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On the Joint Price and Replenishment Decisions for Perishable Products

Permanent Link: http://ufdc.ufl.edu/UFE0024368/00001

Material Information

Title: On the Joint Price and Replenishment Decisions for Perishable Products
Physical Description: 1 online resource (115 p.)
Language: english
Creator: Chen, Li-Ming
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: inventory, perishable, price
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A perishable item is characterized by its usefulness over a limited period of time, known as life. Once the life is over, these items spoil, which obviously is a loss. The bottom line of a firm can improve significantly if some of this spoilage is prevented, i.e., if the perishable nature of products is managed properly. One mechanism by which this may be achieved is demand management using price. Through an appropriate selection of price, demand can be modulated to improve profit. The modulation of demand can not only increase revenue but also reduce shortage, holding, and spoilage costs. Potential spoilage due to limited life-time of the perishable products is the main reason demand management of perishable products is even more important than for non-perishable products. Whereas for non-perishable products the only cost of unsold inventory is the cost for holding inventory, for perishable products the unsold inventory not only incurs inventory holding cost but, in addition, with increasing age of the inventory the risk of it remaining unsold by the end of its lifetime increases. Therefore, this dissertation explores joint demand and replenishment decisions on the inventory control of perishable items with random demand. The first part of the dissertation is primarily motivated by a dilemma routinely faced by food retailers: when to replace old inventory of perishable products with fresh units when economies of scale exist in order placement. On one hand, economies of scale make it more attractive to place orders for large quantities. On the other hand, the demand for perishable products declines as their age approaches their lifetime; the reduction occurs since customers prefer fresh units and/or avoid units that are close to expiry. To answer the above question, we develop three models with increasing flexibility. In each model, we consider a finite horizon periodic review system for a single product at a single retailer with a fixed cost of order placement. The retailer faces price-dependent stochastic demand and loses excess demand. The goal of each model is to identify when to place an order, the quantity whenever an order is placed, and the price in each period, regardless of whether an order is placed or not. The second part of the dissertation examines joint replenishment and price decisions when economies of scale in order placement do not exist and demand is age-independent. We consider a periodic review model over finite horizon for a perishable product with fixed lifetime equal to two review periods such that excess demand in a period is backlogged. The optimal replenishment and demand management decisions for such a product depend on the issuing rule. For both first-in, first-out (FIFO) and last-in, first-out (LIFO) issuing rules, we obtain insights on the nature of these decisions. We find that the insights on the optimal policies for non-perishable products do not extend to perishable products with multi-period lifetime. For the FIFO rule, we also obtain bounds on both the optimal replenishment quantity as well as expected demand. Taking a weighted average of these bounds, we propose an approximate policy; computational experiments indicate that the policy performs within 2% of the optimal profit for a wide range of system parameters. We also conduct experiments to understand whether demand management bridges the profit gap between the FIFO and LIFO rules as well as to explore whether the profit improvement due to demand management favors one issuing rule over another.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Li-Ming Chen.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sapra, Amar.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024368:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024368/00001

Material Information

Title: On the Joint Price and Replenishment Decisions for Perishable Products
Physical Description: 1 online resource (115 p.)
Language: english
Creator: Chen, Li-Ming
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: inventory, perishable, price
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A perishable item is characterized by its usefulness over a limited period of time, known as life. Once the life is over, these items spoil, which obviously is a loss. The bottom line of a firm can improve significantly if some of this spoilage is prevented, i.e., if the perishable nature of products is managed properly. One mechanism by which this may be achieved is demand management using price. Through an appropriate selection of price, demand can be modulated to improve profit. The modulation of demand can not only increase revenue but also reduce shortage, holding, and spoilage costs. Potential spoilage due to limited life-time of the perishable products is the main reason demand management of perishable products is even more important than for non-perishable products. Whereas for non-perishable products the only cost of unsold inventory is the cost for holding inventory, for perishable products the unsold inventory not only incurs inventory holding cost but, in addition, with increasing age of the inventory the risk of it remaining unsold by the end of its lifetime increases. Therefore, this dissertation explores joint demand and replenishment decisions on the inventory control of perishable items with random demand. The first part of the dissertation is primarily motivated by a dilemma routinely faced by food retailers: when to replace old inventory of perishable products with fresh units when economies of scale exist in order placement. On one hand, economies of scale make it more attractive to place orders for large quantities. On the other hand, the demand for perishable products declines as their age approaches their lifetime; the reduction occurs since customers prefer fresh units and/or avoid units that are close to expiry. To answer the above question, we develop three models with increasing flexibility. In each model, we consider a finite horizon periodic review system for a single product at a single retailer with a fixed cost of order placement. The retailer faces price-dependent stochastic demand and loses excess demand. The goal of each model is to identify when to place an order, the quantity whenever an order is placed, and the price in each period, regardless of whether an order is placed or not. The second part of the dissertation examines joint replenishment and price decisions when economies of scale in order placement do not exist and demand is age-independent. We consider a periodic review model over finite horizon for a perishable product with fixed lifetime equal to two review periods such that excess demand in a period is backlogged. The optimal replenishment and demand management decisions for such a product depend on the issuing rule. For both first-in, first-out (FIFO) and last-in, first-out (LIFO) issuing rules, we obtain insights on the nature of these decisions. We find that the insights on the optimal policies for non-perishable products do not extend to perishable products with multi-period lifetime. For the FIFO rule, we also obtain bounds on both the optimal replenishment quantity as well as expected demand. Taking a weighted average of these bounds, we propose an approximate policy; computational experiments indicate that the policy performs within 2% of the optimal profit for a wide range of system parameters. We also conduct experiments to understand whether demand management bridges the profit gap between the FIFO and LIFO rules as well as to explore whether the profit improvement due to demand management favors one issuing rule over another.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Li-Ming Chen.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Sapra, Amar.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024368:00001


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Iwouldliketoexpressmydeepestappreciationtomycommitteechair,Dr.AmarSapraforhispersistentguidanceoverlastfouryears.HehasnotonlyinstilledmewithcomprehensiveknowledgebutalsogivenmeaninsightintotheproblemsthatIhaveexamined.Hehaschallengedandmotivatedmetopursuequalityresearch,andhasmademeadeepthinker.Ialsowanttothankmydoctoralcommitteemembers,Dr.JosephGeunes,Dr.FaridAitSahlia,andDr.AydinAlptekinoglu,whohaveprovidedvaluablesuggestionsandconstructiveadviceduringthepreparationofthisworkandhavemademyresearchmorecomplete.Lastbutnotleast,Ican'tthankmyparentsenough.Theyhavealwayssupportedandencouragedmeinthepursuitofhighereducation.Withoutthem,Iwillnothavebeenabletoaccomplishthis. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2LITERATUREREVIEW .............................. 13 3INVENTORYRENEWALANDDEMANDMANAGEMENTFORAPERISHABLEPRODUCT:ECONOMIESOFSCALEANDAGE-DEPENDENTDEMAND 17 3.1FixedCycleModel ............................... 19 3.2FlexibleReplenishmentModel ......................... 25 3.3PartialSalvageModel ............................. 32 3.4NumericalExperiments ............................. 38 4JOINTINVENTORYANDDEMANDMANAGEMENTFORPERISHABLEPRODUCTS:FIFOVERSUSLIFO ......................... 44 4.1NotationandCommonAssumptions ..................... 46 4.2First-In,First-Out ............................... 48 4.2.1Analysis ................................. 48 4.2.2BoundsonOptimalReplenishmentQuantityandDemand ..... 53 4.2.3RelationshipwithOne-PeriodandInniteLifetimeSystems ..... 53 4.3Last-In,First-Out ................................ 56 4.4ComputationalExperiments .......................... 59 4.4.1BenetsofDemandManagement ................... 59 4.4.1.1Non-homogeneousdemand .................. 59 4.4.1.2Capacityconstraint ...................... 62 4.4.2PerformanceofApproximatePolicy .................. 64 5CONCLUSIONSANDFUTURERESEARCH ................... 68 5.1InthePresenceofEconomiesofScaleandAge-dependentDemand .... 68 5.2WithouttheEconomiesofScaleandAge-dependentDemand ........ 69 APPENDIX APROOFSINCHAPTER3 .............................. 71 BPROOFSINCHAPTER4 .............................. 85 5

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....................................... 112 BIOGRAPHICALSKETCH ................................ 115 6

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Table page 3-1Parametervalues. ................................... 39 4-1Parametervalueswithnon-homogeneousdemand. ................. 60 4-2Parametervaluesfortheapproximatepolicy. .................... 65 7

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Figure page 3-1Theprotsforstrategiesandtheoptimalprotwithdemandrandomness. ... 27 3-2Theprotsforstrategiesandtheoptimalprotwithoutdemandrandomness. 29 3-3Optimalpriceasafunctionofon-handinventory. ................. 33 3-4Optimalpricewithdierentdemandvolatility. ................... 38 3-5Protimprovementduetodemandmanagement,exiblereplenishment,andpartialsalvagestrategies. ............................... 43 4-1Optimalorderquantityandpriceasafunctionofnetinventory. ......... 50 4-2PercentprotadvantageofFIFOwithrespecttoLIFOinthepresenceoftimevaryingdemand. ................................... 63 4-3PercentprotadvantageofFIFOwithrespecttoLIFOinthepresenceofcapacityconstraint. ....................................... 65 4-4Percentprotdierencebetweenapproximatepolicyandoptimalprot. .... 67 8

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Webb ( 2006 )).Similarly,itisestimatedthatthetop40retailersintheU.S.dumpasmuchas500millionpoundsoffoodeveryyearduetospoilage( Gallagher ( 2008 )).However,spoilageisnotlimitedtoproduceorconsumergoodsalone;severalindustrialproductsalsohavealimitedlifetime.Forexample, Chen ( 2007 )mentionedthatadhesivematerialsusedforplywoodpanelslosetheirstrengthwithin7days.Obviously,spoilageisaloss,andthebottomlineofarmcanimprovesignicantlyifsomeofthisspoilageisprevented,i.e.,iftheperishablenatureofproductsismanagedproperly.Onemechanismbywhichthismaybeachievedisdemandmanagementusingprice.Throughanappropriateselectionofprice,demandcanbemodulatedtoimproveprot.Themodulationofdemandcannotonlyincreaserevenuebutalsoreduceshortage,holding,andspoilagecosts.Potentialspoilageduetolimitedlife-timeoftheperishableproductsisthemainreasondemandmanagementofperishableproductsisevenmoreimportantthanfornon-perishableproducts.Whereasfornon-perishableproductstheonlycostofunsoldinventoryisthecostforholdinginventory,forperishableproductstheunsoldinventorynotonlyincursinventoryholdingcostbut,inaddition,withincreasingageoftheinventorytheriskofitremainingunsoldbytheendofitslifetimeincreases.Thisdissertationiscomposedoftwoparts.Intherstpart,wemodelascenarioinwhichdemandforaperishableproductisage-dependentandthereexisteconomiesofscaleinorderplacement.Wedevelopthreemodelswithincreasingexibility.Ineachmodel,weconsideranitehorizonperiodicreviewsystemforasingleproductatasingleretailer 11

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Dave ( 1991 ), GoyalandGiri ( 2001 ),and Raafat ( 1991 )forareviewofsuchmodels.Theanalysisofmodelsthatconsiderrandomnessinbothdemandandlifetimehasnotreceivedmuchattention,withthenotableexceptionof Nahmias ( 1977b ).Ontheotherhand,mostofthepapersthatadoptthesecondapproachutilizeperiodicreviewmodelswithrandomdemand.Unlikenon-perishableproducts,theoptimalreplenishmentpolicydependsontherelativeorderofinventoryarrivalandconsumption.Itisobviousthataninventorymanagershouldpreferarst-in,rst-out(FIFO)issuingpolicysinceitminimizesinventorywastage.(Itisimplicitlyassumed 13

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Prastacos ( 1984 )),theassumptionofaFIFOissuancepolicywasstandard.TheoppositeofFIFOisthelast-in,rst-out(LIFO)policy.Iftheuser/customerdeterminestheissuancepolicyandhederivesgreaterutilityfromnewerunits(asisthecaseforfoodproductssuchasmilk),thentheinventoryissold/issuedaccordingtotheLIFOpolicy.(Asbefore,theimplicitassumptionhereisthatallunitshavethesameageuponarrival.Otherwise,theoptimalissuancepolicyfromacustomer'sperspectiveisthelast-expiring,rst-outpolicy.)Theanalysisofinventorycontrolusingeitherissuancepolicyisdicult.Theprimaryreasonistheneedtocarryaninventoryvectorcorrespondingtounitsofdierentages.Inparalleleorts, Fries ( 1975 )and Nahmias ( 1975c )characterizetheformoftheoptimalpolicyforthelost-salesandbackloggingcase,respectively.Usingthespecialcharacteristicsoftheoptimalsolution,manypapershavedevelopedmyopicornear-myopicpoliciesthatignoretheage-distributionoftheon-handinventory( Brodheimetal. ( 1975 ), Nahmias ( 1975a ), Nahmias ( 1975b ), Nahmias ( 1976 ), Nahmias ( 1977a ),and NandakumarandMorton ( 1993 )).TheresearchrelatedtotheinventorypolicyfortheLIFOruleisespeciallylimited.Asanexampleofresearchinthisarea, CohenandPekelman ( 1978 )developage-distributionsinaperiodicreviewinventorysystemwithlost-salestodeterminetheorderpolicy.Onceagain, Nahmias ( 1982 )and Karaesmenetal. ( 2008 )summarizemanyofthesepapers.Tworeviewpapersthatexclusivelyfocusontheinventorycontrolofbloodare Prastacos ( 1984 )and Pierskalla ( 2004 ).Themajorcontributionoftherstpartofthedissertationistocaptureanotherfeatureofperishableproducts,age-dependentdemand,whichhasnotbeenyetexaminedintheexistingliterature.Theearliermentionedresearchonperiodicreviewmodels 14

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3.1 .)Thisassumptionisamarkeddeparturefromseveralexistingmodelsinperishableinventorytheorythatassumethatinventoriesofdierentagesareperfectlysubstitutable.Thesubstitutabilityassumptionoftenresultsincomplicatedanalysisduetothepresenceofaninventoryvectorcorrespondingtostocksofdierentages( Nahmias ( 1982 )).Eventhoughouranalysisavoidssomeofthesediculties,theassumptionthatorderplacementresultsinsalvageoftheoldinventoryposesitsownchallenges.Naturally,ourworkinthesecondpartalsocontributestothisstreamofliteraturegivenourconsiderationofaproductwithtwo-periodlifetime.Tothebestofourknowledge,thisistherstworkthatexaminesthescenarioinwhichdemandisendogenousforaperishableproductwithxedandmulti-periodlifetime.Wewouldliketopointoutthat Chandeetal. ( 2004 ), Chandeetal. ( 2005 ),and Chandrashekaretal. ( 2003 )alsoconsiderreplenishmentanddemandmanagementdecisionsforperishableproducts,butthesepapersmodelonlyasinglepromotion(theonlydemand-relateddecision)duringthehorizon.Relativetothesepapers,ourcontributionistooptimizedemanddynamically.Wealsodevelopanalyticalresultstoderiveinsights,whereasthesepapersuseacomputationalapproachtoobtaininsights.Wewouldalsoliketomentiontherecentworkincoordinationofprice(ordemand)andproduction/orderingdecisionsfornon-perishableproducts.Usingaperiodicreview 15

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ChenandSimchi-Levi ( 2004 )and FedergruenandHeching ( 1999 )developoptimalpolicieswithandwithoutxedcostoforderplacementandbacklogging. Chenetal. ( 2006 )considertheproblemwithxedcostbutassumethatexcessdemandislost. HuhandJanakiraman ( 2006 )generalizetheseresultsbyusingalternativeprooftechniques.Forshort-lifecycleproductswithasinglereplenishmentopportunity, PetruzziandDada ( 1999 )analyzeaNewsvendormodelwithprice-dependentdemand.Anexcellentreviewofliteratureonthecoordinationofpricingandinventorydecisionsis YanoandGilbert ( 2006 ). 16

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Adamy ( 2004 )).RecentinnovationsinthebakingindustryhaveincreasedthelifetimeofseveralbakeryproductsandInterstatesoughttotakefulladvantageoftheseinnovationsbyincreasingtheshelf-durationofitsproducts.Themajoradvantageofincreasedshelf-durationislessspoilage;itisestimatedthattop40retailersintheUSdumpasmuchas500Millionpoundsoffoodeveryyearduetospoilage( Gallagher ( 2008 )).Yetcontrarytothecompany'sexpectation,thestrategydidnothelp.Thelongerbreadstayedontheshelf,themorefrequentlyitwasmovedbystorepersonnelandcustomers.Thismadethebreadlookshelf-worneventhoughitwasstillconsumable.Thenetresultwasareductionindemandofmanypopularproducts,whichplayedasignicantroleinInterstate'slingforbankruptcyprotection.AcompetitorofInterstate,FlowerFoods,utilizingthesametechnologychosetoincreasetheshelf-durationofbreadfrom3to4daysinsteadoffull7days,yetitssalesincreasedby5%( Adamy ( 2004 )).Thesecontrastingexamplesillustratetheimportanceofthedecisionfoodcompaniesroutinelyface:Whentotake-otheoldinventoryfromtheshelfandreplaceitwithfreshinventory.Inthiswork,wedevelopandanalyzethreemodelstoderiveinsightsonthisproblem.Thedecisionregardingwhentorenewinventoryisdrivenbythetotalvalueofinventory,whichisequaltothepotentialrevenueearnedthroughthesaleoftheinventoryuntilitexpiresplusthesavingsinshortagecostlesstheholdingcost.Theinventoryisrenewediffreshunitsprovidemorevaluethantheexistinginventory.Asinventoryages,thepotentialrevenuedeclinessincethedemanddeterioratesasthelifetimeapproaches.Therearethreereasonsforthedeteriorationofdemandwithageforperishableproducts. 17

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3.1 ,wedescribethenotationandformulatetherstmodel.Intherstmodel,whichwerefertoasthexedcyclemodel,weassumethatthereorderintervalisxed;thereorderintervalisatmostthelifetimeoftheproduct.Theprice,however,maybeadjustedeveryperiod.Thismodelissuitableforproductswitharelativelylessvolatiledemand.Usingdynamicprogramming,weshowthattheoptimalprotfunctionisconcaveintheon-hand 18

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3.2 ,weeliminatetherestrictionthatthereorderintervalisxed,andwerefertothismodelastheexiblereplenishmentmodel.Thismodelisthusmoresuitableforproductswithvolatiledemand.Theanalysisofthemodeldemonstratesthattheoptimalprotfunctionhasacomplicatedstructure.However,weareabletoshowtheexistenceoftwothresholdlevelsofinventorysuchthatanorderisplacedwhenevertheinventoryislessthanthesmallerthresholdlevelorgreaterthanthelargerthresholdlevel.InSection 3.3 ,wedescribethethirdmodel,whichwerefertoasthepartialsalvagemodel.Inthismodelweexamineapartialsalvagestrategyinthepresenceofexiblereplenishmentintervals.Themodelallowsforsituationsinwhichtheretailerhasanoptiontopartiallysalvagetheinventorywhenshehasanexcessamountofit.Thepartialsalvagemodel,despitebeingmoreexible,hassimplerstructurecomparedtotheexiblereplenishmentmodel.Wedemonstratetheexistenceofathresholdinventorylevelsuchthatanorderisplacedifandonlyiftheinventoryislessthanthethresholdlevel.InSection 3.4 ,weconductnumericalexperimentstoidentifythemarginalprotimprovementsduetothedemandmanagement,exiblereplenishment,andpartialsalvagestrategiesasafunctionofseveralmodelparameterssuchasthepricesensitivityofdemandandvarianceofdemand.Wendthatwhilethedemandmanagementandexiblereplenishmentstrategyyieldprotimprovementsof2-7%and3-8%,respectively,thepartialsalvagestrategyaddslittlebenet.Thisimpliesthattheretailercangarnermostofthebenetbyacombinationofexiblereplenishmentanddemandmanagementstrategies.WeconcludeandsummarizeourndingsinSection 5.1 19

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wherepisthepriceinperiodt,sistheageoftheinventory,and1and2arerandomvariablessuchthatE[1]=1andE[2]=0.Further,1and2areassumedtobecontinuousrandomvariableswithdensitiesandIIDfordierentperiods.SinceE(1)=1andE(2)=0,theexpectationofDt(s;p)isequaltodt(p)f(s).When11,werefertothedemandfunctionasadditive.Weassumef(s)isanon-increasingfunctionofsandtakesvaluesintheinterval[0;1]suchthatf(0)=1.Wealsotakedt(p)tobealinearfunctionofpricep,thatis,dt(p)=abp.Whilemostofourresultsonlyrequirethattheexpectedrevenuep:dt(p)beconcaveinp(asin FedergruenandHeching ( 1999 )and ChenandSimchi-Levi ( 2004 )),thelinearityassumptionprovidesaconcreteinterpretationofcustomerbehavior.Toseethis,wethinkofatobetheexpectednumberofcustomerswhowouldpurchasetheproductifthepricewere0andtheproductwerefresh(age=0).Whenpricepisset,thenumberofinterestedcustomersinexpectationdwindlestoabp.Further,iftheproductissperiodsold,anotherfraction(1f(s))ofcustomersloseinterest.Thus,theexpectednumberofcustomersthatwanttopurchaseaunitthatissperiodsoldis(abp)f(s).Wepointoutthatitispossibletoextendourresultstomoregeneraldemandfunctions:MostofourresultsonlyrequirethatDt(s;p)benon-increasingins.Forexample,thedemandfunctionmayalsobemodeledasDt(s;p)=((abpf1(s))1+2)f(s).Giventheaboveinterpretationofa,thetermf1(s)mayrepresenttheexpectednumberofcustomerswhowalkawayafterobservingtheageoftheinventorywithoutevencheckingtheprice.Thetermf(s)wouldthenrepresentthefractionofcustomersthatafterobservingthepriceandagedecidetopurchasetheproduct. 21

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3.1.1 ,Dsisidenticalforalls.Asaresult,wewilldropsubscriptsfromDsinthesubsequentanalysis.Letx>0betheavailableinventorytosatisfydemandinaperiod.Ifdistheexpecteddemandandsistheageoftheinventory,theexpectedone-periodprotinthatperiodisequalto wherehistheunitholdingcost,istheunitlost-salescostand[]+=max[;0].Thelost-salescostnotonlyincludeslostmarginforanyunsatiseddemandbutalsothecostofnotfulllingdemand.Sincepriceisavariableinourmodel,thelostmarginisalsoendogenoustothemodel.Theassumptionoflost-salescostthatisindependentofthepriceisapplicablewhenthelost-marginismuchsmallercomparedtothecostofnotfulllingdemand.Thisshouldholdtrueinthegroceryindustrywheremarginsareoftensmall.Further,thegoodwillcostofnotprovidinganessentialfooditemcouldbesignicantsincethecustomermaytakeherentirebusinesselsewhere.Wealsonotethata 22

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PetruzziandDada ( 1999 )inthecontextofaNewsvendormodelwithprice-dependentdemand.Onecrucialmodelingdicultywithalost-salesmodel,unlikeabackloggingmodel,istoapproximatethelost-salescostinaperiodwhenthereisnoon-handinventoryatthebeginningofaperiod.Clearly,theretailercannotsetahighpricetodrivedownthedemandtosaveonherlost-salescostasshecanwhenthereispositiveon-handinventory.Thereasonisthatthedemandmanagementusingpricecannotworkwhenthereisnothingonhand.Wehandlethisdicultbyassumingthatexpectedone-periodprotcorrespondingtox=0isstrictlylessthanlimx!0+Ls(x;d).Wedenotetheexpectedone-periodprotwhenx=0byA.ThisassumptionmakesLs(;d)functiondiscontinuousatx=0.Wearenowreadytoformulatetheproblem.Letvt(s;x)betheoptimalprotfromperiodtthroughtheendofhorizonwhenxunitsofagesareonhand.Ifnoorderisplacedinperiodt,vt(s;x)=maxd2DLs(x;d)+E[vt+1(s+1;(xd12)+)];x>0;=(Rs)A;x=0;whereisthediscountingfactor.Theonlydecisionvariablehereistheexpecteddemand.Ontheotherhand,ifanorderisplacedinperiodt,theoptimalprotfunctionisequaltovt(s;x)=wRx+maxd2D;y0fK(y)+L0(y;d)cy+E[vt+1(1;(yd12)+)]g;suchthatvT+1(s;x)=wsx:Intheaboveformulation,(y)=1ify>0and0otherwise.ObservethattheabovemodelbecomesaNewsvendormodelwithprice-dependentdemandwhenS=1.Weassumethatthereexistsay>0suchthatK+L0(y;d(y))+w1E(yd(y)12)+cy>A,whered(y)istheoptimaldemandcorrespondingtoyforaone-periodmodel(Newsvendormodelwithendogenousdemand),sothatitissub-optimaltonot 23

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Theoptimalprotfunctionvt(s;x)isconcaveinxforanys. 2. Thereexistsauniquedt(x)foreachx,whichisnon-decreasinginx. 3. Whenanorderisplaced,thereexistsauniqueorderquantityy.Further,whenthedemandfunctionisadditive,thatis,11,theoptimaldemandisequaltoargmaxdd(D10(d)c).TheproofoftheaboveresultaswellasotherresultsareavailableintheAppendix.Inmulti-perioddynamicinventorysystems,thelinkthatconnectsdecisions(suchastheprice)indierentperiodsistheinventorythatiscarried.IntheabovemodeltheinventoryisrenewedeveryRperiods,sotheorderplacementdecisiontakeninaperiodaectstheprotofonlythefollowingR1periods.WerefertothisgroupofRperiodsasacycle.LetV(R)bethetotaloptimalprotoveracycleandy(R)betheoptimalorderquantityinacycleoflengthR.Inthefollowingproposition,weshowhowV(R)andy(R)changewiththecyclelengthRwhenthedemandisdeterministic. 24

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@d(dD1n1(d))@ @d(dD1n(d)).Further,itcostshmoretocarryaunitintoperiodn+1comparedtoaunitcarriedintoperiodn.Sinceeachunitforthelastperiodinn+1-periodcycleimprovesthecycleprotbyasmalleramount,fewerunitsarekeptforthelastperiodinn+1-periodcyclecomparedtothelastperiodinn-periodcycle.Bythesamelogic,themarginalprotincreaseduetothe(n+1)-thperiodissmallercomparedtothen-thperiod.Ingeneral,webelievethatthisargumentshouldholdfortherandomdemandcaseaswell.ThisiswhywethinkthataresultanalogoustoProposition 3.1.2 shouldexistwhendemandisuncertainalthoughweareunabletoestablishit.TheresultinProposition 3.1.2 isusefulsinceitleadstoanapproachtocomputeadesirablecyclelength.Inthisapproach,wecanchooseacyclelengththatmaximizestheoptimalprotperperiod,V(R)K R.ThefunctionV(R)K Risquasi-concaveifV(R)isconcave( BoydandVandenberghe ( 2004 )).Weformallystatethisresultinthefollowingcorollary. Rinacycle. 25

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istheoptimalprotfromperiodtthroughtheendofhorizonwhenStrategy1isusedinperiodtand istheoptimalprotfromperiodtthroughtheendofhorizonwhenStrategy2isfollowedinperiodt.ThetermLs(x;d)inEquations 3.2.3 and 3.2.4 isasdenedin 3.1.2 .Tomakeanalysismoreinteresting,weassumethatv2t(s;0)K>v1t(s;0).Thisassumptionensuresthatanorderisnecessarilyplacedwhenthereisnothingon-hand.TheremainingassumptionsremainthesameasinSection 3.1 .ObservethatsimilartothemodelinSection 3.2 ,theabovemodelbecomesaNewsvendormodelwithprice-dependentdemandwhenS=1.Further,whenthedemandisdeterministic(thatis,1=1and2=0),thexedcycleandexiblereplenishmentmodelsproduceidenticalresults.Sincetheexibilitytoplaceordersismoreusefulwhendemandismorevolatile,theexiblereplenishmentislikelytobemoreusefulinvolatiledemandenvironments.Fortheexiblereplenishmentmodel,theoptimalprotfunctionvt(s;x)isnotnecessarilyconcaveinx.Toseethis,observethattheoptimalprotforeitherstrategyisconcavefort=T(assumings
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Figure3-1. Theprotsforstrategies1and2andtheoptimalprotasafunctionofon-handinventorywhenK=200,h=1,c=5,=50,Dt(s;p)=(40p+2)f(1),f(s)=max(0;2exp(0:1s)),t=4,T=8,2N(0;100)truncatedat10. Webeginbystatinginthefollowingpropositiontheformoftheoptimalpolicywhenthelifetimeisequalto2periods. 1. ForStrategy1,v1t(s;)isconcave.Further,thereexistsauniquevalueoftheoptimaldemand,wheneverStrategy1isused. 2. Eitherv2t(s;x)>v1t(s;x)8x,inwhichcaseStrategy2isfollowedforeveryx,orthereexisttwothresholdsxlt(s)(possibly0)andxut(s)suchthatStrategy1isfollowedforallvaluesofx2(xlt(s);xut(s))andStrategy2otherwise.

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2. 3. 28

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Ifv1t(s;x)>v2t(s;x)forsomex,thenthereexistsalowerthresholdxlt(s)andupperthresholdxut(s)suchthatStrategy2isfollowedwhenxxlt(s)andxxut(s). 2. Ifxlt(s)andxut(s)existandthesalvagevaluewsisindependentofs,thenxlt(s)increasesinsandxut(s)decreasesins.Aswestatedabove,weareunabletoprovethatStrategy1isnecessarilyfollowedbetweenxlt(s)andxut(s),thoughourextensivenumericalexperimentsshowthistobethecaseusually.Ourexperimentsindicatethatonlywhendemandisdeterministicorhasverylowvolatilitythatanordermightbeplacedwhentheinventoryfallsbetweenthetwothresholds.Inotherwords,whendemandissucientlyvolatile,onlyStrategy1islikelytobeusedbetweenxlt(s)andxut(s). Figure3-2. Theprotsforstrategies1and2andtheoptimalprotasafunctionofon-handinventorywhenK=200,h=1,c=5,=50,Dt(s;p)=(40p)f(1),f(s)=max(0;2exp(0:1s)),p2[1;10],t=4,T=8. Toexplainthisobservation,considerFigure 3-2 .Inthisgure,wehaveplottedtheoptimalprotscorrespondingtoStrategy1,Strategy2,andtheirmaximumasa 29

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3-2 ,multiplepointsofintersectionbetweenv1t(s;)andv2t(s;)mayariseduetothezigzagnatureoftheplotforv1t(s;).Sharppeaksandtroughsarisesincetheoptimaldemandfornextfewperiodscanbecomputedexactly.Thisceasestobetruewhendemandisnotdeterministic.Asaresult,thecurvebecomesmore\smooth"andpeaksandtroughsgraduallyvanish.Ourexperimentsindicatethatrelativelymildvaluesofdemanduncertaintyresultinasmoothcurvewithmildorno 30

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3-3 forasampleplotoftheoptimalpriceasafunctionoftheinventory.Asonewouldexpect,theoptimalpriceischosentomaximizetheprotfromtheavailableinventory.Ournumericalexperimentsindicatethatthismayhappenintwoways.Firstly,whenthereissucientinventorytosatisfydemandsofkfutureperiods,includingthecurrentperiod,thoughitislessthanideal,thepricemaybeincreasedtoreducethecostoflost-sales.Forexample,inFigure 3-3 ,forx=xlt(s)=10tox=24thepriceisusedtoreducelost-salessincetheamountoftheinventoryislessthanidealtosatisfyoneperiod'sdemand.Thepricestabilizesbetweenx=24andx=35oncetheinventorybecomessucientlylarge.Betweenx=35andx=59,thevariationinpriceismainlytoensurethattheinventorylasts2and3periodsdependinguponthevalueofx.Secondly,priceisusedwhentheinventoryishighsothattheinventoryissalvaged(withahighprobability)nextperiod.InFigure 3-3 ,forx72theinventoryisexcessiveandislikelytobesalvagednextperiod.Thepriceisnowchosensothatthecombinedprotfromthesaleoftheinventorythisperiodaswellassalvagenextperiodismaximized. 32

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Optimalpriceasafunctionofon-handinventorywhent=4,T=8,K=200,h=1,c=5,=50,Dt(s;p)=(40p+2)f(s),f(s)=max(2exp(0:1s);0:75exp(0:1s)),2N(0;100)truncatedat10. solutionofpartialsalvagehasseveralbenets.Ifsomeunitsareunlikelytobesoldbeforetheendoftheirlifetime,salvagingtheminadvancemayfetchhighersalvagevalue.Theretaileralsosavescostsrelatedtocarryinginventory.Inthissection,weanalyzeamodelinwhichsomeinventorymaybesalvagedwhenStrategy1isfollowed,thatis,whennoorderisplaced.Inventorywrite-osordisposalsforbothperishableornon-perishableproductsarequitefrequentinthereal-world.Yetfewsupplychainpapershaveexaminedsuchadecisionintheirmodels.Wehavecomeacrossonlytwopapersthathaveconsideredthisphenomenonforrandomdemand, Roseneld ( 1989 )and Roseneld ( 1992 ).(Anumberofpapershaveexaminedthesalvageofexcessinventoryfordeterministicdemand.See Roseneld ( 1989 )fordetails.)Itisoptimaltodisposesomeoftheinventoryifthecurrentsalvagevalueoftheinventoryisgreaterthanitsfuturesale,whichincludestherevenueaswellassavingsinlost-salescostduetoitbeforetheinventoryexpires,lesstheholding 33

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Roseneld ( 1989 )developsanapproachtoidentifywhenandinwhatquantitytosalvagesomeoftheinventory.Healsolooksatthecasewhentheproductisperishable.Comparedtohim,wedierinthemodelingapproachinthatweuseaperiodicreviewmodel,whereasheutilizesacontinuousreviewmodelwithPoissondemand.Wealsoincludethepossibilityofdemandmanagementandconsiderage-dependentdemandwhileidentifyingtheamounttobesalvaged.Tomodelpartialsalvage,weincludeanewdecisionvariablezintotheformulationwhichrepresentsthepartialsalvagequantity.Themodiedexpressionforone-periodexpectedprotisasfollows:Ls(x;z;d)=dD1s(d)+wszhE[xzd12]+E[d1+2x+z]+;wherewehaveincludedzasanargumentinthedenitionofLs.Weassumethattheunitsalvagevaluewsforpartialsalvageisthesameasforcompletesalvage.Thisassumptionisnotcriticalfortheanalysis;ouranalysiscanbeeasilyextendedtothecaseinwhichtheunitsalvagevaluesforthepartialandcompletesalvagesaredierent.Fromapracticalperspective,forobviousreasonstheunitsthatarepartiallysalvagedcannotbesoldwithinthesamestoreatadiscountunlikethecasewhencompletesalvageoccurs.Inotherwords,thesalvagecannotoccurintheformofapricediscountwithinthesamestore.GiventhedenitionofLs(x;z;d),wenowstatetheformulationsofthetwostrategiesasfollows: 34

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3.2 ,thatis,vt(s;x)=8>>>><>>>>:maxfv1t(s;x);v2t(s;x)g;forsv1t(s;0)toensurethatanorderisnecessarilyplacedwhenthereisnothingon-hand.Theremainingassumptionsremainthesameasforthexedcyclemodel.Comparedtotheexiblereplenishmentmodel,theoptimalprotfunctionhassimplerstructure.ItstilllacksconcavityforthesamereasonsasinSection 3.2 ,butitnowisnon-decreasinginx.Thereasonitisnon-decreasingisthattheprotfunctioncorrespondingtoStrategy1isnownon-decreasinginx.SincetheprotfunctioncorrespondingtoStrategy2,whichissameasinSection 3.2 ,isalsonon-decreasing,theoptimalprotfunctionisalsonon-decreasinginx.Themonotonicityofvmakesanalysisconsiderablyeasier.Asaconsequence,weareabletocharacterizetheoptimalpolicyforthismodelalmostcompletelyeventhoughithasmoredecisionvariablescomparedtothemodelinSection 3.2 .Webeginbyshowinginthefollowingcorollarythattheexpectedprotinaperiodisnon-increasingintheageoftheinventoryforanygivenlevelsoftheinventory, 35

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2. 2.

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1. Thereexistsathresholdxlt(s)suchthatanorderisplacedifandonlyifxxlt(s). 2. SupposeStrategy1isfollowed.Then,foranyxqx,thequantityxqxissalvaged.Further,forallxq,thequantityxqissalvaged.Theabovetheoremshowsthatthereexistsathreshold(xlt(s))suchthatanorderisplacedifandonlyiftheinventoryislessthanthethreshold.Otherwise,noorderisplaced.Whennoorderisplacedandtheinventoryislarge,theexcessinventoryissalvaged.Infact,wheneverthesystemexceedstheinventorybeyondacertainthreshold(q),itwillsalvageeveryunitinexcessofthethreshold.Evenwhentheinventoryliesbetweenxlt(s)andq,salvagemayoccurthoughitisnotnecessary.Insuchacase,salvagewilloccurtobringtheinventoryleveltoalocalmaximumthatlieswithintheinterval[xlt(s);x]andthathasthelargestprotvalue.Inthefollowingproposition,weshowthatfortheadditivedemandmodel,thatis,when11,theoptimaldemandhasasimplestructure.Further,thecomputationmerelyinvolvesoptimizingasingledimensionalfunctionoverdemandd.Note,however,thattheformofthisfunctiondependsonthestrategy. 1. Theoptimaldemandisargmaxdd(D10(d)c)whenanorderisplaced.

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Whennoorderisplacedandapositiveamountofinventoryissalvaged,theoptimaldemandisargmaxdd(D1s(d)ws)whentheageoftheinventoryiss.Theoptimalpricedoesnothaveasimplestructurewhenthereisnosalvage.Figure 3-4 showstwosampleplotsoftheoptimalpriceasafunctionoftheinventory.Inthismodel,priceisusedonlywhenthereissucientinventorytosatisfydemandsofkfutureperiodsthoughtheamountislessthanideal.Inthatcase,pricemayberaisedtosaveonthelost-salescost.Unliketheexiblereplenishmentmodel,priceisnotutilizedwhenthereisexcessiveinventorysinceitcanbesalvaged.Figure 3-4 suggeststhattheoptimalpricemayhaveasimplerstructurewhendemandismorevolatile. BFigure3-4. Optimalpriceasafunctionofon-handinventorywhent=5,T=20,K=200,h=1,c=5,=50,Dt(s;p)=(40p+2)f(s),f(s)=max(2exp(0:1s);0:75exp(0:1s)).A)2N(0;1)truncatedat10.B)2N(0;100)truncatedat10. 38

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Parametervalues. a K c h U w 40 0.75 200 5 1 10 0.6 pricesensitivityofdemandb,thevolatilityofdemandandtherateofdemandreductionwithrespecttotheageoftheinventory.Weconsideranadditivedemandmodelforourexperiments,andchoosef(s)=max(2exp(s);exp(s)).Forsmallvaluesofs,thisfunctionisconcave.Thismeansthatthereductioninthefractionofcustomerswhodonotwanttopurchasetheproductduetoitsageacceleratesastheageincreases.Whensbecomeslargeenough,thisfunctionisconvex,whichmeansthattherateofdeclinestabilizesassbecomeslargeimplyingthattherewillalwaysbesomecustomerswhowouldbuytheproduct.Intheabovefunction,theparameterdeterminestherateatwhichdemanddeclineswithrespecttotheage.Forsmallvaluesof,thedemandreductionastheageincreasesisrelativelysmall.Thisreductionoccursmoreandmorerapidlyasincreases.Thisiswhywewillrefertoastherateofdemandreductionwithrespecttotheageintherestofthissection.Notethatintheexperiments,wedonotimposeanexplicitupperlimitons.Puttingeverythingtogether,thedemandmodelusedinourexperimentsisasfollows:Dt(s;p)=(abp+2)max(2exp(s);exp(s)):Wetake2tohaveatruncatednormaldistributionwithmean0andvariance2;thetruncationoccursatU.Wesetthesalvagevalueinperiodstobewf(s).Thus,thereductioninsalvagevaluewithrespecttotheagemimicsthereductionindemandwithrespecttotheage.Thevaluesoftheparametersusedinourexperimentsarereportedinthefollowingtable.InFigures 3-5 ,weplotthepercentageimprovementintheoptimalprotduetothedemandmanagement,exiblereplenishmentandpartialsalvagestrategiesasafunction 39

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Protofxedcyclemodelwithoutdemandmanagement100%:Tocomputetheprotofthexedcyclemodel,wecomputetheoptimalcycleprotfordierentcyclelengths.Subsequently,wechoosetheoptimalcyclelengthRastheonethatproducesthehighestaverageprotperperiod.Sincetheremaynotbeanintegernumberofcyclescorrespondingtotheoptimalcyclelengthwithintheplanninghorizon,weapproximatethetotalprotoverthehorizonbyT(OptimalCycleProtCorrespondingtoR) Protofxedcyclemodel100%:Similarly,thepercentageimprovementduetopartialsalvageiscomputedasfollows:Protofpartialsalvagemodel-Protofexiblereplenishmentmodel Protofexiblereplenishmentmodel100%:Figures 3-5A 3-5D demonstratethatthebenetofbothdemandmanagementandexiblereplenishmentstrategiesincreasesasanyofthepricesensitivityofdemand,lost-salescost,andvarianceof2;thevariationwithrespecttoisnotmonotonicthoughitoccursinanarrowrange.Broadlyspeaking,thesestrategiesprovidebenetbyhelpingmanagethedemandrisk.(Wenotethatwhilethedemandmanagementstrategyisbenecialevenwhendemandisdeterministic,theexiblereplenishmentstrategyimprovesprotonlywhenthereisdemanduncertainty.)Thiskeyideawillformthecoreofour 40

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3-5 showsthatthedemandmanagement,exiblereplenishmentandpartialsalvagestrategiesprovideaprotimprovementof2-7%(withoutincludingtheeectofb),3-8%,andlessthan0.1%,respectively.(Theprotimprovementduetothepartialsalvagestrategyinthegureismultipliedby10sothatitcanbeplottedonthesamegure.)Moredetailsoftheinsightsfromthegureareasfollows.Figure 3-5A showsthattheprotimprovementduetothedemandmanagementstrategyacceleratesasthepricesensitivityofdemandbincreases.Asbincreases,thedemand-lossduetoasub-optimalpriceincreases.Asaresult,demandmanagementyieldsgreaterbenetcomparedtothestaticpricingstrategyasbincreases.Theprotimprovementduetotheexiblereplenishmentstrategyalsoincreasesasbincreases.Withtheincreaseinthepricesensitivityofdemand,themeandemanddecreasesforagivenvalueofpriceforeitherofthexedorderandexiblereplenishmentmodels,whichresultsinlowerpricesandlowerdemandinoptimalityforanygivencombinationofx;sandt.Duetolowerdemandmean,theoptimalnumberofperiodsinacycleincreasesinthexed 41

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42

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B C DFigure3-5. Protimprovementduetodemandmanagement,exiblereplenishment,andpartialsalvagestrategiesasafunctionofmodelparameters.A)Dierentpricesensitivityofdemand.B)Dierentunitcostoflost-sales.C)Standarddeviationof2.D)Rateofdemandreductionwithrespecttoage. 43

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Karaesmenetal. ( 2008 )).Thisworkaimstofulllthisimportantgapbyconsideringaperishableproductwhoselifetimeisequaltotworeviewperiods.Forsuchaproduct,itiswell-understoodthattheprotduringtheplanninghorizon(andhencetheoptimalreplenishmentpolicy)dependsontheissuingrule,thatis,therelativeorderinwhichunitsarriveandareconsumed( Nahmias ( 1982 )).Naturally,thisobservationcontinuestoholdevenwhendemandisendogenous.Intheexistingliterature,twocommonlymodeledissuingrulesarerst-in,rst-out(FIFO)andlast-in,rst-out(LIFO).UndertheFIFOrule,asthenamesuggests,whenunitsofdierentagesarepresentinthesystem,theyareconsumedintheorderofoldestrst.TheLIFOrule,ontheotherhand,isexactlyopposite;theunitsareconsumedintheorderofnewestrstwhenunitsofdierentagesarepresentinthesystem.Clearly,theFIFOruleminimizesinventorywastageandisthusthefavoriteofaninventorymanager.(Itisimplicitlyassumedthatallunitshavethesameageuponarrival.Otherwise,theoptimalissuingruleisrst-expiring,rst-out.)TheFIFOrulecanbeeasilyimplementedwhentheinventorymanagerdeterminestheorderinwhichinventoryissoldorconsumed(asisthecaseinhospitalbloodbanks).However,iftheuser/customerdeterminestheissuingruleandshederivesgreaterutilityfromnewerunits(asisthecaseforfoodproductssuchas 44

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4.1 ,wediscussthebasicassumptionsandnotation.InSections 4.2 and 4.3 ,wedevelopandanalyzeperiodic 45

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4.2.2 .WenumericallyexaminetheperformanceofanapproximatepolicyobtainedbytakingaweightedaverageoftheseboundsinSection 4.4 .Inthesamesection,wealsodiscussothercomputationalexperiments.Finally,weconcludeinSection 5.2 .Webeginbypositioningourworkinthebasicnotationandcommonassumptions. 46

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NahmiasandPierskalla ( 1973 );themajordierenceisourconsiderationofendogenousdemand. Nahmias ( 1982 )).Naturally,aninventorymanagerwouldprefertosellinventoryaccordingtothisallocationrule.Therefore,ifaninventorymanagercontrolstheorderinwhichinventoryissold,thentheinventoryislikelytobesoldintheFIFOorder.Thisconditionissatisedinabusiness-to-businesssettingifthemanufacturerselectstheunitstoshiptoacustomer.Forinstance,inavendormanagedinventorysystem,themanufacturerselectswhichunitstodelivertoacustomer.However,aninventorymanageralsocontrolstheorderofinventoryconsumptioninonlinegrocerystores,suchasNetGrocer.com,wherethegrocerpicksinventoryanddeliversittotheconsumers'homes.Eventhoughexcessdemandisusuallylostinaretailsetting,inanonlinegrocerystoreexcessdemandmaybebackloggedforfooditems.Unlikeaconsumerwhovisits 48

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whereisthediscountfactor.WetaketheendofhorizonprotvT+1(x)tobeequaltosx+cx,wheresisthesalvagevalue.Observethattheargumentofvt+1,(q(Dx)+)+,istheamountofinventorythatisone-periodoldatthebeginningofperiodt+1.Also,dene sothatvt(x)=maxd2D;q0Gt(x;q;d).Wenextdiscussanobservationregardingtheformoftheoptimalpolicy.Fornon-perishableproducts,itiswell-knownthatwhenthedemandfunctionisadditiveandexcessdemandisbacklogged,theoptimalvalueofd(orp)wheneveranorderisplacedcanbeobtainedbymaximizingasingle-dimensional,concavefunctionofexpecteddemand,R(d)cd(or,equivalently,price).Oneconsequenceofthisresultisthatthereplenishmentanddemanddecisionsbecomeseparable.Thatis,giventheoptimalvalue 49

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4-1A and 4-1B ,inwhichweplottheoptimalorderquantityandexpecteddemandasafunctionofnetinventory,clearlyillustratethattheoptimalorderquantityandexpecteddemandmaydependonnetinventory.Further,thelackofseparabilitybetweenthereplenishmentanddemanddecisionscanbeeasilydeducedusingEquation 4.2.1 BFigure4-1. Trendsofoptimalorderquantityandpricewhen=1,D=(422p+),truncatedN(10;6)over[10;10],t=1,T=4.A)Netinventoryvsoptimalorderquantity.B)Netinventoryvsoptimalprice. Westatetheseobservationsformallyasfollows. 50

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NahmiasandPierskalla ( 1973 )).Naturally,thepresenceofanothervariable(expecteddemand)complicatesanalysis,sotheproofsrequireadditionalargumentscomparedtothatpaper.Anotherdimension,albeitminor,alongwhichwecontributerelativetoNahmiasandPierskallaisthatweallowtheendofhorizonsalvagevaluestobelessthancunlikethem,whoassumethatthetwoareequal.Letd(x)andq(x)betheoptimalexpecteddemandandorderquantitywhennetinventoryisequaltox.Thetheoremisasfollows. 1. Foreacht,Gt(x;q;d)isajointlyconcavefunctionofqandd. 2. Foreacht,thereexistsauniquext>0suchthatx1=x2=:::=xT1xT.Forx0. (b) (c) (d) 3. Ontheotherhand,whenxxt, (a) (b) (c)

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52

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4.2.2 4.2.3 .Forinstance,inidentifyinganupperboundontheorderquantity,weutilizedtheresultthatv0tisboundedfromabovebyc.Theresultisformallystatedasfollows. Whenxx,theexpecteddemandisboundedfromabovebymaxf0;xygandwhenxx,itisboundedfromabovebymaxf0;xygwherey=F1[c+c +h++c]. 2. Foranyvalueofx,theexpecteddemandisboundedfrombelowbydcsuchthatR0(dc)=c. 3. Whenxx,theorderquantityisboundedfromabovebymaxf0;r+xyxgsuchthatr=F1[c+c +h]andboundedfrombelowbymaxf0;r 4.4.2 ,wenumericallyevaluatetheeectivenessofapproximatepoliciesthatarederivedbytakingaweightedaverageoftheabovebounds. 4.2.1 tothatofasysteminwhichtheproducthasasingle-periodlifetimeaswellasa 53

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4.2.1 ;thesoledierenceisthatanyunsoldunitsattheendofeachperiodarediscarded.Theoptimalprotfromperiodtthroughtheendofhorizonis wherevT+1(x)=cx.Observethatonlybacklogsarecarriedfromoneperiodtothenextperiod.Asaresult,xcantakeonlynon-positivevalues.Similarly,wecandeneamodelfornon-perishableitems.Theformulationisasfollows: wherevT+1(x)=sx+cx.Sincenothingeverperishes,thereisnoterminvolvingintheaboveformulation.Asnotedbefore,itiswell-knownthattheoptimaldemandwhentheproductisnon-perishablemaximizesR(d1)cd1wheneveranorderisplaced.Thesamecanbealsobeeasilyshownforasysteminwhichtheproductlifetimeisequaltooneperiod;weomitthedetails.CouplingtheseobservationswithTheorem 4.2.3 ,inwhich 54

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2. 55

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2. Whent<
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suchthatvT+1(x)=sx+cx,wheresisthesalvagevalueofanyinventoryleftattheendofhorizon.Theargumenttovt+1appearscomplex,butitcanbederivedinasimplemanner.Itisequaltonetinventoryattheendofperiodt(x+qD)lesstheamountofinventoryspoiled((x(Dq)+)+).SimilartothemodelinSection 4.2 ,weconsideranadditivedemandmodel,thatis,D=d+,wheredtakesvaluesinD.RecallthatfortheFIFOrule,theoptimaldemandpolicycannotbeobtainedbymaximizingasingle-dimensionalfunctionofd,R(d)cd,wheneveranorderisplaced,unlikefornon-perishableproducts.Infact,theoptimalvalueofddependsonthevalueofnetinventory.Fortunately,theLIFOrulebehavesdierently,andtheoptimalexpecteddemandcan,onceagain,byobtainedbymaximizingR(d)cdwheneveranorderisplaced.Thisalsoensuresthattheobjectivefunctionisseparableinqandd.Westatetheresultformallyinthefollowingproposition. Nahmias ( 1982 )).Naturally,theadditionofanothervariable,expecteddemand,canonlycomplicatetheanalysis,sotheprotfunctioncontinuestolackasimplestructure.Forinstance,theoptimalprotfunctiondenedinEquation 4.3.5 isnotnecessarilyconcaveinnetinventory.Westatethisobservationformallyasfollows.

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1. Thereexistsauniquex,whichisindependentoft,suchthatanorderisplacedonlyifx
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4.2.3 Chanetal. ( 2006 )alsoconductcomputationalexperimentsunderthesescenarios,albeittheirfocusisonnon-perishableproductswithlost-sales. 59

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Parametervalueswithnon-homogeneousdemand. c h s al 5 1 40 1.5 0 1.2 1.2 20 consistofweekendswouldseefarmoredemandthantheotherperiods.Wenotethatthenon-homogeneityofdemandinourmodelmayalsobeconstruedasseasonalitywithaperiodicityoftwoperiods.ThecomputationoftheoptimalprotovertheplanninghorizoncorrespondingtotheFIFOandLIFOruleswhendemandisadecisionvariableisaccomplishedusingEquations 4.2.1 and 4.3.5 ,respectively.Toidentifytheinuenceofdemandmanagement,wecomparetheseprotswiththeprotcorrespondingtoaxeddemand(orprice)strategyinwhichtheexpecteddemand(orprice)isoptimizedatthebeginningofplanninghorizonbutisnotchangedafterwards.FortheFIFOrule,thecorrespondingdynamicprogramisasfollows: whereVT+1(x;d)=sx+cx.Theoptimalprotovertheplanninghorizonisequaltomaxd2DV1(0;d).Similarly,fortheLIFOrule,thedynamicprogramcorrespondingtothexedpricestrategyisasfollows: suchthatVt+1(x;d)=sx+cx.SimilartotheFIFOrule,theoptimalprotovertheplanninghorizonisequaltomaxd2DV1(0;d).Toruntheexperiments,wetake1tobeBetadistributedwithparametersand,and2tobetruncatedNormaldistributedwithparametersand.Thevaluesofthemodelparametersaresummarizedinthefollowingtable. 60

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PercentProtImprovement=OptimalProtofFIFO-OptimalProtofLIFO OptimalProtofLIFO100%: InFigure 4-2 ,weplottheprotimprovementduetotheFIFOrulebothinthepresenceandabsenceofdemandmanagementwhendemandistime-heterogeneous.ThemaininsightfromthegureisthattheprotadvantageoftheFIFOrulecomparedtotheLIFOrulemaydiminishinthepresenceofdemandmanagement,althoughtheFIFOrulecontinuestodominatetheLIFOrule.It,however,ispossiblethatforsomeparametervaluesthattheprotadvantageoftheFIFOrulemayincreaseevenfurtherinthepresenceofdemandmanagement.Ourexperimentsindicatethatforhighvaluesofah(relativetoal),theprotadvantageoftheFIFOruleincreasesevenfurtherwhenb,,andtakerelativelysmallvalues.Ontheotherhand,whenahisclosetoal(ah=al1:8),theprotadvantageoftheFIFOrulediminishescomparedtotheLIFOruleforalltheparametervaluesthatweconsidered.TounderstandwhytheprotadvantageoftheFIFOrulediminishes,webreakdowntheprotforallthefourcasesintorevenueanddierenttypesofcosts.WendthatdemandmanagementallowsLIFOtobetterutilizeinventoryandreduceshortagecostsmorecomparedtotheFIFO.ThisresultsinlowerprotdierencebetweentheFIFOandLIFOwhendemandismanaged.OurcomputationsalsoshowthattheprotadvantageoftheFIFOrulealwaysreduceswhendemandismanagedasb,,orincreases.(SeeFigures 4-2B 4-4C ,and 4-4B asexamples.)Onceagain,thisismainlycausedbybetterutilizationofinventorybytheLIFO,whichreducesshortageandwastagecosts.Thesamesetofcomputationsalsoallowustoexamineifdemandmanagementfavorsoneissuingruleoveranother.Tofurtherexplorethisquestion,wecomputevaluesofthe 61

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PercentProtImprovement=OptimalProt-ProtofFixedDemandStrategy ProtofFixedDemandStrategy100%: forbothFIFOandLIFOrules.WendthatdemandmanagementfavorstheLIFOrulemorethantheFIFOrulewhenah=alislowandb;;andarerelativelyhigh.Ontheotherhand,whenah=alishighandb,,andarerelativelylow,theFIFOrulebenetsmorefromdemandmanagement.(Weomittheplots.)Inparticular,wendthattheparametervaluesforwhichtheprotadvantageoftheFIFOrulediminishes(enhances)comparedtotheLIFOrulearethesameforwhichdemandmanagementfavorstheLIFOrulemore(less). 4.2.1 4.3.5 4.4.6 ,and 4.4.7 canbeeasilymodied,andthedetailsareomitted.InFigure 4-3 ,weplotthemetricdenedinEquation 4.4.8 ,PercentProtImprove-ment,whentheorderquantityiscapacityconstrained.ContrarytoFigure 4-2 ,demandmanagementincreasestheprotadvantageoftheFIFOruleevenfurtherinthepresenceofacapacityconstraint.AbreakdownofprotintorevenueandcostsindicatesthatthemaindriveroftheimprovementinperformancefortheFIFOruleisrevenue;allthecostschangebyroughlythesameamountforthetworuleswhendemandismanaged.Thereasonforthisobservationliesininventoryavailability.Whiletheinventoryavailabilityis 62

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B C DFigure4-2. PercentprotadvantageofFIFOwithrespecttoLIFOinthepresenceoftime-varyingdemand.A)Dierentahwithb=1:5;=1;=6.B)Dierentpricesensitivityofdemandwith=1;=6;ah=32.C)Dierentsalvagecostwithb=1:5;=6;ah=32.D)Dierentstandarddeviationwithb=1:5;=1;ah=32. 63

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4.4.9 .Uponcomputationofthismetric,wendthattheprotimprovementfortheFIFOruleisgreaterthantheprotimprovementfortheLIFOruleforallthemodelparametersthatweconsidered.Onceagain,thisresultisnotsurprisinggiventhattheprotdierencebetweentheFIFOandLIFOrulesincreaseswhendemandisendogenousinthepresenceofacapacityconstraint. 4.2.3 withrespecttotheoptimalpolicy.Wewillperformthismeasurementforawiderangeofvaluesofthreeparameters,b,,and.Theexperimentsalsoillustratehowtheoptimalweightschangewiththeseparameters.Thesetupforthissetofexperimentsisslightlydierentfromtheprevioussubsection,andtheassumptionsforthissubsectionareasfollows.Firstly,weconsideranadditivedemandmodelsinceTheorem 4.2.3 assumesthatmodel;thus,D=d+,whered=abp.Therandomvariableisnormallydistributedwithmeanandstandarddeviation.Secondly,demandistime-homogeneousandthereisnocapacityconstraint.Thevaluesofmodelparametersareasfollows:Inalltheexperimentsthatweconducted,wefoundthattheoptimalweightsfordemandareequalto0fortheupperboundand1forthelowerbound.Thatis,whenthe 64

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B C DFigure4-3. PercentprotadvantageofFIFOwithrespecttoLIFOinthepresenceofcapacityconstraint.A)Dierentcapacitywithb=1:5;=1;=6.B)Dierentpricesensitivityofdemandwith=1;=6;capacity=14.C)Dierentsalvagecostwithb=1:5;=6,capacity=14.D)Dierentstandarddeviationwithb=1:5;=1;capacity=14. Table4-2. Parametervaluesfortheapproximatepolicy. c h s a 5 1 40 1.5 0 40 65

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4-4 ,weplotthePercentProtDierence,whichisdenedbelow,asafunctionoftheweightontheupperbound.PercentProtDierence=OptimalProt-ProtofApproximatePolicy OptimalProt100%:Thegureshowsthatforlowvaluesofb,theapproximatepolicyperformsquitewell.Forinstance,forb2,theprotoftheapproximatepolicyiswithin2%oftheoptimalpolicy.However,asthevalueofbincreases,theperformanceoftheapproximatepolicyworsens.Forinstance,forb=3,theProtDierencebecomesequalto10%.Thegurealsoillustratesthattheperformanceofthepolicyappearstoberelativelyinsensitivetotheothertwoparameters,and. 66

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B CFigure4-4. Percentprotdierencebetweenapproximatepolicyandoptimalprot.A)Dierentpricesensitivityofdemandwith=0;=6.B)Dierentsalvagecostwithb=1:5;=6.C)Dierentstandarddeviationwithb=1:5;=0. 67

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3.1.1 ProofofPart1Werstdenegt(s;x)=vt(s;x)xforx>0andgt(s;0)=limx!0+vt(s;x).Notethatunlikevt(s;x),gt(s;x)isacontinuousfunction.Withthisdenition,gt(s;x)=8>>>>>>>>>>><>>>>>>>>>>>:maxd2DfLs(x;d)x+E[gt+1(s+1;(xd12)+)+(xd12)+]g;s0;limx!0+vt(s;x);x=0;(wR)x+maxd2D0;y0fL0(y;d)K(y)cy+E[gt+1(1;(yd12)+)]+E[(yd12)]+g;s=R:Let Bythisdenition,gt(s;x)=maxd2DGt(s;x;d)fors0.Wewillprovetheresultbyinductionont.Sincealltheordercyclesareprobabilisticallyidentical,itisenoughtoestablishtheresultforonecycle.Accordingly,weprovethisresultfortherstcyclewhichconsistsofperiods1throughR.WerstshowthatgR(R1;x)isaconcavefunctionofx.Toaccomplishthisobjective,itisenoughto 71

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BoydandVandenberghe ( 2004 )). Clearly,theaboveexpressionisjointlyconcaveinxanddfor>wRforanyR.Next,supposethatgk(s;x)isnon-increasingandconcaveinxfork=t+1;t+2;:::;Randconsiderk=t.Onceagain,consider Asabove,itisenoughtoshowthatGt(s;x;d)isjointlyconcaveinxandd.Toprovethis,itissucienttoshowthatE[gt+1(s+1;(xd12)+)]isjointlyconcaveinxandd.Byinductionassumption,gt+1(s+1;x)isanon-increasingconcavefunctionofx.Consideranytwofeasiblepoints(x1;d1)and(x2;d2).Foranygivenrealizationsof1and2andany2(0;1),gt+1(s+1;[(x1+(1)x2)(d1+(1)d2)12]+)gt+1(s+1;[x1d112]++(1)[x2d212]+)gt+1(s+1;[x1d112]+)+(1)gt+1(s+1;[x2d212]+):Therstinequalityholdsduetothenon-increasingnatureofgt+1.Thesecondinequalityholdssincegt+1(s;)isconcave.Thus,gt+1(s+1;(xd12)+)isjointlyconcaveinxandd. 72

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A.0.2 and A.0.3 ).WerstprovethatGt(s;x;d)issupermodularforanygivens.AfunctionF(u;v)issupermodularifF(u1;v)F(u2;v)isnon-decreasinginvforu1>u2( HeymanandSobel ( 1984 )).Weonlyshowtheresultforsx2andd1>d2.TherstandthirdtermsinEquation A.0.3 aresupermodularsincetheyonlydependond.ConsidernowthesecondtermhE[xd12]+.Foranyrealizationof1and2,[x1d112]+[x2d112]+=[x2d112+(x1x2)]+[x2d112]+[x2d212+(x1x2)]+[x2d212]+=[x1d212]+[x2d212]+;whichmeansthath[xd12]+isasupermodularfunctionforanygiven1and2,implyingthatE[xd112]+isalsoasupermodularfunctionsincesupermodularityholdsunderexpectation. 73

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3.1.2 75

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@ddD1k(d)@ @ddD1k1(d),@ @dd(D1k(d)ckh)<@ @dd(D1k1(d)c(k1)h).Thisimpliesthatdk
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A.0.4 ,v1t(1;x)isaconcavefunctionofx.Further,v2t(1;x)isalinearandnon-decreasingfunctionofx.Therefore,eitherv2t(1;x)isgreaterthanv1t(s;x)forallxorthetwofunctionsintersectatexactlytwopoints,xltandxut.Inthiscase,itisoptimaltoemployStrategy2forallxlessthanxltaswellasforallxgreaterthanxut.ProofofProposition 3.2.2 ProofofPart1SinceD1s(d)D1s1(d),Ls(x;d)=dD1s(d)hE[xd12]+E[d1+2x]+dD1s1(d)hE[xd12]+E[d1+2x]+=Ls1(x;d)ProofofPart2Theresultisestablishedbyinduction.Whent=T+1,vT+1(s;x)=wsxws1x=vT+1(s1;x)Supposenowthattheresultholdsforperiodst+1;t+2;:::;T.Thisimpliesthatvt+1(s;x)vt+1(s1;x).Considerperiodtsuchthats
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3.2.3 ProofofPart1Sincev2t(s;0)K>v1t(s;0),v2t(s;0)>v1t(s;0). 79

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@x[v1t(s;x)]=h+ws+1v1t(s;0)andv1t(s;x)v2t(s;x)forlargevaluesofx,v1t(s;)andv2t(s;)mustcrossatleasttwiceprovidedtheycrossatleastonce.Thesmallestandlargestsuchcrossingpointsarexlt(s)andxut(s).ProofofPart2Sincethesalvagevalueisindependentofs,v2t(s;x)=v2t(s1;x):FromProposition 3.2.2 ,wealsoknowthatv1t(s;x)v1t(s1;x):

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3.2.4 81

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3.3.2 ProofofPart1Letx1
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3.3.3 83

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3.3.4 3.1.1 ,Part3.ProofofPart2ForStrategy1,letq=xz.Thus, UsingthesamelogicasinPart1,theoptimalqwillbegreaterthanorequaltozero.Wewanttoshowthatwhenevertheoptimalqover[0;x],q,isstrictlylessthanx(andsothedierenceofxandqissalvaged),theoptimaldemandisdeterminedbysolvingmaxd2DfdD1s(d)wsdg.Lettheoptimalsolutiontomaxd2DfdD1s(d)wsdgbeequaltod.Supposebywayofcontradiction,theoptimaldemanddisnotequaltodeventhoughq2(0;x).Withoutlossofgenerality,letd0toq+andd+suchthatmin(xq;dd).Withthischange,themaximandinEquation A.0.5 changesby(d+)D1s(d+)(d)D1s(d)ws,whichisidenticaltothechangeinthevalueofdD1s(d)wsdwhendisincreasedfromdtod+.SincedD1s(d)wsdisconcaveindandd
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4.2.2 andso@2GT(x;q;d)

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wheretheinequalityfollowssincebyinductionhypothesisv00t+1()0andv0t+1()h.Theexpressionin B.0.5 isclearlynon-positiveif(1)h.When(1)
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B.0.5 isnon-positive.Therefore,Gt(x;q;d)isajointlyconcavefunctionofdandqforq0andd2D.ProofofPart2Werstconsiderperiodst2f1;2;:::;T1g.Givenanyx,supposethattheorderquantityis0.Todeterminetheoptimalexpecteddemandinthisscenario,notethat whereweusev0t+1(x)=cforx0byinductionhypothesis.Ifthereexistsad02DforwhichtheRHSisequalto0,thend0istheoptimaldemand.BysettingEquation B.0.6 equalto0,wenethatd0satisesthefollowingrelationship:F(xd)=+cR0(d) B.0.6 isnegative(positive)foralld2D,thentheoptimaldemandisequaltominD(maxD).Whentheorderquantityis0,thevaluesofnetinventoryxcanbesegmentedintothreemutuallydisjoint(oneormoreofwhichcouldbepossiblyempty)setsdependinguponthevalueofcorrespondingoptimalexpecteddemand.Letthethreesetsbedenoted 87

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@2Gt=@d2=(+h++c)f(xd) (+h++c)f(xd)R00(d):Asaresult,xd(andhenceF(xd))isincreasinginx.Ontheotherhand,forx2A;C,d=0.Onceagain,xd(andhenceF(xd))isincreasinginxforallx2A;C.Now,observethatanyvalueofxforwhichtheunconstrainedoptimalorderquantityis0mustsatisfythefollowingequationforgivenoptimaldemandd: whereweusev0t+1(x)=cforx0byinductionhypothesis.SinceF(xd)increaseswithx(asweprovedabove),thereexistsauniquevalueofxforwhichtheRHSisequalto0.Thisvalueisdenotedbyxt.Further,sinced(xt)isindependentoft,itisobviousthatxtisalsoindependentoft.ForperiodT,theanalysisissameasaboveexceptthat Asaconsequence,@GT(x;q;d) 88

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B.0.7 ,thismeansthat@Gt(x;q;d) B.0.1 and B.0.2 toobtain@GT(x;q;d) @2GT(x;q;d)=@d2jq=q;d=d=(+s)f(xd) @2GT(x;q;d)=@q2jq=q;d=d=d(x)1: Sinced(x)2(0;1),q(x)2(1;0).Considernowthecaseinwhich@GT B.0.1 ,whichisequalto0,inEquation B.0.2 foranygivend,weget@GT(x;q;d) B.0.9 ,q(x)=1. 89

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B.0.3 andEquation B.0.4 resultsin@Gt(x;q;d) @2GT(x;q;d)=@d2jq=q;d=d=[+v0t+1(q)]f(xd)+v00t+1(q)q(x)F(xd) Ontheotherhand,wedierentiateEquation B.0.3 withrespecttoxandget @x[@Gt(x;q;d) Toestablishtheresultbycontradiction,supposeq(x)>0.Threecasesarise.Case1:q(x)xt+1.Inthiscase,+v0t+1(q(x))0.Weprovetheresultinthefollowinglemma. Proof.

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B.0.16 .)Asimilarargumentcanbedevelopedwhent+1=TusingEquation B.0.15 ;thedetailsareomitted. Now,sinceq(x)>0,Equation B.0.11 canonlybetrueif1+q(x)d(x)0.Thatis,d(x)1+q(x)>1.Ontheotherhand,thedenominatorinEquation B.0.10 isnotonlypositiveandbutalsolargerthanthenumerator,whichmeansthatd(x)1.Butthisisacontradictionandhenceq(x)0.Givenq(x)0,weobtaind(x)0usingEquation B.0.10 sinceboththenumeratoranddenominatorarenon-negative.Thenon-positivityofqalsorequiresthat1+q(x)d(x)0forEquation B.0.11 tobetrue.Thatis,d(x)1+q(x)1.Thesameinequalityalsoimpliesthatq(x)d(x)11.Case2:q(x)xt+1and+v0t+1(q(x))0.Inthiscase,theargumentinCase1canberepeatedtoobtain0d(x)1and1q(x)0.Case3:q(x)xt+1and+v0t+1(q)<0.Inthiscase,thedenominatorinEquation B.0.10 isequaltoR00(d)+[+v0t+1(q)]f(xd)R00(d)+[(h+)]f(xd)wheretheinequalityfollowssincev0t+1()hbyinductionhypothesis.SinceR00(d)handf()1,theaboveexpressionispositive.Sincethenumeratorin B.0.10 isnegativeaswell,d(x)0.Sinceq(x)>0andd(x)0,1+q(x)d(x)>0.However,if1+q(x)d(x)>0,Equation B.0.11 cannotbetrue.Thus,wehaveacontradictionhere.Hence,q(x)0. 91

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+h: Sincev0t+1()c,F(q(0)d(0))c+c +h:Ontheotherhand,fromtheproofofPart2,weknowthatF(xtd(xt))=c+c +h:Clearly,q(0)d(0)xtd(xt).Thus,itsucestoshowthatd(0)d(xt)toshowthatq(0)xt.Bysubstituting@Gt(0;q;d(0)) 92

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+h:Now,sincexT1xT,v0T(xT1)=(h+++c)F(xT1d(xT1))++c;whereweuseEquation B.0.17 ,whichisderivedintheproofofPart3(b).SubstitutingforF(xT1d(xT1)),wegetv0T(xT1)=(h+++c)c+c +h++c:Therefore,+v0T(xT1)>0:Finally,weconsiderthecaseinwhich@Gt(x;q;d @djq=q;d=disnotnecessarilyequalto0forallxxt.Recallfromourargumentabovethatd(0)isintheinteriorofD.Thus,@Gt(x;q;d @djq=q;d=disequalto0atx=0.Asxincreases,d(x)alsoincreasessinceitsslopeisnon-negative.Supposenowthatthereexistsax 93

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B.0.3 ,whichisequalto0,toEquation B.0.4 ,weget@Gt(x;q;d) B.0.11 ,wegetq(x)=(h+)f(x+qd)R1xdv00t+1(x+qd)f()d wherethesecondtermisequaltozeroatq=qsince@GT(x;q;d) 94

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=c(h+s+c)F(x+qd)+(+s)F(xd)=c(+s)F(xd)+@GT(x;q;d) where@GT(x;q;d) Now,fromEquation B.0.15 ,cv0T(x)c(+s):Ontheotherhand,usingEquation B.0.16 ,v0t(x)c(+c)F(xd)c(1);wheretheinequalityholdsbyusingtheinductionhypothesis,v0t+1()c.Further,since+vt+1(q(x))0forxxt,asweprovedintheproofofPart2(b),v0t(x)c.Now,whenx0v0T(x)=csinceF(xd)=0forx0.Usingthesameargument,v0t(x)=c:forx0.ProofofPart2(d)UsingEquation B.0.16 ,v00t(x)=v00t+1(q)q(x)F(xd)(+v0t+1(q))f(xd)(1d(x))0;whereweusetheconcavityofvt+1,thenon-positivityofq(x),thenon-negativityof(+v0t+1(q)),asprovedintheProofofPart2(b),andd(x)2[0;1].TheproofforperiodTissimilar,andthedetailsareomitted. 95

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@2Gt(x;q;d)=@d2jq=0;d=d(x)=(h+++c)f(xd) whereeither@GT(x;q;d) B.0.13 ,andthedetailsareomitted.Next,we 97

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B.0.13 ,weget Fromequations B.0.17 and B.0.18 foranygivenperiodt,v0t(x)(h+++c)++c(h+)whichprovidesalowerbound.Also, wheretheinequalitiesfollowfromtheProofofPart3(a).Using B.0.19 and B.0.20 ,F(xd)c+c +hs+cfort=T,andF(xd)c+c +hfort
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B.0.17 and B.0.18 ,wegetv0T(x)c(c+c)(s) B.0.17 and B.0.18 ,v00t(x)=(h+++c)f(xd(x))[1d(x)]inanygivenperiodt.Sinced(x)2[0;1],v00t(x)0.ProofofTheorem 4.2.3 ProofofPart1Werstderivetheupperboundontheoptimalexpecteddemand.Whenx>xandtT,usingEquations B.0.17 and B.0.18 ,v0t(x)=(h+++c)F(xd(x))++cc;wheretheinequalityfollowsfromTheorem 4.2.2 Part3(b).Therefore,d(x)xF1[c+c h+++c]=xy;wherewedeney=F1[c+c h+++c].Thus,theexpecteddemandcorrespondingtonetinventoryxisboundedfromabovebyxy.Ontheotherhand,whenxxt,d(x)d(xt)xty;sinced(x)isincreasinginx.ProofofPart2Letdcbedenedsuchthat@[R(d)] 99

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4.2.2 .Therefore,x+qdF1[c++c h+]=:r:Consequently,x+qF1[c++c h+]+dr+xty;wherethesecondinequalityfollowssinced(x)xtyfromPart1above.Thus,theorderquantityisboundedfromabovebyr+xtyxwhenxxt.Thelowerboundonorderquantityisobtainedbyusingv0t+1(x)h.Asaconsequence,x+qdF1[c(h+)] ;whichimpliesqr 4.2.5 B.0.12 thatF(q2(0)d2(0))=c+R1av0t+1(q2(0)d2(0))f()d h+:Further,sincext0,usingLemma B.0.1 ,+v0t+1(q2(0)d2(0))0:Bysubstitutingv0t+1(q2(0)d2(0)) andv0t+1(z)=cforz0intheaboveequation,weget h+++c: 100

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4.2.3 .Now, (B.0.22) whereweusev0t+1(z)=c(withoutproof)forz0.Thus,theoptimalvalueofq1satises +h++c: ComparingEquations B.0.23 and B.0.21 ,q1(0)q2(0)whereweused1(0)=d2(0).Next,usingEquation B.0.6 ,F2(x2td2(x2t))=+cc h+:ComparingtheaboveequationwithEquation B.0.23 ,q1t(0)d1t(0)x2td2(x2t):Sincetheoptimalvalueofd2(x)isincreasinginx,d2(x2t)d2(0)=d1(0).Therefore,q1t(0)x2t.ProofofProposition 4.2.6 ProofofPart1LetG1(x;q1;d1)beusedtodenotethemaximandinEquation 4.2.4 .Fort=T,@G1T(x;q1;d1) B.0.1 .Henceforanyx,q1(x)d1(x)=q2(x)d2(x).Sinced2(x)d2(0)=d1(x),q1(x)q2(x).ProofofPart2

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+h: Ontheotherhand,usingEquation B.0.12 ,F(x+q2(0)d2(0))c+c +h;whereweusetheupperboundonv0t+1(),whichisequaltoc,toobtaintheupperbound.ComparingtheaboveequationwithEquation B.0.24 ,q2(0)d2(0)q1(0)d1(0):Sinced2(0)=d1(0),q2(0)q1(0).ProofofPart3FollowsdirectlyfromParts1and2.ProofofPart4RecallfromtheproofofProposition 4.2.5 thatxtsatisesF2(x2td2(x2t))=+cc h+:ComparingtheaboveequationwithEquation B.0.24 ,weseethatx2td2(x2t))=q1(0)d1(0):Sinced2(x2t))d2(0)=d1(0),x2tq1(0). 102

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4.3.1 4.3.5 .Thatis,Gt(x;q;d)=R(d)cqhE[x+qd]+E[d+xq]+E[x[d+q]+]++Evt+1(x+qd(x(d+q)+)+):Letz=qd.ThentheoptimizationprobleminanyperiodtbecomesGt(x;q;d)=maxd2D;zdR(d)c(z+d)hE[x+z]+E[xz]+E[x[z]+]++Evt+1(x+z(x(z)+)+):Considersomexsuchthatq(x)>0.Inthatcase,theconstraintzdbecomesredundant.Thus,theaboveoptimizationproblembecomesGt(x;q;d)=maxd2DfR(d)cdg+maxzfczhE[x+z]+E[xz]+E[x[z]+]++Evt+1(x+z(x(z)+)+)g:Clearly,theoptimalvalueofdisthemaximizerofR(d)cd.JusticationBehindObservation 4.3.2

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4.3.1 ,dmaximizesR(d)cdandthusliesintheinteriorofD.)Now,@vT(x) {z }=0@q(x) {z }=0@d(x) whered=0sincedisaconstant.ToshowthatvTisnotnecessarilyconcaveinx,itsucestoshowthat1+qisnotnecessarilynon-negative.UsingtheImplicitFunctionTheorem,q(x)=@2GT(x;q;d)=@x@q @2GT(x;q;d)=@q2jq=q;d=d=(+c+h+)f(x+qd) (s+)f(qd)(+c+h+)f(x+qd);whichisnotnecessarilygreaterthan-1ifsorarestrictlypositive.ProofofTheorem 4.3.3 ProofofPart1InperiodT, Denex=xsuchthattheconstrainedoptimalvalueofq,q(x)=0.Ifthereexistmultiplesuchvaluesofx,thenwetakethemaximumofthosevalues.Itcanbeeasily 104

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Onceagain,setx=xsuchthattheunconstrainedoptimalvalueofq(x)=0.Incaseofmultiplesuchvalues,choosethehighestone.Usingtheinductionhypothesis,v0t+1(x)=cforx0andforanygivend, (B.0.30) (B.0.31) 107

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{z }=0:Whiletherstequationshowsthatv0T(x)h,thesecondequationestablishesthatv0T(x)isboundedfromabovebyc.Similarly,forperiodt, =+c(h+++c)F(x+qd);whereweusev0t+1(x)=cforx0.Itcanbeeasilyseenthatv0t(x)h.Now,usingEquation B.0.28 ,Equation B.0.32 canalsobewrittenasv0t(x)=cF(qd)Zqdav0t+1(qd)f()d+Gt(x;q;d) {z }=0:Usingtheinductionhypothesis,v0t+1(x)h.Therefore,v0t(x)c+h(1).ProofofPart3Whenxx,q(x)=0.Asaresult,@GT(x;0;d) @2Gt(x;0;d)=@d2jd=d=(h+++c)f(xd)

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SimilartoProposition 4.3.1 ,itcanbeeasilyshownthattheoptimalvalueofdsatisesR0(d)=cforallsuchx.Now,weclaimthatthereexistsavalueofy>xthatproducesstrictlygreaterprotthany=x.Toseethis,wecomputethederivativeofthemaximandwithrespecttoyaty=xasfollows:c(h+)F(xd)++c;whereweusev0t+1(x)=cforx0byinductionhypothesis.Sincexda,F(xd)=0.Thus,thederivativebecomesc+c>0.Hence,increasingyfromxtox+willincreaseprot,soy>x.Withthisobservation,wecannowdroptheconstraintyxfromtheRHSinEquation B.0.33 .Itcannowbeeasilyseenthatv0t(x)=c.ProofofParts4and5Forxx,q(x)=0.@vT(x) {z }=0=(+h++c)F(xd)++c:

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Adamy,Janet.2004.Halfaloaf:Atgiantbaker,freshnessprojecttakessourturn{interstate'sdrivetoextendshelflifehasproblems;now,achapter11ling{disarrayinwonderbreadrack.WallStreetJournal23SeptemberA1. Boyd,Stephen,LievenVandenberghe.2004.ConvexOptimization.CambridgeUniversityPress,NewYork. Brodheim,E.,C.Derman,G.P.Prastacos.1975.Ontheevaluationofaclassofinventorypoliciesforperishableproductssuchasblood.ManagementScience221320{1325. Chan,LapMuiAnn,DavidSimchi-Levi,JulieSwann.2006.Pricing,production,andinventorypoliciesformanufacturingwithstochasticdemandanddiscretionarysales.ManufacturingandServiceOperationsManagement8(2)149{168. Chande,A.,S.Dhekane,N.Hemachandra,N.Rangaraj.2004.Fixedlifeperishableinventoryproblemandapproximationunderpricepromotion.Tech.rep.,IndustrialEngineeringandOperationsResearch,IITBomaby,Mumbai,India. Chande,A.,S.Dhekane,N.Hemachandra,N.Rangaraj.2005.Perishableinventorymanagementanddynamicpricingusingrdtechnology.Sadhana30445{462. Chandrashekar,K.,N.Dave,N.Hemachandra,N.Rangaraj.2003.Timingofdiscountoersforperishableinventories.ProceedingsofSixthAsiaPacicOperationsResearchSociety,AlliedPublishers,NewDelhi. Chen,Xin,DavidSimchi-Levi.2004.Coordinatinginventorycontrolandpricingstrategieswithrandomdemandandxedordercost:Thenitehorizoncase.OperationsResearch52(6)887{896. Chen,Yonhua(Frank),SaibalRay,YuyueSong.2006.Optimalpricingandinventorycontrolpolicyinperiodic-reviewsystemswithxedorderingcostandlostsales.NavalResearchLogistics53117{136. Chen,Zhi-Long.2007.Integratedproductionandoutbounddistributionschedulinginasupplychain:Reviewandextensions.WorkingPaper,RobertH.SmithSchoolofBusiness,UniversityofMaryland. Cohen,MorrisA.,DovPekelman.1978.Lifoinventorysystems.ManagementScience24(11)1150{1162. Dave,Upendra.1991.Surveyofliteratureoncontinuouslydeterioratinginventorymodels-arejoinder.TheJournalofOperationalResearchSociety42(8)725. Federgruen,Awi,AlizaHeching.1999.Combinedpricingandinventorycontrolunderuncertainty.OperationsResearch47(3)454{475. 112

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Li-MingChenwasborninTaipeiTaiwanonNovember5,1979.Uponcompletionofhighschool,heattendedNationalChiaoTungUniversityinHsinChu,earningadegreeinindustrialengineeringandmanagement.Afterheobtainedhisbachelor'sdegreeinJune2001,heservedasasecondlieutenantinthemilitaryfor2years.InJune2003,Li-MingnishedhismilitarycareerinTaiwan.Realizingthedesireforknowledge,hewenttotheUnitedStatesforthepursuitofhighereducation.HebeganhisgraduatestudiesattheUniversityofMichiganinthefallof2003,andgainedhismaster'sdegreeinindustrialandoperationsengineeringinJanuary2005.Li-Mingbegantondenjoymentinexploringnewideas,resolvingcomplexmathematicalproblems,andseekingecientsolutions.Therefore,hedecidedtopursuehisdoctorialdegreeinindustrialandsystemsengineeringattheUniversityofFlorida,whichoeredprogramsalignedwithhisinterestsinOperationsResearchandSupplyChainManagement.WhileattheUniversityofFlorida,hereceiveddepartmentfundingasafull-timeresearchassistant,andduringwhichtime,healsoservedasateachingassistantforseveralcoursesinsupportofcompletingcourseobjectives.HeearnedtheDoctorofPhilosophydegreeinAugust,2009Dr.Chen'srecentworkdiscussedthereplenishmentdecisionfrommultiplesourcingwhenSupplyChainDisruptionhappens. 115