to the OSPNDC LP relaxation, and these times are therefore shown in Table 26. Unlike our previous computational results for the OSP and the OSPNDC, we found several problem instances of the OSPAND 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 OSPAND problem. Interestingly, however, in contrast to our previous results for the OSP, the average and maximum MIP solution times for the OSPAND were smaller for the 200order per period problem set than for the 25 and 50order 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 28: OSPAND optimality gap measures. Orders per Period Gap Measurement 25 50 200 Overall Average OSPLP 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(OSPLP MIP)/MIP x 100%. b (ASF MIP)/MIP x 100%. S(DASF MIP)/MIP x 100%. d % of problems for which CPLEX branchandbound found an optimal solution. Table 29: OSPAND 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 210 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 OSPNDC, 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 OSPNDC and the OSPAND occurred using the LP rounding heuristic applied to the DASF LP relaxation solution. As Table 26 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 OSPNDC and OSPAND, 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, OSPNDC, and OSPAND 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 210: OSPAND heuristic solution performance measures. OSPAND 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 WagnerWhitin [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 allornothing 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 capacityfeasible solution. Note that we present the FSG algorithm as it applies to the OSP, and that certain straightforward modifications must be made for the OSPAND 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 capacityinfeasible 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 capacityinfeasible) 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 perunit profitability. Perunit 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 underutilized periods 0) Create a new list for each period of all profitable orders not fulfilled. Each list is indexed in nonincreasing order of perunit 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 madetoorder 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 demandbased 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 maketoorder 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 singlestage systems. Most of the requirements planning literature focuses on production require ments based on prespecified 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 pricedemand 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 piecewiselinear concave revenue functions. Given fixed plus linear production costs and piecewiselinear concave revenue functions, we also provide a 'tight' linear programming formulation of our model, using a dualbased 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 timeinvariant production capacity limits and piecewiselinear 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 downwardsloped 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 prespecified 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, SimchiLevi, and Swann [14] analyzed a model similar to ours under strictly linear production costs (i.e., no fixed setup costs, and assuming timevarying 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, SimchiLevi, 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 WagnerWhitin [83] shortest path solution method (discussed in Chapter 2) to contexts with general concave revenue functions and fixedcharge production costs. Assuming piecewise linear concave revenue functions, we then provide a dualbased polynomial time algorithm for solving the uncapacitated problem. This dualbased 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 pricedemand curves in any given period. In addition to presenting solution approaches to several incapacitated versions of the problem, we provide an analysis of the equalcapacity version of the model under piecewiselinear 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 onetoone correspondence between price and market demand volume in any period, where market demand is a downwardsloping 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 endofperiod 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 fixedcharge 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 piecewiselinear 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 WagnerWhitin [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 fixedcharge 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 piecewiselinear 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 fixedcharge 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 WagnerWhitin [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 Tperiod RPP problems into a set of smaller contiguous interval subproblems, using the shortest path graph structure previously shown in Figure 22. 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 (nonnegative) 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 t71 tI1 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 piecewiselinear 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 k1 ki 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 piecewiselinear 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 Dualascent Method for the Uncapacitated RPP When the revenue functions are piecewiselinear and concave in every period, and production costs contain a fixed plus variable cost structure, we can also use a dualbased 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 piecewiselinear 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 piecewiselinear 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 piecewiselinear concave revenue functions, which we refer to as the RPPPLc, as follows. [RPPPLC] maximize: _t (Ejl 1 rjtdjtzjt Stt ptxt htIt subject to: It1 + = 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 equalcapacity 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 piecewiselinear 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 piecewiselinear 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 jit 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 dualascent 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 iil, 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 dualascent 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 piecewiselinear 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 dualbased 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= flr ',, < 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 rewritten 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 segmentperiod 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 dualascent 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 (Pc2,0) 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(pQci,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 DualAscent Solution Algorithm 0. Delete any segmentperiod 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 k1 S (4) E E djt max 0,pjtc((k)) E E djt max(pl c(),0) t(2) 2tmxJ t=()jJ (If all C7(k) + 2 dk 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 dualascent algorithm by (JmaxT2), the same as that of the shortestpath algorithm in the previous section. We next show that the dualascent 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 dualascent 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 dualascent 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+ jj+ 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 dualascent ,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 dualascent 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 PriceDemand Curves To this point we have made two sets of key assumptions that have facilitated providing polynomialtime solution methods for the uncapacitated RPP. The first of these assumptions relies on the production cost function taking a fixedcharge structure in each period, while the second assumes that a single pricedemand curve exists in each period. We next explore the degree to which we can relax these assumptions, while retaining our ability to apply the polynomialtime 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 piecewiselinear concave production cost structure (which may include a fixed setup cost) in each period. 3.2.3.1 Multiple pricedemand 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 piecewiselinear 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 + It1 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 + It1 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 piecewiselinear and concave (and Rt(0) = 0) for all m and t, Rt() is piecewiselinear 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 Piecewiselinear concave production costs We next consider the case in which the production cost function in each period is piecewiselinear concave and nondecreasing in the production volume in the period. Note that any nondecreasing piecewiselinear concave function can be viewed as the minimum of a number of fixedcharge functions. Therefore, if the production functions are piecewiselinear 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 tf1 ,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 22) 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 piecewiselinear concave production cost functions. As this analysis shows, the case of piecewiselinear 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 timevarying finite production capacities is NPHard 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 equalcapacity 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 equalcapacity CLSP relies on characterizing socalled 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 equalcapacity 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 equalcapacity CLSP; thus, an optimal solution exists for the equalcapacity 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 capacityconstrained 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 shortestpath graph that contains a path for every feasible capacityconstrained production sequence between periods T + 1 and r'. Solving this shortestpath 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 equalcapacity 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 lotsizing 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 segmentperiod 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 equalcapacity 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) equalcapacity lotsizing 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 onetoone 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 pricedemand 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 vv depending on the model's intended application context. We use Figure 31 to illustrate two such interpretations. Figures 31(a) and 31(b) show identical piecewiselinear 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 31(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 31(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 31: Pricing interpretations based on total revenue and demand. In Figure 31(b), on the other hand, we assume that we have only a sin gle market available, and all satisfied demands provide the same perunit 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 31(a), we can now view the individual segments of the piecewiselinear 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 dualascent 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 equalcapacity case. Since the [RPPPLc] formulation served as our starting point for the analysis of the equalcapacity 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 equalcapacity version of the order selection problem. If, however, customers do not permit partial order satisfaction, the problem is NPHard. To demonstrate the difficulty of the problem when partial order satisfaction is not allowed, we next briefly consider the singleperiod 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 singleperiod 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 singleperiod 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 allornothing order satisfaction version of the capacitated problem problem is NPHard, even in the singleperiod special case (although the singleperiod version is not strongly NPHard). This problem is therefore clearly NPHard for the multipleperiod case with or without equal capacities in all periods. 3.4 Conclusions Allocating appropriate amounts of resources to anticipated demand sources has been a wellresearched 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 timeinvariant 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 dualascent 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 shortestpath 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 Tperiod 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 t1 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). t1 jGcJ If J, is empty, then we have a singleperiod 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 dualascent 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 T1 Sk E E dj max (0, pt Ck) djl max (p~ t=kjGO t=k ji: C dj, j GJ" The Tperiod objective function then becomes T Z d t (/ 1 pit) t 1 jeJl k1 E d (/ 4 t 1 j~Jl 71 P t= j+ t ( tk 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 tk jyGo 71 Ck) djl max (p t k jcG~ Ck,O) Y pjdj icJ Returning to the Tperiod objective function, and using our induction hypothesis, we now have 71 T Zj= F(k 1)+5djCk+Sk+ di t Pjt maxQ t Ck, 0)) djti max (0, pit Ck) Pj Tdj tkc 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) + Tl 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 tk 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 subplan 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 capacityconstrained 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 everdecreasing 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 socalled 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 shortlifecycle 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 pointofview. 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 Tzv, 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 singleperiod i,. v1., i'dor" setting. Additional literature considers the multiitem newsvendor problem as well, for which we can draw some similarities to our "multiplei1 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 multipleproduct 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 multiitem newsvendor problem with a budget constraint. Other researchers have investigated how production capacity can be adjusted within the framework of a newsvendortype problem. Fine and Freund [25] consider costflexibility tradeoffs in investing in productflexible manufacturing capacity. They formulate the capacity investment decision as a twostage 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 