Improving Supplier Yield under Knowledge Spillover

Transcription

1 Submtted to manuscrpt August 2011 Improvng Suppler Yeld under Knowledge Spllover Ymn Wang W. P. Carey School of Busness, Arzona State Unversty, Tempe, AZ 85287, ymn Yxuan Xao Oln School of Busness, Washngton Unversty n St. Lous, St. Lous, MO 63130, xaoy@wustl.edu Nan Yang Oln School of Busness, Washngton Unversty n St. Lous, St. Lous, MO 63130, yangn@wustl.edu We study the effect of knowledge spllover on OEM manufacturers efforts to mprove supplers process yeld. The OEM manufacturers compete wth mperfectly substtutable products, and they share a common component suppler whose producton process s subject to random output due to varatons n qualty or producton yeld. We characterze the manufacturers equlbrum mprovement efforts and provde manageral nsghts on market characterstcs that nfluence the manufacturers equlbrum mprovement efforts. Wth spllover effect, each manufacturer s feasble strategy set of mprovement effort s dependent on the strategy adopted by ts compettor, and the set of jont feasble strateges s not a lattce. As such, the exstence of equlbrum on mprovement effort s not guaranteed. Nevertheless, we are able to characterze suffcent condtons for the exstence of the manufacturers equlbrum mprovement efforts wth a surrogate game. From a manageral pont of vew, we fnd that, contrary to common wsdom, spllover effect often brngs favorable mprovement to the manufacturers expected profts, regardless of whether the manufacturers are asymmetrc or symmetrc. We also fnd that manufacturers mprovement efforts typcally declne n market competton or market uncertanty, although, paradoxcally, both manufacturers beneft from ncreased spllover effect. Key words : knowledge spllover; random yeld; suppler mprovement 1. Introducton Across ndustres (e.g. automoble, electroncs, and aero spaces), manufacturers ncreasngly depend on ther supplers for compettve advantage. To ensure that supplers are capable, manufacturers often engage n suppler mprovement ntatves, for example, to mprove suppler qualty or yeld. Unted Technologes Corporaton (UTC), for example, treat suppler mprovement as an essental element of ther corporate strategy, and they provde numerous resources for supplers to support contnuous mprovement and lean manufacturng. (Unted Technologes 2011) Suppler 1

2 2 Artcle submtted to ; manuscrpt no. August 2011 mprovement s also prevalent n the automoble ndustry, where major manufacturers often form close relatonshp wth supplers (Lker and Cho 2004). In fact, many manufacturers, and notably Japanese manufacturers, have long engaged n actve suppler mprovement, e.g. nvestng n technology knowhow, to enhance the suppler s operatonal performance, such as delvery, average cost, producton yeld, and qualty. Supplers, however, often work wth multple manufacturers (Markoff 2001), whch can create ntrcate ncentve conflcts n manufacturers suppler mprovement efforts. In partcular, when one manufacturer engages n suppler mprovement actvtes, the knowledge ganed n the suppler s manufacturng process may spllover to other manufacturers. In ths case, a manufacturer s suppler mprovement effort may nadvertently beneft ts drect compettors. Emprcal evdence suggests that manufacturng OEMs are often aware of, and also wary of, the potental knowledge spllover, but dfferent manufacturers seem to take the ssue dfferently. Farney (2000) descrbes the suppler mprovement effort by UTC s Pratt & Whtney Arcraft unt (P&W) wth Dynamc Gunver Technologes (DGT) (now a unt of Brtan s Smths Group). UTC plan to ntate suppler mprovement wth DGT despte the fact that (a) DGT has close relatonshp wth a P&W compettor, Rolls Royce, and (b) Rolls had engaged n projects to help DGT mprove delvery relablty through mproved manufacturng flexblty. (p.4). More troublng, perhaps, s the fact that, hstorcally, by leveragng ts expertse and knowledge ganed from workng wth UTC, Dynamc successfully won bds from Alled Sgnal, Rolls Royce, and General Electrc. (p.5). Although beng aware of the compettve concerns above, UTC had attempted to brng process mprovements to DGT operatons. DGT had receved substantal assstance from P&W n the form of workshops n Kazen process mprovement, lean manufacturng, and UTC s ACE program for qualty mprovement.. (p.5). UTC were, however, wary of the potental spllover: Ther [suppler mprovement] goal dd not nclude the scenaro where the supply chan partner sold parts nto the aftermarket, competed wth UTC for other busness, and partnered wth UTC s compettors.

3 Artcle submtted to ; manuscrpt no. August (p.8). In ths case, UTC seem to reluctantly accept the exstence of knowledge spllover from ther suppler mprovement effort. Toyota, on the other hand, seems to have a more open mnd towards knowledge spllover: Producton processes are smply not vewed as propretary and Toyota accepts that some valuable knowledge wll spll over to beneft compettors. (Dyer and Nobeoka 2000, p.358). In addton, Toyota provde gudance, qute conscously brngng our attenton to other aspects from whch we can gan for ourselves regardless of Toyota. They tell us from tme to tme to drect our kazen effort to these aspects. (p.291) Yet n Honda s case, ts suppler mprovement program mplctly encourages the supplers to work wth compettors: Honda espouses supplers self-relance,.e. balancng responsveness to Honda s needs wth a suffcently dversfed customer base. (Sako 2004, p.296). In fact, encouragng supplers to work wth multple manufacturers s often a polcy of many frms to avod the captve suppler ssue (Lascelles and Dale 1990, p.53). To study the effect of potental knowledge spllover on manufacturers suppler mprovement efforts, we consder the scenaro where two manufacturers share a common component suppler, whch has a producton process that s subject to random output due to varatons n qualty or producton yeld. We adopt the commonly used stochastcally proportonal yeld model, as surveyed n Yano and Lee (1995), to descrbe the suppler s random producton process. The stochastcally proportonal yeld model s not as restrctve as t mght appear n the context of suppler development: lean manufacturng effort, for example, s often amed at reducng producton waste whch s drectly related to yeld. In addton, process qualty and capablty, as reflected by C p and C pk ndces (Montgomery 2004), can also be represented by producton yeld. On demand sde, we consder a class of demand functons under wdely satsfed condtons (ncludng general attracton model, lnear model and log-separable model), that allow us to ncorporate servce levels (n-stock probablty) offered by the two manufacturers. Both manufacturers may exert effort to mprove the relablty (e.g. expected yeld) of the suppler s producton process.

4 4 Artcle submtted to ; manuscrpt no. August 2011 We characterze the manufacturers equlbrum mprovement efforts and provde manageral nsghts on the market characterstcs that nfluence ther equlbrum mprovement efforts. The presence of spllover effect makes the manufacturers feasble strategy sets (n mprovement effort) dependent on each other, and the set of jont feasble strateges s not a lattce. As such, the exstence of equlbrum on mprovement effort s not guaranteed. In fact, we have observed numercal nstances of no equlbrum, unque equlbrum and multple equlbra n the manufacturers mprovement efforts. Nevertheless, under varous condtons, we are able to characterze the manufacturers equlbrum mprovement efforts wth a surrogate game. From a manageral pont of vew, we fnd that, contrary to common wsdom, the spllover effect often brngs favorable mprovement to the manufacturers expected profts, regardless of whether the manufacturers are asymmetrc or symmetrc. When manufacturers are asymmetrc, we fnd that the ex ante more advantageous manufacturer typcally exerts a hgher proporton of mprovement efforts, especally when the spllover effect s large. Whle such hgher proporton of mprovement efforts benefts the focal manufacturer, t may also beneft, n terms of relatve proft, ts compettor more. Thus, the ex ante less advantageous manufacturer may prefer a hgher spllover effect than the ex ante more advantageous one does. When manufacturers are symmetrc, we fnd that both manufacturers mprovement efforts declne n market competton or market uncertanty, although, paradoxcally, both manufacturers beneft from ncreased spllover effect. In the context of our modelng framework, our fndngs seem to support Toyota s (and Honda s) atttude on suppler mprovement effort: one should engage n suppler mprovement wthout worryng too much about the spllover effect. 2. Lterature Ths research s closely related wth several streams of exstng lterature n suppler development, nformaton spllover, random yeld, and market competton. In what follows we examne these four streams of lterature n turn. A large body of extant lterature examnes frms suppler development efforts, wth a majorty of them emprcally nvestgatng the antecedents of successful suppler development effort as well as

5 Artcle submtted to ; manuscrpt no. August key elements for success n suppler development. Some of these studes rely upon prmary survey data to solate mportant common factors that nfluence suppler development, e.g. Krause and Ellram (1997), Krause (1997), Humphreys et al. (2004), and Mod and Mabert (2007). Usng a sample of 215 manufacturng frms n the US, for example, Mod and Mabert (2007) fnd that pror evaluaton and certfcaton are two mportant prerequstes for manufacturers to ntate suppler development program. They also fnd that collaboratve communcaton s one of the key factors for the suppler development program to be successful. In a smlar ven, surveyng 142 electronc manufacturers n Hong Kong, Humphreys et al. (2004) dentfy seven factors that nfluence the performance of suppler mprovement effort, and they fnd that trust, strategc objectves, and effectve communcaton are among the most crtcal antecedents for successful suppler development. Smlarly, usng surveys of 527 frms, Krause (1997) emprcally examnes dfferent approaches a buyng frm may use to engage n suppler mprovement, where the top three methods used nclude performance feedback, nvtng suppler s personnel, and ste vsts (p.14). Krause (1997) concludes that suppler mprovement s generally vewed favorably by the manufacturers but further mprovements are stll possble. A concurrent study n Krause and Ellram (1997) dentfes addtonal elements that contrbute to successful suppler development, ncludng top management nvolvement and buyer s clout. Some other emprcal studes rely on more n-depth case studes to contrast suppler development practces at dfferent frms. A notable example s Sako (2004), who compares suppler development effort usng case studes of three major automoble manufacturers: Toyota, Honda, and Nssan. Sako (2004) fnds that shared practce s often more effectve than standard tranng sessons where tact knowledge cannot be easly transmtted. In addton, suppler development requres a certan level of the manufacturer s nterventon on the suppler s nternal process, whch requres a proper form of corporate governance to make the suppler development program effectve. Smlar studes also nclude Dyer and Nobeoka (2000) as well as references n Krause and Ellram (1997) for earler case studes.

6 6 Artcle submtted to ; manuscrpt no. August 2011 All of the above study (except for Dyer and Nobeoka (2000)) focus on the dyadc buyer-suppler relatonshps and do not explore the knowledge spllover. In contrast, Stump and Hede (1996), Bensaou and Anderson (1999), and Kang et al. (2007) explore general opportunsm n the buyersuppler relatonshp and the related control mechansm, such as screenng and montorng, to mtgate such opportunstc behavors. For example, Kang et al. (2007) note that the exchange relatonshp between an OEM suppler and ts buyer enables the suppler to develop capabltes that over tme enable ths OEM suppler to gan economcally proftable busness from other buyers. (p.14). Note that n these studes the opportunsm ncludes knowledge spllover but also ncludes many other aspects such as drect competton and msapproprate valuable nformaton between the buyer and the suppler. Recently, Bernsten and Kök (2009) study supplers own mprovement efforts n an assemble to order system. They consder spllover effects as an extenson, but they manly focus on two dfferent contractng mechansms to motve the supplers to exert mprovement efforts. There s a large stream of lterature that studes pure nformaton exchange and nformaton spllover n dfferent settngs, e.g. nformaton sharng to reduce cost (Bönte and Wethaus 2005), nformaton msappropraton (Baman and Rajan 2002), demand nformaton sharng (Lee et al. 2000, L 2002, Zhang 2002), and nformaton sharng n an R&D settng (Harhoff 1996). None of these studes consder suppler mprovement, nor do they consder suppler process relablty ssues, such as random yeld or qualty. Our research s related wth the vast lterature n random yeld, e.g. Gerchak and Parlar (1990), Anupnd and Akella (1993), Federgruen and Yang (2009a) and Dada et al. (2007). None of these papers consder suppler mprovement effort. A recent paper by Wang et al. (2010) contrasts suppler mprovement strategy wth dversfcaton strategy under random capacty (yeld). They do not, however, consder knowledge spllover, market competton, or servce level effects consdered n ths paper. We refer nterested readers to Yano and Lee (1995) for an excellent revew of the lterature n random yeld.

7 Artcle submtted to ; manuscrpt no. August Our research s also related wth the stream of lterature that consders market competton on the bass of strategc choces other than prce (alone). Bernsten and Federgruen (2004, 2007) model an ndustry n whch make-to-stock supplers of goods that are (perfect or mperfect) substtutes, compete by selectng both a prce and a servce level, defned as the tem s fll rate,.e. the fracton of the demand whch can be met mmedately from stock. In ther nfnte horzon models, the expected per perod demand volume of each suppler s gven by a general functon of all prces and all servce levels offered n the ndustry. As an alternatve to supplers commttng themselves to a gven fll rate, Bernsten and de Vércourt (2008) assume the frms offer a buyer-specfc drect penalty for each unt and each unt of tme a buyer needs to wat for ts demand to be flled. De Vany and Savng (1983), So (2000), Cachon and Harker (2002) and Chayet and Hopp (2005) model ndustres wth supplers of make-to-order goods or servces, who compete by selectng a prce as well as ether a capacty level or a watng tme guarantee. These authors assume that all buyers or consumers aggregate the offered prce and watng tme nto a sngle full prce measure selectng the suppler whose full prce s lowest. Allon and Federgruen (2008, 2009, 2007) model settngs where the supplers goods or servces are mperfect substtutes and the demand volumes faced by each depend on all prces and watng tme standards n the ndustry, accordng to a general set of demand functons. A few recent papers analyze game theoretc models for ndustres n whch the supplers face uncertan yelds. Corbett and Deo (2009) assume that an arbtrary number of supplers, offerng a homogenous good, engage n Cournot competton, where the (common) per unt prce s gven by a lnear functon of the total actual supply offered to the market. (A frm s actual supply s gven by ts ntended producton volume multpled wth a random yeld factor, generated from a common Normal dstrbuton.) The frms compete by selectng ther ntended producton volumes. As standard Cournot competton models, ths model does not capture demand uncertanty. Corbett and Deo (2009) use ther model to explan the number of flu vaccne supplers n the Unted States. Also motvated by the flu vaccne supply problem, Chck et al. (2008) consder a supply chan wth a sngle buyer and a sngle suppler, whose random yeld factor follows a general dstrbuton. The

8 8 Artcle submtted to ; manuscrpt no. August 2011 buyer derves a beneft from ts order, the magntude of whch grows as a concave functon of the order sze. The authors characterze how the buyer and the suppler sequentally determne ther order sze and ntended producton volume, respectvely. Federgruen and Yang (2009b) analyze a supply chan n whch supplers compete by targetng specfc key characterstcs of ther uncertan yeld processes. Ther analyss of the Stackelberg game starts wth a characterzaton of the optmal procurement polces of the M buyers, n response to chosen yeld characterstcs by the N supplers. To the best of our knowledge, our research s the frst to consder suppler mprovement under random yeld, knowledge spllover, stochastc demand and mperfectly substtutable products. To sum up, exstng lterature has ether adopted the case or survey based approach to dentfy mportant antecedents for suppler development, or adopted the modelng approach to study the random yeld and/or market competton wthout consderaton of suppler mprovement. Our research provdes the frst model to study suppler mprovement effort under both knowledge spllover and market competton. The rest of ths paper s organzed as follows. We descrbe the model n 3 and present some prelmnary analyss n 4. The manufacturers equlbrum mprovement efforts are characterzed n 5, and manageral nsghts are dscussed n 6. We then conclude n 7. All proofs are contaned n the Appendx. 3. Model We consder a newsvendor model wth two manufacturers that source a common component from an unrelable suppler,.e. the suppler s delvery may be less than the quantty ordered. The manufacturers compete aganst each other wth mperfectly substtutable products, and ther market demand s determned by a general demand model descrbed below Demand Each manufacturer s market demand s stochastc, mperfectly substtutable, and nfluenced by both manufacturers product servce levels. The servce level here refers to n-stock probablty (Type-1 servce level), as market demand s stochastc. If a manufacturer has a low servce level, t

9 Artcle submtted to ; manuscrpt no. August may experence a product shortage whch can have a negatve mpact on ts demand. Conversely, a hgh servce level often has a postve mpact on market demand. Let D (f) denote the random demand of manufacturer, gven the vector of the servce levels f provded by both manufacturers. We assume that demand uncertanty s of the multplcatve form, that s, D (f) = d (f)ϵ, (1) where ϵ s a contnuous random varable on the support [0, + ). We assume the dstrbuton and densty of ϵ exst and are gven by G ϵ ( ) and g ϵ ( ), respectvely. The multplcatve form of demand uncertanty n (1) mples that the market uncertanty (measured by coeffcent of varaton) s not affected by the servce levels f, but the expected demand s. Wthout loss of generalty, we can normalze ϵ such that E[ϵ] = 1, and nterpret d (f) as the expected market demand of manufacturer. We requre the followng two assumptons on demand. Assumpton 1. G ϵ (ϵ) s log-concave. Assumpton 2. d (f) f 0, d (f) f j 0, 2 log d (f) f f j 0, and 2 log d (f) f log d (f) f f j 0. Assumpton 1 s satsfed by many common dstrbutons, ncludng unform, normal, exponental, and gamma. In addton, any concave functon s also log-concave. Assumpton 2 s satsfed by many dfferent forms of demand functons, ncludng the lnear demand functon, the log-separable demand functon, and the general class of attracton models (e.g. Anderson et al. (1992), Federgruen and Yang (2009b)) that s wdely used n the operatons and marketng lterature. Next, we prove, under mld condtons, that these three types of demand functons satsfy Assumpton 2. Lemma 1 (Attracton Model). Suppose d (f) s gven by the followng attracton model d (f) = M x (f ) 2 j=0 x j(f j ), (2)

10 10 Artcle submtted to ; manuscrpt no. August 2011 where M s the average total market sze. Here, we assume that x 0 s a constant, whch s the value of the no-purchase opton; and x (f ) ( = 1, 2) s manufacturer s product attracton value whch s ncreasng and log-concave n ts servce level,.e. dx (f )/df 0, d 2 log x (f )/df 2 0. If then d (f) satsfes Assumpton 2. 2 j=1 d (f) f j 0 for all, (3) The techncal condton (3) mples that a unform ncrease n servce levels by both manufacturers wll not decrease any manufacturer s demand. Lemma 2 (Lnear Model). Suppose d (f) s gven by the followng lnear model d (f) = a + b f c j f j, (4) wth a, b, and c j beng nonnegatve constants. If 2 j=1 d (f)/ f j 0 for all,.e., b c j for all, then d (f) satsfes Assumpton 2. Lemma 3 (Log-Separable Model). Suppose d (f) s gven by the followng log-separable model d (f) = ψ (f )h (f j ) (5) wth dψ (f )/df 0 and dh (f j )/df j 0. If ψ (f ) s log-concave n f,.e. d 2 log ψ (f )/df 2 0, then d (f) satsfes Assumpton 2. A specal form of log-separable functon that satsfes Assumpton 2 s the Cobb-Douglas functon: d (f) = γ f β f β j j (6) wth γ > 0, β > 0 and β j > 0 for all and j Supply The suppler s not perfectly relable, and ts producton process can be characterzed by the standard proportonal random yeld process. In ths settng, for any gven order Q placed wth manufacturer, the delvered quantty s q = Y Q, where Y s the random yeld factor.

11 Artcle submtted to ; manuscrpt no. August The dstrbuton of the random yeld factor Y s nfluenced by the suppler s relablty ndex p, an aggregate ndex reflectng factors (e.g., equpment, techncal knowhow, and tranng) that may nfluence suppler yeld. Let Φ Y (, p ) denote the dstrbuton of Y. We adopt the conventon that a hgher p s assocated wth a more relable producton process,.e., Y s frst-order stochastcally ncreasng n p. Thus, for any gven p > p, Φ Y (y, p ) Φ Y (y, p ) for any y. As common n the operatons lterature, we assume that the manufacturer pays only for quantty delvered, and manufacturer offers a unt payment of w to the suppler Improvement Effort and Knowledge Spllover Both manufacturers may exert efforts (e.g., knowledge transfer, techncal assstance, equpment nvestment) to mprove the suppler s relablty. If manufacturer exerts mprovement effort level at z and let z = (z 1, z 2 ), then the suppler s producton relablty ndex for manufacturer ncreases from p 0 to p (z) = p 0 + z + αz j, = 1, 2, j = 3, (7) where 0 α 1 capturers the knowledge spllover effect n the suppler s producton process. The extent of the knowledge spllover effect depends on the confguratons of the suppler s producton process. Sometmes the suppler may use the same producton lne for both manufacturers f the components are dentcal. In ths case, p = p j and α = 1. In contrast, the suppler oftentmes may use somewhat dfferent producton lnes f the components are customzed for each manufacturer. For example, the suppler may use a shared producton lne for the early steps of the producton process, and then perform addtonal customzaton steps to satsfy each manufacturer s specfc requrement. In ths settng, the knowledge ganed from mprovement n the producton process for manufacturer can partally spllover to the producton process for manufacturer j. In ths case, p p j and α s n general less than 1. It s also possble that the suppler mantans completely separate producton lnes (focused factory wthn factory). In ths case, there s no knowledge spllover effect, and hence α equals zero.

12 12 Artcle submtted to ; manuscrpt no. August 2011 It s natural to assume that manufacturer s mprovement cost m (z ) s ncreasng n ts effort level z,.e. dm (z )/dz 0. Note that we do not consder other benefts assocated wth the manufacturer s mprovement effort, such as mproved effcency or mproved contract terms Problem Formulaton Sequence of events. Frst, each manufacturer decdes ts level of suppler mprovement effort. Second, after exertng suppler mprovement effort but before yeld and demand uncertanty s resolved, each manufacturer decdes ts servce level f and ts order quantty Q from the suppler to meet the servce level. Last, after observng realzed producton yeld and demand uncertanty, each manufacturer satsfes ts demand as much as possble. Gven the above sequence of events, the manufacturers decson problem can be naturally formulated as a two-stage stochastc program. Let r, s, and π denote manufacturer s unt prce (charged n the market), unt salvage value (for leftovers), and unt penalty cost (for shortages), respectvely. Defne H (u, v) = r mn{u, v} + s (u v) + π (v u) +. The second stage problem. Because the suppler s producton process s random, manufacturer wll choose an order quantty Q such that the manufacturer wll acheve a servce level f. Let Y = (Y 1, Y 2 ) and p = (p 1, p 2 ), manufacturer s second stage problem s to fnd the optmal {f, Q } that maxmzes v (f, Q p, f j ) = w E Y [Y Q ] + E Y,ϵ [H (Y Q, D (f))], (8) s.t. Prob (D (f) Y Q ) f. (9) The servce level f defned n (9) s the n-stock probablty, commonly known as Type-1 servce level n the lterature. Wth random yeld, the servce level vares wth dfferent yeld realzatons, and therefore the n-stock probablty needs to be condtoned and uncondtoned over all possble yeld realzatons. The servce level defned n (9) mples that manufacturers can credbly commt on ther servce levels, whch s true n most busness to busness markets. Such servce level constrant s also applcable n consumer markets wth short lfe cycle products, where commtment to servce levels can attract potental customers (Seth et al. 2007, p ).

13 Artcle submtted to ; manuscrpt no. August Because manufacturer decdes ts servce level f and orderng quantty Q jontly, one can vew f as a surrogate for the classc quantty competton. Introducng the servce level f allows us to generalze the classc quantty competton consderably,.e. ncorporate stochastc demand, random supply, and mperfectly substtutable products. The frst stage problem. Let the suppler s ntal producton relablty ndex be p 0 = (p 0 1, p 0 2), the frst stage problem for manufacturer can be formulated as J (z ) = m (z ) + v (f, Q p(z), f j ), (10) where p(z) = (p 1 (z), p 2 (z)) and p (z) s gven by (7). We note that t s farly straghtforward to ncorporate probablstc suppler mprovement results, e.g. an mprovement effort may not always be fully successful. In ths case (10) can be modfed by mposng an expectaton operator on v (f, Q p(z), fj ) over all possble realzatons of p(z). 4. Prelmnares In ths secton, we characterze the manufacturer s second stage problem, where manufacturer chooses servce level f and order quantty Q to maxmze ts expected proft gven by (8) Stockng Factor For analytcal convenence and expostonal ease, defne q (f) = Q /d (f). Hereafter we refer to q (f) the stockng factor, whch captures the extent by whch the manufacturer nflates ts order quantty relatve to the expected demand for any gven vector of servce level f. Usng the stockng factor q (f), we can rewrte (8) and (9) as v (f, q (f) p, f j ) = w d (f) E Y [Y q (f)] + E Y,ϵ [H (Y q (f)d (f), d (f)ϵ)], (11) s.t. E Y [G ϵ (Y q (f))] f, (12) where the servce level constrant (12) follows from the fact that Prob (D (f) Y Q ) f Prob (ϵ Y q (f)) f E Y [G ϵ (Y q (f))] f.

14 14 Artcle submtted to ; manuscrpt no. August 2011 Note that constrant (12) s n general not a convex set, and therefore one cannot use the standard Lagrangan approach n constraned optmzaton to solve (11). However, for any gven f, v ( ) s concave n q (f); and for any gven q (f), v ( ) s log-concave n f. We therefore adopt a two-stage approach by frst characterzng the optmal q as a functon of f, and then characterzng the optmal servce level f. Smplfyng (11), we have { ( ) v (f, q (f) p, f j ) = d (f) q (f) (r w + π ) E Y [Y ] (r s + π ) E Y [Y G ϵ (Y q (f))] + (r s + π ) E Y [ Y q (f) 0 ] } ϵg ϵ (ϵ)dϵ π. (13) Let u (f, q (f) p, f j ) denote the terms n the curly bracket n (13),.e. manufacturer s expected proft per unt of demand. The followng proposton characterzes the optmal stockng factor. Proposton 1. For any gven vector of servce level f, (a) u (f, q (f) p, f j ) s concave n the stockng factor q (f). (b) The optmal stockng factor q (f) satsfes one of the followng condtons: E Y [Y G ϵ (Y q (f))] = r w + π r s + π E Y [Y ]; (14) E Y [G ϵ (Y q (f))] = f. (15) Notce that (14) gves an nteror soluton to u (f, q (f) p, f j )/ q (f) = 0, and (15) gves a corner soluton constraned by (12). Leveragng Proposton 1, the followng theorem completely characterzes the propertes of optmal stockng factor for any gven servce level f. Theorem 1. (a) Manufacturer s optmal stockng factor q (f) s ndependent of ts compettor s servce level f j and non-decreasng n ts own servce level f,.e., q s a functon of f only, and dq (f )/df 0. (b) There exsts a unque threshold servce level f such that f f f, q (f ) unquely solves (14), and f f > f, q (f ) unquely solves (15). (c) (r w + π )/(r s + π ) E Y [Y ] f (r w + π )/(r s + π ). (d) f can be unquely obtaned by solvng the system of equatons specfed by (14) and (15) smultaneously.

15 Artcle submtted to ; manuscrpt no. August (e) 0 = dq (f )/df f f < dq (f )/df f >f. Theorem 1 shows that f the servce level s less than a crtcal servce threshold f, the optmal stockng factor, q (f ), s a constant that s ndependent of the servce level chosen. 1 In contrast, f the servce level f > f, the optmal stockng factor s no longer constant n servce level. Instead, q (f ) ncreases n f,.e., a larger nflaton of the expected demand s requred to satsfy a hgher servce level. The servce level threshold f plays an mportant role n characterzng the optmal servce level, and, as we wll see shortly, t also has an mportant practcal nterpretaton Equlbrum Servce Level We now turn our attenton to manufacturer s optmal servce level decson, gven ts compettor s servce level. It s mmedate from Theorem 1(a) that u s ndependent of f j, so we can rewrte (8) as a unvarate functon,.e., v (f p, f j ) = d (f p, f j )u (f p), where (16) u (f p) = q (f ) ( (r w + π ) E Y [Y ] (r s + π ) E Y [Y G ϵ (Y q (f ))] ) [ Y q (f ) + (r s + π ) E Y The followng theorem establshes a lower bound on the optmal servce level. 0 ] ϵg ϵ (ϵ)dϵ π. (17) Theorem 2. The optmal f f. Theorem 2 shows that the optmal servce level f can never be lower than the threshold servce level f. One can therefore nterpret f as the mnmum servce level requred for manufacturer to compete n the market place. The manufacturer ether enters the market and acheves a servce level at least as hgh as f, or exts the market altogether. The followng theorem ensues. Theorem 3. A necessary and suffcent condton for manufacturer to enter the market s [ Y q (f f =f ) (r s + π ) E Y 0 ] ϵg ϵ (ϵ)dϵ > π. (18) Hereafter we focus on the case where the market entry condton (18) holds for both manufacturers, and do not study the degenerate case where only one manufacturer enters n the market.

16 16 Artcle submtted to ; manuscrpt no. August 2011 Now we are ready to study the second stage problem on manufacturers equlbrum servce levels. Gven the relablty ndex p for both manufacturers, manufacturer s second stage expected proft v depends on both manufacturers servce levels. For the ease of exposton, we rewrte v as a functon of f,.e., v (f p) = d (f p)u (f p). For general yeld dstrbutons, the expected revenue (per unt demand), u ( ), s not necessarly concave n f, hence v (f p) s not necessarly log-concave n f. Nevertheless, by Assumpton 2, v (f p) s log-supermodular n f, and thus the followng theorem ensues. Theorem 4. The servce level equlbrum f exsts. Whle Theorem 4 shows the exstence of servce level equlbrum, t s challengng to prove the unqueness of the servce level equlbrum f under general yeld dstrbutons. Wth the followng assumpton, however, we are able to prove the unqueness of servce level equlbrum. We thus make the followng assumpton for the rest of the analyss. Assumpton 3. Φ Y (, p ) follows a Bernoull dstrbuton wth parameter p. Theorem 5. There exsts a unque Nash equlbrum on the servce levels f under Bernoull yeld dstrbuton. Furthermore, the equlbrum f satsfes q (f ) d (f p)/ f + d (f p)q (f ) d (f p)u (f p) + d (f p)/ f d (f p)u (f p) {(r s + π ) E Y [ Y q (f ) { } (r w + π ) E Y [Y ] (r s + π ) E Y [Y G ϵ (Y q (f ))] 0 ] } ϵg ϵ (ϵ)dϵ π = 0. (19) Theorem 5 demonstrates that although each manufacturer s objectve functon s constraned by a non-convex set (n servce level and stockng factor), there exsts nevertheless a unque Nash equlbrum such that both frms acheve the unque, stable equlbrum servce levels. It s worth notng that all analyss so far drectly extends to the N-manufacturer case. In closng, we note that Propostons A1 and A2 n Appendx A provde addtonal senstvty analyss on the mnmum servce level f and the equlbrum servce level f.

17 Artcle submtted to ; manuscrpt no. August Suppler Improvement In ths secton, we characterze the manufacturers frst stage problem, where each manufacturer chooses ts mprovement effort to maxmze ts objectve functon gven by (10) Do Manufacturers Beneft From Hgher Suppler Relablty? In order to characterze the mprovement effort z, t s helpful to frst understand the effect of target relablty ndex p (after mprovement effort) on each manufacturer s expected proft. Whle one mght expect that a hgher p s always benefcal to manufacturer, t s unclear whether such conjecture holds n the presence of spllover effects. In what follows, we frst study cross effect and drect effect of target relablty ndex p, and then explore the role of spllover ndex α on the combned effect of p. Note that under Bernoull yeld dstrbuton, the second-stage expected proft at the equlbrum servce levels s gven by v (f p) = d (f p)u (f p), where f = (f, fj ) = (f (p, p j ), fj (p, p j )) are mplctly gven by (19). We can therefore treat v (f p) as a functon of target relablty ndex p only, where p explctly appears n u ( ) and mplctly appears n f and fj (see (17) under Bernoull dstrbuton), but p j only mplctly appears n f and fj. For expostonal ease, we denote the second-stage expected proft at the equlbrum servce levels as a functon of p,.e., v (p) v (f p). The cross effect. The followng proposton proves that, as one mght expect, the focal manufacturer s expected proft declnes as ts compettor s target relablty ncreases. Proposton 2. Manufacturer s expected proft s (weakly) decreasng n suppler s relablty ndex p j,.e. v (p)/ p j 0. The ntuton for Proposton 2 s as follows. An ncrease n p j affects manufacturer s expected proft v by affectng both manufacturers equlbrum servce levels f and fj. However, v s not affected by ts own equlbrum servce level at f. Hence, p j nfluences v through ts mpact on f j only. Snce manufacturer s demand s decreasng n f j, the ncrease n p j reduces manufacturer s expected proft. Leveragng Proposton 2, the followng corollary (proof omtted) s ntutve.

18 18 Artcle submtted to ; manuscrpt no. August 2011 Corollary 1. If suppler mprovement cost s fxed, then manufacturer s frst stage objectve J s (weakly) decreasng n p j regardless of the spllover effect α,.e. J / p j 0. The drect effect. A more nterestng queston s whether manufacturer always benefts from an ncrease n ts own target relablty ndex p, and how knowledge spllover ndex α affects such benefts. Applyng the envelope theorem to v, we have v (p) p = dv (f p ) dp = v (f p) f + v (f p) fj + v (f p) f }{{ p } f j p p =0 = d (f p) f j } {{ } f j p }{{} + u + d u (f p) p } {{ } + (20) where d (f p)/ f j 0 follows from Assumpton 2, f j / p 0 follows from part (a) of Proposton A2 n Appendx A, and v (f p)/ f = 0 follows from (19). In general, (20) can be postve or negatve, so manufacturer may not necessarly beneft from an mprovement n the suppler s relablty ndex p. Under certan condtons, however, we are able to unambguously sgn (20)., Proposton 3. If demand functon s log-separable and satsfes Assumpton 2, then each manufacturer s second stage expected proft s strctly ncreasng n ts own relablty ndex,.e., v (p)/ p > 0 ( = 1, 2). Proposton 3 shows that, wth log-separable demand, manufacturer s expected proft v (p) ncreases n ts own target relablty ndex p, and the rate of ncrease v (p)/ p = d ( u / p ) depends on ts own demand. The combned effect. Combnng the cross effect (Proposton 2) and drect effect (Proposton 3), we have J (p) z = v (p) p }{{} + +α v (p) p j }{{} dm (z ) dz } {{ } +. (21) Snce the spllover ndex α lnks the cross effect (a negatve mpact) wth the drect effect (a postve mpact) of an ncrease n p, the net effect (.e., whether the focal manufacturer s better off wth hgher suppler relablty) can be ambguous. In fact, we have observed numercal examples n whch the focal manufacturer s better off (or worse off) wth a hgher relablty ndex of the

19 Artcle submtted to ; manuscrpt no. August suppler. We thus conclude that the manufacturers may not necessarly beneft from hgher suppler relablty due to the spllover effect. It s natural to conjecture, however, that f there s no spllover effect (.e., α = 0) the manufacturers should beneft from hgher suppler relablty and therefore have ncentves to mprove suppler relablty p The Case of No Spllover Effect (but wth Demand Competton) The followng proposton proves that, when there s no spllover effect and the mprovement cost s fxed, each manufacturer s mprovement effort s of the all-or-nothng type. Proposton 4. If there s no spllover effect (.e., α = 0) and suppler mprovement cost s fxed, then t s optmal for each manufacturer to ether target a perfect relablty or not to mprove the suppler at all. Proposton 4 shows that f the gan from mprovng suppler relablty (to perfect yeld) offsets the fxed mprovement cost, then the manufacturer wll target a perfect yeld; otherwse t wll exert no mprovement effort. Note that f the mprovement cost vares wth mprovement effort, then the manufacturer s mprovement effort no longer exhbts ths all-or-nothng type of behavor. Next we study the equlbrum behavor of the manufacturers mprovement efforts. Snce each manufacturer wll not exert more effort than necessary to brng the target relablty to 1 (under Bernoull yeld dstrbuton), the set of feasble jont strateges s gven by the followng nonempty convex and compact set: z [0, 1 p 0 ], = 1, 2. Theorem 6. If there s no spllover effect (.e., α = 0), demand functon s log-separable and satsfes Assumpton 2, then each manufacturer s expected proft J ( ) s submodular n mprovement efforts z, and there exsts a Nash equlbrum on the optmal mprovement efforts z. The settng of α = 0 can be loosely nterpreted as a market settng where two manufacturers compete on demand but wth dstnct technologes such that there s no collaboraton on suppler mprovement. In ths specal settng, Theorem 6 shows that the two manufacturers settle n a stable equlbrum n ther mprovement efforts. When α > 0, however, snce each manufacturer s

20 20 Artcle submtted to ; manuscrpt no. August 2011 mprovement effort nevtably affects the other s relablty, the spllover effect may dsturb the above equlbrum or even lead to non-exstence of Nash equlbrum The Case of Postve Spllover Effect (but wthout Demand Competton) For the more general case wth postve spllover effect, the set of feasble jont strateges s z [0, 1 p 0 αz j ], = 1, 2, j = 3. In ths case, each manufacturer s feasble strategy set s dependent on the strategy adopted by ts compettor, and the set of jont feasble strateges s not a lattce. To tackle ths challenge, we consder the surrogate game wth the followng change of varables: (ẑ 1, ẑ 2 ) = (1 z 1, z 2 ). The set of feasble jont strateges n the surrogate game s the followng: ẑ 1 [p αẑ 2, 1], ẑ 2 [0, 1 α p αẑ 1 ], whch s a lattce. We wll utlze ths surrogate game to show exstence of equlbrum mprovement efforts n ths subsecton and next ( 5.3 and 5.4). A specal case s when each manufacturer s demand depends on ts own servce level only. Ths can be the case, for example, when the two manufacturers serve two geographcally separate markets. Ths specal case can be modeled by settng h (f j ) = 1 n the log-separable demand model defned n (5). In ths case, Assumpton 2 s reduced to d (f ) 0 and d 2 log d (f )/df 2 0. The followng corollary (proof omtted) s mmedate from Proposton 3 and (21). Corollary 2. If each manufacturer s demand s a weakly ncreasng and log-concave functon of ts own servce level only, then each manufacturer s second stage expected proft s strctly ncreasng n ts own relablty ndex,.e. v (p)/ p > 0 ( = 1, 2). If suppler mprovement cost s fxed, we can further characterze each manufacturer s best response functon n optmal mprovement effort. Proposton 5. If suppler mprovement cost s fxed, and each manufacturer s demand s a weakly ncreasng and log-concave functon of ts own servce level only, then both manufacturers would ether target perfect relablty or exert no mprovement effort at all.

21 Artcle submtted to ; manuscrpt no. August (a) If the fxed mprovement cost s greater than the maxmum gan from mproved supply relablty,.e., m > v (p = 1) v (p = p 0 ), then manufacturer exerts no mprovement effort. (b) Otherwse, there exsts a unque threshold z j [0, (1 p 0 )/α] such that manufacturer would target perfect relablty f z j < z j, exert no mprovement effort f z j > z j, and be ndfferent between the two optons f z j = z j. (c) Moreover, f z j > 1 p 0 j, then manufacturer wll always target perfect relablty under any feasble strategy of manufacturer j. Leveragng Proposton 5, we can now characterze the equlbrum behavor of the manufacturers mprovement efforts. Theorem 7. If suppler mprovement cost s fxed, and each manufacturer s demand s a weakly ncreasng and log-concave functon of ts own servce level only, then each manufacturer s optmal mprovement effort n the surrogate game s ncreasng n ts compettor s mprovement effort. There exsts at least one Nash equlbrum n the surrogate game. The set of equlbra s a lattce; there exsts a componentwse smallest equlbrum ẑ and a componentwse largest equlbrum ẑ. Theorem 7 shows that, f there are (unntentonal) collaboraton on suppler mprovement (.e., postve spllover effect) but no drect competton on demand (.e., demand depends on each manufacturer s own servce level only), then agan, n the orgnal game, the best response of one manufacturer s supply mprovement effort s decreasng n the other manufacturer s suppler mprovement effort and there exsts one or more equlbra. Ths s the case when two manufacturers share a common suppler wth smlar technologes but they serve geographcally separate markets. Next we study the most general settng wth postve spllover effect and drect market competton between two manufacturers The Case of Postve Spllover Effect Wth Demand Competton When two manufacturers engage n general demand competton under postve spllover effect n mprovement efforts, t s very challengng to characterze the equlbrum behavor. We therefore assume the Cobb-Douglas demand functon, see (6). The followng theorem gves suffcent condtons for the exstence of Nash equlbrum n the manufacturers mprovement efforts.

22 22 Artcle submtted to ; manuscrpt no. August 2011 Theorem 8. Assume that there s no shortage penalty cost (.e., π = 0) and each manufacturer s demand functon s a Cobb-Douglas functon (see (6)). (a) The equlbrum servce level satsfes f = K p, where K depends on cost parameters r, w, s and the dstrbuton of ϵ only. (b) Defne { β p 0 j M = max + β j + 1, β j (β + 1)p 0 j β β j p 0 + (β j + 1)p 0 β + 1 }, = 1, 2, and M = max{m 1, M 2 }. If M 2, the frst stage surrogate game s a supermodular game for all α [0, 1]; otherwse M > 2 [ ] 2 and the frst stage surrogate game s a supermodular game for α 0,. (c) When the frst stage surrogate game s a supermodular game, t has at least one equlbrum. The set of equlbra s a lattce; there exsts a componentwse smallest equlbrum ẑ and a componentwse largest equlbrum ẑ. Theorem 8 shows that under certan techncal condtons, the best response of one manufacturer s suppler mprovement effort s agan decreasng n hs compettor s suppler mprovement effort n the orgnal game, and there exst one or more equlbra n the manufacturers mprovement efforts. An mportant mplcaton of ths result s that a stable balance (n mprovement effort) can be acheved, despte a sgnfcant presence of spllover effect (α) and a farly ntense market competton (Cobb-Douglas demand) Summary In ths secton, we frst show that manufacturer may not necessarly beneft from ts suppler mprovement effort, even f the effort s costless, because, although t benefts from a hgher p, t suffers from a hgher p j due to spllover effect. The net effect can therefore be ambguous ( 5.1). Ths s a statc vew of the spllover effect for a gven vector of manufacturers mprovement efforts. When manufacturers optmally respond to each other s mprovement effort, however, we wll see n 6 that the spllover effect has a qute surprsng mpact on manufacturers expected profts. We then characterze the suffcent condtons for the exstence of manufacturers equlbrum mprovement efforts n three dfferent cases: () the manufacturers use dstnct or propretary M+ M 2 4

23 Artcle submtted to ; manuscrpt no. August technologes (.e., no spllover effect) and compete on demand, () the manufacturers use smlar technologes (.e., postve spllover effect) but they serve dfferent markets, and () the manufacturers use smlar technologes (.e., postve spllover effect) and they compete n the same market. In all three cases ( ), the best response of one manufacturer s suppler mprovement effort s decreasng n hs compettor s suppler mprovement effort (n the orgnal game),.e., the manufacturers mprovement efforts are substtutes. In closng, we note that under general demand functons we numercally observed nstances where there exsts no equlbrum, unque equlbrum or multple equlbra n the manufacturers mprovement efforts. The no-equlbrum case typcally arses when spllover effect s large, demand s hghly senstve to servce levels and the manufacturers dffer sgnfcantly n ther market characterstcs. 6. Manageral Implcatons In 5, we characterze manufacturers equlbrum mprovement efforts under certan condtons. It s of nterest to further explore the mpact of market characterstcs on manufacturers optmal mprovement efforts and the resultng mplcatons on manufacturers expected profts Endowed Market Advantage Oftentmes manufacturers dffer from one another n the expected market demand even f they set dentcal servce levels. Such persstent dfference n market demand can often be attrbuted to exogenous factors such as brand mage, product desgn, and dosyncratc consumer tastes. We refer to a manufacturer that all else beng equal enjoys a hgher market demand as the one that has endowed market advantage (EMA). In the context of our demand model, for example, manufacturer has EMA f (all else beng equal) x (f) > x j (f) for any f n attracton model (see (2)), or a > a j n lnear model (see (4)), or γ > γ j n log-separable model (see (6)). It s unclear a pror whether the manufacturer wth EMA would exert a hgher or lower mprovement effort relatve to ts compettor. On one hand, ntuton suggests that t should exert less effort: snce t already enjoys a market advantage, t does not have to compete as aggressvely.

24 24 Artcle submtted to ; manuscrpt no. August 2011 On the other hand, ntuton also suggests that t should exert more effort to explot ts market advantage more effectvely. Fgure 1 llustrates the effect of EMA on manufacturers equlbrum mprovement efforts. (In Fgure 1, yeld s unform Y [1 0.6/p, 1], where p 0 = 1.0; demand s lnear (as n (4)), where the base values are a = 0.5, b = 1.0, and c j = 0.5; demand uncertanty ϵ s normal wth µ = 1.0 and σ = 0.3; the other base values are r = 1.0, w = 0.5, s = 0.2, and π = 0.3; mprovement cost s m (z ) = z 2. Fgure 1 s obtaned by varyng α and a 1 whle fxng all other parameter values. The rest of fgures are obtaned usng smlar settngs.) 40% (a) Effect of Knowledge Spllover (α) on Equlbrum Improvement Effort 40% (b) Effect of Endowed Market Advantage (a 1 a 2 ) on Equlbrum Improvement Effort 30% 30% Rato of Optmal Improvement Effort (z 1 /z 2 1)x100% 20% 10% 0% 10% Manufacturer 1 enjoys ncreasng endowed market advantage 20% 10% 0% 10% Increasng Spllover α 20% Manufacturer 1 ncreasngly suffers from market dsadvantage 20% Increasng Spllover α 30% Spllover α 30% Endowed Market Advantage (a 1 a 2 ) Fgure 1 Effect of endowed market advantage on manufacturers equlbrum mprovement efforts Fgure 1 ndcates that the manufacturer wth EMA tends to exert a hgher effort level relatve to ts compettor, and ths phenomenon becomes more pronounced as the spllover ndex α ncreases. We note that, as the spllover effect ncreases, both manufacturers mprovement effort declnes, but the declnng speed s slower for the manufacturer wth EMA. As a result, ts relatve mprovement effort ncreases n α.

25 Artcle submtted to ; manuscrpt no. August One naturally wonders whether the fact that the manufacturer wth EMA exerts an ncreasngly hgher proporton of mprovement efforts also translates nto a relatve proft advantage,.e. whether ts relatve fnancal performance s also enhanced by the spllover effect. Fgure 2 llustrates the effect of spllover α and the EMA on manufacturers expected profts. 50 (a) Effect of Knowledge Spllover (α) on Optmal Expected Proft 4 (b) Effect of Knowledge Spllover (α) on Relatve Expected Proft 45 Manufacturer 1 wth endowed market advantage 3.5 Optmal Expected Proft Manufacturer 1 & 2 wth dentcal market advantage Manufacturer 2 wth market dsadvantage Rato of Optmal Expected Proft (J 1 /J 2 ) Rato of expected proft when manufacturer 1 has endowed market advantage Rato of expected proft when Manufacturer 1 & 2 have dentcal market advantage Spllover α Spllover α Fgure 2 Effect of endowed market advantage on equlbrum expected profts Fgure 2(a) shows that both manufacturers expected profts ncrease n the spllover effect α, but the relatve proft advantage of the manufacturer wth EMA (manufacturer 1) declnes as the spllover effect ncreases (Fgure 2(b)). Thus, the manufacturer wth EMA may vew the spllover effect wth some sort of dlemma: n absolute proft terms, ts ncreasngly hgher proporton of mprovement effort pays off because ts expected proft ncreases n α; n relatve proft terms, however, ts proft advantage (over ts compettor) declnes, suggestng that ts ever larger proporton of mprovement effort benefts ts compettor more than tself. Back to the UTC s example, ts Pratt & Whtney s ambvalence about the spllover effect s