Performance-based contracting is reshaping service support supply chains in capital-intensive industries such

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1 MANAGEMENT SCIENCE Vol. 53, No. 12, December 2007, pp ssn essn nforms do /mnsc INFORMS Performance Contractng n After-Sales Servce Supply Chans Sang-Hyun Km, Morrs A. Cohen, Sergue Netessne The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana {shkm@wharton.upenn.edu, cohen@wharton.upenn.edu, netessne@wharton.upenn.edu} Performance-based contractng s reshapng servce support supply chans n captal-ntensve ndustres such as aerospace and defense. Known as power by the hour n the prvate sector and as performance-based logstcs (PBL) n defense contractng, t ams to replace tradtonally used fxed-prce and cost-plus contracts to mprove product avalablty and reduce the cost of ownershp by tyng a suppler s compensaton to the output value of the product generated by the customer (buyer). To analyze mplcatons of performance-based relatonshps, we ntroduce a multtask prncpal-agent model to support resource allocaton and use t to analyze commonly observed contracts. In our model the customer (prncpal) faces a product avalablty requrement for the uptme of the end product. The customer then offers contracts contngent on avalablty to n supplers (agents) of the key subsystems used n the product, who n turn exert cost reducton efforts and set spare-parts nventory nvestment levels. We show that the frst-best soluton can be acheved f channel members are rsk neutral. When channel members are rsk averse, we fnd that the second-best contract combnes a fxed payment, a cost-sharng ncentve, and a performance ncentve. Furthermore, we study how these contracts evolve over the product deployment lfe cycle as uncertanty n support cost changes. Fnally, we llustrate the applcaton of our model to a problem based on arcraft mantenance data and show how the allocaton of performance requrements and contractual terms change under varous envronmental assumptons. Key words: games; prncpal-agent; replacement-renewal; mltary; logstcs; nventory-producton; mantenance-replacement; government; defense Hstory: Accepted by Ananth Iyer, operatons and supply chan management; receved January 11, Ths paper was wth the authors 4 months for 2 revsons. 1. Introducton Support and mantenance servces contnue to consttute a sgnfcant part of the U.S. economy, often generatng twce as much proft as do sales of orgnal products. For example, a 2003 study by Accenture (see Denns and Kambl 2003) found that $9B n after-sales revenues produced $2B n profts for General Motors, whch s a much hgher rate of proft than ts $150B n car sales generated over the same tme perod. Accordng to the same study, after-sales servces and parts contrbute only 25% of revenues across all manufacturng companes, but are responsble for 40% 50% of profts. Because after-sales support servces are often provded and consumed by two dfferent organzatons (.e., the OEM and the customer), the ssue of contractng between them becomes mportant. Although contracts for mantenance servces of smpler products (electroncs, automobles) nvolve fxed payments for warrantes, there are many nstances of complex systems that requre more sophstcated relatonshps between servce buyers and supplers. For example, n captal-ntensve ndustres such as aerospace and defense, sgnfcant uncertantes n cost and repar processes make t very hard to guarantee a predetermned servce level or quote a prce for provdng t. Therefore, mantenance support n these ndustres typcally nvolves cost-sharng arrangements, whch nclude fxed-prce and cost-plus contracts. Under the former, the buyer of support servces (henceforth called customer ) pays a fxed fee to the suppler to purchase necessary parts and support servces; under the latter, the customer compensates the suppler s full cost and adds a premum. Through our work wth aerospace and defense contractors we have observed a major shft n support and mantenance logstcs for complex systems over the past few years. Performance-based contractng, a novel approach n ths area, s replacng tradtonal servce procurement practces. Ths approach s often referred to as power by the hour or performancebased logstcs (PBL) n, respectvely, the commercal arlne and defense ndustres. The premse behnd performance-based contractng s summarzed n the offcal U.S. Department of Defense (DoD) gudelnes ( 5.3 n Defense Acquston Unversty 2005a): The 1843

2 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1844 Management Scence 53(12), pp , 2007 INFORMS essence of Performance Based Logstcs s buyng performance outcomes, not the ndvdual parts and repar actons Instead of buyng set levels of spares, repars, tools, and data, the new focus s on buyng a predetermned level of avalablty to meet the [customer s] objectves. In 2003, the DoD ssued Drectve (U.S. Department of Defense 2003), whch requres program managers to develop and mplement PBL strateges that optmze total system avalablty. Hence, all future DoD mantenance contracts are mandated to be performance based. A crtcal element of performance-based contractng s the clear separaton between the customer s expectatons of servce (the performance goal) and the suppler s mplementaton (how t s acheved). In the words of Macfarlan and Mansr (2004, p. 40), The contract explctly dentfes what s requred, but the contractor determnes how to fulfll the requrement. As a consequence of ths flexblty, PBL contractng should promote new and mproved ways to manage spare-parts nventory and reduce admnstratve overhead, negotate contracts, and make resource allocaton decsons. For example, under the tradtonal cost-plus contract, the suppler of a servce must truthfully report ts detaled cost structure to the customer to determne whch expenses are elgble for rembursement. Under a PBL arrangement, the suppler does not have to share cost nformaton at ths level of detal. Moreover, the customer no longer drectly manages or possbly even owns resources such as the nventory of spares. Fnally, n the long run supplers may fnd t n ther nterest to nvest n desgnng more relable products and more effcent repar and logstcs capabltes. Not surprsngly, such a radcal change n the approach to contractng has caused confuson among supplers of after-sales support servces. The academc lterature, however, offers lttle gudance wth respect to how such contracts should be executed. In ths paper we am to take a frst step toward fllng ths vod by proposng a model of performance-based contractual relatonshps that arse n practce when procurng repar and mantenance servces. We embed a classcal sngle-locaton spare-parts nventory management problem nto a prncpal-agent model wth one prncpal (representng the end customer) and multple nterdependent agents (representng supplers of the key product subsystems). Each agent performs two tasks that are subject to moral hazard: spare-parts nventory management and cost reducton. We use ths model to analyze three types of contracts (and any combnaton thereof) that are commonly encountered n aerospace and defense procurement and hgh-technology ndustres, namely, fxed-prce, cost-plus, and PBL. In analyzng these contracts we ask the followng questons. (1) What s the optmal combnaton of contractual levers that acheves the best possble outcome for the customer? (2) How should a performance requrement for the fnal product be translated nto the performance requrements for the supplers who provde crtcal components? (3) How should the rsk assocated wth the mantenance of complex equpment be shared among all supply chan members? We show that f supplers decsons are observable and contractble, the contract that acheves the frst-best soluton s a nonperformance arrangement that combnes partal cost rembursement wth a fxed payment. If suppler actons are unobservable and the partes are rsk neutral, we show that the frstbest soluton can stll be acheved by usng a contract that combnes a performance ncentve wth a fxed payment (but no cost sharng). However, when even one of the partes s rsk averse, the frst-best soluton cannot be acheved. In ths case, we show that pure fxed-prce, cost-plus, or performance-based contracts (or any parwse combnaton of them) are not sutable because they do not provde the necessary ncentves. Thus, we show that the second-best contract nvolves all three elements: a combnaton of a fxed payment, a cost-sharng payment, and a performance-based payment. For any such contract proposed by the customer, we fnd analytcally optmal decsons for all supplers. Unfortunately, the customer s problem s nether well-behaved nor admts tractable analytcal solutons (the latter s true even n the centralzed supply chan). Usng a combnaton of analytcal results for specal cases and numercal analyss performed on a data set that s representatve of a supply chan supportng a fleet of mltary arplanes, we obtan nsghts nto the structure of the optmal contract. In partcular, we study the senstvty of the optmal contract to cost uncertanty, and nfer that when the customer s less (more) rsk averse than the supplers, the performance ncentve ncreases (decreases), whereas the cost-sharng ncentve decreases (ncreases) as tme progresses. Fnally, we analyze the mpact of problem parameters on contractual terms, performance, and proftablty. To the best of our knowledge, ths paper represents the frst attempt to embed the after-sales servce supply chan model nto the prncpal-agent framework n whch the supply chan members behave n a self-nterested manner. Our results are consstent wth the observed practce of usng multple contract types whose mx evolves over tme. The rest of the paper s organzed as follows. After a bref revew of related lterature n 2, we present modelng assumptons and notatons n 3. In 4 we analyze the frstbest soluton as well as dervng solutons for the general second-best case. In 5 we analyze specal cases, begnnng wth the rsk-neutral case; then we

3 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1845 study an envronment n whch the supplers actons are partally observable, and, fnally, we study a stuaton wth one suppler. A numercal example that s based on the aforementoned mltary arcraft data set s presented n 6. Fnally, n 7 we dscuss manageral mplcatons of our study. 2. Lterature Revew Two dstnct models are blended together n our paper: a classcal nventory-plannng model for reparable tems, well known n operatons management; and the moral hazard model, whch has been an area of actve research n economcs. The theory of reparable parts nventory management dates back to the 1960s, when Feeney and Sherbrooke (1966) ntroduced a stochastc model of the reparable nventory problem whose steady-state soluton reles on the applcaton of Palm s Theorem. Sherbrooke s METRIC model (Sherbrooke 1968) establshed the basc modelng framework and heurstc optmzaton algorthms for allocatng nventory resources n multechelon, mult-ndentured envronments. Subsequent models have led to notable success n enablng the management of multmllon-dollar servce parts nventory resources n both commercal and government applcatons (e.g., see Cohen et al for a dscusson of a successful applcaton of multechelon optmzaton by IBM s servce support dvson). Research n ths area has largely focused on mprovng computatonal effcency and on ncorporatng more realstc modelng assumptons, such as allowng for capactated supply or nonstatonary demand processes. For a recent comprehensve account of developments n ths feld, see Muckstadt (2005), who revews the underlyng theory, Sherbrooke (1992), whch focuses on aerospace and defense ndustry applcatons, and Cohen et al. (2006), whch ntroduces a modelng framework that has been used to gude the development of state-of-the art software solutons n varous ndustres. In bref, reparable nventory models are concerned wth fndng optmal (cost-mnmzng) nventorystockng targets for each product component subject to a predefned servce constrant. Servce (performance) requrement can be defned n terms of ether tem fll rates or end product avalablty (.e., system uptme ). The latter s the preferred choce n aerospace and defense envronments, and we adopt t n our paper (for a dscusson of comparson of these metrcs, see Sherbrooke 1968). Numerous papers study the prncpal-agent models, and a comprehensve revew can be found n Bolton and Dewatrpont (2005). The buldng block for our paper s the moral hazard model, n whch actons of agents (supplers) are unobservable to the prncpal (customer). Moreover, our model ncludes elements of multtaskng (Holmström and Mlgrom 1991), because the two decson varables for supplers (the cost reducton effort and the nventory poston) nteract wth each other. An addtonal complcaton s the presence of multple agents whose decsons together mpact the performance constrant that the prncpal faces. A number of economcs papers dscuss cost rembursement contracts n the presence of moral hazard. For example, Scherer (1964) consders optmal cost sharng and the mpact of rsk averson n defense procurement. McAfee and McMllan (1986) presents a model n whch frms bd for government contracts under sgnfcant cost-related rsks. Inspred by ths research, we allow for rsk averson, and study cost-plus and fxed-prce contracts n the context of after-sales support and compare them wth performance-based contracts. In the operatons management lterature, a work by So and Tang (2000) s closely related to our paper n that they also consder outcome-based rembursement polces, but ther focus s on the healthcare ndustry. Incentve algnment n supply chans through contracts has been a topc of great nterest n operatons management over the past decade (see Cachon 2003 for a comprehensve survey). Recently, the role of nformaton asymmetry has receved consderable attenton both n the adverse selecton settng (representatve artcles nclude Corbett 2001, Iyer et al. 2005, Lutze and Ozer 2006, and Su and Zenos 2006) and n the moral hazard settng (for example, see Plambeck and Zenos 2000, Chen 2005, Plambeck and Taylor 2006). The current paper contrbutes to ths growng area as well. As s evdent from our survey, the volumnous lterature to date on reparable nventory management has been confned to sngle-frm models and hence does not address ssues that arse n decentralzed supply chans. Furthermore, although the extensve lterature n economcs ams to model contractual relatonshps among dfferent partes, t does not address the complextes of repar and mantenance contractng envronments. To our knowledge, our paper s the frst to put a reparable nventory model nto the decentralzed framework and to study the ssue of contractng n after-sales servce supply chans. 3. Modelng Assumptons The prncpal s the customer for N dentcal assembled products ( systems, whch can be arplanes, computers, manufacturng equpment, etc.). Each system s composed of n dstnct major parts ( subsystems that, n the case of an arplane, can represent avoncs, engnes, landng gear, weapons systems, etc.), each produced and mantaned by a unque suppler. We use subscrpt 0 to denote the customer and subscrpt for subsystem suppler, = 1 2 n.we

4 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1846 Management Scence 53(12), pp , 2007 INFORMS gnore the ndenture structure n the subsystem s bll of materals, treatng each subsystem as a sngle composte tem. In the followng subsectons, we descrbe the repar process and suppler cost structure, explan how rsk averson s represented, defne the performance measure, specfy contract terms, and derve the utltes of the customer and the supplers. Fgure 1 Closed-Loop Cycle for Reparable Items Owned by a suppler (s unts) Inventory 3.1. Repar Process Falure of the subsystem s assumed to occur at a Posson rate, ndependently from falures of other subsystems. Each suppler mantans an nventory of spares and a repar faclty. A one-for-one base stock polcy s employed for spares nventory control. That s, a faled unt s mmedately replaced by a workng unt (f t s avalable) from the suppler s nventory. If a replacement s unavalable, a backorder occurs, and the affected system becomes noperable. As a result, downtme n any subsystem leads to downtme of the system. Upon falure, the defectve unt mmedately enters the repar faclty, modeled as an M/G/ queue. We assume ample capacty (.e., nfnte number of servers), whch s an dealzaton of realty, but t s consdered a reasonable approxmaton n many crcumstances (Sherbrooke 1992). Ths assumpton leads to the desrable property that repar lead tmes of dfferent tems are ndependent. It takes, on average, L tme unts to repar the subsystem, and once the task s completed the subsystem s placed n the suppler s nventory. Forward and return transportaton lead tmes are ncorporated nto the repar lead tme and are assumed to be ndependent of the customer locaton (Wang et al relax ths assumpton). The number of backorders of subsystem, B,sa random varable that s observed at a random pont n tme after steady state s reached. Suppler chooses a target spare stockng level s for subsystem. B and s are related to each other through B = O s +, where O s a statonary random varable representng the repar ppelne (on-order) nventory. Palm s Theorem states that O s Posson dstrbuted for any repar lead tme dstrbuton, wth the mean L (Feeney and Sherbrooke 1966). The repar process forms a closed-loop cycle. Because the subsystems are typcally very expensve and ther lfetmes are very long, we assume that no subsystem s dscarded durng the entre support perod. Fgure 1 llustrates ths process. Thus, there s a total of N + s unts of subsystem n the supply chan, but only s of them are owned by the suppler. The fxed falure rate assumpton s n fact an approxmaton, because the closed-loop cycle wth fnte populaton means s a functon of the number of workng unts. However, ths approxmaton s reasonable n our problem context because the condton E B s L N s satsfed n practce for Subsystems n deployment (N unts) Repar faclty most spare subsystems. Ths condton ensures that, on average, the number of subsystems beng repared at any gven tme s relatvely small, and the correcton due to state dependency can be safely gnored. Although the Posson dstrbuton arsng from Palm s Theorem s appealng, workng wth ntegervalued random varables O and B as well as the dscrete decson varable s sgnfcantly complcates our analyss of game-theoretc stuatons assocated wth varous contractng optons. In partcular, dervng tractable mathematcal expressons to gan nsghts nto frms behavor becomes prohbtvely complex. For ths reason, we depart from the usual dscrete dstrbuton assumpton and model O, B, and s as contnuous varables. Ths approach s reasonable n our context because each unt of a suppler s nventory represents a composte of the varous components assocated wth ther partcular subsystem. Such aggregaton results n suffcently hgh values for so that normal approxmaton for the Posson dstrbuton can be appled (see Zpkn 2000, pp , for examples that show extremely accurate approxmatons of E B s for 10). However, the normalty assumpton s not essental: We derve all our results for an arbtrary dstrbuton. To ths end, we let O be dstrbuted contnuously wth cdf F and pdf f, whch have nonnegatve support 0 and F 0 0. The dstrbuton of B s obtaned from Pr B x s = Pr O x +s. Furthermore, E B s = 1 F x dx, so we obtan s de B s /ds = 1 + F s 0 d 2 E B s /ds 2 = f s 0 Hence, we see that the expected backorder s decreasng and convex n s Suppler Cost Suppler s total cost to mantan ts subsystem, C, has fxed and varable components. The fxed cost (1)

5 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1847 contans an addtve stochastc term havng zero mean and fnte varance. The expected fxed cost can be normalzed to zero wthout affectng any of our results because t does not play a role n determnng optmal suppler decsons and contract terms, as wll become evdent n the next secton. The varable cost s equal to c, the unt cost of a spare subsystem, tmes s, the number of unts n the base stock. represents the uncertanty n total cost that s beyond suppler s control, and are assumed to be uncorrelated across supplers,.e., Cov j = 0 for j. Furthermore, we assume that Cov B = Cov B j = 0 holds for all and j. The uncertanty n the unt cost c s assumed to be neglgble compared to. Ths assumpton s based on our dscussons wth practtoners who ndcated that the uncertanty wth respect to fxed costs s of greater mportance durng the support stage. The unt cost uncertanty may be sgnfcant durng the product development stage, but we do not model t n ths paper. In addton, we assume that c and the dstrbutons of and B are common knowledge. 1 The fxed cost of support can be reduced by the dollar amount a, whch s nterpreted as the suppler s cost reducton effort. We assume that the varable cost s unaffected by a. Hence, C = c s a +. By exertng effort, the suppler ncurs dsutlty a, whch s convex ncreasng ( a >0 a >0), and 0 = 0. As s the case for c, we assume that a s known to the customer. Note that the suppler s not compensated for hs dsutlty of effort a. Wth ths conventon, we effectvely assume that the effort a s the suppler s own dscretonary decson and, hence, the customer does not subsdze the suppler s nternal cost for t. In other words, the customer remburses only the undsputable drect cost of mantenance that would wthstand the scrutny of a possble audt. In the sequel, we assume a quadratc functonal form a = k a 2 /2 wth k > 0. Ths assumpton generates compact expressons wthout fundamentally changng the nsghts of our model (see, for example, Chen 2005). We take the accountng conventon that C s observable by the customer and s the bass of rembursement (see Laffont and Trole 1993, p. 55). The crucal dstncton between the suppler s actons a and s s the way each varable contrbutes to the performance outcomes. The backorder functon s nfluenced by s only, because B = O s +, whereas the total cost C = c s a + s affected by both decson varables. Ths nteracton creates asymmetry n how the supplers actons nfluence outcomes B and C. Rasng a reduces the total cost but has no mpact on avalablty, whereas rasng s mproves avalablty but ncurs a hgher cost. The latter s the classcal cost-avalablty trade-off seen n reparable nventory models. We do not consder an alternatve formulaton, whereby suppler effort mpacts product relablty and/or repar capabltes (thus mpactng and L ) Rsk Averson We assume that all members of the supply chan are rsk averse, wth expected mean-varance utlty E U X = E X r Var X /2 (2) The constant r 0 s the rsk averson factor, representng member s nherent atttude toward uncertanty. Great uncertantes that pervade product development, producton, and mantenance mean sgnfcant rsks for the frms, and ther rsk-averse perspectve s commonly observed (see Scherer 1964 for dscusson and references). The larger the value of r, the more rsk averse a frm s, whereas rsk neutralty s a specal case wth r = 0. Ths form of utlty functon has been wdely used n recent operatons management lterature because of ts tractablty (Chen and Federgruen 2000, Van Meghem 2007) Performance Measure The performance metrc for suppler n our problem s subsystem avalablty A = 1 B /N, the fracton of deployed systems that have a functonal subsystem at a random pont n tme. Note that each backorder of subsystem results n a downed system. Smlarly, we defne the system avalablty A 0 as the fracton of deployed systems that are fully functonal. One can see that 1 n =1 B /N A 0 1 max B /N. However, a common assumpton n the lterature (for example, see Muckstadt 2005) s that the probablty of two or more subsystems beng down wthn the same system at any pont n tme s neglgble. Ths s a reasonable assumpton n our case because the falures of deployed subsystems typcally occur very nfrequently. Thus, the relaton A 0 = 1 B 0 /N, where B 0 = n =1 B s assumed to hold for our problem, and no ambguty exsts n assgnng accountablty for system downtme to a specfc suppler. Because of the one-to-one correspondence between A 0 and B 0 resultng from the above assumpton, the system avalablty requrement E A 0 s 1 s 2 s n A 0 1 That the unt cost s known to the customer s plausble n the defense ndustry, because most of the current PBL contracts apply to exstng subsystems whose unt cost had to be revealed under pre-pbl relatonshps. In tradtonal defense contractng, the DoD negotates the prce of a spare part or a subsystem based on the reported unt cost. 2 Havng fxed and L s a reasonable representaton of realty n the defense ndustry. At present, most PBL contracts are awarded for exstng systems whose subsystem specfcatons (hence relablty) and repar facltes (e.g., specalzed equpment, tools) are already set and cannot be easly altered. Endogenzng relablty mprovement s consdered n Km et al. (2007).

6 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1848 Management Scence 53(12), pp , 2007 INFORMS (e.g., expected system avalablty has to exceed 95% ) s equvalent n our model to a system backorder constrant E B 0 s 1 s 2 s n = n =1 E B s B 0. We call B 0 the system backorder target. Addtonally, we assume that n =0 > B 0 to rule out the trval case n whch s 1 = s 2 = =s n = 0 s optmal. We note that our focus on performance ncentves rases the queston of how our defnton of subsystem avalablty can be used to quantfy suppler performance. One approach would be to compute the backorder metrc as the tme average of the statonary backorder process (ndexed by tme t) and not the statonary varable tself. That s, the random varable B 1/ B 0 t dt could be used, nstead of B, as the performance metrc. s the horzon over whch the number of backorders are counted and averaged. The dstncton between these two measures s nconsequental n a rsk-neutral settng because the utltes of the customer and the supplers are functons of expectaton only, and E B = E B. For the case of rsk averson, however, these two measures dverge because Var B s s ndependent of, whereas Var B s decreases wth. The latter s a consequence of the ergodc property of B t. Therefore, f we were to adopt B as our performance measure, uncertanty wth respect to avalablty may become nsgnfcant wth suffcently large. However, ths s not a good representaton of realty because the customer and the supplers alke express major concerns about performance varablty at any pont n tme rather than tme-averaged performance. For example, The DoD Gude for Achevng Relablty, Avalablty, and Mantanablty (U.S. Department of Defense 2005, p. 1-1) defnes avalablty as a measure of a degree to whch an tem s n operable state and can be commtted at the start of a msson when the msson s called for at an unknown random pont n tme. In lne wth ths defnton, we choose avalablty defned n terms of steady-state backorders as the approprate performance measure. 3 We note, however, that there s an ongong debate on ths very ssue among the practtoners n the aerospace and defense ndustres (see the onlne techncal appendx for further detals) Contract Terms and Utltes The customer s payment (transfer) to the suppler s comprsed of three terms: (1) a fxed payment, (2) rembursement for the suppler s cost, and (3) a 3 For completeness, we have nvestgated the mpact of choosng the alternatve measure B and found that none of the qualtatve results n ths paper change, except for numercal adjustments on the optmal contract parameters; see the onlne techncal appendx (provded n the e-companon). The e-companon to ths paper s avalable as part of the onlne verson that can be found at backorder-contngent ncentve payment. Specfcally, t has the form T C B = w + C v B (3) where w,, and v are the contract parameters determned by the customer. w s the fxed payment, s the customer s share of the suppler s costs, and v s the penalty rate for backorders B ncurred by the suppler. Wth v = 0 and = 0, we obtan a fxedprce (FP) contract; wth = 1 and v = 0 we obtan a cost-plus (C+) contract wth full rembursement. Under the assumptons we have lad out so far, suppler, who s gven a contract T C B, has the followng expected utlty: E U T C B C a a s = w 1 c s a v E B s k a 2 /2 r 1 2 Var /2 r v 2 Var B s /2 (4) The frst three terms together represent the expected net ncome of the suppler, whereas the fourth term s nternal dsutlty for exertng cost reducton effort. The last two terms, respectvely, represent rsk premums assocated wth cost and performance uncertantes. Smlarly, the customer s expected utlty s ( ] n E [U 0 T C B ) a s = =1 n w + c s a v E B s =1 + r 0 2 Var /2 + r 0 v 2 Var B s /2 (5) That s, the customer s utlty s a functon of her total expendture only. Lastly, each suppler s assumed to have fxed reservaton utlty, whch he can gan by not partcpatng n the trade wth the customer. Wthout loss of generalty, we can normalze ts value to zero Sequence of Events Our representaton of the after-sales support relatonshp s based on the standard sngle-locaton, steady-state reparable model wth a take-t-or-leavet contract. We do not consder ssues arsng from repeated nteractons between the customer and the supplers n ths paper. 4 Under the assumptons of the model, the sequence of events s as follows. (1) the customer offers the 4 Due to uncertantes n fleet deployment schedules and future support budgets, the DoD s unwllng to sgn long-term contracts (.e., for the lfe of the program) and nstead typcally contracts on a shorter-term bass wth annual adjustments. Supplers typcally conduct multperod budget plannng usng a short-term, steady-state model on a rollng horzon bass. Therefore, makng a sngle-nteracton assumpton s approprate. Although precontractual barganng or renegotaton may exst n practcal stuatons, we do not formally model them n ths paper.

7 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1849 supplers take-t-or-leave-t contracts; (2) the supplers accept or reject the contracts; (3) the supplers who have accepted the contracts take cost reducton measures and set the base stock levels of ther spares nventory; (4) realzed costs and backorders are evaluated at the end of the contract horzon; and fnally, (5) supplers are compensated accordng to the contract terms. 4. Analyss In the performance-based contractng envronment, nether the detals of suppler cost nor how the suppler meets ther performance objectves s revealed to the customer. Instead, each suppler s compensated based on hs total realzed cost C and hs realzed backorder level B. The fact that both of these contractble varables nclude uncertanty rases the ssue of ncentves. Because C and B are functons of the suppler s cost reducton effort a and base stock level decson s, the suppler can partally control the performance related to hs subsystem and hs compensaton by settng a and s. However, he may choose a par a s that s not optmal from the customer s pont of vew. For example, an opportunstc suppler may choose to mnmze hs own dsutlty of efforts by shrkng (.e., choosng low a and s, hopng that a fortutous state of the world s realzed. The customer s task s then to provde approprate ncentves through contract terms that would nduce the suppler to make the desred decsons. The customer s objectve s to maxmze her expected utlty (or mnmze her negatve utlty) subject to the system avalablty requrement, or equvalently, the backorder requrement. In the followng dscusson, we use the term observable to mean that a varable s both observable and verfable and hence can be specfed n a contract. We frst present the benchmark case wth complete observablty, and then consder the prvate acton case. Ths secton concludes wth a comparson of common contractng optons Frst-Best Soluton: Complete Observablty of Supplers Actons In ths subsecton we analyze the problem under the assumpton that supplers actons a s are both observable, a stuaton often referred to as the frstbest soluton because the customer avods ncentve problems by dctatng a s to the supplers. Ths s the benchmark case aganst whch we can evaluate the effcency of other contracts. The customer s problem s FB mn w v a s s.t [ ( n ] E U 0 T C B ) a s =1 n E B s B 0 =1 (AR) E U T C B C a a s 0 (IR ) 0 1 a s 0 The expected utlty expressons are gven by (4) and (5). (AR) s the system avalablty requrement constrant expressed n terms of backorders, and (IR ) s the ndvdual ratonalty constrant that ensures suppler s partcpaton. As s typcal n moral hazard problems, each (IR ) constrant bnds at the optmal soluton. That s, the customer s able to extract all of the surplus from the supplers by settng approprate fxed payments w. Let be the Lagrangan multpler assocated wth the constrant (AR). The followng proposton specfes the frst-best soluton. Proposton 1. When the supplers decsons are observable and contractble, the optmal contract specfes the followng suppler decsons a s : a = 1/k (6) s = F 1 max 1 c / 0 (7) n E B s = B 0 (8) =1 The soluton a FB, FB and s FB = s FB s unque and s obtaned by offerng a non-performance-based, rsksharng contract such that v FB = 0 and FB = r / r 0 + r (9) provded that r 0 > 0 or r > 0. Suppler s expected utlty s zero, whereas the customer s expected utlty s n =1 c s FB + 1/ 2k 1/2 r 0 r Var / r 0 + r. We note that s FB and FB are determned smultaneously from Equatons (7) and (8). The optmal rsk sharng of cost, represented by (9), s a modfed verson of the Borch rule (see Bolton and Dewatrpont 2005). To gan nsghts, we consder extreme cases. If r 0 > 0 and r = 0,.e., f suppler s rsk neutral but the customer s not, = 0. Ths outcome corresponds to an FP contract; because the customer s rsk averse whle the suppler s not, the customer transfers all rsks to the suppler. At the opposte end, consder r 0 = 0 but r > 0,.e., only the customer s rsk neutral. In ths case = 1, meanng that the C+ contract wth full rembursement s used. Although t may sound counterntutve that the C+ contract acheves the frst-best soluton, we should recall that ncentves are not an ssue n the current settng because the supplers actons are observable and contractble. The role of the C+ contract s merely to mtgate the supplers reluctance to partcpate n the trade (the (IR ) constrant). The rsk-neutral customer can absorb

8 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1850 Management Scence 53(12), pp , 2007 INFORMS all rsks wthout effcency loss. When both r 0 and r are postve, the customer and the suppler share the rsk accordng to (9),.e., based on the suppler s rsk averson relatve to that of the customer. For the remanng case r 0 = r = 0 (whch s not covered by Proposton 1), rsk sharng s not an ssue, and an nfnte number of w v combnatons are optmal. We now focus on the customer s frst-best expected utlty, whch contans three terms for each suppler. The frst term ( c s FB s the cost of ownng s FB unts n the suppler s spares nventory. The second term 1/2k s the net savngs due to the suppler s cost reducton efforts. The last term 1/2 r 0 r / r 0 +r Var can be nterpreted as the jont rsk premum between suppler and the customer, and t s postve only f they are both rsk averse. It represents the neffcency created by a trade-off between the customer s desre to protect herself from rsk and to facltate the supplers partcpaton, whch requres some degree of rsk sharng through cost rembursement. Unlke cost rsk Var ], performance rsk Var B s poses no trade-off between the customer and the supplers; t can be elmnated by settng v = 0. In other words, all partes mutually beneft wthout the performance clause n the frst-best case. If v > 0, a rsk-averse suppler demands a premum due to the possble penalty assocated wth the stochastc realzaton of backorders. Ths leads to ncome fluctuatons for a rsk-averse customer. Both concerns dsappear when v = 0 wthout ncurrng extra cost because the contractblty of the supplers actons s mples that the actons can be perfectly enforced even wthout performance ncentves. Thus, the customer s atttude toward cost uncertanty and performance uncertanty dffer. Ths key observaton wll contnue to hold even when the supplers actons are unobservable and, hence, not contractble Prvate Actons: The Supplers Problem We now turn to the stuaton n whch supplers actons are unobservable to the customer, as s expected n a PBL envronment. Gven the contract parameters w v, suppler chooses a s that maxmze hs expected utlty (4). That s, he solves max w 1 c s a v E B s a s k a 2 /2 r 1 2 Var /2 r v 2 Var B s /2 A dstnctve feature of ths problem s that Var B s s a functon of the decson varable s. Ths s a departure from most moral hazard models, n whch only the mean of the performance measure s affected by the decson varable. In our model the dependence of Var B s on s s unavodable. As wll become evdent, ths dependency complcates the analyss sgnfcantly, but at the same tme creates new dynamcs. The suppler s problem s generally not quasconcave n s, but unmodalty can be guaranteed under mld parametrc assumptons. Proposton 2. Suppose < 1 and v 1 F 0 1 c. Then there s a unque nteror optmal soluton to the suppler s problem n whch suppler chooses a and s such that a = 1 /k (10) v 1 F s +r v 2 F s E B s = 1 c (11) The condtons we specfy n Proposton 2 ensure that the suppler s utlty functon s ncreasng at s = 0. These condtons depend on and v, whch are determned by the customer (optmal solutons for these are presented n the next subsecton). We have verfed through numercal examples that the condtons are mld n the sense that they are volated only under extreme parameter settngs (e.g., when the customer s rsk averson factor r 0 s orders of magntude greater than that of the suppler, r ). We henceforth assume that the condtons of Proposton 2 are always satsfed. From the proposton we obtan the followng results, whch offer nsghts nto the mpact of contract parameters on optmal decsons. Corollary 1. Suppose the condtons n Proposton 2 hold. Then () s / r > 0, a / r = 0, () s / > 0, a / < 0, and () s / v > 0, a / v = 0. From () we see that the more rsk averse the suppler, the greater the optmal nventory poston he chooses. By nvestng n more spares, the suppler reduces the lkelhood of backorder occurrences, thereby reducng the varance assocated wth performance. Hence, ncreasng s s a preventve measure that can be taken by the suppler to avod performance rsk. A smlar mechansm, however, does not exst for avodng cost rsk, as evdenced by the fact that the optmal cost reducton effort a s ndependent of the degree of rsk averson r (see Equaton (10)). 5 Parts () and () of Corollary 1 explan optmal suppler responses to the contract terms and v.if the customer ncreases the rembursement rato, the suppler becomes less concerned wth cost overruns, and hence does not exert as much cost reducton effort as he mght otherwse ( a / < 0). At the same tme, 5 Ths result s due to the assumpton that the stochastc term enters addtvely nto the suppler s total cost C = c s a + ; the effort reduces the mean of C, but not the varance. Under ths standard assumpton the suppler has no control over the varablty of cost, so hs atttude toward rsk does not factor nto the decson about a.

9 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1851 hs perceved effectve unt cost of nventory ( 1 c on the rght-hand sde of (11)) decreases, makng t desrable to stock more. Wth respect to the backorder penalty v, a larger v means a stronger ncentve to decrease backorders, so s ncreases. However, v does not affect a because t serves only as an ncentve to reduce backorders and not the total cost. Ths behavor s, n part, a consequence of our modelng assumptons that the suppler s effort a affects only the cost, and that the uncertantes n cost and n performance are ndependent of each other Prvate Actons: The Customer s Problem Antcpatng that the supplers wll respond by choosng a s accordng to (10) and (11), the customer selects contract terms w v that acheve mnmal total dsutlty subject to the backorder constrant. Wth the rght ncentves, each suppler wll voluntarly choose a s that match the customer s expectaton, even though the supplers decsons are not drectly verfed. Ths voluntary acton s expressed n terms of ncentve compatblty (IC) constrants a s argmaxe U T C B C a a s 0 (IC ) The customer s problem formulaton s smlar to FB, except that the contract space s reduced to w v because a s are now determned by those terms, and except that (IC ) constrants are added. As n the frst-best case, t can be demonstrated that the (IR ) constrants bnd at the optmum, so that we can smplfy the problem by solvng for values of w that leave the supplers wth zero expected utlty. Usng the Lagrange multpler for the backorder constrant (AR), we can wrte n ndvdual Lagrangan functons. Moreover, t s convenent to convert the Lagrangan nto a functon of s rather than a functon of v, usng the monotoncty result s / v > 0 from Corollary 1. Usng (10), we obtan L s = c s + E B s 1 /k / 2k + r 0 2 +r 1 2 Var /2 + r 0 +r v s 2 Var B s /2 (12) whereby v s 1 c 1 F s f r = 0 1 F s = 2r F s E B s (13) ( ) r c 1 F s E B s 1 F s 2 f r > 0 from (11). We readly notce that the optmal performance ncentve v s s a decreasng functon of ; to have the suppler choose s, the customer may decrease v whle ncreasng, or vce versa. Thus, v, the ncentve to ncrease the stockng level, and 1, the ncentve to reduce costs, are complements. Ths observaton plays a key role n a later analyss and wll be dscussed further. We denote the optmal soluton pars wth superscrpts SB, SB s SB. Unfortunately, (12) s not generally quasconvex, and hence s not necessarly unmodal. The analytcal specfcaton of s SB s ntractable even wth fxed, thereby requrng numercal analyss. To gan addtonal nsghts whle crcumventng ths dffculty, n the next secton we focus on several specal cases and later analyze the orgnal problem numercally Cost Plus (C+) vs. Fxed Prce (FP) vs. Performance-Based Contracts Before delvng nto the analyss of optmal contracts for specal cases, we pause here to evaluate the effectveness of the most wdely used contract forms, C+ = 1 v = 0) and FP ( = v = 0), and compare them wth performance contracts (v > 0). Consstent wth other lterature analyzng and comparng these contracts (see, for example, Scherer 1964), our model ndcates that C+ and FP are polar oppostes when t comes to provdng cost reducton ncentves. Wth an FP contract a suppler becomes the resdual clamant, and hence t s n hs nterest to reduce costs as much as possble. In terms of cost rsk, the FP contract gves perfect nsurance to the customer because the suppler bears all rsks from cost under- or overruns. In contrast, the C+ contract shfts all rsks to the customer, because she has to remburse whatever the suppler s realzed cost may be. At the same tme, the C+ contract provdes no ncentve for the suppler to reduce costs. Despte the prevalence of C+ and FP contracts n practce, they do not nduce the desred suppler behavor when a performance constrant s present and the customer cannot observe supplers actons. Ths becomes clear after nspectng the suppler s utlty functon (4). Wth the FP contract, t s n the suppler s nterest to reduce not only effort a, but also spares nventory s as much as possble, thus volatng the mnmum avalablty desred by the customer. A C+ contract, on the other hand, has the effect of makng the suppler ndfferent to the choce of s. Clearly, nducng proper actons requres performance ncentves. The smplest contract n ths category (the pure performance contract ) has = 0 and w = v N wth v > 0. By settng w = v N, the payment to suppler becomes T = v N 1 B /N = v NA, so n ths case v s nterpreted as the prce for

10 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1852 Management Scence 53(12), pp , 2007 INFORMS Table 1 Incentve Effects of Varous Contract Combnatons Contract type No performance-based compensaton (v = 0) Performance-based compensaton (v > 0) Pure performance N/A The customer s unable to extract all ( = 0, w = vn suppler surplus. Fxed prce ( = 0) Whle achevng the frst-best cost reducton effort a FB, Frst best can be acheved wth the approprate the suppler s ncentvzed to reduce s as much as possble. choce of w and v under rsk neutralty. Frst best s not acheved under rsk averson ( >0) n general. Cost plus ( = 1) The suppler exerts no cost reducton effort (a = 0) The suppler exerts no cost reducton effort (a = 0) and s ndfferent toward s. and tres to ncrease s as much as possble. a percent of avalablty that the suppler s able to provde to each deployed system. Indeed, such a contract can nduce the suppler to choose the optmal nventory level s. However, t s neffcent because t leaves a postve resdual surplus to the supplers (.e., the (IR) constrant does not bnd n general). Interestngly (to be demonstrated n the followng secton), a contract wth w > 0 and v > 0 that are determned ndependently can acheve the frst-best soluton f all partes are rsk neutral. However, proper rsk sharng requres > 0, so the optmal contract wll have all three components: a fxed payment, a cost-sharng clause, and a performance ncentve. Table 1 summarzes suppler behavor under all of these contract combnatons. 5. Specal Cases 5.1. Rsk-Neutral Frms Many dffcultes assocated wth the analyss dsappear f all supplers and the customer are rsk neutral, whch may be the case n practce f the customer and the supplers are all very large, well-dversfed corporatons. In ths case, as we show below, even when actons are unobservable, the frst-best soluton s acheved wth a contract that s a smple combnaton of a fxed payment and a performance component (henceforth called FP/performance). Ths soluton hghlghts the performance allocaton aspect of our problem at the expense of gnorng the ssue of rsk sharng. Proposton 3. Wth r 0 = r 1 = =r n = 0, the frstbest soluton s acheved f and only f () 1 = 2 = = n = 0 () w = c s FB + FB E B s FB 1/2k () v 1 = v 2 = =v n = FB where s FB and FB are computed from (7) and (8). The suppler s expected utlty s zero, whereas the customer s expected utlty s n =1 c s FB + 1/2k. The precedng result s not entrely new: It s often the case n other prncpal-agent models that the frstbest soluton s acheved wth an FP/performance contract between two rsk-neutral frms when there s only one effort varable (for example, see Bolton and Dewatrpont 2005). Havng two effort varables a and s as well as multple supplers does not change ths basc result. Frst best s obtaned because and v under rsk neutralty serve only as ncentves and not as nstruments for provdng nsurance aganst rsk, elmnatng the trade-off between the two factors. The presence of the system avalablty constrant (AR), however, offers an nterestng devaton from the standard prncpal-agent analyss. It s captured n part (), whch can be nterpreted to mean that every backorder from heterogeneous subsystems has equal mportance regardless of the subsystem unt prce c. Thus, performance ncentves are equal across supplers. In our addtvely separable backorder model B 0 = n =1 B ths makes ntutve sense, because the customer does not dscrmnate between a backorder of a $1,000 tem and that of a $10 tem; each tem contrbutes equally to the downtme of the system. However, t would be erroneous to conclude that tem unt costs c have no effect on determnng the unform performance ncentve v 1 = =v n = FB because they determne FB ndrectly through the jont satsfacton of (7) and (8). The fact that penalty rates are lnked across supplers contnues to hold n the rsk-averse case, although the equalty as n () can no longer be sustaned because of the supplers varyng atttudes toward rsk. The polcy mplcaton of ths result s to treat all supplers equally wth respect to the performance ncentve as long as rsk averson s not present Rsk-Averse Frms: Cases wth Partal Observablty As the next step n ganng nsghts, we now analyze the problem under a smplfyng assumpton that ether s or a are observable and contractble, but not both. As wll become evdent, these specal cases serve as bounds on the optmal contract parameters under condtons of complete unobservablty, and hence are useful n understandng the structure of the problem. We shall frst consder the case when s are observable but a are not. Ths may happen f the supplers utlze consgnment nventory management for all subsystems (whch s sometmes the case

11 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1853 n practce) so that nventores are vsble to the customer. As s can now be dctated by the customer, the performance ncentve v s unnecessary,.e., the optmal contract has v = 0 for all. The optmal contract (denoted by the superscrpt SO) s as follows. Proposton 4. When s of all supplers are observable to the customer but a are not, t s optmal for the customer to specfy the contract terms accordng to () SO = k r / 1/Var + k r 0 + r < FB, () w SO r 1 SO () v SO = 1 SO 2 Var /2, and c s FB 1 SO 2 / 2k + = 0. s SO = s FB s mposed on suppler whle the contract terms nduce the cost reducton effort a SO 1 + k = r 0 Var k + k 2 r 0 + r Var Even though one of the suppler s actons s observable to the customer, we see that the frst-best soluton cannot be acheved, and, hence, ncentve ssues create neffcences. Namely, cost sharng s less than optmal under the frst-best soluton ( SO < FB ). The customer has to provde more ncentve to reduce costs than would have been the case f she dctated a, and ths s acheved by exposng the suppler to more rsk (smaller ). We see that SO exhbts ntutve propertes: As Var approaches nfnty, SO ncreases asymptotcally to the frst-best optmal rsksharng rato FB because the suppler s effort a becomes overshadowed by huge cost uncertanty. It s also clear that SO moves toward zero (toward an FP contract) as Var decreases. The relatve rsk averson rato r 0 /r s another major determnant of SO, whch s smlar to the frst-best case. If the rato s small, SO s on the C+ sde (closer to one), whereas a large rato mples that SO s on the FP sde (closer to zero). The other possblty s when a of all supplers are observable but s are not. Ths stuaton could arse n government contractng, where a sgnfcant amount of nformaton on suppler costs must be dvulged to the customer. 6 We denote the optmal soluton n ths case wth the superscrpt AO. Its easy to show that a AO = a FB as n (6), but tractable expressons for AO and s AO do not exst. Despte ths shortcomng, AO can be evaluated analytcally n the specal case wth only one suppler, a scenaro we present next Sngle Rsk-Averse Suppler In ths subsecton we assume that there s only one suppler, so we drop the subscrpt. Not only s such a 6 The Truth n Negotatons Act (TINA) has been appled to many government contracts snce the 1960s. It requres supplers to reveal cost data to the government (customer) to avod excessve payments to the supplers. In most PBL contracts, however, TINA s waved. frm-to-frm settng consstent wth a majorty of supply chan contractng models n the lterature, but t s also a commonly observed stuaton n PBL practce. For example, a settng n whch mantenance of a sngle key component s outsourced or a mltary customer contracts drectly wth a subsystem suppler fts ths descrpton (e.g., the U.S. Navy s PBL contract wth Mcheln for tres or commercal arlne power by the hour contracts wth engne manufacturers lke GE and Rolls Royce). As we wll see shortly through numercal experments, nsghts from ths smpler model contnue to hold for the general assembly structure wth multple supplers. Wth a sngle suppler, t may appear that the customer should set ncentves so that E B s SB = B 0 holds. In partcular, ths would be the case f the customer s objectve functon were ncreasng monotoncally n s, whch s an ntutve property. Unfortunately, ths ntuton s not entrely correct. As noted n the prevous secton, the analyss of rsk-averse frms s complcated by the nonquasconcavty, mplyng that the Lagrangan (12) can be bmodal. Thus, the customer may prefer for the suppler to have more nventory than follows from E B s SB = B 0. Ths, however, happens only n extreme cases n whch the customer s several orders of magntude more rsk averse than the suppler and, therefore, wants to protect herself from performance rsk wth a very large nventory. In most of our numercal examples, whch cover a wde range of parameter combnatons, the customer s objectve functon s, ndeed, ncreasng monotoncally n s. Therefore, we wll henceforth assume that the problem parameters are such that the backorder constrant s bndng, so the optmal nventory poston s SB satsfes E B s SB = B 0. Gven that v s completely determned by and s accordng to (13), the only varable to be determned s the cost-sharng parameter, so our problem s smplfed to a onedmensonal optmzaton. Lemma 1. The customer s Lagrangan (12) s convex n when s s fxed. It follows that there s a unque SB that mnmzes the customer s objectve functon. A closed-form soluton exsts, but t s qute complex (see the proof n the onlne techncal appendx), and nspecton alone does not provde ready nsghts. Instead, we focus on understandng how the parameters of the contract change when cost uncertanty Var changes. There are several motvatons behnd ths analyss. Frst, cost uncertanty s of prmary mportance n practce because t s often harder to estmate than performance uncertanty. Second, over the product lfe cycle, sgnfcant changes occur n cost uncertanty (whereas performance uncertanty can be relatvely more stable), so understandng how contractual terms should

12 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1854 Management Scence 53(12), pp , 2007 INFORMS change n response becomes necessary. Fnally, as wll be seen shortly, by varyng cost uncertanty we are able to obtan nsghts that under some condtons dffer fundamentally from nsghts n the classcal lterature on moral hazard problems wth multtaskng. Proposton 5. Suppose r 0 r >0 and that s SB s fxed by the backorder constrant E B s SB = B 0. Then SO < SB < AO and v SB >v AO >v SO = 0. Further, let l r 0 r = L/ = FB where L s the customer s Lagrangan defned n (12). Functon l r 0 r ncreases n the rato r/r 0 and crosses zero exactly once. The optmal contract parameters SB and v SB are related to FB and v FB as follows. () If l r 0 r >0, SB < FB, d SB /d Var > 0, and dv SB /d Var < 0. () If l r 0 r = 0, SB = FB v SB = v FB, and d SB /d Var = dv SB /d Var = 0. () If l r 0 r <0, SB > FB, d SB /d Var < 0, and dv SB /d Var > 0. Frst, we note that the optmal cost-sharng rato SB s bounded above by AO, the optmal rato when the cost reducton effort a s observable. In the current case the effort s not observable, and therefore the customer has to reduce to provde more ncentves to reduce costs. The sde effect s that the suppler s effectve unt cost 1 c ncreases, thus requrng a hgher performance ncentve v to nduce the desred nventory poston s SB. Therefore, v SB >v AO. Second, we note that SB s bounded below by SO, whch we derved by assumng that the nventory poston s s observable. When s s not observable, the customer needs to provde a hgher performance ncentve, v SB >v SO = 0. However, dong so exposes both the customer and the suppler to performance rsk (recall that the performance rsk premum s ncreasng n v for both the customer and the suppler; see (4) and (5)), thus creatng neffcency that can be mtgated by ncreasng. Hgher reduces the effectve unt cost 1 c for the suppler and allows hm to acheve the nventory poston s SB wth a smaller v. Hence, ncreasng above SO s optmal. A comparson of the second-best soluton wth the frst-best soluton s more complex. It s nstrumental to consder two cases based on the relatve rsk averson of the customer and the suppler separately. Because functon l r 0 r ncreases n the rato r/r 0 and crosses zero exactly once, the condton l r 0 r > 0 n () can be nterpreted as r r 0, where the symbol means that the suppler s relatvely more rsk averse than the customer. Smlarly, l r 0 r < 0 can be nterpreted as r r 0, whereby the customer s relatvely more rsk averse than the suppler. We frst consder the former stuaton (whch may arse f the customer s a bgger and more dversfed company than the suppler). We beleve that ths case s more natural n practce. Fgure 2 llustrates the results n (). We make the followng observatons from these fgures. Frst, SB < FB, and the unobservablty of effort and nventory results n less cost rembursement than under the frst-best soluton. Second, SB ncreases wth Var and asymptotcally approaches FB. Wth large cost uncertanty, the rsk-averse suppler s reluctant to partcpate n the trade, so the customer has to provde nsurance by rembursng a large proporton of the suppler s costs. Thus, the suppler has less ncentve to make efforts to reduce costs. On the other hand, when Var s small, provdng cost reducton ncentves becomes more mportant. Thrd, the gap between SB and SO decreases n Var. Ths gap can be nterpreted as the addtonal neffcency attrbuted to performance rsk. When cost uncertanty s large, performance uncertanty Var B s SB s neglgble and the gap between SB and SOdsappears. The gap between SB and AO s nterpreted smlarly. Fnally, v SB decreases wth Var, asymptotcally approachng v FB s FB. Wth hgher cost uncertanty, the performance ncentve s lowered. Fgure 2 Less rsk and cost reducton ncentve to suppler; more rsk to customer More rsk and cost reducton ncentve to suppler; less rsk to customer Optmal Contract Parameters as a Functon of Var When the Suppler Is Relatvely More Rsk Averse than the Customer l r 0 r >0 α 1 (C+) (FP) 0 α AO α FB = r/(r 0 +r) α SB α SO Cost reducton mportant Var[ε] Rsk sharng mportant More performance ncentve 0 v v SB v AO v FB = 0 v SO = 0 Var[ε]

13 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1855 Overall, we observe that SB and v SB move n opposte drectons as Var ncreases because the customer ncreases to mtgate the suppler s cost rsk (we recall that the suppler s more rsk averse than the customer n the current settng). As a result, the suppler s effectve unt cost 1 c s smaller, makng t less expensve to stock nventory and allowng for a smaller ncentve v. Therefore, ncreasng 1 has the same effect on nventory as ncreasng v; these two ncentves are complements wth respect to s. Ths concluson s smlar to the one presented n Holmström and Mlgrom s (1991) orgnal multtask prncpal-agent model n whch ncreasng varablty n one output leads to weaker ncentves for all outputs. However, the mechansm by whch we arrve at our concluson s dfferent. Specfcally, n Holmström and Mlgrom (1991), rasng one effort rases the margnal dsutlty of rasng another effort, whch s not the case n our model (because the suppler s dsutltes 1 cs and ka 2 /2 are ndependent of each other). Another mportant assumpton n ther model s that the outcomes are affected by exactly one effort each, so there s a one-to-one correspondence between an ncentve and an effort. In contrast, our model has an outcome C that s a functon of both varables a and s va C = cs a+. In ths respect, the model closest to ours s found n Bolton and Dewatrpont (2005, pp ) where there s drect conflct between the tasks, because exertng one effort postvely affects one outcome but negatvely affects the other. Next, we consder the case n whch the customer s relatvely more rsk averse than the suppler, r r 0 (case () n Proposton 5). Fgure 3 s an analog of Fgure 2. Compared to the prevous dscusson, SB and v SB exhbt exactly opposte behavor. Now SB > FB and SB decreases n Var whereas v SB ncreases n Var. Ths fundamental dfference arses because, unlke n the prevous case, the customer now needs more protecton from cost rsk. In the presence of large cost uncertanty, ths can be acheved by choosng small, thereby transferrng most of the rsk to the suppler. A nonntutve consequence of ths outcome s that the suppler s ncentvzed more to reduce hs cost and ncrease hs stockng level when cost uncertanty s great. Therefore, the customer s concern for her own rsk protecton reverses contractual terms and comparatve statcs. The complementarty between 1 SB and v SB stll remans, however: As 1 SB ncreases, so does v SB. We note that results when the customer s more rsk averse than the suppler are somewhat contrary to what we have come to expect from the exstng lterature on multtaskng where the customer s often assumed to be rsk neutral. 6. Example wth Multple Rsk-Averse Supplers In ths secton we present a numercal analyss of the problem wth multple supplers. We llustrate our fndngs through an example based on real-lfe mantenance data from a fleet of mltary fghter arcraft. Ths example shows how our model can be appled n practce to support long-term strategc plannng and contract negotatons (note that we also consdered a smpler case where there are two supplers that dffer by at most one of the parameters r Var, thereby solatng the trade-off between ncentves and rsk; see the onlne techncal appendx). A total of N = 156 arcraft are deployed n the fleet. We obtaned data on unt costs, daly falure rates, and repar lead tmes for a representatve collecton of 45 lne-replaceable unts ( parts ) that are unque to the arcraft and covered by a PBL contract. To utlze our model, we aggregate data nto fve subsystem groups: avoncs (a), engnes (e), landng gear (l), mechancal (m), and weapons (w), based on descrptons of each part. We employ the followng technque to obtan unt costs, falure rates, and lead tmes for these subsystems. Frst, we assgn each part Fgure 3 Optmal Contract Parameters as a Functon of Var When the Customer Is Relatvely More Rsk Averse than the Customer l r 0 r <0 α v 1 (C+) α AO v SB α SB v AO α FB = r/(r 0 +r) (FP) 0 α SO Var[ε] 0 Var[ε] v FB = 0 v SO = 0

14 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans 1856 Management Scence 53(12), pp , 2007 INFORMS Table 2 and c for Each Subsystem Subsystem Avoncs (a) Engne (e) Landng gear (l) Mechancal (m) Weapons (w) c (n $1,000) to one of the groups, and compute the subsystem s mean nventory on order as = n j=1 jl j, where = a e l m w and n = the number of parts wthn subsystem. Thus, we treat each subsystem as a kt that s replaced whenever any part wthn t fals. Subsystem unt costs are nferred from an output generated by the propretary commercal software from MCA Solutons, Inc. ( Gven a system avalablty target, ths software calculates optmal stockng levels over multple echelons and ndentures whle drectly consderng each part locaton. By aggregatng ts output, we can nfer the effectve subsystem unt costs by dvdng the dollar amount nvested n nventory resources for each subsystem n j=1 c js j by the total number of stockng unts wthn t n j=1 s j. For ths example the avalablty target of 95% was chosen. Table 2 summarzes the nferred values of and c usng ths heurstc. 7 We note that are an order of magntude smaller than N, thus satsfyng the condton E B s N needed to apply the fxed falure rate approxmaton. To determne values of parameters k and Var, we use the followng approach. Let K be the suppler s fxed cost such that K = E K +. For each suppler, we assume that the expected fxed cost s 50 tmes hgher than the unt cost c. The maxmum dollar amount of cost reducton a FB = 1/k s assumed to be 0 2E K. Thus, k = 1/ 10c. For the sake of smplcty, we also assume that the coeffcent of varaton Var K /E K s the same across supplers. We nfer the rsk averson coeffcent for each suppler from the market captalzaton of a representatve manufacturer of such a subsystem. For example, f Boeng s chosen as the customer and GE as the engne suppler, we calculate the rsk averson rato of r 0 /r e 7 because GE s market captalzaton s roughly seven tmes that of Boeng (see justfcaton for usng company sze as a proxy for rsk averson n Cummns 1977). Ths approach s, of course, qute smplstc, but t fts our am to llustrate the model. Usng ths methodology, we choose r a /r 0 = 1 79, r e /r 0 = 0 15, r l /r 0 = 11 76, r m /r 0 = 1, and r w /r 0 = 3 33, and we select r 0 = The optmal contract terms and the supplers actons are presented n Table 3. 7 Note that our data are restrcted to the subset of unque parts under a PBL contract. Consequently, the nferred values of and c are not representatve of the values assocated wth all of the parts used to support the subsystem. We consder two scenaros: wth small and hgh cost uncertanty (as captured by the coeffcent of varaton ). For smplcty, assume that all supplers have the same value of. Table 3 summarzes optmal contract parameters and the mpled cost terms, ncludng the cost and performance premums. In the case of hgh uncertanty, observe that the cost premum s hgher than the performance premum for all supplers except for the engne suppler (e). Ths asymmetry arses because he s the only suppler who s less rsk averse than the customer. On the other hand, the performance premum becomes more salent when cost uncertanty s small. We also observe that SB ncreases and v SB decreases wth for all supplers except (e), whch s consstent wth our results for a sngle suppler. 7. Concluson The goal of ths paper s to ntroduce contractng consderatons nto the management of after-sales servce supply chans. We do so by blendng the classcal problem of managng the nventory of reparable servce parts wth a multtask prncpal-agent model. We use ths model to analyze ncentves provded by three commonly used contractng arrangements, fxed-prce, cost-plus, and performance-based (FP, C+, and PBL). By dong so, we analyze two practcally mportant ssues of contractng n servce supply chans performance requrement allocaton and rsk sharng when a sngle customer s contractng wth a collecton of frst-ter supplers of the major subsystems used by an end product/system. When performance s defned as overall system avalablty, the answer to the former can be found from the soluton of the classc servce part resource allocaton problem. Our nnovaton s n explctly modelng decentralzed decson makng and consderng how frms behave when they face uncertantes arsng from both support costs and product performance. The noton of rsk sharng found n the prncpal-agent lterature s ncorporated nto our model, provdng nsghts nto what types of contracts should be used under varous operatng envronments. Specfcally, we have dscovered that ncentve terms n the contract exhbt complementarty,.e., ncentves for both cost reducton and hgh avalablty move n the same drecton as cost uncertanty changes. Furthermore, our analyss allows us to make normatve predctons wth respect to how contracts are

15 Km, Cohen, and Netessne: Performance Contractng n After-Sales Servce Supply Chans Management Scence 53(12), pp , 2007 INFORMS 1857 Table 3 Optmal Contract Terms and Supplers Actons = 0 02 = 0 1 a e l m w a e l m w SB v SB a SB s SB A (%) IIR NCR CRP PRP Notes. The dollar fgures are n thousands. IIR stands for nvestment n resources and s equal to c s SB 1/2 k a SB 2, the net cost reducton. CRP s the resdual cost rsk premum, 1/2 r 0 SB the resdual performance rsk premum, 1/2 s r 0 + r v SB. NCR s a SB r 1 SB 2 Var, and PRP s 2 Var B s SB. System avalablty target s 95%. lkely to evolve over the product lfe cycle. In our model, we assumed that suppler effort reduces support costs but does not mprove product performance relablty or repar capabltes. Ths s consstent wth the observaton that performance uncertanty s relatvely stable throughout the repar and mantenance process, whereas cost uncertanty s lkely to be reduced over tme by learnng about costs through the deployment of a larger fleet of systems. Thus, f a seres of performance contracts are sgned over the product lfetme, our analyss ndcates that the cost rembursement rato wll decrease (ncrease) over tme f the suppler s relatvely more (less) rsk averse than the customer. For the performance ncentve v the drecton s reversed. Because larger, more dversfed customers are more common n practce, our results predct that the optmal contract wll typcally assume less cost sharng and more performance ncentve as the product matures. Indeed, ths predcton s confrmed by practtoners and from the DoD publcatons: PBL strateges wll generally have a phased contractng approach, ntated by Cost Plus cost rembursement type contracts to Cost Plus ncentve contracts to Fxed Prce ncentve contracts, over tme (Defense Acquston Unversty 2005b). We fnd that, n the presence of great resdual uncertanty assocated wth performance, cost sharng s stll an effectve tool even f cost uncertanty s small. That s, the combnaton FP/performance-based contract s not optmal n such nstances (notce the gap between zero and SB at Var = 0 n Fgure 2), because the cost rembursement can be used as a rsk protecton mechansm even for the rsk borne by the performance. Although nventory s can be used as an nstrument to hedge aganst performance rsk, adjustng s more effectve for ths purpose because the prmary role of s s controllng for the backorder level to acheve the avalablty target. Hence, some degree of cost sharng s recommended n a performance-contractng envronment even when cost uncertanty s low. Our numercal study shows that the optmal nventory poston profle s SB s qute nsenstve to changes n rsk-related parameters such as r 0, r, and Var. Ths happens because the presence of a strngent backorder constrant lmts the range n whch s SB can be vared. Performance-based contractng n servce supply chans offers fertle ground for research where economcs and classcal nventory theory converge naturally. Not only does t pose theoretcally challengng questons, but nsghts ganed from the analyss are of great nterest to practtoners who are currently undergong major busness process changes due to the move toward PBL contractng. Our paper analyzes several major ssues n performance contractng, but many open questons reman. Follow-up studes may address such topcs as the free-rdng problem arsng from overlappng downtmes across parts; gamng among supplers and the consequences to realzed performance; long-term, strategc product relablty nvestment versus ntermedate-term, tactcal nventory decsons; nvestment n enhanced repar and logstcs capabltes that would reduce lead tmes; alternatve ownershp and management scenaros; and many more. We are currently workng on some of these ssues (for example, see Km et al. 2007). Fnally, emprcal verfcaton of the nsghts ganed from ths paper wll lead to more effectve mplementaton of contract desgn, and ad contract negotatons. 8. Electronc Companon An electronc companon to ths paper s avalable as part of the onlne verson that can be found at mansc.journal.nforms.org/.

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