Optimal Service-Based Procurement with Heterogeneous Suppliers

Transcription

1 Optmal Servce-Based Procurement wth Heterogeneous Supplers Ehsan Elah 1 Saf Benjaafar 2 Karen L. Donohue 3 1 College of Management, Unversty of Massachusetts, Boston, MA Industral & Systems Engneerng, Unversty of Mnnesota, Mnneapols, M Carlson School of Management, Unversty of Mnnesota, Mnneapols, M Abstract We nvestgate how a buyer can desgn an optmal mechansm to select supplers, allocate demand and set procurement prces when both the buyer s revenue and the supplers costs depend on the servce levels the supplers provde. The supplers dffer n ther servce costs, producton costs, and capactes. We consder three procurement mechansms that buyers often utlze: take-t-or-leave-t contract, ncentve contract, and competton. We characterze the optmal desgn for each mechansm from the buyer s pont of vew and show that all three can acheve the maxmum feasble proft for the channel and extract all proft for the buyer. However, ths s true only when the buyer has full flexblty n settng all the contract and competton parameters and customzng them for each suppler. Through numercal examples, we show that when the buyer can decde on the value of procurement prces and ncentve payments but cannot customze them for dfferent supplers, then generally ncentve contracts perform better than the other two mechansms. When the procurement prces are set exogenously, the preferred mechansm for the buyer depends on the value of the exogenously set procurement prce. The buyer s better off usng an ncentve contract f the procurement prce s relatvely low, and better off usng a take-t-or-leave-t contract or competton when the prce s relatvely hgh. Keywords: procurement and suppler selecton, servce qualty, suppler competton, ncentve contracts, take-t-or-leave contracts 1

2 1 Introducton Whle some procurement decsons are stll drven prmarly by prce, the servce capablty of supplers s becomng an ncreasngly mportant factor for many manufacturng and servce frms. A recent ndustry survey reveals that the most mportant factor n outsourcng decsons s process effcency and qualty whle cost reducton s ranked thrd (Mazars Annual Outsourcng Survey 2010). Large retalers, such as Wal-Mart, and manufacturers, such as Dell, place a premum on the servce levels ther supplers provde and use sophstcated suppler ratng systems for trackng and rewardng suppler performance. Servce qualty features promnently n the suppler ratng systems used by other frms as well. For example, Raytheon, a major aerospace and defense systems suppler, places the largest weght n ts suppler selecton score (35%) 1 on responsveness and schedule/delvery performance whle assgnng a relatvely small weght (10%) to prce. Saturn Electroncs, a global suppler to orgnal equpment manufacturers (OEMs), also uses a ratng system 2 that weghs on-tme delvery (20%) and qualty complance (30%) more sgnfcantly than cost (15%). Ths ncreased focus on suppler servce level s drven, n part, by the avalablty n many ndustres of multple qualfed supplers. The relatvely weak barganng poston of these supplers, partcularly when the buyng frm s large, allows the buyng frm to set the prce, wth servce level becomng a prmary factor n dfferentatng between supplers. Concern about servce level s also drven by the operatonal polces adopted by many frms whch emphasze on-demand producton and on-tme delvery. Such polces make frms partcularly vulnerable to poor suppler performance because of the lmted safety stocks and safety lead-tmes these frms mantan, wth qualty of servce from the supplers drectly affectng ther revenue. Hgh suppler servce level s, of course, crtcal to frms that have chosen to compete on the bass of customer servce or that can extract a prce premum for hgher servce levels. In some cases, the supplers deal drectly wth the buyng frm s customers, such as n call centers. In those cases, the servce level the supplers provde drectly affects the servce levels end customers receve. Servce level s typcally measured n terms of the avalablty of the demanded good or servce at the tme t s requested. For physcal goods, typcal measures of servce qualty nclude fll rate, expected order delay, the probablty that order delay does not exceed a quoted lead-tme, and the percentage of

3 orders fulflled wthn specfcaton. For servces, servce level measures nclude expected customer watng tme, the probablty that the customer receves servce wthn the specfed tme wndow, and the probablty that a customer does not renege before beng served. Whle the mportance of havng supplers provde hgh servce levels s clear, t s less clear how frms should go about nducng ther supplers to nvest n servce qualty, partcularly when supplers vary n ther capacty levels and cost structures. In settngs where the buyer has also the power to set the procurement prce, t s not clear how these prces should be set to entce hgh servce levels wthout compromsng proft. Smlarly, when the buyer has the flexblty to allocate demand among more than one suppler, t s not clear how such allocaton should be carred out to nduce maxmum servce qualty and how these allocatons are affected by the procurement prces. One approach s to smply select those supplers that promse to offer the hghest servce level and accept the lowest prce. However, when supplers vary n ther capabltes, and these capabltes are common knowledge among all partes, there may not be enough ncentve for the more capable supplers to mantan ther maxmum feasble servce levels. In ths paper, we nvestgate three mechansms that could be used to select supplers, allocate demand, and choose procurement prces when supplers vary n ther capacty and cost effcency. Snce our settng s one of complete nformaton, we frst consder take-t-or-leave-t contracts that specfy the procurement prces, demand allocatons, and requred servce levels. We show how the supplers can be selected optmally and how the correspondng contract can be desgned to acheve the maxmum feasble proft for the buyer, ncludng when the buyer s proft depends drectly on servce levels. We also show that ths soluton s frst-best n the sense that t maxmzes total expected proft for the channel. The second mechansm conssts of offerng ncentve contracts to selected supplers. These contracts specfy the demand shares for each suppler and defne assocated procurement prces and ncentve functons that determne fnancal transfers between the buyer and suppler based on the servce level provded. In contrast to take-t-or-leave-t contracts, the servce levels are not specfed and are left to the supplers to determne n response to the terms of the ncentve contracts. We show how ncentve contracts can be desgned to match the performance of the take-t-or-leave t mechansm. The thrd mechansm s a suppler competton where the buyer allows all supplers to compete for a share of demand and specfes only procurement prces and a demand allocaton functon. The demand allocaton functon specfes shares of demand allocated to the supplers based on the relatve servce 3

4 levels the supplers commt to offer. We show how the buyer can choose the procurement prces and desgn the allocaton functon to agan match the performance of the take-t-or-leave t mechansm. Whle the three mechansms dffer n ther form, our results show that each s capable of maxmzng the buyer s proft, makng them equvalent n performance. However, ths s the case only when the buyer has the flexblty of decdng on the amount of demand allocated and talorng the procurement prce or servce ncentve pad to each suppler. If the buyer does not have flexblty on ether demand allocaton or payments, the mechansms can behave dfferently and result n dfferent servce levels and buyer profts. We provde nsght nto these trade-offs through a seres of numercal examples. An mportant contrbuton of ths paper s n showng how allocaton functons can be desgned to nduce supplers to provde the maxmum feasble servce level. We do so regardless of whether or not there s flexblty n prcng and regardless of the heterogenety n the cost structures of the supplers and ther capactes. Moreover, n settngs where t s desrable to set the demand allocaton n a specfc way, we show that t s possble to desgn the allocaton functon to nduce ths desred allocaton as an outcome of the competton. The suppler competton we descrbe here s smlar to the SA competton dscussed n Benjaafar et al. (2007). However, that paper s not concerned wth determnng optmal allocaton functons. Instead, the focus s on studyng the behavor of supplers who are engaged n a suppler competton orchestrated by a sngle buyer under a specfc servce-proportonal allocaton, exogenously determned. In that paper, the analyss s lmted to dentcal supplers wth dentcal costs and revenue structures and wth no constrants on capacty. Also n ther case, procurement prces are exogenously determned and the buyer measures the performance of the procurement mechansm through (demand-weghted) average servce level. In ths paper we evaluate the performance of dfferent mechansms through a more drect measure: buyer s proft. In addton to Benjaafar et al. (2007), the other paper that s most related to our competton settng s Cachon and Zhang (2007). They consder a specfc context where supplers are modeled as sngle server queues and compete n terms of nvestment n servce rates. Hgher servce rates translate nto hgher servce levels n the form of lower queueng delays for the buyer. Smlar to Benjaafar et al. (2007), they treat the case of homogeneous supplers wth dentcal revenue and cost structures. They compare dfferent demand allocatons and show that a lnear allocaton functon leads supplers to nvest n the maxmum feasble servce rates for the fracton of demand they are allocated (see secton 4 for further 4

5 dscusson). However, because they consder only symmetrc allocatons (supplers that provde the same servce rates are allocated the same amount of demand), the proposed allocaton does not necessarly maxmze overall qualty-of-servce. As wth Benjaafar et al. they do not consder other procurement mechansms, capacty lmts, and procurement prce selecton. A general revew of the lterature on servce-based suppler selecton and procurement s ncluded n Benjaafar et al. (2007). For the sake of brevty, we wll not reproduce t here. For more recent papers see Jn and Ryan (2009), Xaoyuan Lu et al (2009), and Zhou and Ren (2010). However, we should note that much of the exstng lterature has focused on schemes nvolvng competton among dentcal supplers or competton nvolvng specfc allocaton functons, typcally proportonal allocaton functons. Few results exst for settngs wth heterogeneous supplers or supplers wth capacty constrants. We are not aware of any results on the jont optmzaton of demand allocaton and procurement prces. We are also not aware of any results comparng procurement mechansms nvolvng competton wth other noncompettve schemes, ncludng take-or-leave-t or ncentve contracts for settngs where proft s affected by servce levels. There s related lterature n economcs on optmal mechansm desgn (revews can be found for example n Klemperer 1999 and Kalra and Sh 2001). Papers n ths area focus on fndng the optmal method for a prncpal to elct maxmum effort or mnmum prce from a set of agents. Ths lterature spans dverse applcaton areas, ncludng desgnng employee ncentves, ncentves for sales agents, contest rules among ndependent agents to elct maxmum effort, and auctons. In ths lterature, emphass s placed on agents competton whle there s nformaton asymmetry regardng agent characterstcs or effort. Our paper can be vewed as nvolvng mechansm desgn. However, n our case, we assume complete nformaton among all partes. There s also related lterature n economcs on rent-seekng contests. In a rent-seekng contest, there are contestants who compete for a prze. The probablty that a contestant wns the prze (the rent) ncreases wth hs expendtures and decreases n the expendtures of other contestants. A revew of mportant results from ths lterature can be found n Mueller (2003, Ch. 15), Congleton et al. (2008), and Konrad (2009). A focus of ths lterature s on documentng the so-called neffcency of rent-seekng contests. Rent-seekng s vewed as wasteful snce the total expendtures by the contestants can equal the value of the prze tself, a phenomenon called rent dsspaton. However, n systems wth non-dentcal contestants, t has been shown that there may not be complete rent dsspaton; see for example Hlman 5

6 and Rely (1989), Paul and Wlhte (1990), t (1999), and Dxt (1987). That s, heterogenety n the contestants characterstcs can dmnsh the ntensty of the competton. In our research, we show that by carefully desgnng an allocaton functon (the equvalent of a selecton probablty n the rent seekng context), one may nduce the contestants to exert the maxmum feasble effort even when they have nondentcal characterstcs. The rest of the paper s organzed as follows. Secton 2 descrbes our problem settng and characterzes the optmal take-t-or-leave-t contract. Secton 3 focuses on ncentve contracts and descrbes the structure of optmal ncentve contracts. Secton 4 provdes smlar analyss for the servce competton mechansm. Secton 5 compares the three mechansms and hghlghts the mpact of dfferent market constrants on performance. Secton 6 offers concludng remarks. 2. Prelmnares and the Take-t-or-Leave-t Contract Our supply chan conssts of a buyer and potental supplers who dffer n ther producton and servce capabltes. The buyer wshes to allocate her expected demand quantty,, across these supplers n a manner that maxmzes expected proft. 3 Our purpose n ths secton s to descrbe the revenue and cost functons that underle the buyer s problem and uncover the structure of the buyer s optmal take-t-orleave-t contract. Unlke prevous studes, we assume the buyer s revenue depends drectly on the servce level. The optmal take-t-or-leave-t contract serves as a benchmark for other contract mechansms snce, as we wll show, t coordnates the channel whle extractng all profts for the buyer. In other words, t provdes a frst-best soluton to the buyer s problem. However, t does so by havng the buyer strctly dctate the demand allocaton and servce levels for each suppler and offerng a procurement prce that leaves no surplus to the suppler. In determnng the optmal demand allocaton, procurement prces, and servce levels for each suppler, the buyer must evaluate and compare the supplers capabltes 4. Each suppler s unquely 3 Demand can be nterpreted as ether a sngle quantty coverng one sales perod or a demand-rate mantaned over multple perods. 4 The settng we consder s one where there s full nformaton regardng the cost structure of the supplers. Ths would be the case when the prmary cost drvers of the supplers are n the publc doman (e.g., settngs where costs are determned by regonal factors such as labor costs; taxes and regulatons; cost of materals and energy; type of producton technology used; and transportaton costs). The assumpton of full nformaton bulds on assumptons made prevously n the lterature and serves as an upper bound on performance for the buyer and the supply chan. It also provdes nsght n to the desgn of procurement mechansms f the buyer s to leverage knowledge of cost and capacty dfferences among the suppler and to assess the beneft derved from ths knowledge. 6

7 characterzed by ts capacty level,, unt operatng cost, c, and servce related costs, f( s, ), 1,...,. We assume that suppler s servce related cost s a functon of the proporton of demand allocated to the suppler,, as well as the suppler s servce level, s, both dctated by the buyer, where 1 1 and s 0. The actual demand allocated to suppler wll then be. We focus on a partcular class of plausble servce cost functons of the form where f ( s, ) k v ( s ), (1) k s a postve constant and v ( s ) s a contnuous, ncreasng and convex functon n s, wth v (0) 0, for 1,...,. We also assume that v ( s ) s twce dfferentable. Ths functon generalzes servce cost functons used n pror research (e.g., Benjaafar et al. 2007, Cachon and Zhang 2007) by accountng for suppler heterogenety. The frst term, k, s the demand-dependent servce cost, whch vares lnearly wth the demand allocated to the suppler. The second term, v ( s ), captures suppler-specfc costs that ncrease only wth the servce level tself. Ths demand-ndependent cost s not affected by the amount of demand allocated 5. Examples of servce related costs that ft ths model nclude nvestments n capacty, nventory, transportaton, and/or contnuous mprovement efforts. We wll elaborate on one specfc example n secton 5, as part of our numercal study. Takng nto account these unque aspects of her supply base, the buyer would lke to set demand allocatons, acquston prces, and servce levels n a manner that maxmzes her expected proft. The buyer s proft s determned by the revenue receved from her customers mnus the procurement prces pad to her supplers to cover ther producton and servce costs. We assume the buyer s drectly rewarded for hgh servce qualty through the revenue she receves from her own customers. More specfcally, we characterze the buyer s revenue as the sum of ncreasng concave functons of the servce levels provded by each suppler, hs ( ), 1,,, weghted by the proporton of demand recevng that servce. That s, R( δ, s) ( ) 1 hs (2) where δ ( 1,..., ) and s ( s1,..., s ) are the demand allocatons and servce levels dctated by the buyer. Because the revenue functon s concave n all elements of s, t has the appealng property of 5 A more general form of f( s, ) can be defned n whch the demand-dependent cost s also a functon of the servce level. That s, f ( s, ) u ( s ) v ( s ), n whch u ( s ) s a contnuous, non-decreasng, and convex functon of s. Most of our results also hold for ths more general form of cost functon. 7

8 decreasng returns to servce. Ths revenue structure captures stuatons where the buyer s customers observe the servce level provded by the supplers whle the buyer s responsble for payng the penalty for poor servce levels. Applcatons where the supplers servce level s observable by the end customer nclude outsourcng after-sales servces, call centers, roadsde assstance, or when the suppler drectly shps to the end customer. In these stuatons, hs ( ) s smply the buyer s orgnal sellng prce mnus any servce penalty she pays the end customers 6. The buyer s problem under a take-t-or-leave-t contract can now be wrtten as follows, subject to: B 1 max ( s, δ, p) R( s, δ ) p (3) s,δ,p S ( s,, p) 0, 1,...,, (4) S, 1,...,, (5) 1, (6) 1 0 1, s 0, 1,...,. (7) where ( s,, p) ( p c) f ( s, ) denotes the expected proft of suppler 1,...,, and p = (p 1, p 2,,p ) s the vector of offered procurement prces. As mentoned earler, the buyer s proft functon (3) s smply the revenue receved from her customers mnus the procurement charges pad to each suppler. Constrant (4) s the supplers partcpaton condton. It guarantees that the supplers receve at least ther reservaton proft (assumed to be zero wthout loss of generalty). Constrant (5) guarantees that no suppler receves more demand than hs capacty can support. We also assume that to avod the trval case where the suppler s optmal servce level s hs ( )/ s v( )/ s 0 s s s 0 always zero. Moreover, we assume the amount of capacty avalable across supplers s large enough to meet all demand,.e.. 1 To characterze the structure of an optmal take-t-or-leave-t contract, we ntroduce the functon ( ), whch s defned to be the maxmum servce level suppler (for = 1,, ) can provde when the demand share t receves s and the proft he earns s not less than hs reservaton proft. It s easy to verfy that 6 For example, when make-to-order supplers drectly shp to the buyer s customers and servce level s measured by the probablty of meetng a quoted lead tme, we have hs ( ) p0 a(1 s), where p 0 s the orgnal buyer s sellng prce per unt (the prce that the buyer charges when the servce level s as promsed to her customers) and a s the penalty pad for each unt delvered later than the quoted lead tme. 8

9 arg max x hx ( ) ( c f( x, ) / ) f 0 ( ), 1,..., 0 otherwse. Proposton 1 descrbes the structure of the optmal take-t-or-leave-t contract that maxmzes the buyer s Proft. Proposton 1. The buyer problem n (3)-(7) always has a fnte soluton, p1 p p (,..., ), and s ( s,..., s ). The optmal set of demand shares, δ the followng problem 1 1 δ 1 (,..., ), (,..., ), s a soluton to B max ( σδ ( ), δ ) ( ) ( ( ( ), ) / ) 1 h c f, (8) δ subject to constrants (5)-(7), where procurement prces can then be obtaned as s ( ), 1,..., and σδ () ( ),..., ( ). The optmal set of servce levels and 1 1 p c f ( s, ) /, 1,...,. In proposton 1, ( ) represents the servce level that maxmzes the buyer s proft contrbuton from suppler for a gven demand allocaton. Havng ths optmal servce level, the buyer s problem smplfes to (8), whch s an optmzaton problem wth only one set of decson varables,. Although the proposton does not provde a closed form soluton for δ, t guarantees the exstence of a fnte soluton whch can be computed numercally usng a standard multvarable optmzaton algorthm. The buyer s optmal take-t-or-leave-t contract also turns out to be effcent n the sense that t yelds the maxmum possble expected proft for the supply chan as a whole. In other words, t maxmzes the combned proft of the buyer and supplers. Proposton 2. The soluton to the buyer s take-t-or-leave-t problem, as defned n proposton 1, maxmzes channel proft for the supply chan and extracts all profts for the buyer. As a result, the soluton to the buyer s take-t-or-leave-t problem serves as an upper bound on the expected proft level the buyer can acheve under any contract mechansm. To provde more nsght nto the structure of the optmal soluton, we examne a seres of specal cases where one or more aspects of the problem are smplfed. In each case we determne the optmal demand allocaton vectorwhch, along wth the results of proposton 1, defnes the optmal take-t-orleave-t contract. The followng noton of suppler effcency wll help n our analyss. A suppler s effcency level e s defned as the maxmum proft per unt demand that the suppler can generate for the buyer when the suppler s gven the hghest feasble demand allocaton. Ths hghest feasble demand 9

10 allocaton depends on the suppler s capacty and the total demand avalable; t s gven by mn( /,1). The effcency of suppler can then be computed as e h( ( )) c f( ( ), )/, (9) Wthout loss of generalty, for the remander of the paper we rename the supplers n descendng order of ther effcences such that e 1 e 2... e. As we wll see n the followng cases, the buyer usually (but not always) allocates as much demand as possble to the most effcency supplers. Once the optmal allocaton of demand shares s known, proposton 1 can be nvoked to determne the servce levels and procurement prces that maxmze the buyer s proft. Case 1: o Bndng Capacty Constrant When the capacty of the most effcent suppler s at least as large as the buyer s demand, t s optmal for the buyer to allocate all demand to ths suppler. Ths result s stated n proposton 3. Proposton 3. When the most effcent suppler s capacty s larger than the buyer s demand, the optmal set of demand shares s 1 1 and 0, 2,...,. If more than one suppler shares the hghest effcency, that s e 1 e 2... em, where 1M, then the entre demand can be allocated to any of the frst M supplers. Ths result s consstent wth ntuton. Snce the supplers cost structure contans a demand-ndependent term, due to economy of scale, each suppler can provde a hgher servce level when recevng a hgher demand share. Hence, when the most effcent suppler can process the entre buyer s demand, the buyer allocates only to ths suppler to gan the hghest level of servce. Case 2: Bndng Capacty Constrant(s) and Homogenous Costs In ths case, supplers have dentcal cost structures, mplyng that c c and f ( s, ) f ( s, ) k v( s ) for 1,...,. (10) Ths symmetrc cost structure s typcal, for example, of ndustres where the supplers use smlar processes and technologes. Although the supplers have dentcal cost structures, they are stll heterogeneous n terms of ther capactes. The dfference n the supplers effcences s solely due to dfference n the values of, whch n turn depend only on the supplers capactes,. Snce s 10

11 non-ncreasng n t s easy to verfy that the effcency of each suppler s also non-decreasng n hs capacty. In ths case, orderng supplers by decreasng effcency levels s equvalent to orderng them by descendng capactes. The structure of the optmal soluton n ths case depends on whether the buyer s proft functon (8) s convex n. We focus frst on the sub-case where ths assumpton holds. Proposton 4 characterzes the correspondng optmal set of demand shares. Proposton 4. For supplers wth dentcal cost structures, when the buyer s proft functon (8) s convex n, the optmal set of demand shares whch solves problem (3) subject to (4)-(7) s ˆ for 1,..., 1 ˆ 1 ˆ, 1,..., 1 for, (11) 0 for ˆ ˆ 1 where ˆ s the smallest nteger such that. We learned from proposton 3 that f capacty were unlmted, t would be optmal to allocate all demand to the most effcent suppler (suppler 1 n our orderng). However, because supplers have lmted capactes, the optmal soluton n proposton 4 s to allocate demand to ˆ supplers. More specfcally, we should assgn full capacty to the frst ˆ 1 most effcent supplers and then assgn the remanng demand to the ˆ th most effcent suppler. The soluton defned n (11) s farly ntutve. We choose the most effcent suppler and allocate as much as possble to hm. If there s stll unallocated demand, we choose the second most effcent suppler and, agan, allocate as much as possble to hm. We contnue ths process untl we allocate the entre demand. Ths process, however, s optmal only when we have a convex proft functon wth respect to. The convexty condton, whch depends on both the servce cost functon and the buyer s revenue functon, holds for many practcal applcatons, ncludng the example applcaton ntroduced n secton 5. When the buyer s proft s not convex n, ths structure may no longer hold. To provde more nsght nto how the soluton may dffer, consder the followng example. Suppose the supplers servce cost functon and the buyer s revenue functons are vs ( ) a b1 s n for 1,...,, and h( s ) p t(1 s ) m, respectvely. For the case of two supplers, as long as m n we have a convex s buyer s proft functon and so the result of proposton 4 holds. However, f m n, then the buyer s proft functon become non-convex and a partal allocaton s optmal. Fgure 1 llustrates how the buyer s proft 11

12 s mpacted by changes n the demand allocaton between two supplers under a convex proft functon (m = 3, n = 2), and a non-convex proft functon (m = 2, n = 3). In both cases, the buyer s demand s. ow consder, for nstance, the case where the capactes of the supplers are w and w. Fgure 1 shows, under the convex proft functon, the buyer optmzes her proft by allocatng as much demand as possble to the more effcent suppler ( or. Under the non-convex proft functon the buyer s better off allocatng half of ts demand to each suppler ( or, whch means partal allocatons to both supplers. Mathematcally, when the objectve functon of our optmzaton problem s convex wth respect to the decson varables, the maxmum pont happens on the boundares of a convex doman. That s, each suppler ether receves full allocaton or no allocaton, except for the suppler whch receves the leftover to satsfy constrant (6), whch s what we have n proposton 4. When the objectve functon s not convex t s not possble to derve general analytcal conclusons, but our example shows that the maxmum can happen n an nteror pont, whch means all supplers can receve partal allocaton. Buyer's Proft (1000) m=3, n=2, b= m=2, n=3, b= Fgure 1 The mpact of revenue and servce cost functon on the Buyer s proft, vs ( ) a b1 s h( s ) p t(1 s ) m s n, p 1900, t 1000 and s Case 3: Bndng Capacty Constrant(s), Homogeneous Capacty and Heterogeneous Costs In ths case, supplers have dentcal capacty, or the buyer s restrcted not to allocate more than a gven value to each suppler, mplyng w w and w/ for 1,...,. The effcency of suppler then becomes 12

13 e h( ( w )) c f( ( w ), w )/ w (12) for 1,...,. Intutvely, the effcency measure s evaluated at the same maxmum level of capacty for each suppler. In ths case, when we order supplers accordng to ther effcency levels we are smply orderng them by ncreasng order of ther costs. The structure of the optmal soluton s the same as that descrbe n Case 2, expect that effcency s now defned n terms of costs rather than capactes. Specfcally, the structure stated n proposton 4 holds when the buyer s proft s convex n, but the optmal allocaton scheme may contan partal allocatons to supplers otherwse. 3. Incentve Contracts We now consder an alternatve contract mechansm, whch s less restrctve n terms of what s dctated to the supplers. A take-t-or-leave-t contract smultaneously dctates the demand allocaton, servce levels, and prce. An ncentve contract, on the other hand, only dctates the demand allocaton and prce whle leavng the decson about the servce level to the suppler. Ths type of contract may be more attractve to supplers who desre some decson freedom. It s also less burdensome for the buyer snce she only has to determne demand allocatons and prces. Because the buyer cannot dctate servce levels drectly, she must use an ncentve to encourage the supplers to nvest n servce qualty. Let D ( s ) denote the fnancal ncentve that the buyer agrees to pay suppler f the suppler promses to provde servce level s, where D( s ) s ncreasng n s. We wll focus on the smple lnear case, D ( s ) d s, (13) where d 0, s a scalar and d ( d1, d2,..., d ). As we wll see shortly, ths smple ncentve scheme s suffcent to provde an optmal soluton for the buyer and the channel as a whole. The buyer s problem under an ncentve contract can now be wrtten as subject to: max ( d, δ, p ) R ( d, δ ) ( p d s ) (14) d, δ, p BI 1 arg max ( ) ( ), 1,...,, (15) s p c k v x d x x SI ( s ) ( p c k ) v ( s ) d s 0, 1,...,, (16), 1,...,, (17) 13

14 1, (18) 1 0 1, 1,...,. (19) The buyer s problem under an ncentve contract has one addtonal constrant compared wth the buyer s proft under a take-t-or-leave-t contract. The new constrant (15) captures the supplers decsons where each suppler chooses a servce level to maxmze hs own proft. The followng proposton characterzes the buyer s optmal soluton under ths ncentve contract. Proposton 5. When the buyer outsources to suppler, the optmal ncentve contract that maxmzes the buyer s proft s p ( d,, ), n whch ( )/ s s d v s s and where, s arg max h( x) c f ( x, ) / x ds f ( s, ) p c, s the optmal servce level provded by the suppler. SI Moreover, ths contract extracts all profts for the buyer, ( s ) 0. Proposton 5 characterzes the optmal ncentve term and procurement prce for the buyer when dealng wth a sngle suppler who s gven an arbtrary allocaton /. To fully defne the buyer s optmal soluton, we need to also characterze the optmal allocaton of demand across the supply base. The optmal allocaton n ths case s dentcal to the optmal allocaton defned for the take-t-or-leave-t contract. Proposton 6 states ths results. Proposton 6. The soluton to the buyer s problem under an ncentve contract (14)-(19), s acheved when the ncentve terms and prces are set as n proposton 5, and demand shares are set as n proposton 1. Ths soluton provdes the maxmum feasble proft for the buyer whch s the same as the buyer s proft under an optmal take-t-or-leave-t contract. Proposton 6 asserts that an optmal set of ncentve contracts can acheve the maxmum feasble channel proft whch s the same as the proft of an optmal take-t-or-leave-t contract. The supplers actual procurement prces, however, are lower under the optmal ncentve contract because of the addtonal ncentve term pad by the buyer. 4. Servce-Based Competton The fnal mechansm we consder s the least restrctve. The buyer dctates nether the demand allocatons nor the servce levels. Rather, the buyer allows the supplers to compete wth each other for 14

15 demand through ther servce level decsons. We examne a general form for ths servce-based competton where ( s, s ) denotes a functon specfyng the fracton of demand allocated to suppler gven the suppler s servce level s and the servce levels s ( s1,..., s 1, s 1,..., s) offered by her compettors. We assume ( s, s ) s non-decreasng n s, equal to zero when s and 0 ( s, s ) 1, for = 1,,. The competton begns wth the buyer, who announces the competton terms ( α( s), p ), where = ( 1,, ) denotes a vector of allocaton functons and p s the vector of unt procurement prces defned prevously. Each suppler responds to the buyer s announcement by choosng a servce level that maxmzes her expected proft subject to the behavor of other supplers. The buyer s problem under ths competton settng can be stated as subject to: max ( s, α( s), p) R( s, α( s)) ( ( ) ) 1 s p, (20) s F 1( s),..., ( ) ( p1,..., p ) R BC s arg max x ( x, s ) [ p c k] v( x), 1,..., (21) SC ( s, s ) ( s, s ) p c k v ( s ) 0, 1,...,, (22) ( s ), 1,..., (23) where F 1 ( x ),..., ( x ) s the set of all -dmensonal vectors of functons wth : R [0,1] and ( ) 1 1 x. Unlke the prevous buyer problems, the optmzaton s carred out over all vectors of functons n F, n addton to all vectors of postve real numbers n R. The frst set of new constrants (20) reflect the supplers subgame, where each suppler chooses a servce level to maxmze her own proft for any gven set of servce levels chosen by her compettors. Each suppler s decson n ths subgame s affected not only by hs compettors decsons, but also by the form of allocaton functon set by the buyer. Constrant (22) guarantees non-negatve profts for supplers. The other new constrants (23) defne capacty lmts on the allocaton functon values. As we mentoned above, n ths procurement mechansm, the buyer needs to fnd the best form of allocaton functon as well as the optmal set of procurement prces. We wll show below that ths can be done n a two step process. In the frst step, the buyer desgns allocaton functons whch nduce maxmum feasble servce levels through suppler competton for any gven set of procurement prces and target demand shares. In the second step, the buyer seeks the optmal procurement prces as well as the optmal set of target demand shares whch she wshes to nduce n the supplers competton. 15

16 Step 1: Fndng a Servce-Maxmzng Allocaton Functon Fndng an allocaton functon that solves the buyer s problem (20) nvolves searchng for optmal functons (not optmal quanttes) and so s dffcult to solve drectly. Our goal, n ths step, s to characterze a famly of allocaton functons that maxmzes the buyer s proft for a gven set of procurement prces p and targeted demand shares. So, for now we assume that p and are pre-specfed and therefore fxed. Throughout our analyss, we assume that the gven target demand shares and procurement prces are feasble, mplyng that 1, and p c k for 1,...,. 1 We begn by defnng a specfc type of allocaton functon, whch we refer to as servce-maxmzng. When (p, s fxed, the buyer s proft ncreases n each suppler s servce level and so the buyer s problem reduces to fndng an allocaton functon whch nduces the maxmum feasble servce levels. In other words, we are lookng for an allocaton functon that ntensfes the competton to a level where each suppler apples all the ncome ganed from hs assocated revenue (.e., p ) to cover hs producton cost and maxmze servce. Let s max (, p) denote the maxmum servce level assocated wth (, p ) p c f ( s )/ 0 for 1,...,. We are now ready to, whch s the soluton to defne our specfc type of allocaton functon. Defnton 1: An allocaton functon ( s, s ) s servce-maxmzng, wth respect to (, δ p ), f t nduces a ash equlbrum servce level vector s 1,..., s ( s, s ), 1,...,. s for whch s s max (, p) and Ths property s mportant because f an allocaton functon s shown to be servce-maxmzng, ths ensures that t maxmzes the buyer s problem when (, p) s gven. It s a suffcent condton for optmalty n ths specal case. In our search for a servce maxmzng allocaton functon, we focus on proportonal allocaton functons, snce they are commonly used n the lterature. In partcular, we consder the followng general characterzaton of proportonal allocaton functons that allows for heterogenety across supplers: g( s) ( s, s), (24) g ( s ) for 1,...,, where g ( s ) s a non-decreasng functon of s wth g (0) 0. Unlke pror lterature whch focuses almost exclusvely on symmetrc functons 7 (e.g., Benjaafar et al. 2007, Cachon and j1 j j 7 Symmetrc functons mply that f two or more supplers choose the same servce nvestment, they wll receve the same proporton of demand. Ths type of allocaton functon s known to be servce-maxmzng n some specal cases when supplers 16

17 Zhang 2007, Allon and Federgruen 2005, and the references theren), we allow the parameters of the allocaton functon to dffer by suppler. Symmetrc functons are a subset of ths famly, wth the restrcton that g ( s ) g( s ) for 1,...,. We know from pror research that a proportonal allocaton functon does not always guarantee a unque ash equlbrum soluton. For example, Benjaafar et al (2007) show that a symmetrc proportonal allocaton functon n a system wth dentcal supplers does not guarantee unqueness of the ash equlbrum when g( s ) s not concave (see also Cachon and Zhang (2007) for a smlar result). It follows that n our more general case of heterogeneous allocaton functons and heterogeneous supplers, a unque ash equlbrum wll not be possble for all forms of g ( s ), 1,...,. The followng proposton defnes a specfc form for g ( s ) that guarantees both a servcemaxmzng allocaton and a ash equlbrum. The functon s parameterzed for each suppler by ( s ) v ( s )/ ( p c k ), whch s the rato of a suppler s demand ndependent cost to the suppler s revenue. Ths functon can be nterpreted as a measure of the relatve mpact of servce level on the suppler s revenue. Here, we defne ( s ) as a functon of only s snce we are currently holdng p and fxed. Proposton 7. The proportonal allocaton functon defned n (24) s servce-maxmzng for a gven (p, ) when 1/(1 ) ( ) ( s) g s, 0 1, 1,...,. Furthermore, f ths allocaton functon s used n the buyer s problem for a gven (p, ) then the followng propertes hold. (a) A ash equlbrum exsts, wth the supplers servce levels and profts gven by s s max ( ) and ( s ) 0 for 1,...,. (b) Allocaton levels at ths equlbrum are gven by ( s ) for 1,...,. (c) Ths ash equlbrum s unque f supplers are constraned to provde a strctly postve servce level (.e., s 0 for 1,..., ). Unlke prevous research nvolvng symmetrc allocaton functons, t s worth notcng that proposton 7 does not requre convexty of the cost functons. To ensure that s max ( ) s fnte, we only need to restrct our analyss to the non-trval case where sup x 0 v( x) s strctly greater than ( p c k ). Otherwse, the suppler s servce cost would never exceed hs potental revenue and the are dentcal (Benjaafar et al. 2007, Cachon and Zhang 2007). However, a symmetrc functon s not servce-maxmzng when supplers are heterogeneous. 17

18 suppler would ncrease hs servce level unlmtedly. Ths condton could prevent us from achevng a fnte ash equlbrum. The mplcatons of proposton 6 are rather remarkable. By smply manpulatng the parameters of the allocaton functon, the buyer can orchestrate the competton so that each suppler, regardless of hs effcency, has the ncentve to spend all hs revenue to provde the maxmum feasble servce level. In addton, the allocaton functon ( s ) results n the predefned demand share at the ash equlbrum. Step 2: Characterzng the Optmal Competton Mechansm We now turn to the more general problem where p and are no longer fxed, but are decson varables for the buyer. Snce for any gven set of p and, the buyer s proft s ncreasng n all s, 1,...,, we can conclude that t s always optmal for the buyer to use a servce maxmzng allocaton functon. Usng a servce maxmzng allocaton functon, the buyer can then nduce equlbrum servce levels s max (, p ) whch zero-out supplers proft. That s, or equvalently max max s p p c f s p SC (, ) (, ) / 0, 18 p c f s max (, p ) /. (25) Therefore, when the buyer s usng a servce maxmzng allocaton functon to nduce the targeted set of demand shares, we can rewrte the buyer s proft functon as BC max max ( δ, p ) 1 hs (, p) c fs (, p) /. (26) Ths proft functon has a smlar form to the proft functon of the take-t-or-leave-t contract (8), wth s max (, p) substtuted for ( ). ote that we can set the value of s max (, p ) by choosng the proper value of p. Proposton 8 uses ths smlarty to characterze the optmal set of p and. Proposton 8: The buyer s problem under a competton contract (20)-(23) can be solved by usng a servce maxmzng allocaton functon along wth optmal sets of procurement prces, p ( p 1,..., p ), and demand shares, (,..., δ ), where p c f ( s, )/ for 1,...,, and δ and s are 1 max as defned n proposton 1. Moreover, s (, p ) s, for 1,...,. Ths mmedately mples the followng corollary. Corollary 1: The soluton to the buyer s competton problem, as defned n proposton 7, extracts all SC profts for the buyer, ( s ) 0, 1,...,. Furthermore, when the optmal demand shares result n an allocaton to more than one suppler, the soluton maxmzes channel proft.

19 When the capacty of the most effcent suppler s suffcent to fulfll the buyer s entre demand, we cannot drectly dentfy a servce-maxmzng allocaton functon that can nduce 1 1, 0, 2,...,, snce ths s nconsstent wth the requrement of 0 for the servce-maxmzng allocaton functons ntroduced n proposton Systems wth Restrctons on Procurement Prces and Allocaton Functons We have shown that the take-t-or-leave-t, ncentve, and competton contract mechansms can each be desgned to acheve the maxmum feasble proft for the buyer and the channel as a whole. These results requre that the buyer has full power to set the allocaton rule, procurement prces, and ncentve terms and, f necessary, to make these suppler-specfc. In practce, ths s not always the case. Regulated markets mght prevent the buyer from dscrmnatve treatments of dfferent supplers. In ths case, the buyer would have to offer the same terms to all supplers regardless of ther capabltes (e.g., regardless of dfferences n cost effcences or capacty levels). In some ndustres, the buyer mght lack the power to set the procurement prces (e.g., procurement prces are set exogenously by the market or by a handful of powerful supplers). In some settngs, smple allocaton schemes mght also be the accepted norm because of ther smplcty, wdespread use, or perceved farness. In ths secton, we nvestgate how such restrctons would change the buyer s optmal decson under the three contract mechansms. We do so n the context of a specfc applcaton settng whch we descrbe below. Ths s an extenson of an example orgnally ntroduced n Benjaafar et al. (2007), but where we now allow supplers to be heterogeneous n cost and capacty. Although the numercal results we present are specfc to ths example applcaton, the general nsghts apply more broadly to other applcatons that ft our model assumptons. Example descrpton: Consder a system wth one buyer and two supplers. Demand arrves accordng to a Posson process wth mean λ and the processng tme at suppler s exponental wth mean 1/μ. Each suppler s servce level s defned by hs probablty of meetng a gven lead-tme target τ,.e., s Pr( W ) where W s a random varable representng lead-tme for suppler. Supplers have the ablty to ncrease ther servce level by nvestng n capacty at an amortzed capacty cost of k per unt of 19

20 servce rate. In addton, suppler ncurs producton cost c for each tem produced. The above mples that the supplers servce cost functons are gven by where ( ) s Pr( W ) 1 e, f ( s, ) k v ( s ) k k ln[1/(1 s )]/, 1,...,, see Benjaafar et al. (2007) for further detals and motvaton. The buyer s revenue functon s gven by the followng relatvely general quadratc concave functon: where p s and t are postve constants. 2 hs ( ) p t1 s, s Scenaro A: on-dfferentated Procurement Prces and Incentves When the buyer has the power to set the procurement prces and ncentves, but cannot talor them to specfc supplers, the buyer s decson vectors for each mechansm (n terms of procurement prces and ncentve terms) collapse to sngle varables. The mpact of ths restrcton on the performance of each mechansm depends on the level and type of heterogenety across the supply base. Consder the cases outlned n secton 2. If there are no bndng capacty constrants (case 1), only the most effcent supplers are relevant. In ths case both ncentve and take-t-or-leave-t contracts can stll provde an optmal soluton. However, we know from Corollary 1 that the competton mechansm wll acheve a lower proft level n ths case. When the optmal soluton contans bndng capacty constrant(s), the performance of the mechansms depends on the type and level of suppler heterogenety. Fgure 2 llustrates how the mechansms compare when the supplers have dfferent capacty levels but homogeneous costs (case 2). For ths example, we assume 1, p 1900, t 1000, c1 c2 450, k1 k2 500, and vary the rato of s capacty levels whle keepng the total capacty avalable equal to demand, w1 w2 50. In general, performance degrades wth hgher levels of heterogenety for all three mechansms. However, the mpact s less severe under the ncentve contract. 20

21 98.3 Buyer's Proft (1000) Optmal Competton and Take-t-orleave-t: Equal p Incentve: Equal p Incentve: Equal p and d w 1 w 2 Fgure 2 Performance of mechansms: heterogeneous capacty and homogeneous costs (case 2), The optmal soluton n fgure 2 reflects the soluton acheved when the buyer has full flexblty n customzng the contract terms by suppler (dfferentated p and d). Ths serves as an upper bound on performance and a benchmark for measurng the mpact of usng non-dfferentated contract terms. The other lnes on the graph llustrate the performance of our three mechansms under non-dfferentated terms, wth the terms ncludng ether dentcal procurement prces or both dentcal procurement prces and ncentve terms (n the case of the ncentve contract). The ncentve contract clearly outperforms the other two mechansms for all capacty ratos. Ths holds true regardless of whether the ncentve terms are also forced to be dentcal for all supplers. Whle performance for the ncentve contract does degrade wth the capacty rato, the mpact s farly mnor (a 0.02% reducton n performance when capactes vary by a factor of 4 n ths example). In contrast, the performance of the competton and take-t-or-leave-t contract degrade at a faster rate wth the capacty rato (a performance reducton of 1% for a capacty rato of 4 n ths example). The reason an ncentve contract yelds hgher buyer proft s n that ths contract ncludes an extra payment transfer term, the ncentve payment, whch gves the buyer more power to nduce servce levels. Snce the form of heterogenety s n capacty rather than cost, and the fact that the buyer can drectly dctate the demand shares, the optmal parameter values of the ncentve contract do not vary sgnfcantly by suppler. 21

22 The competton and take-t-or-leave t contracts, n contrast, sgnfcantly degrade n performance as the dfference n the supplers capacty levels ncrease. Both mechansms contnue to nduce the supplers to fully nvest n servce level, resultng n zero proft for the supplers. However, there s not suffcent leverage to optmally dfferentate servce levels snce the supplers cannot be rewarded based on ther relatve capacty levels (and thus dfferences n economes of scale). We see a smlar phenomenon when supplers are heterogeneous n ther cost structures rather than ther capacty levels (case 3). Fgure 3 llustrates how the prevous example changes when w1 w2 25, c1 $450, and the rato of servce cost parameters c 2 / c 1 vares. Once agan, the ncentve contract wth equal procurement prces performs reasonably well, whle the gap n performance for the competton and take-t-or-leave-t contract mechansms ncreases as the cost dfferental between supplers grows. Surprsngly, the ncentve contract wth both dentcal procurement prces and ncentve terms now performs much worse than when only the procurement prces are constraned. Its performance s now smlar to the competton and take-t-or-leave t mechansms. Ths degradaton n performance occurs because the ncentve contract now needs the flexblty of settng dfferent ncentve terms among the supplers to counter the effect of the supplers dfferent producton costs. The buyer cannot set the sngle ncentve term hgh enough to extract all of the most effcent suppler s proft, whle also allowng the 98 Buyer's Proft (1000) Optmal Competton and Take-tor-leave-t: Equal p Incentve: Equal p Incentve: Equal p and d c 2 / c 1 Fgure 3 Performance of mechansms: heterogeneous costs and homogeneous capactes (case 3) 22

23 less cost effcent suppler to mantan hs reservaton proft. Ths results n lower proft for the buyer but postve proft for the more effcent suppler. In fact, ths s the only example (of all the examples llustrated n fgures 2 and 3) where the buyer does not extract all the supply chan proft. Scenaro B: Exogenous Procurement Prces ow suppose that rather than constranng the buyer to choose dentcal contract parameters for all supplers, some subset of these parameters s dctated by the market. For example, n some settngs the procurement prces may be set exogenously by market mechansms, whle other terms of the contract are more flexble. Fgure 4 shows how the buyer s expected proft vares across the three contract mechansms for dfferent exogenously determned procurement prces when supplers are dentcal (.e., w 1 w2 25, c1 c2 450, k1 k2 500 ). Each pont n the fgure represents the optmal soluton for the buyer gven the mechansm and dctated procurement prce. We know from the results n the prevous sectons that there exsts an optmal procurement prce for each mechansm where the resultng proft for the buyer and supply chan s maxmzed. These prces are ndcated n the fgure. The optmal procurement prce for the ncentve contract s lower than the others snce the ncentve contract provdes the supplers wth both a revenue payment (through the procurement prce) and an ncentve payment, whle the competton and take-t-or-leave-t mechansms only provde a revenue payment. More nterestng s how the buyer s proft changes as the prces move away from these optmal values. Under the ncentve contract, performance s mpacted more severely f the procurement prces are set too hgh rather than too low. Fgure 4 shows that the buyer s better off under the ncentve contract when procurement prces are low. However, f the prce suffcently ncreases, the competton and take-t-or-leave-t mechansms provde hgher profts for the buyer. For the competton and take-tor-leave-t mechansms, there s also a mnmum procurement prce p c k requred for the supplers to engage n these mechansms. If the procurement prce s set below ths mnmum, the supplers wll earn negatve proft and so wll not accept the buyer s contract. Fgure 5 llustrates how the comparson dffers when supplers are heterogeneous n cost. In ths example, the rato of costs s 500/450 = As noted prevously, the competton and take-t-or-leave-t mechansms are less effectve n ths envronment; an optmal soluton for the buyer s no longer possble for any procurement prce. We also know from prevous dscusson that the effectveness of the ncentve 23