RISK POOLING AND COST ALLOCATION IN SPARE PARTS … · 2016-01-07 · Although the so stated relevance of spare parts inventories and risk pooling, the literature lacks of evidence

September 24-28, 2012Rio de Janeiro, Brazil

RISKPOOLINGANDCOSTALLOCATIONINSPAREPARTSINVENTORIES

MarioGuajardoDepartmentofFinanceandManagementScience,NHHNorwegianSchoolofEconomics

Helleveien30,N‐5045Bergen,[email protected]

MikaelRönnqvist

DepartmentofFinanceandManagementScience,NHHNorwegianSchoolofEconomicsHelleveien30,N‐5045Bergen,Norway

[email protected]

ABSTRACT

Wepresent a problem on risk pooling and cost allocation in inventory of spareparts motivated in the oil and gas sector. Spare parts are held to facilitate operationalconditions.Multipleinventoryplantscontroltheirstockperseparate.Inpractice,however,theysupplyeachotherwhenneeded.Riskpooling isasourceofsavings in thissituation.Underacentralized inventorysolution, theplantsshouldagreeonhowtosharecosts fortakingadvantageofthebenefitsofpooling.Wereportcomputationswherethecentralizedinventoryachievessignificantsavingsoverthedecentralizedsolution,consideringorderingand holding costs subject to a service level constraint. We then test five cost allocationmethods and analyse their outcomes, using game theory principles. We show exampleswhereamethodresults in stableallocationswhen theplantsuse thesame target servicelevels,butnotwhenatleastoneplantuseadifferenttarget.

KEYWORDS.Inventoryofspareparts.Riskpooling.Costallocation.

P&G‐POnaÁreadePetróleo&Gás

L&T‐LogísticaeTransportes

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1.IntroductionRiskpoolinghasbeen traditionallyusedasa formofprotectingagainstdemand

variability in inventorymanagement.Theprinciple is that by aggregatingdemandacrossdifferent locations, highdemand fromone customer is likely tobeoffset by lowdemandfromanother. This reduction in variability allows a decrease in safety stock and averageinventory.While risk poolingmay result in lower total inventory costs comparing to thecase when the locations act per separate, it is not straightforward to answer how theparticipantsinthepoolshouldsharecostsinordertoachievethesesavings.Thedifficultyarises, forexample,becausethe locationscanbeexposedtodifferentdemandbehavioursand,therefore,someofthemmayinpracticeperceivemorebenefitsfromtheriskpoolingsituationthanothers.

The cost allocation problem in multi‐location inventory systems has receivedattentionduringquiteafewyears(e.g.GerchakandGupta,1991;Robinson,1993;Hartmanand Dror, 1996, 2000). However, in the specific context of spare parts inventories, theresearchincostallocationismoresparse.ThefirstrelatedworkwaspublishedbyWongetal. (2007),whousegame theoryprinciples toanalysecostallocationrules insparepartsinventories.Kilpietal.(2009)discusscooperativestrategiesforriskpoolingofsparepartsfor aircrafts.More recently, Kartsen et al. (2009, 2011a, 2011b) andKarsten andBasten(2012)havestudiedaseriesofrelatedproblems.

Spare parts are relevant to facilitate operational conditions in contexts whereequipments are subject to maintenance and failure. The consumption of spare parts istriggered by events subject to variability and, therefore, holding inventory to fulfil thedemandrequirementsmaybecrucial.Alowstockofsparepartscouldimplystockoutand,in consequence, production downtime until the part is repaired or replenished. On theotherhand,ahighstockof sparepartscould implyan importantbindingcost.There isahugebodyofliteraturedealingwithspareparts,asreviewedbyKennedyetal.(2002).Animportantpartofthis literaturehasbeenmotivatedinair forceandaircraftcontexts(e.g.Muckstadt, 2005; Sherbrooke, 2004). Applications in other contexts include the servicesupportof IBM in theUS (Cohenet al.1990), a chemicalplant inBelgium(VereeckeandVerstraeten,1994), awhitegoodsmanufacturer in Italy (Kalchschmidtet al.2003) andadistributorofcastorsandwheelsinGreece(Nenesetal.2010).Ourresearchismotivatedinthe oil and gas sector, where we have found only one recent article on spare partsinventories,publishedbyPorras andDekker (2008).Characterizedbyhigh service levels(becauseofsafetyandproductionfactors),customizedequipmentspecificationswithlonglead times,and facilitynetworksspreadonshoreandoffshore,webelieve theproblemofinventorymanagementofspareparts in this industrydeservesspecialattention fromtheresearchcommunity.

Togetherwiththeinterestoninventoriesofspareparts,thereisanextensivebodyofliteraturedealingwithlateraltransshipmentandriskpooling,asreviewedbyPatersonetal.(2011).

Although thesostatedrelevanceof spareparts inventoriesandriskpooling, theliterature lacks of evidence on how risk pooling in spare part inventories should beimplementedandhowthecostshouldbesharedamong thedifferentplayers, inorder totakeadvantageofthebenefitsofpooling.Inthisarticle,wedescribeareal‐worldsituationwhere risk pooling may represent an important source of savings and we discuss thesuitabilityoffivecostallocationmethods.Wereportnumericalexampleswherethesecostallocationsmethodscanprovidestableallocationsinsituationswherethedifferentplayersuse the same target service levels. The same methods in the same numerical examples,however,mayfailonthatpurposewhenatleastoneoftheplayershasatargetserviceleveldifferentthantheotherplayers.

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2.Background Our motivation comes from a real‐world problem faced at an energy companyproducerofoilandgas.HeadquarteredinNorwayandwithpresencein42countries,thiscompany is one of the world's largest net sellers of crude oil and the second largestexporterof gas toEurope. It actsasoperatorof severaloffshoreplatformswithdifferentownershipstructureanditholdsinventoryofabout200,000sparepartitemsinanumberof locations spread in the Scandinavian region. Some of these parts are highly critical toassuresafetyandproduction.Atthesametime,thestoreofsparepartsmeansanimportantbinding cost. The company has operational responsibility for seven warehouses locatedalong the Norwegian coast, which serve offshore installations. Within this structure ofwarehouses, there are 24 inventory plants. Themajority of theseplants is ownedby thecompanyunderstudy,but there isalsoasetofothercompaniesowningashareof them.Each of the 24 plants holds its own inventory of spare parts and they aremanaged perseparate.Eachof theplantsdetermines itsown inventory controlparameters separately,andislinkedtooneproductivelocation(andvice‐versa).Whenasparepartisrequired,itisprovided from the corresponding plant. However, when a productive location requirescertain spare part in such amount that it is not available at the corresponding plant, therequirementcanbefulfiled(ifagreed)fromotherplantthathasavailabilityonstock.Thisisespeciallyrealizedwhentherequiredpart ishighlycritical.This typeof fulfilmentoccursbasedonan informalagreementbetween inventoryplants,but it isnotconsideredwhendecidingthecontrolparameters. It is, therefore,relevanttostudythepotentialofsavingsthatariskpoolingapproachwouldgenerateiftherewouldbeacentralizeddecisionontheinventory control parameters instead of the separate planning that the plants carry outtoday. The importanceof lateralresupplyandpooling is illustrated inMuckstadt(2005),whogivesexamplesofsystemsinwhichpoolingexiststhatrequireroughlyathirdofthesafety stock required when operating a completely decentralized system. A recentapplication byKranenburg and vanHoutum (2009) reports that under full poolingmorethan50%of thenopoolingcostcanbesaved in in thecaseofASML,aDutchequipmentmanufacturer.Before their implementation,ASMLdidnot take lateral transshipment intoaccount in the planning phase, i.e., the inventory in each local warehouse was plannedseparately. Nevertheless, in daily practice lateral transshipment was used, the samepracticeasrealizedatthecompanyinourcase.AsimilarsituationispresentedbyKukrejaandSchmidt(2005),whodealwithalargeutilitycompanyhaving29generatingplants.Theplantsdonotconsidertheeffectofbeingsuppliedfromeachotherwhendecidingtheirowninventory control policies, but in practice they also collaborate. Experimentation by theauthorsincludingfromtwotofiveplantsshowthatpoolingleadtosavingsbetween31%and 58.78%. Other works in lateral transshipments under emergency situations arereportedinArchibald(2007),Wongetal.(2006)andAlfredssonandVerrijdt(1999). AmoredetaileddescriptionaboutthecontextinourcasecanbefoundinGuajardoetal.(2012).Notethat,whileriskpoolingcanconducetogreatsavings,itisnotclearhowthedifferentshareholdersattheplantsshouldsharethecosts.Exploringthisproblemisthemainpurposeof thispaper. In the following section,wewill defineourproblemsettingsandaproceduretofindtheoptimalpolicy.3.Problemsettingandoptimalbase‐stockpolicy Weconsiderasingle‐itemproblemundercontinuousreview(S‐1,S)orbase‐stockpolicy. Inthispolicy,everydemandeventoriginatesanorderforanamountsuchthatthebase‐stock levelS is reached.For this reason, it is also calledaone‐for‐onereplenishmentpolicy.The(S‐1,S)policyiscommonlyassumedininventoryofsparepartproblems,wheretheitemsareusuallyexpensiveandslow‐moving(Muckstadt,2005;Sherbrooke,2004).We

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assumefixedleadtimeanddemandperunitoftimefollowingaPoissondistribution.Thisdistributionisalsocommonlyassumedinsparepartsinventorymodelling. There are traditionally three type of costs involved in inventory problems: ordercost,holdingcostanddowntimecost.Theordercostisafixedcostperorder,theholdingcost is a variable cost of carrying inventory and the downtime cost is the cost incurredwhenademandorderisnotsatisfiedbecauseofastockout.FindingtheoptimalinventorycontrolparameterS couldbepursuedbyminimizing thewholecost functionconsideringthese three cost elements. However, in practice, the downtime cost is usually hard toestimate.Apopularstrategyistominimizecarryingplusorderingcosts,subjecttoaservicelevel constraint. The service level criterion is generally easier to state and interpret bypractitioners.Anumberof referencespointout this fact, suchasChenandKrass (2001),BashyamandFu(1998)andCohenetal.(1989). Asservicelevelmeasure,weutilizethefillrate,definedastheexpectedfractionofdemand satisfied directly from stock on hand, which is usually utilized in spare partsinventoryproblems.Hence,weutilizeatargetservicelevelβaslowerboundthatmustbesatisfiedbythefillrateachievedforthecontrolparameterS.WereferthereadertoSilveret al. (1998) for more background on inventory costs and service levels, and for thefollowingdefinitionsandformulae. Let A be the cost for an order and λ the average demand per unit of time. WecomputetheexpectedordercostasCorder=A/λ. Letrbethecarryingcharge,vtheunitvalueoftheitemandĪtheaverageinventoryonhand.We compute the expectedholding cost asChold= Īvr.We calculate Ī as κ+1/2,whereκisthesafetystockcalculatedasthereorderpointS‐1minustheexpecteddemandduringthe leadtimeλL.LetXLbethedemandduringthe leadtime.Sincethedemandperunit of time follows a Poisson distribution with mean λ, then XL follows a PoissondistributionwithmeanλL=λL,whereListheleadtime. ThetotalexpectedcostperitemisC=Corder+Chold=A/λ+Īvr=A/λ+(S‐λL‐½)vr. Foragivenbase‐stocklevelS,thefillrateβSinoursettingcanbecalculatedasthe

probabilityofstockout1‐P(XL≤S‐1).Thus, 1 ∑!

. TheoptimizationproblemconsistsoffindingthecontrolparameterSthatsolvestheproblembelow.

MinC(S) s.t. Since the cost increases inS (in fact,note thatC canbewrittenasC(S)=K+Svr,whereKdoesnotdependonS),theproblemreducestofindtheminimumSsuchthattheservicelevelconstraintβS≥β isfulfiled.Asimpleandusualapproachtofindsuchoptimalbase‐stock level S is to evaluate the fill rate βS starting from S = 1 and consecutivelyincreasingSbyone,untilthetargetβisfulfiled.Hence,iftheservicelevelachievedbyS=1is less thanβ, thenS=2 is triedand soon,until the lowestS satisfying the constraint isfound.Forcomputationalpurposes,anupperboundonScouldbepredefined,sothattheevaluation procedure is assured to finish. However, in practice the procedurewill finishquickly,becauseinsparepartproblemsusuallyalowvalueofSwillbeoptimal.4.Costallocation We consider a game consisting of a set N of all players (inventory plants), whodecidewhethertoplantheir inventoryseparatelyortoplantheir inventoryinacoalitionwith other players based on a risk pooling strategy. If a player stays alone, itwill set itsbase‐stocklevelSandrunitsinventoryindependentfromtheotherplayers.Ifplayersactin

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acoalition, theywill setabase‐stock levelSunder thesameobjective functionandrunasingle inventory serving their whole demand. This last case corresponds to a purecooperative strategy which in operational terms translates into a centralized inventorysystem.Foracoalition ⊆ ,wereferbyCMtotheoptimalexpectedcostifallplayersincoalitionM would implement risk pooling.N is the grand coalition and CN is its optimalexpectedcost,asiftherewouldbeacentralizedinventoryplantplanningtheinventorytosatisfydemand fromall theplayers.Foragivencostallocationrule,wereferbyuj to thecostallocatedtoplayerj.4.1Thecoreofthegame Acostallocationvectoru=(u1,u2,...,un)issaidtobeinthecoreofthegame(Gillies,1959)ifitsatisfiesconstraints(1)‐(3)below.

∀ ∈ (1)

∑ ∈ ∀ ⊂ (2)

∑ ∈ (3) Constraint(1)correspondstothe individuallyrationalcondition,whichstatesthatthecostallocatedtoeachplayerjmustnotbegreaterthanitsstand‐alonecost.Constraint(2)correspondstoastabilitycondition,whichstatesthatthereisnosubsetofplayerssuchthatiftheywouldformacoalitionseparatefromtheresttheywouldperceivelesscostthantheallocationu.Constraint(3)correspondstotheefficiencycondition,whichstatesthatthesumofthecostsallocatedtoalltheplayersequalstheoptimalcostofthegrandcoalition,andthusittakesfulladvantageofriskpooling.Thecoreofthegameisthesetofallvectorsusatisfyingconstraints(1)‐(3).Inotherwords,acostallocationvectorinthecoreassuresthat the savings of risk pooling are achieved andmakes all players to stay in the grandcoalition,withoutincentivesforaplayertostayaloneorwithinasmallercoalition.4.2Allocationmethods

Wewillusethefivecostallocationmethodsdescribedbelow. Costallocationmethod1:Egalitarian

Thismethodsimplyassignsequalcostsharestoalltheplayers,asfollows:

| |∀ ∈

Costallocationmethod2:Proportionaltodemand(orαDemand)

Foraplayerj,whosedemandrateisλj,thismethodassignsacostshareproportionaltotheweightthatthisdemandraterepresentsoverthetotaldemandrateofthepool.

∑ ∈∙ ∀ ∈

Costallocationmethod3:Altruistic

Foraplayerj,whosestand‐alonecostisC{j},thismethodassignsacostshareproportionaltothecostthatC{j}representsoverthetotalcostoftheplayersiftheyallwouldactalone.

∑ ∈∙ ∀ ∈

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Costallocationmethod4:ShapleyvalueShapley(1953)introducedanallocationbasedonthemarginalcostsinenteringcoalitions.ThemarginalcostofplayerjenteringcoalitionMis \ .TheShapleycostallocationmethod assigns player j the average of the marginal costs he implies when entering allcoalitionshecanenter.Themethodconsidersallsizesofacoalitionequallylikely,thusthesize|M|isassignedaprobability1/|N|.Considerplayer jandthe|N|‐1remainingplayers.

Foragivensize|M|,thereare | || | waysofchoosing|M|‐1playersincoalitionMfromthe

|N|‐1 remainingplayers.Therefore, theprobability of a particular coalitionM is| | | |

| |

.

RegroupingtermsandconsideringallpossiblesubsetsofN,thismethodallocatesthecostsasfollows:

| | | | ! | | 1 !| |!

∙ \

⊆ : ∈

∀ ∈

Costallocationmethod5:EqualProfitMethod(EPM)

EPMlooksforastableallocationsuchthattherelativesavingsareassimilaraspossibleforallparticipants.Suchallocationuissolutionofthefollowinglinearprogrammingmodel:

min s.t.

∀ , ∈

∈

∀ ⊂

∈

0∀ ∈ ; ∈ .

TheegalitarianmethodisreferredbyTijsandDriessen(1986)asthesimplestcostallocationmethod.TheαDemandandtheShapleymethodscorrespondtowhatWongetal.(2007) refer to ascostallocationpolicies3 and4, respectively.The altruisticmethodhasbeenusedbyAudyetal.(2012)inaforestryproblem.TheEPM,recentlyproposedbyFrisketal. (2010),hasbeenused incollaborative forest transportationandalwaysresults inastableallocation,ifthecoreisnotempty.Inadditiontothesemethods,ofcourse,anumberofothermethodshavebeenproposedintheliterature(see,forinstance,TijsandDriessen(1986)whopresentasurveyofcostallocationmethodsandgametheory,andproposetheirownmethodtoo).Wongetal.(2007)usedfourmethodsincomputationalexperiments,butin their problem they consider a downtime cost, while we explicitly use a service levelconstraint instead. Other difference in our setting is that we do not consider lateraltransshipmentcosts that theydo. Inouroilandgascontext,all theplantswearedealingwithlierelativelyclosetoeachother.Moreover,someoftheinventoryplantsarephysicallylocated within the same warehouse. Also, the most extreme emergency cases are thosewhen amain part in a platformoff‐shore need to be replaced and its corresponding on‐shoreplantdoesnothavethepartonstock.Ifthepartisavailablefromotherplant,thepartdoesnotneedtobesentfromonetoanotheron‐shoreplantbeforebeingsentoff‐shore,butitisjustsentstraightfromtheon‐shorepantsupplyingittotheoff‐shorelocation.

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5.Numericalexamples Next,we illustrate theallocationsgeneratedby the fivemethodsusing real‐worlddataofthreeitemsinthreeplants.Thedatahasbeentakenfromthecompanyintheoilandgas industrythatweintroducedinthebackgroundsection.Weusethehistoricaldemanddata to estimate thedemanddistributionparameters in eachplant.When computing theparameters for a coalition of plants, we aggregate the historical demand data of thecorrespondingplants.5.1.Equaltargetservicelevels Wefirstcomputeresultsforthecasewhen, forasameitem,thethreeplantshavethe same target service levels. The targets are βitem1=0.97, βitem2=0.95, βitem3=0.99. Otherparameter values are shown in Table 1. For confidentiality reasons, in the followinganalysiswedonotrevealdetailsonthecostdataandwhenshowedintheresults,currencyunitsandconversionfactorshavebeenomitted.

Table1:Dataonleadtimesandaveragedemandforeachitemateachplant.Plant Item1 Item2 Item3

Leadtime 6.67 1.17 0.80

λPlantP1 0.0448 0.0448 0.0597

λPlantP2 0.0149 0.0448 0.0149

λPlantP3 0.0299 0.0149 0.0597 Foralltheinstancessolvedinthisarticle,theoptimalbase‐stocklevelswerequicklycomputed,inlessthanasecond.TheresultsobservedforSareusuallyintherangefrom1to3.Asummaryofresultsonsafetystocks,averageinventoryonhandandtotalexpectedcostsinthecaseofequaltargetservicelevelsforalltheplantsisshowninTable2.Itcanbeobservedthatriskpoolingimpliesdecreasesofmorethan50%insafetystocksandaverageinventoryonhandforallitems,whencomparingtothesumoftheresultswhentheplantscontroltheirinventoryperseparate.Thecostreductionis51%,38%and45%foritems1,2and3,respectively. Table2:Safetystock(κ),averageinventoryonhand(Ī)andtotalcost(C)resultsforthreeitemsinthreeequalservicelevelplants(thefirstthreerowsshowstand‐alonecosts,thefourthrowtheirsumandthelastrowthe

costwhenpooledinthegrandcoalitionN).Item1 Item2 Item3

Plant κ Ī C Κ Ī C Κ Ī C

P1 1.70 2.20 1,113 0.95 1.45 562 0.95 1.45 4,824

P2 0.90 1.40 646 0.95 1.45 562 0.99 1.49 4,619

P3 0.80 1.30 673 ‐0.02 0.48 187 0.95 1.45 4,824

P1+P2+P3 3.40 4.90 2,432 1.88 3.38 1,310 1.94 2.94 9,443

N 1.40 1.90 1,196 0.88 1.38 810 0.89 1.39 5,167 Table3presentsthecostallocationresultsobtainedbythefivedifferentmethods.Foritems1and3,thefivemethodsconducedtocostallocationsinthecore.ThisiseasilyverifiablewhenusingthecostallocationsvaluesfromTable3 inconstraints(1)‐(3).Foritem2,all themethodsconducedtocostallocations inthecore,except fortheegalitarianone.Thecostallocatedtoplant3bythismethodis270(highlightedcellinTable3),whichisgreaterthanthecost187ofthisplantforitem2whenitwasplannedperseparate,thusconstraint(1)isviolatedandtheresultingallocationdoesnotbelongtothecore.

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Table3:Costallocationsbythefivemethodsforthreeitemsinthreeplants,sametargetservicelevels.

Notethattheupperboundsforconstraints(1)correspondtotheCTvaluesinthefirstthreerowsofTable2,whiletheright‐handsideofequation(3)correspondstotheoptimalcostofthegrandcoalitiongiveninthelastrowofTable2.Theupperboundsforconstraints(2),namelytheoptimalcostfortwo‐playerscoalitionsaregiveninTable4.

Table4:Optimalcostsforcoalitionsoftwoplayers,sametargetservicelevels.Coalition Item1 Item2 Item3

{P1,P2} 1,141 748 4,893

{P1,P3} 1,168 624 5,098

{P2,P3} 1,113 624 4,8935.2.Differenttargetservicelevels Nowweconsiderthecasewhenthethreeplantshavedifferenttargetservicelevelsfor a same item. This may happen, for instance, because the same items are utilized indifferent contexts in the different plants, or because the service levels at different plantshave been set regarding different evaluation basis.Moreover, Porras and Dekker (2008)andGuajardoetal.(2012)havereportedoilrelatedproblemswherethesameitemcanbeutilizedwithdifferentservicelevelsevenatasingleinventorylocation. WedefineplantP1asahighsafeplant,inthesensethatitrequiresrelativelyhightargetservicelevels,whileplantP2ismediumsafeandplantP3lowsafe. ForplantP1,thetargetsareβP1item1=0.97,βP1item2=0.95,βP1item3=0.99. ForplantP2,thetargetsareβP2item1=0.87,βP2item2=0.85,βP2item3=0.89. ForplantP3,thetargetsareβP3item1=0.77,βP3item2=0.75,βP3item3=0.79. When plants act in coalition, the target service level that we consider is themaximumof the target valuesof the plants involved in the coalition.Rounding‐up targetservice levels inthisway isacommonpractice(althoughnotnecessarilyoptimal,seee.g.threshold rationing policies by Dekker et al. 1998 and Deshpande et al. 2003; and acomputational studybyGuajardo andRönnqvist 2012). In our context, it is arguable if aplayerwouldbewilling to joinacoalitionundera targetservice level lowerthan itsowntarget; on the other hand, a playerwith a lower targetwould still satisfy its own targetwhenjoiningacoalitionwithahighertargetvalue.

Table5:Safetystock(κ),averageinventoryonhand(Ī)andtotalcost(C)resultsforthreeitemsinthreedifferentservicelevelplants(thefirstthreerowsshowstand‐alonecosts,thefourthrowtheirsumandthelast

rowthecostwhenpooledinthegrandcoalitionN).Item1 Item2 Item3

Plant κ Ī C Κ Ī C Κ Ī C

P1 1.70 2.20 1,113 0.95 1.45 562 0.95 1.45 4,824

P2 ‐0.10 0.40 234 ‐0.05 0.45 311 ‐0.05 0.45 1,791

P3 ‐0.20 0.30 261 ‐0.02 0.48 187 ‐0.01 0.49 1,585

P1+P2+P3 1.40 2.90 1,608 0.88 2.38 1,060 0.89 2.39 8,200

N 1.40 1.90 1,196 0.88 1.38 810 0.89 1.39 5,167

Plant Egalit. αD Altruistic Shapley EPM Egalit. αD Altruistic Shapley EPM Egalit. αD Altruistic SAM EPM

P1 399 598 547 564 547 270 347 347 353 347 1,722 2,296 1,747 1,791 1,747

P2 399 199 317 302 317 270 347 347 353 347 1,722 574 1,673 1,585 1,673

P3 399 399 331 330 331 270 116 116 104 116 1,722 2,296 1,747 1,791 1,747

Total 1,196 1,196 1,196 1,196 1,196 810 810 810 810 810 5,167 5,167 5,167 5,167 5,167

Item1 Item2 Item3

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Asitwasinthecaseofequalservicelevelplants,Table5revealsthatriskpoolingimplies important savings in total costs.Although in this case safety stocks are the samecomparingtowhentheplantsactperseparate,thereductionofaverageinventoryonhandtranslatesintocostreductionsof26%,24%and37%foritems1,2and3,respectively. Table6presentsthecostallocationresultsobtainedbythefivedifferentmethods.Incontrasttothecasewhenallplantsusedthesameservicelevel,nowforallitemsmostofthecostallocationderivedfromthemethodsdonotbelongtothecore.Wehavehighlightedallthecaseswhereconstraint(1)isviolated.Table6:Costallocationsbythefivemethodsforthreeitemsinthreeplants,differenttargetservicelevels.

The only methods that produce allocations in the core for all the items are theShapleyvaluemethodandtheEPM. Theegalitarianmethodforall items leaveseitherplantsP2orP3orbothof themwithamoreexpensiveallocationthanifeachofthemwouldactalonewithoutanypoolingstrategyand,therefore,theresultingallocationviolatestheindividuallyrationalcondition(1)thusitdoesnotbelongtothecore. Forthesamereason,thecostallocationproducedbytheαDemandmethoddoesnotbelongtothecore. Thealtruisticmethodconducestoacostallocationthatsatisfiesconstraints(1)forallitems.However,onlyforitem1constraints(2)and(3)alsoholdand,therefore,forthisitemtheresultingallocationbelongstothecore.Foritems2and3,thestabilityconstraint(2) for coalition {P2, P3} is violated.Wepresent the upper bounds for constraints (2) inTable7.Itcanbeverifiedthatforitem2thealtruisticmethodproducesu2+u3=381,whichisgreaterthanthecost373ofcoalition{P2,P3}forthisitem.Itmeansthatunderthiscostallocation, plants P2 and P3would have incentives to apply risk pooling between them,excludingplantP1and,therefore,thecostallocationuisnotinthecore.Thesamesituationisobservedforitem3,wherethealtruisticmethodproducesu2+u3=2,127,whichisgreaterthanthecost1,859ofcoalition{P2,P3}forthisitem.

Table7:Optimalcostsforcoalitionsoftwoplayers,differenttargetservicelevels.Coalition Item1 Item2 Item3

{P1,P2} 1,141 748 4,893

{P1,P3} 1,168 624 5,098

{P2,P3} 701 373 1,859 Inaddition,wehavebuiltother instanceswithdifferent targetservice levelssuchthat the Shapley method, as well as the proportional methods, produce non‐stableallocations.Table8showstheresultsofthecostallocationsderivedfromtheShapleyvaluemethodforaninstancewithtargetservicelevelssetasfollows: ForplantP1,thetargetsareβP1item1=0.99,βP1item2=0.9999,βP1item3=0.9999. ForplantP2,thetargetsareβP2item1=0.89,βP2item2=0.90,βP2item3=0.59. ForplantP3,thetargetsareβP3item1=0.79,βP3item2=0.90,βP3item3=0.59. NotethatthisservicelevelsettingconsidersveryhightargetsforplantP1andlowertargetsforplantsP2andP3.

Plant Egalit. αD Altruistic Shapley EPM Egalit. αD Altruistic Shapley EPM Egalit. αD Altruistic SAM EPM

P1 399 598 828 838 828 270 347 429 478 436 1,722 2,296 3,039 3,813 3,307

P2 399 199 174 165 174 270 347 238 228 233 1,722 574 999 574 986

P3 399 399 194 193 194 270 116 143 104 140 1,722 2,296 1,128 780 873

Total 1,196 1,196 1,196 1,196 1,196 810 810 810 810 810 5,167 5,167 5,167 5,167 5,167

Item1 Item2 Item3

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Table8:Verificationofconstraints(1)‐(3)forcostallocationsgeneratedbyShapleyvaluemethodinadditionalinstanceswithdifferenttargetservicelevels.Item1 Item2 Item3

u1 1,044 ≤ 1,113 854 > 812 8,363 > 7,857

u2 165 ≤ 234 353 > 311 574 ≤ 1,585

u3 399 > 261 104 ≤ 187 2,296 > 1,791

u1+u2 1,209 > 1,141 1,207 ≤ 1,248 8,937 > 7,926

u1+u3 1,443 ≤ 1,580 957 > 874 10,659 ≤ 11,165

u2+u3 564 ≤ 701 457 > 373 2,870 > 1,859u1+u2+u3 1,608 = 1,608 1,310 = 1,310 11,233 = 11,233

InTable8,wehavehighlightedall constrainsof type (1)and (2)violatedby theShapleyvalueallocation.Notethatforitems2and3,wesetequaltargetservicelevelsforplantsP2and P3. Still in these cases, where only plant P1 used different targets, the resultingallocationsarenotstable.Moreover,byrunningthelinearprogrammingassociatedtotheEPMweobtaininfeasibility,thusweareabletoverifythatthecoreinthesethreeinstancesisempty.6.Concludingremarks Motivatedfromacaseintheoilandgassector,wehavepresentedaproblemwhererisk pooling has potential of significant savings. Our work highlights the importance offinding cost allocations which would allow the implementation of risk pooling ininventories of spare parts. Although risk pooling is a traditional approach in inventorymanagement,inthespecificcontextofsparepartstheproblemabouthowtoallocatecostsbetweenthedifferentplayersinthepoolhasreceivedlittleattentionfromtheliterature. Wehaveperformeda seriesof computations,using five cost allocationsmethods.Weshowedexampleswhereasamecostallocationmethodcould lead tosolutions in thecorewhentheplayershavethesametargetservice levels,whileunderaround‐uppolicyand the same setting but at least one playerwith different target service level the samemethod leads to an allocationwhich does not belong to the core or,moreover, the corebecomesempty.Wongetal.(2007)alsopresentedillustrativeexamplesofcostallocationsin spare part inventories and analysed them using game theory principles, but a maindifferenceofoursettingisthatweconsideratargetservicelevelexplicitlywhenfindingtheoptimal base‐stock levels instead of a downtime cost. Aswell as in theirwork though, itremainsaspendingworkfindingstructuralresultsinordertocharacterizetheallocationsand the core of the game. This is not a straightforward task, however, because of theuntractabilityof the expressions involved in thecomputation.Specificallywhentryingtofind the optimal base‐stock level S subject to a service level constraint, an iterativeprocedureastheoneusedinourarticleisthecommonapproach,whichdoesnotfacilitateclosed forms to characterize S in terms of the parameters of the problem. Structuralproperties for cost allocation in spare parts problems have been found by Karsten et al.(2009,2011a,2011b)andKarstenandBasten(2012),usingpenaltycosts forbackordersinsteadofservicelevelconstraints.Inpractice,aservicelevelconstraintsinsteadofpenaltycosts is generally easier to state and interpret by practitioners (Chen and Krass, 2001;BashyamandFu,1998;Cohenetal.1989).

As an extension to our article, we are designing a new allocation method whichdeals with diferent target service levels. It is based on computing a referential costallocation for each player derived from the scenario where its target is the maximumallowed for all players. Then, in the actual scenario,where the targets of all players areallowed, we use a linear programmingmodel whichminimizes the maximum differencebetween the actual allocation and the referential cost. We are also studying the cost

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allocation problem in threshold rationing policies (Dekker et al. 1998, Deshpande et al.2003)asanalternativetotheround‐uppolicy. Infutureresearch,itwouldbeinterestingtotestmorecostallocationmethodsandtoapproachetheproblemwithotherdemanddistributionsorothercontrolpolicies,suchas the(s,S)policy.Finally, incorporatingotherpractical issues, suchaswhere toactuallylocate the parts when implementing a centralized solution for players originallydecentralizedortheimplicationsofcostsinanetworkwithlateraltransshipmentsinsteadofacentralizedsolutioncomplementtheagendaforfuturework.Atthetimeofwriting,wearecollaboratingwiththecompanypresentedinthisarticleinordertoextendouranalysisintheseandotherissuesbyincorporatingalargesetofplantsanditems.

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RISK POOLING AND COST ALLOCATION IN SPARE PARTS … · 2016-01-07 · Although the so stated relevance of spare parts inventories and risk pooling, the literature lacks of evidence