Values of the balanced decision making between supply chain partners

Transcription

1 Intl. Trans. in Op. Res. 15 (2008) INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH Vales of the balanced decision making between spply chain partners Bowon Kim a,b and Jongjoo Kim a a Gradate School of Management, Korea Advanced Institte of Science and Technology (KAIST), Cheongryangri, Dongdaemoon, Seol , Korea, b Sader School of Bsiness, University of British Colmbia, Vancover, BC V6T 1Z2, Canada BowonKim@kgsm.kaist.ac.kr [Kim] Received 23 September 2005; received in revised form 6 December and 20 September 2007; accepted 5 Janary 2008 Abstract Coordination between spply chain partners is critical to effective spply chain management. In this paper, we consider two cases of decision-making strctre for a simple spply chain consisting of two players: the first case in which a spply chain partner dominates the decision-making process, and the other in which two players share the decision-making process eqally. According to the literatre, we know a priori that the balanced decision making forges a better (financial) otcome than the dominating case does. Therefore, or primary research objective is to analyze detailed dynamics of resorce allocation and sorces of the benefit of the balanced decision making. Or analysis indicates that the vale of the balanced decision making arises from more effective resorce tilization than the dominating case does: each of the cooperative partners knows how to optimize its resorce tilization better than the other does. Keywords: balanced decision making; spply chain cooperation; optimal control theory model 1. Introdction For effective spply chain management, coordination between spply chain partners is critical (Kim, 2000). Bt, there is a sbtle isse: depending on the bargaining power balance between the spply chain participants, it is determined who exerts more inflence in making decisions related to sch coordination. For example, Mnson et al. (1999) postlated that often one company in a chain might attempt to inflence other members in order to achieve its own goals and promote its own interests. In this paper, we consider two cases of decision-making strctre in the context of a simple spply chain consisting of two players: the first case in which a spply chain partner dominates the decision-making process and the other partner passively follows the dominant player s decision, Pblished by Blackwell Pblishing, 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350 Main St, Malden, MA 02148, USA.

2 624 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) and the other case in which the two players share the decision-making process eqally in a balanced way. For instance, in the atomobile indstry, a spplier that does not have a strong bargaining power cannot help bt act passively vis-a` -vis its stronger partner, e.g. the carmaker. In the game theory literatre, it is well established that a collsive or a cooperative case generates a better reslt than a competitive case from the entire system s perspective: this reslt is mainly de to the existence of free riding (Fershtman and Nitzan, 1991). Applying the reasoning to the crrent context, we know a priori that the balanced decision making forges a better otcome than the dominating case does for the spply chain system as a whole. In this paper, therefore, rather than asking abot which decision-making strctre is better for the spply chain, or primary qestions are How do the resorce allocation dynamics between the spply chain partners behave? and From where comes the benefit of the balanced case? To answer the qestions, we set p optimal control theory models and derive analytical soltions. In the literatre, many of the stdies on a similar isse have sed static game-theoretic approaches. Althogh a game-theoretic approach is sefl in examining competitive interactions between players participating in a static context (e.g. a single- or two-period time horizon), it is not sally effective in analyzing dynamic changes over time. That is, the game-theoretic methodology analyzes static, i.e. snapshot, portrayals, whereas the optimal control theory model can captre dynamic changes occrring over an entire decision-time horizon. Or optimal control analysis otcome indicates that the vale of the balanced decision making arises from more effective resorce tilization than in the dominating case. Or paper is strctred as follows: in the next section, we review the relevant literatre. In Section 3, we develop optimal control models to answer or main research qestions. Analysis of these models is carried ot in detail in Section 4, focsing on where the benefit of the balanced decision making arises from. In Section 5, we present nmerical examples to show the dynamics of resorce allocation and profit increase throghot the decision horizon: the nmerical analysis is based on a real-world case in the atomobile indstry. Finally, the conclding section smmarizes or findings and discsses key managerial implications. Detailed analytical procedres to solve the optimal control models are smmarized in Appendix A. 2. Literatre review In the literatre, there is an abndance of relevant references, which can be groped nder several headings Spply chain coordination Kim (2000) emphasized the importance of a long-term strategic relationship between a manfactrer and its sppliers in order to create more vale in the spply chain (Spekman, 1988; Doyle, 1989; Choi and Hartley, 1996). It is fndamentally based on the belief that sch a long-term relationship wold eventally benefit all of the spply chain participants (Iyer and Bergen, 1997; Parlar and Weng, 1997). Withot sch a mtal benefit, spply chain collaboration may not be sstainable (Crewe and Davenport, 1992; Sparks, 1994; Ellram and Edis, 1996; Boddy

3 et al., 1998; Stank et al., 1999; Ireland and Brce, 2000; Lee and Whang, 2000; McIvor and McHgh, 2000; Barratt, 2004). Jorgensen and Zaccor (2003) modeled a two-member channel of distribtion, consisting of a manfactrer and a retailer (Chintagnta and Jain, 1992; Jorgensen and Zaccor, 1999). They postlated that in the absence of coordination, channel members determine their decision variables independently and non-cooperatively, bt sch ncoordinated decision making creates channel inefficiency, that is, channel members strategies are not at their joint profitmaximization levels and their profits are inferior to what cold be achieved with coordinated behavior. They implied that there exist incentives to coordinate among spply chain partners so as to overcome sch channel inefficiencies. In the literatre, some arged that sccessfl SCM improves financial performance for all chain members (Wisner and Tan, 2000; Tan, 2002; Crook and Combs, 2007). Others dobted it: Ketchen and Ginipero (2004) sggested a chain member may exploit its partners for its own gain. At the essence of any strategic spply chain relationship, there mst be effective coordination between spply chain partners: Weng (1999) pt forth that effective coordination can lead to either increasing expected joint profit, increasing the probability of achieving the desired profit level, or a combination of both (Reyniers, 1992; Whang, 1995; Sox et al., 1997) Coordination mechanisms B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Coordination has been an important research isse not only in operations bt also in marketing (Monahan, 1984; Chandra and Fisher, 1994). However, despite the weighty emphasis on the advantages of close spply chain coordination, Goffin et al. (2006) pointed ot that there has been little research carried ot on identifying the specific attribtes of sch relationships. As Jeland and Shgan (1983) sggested, coordination or cooperation is not a natral behavioral pattern for the channel members (Coghlan, 1985). That is, in order to accomplish effective coordination, the spply chain partners mst design appropriate contractal arrangements (Cachon and Lariviere, 2005). These contracts need to specify sch essential aspects as how to share the revenes as well as the costs (Taylor, 2000; Dana and Spier, 2001; Pasternack, 2002; Gerchak and Wang, 2004; Cachon and Lariviere, 2005), how to allocate decision rights among the participating spply chain members (Whang, 1995; Tsay et al., 1999), and so forth. Now rather than jst highlighting the importance of coordination itself, finding ot effective mechanisms, i.e. coordination strctres, to achieve high-performing coordination has become the focs in the literatre (Anand and Mendelson, 1997; Moses and Seshadri, 2000). Narayanan and Raman (2004) sggested that companies have to create monetary incentive systems to indce the spply chain partners to behave in ways that are best for everyone in the chain. Sahin and Robinson (2005) highlighted information sharing and physical flow coordination for tighter spply chain integration in make-to-order spply chains, which is spposed to improve the spply chain s economic performance. Throgh logitdinal case stdies, Spina and Zotteri (2000) showed that negotiations on how the benefits of joint cost-saving efforts between spply chain partners will be shared shold be condted p-front, sggesting Once a player knows how the cake will be split, all he can do to get a larger slice is to contribte to making a larger cake; otherwise players will spend a lot of time fighting for a relatively larger slice rather than working to prodce a larger cake.

4 Decision-making strctres B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Sahin and Robinson (2002) sggested that there are two decision-making strctres for accomplishing coordination, i.e. centralized and decentralized decision making. Mnson et al. (1999) investigated the isse from a bargaining power perspective, postlating that althogh firms in a spply chain depend on each other and work together for mtal benefit, the relationships among them are rarely symmetrical. That is, in reality, there can be an imbalance of bargaining power among the spply chain partners. This neqal distribtion of bargaining power might case a spply chain partner with a larger bargaining power to impose its own decisions on the other weaker partners in the spply chain. Similarly, Crook and Combs (2007) noted that the benefits of spply chain coordination vary according to how bargaining power is sed: a stronger member might calclate other weaker members dependency and exert its power dring negotiation to appropriate a larger percentage, if not all, of the gains from spply chain coordination. At the extreme, the stronger member might even negotiate all spply chain gains pls some of the weaker members pre-scm profits. In a similar vein, Dyer (1996b) arged that de to the bargaining power bias or imbalance, US atomobile manfactrers and their sppliers had higher procrement costs than their Japanese conterparts. Under what circmstances, then, is a spply chain member powerfl? Pfeffer (1992) advocated five different sorces of a stronger power: a spply chain member has larger bargaining power if (1) other members of the chain depend on it for their essential needs, (2) it has control over financial resorces, (3) it plays a central role in the chain, (4) it is not sbstittable, and (5) it has the ability to redce critical ncertainty. We already pointed ot that it is ineffective or sboptimal for a single player in the spply chain to dominate the decision-making process. When they stdied a spply chain system where the manfactrer dominated the spply chain decision making, X et al. (2001) implied that the manfactrer was the main beneficiary of coordination in terms of safety stock and resorce waste redction, while there was little effect on the retailer in these two areas. We can infer that when one member in the spply chain wields an nconstrained decision-making power, it can be difficlt to avoid resorce wastes in the spply chain system as a whole. Consistent with this inference, Mnson et al. (1999) postlated that in order to maximize the effectiveness of coordination, strong firms can often maintain solid channel relationships by inclding other channel members in the decision-making process and/or providing concessions to eliminate potential losses. That is, allowing other spply chain partners to be involved in the decisionmaking process, it will be possible for the spply chain system to achieve effective coordination by eliminating possible wastes of resorces Methodology Varios research methods for stdying isses related to spply chain coordination have been sed in the literatre. There are researchers who have sed traditional inventory models, sometimes with nmerical analysis: for references, see Iyer and Bergen (1997), Parlar and Weng (1997), Chen (1998), Cachon and Fisher (2000), Donohe (2000), Chen et al. (2001), Fisher et al. (2001). Others adopted more complicated analytical models based on game theory: for a few notable examples,

5 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) see Spengler (1950), Pasternack (1985); Parlar (1988), Ernst and Cohen (1992), Christy and Grot (1994), Wang and Parlar (1994), Ch and Desai (1995), Moses and Seshadri (2000), Anpindi et al. (2001). There have been more empirical attempts as well: see Frohlich and Westbrook (2001), Griffith et al. (2006). There are researchers who favored sing simlation techniqes: Waller et al. (1999), Towill (1991), Wikner et al. (1991), Towill et al. (1992). In the literatre, it is easy to find papers tilizing sal game-theoretic models. The gametheoretic approach has advantages, one of which is to enable the researcher to analyze competitive interactions between players (e.g. spply chain partners) rigorosly. However, with the optimal control theory model, we are more capable of analyzing the optimal dynamics and nderstanding intricate changes of the fndamental forces that drive sch dynamics, i.e. analyzing the competitive sitation dynamically over a longer period of time in a continos time line. 3. Optimal control theory models There are many areas of spply chain coordination (Kim, 2005). Spply chain partners can coordinate for sharing information abot market demand, improving the qality of an existing prodct, condcting a joint marketing and/or promotion, making joint decisions on plant capacity or the inventory level, and the like. In this paper, we focs on a specific area of coordination: new prodct development (NPD). We consider a simple spply chain consisting of two players, e.g. a spplier and a manfactrer (Jorgensen and Zaccor, 2003). To highlight the difference in decision-making strctres, two cases are stdied. The first case portrays a sitation where the manfactrer dominates the decision-making process (X et al., 2001). The manfactrer s domination is embodied in its ability to make se of the entire resorces in the spply chain inclding the spplier s as well as its own. In the second case, on the other hand, the spplier is also actively involved in the decision-making process, i.e. participating in the decision making for resorce allocation Research motivation the atomotive indstry Or research was motivated by the atomobile indstry, where it is easy to find the sitations described above. In the atomobile indstry, it is well known that there sally exists a hge imbalance in bargaining power between the spply chain partners, in particlar, a carmaker (manfactrer) and its spplier(s): in general, it is the manfactrer that has the larger bargaining power. Sometimes the player with a dominating bargaining power wields its excessive inflence on the weaker player to gain its own profit at the cost of the other. In Liker and Choi (2004), there is a qote from a director of a major spplier to the big three US carmakers, saying The Big Three (US atomakers) set annal cost-redction targets (for the parts they prchase). To realize those targets, they ll do anything. (They ve nleashed) a reign of terror and it gets worse every year. This is a sitation comparable to the manfactrer-dominating decision-making strctre mentioned above, where the manfactrer exerts its excessive bargaining power to coerce the weaker spplier into passively accepting the terms and conditions withot any meaningfl

6 628 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) bilateral negotiations. Therefore, the manfactrer obtains a de facto right to tilize the spplier s resorces at its own arbitrary discretion. The qestion is, Is this type of manfactrerdominating decision strctre in a spply chain good even for the manfactrer itself? Bt this kind of coercive relationship is not always the rle in the indstry. In the same reference, there is another qote from a spplier to Toyota, saying Toyota helped s dramatically improve or prodction system. We started by making one component, and as we improved, [Toyota] rewarded s with orders for more components. Toyota is or best cstomer. Dyer (2000) described a sitation that exactly matches the balanced decision-making strctre in a spply chain mentioned above: Instead of relying solely on its own engineers to create the concept for a new car and then to design all the car s components, Chrysler now involves sppliers. And instead of Chrysler dictating prices to sppliers, regardless of whether the prices are realistic or fair, the two sides now strive together to find ways to lower the costs of making cars and to share the savings.... The reslts have been astonding. The time Chrysler needs to develop a new vehicle is approaching 160 weeks, down from an average of 234 weeks dring the 1980s. The cost of developing a new vehicle has plnged an estimated percent dring the last decade. And, at the same time, Chrysler has managed to prodce one consmer hit after another The single player-dominating decision-making strctre (coercive case) For or first case, we develop an optimal control theory model that describes the manfactrerdominating spply chain. In general, the manfactrer has larger bargaining power vis-à-vis its spplier, implying that the manfactrer can exert a mch larger decision-making power than its spplier (Mnson et al., 1999). We embody the manfactrer s domination in the objective fnction of the optimal control model in P1-M. Table 1 explains the variables and parameters in the model Z T P1-M: Maximize ðx 1 þ x 2 Þða 1 ðx 1 þ x 2 ÞÞ r dt; ð1þ 0 Sbject to _x 1 ¼ 1 ; ð2þ 1 ) 1 ; ð3þ _x 2 ¼ 2 ; ð4þ Table 1 Notation x i : Firm i s knowledge/technology stock for new prodct development (state variable), i 5 1, 2; the manfactrer, i 5 1 i : Firm i s investment rate (control variable) at t, i 5 1, 2 i : Limit on firm i s resorce investment rate at t a i : Parameter that determines firm i s profitability r i : Parameter that determines the cost to se firm i s resorces

7 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) ) 2 ; x 1 ð0þ ¼a 1 ; x 2 ; ð0þ ¼a 2 : ð5þ ð6þ P1-M is the manfactrer s decision problem: this optimal control model consists of an objective fnction and seven constraints. We explain each of the components in detail. R T 0 ðx 1 þ x 2 Þða 1 ðx 1 þ x 2 ÞÞ r dt: In the objective fnction (1), the profit maximization is carried ot from the manfactrer s perspective and only the cost of sing the manfactrer s resorce is inclded in the model. That is, when the manfactrer can wield its dominating bargaining power, it can make a decision on how the spplier s resorce shold be tilized, bt does not take into accont the cost of sing it. Althogh this kind of decisionmaking strctre or arrangement is nfair from the spplier s viewpoint, in reality, it can happen when the spply chain partners have extremely skewed bargaining powers: refer to an example in the atomobile indstry mentioned in the previos section. ðx 1 þ x 2 Þða 1 ðx 1 þ x 2 ÞÞ: The first part in the integrand of the objective fnction is the manfactrer s profit, jointly determined by the manfactrer s knowledge stock and the spplier s at the same time, and concave in the combined knowledge stock, x 1 1x 2 : this format of profit fnction is widely sed, in particlar when competing or coordinating players have to devote effort jointly in order to create a common grond (e.g. a pblic good) on which each of them can earn its own profit (Fershtman and Nitzan, 1991). See Fig. 1. That is, the manfactrer needs the spplier s knowledge stock, i.e. investment in NPD, in order to generate its own profit, and vice versa. In this sense, it is important for the spply chain partners to coordinate with each other. As we can see in Fig. 1, there might be a technical isse: there is an effective range of the combined knowledge stock, dring which an SC partner s profit increases as the knowledge stock increases. Once the combined knowledge stock exceeds the range, the profit decreases even when the knowledge stock increases. Althogh this kind of phenomenon can happen if we allow diseconomies of scale, to avoid any nnecessary (and nrealistic) Payoff a 4 a 1 2 Combined Knowledge Stock x + x An effective range of joint effort Fig. 1. Profit fnction.

8 630 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) complications in the ensing discssion, we wold like to consider only the cases where the combined knowledge stock is within the effective range. Using the example of the atomobile indstry (as in this paper), for example, we propose that a carmaker needs knowledge or technology to develop new models continosly. The carmaker invests in improving sch knowledge stock, x 1, and the spplier also invests in enhancing its own knowledge stock, x 2. Finally, the carmaker s NPD capability is determined by the combined knowledge stock of both the carmaker and the spplier, i.e. X 5 x 1 1x 2. Now we assme that the carmaker s profit or payoff is dependent on its NPD capability, i.e. the higher the NPD capability, the larger the potential payoff. Based on empirical observations in the literatre, we frther pt forth that the relationship between the payoff and the NPD capability is concave, implying an existence of diminishing retrns to scale as the firm s knowledge stock accmlates. More specifically, we express this relationship as follows: the carmaker s payoff at t 5 kx(a 1 X) and withot loss of generality, we normalize so as to have k 1. The relationship between the firm s NPD capability (i.e. knowledge/technology stock) and its payoff (related with NPD) is based on the literatre as well as empirical findings. As the firm invests in enhancing NPD knowledge, its NPD knowledge stock increases and so does the firm s NPD capability. As the firm s NPD capability grows, it will be able to speed p its NPD process, i.e. to redce the NPD lead-time. Redced NPD lead-time implies faster introdction of new models into the market, and therefore increased market share, which helps the firm achieve higher profit (Datar et al., 1997; Hendricks and Singhal, 1997). r : Developing the knowledge stock is costly, i.e. each SC partner needs to pay its share of cost. However, as allded to in the atomobile indstry, it does happen that the manfactrer makes decisions on how to tilize the spplier s resorces when it has dominating bargaining power vis-à-vis the spplier. When this kind of sitation indeed happens, from the manfactrer s perspective, the spplier s resorces are free and the cost to tilize them is not inclded in the manfactrer s decision problem. That is why there is only one cost element, r 1 2 1,inP1-M. _x i ¼ i : In P1-M, (2) and (4) show the dynamics of the knowledge/technology stock. Withot compromising generalizability, we assme that the investment is scaled so that investing one nit of resorce increases one nit of knowledge. 1 ) 1 : (3) and (5) specify the resorce availability that can be tilized at t for each of the players. For instance, the manfactrer cannot spend more than 1 at t on developing its knowledge stock for NPD. x i ð0þ ¼a i : (6) presents the initial vales of the state variables. With P1-M, we formlate the Hamiltonian as in (7) and solve the optimal control model as in Appendix Section A.1. The problem s Hamiltonian or Lagrangian is strctred as follows: L ¼ H ¼ðx 1 þ x 2 Þða 1 ðx 1 þ x 2 ÞÞ r 2 1 þ l 1 1 þ l 2 2 þ w 1 ð 1 1 Þþw 2 ð 2 2 Þ: ð7þ Now the profit of the weaker player in the spply chain, i.e. the spplier, is completely determined by the soltions to P1-M: indicates the vale is part of the optimal soltion determined in the manfactrer s problem.

9 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Z T n P1-S: Spplier s profit ¼ ðx 1 þ x 2 Þ a 2 ðx 1 þ x 2 Þ o r2 ð 2 Þ2 dt: ð1 1Þ 0 In the single player-dominating case, where the manfactrer wields an excessively large bargaining power, the spplier can only react passively, allowing the manfactrer to decide how to tilize the spplier s resorces. Therefore, the spplier s decision problem is simply to calclate its profit by plgging in its profit fnction the solved decision variables, both state and control variables Balanced decision-making strctre (balanced case) In this section, we develop an optimal control theory model for the second case, the balanced decision-making strctre. For this case, a shared decision making or a balanced allocation of bargaining power between SC partners is assmed: what we mean by balanced in this paper is that the spplier s profit and cost are taken into accont eqally or eqitably when the resorce allocation decision is made. That is, the decision maker adopts a balanced (i.e. not biased towards one side s profit maximization only) perspective when making the resorce allocation decision. We already discssed cases abot Toyota s relationship with its sppliers and Chrysler s coordination in developing new models with its sppliers. Even if the manfactrer can wield a dominating bargaining power, it might not want to make a decision nilaterally becase it knows that sch a biased decision making will not improve the profitability not only for the spplier bt also for the manfactrer itself. The most significant difference from the single player-dominating case is the objective fnction. That is, in the objective fnction of the balanced decision-making strctre, the profit to be maximized is the sm of those for the spplier and the manfactrer. In addition, the cost to tilize the spplier s resorces is also inclded in the objective fnction, implying that the spplier s resorces are not free any longer even to the manfactrer. This arrangement in the objective fnction implies that the manfactrer and the spplier are now carrying ot joint decision making, i.e. it is a balanced decision-making strctre. That is, we embody the balanced decision-making scheme by taking into accont the costs and benefits of the spplier flly in the objective fnction. P2 is the optimal control theory model for this case. Z T P2: Maximize ðx 1 þ x 2 Þða 1 þ a 2 2ðx 1 þ x 2 ÞÞ r r dt; ð8þ 0 Sbject to _x 1 ¼ 1 ; ð9þ 1 ) 1 ; _x 2 ¼ 2 ; 2 ) 2 ; x 1 ð0þ ¼a 1 ; x 2 ð0þ ¼a 2 : ð10þ ð11þ ð12þ ð13þ

10 632 The relevant Hamiltonian or Lagrangian is as follows: L ¼ H ¼ðx 1 þ x 2 Þða 1 þ a 2 2ðx 1 þ x 2 ÞÞ r r þ l 1 1 þ l 2 2 þ w 1 ð 1 1 Þ þ w 2 ð 2 2 Þ: B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) In P2, (9) (13) are exactly the same as (2) (6). Using the Hamiltonian (14), we derive the analytical soltion to P2, the procedres of which are well docmented in Appendix Section A.2. ð14þ 4. Analysis of the model In Section 3, we have developed optimal control theory models for two cases: the coercive (single player-dominating) and the balanced decision-making sitation. Becase the detailed mathematical derivations are completely listed in Appendix Sections A.1 and A.2, we do not have to repeat the problem-solving procedres here. Rather, in this section, we wold like to discss the dynamics of the soltions by depicting them in figres: we focs on how to analyze the patterns of resorce allocation dynamics and where the benefits of the balanced decision making arrive from. For the single player-dominating case, we show why the manfactrer (with the sperior bargaining power) always ses p the spplier s resorce first before it starts sing its own resorce: we prove the SC participants behavioral dynamics by analyzing their marginal valations reflected in co-state variables and Lagrangian mltipliers. We condct a similar analysis for the balanced case. More importantly, with the soltions obtained from this section, we will be able to perform the nmerical analysis in the next section Dynamics of the single player-dominating case (the coercive case) In Appendix Section A.1, we have detailed the soltion procedres for the single playerdominating case: we assme the dominant player is the manfactrer, reflecting the reality in general (Mnson et al., 1999). We have paid special attention to the behaviors of l i and w i in order to nderstand nder what circmstances the sole decision maker, e.g. the manfactrer in or case, ses the resorces to their limits, i.e. the dynamics of i ) i. Or analysis indicates that becase the spplier s resorces are essentially free to the manfactrer, assming the resorces are valable, the manfactrer always consmes the spplier s resorces to the limit, i.e. 2 ¼ 2. In addition, the reslt shows that the relative magnitde between l 1 (0) and 2r 1 is critical to determining whether the manfactrer ses p its own resorces completely or not: l 1 (0) is the shadow price or the marginal vale of the nit resorce at the beginning of the decision time horizon; becase r is a cost coefficient and 1 is the limit of the manfactrer s resorces, 2r 1 somehow represents the total cost to tilize its own entire resorces. There are two cases: l 1 ð0þ < 2r 1, i.e. the marginal vale of nit resorce is less than a measre related to the total cost to se p the manfactrer s entire resorces.

11 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) ρ 1 λ 1 (0) λ = λ = w w 1 0 T * * 1 < 1 and 2 = 2 Fig. 2. Dynamics of resorce tilization when l 1 ð0þ < 2r 1. Figre 2 describes this sitation. As in Appendix Section A.1 and Fig. 2, we see that nder this condition, the manfactrer never consmes its own resorces completely, while it always ses p the spplier s: 1 < 1 and 2 ¼ 2 throghot 0)t)T. It is, therefore, reasonable to infer that nder this circmstance, it might be navoidable for the resorces to be tilized inefficiently and ineffectively. Sch inefficient resorce tilization easily cases sb-optimality of performance in the spply chain as a whole. l 1 ð0þ > 2r 1, i.e. the marginal vale of nit resorce is larger than a measre related to the total cost to se p the manfactrer s entire resorces. In the second case, there can exist ^t sch that 1 ¼ 1 for 0)t < ^t and 1 < 1 for ^t)t)t: see Fig. 3. Appendix Section A.1 proves that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 1 ð0þ 2r 1 ^t ¼ : ð15þ 1 þ 2 (A22) or (15) indicates the following: d l 1ð0Þ 2r 1 1 þ 2 ðaþ ¼ l 1ð0Þþ2r 2 d 1 ð 1 þ 2 Þ 2 < 0; implying that the larger the manfactrer s resorce limit, the shorter ^t, i.e. the shorter the period dring which 1 ¼ 1, other things being eqal. That is, when the manfactrer s resorces are abndant, it becomes less likely that the manfactrer consmes its resorces flly. ðbþ d l 1ð0Þ 2r 1 1 þ 2 ¼ l 1ð0Þþ2r 1 d 2 ð 1 þ 2 Þ 2 < 0;

12 634 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) λ 1 (0) λ = λ = w ρ 1 w1 (0) 0 ˆt T * 1 = 1 * 1 < 1 * 2 = 2 Fig. 3. Dynamics of resorce allocation when l 1 ð0þ > 2r 1. implying that the larger the spplier s resorce limit, the shorter ^t, i.e. the shorter the period dring which 1 ¼ 1, other things being eqal. That is, when the spplier s resorces are abndant, it becomes less likely that the manfactrer consmes its resorces flly becase it wold be sfficient to se p the spplier s resorces first. Which sitation is more likely to occr in reality: l 1 ð0þ < 2r 1 or l 1 ð0þ > 2r 1? It might depend on the specific context in point. Bt l 1 (0) is the shadow price or the marginal vale of the nit resorce, while 2r 1 somehow represents the total cost to tilize the fll resorces, and therefore, in general, l 1 ð0þ < 2r 1 seems more likely to occr. This observation implies that it is more likely to have 1 < 1 and 2 ¼ 2 throghot 04t4T, i.e. the manfactrer always consmes the spplier s resorces completely, whereas it never ses p its own resorces, regardless of the relative efficiency or effectiveness between the two different types of resorces. Hence, it is more likely to waste valable resorces Dynamics of the balanced decision-making case (the balanced case) Now, we wold like to analyze the dynamics of resorce allocation for the balanced decisionmaking case. Again, the details of mathematical derivations are well docmented in Appendix Section A.2. Similar to or analysis for the single player-dominating case, the precise dynamics of resorce allocation for the balanced case are dependent on the relative magnitdes among r 1 1, r 2 2, and l(0): see Fig. A1 and Table A1. Figre 4 depicts the case where 2r 2 2 < 2r 1 1 < lð0þ. Under this condition, ^t 1 and ^t 2 can be determined as in (A43) and (A48) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 1 ð0þ 2r ^t 1 ¼ 1 1 ; ð16þ 2ð 1 þ 2 Þ

13 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) λ (0) λ = λ ρ11 w 2 (0) 2 ρ 22 w 1 (0) 0 ˆt 1 ˆt 2 T * 1 = 1 * 2 = 2 * 1 < 1 * 2 < 2 Fig. 4. Dynamics of the resorce tilization for the balanced case. ^t 2 ¼ T r 2 2 ðt ^t 1 Þ: ð17þ r 1 1 We can make a similar interpretation for ^t 1 to that for ^t in (15). It is noteworthy that ^t 2 is determined mainly by two elements: the difference between ^t 1 and T as expressed in ðt ^t 1 Þ and the relative efficiency between the two different resorce tilizations as represented in r 2 2 =r 1 1. The biggest difference between the single player-dominating and the balanced case is that the decision makers in the balanced case se resorces by taking into accont their relative efficiency, whereas in the coercive case the resorces owned by the player with less bargaining power are sed p completely regardless of their efficiency vis-a` -vis that of the resorces possessed by the dominant decision maker. 5. Nmerical examples To cast or analysis in the real-world context, we condct a nmerical analysis. We already mentioned two real-world cases in the atomobile indstry, i.e. GM and Chrysler: we compare the GM case to the single player-dominating strctre and Chrysler to the balanced decision-making strctre. The setting can be described as follows. The carmaker gains benefit as it becomes more capable of developing a new model faster, i.e. redcing the NPD lead-time. In order to improve its NPD speed, the carmaker needs spport from its spplier. Now we have the following one-to-one relationship. x 1 : the carmaker s knowledge stock for speeding p its NPD process, x 2 : the spplier s knowledge stock for speeding p the NPD process,

14 636 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Table 2 Parameter vales a i (i 5 1, 2) r 1 r a 1 a 2 T x 1 1x 2 : the combined knowledge stock developed by both the carmaker and its spplier, which reflects (e.g. proxies) the spply chain s overall NPD capability, ðx 1 þ x 2 Þða i ðx 1 þ x 2 ÞÞ: the payoff the carmaker (i 5 1) or the spplier (i 5 2) earns when the NPD capability (i.e. the combined knowledge stock) is x 1 1x 2 at t. Table 2 presents the parameter vales sed in the nmerical analysis: these parameter vales are based on the information from the Chrysler case (Dyer, 1996a, 2000). The primary resorces are engineering hors. The carmaker can invest as mch as 3000 engineering hors in increasing its NPD knowledge at a time: i.e. 1 ¼ 0:03. Throgh an appropriate adjstment (e.g. normalization), one nit of NPD knowledge stock is eqal to 1 engineering hor, i.e, _x 1 ¼ 1. The carmaker s instantaneos payoff is determined by its NPD knowledge stock developed by both the manfactrer and the spplier. For instance, if the combined knowledge stock for NPD is 0.3 at t, the firm s instantaneos payoff (before sbtracting the investment cost) at t wold be $210 million, i.e. 0.3 (1 0.3) and the nit is $1 billion. As the firm s NPD capability improves, its payoff increases as well. The relationship between the firm s NPD capability (i.e. knowledge/technology stock) and its payoff (related with NPD) is based on the following reasoning. As the firm invests in NPD knowledge/technology development, its NPD knowledge/technology stock increases and so does the firm s NPD capability. As the firm s NPD capability grows, it will be able to speed p its NPD process, i.e. to redce the NPD lead-time. Redced NPD lead-time implies faster introdction of new models into the market, and therefore increased market share, which helps the firm achieve higher profit (Datar et al., 1997; Hendricks and Singhal, 1997). The total cost de to investing 1 engineering hors is r Using r , the relationship between invested engineering hors and total cost resembles that in Fig. 5. The spplier has a similar condition in terms of available engineering hors, althogh the cost strctre is better than the carmaker s (r 1 4r 2 ). That is, it is less expensive to tilize a spplier s engineering hor, possibly becase it pays its engineers less, rns less expensive plant/eqipment, has less costly legacy systems, and so forth. The fnctional relationships are derived from the estimates based on or empirical investigation of the carmaker, sing data from secondary sorces as well. Becase we wold like to see the long-term effect of NPD collaboration, we se a 10-year time horizon, i.e. T Using the parameter vales in Table 2, we condcted nmerical analyses for the two different scenarios. The nmerical analysis reslts are reported in Table 3, which we recapitlate as follows:

15 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Fig. 5. Relationship between engineering hors invested and the cost. Table 3 Nmerical analysis reslts at T Single playerdominating (coercive) Balanced decision making Total payoff Manfactrer s payoff Spplier s payoff Cmlative engineering hors invested: total Cmlative engineering hors invested: manfactrer Cmlative engineering hors invested: spplier Payoffs are in $ billion and engineering hors are in 100,000. Total payoff (combining the manfactrer s and the spplier s together) of the balanced case is mch larger than that of the coercive case, i.e. the single player-dominating case. It is intriging to note that the balanced case total payoff is larger, despite the fact that the total cmlative investment of the balanced case is smaller than that of the coercive case. It implies that the SC partners tilize resorces mch more effectively nder the balanced decision-making strctre than nder the single player-dominating strctre. We infer that when it comes to tilizing its own resorces, each partner in the SC knows better than the other. Under the balanced case, the manfactrer mst increase its investment, while the spplier can redce its spending significantly. When we look at the individal profits, we notice that the spplier s increases significantly on shifting from the coercive to the balanced case, whereas the manfactrer s actally decreases slightly. Here, we can see a potential dilemma faced by the manfactrer. When the manfactrer was exerting its hge bargaining power withot any restraints, it was able to enjoy higher profit, althogh the spply chain as a whole was operating at an inefficient level. In order to make the entire SC more efficient, the manfactrer needs to change its decisionmaking strctre so that a more balanced decision making becomes possible. By doing so, the entire spply chain performs better, bt it appears to be doing that at the cost of the

16 638 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Table 4 Comparison between the two cases [(Balanced case vale Coercive case vale)/coercive case vale]% Total payoff 17 Manfactrer s payoff 6 Spplier s payoff 66 Cmlative investment: total 18 Cmlative investment: manfactrer 67 Cmlative investment: spplier 43 Balanced decision making Single player-dominating Fig. 6. Total payoff. manfactrer s profit. Is there any soltion that can eliminate or at least mitigate sch a dilemma faced by the manfactrer? One possible way is for the SC partners, the manfactrer and its spplier, to agree pon a scheme to share the total profit in a more eqitable manner, e.g. each player s payoff nder the balanced case shold be strictly larger than that nder the coercive case. If they can find a mechanism that measres how mch contribtion was made by each player to enlarging the total profit, the profit-sharing isse nder the balanced case can be mch simpler. Table 4 smmarizes the incremental changes as the SC shifts from a single player-dominating to a balanced decision-making strctre. Figre 6 depicts the dynamics of total payoffs whereas Fig. 7 presents the dynamics of the manfactrer s payoff and Fig. 8 the spplier s. For the manfactrer, it is always better to have a coercive decision-making strctre (of corse, where the manfactrer is the dominant player). On the contrary, for the spplier, it is always better to have a balanced decision-making strctre, where both the manfactrer and the spplier share the decision-making power eqitably. One interesting observation is abot the dynamics of total payoffs in Fig. 6. Althogh the total payoff nder the balanced decision-making strctre becomes mch better than that nder the single player-dominating strctre eventally, it actally remains lower than the coercive payoff almost

17 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) Single player-dominating Balanced decision making Fig. 7. Payoff of the dominating player, i.e. the manfactrer. Balanced decision making Single player-dominating Fig. 8. Payoff of the weaker player, i.e. the spplier ntil the half of the decision-time horizon: that is, the total payoff nder the coercive case is larger than that nder the balanced case in the early period of the decision-time horizon. It is de to this that the balanced decision-making strctre reqires relatively more initial investment from the SC partners, which can be tilized flly in the later period: this kind of early commitment is possible becase the SC partners have trst in each other. As sch, we can infer that if the SC partners have a relatively short decision-time horizon (i.e. a myopic perspective), the balanced decision-making strctre might not be the ideal choice, becase it may generate less total profit than the coercive case does. 6. Managerial implications Or primary research objective in this paper is to nderstand the detailed dynamics of resorce allocation and profit generation nder the two important spply chain decision-making

18 640 B. Kim and J. Kim / Intl. Trans. in Op. Res. 15 (2008) mechanisms, i.e. the single player-dominating (coercive) and the balanced decisionmaking strctre. Essentially, we wold like to answer the research qestion, Where does the benefit of the balanced decision making between spply chain partners come from and how can we analyze its dynamics? by developing optimal control theory models to describe two contrasting cases. By solving the optimal control theory models, we have been able to identify the general sorces and dynamics of inefficiency or efficiency for each of the decision-making strctres. Under the arrangement of a single player-dominating decision making, the dominant partner always consmes the less-powerfl partner s resorces completely before it starts sing its own resorces. This is de to the fact that the less powerfl partner s resorces are essentially free to the dominant player in the spply chain: when something is free, a hman being tends to se it flly withot paying de attention to whether sch sage is efficient or economical from the entire system s perspective. That is, nder the single player-dominating case, it is navoidable to waste resorces to a great extent, and sch resorce waste cases the entire spply chain to perform sboptimally. Under the balanced decision-making strctre, it costs to se not only the dominant partner s resorces bt also the less powerfl player s resorces: in the objective fnction, both the manfactrer s and the spplier s benefits and costs are eqally represented. Or analysis indicates that nder this balanced decision-making strctre, resorce tilization is disciplined in that the decision makers determine how mch of each player s resorces shold be tilized by taking into accont their relative efficiency. Sch disciplined resorce tilization enables the entire spply chain to perform more optimally. To cast or analysis in a more realistic setting, we presented nmerical examples based on the atomobile indstry. Or nmerical examples clearly showed the detailed dynamics of the resorce allocation and profit generation for the two cases. Some of the key reslts are as follows: the total payoff of the balanced case is mch larger than that of the single player-dominating case, despite the fact that the total cmlative investment of the balanced case is smaller, indicating that the SC partners tilize resorces more effectively nder the balanced decision-making strctre, bt the spplier s profit increases significantly on shifting from the coercive to the balanced case, whereas the manfactrer s actally decreases slightly; to fix this dilemma, the manfactrer and its spplier need to agree pon a scheme to share the total profit in a more eqitable manner. If they can find a mechanism that measres how mch contribtion was made by each player to enlarging the total profit, the profit-sharing isse nder the balanced case can be mch simpler. Finding ot sch a mechanism can be a promising area for ftre research. Coordination between spply chain partners becomes an important isse not only from a theoretical standpoint bt also from an empirical perspective. Althogh it seems simple enogh to gess that the performance from close collaboration, i.e. coordinated or balanced decision making, can be higher than withot sch collaboration, research has not been performed extensively to analyze the sorces and dynamics of sch improved performance. The major contribtion made by or paper is that it pinpointed the sorces and dynamics of the benefit from sch a collaboration by developing and solving optimal control theory models. In order to highlight the fndamental natre of sch sorces and dynamics of benefit, we adopted relatively simple models, which can have both merits and demerits, which are potential shortcomings of this research.

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