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Cooperation between multiple news-vendors with transshipments
European Journal of Operational Research 167 (2005) 370–380 www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
Cooperation between multiple news-vendors with transshipments Marco Slikker
a,*
, Jan Fransoo a, Marc Wouters
b
a
b
Department of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Department of Technology Management, University of Twente, P.O. Box 217, 7522 NB Enschede, The Netherlands Received 27 February 2002; accepted 17 March 2004 Available online 17 June 2004
Abstract We study a situation with n retailers, each of them facing a news-vendor problem, i.e., selling to customers over a finite period of time (product with a short life cycle, such as fashion). Groups of retailers might improve their expected joint profit by coordinating their orders, followed by transshipments after demand realization is known. We analyze these situations by defining a cooperative game, called a general news-vendor game, for such a situation with n retailers. We concentrate on whether it makes sense to cooperate by studying properties of general news-vendor games. Our main result states that general news-vendor games have non-empty cores. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Game theory; News-vendor; Transshipments; Balancedness
1. Introduction Consider a group of retailers who buy from the same supplier and who have to place their purchase orders well in advance of receiving customer orders. For example, imagine a supplier located in Asia and retailers located in Europe who have to place orders for a seasonal product. They have to
*
Corresponding author. Tel.: +31-40-247-3940; fax: +31-40246-5949. E-mail addresses: [email protected] (M. Slikker), J.C. [email protected] (J. Fransoo), M.J.F. Wouters@sms. utwente.nl (M. Wouters).
place their orders in advance of knowing actual demand to cover the manufacturing and transportation lead time. After all orders have been received, the supplier has to make a release decision on how much to produce in total (i.e., for all retailers). The retailers might improve their joint expected profit, for example when it is possible to postpone the allocation decision (i.e., which portion of the quantity manufactured to allocate to each of the retailers). This raises the question of whether in such a setting it is always beneficial for companies to cooperate and to order jointly. In this paper we use cooperative game theory to analyze this question. We look at the benefits for the total supply chain and at benefits for the
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.03.014
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individual companies, assuming that under cooperation demand information is shared and orders are based on combined demand. Companies may benefit even more from cooperation by not sharing all information. For example, they may have private information about actual demand as it materializes after placing the order and they may use this information to their own benefit. In case of shortages, they order more than they really need while expecting to actually receive what they anticipate to need (shortage gaming, see Lee et al. (1997)). See, for example, Cachon and Lariviere (1999) for a model in a news-vendor setting where retailers have private information about demand and retailers can influence their allocation through their orders. Cachon and Lariviere (1999) investigate allocation rules that maximize expected total profit and are attractive for the individual companies. They show that supply chains may not benefit from allocation rules that lead retailers to tell the truth (i.e., order exactly their needs). We refer to Cachon (1999) for a review and analysis of non-cooperative game theory in supply chain settings. Further, single period supply chain ordering (usually referred to as contracts), possibly consisting of two consecutive decisions (initial order-reallocation) is reviewed by Tsay et al. (1999). Yet, strategic behavior with non-cooperative gaming is outside the scope of this paper. This paper contributes to the literature that applies cooperative game theory to Operations Research problems. Borm et al. (2001) provide a recent survey of cooperative games associated with operations research games, in which five types of underlying OR-problems are distinguished, one of them being ‘inventory’. We consider a general news-vendor situation in a supply chain consisting of a single supplier (wholesaler) and n retailers. The retailers order the same product from a single supplier and resell the product to consumers. Each retailer i orders a quantity qi at the supplier, who in its turn orders a quantity q at the manufacturer of the goods. Every retailer experiences stochastic demand and realization of demand is not known at the moment of ordering. News-vendor models are single period models, which means that inventory is not carried
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over to another period. Furthermore, any remaining products at the end of the period can be disposed of at a certain expense, or can be sold at a lower price than the market price. Initially, this type of modeling was applied to products with very high perishability, such as newspapers. Later, especially in the fashion industry, news-vendor models were proven to be of use for short life cycle production (see Fisher and Raman (1996) who study the single period setting in the fashion industry), and following the decrease of product life cycles in the high-tech industry, such as personal computers and mobile phones, news-vendor models are now well-accepted to model ordering decisions in these environments (see, e.g., Tayur et al. (1999), for a series of papers using this setting). This means that the news-vendor setting studied in this paper has been widely accepted as one of practical relevance. In the news-vendor situation discussed above, retailers can increase total expected profits (the sum of expected profits of all parties in the supply chain) by combining their orders. Besides price effects, this is because if some companies have ordered more and others have ordered less than they can sell, products are transferred between these companies. This builds on the traditional news-vendor problems (see, e.g., Silver et al. (1998) and many other textbooks; see also Khouja (1999) for a review). Although ordering jointly is collectively rational, the feasibility of such an arrangement depends on whether the expected profits of individual companies increase as a result of cooperation. It has been investigated in several studies whether it makes sense for companies to cooperate. Gerchak and Gupta (1991) compare four simple allocation rules in a continuous review single period inventory model with complete backordering. They show that individual stores may be unhappy. Robinson (1993) reexamines their results in terms of the core and subsequently studies the Shapley value (cf. Shapley, 1953) for these games and an alternative allocation rule for games with a large number of retailers. Hartman and Dror (1996) formulate three criteria for allocation rules in this setting: non-emptiness of the core, computational ease, and justifiability. This last criterion demands the existence of an appealing dual
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allocation rule in case the situation is analyzed in terms of profits rather than in terms of costs. Their results mainly deal with the existence of justifiable pairs, i.e., the analysis of an allocation rule and its dual. Recently, an increasing amount of attention is paid to coopetitive models in which ordering decisions of different retailers are assumed to be made independently/competitively, while allocation or transshipment decisions are assumed to take place cooperatively. We mention Rudi et al. (2001), Anupindi et al. (2001), and Granot and Sosic (2003). They mainly focus on Nash equilibria for the ordering decisions and investigate whether joint optimal solutions can be obtained in equilibrium. The cooperative behavior in the last stage is modeled by assuming or requiring that a coreelement is attained (for this stage only). Finally, we mention Meca-Martınez et al. (1999) who consider cooperation within the field of inventory management. They analyze cooperative games that depend on the information that is revealed by the companies and then focus on proportional costallocation mechanisms. Of specific interest is the work of Hartman et al. (2000) and M€ uller et al. (2002). They consider newsvendor games in which the value of a group of retailers is assumed to be their optimal profit if they coordinate their orders. Hartman et al. (2000) remark that Hartman and Dror (1997) show by means of a simple 3-person example that news-vendor games are not necessarily convex. Furthermore, Hartman et al. (2000) prove that news-vendor games have a non-empty core for specific demand distributions of the retailers. A non-empty core means that no group of players has an incentive not to cooperate, because with cooperation there exists a division of the joint profit that gives each group of players at least as much as they can obtain for themselves. Hartman et al. (2000) end with an open question, which can be reformulated as follows: ‘Do news-vendor games have a nonempty core?’. M€ uller et al. (2002) answer this question affirmatively, restricted to anonymous wholesale and customer prices. 1
1 We remark that the same result was independently derived by Slikker et al. (2001).
Several generalizations, starting from the work of Hartman et al. (2000) and M€ uller et al. (2002), seem relevant. First, the prices that companies pay to the producer and the customers pay to the companies do not need to be identical. Even for identical products, some companies may have more power to negotiate lower prices, for example because they buy many other products from the same supplier, and some companies may be able to charge higher prices to customers, for example because they sell through outlets that have a wider product assortment, better service, nicer stores, and a more expensive image. Furthermore, a reason for different prices may be that the retailers do not order exactly the same product, but products with slight differences, for example regarding packaging and labels. This requires having separate inventories per retailer. It also means that if the product of one retailer is going to be sold by another retailer, some modification might be involved. This leads to a second generalization that is relevant, namely that transshipment costs might be involved when there are separate inventories per retailer and products destined for one retailer are sold by another retailer. Transshipment costs might involve handling and transportation activities if products are identical, but they could also involve costs related to modifying products that have some characteristics that are specific to one retailer, for example repackaging and relabeling. Thirdly, the existing models could be made more general by relaxing the assumption that actual demand is known with certainty at the point in time at which the joint order has to be allocated among the retailers. For each retailer, demand might at that point in time better be modeled by a stochastic variable with less uncertainty (e.g., lower standard deviation) than at the first point in time, instead with no uncertainty. A more fundamental generalization concerns relaxing the assumption that retailers are risk-neutral. This demands an application of stochastic cooperative game theory. For a survey of research in this area we refer to Suijs (1999). Furthermore, strategic behavior of the players could be taken into account. In the models used in the literature, at two moments information is needed to make optimal decisions: At the point in time where the quantity
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
of the joint order is determined and at the moment that retailers have to reveal their actual demand, i.e., the point in time at which their actual demand is known. The events at the second point in time can be modeled as a process of allocation or reallocation. Here, the work of Klaus (1998) can be a guidance. In this paper, we formulate a coalitional game associated with a general news-vendor situation. This coalitional game corresponds essentially to the inventory centralization game introduced by Hartman et al. (2000) and M€ uller et al. (2002). The main difference being that they concentrate on costs of a coalition, whereas we concentrate on profits. We include the first two generalizations mentioned above. Extending the model of Hartman et al. (2000) and M€ uller et al. (2002), we allow that different retailers may experience different wholesale and/or customer prices. We also introduce the possibility of transshipments and make assumptions about costs associated with these. After some remarks about convexity we show that any general news-vendor game has a non-empty core. The plan of this paper is as follows. We start in Section 2 with some preliminaries on cooperative game theory to make this paper self-contained. Then, in Section 3, we introduce general newsvendor situations and associated cooperative games with these situations. These so-called general news-vendor games are studied in Section 4, which concentrates on balancedness. We conclude in Section 5 with some final remarks and discussion.
2. Preliminaries cooperative game theory For reasons of self-containedness, we here give a brief introduction to cooperative game theory, based on the work by Slikker and van den Nouweland (2001). Cooperative game theory covers the problem setting where different parties cooperate to reach a common goal. Let us assume we have n different companies pursuing similar objectives. In the setting studied in this paper, these companies are retailers possibly ordering jointly in order to reach a better overall service level to serve a set of independent markets. In game theory, these companies are referred to as
373
players, with N the set of players. For convenience, we number the players such that the player set is N ¼ f1; 2; . . . ; ng. A subset of N is called a coalition and is denoted by S. We are now interested in various coalitions S, and particularly to what extent a specific coalition can reach the common objective without the players who are not part of the coalition. We assume that the benefits of the cooperation within the coalition S are transferable between the players of S, and denote the benefits of the cooperation as vðSÞ. In the example of the retailers described above, this can be defined as the reduced cost of ordering jointly, or as the increased profits of reaching a higher service level. The function v that assigns to every coalition S N its value or worth vðSÞ, with vð;Þ ¼ 0, is commonly referred to as the characteristic function. A pair ðN ; vÞ consisting of a player set N and a characteristic function v constitutes a cooperative game or coalitional game. For notational convenience we will sometimes write simply game rather than coalitional game or cooperative game. In our analysis, we sometimes focus on only a few of the players involved in a game ðN ; vÞ. For a coalition S N , vjS denotes the restriction of the characteristic function v to the player set S, i.e., vjS ðT Þ ¼ vðT Þ for each coalition T S. The pair ðS; vjS Þ is a game with player set S that is then closely related to the game ðN ; vÞ. Two interesting properties that a game might satisfy are superadditivity and monotonicity. A game (or its characteristic function) is called superadditive if for any two disjoint coalitions S and T of players it holds that vðSÞ þ vðT Þ 6 vðS [ T Þ. An important consequence of a superadditive characteristic function, is that it is always attractive for these two disjoint coalitions to form one big coalition S [ T rather than operating separately. A game is called monotonic if the addition of more players will increase the value obtainable. Formally, ðN ; vÞ is called monotonic if vðSÞ 6 vðT Þ for all S T . Finally, we remark that a game is called additive if for any two disjoint coalitions S and T of players it holds that vðSÞ þ vðT Þ ¼ vðS [ T Þ. In line with any reality involving individual companies and increasingly with papers in the
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Operations Management literature (e.g., Cachon, 1999; Fransoo et al., 2001), players are not primarily interested in the benefits of a coalition, but in the individual benefits they are getting out of the coalition. The allocation is represented as a payoff vector x ¼ ðxi Þi2N 2 RN , specifying for each player i 2 N the benefit (e.g., (extra) profit) xi that this player can expect if he cooperates with the other players. An allocation is called efficient if the payoffs to the various players add up to exactly vðN Þ. The set consisting of all efficient allocations P is the set fx 2 RN j i2N xi ¼ vðN Þg. Note that not all these allocations will be acceptable to the players, as each player will require that he gets at least as much as what he can obtain when staying alone. An allocation x 2 RN with the property that xi P vðiÞ for all i 2 N is called individually rational. The set of all individually rational and efficient allocations is the imputation set IðN ; vÞ ¼ fx 2 RN j P i2N xi ¼ vðN Þ and xi P vðiÞ for each i 2 N g. This type of rationality requirement can be extended to all coalitions, not just individual players, to N P obtain the core CðN ; vÞ ¼ fx 2 R j i2N xi ¼ vðN Þ P and i2S xi P vðSÞ for all S N g. 2 The interpretation of the core is that it consists of all imputations that are such that no group of players has an incentive to split off from the grand coalition N and form a smaller coalition S because they collectively receive at least as much as what they can obtain for themselves as a coalition. A second interpretation of the core is that no group of players gets more than what they collectively add to the value obtainable by the grand coalition N . This follows since forPeach x 2PCðN ; vÞ P and S N it holds that x ¼ x i2S i i2N i i2N nS xi 6 vðN Þ vðN n SÞ. Bondareva (1963) and Shapley (1967) independently identified the class of games that have non-empty cores as the class of balanced games. To describe this class, we define for all S N the vector eS by eSi ¼ 1 for all i 2 S and eSi ¼ 0 for all i 2 N n S. A map jP: 2N n f;g ! ½0; 1 is S N called a balanced map if S22N nf;g jðSÞe ¼ e . Further, a game ðN ; vÞ is called balanced if for every balanced map j : 2N n f;g ! ½0; 1. it holds
2
We define the empty sum to be equal to 0.
P The following that S22N nf;g jðSÞvðSÞ 6 vðN Þ. theorem is due to Bondareva (1963) and Shapley (1967). Theorem 2.1. Let ðN ; vÞ be a coalitional game. Then CðN ; vÞ 6¼ ; if and only if ðN ; vÞ is balanced. A coalitional game ðN ; vÞ is called totally balanced if it is balanced and each of its subgames is balanced as well. The last notion we wish to introduce is the notion of convexity. A coalitional game is convex if a player’s marginal contribution increases if he joins a larger coalition. Formally, coalitional game ðN ; vÞ is convex if for each i 2 N and for all S T N n fig it holds that vðS [ iÞ vðSÞ 6 vðT [ iÞ vðT Þ. We remark that a convex coalitional game has a non-empty core.
3. The model In this section we introduce general news-vendor situations. These are situations where several companies order the same good from a producer. This good can be bought by company i at (wholesale) price ci and subsequently sold to customers at customer price pi . Every company experiences a stochastic demand. As in the standard news-vendor problem it is assumed that the realization of this stochastic demand is not known at the moment of ordering. If several companies cooperate they can, after realizations of demand are known, transship goods. Unit costs for transshipping goods from i to j are denoted by tij . A general news-vendor situation is a tuple ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ, where N denotes a set of companies and Xi the stochastic demand for the good at company i 2 N . Furthermore, ci and pi denote the prices that companies pay to the producer and the customers pay to the companies, respectively, and tij denotes transshipment costs. Throughout this work we assume that pi > ci > 0 for all i 2 N , that tij P 0 for all i, j 2 N , and that each company has a finite expected demand. A general news-vendor situation with one company only corresponds to the standard newsvendor problem.
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
This model extends the model of Hartman et al. (2000), who restrict themselves to anonymous wholesale prices (ci ¼ c for all i 2 N ), anonymous customer prices (pi ¼ p for all i 2 N ), and free transshipments (tij ¼ 0 for all i, j 2 N ). Consider a general news-vendor situation and a collection of retailers S who will jointly determine an order vector qS 2 QS :¼ fq 2 RN jqSi ¼ 0 for all i 2 N n S and qSi P 0 for all i 2 Sg: Suppose coalition S has ordered vector qS 2 QS and subsequently, they face demand vector xS 2 RN with xSi ¼ 0 for all i 2 N n S. A reallocation of qS is a matrix S
S
S
S
RNþN jASij
¼ 0 if A 2 M ðq Þ :¼ fA 2 X S S Aij ¼ qi for all i 2 Sg: i 62 S or j 62 S; j2S
ASij
denotes the amount of products that are Here, transshipped from retailer i to retailer j. For j ¼ i this denotes the amount that is not transshipped. Hence, the total order of a retailer has to be equal to the sum of the transshipments from this retailer. Of course no transshipments with retailers outside S are allowed, i.e., they are equal to 0. The following lemma states that an optimal reallocation exists. Lemma 3.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation, let S N , let qS 2 QS , and let xS be a demand realization vector. Then there exist a reallocation AS 2 M S ðqS Þ that maximizes the expected profit of coalition S. Proof. Let hS ðAS ; qS ; xS Þ be the total revenue minus total transshipment costs if qS is rescheduled according to AS 2 M SP ðqS Þ. Since,PhS ðAS ; qS ; xS Þ P P S S S ¼ i2N j2N Aij tij þ j2S pj minf i2S Aij ; xj g, S S S we conclude that h ð:; q ; x Þ is a continuous function. Since it is defined on M S ðqS Þ, which is compact, we conclude that hS ð:; qS ; xS Þ attains its maximum. If we denote an optimal reallocation of this lemma by AS; then the optimal profit of coalition
375
S with order qS and demand realization xS is described by X pS ðqS ; xS Þ ¼ qSi ci þ hS ðAS; ; qS ; xS Þ: i2S
From now on, we will refer to hS ðAS; ; qS ; xS Þ as rS ðqS ; xS Þ. The expected profit of coalition S depends on their order vector qS and the demand faced by each retailer. We introduce for any vector of demands ðXi Þi2N and any S N the vector of demands X S , where ðX S Þi ¼ Xi for all i 2 S and ðX S Þi ¼ 0 for all i 2 N n S. We will write XiS instead of ðX S Þi . The expected profit of a coalition S that ordered qS and faces stochastic demand vector X S is then given by S ðqS ; X S Þ ¼ E½pS ðqS ; X S Þ: p
ð1Þ
Furthermore, we denote the expected profit that abstains from the initial costs by RS ðqS ; X S Þ ¼ E½rS ðqS ; X S Þ: ð2Þ P S S S S S S S ðq ; X Þ ¼ i2S qi ci þ R ðq ; X Þ. Hence, p A coalition may achieve higher expected profits than the sum of the expected profits of the individual companies. Hence, retailers may have an incentive to cooperate. We will construct a coalitional game with each general news-vendor situation. In this game, the value of a coalition is the maximum expected profit this coalition can obtain if they order jointly. Formally, let C ¼ ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. The associated general news-vendor game ðN ; vC Þ is defined by S ðq; X S Þ for all S N ; vC ðSÞ ¼ max p q2QS
ð3Þ
S are as defined above. 3 In the where X S and p following theorem we will show that each coalition
3 Hartman et al. (2000) and M€ uller et al. (2002) consider for the setting of anonymous wholesale prices ðcÞ, anonymous customer prices ðpÞ and free transshipments the cost game ðN ; cC Þ with cC ðSÞ ¼ ðp cÞE½XS vC ðSÞ for all S N . Additionally, they have a slightly different notation by focusing on oversupplying costs ðcÞ and undersupplying costs ðp cÞ. Since P ðp cÞE½XS ¼ i2S ðp cÞE½Xi , i.e., to get cC , an additive game is added to vC , we have that non-emptiness of the core in our general news-vendor game corresponds to non-emptiness of the (anti-)core in their cost game.
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has an optimal order vector, implying that the general news-vendor game is well-defined. Theorem 3.1. Let C ¼ ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation and let S N . Then there exists an order vector qS; that maximizes the expected profit of coalition S. Proof. We will show that the expected profit S ð:; XS Þ is a continuous function of the order p vector. Additionally we will show that any order outside a specific compact set results in lower expected profits than an order equal to zero. Hence, coalition S optimizes a continuous function over a compact set, which implies that an optimal order vector exists. First, we prove that the optimal profit is a continuous function of the order vector. Therefore, let q 2 QS denote an order vector and let > 0. Define cS ¼ maxi2S ðci ; pS Þ ¼ maxi2S pi , d ¼ jSjcSþpS . Let q0 2 QS with jq q0 j < d, where j j denotes the Euclidean norm. Let qmax and qmin be defined by ¼ maxfqi ; q0i g for all i 2 N ; qmax i min qi ¼ minfqi ; q0i g for all i 2 N : Using the triangle inequality we have for any realization xS , jpS ðq; xS Þ pS ðq0 ; xS Þj X jqi q0i j þ jrS ðq; xS Þ rS ðq0 ; xS Þj: 6 cS
ð4Þ
i2S
Hence, for the expected profit functions we have S ðq0 ; X S Þj j pS ðq; X S Þ p X jqi q0i j þ jRS ðq; X S Þ RS ðq0 ; X S Þj 6 cS i2S
6 cS
X
jqi q0i j þ jRS ðqmax ; X S Þ RS ðqmin ; X S Þj
i2S
6
X
! jqi
q0i j
ðcS þ pS Þ
i2S
6 jq q0 jjSjðcS þ pS Þ < d jSjðcS þ pS Þ ¼ : The first inequality follows by taking expectations over the inequality in (4). The second holds since RS ðqmax ; X S Þ P RS ðq ; X S Þ for q 2 fq; q0 g and re-
verse inequalities for RS ðqmin ; X S Þ. The third inequality holds since RS ðqmin ; XPS Þ 6 RS ðqmax ; X S Þ P max max pP qmin and qmin S i j i j ¼ i2S jqi i2S jqi 0 jq q j. The fourth inequality holds since i i i2S is jq q0 j P jqi q0i j for i 2 S. We conclude that p a continuous function of the order vector. It remains to show that any order vector outside a specific compact set results in lower expected profits than an order equal to zero. Let " # X maxi2S pi X S qm ¼ E½Xi þ E½Xi ; ð5Þ mini2S ci i2S i2S where Xi ¼ maxfXi ; 0g. 4 Then for all q 2 QS with qi > qSm for some i 2 S, say j, we have X ðq; X S Þ 6 qj min ci þ max ðpi Þ p E½Xi i2S
< max ðpi Þ i2S
i2S
X
i2S
S ð0; X S Þ: E½Xi 6 p
i2S
The first inequality follows by taking only part of the wholesale costs and an upperbound of the remaining revenues into consideration. The second inequality follows by definition of qSm . The last inequality follows immediately. Hence, we can restrict attention to order vectors in the compact set fqj0 6 qi 6 qSm for all i 2 Sg. This completes the proof. If S consists of a single retailer or if all retailers in S experience the same wholesale price, the same customer price, and free transshipments, then the determination of an optimal order vector boils down to the determination of a total order (distribution over i 2 S is irrelevant). This, in turn, comes down to a standard P news-vendor problem with stochastic demand i2S Xi , wholesale price c ð¼ ci for all i 2 SÞ and customer price p ð¼ pi for all i 2 SÞ. It is well known that an optimal order size is then given by the ð1 pcÞ-quantile, which need not be a unique order size. For a normal distribution Normðl; rÞ this implies that the optimal order quantity is given by l þ rzc=p ,
4 The introduction of Xi is superfluous if only non-negative demands are possible.
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
where za is defined as the unique real number such that P ðX P za Þ ¼ a if X is a stochastic variable with the standard normal distribution Normð0; 1Þ. A general news-vendor situation with anonymous wholesale and customer prices (ci ¼ c and pi ¼ p for all i 2 N ) and free transshipments is called a news-vendor situation with associated news-vendor game. Consider the following example, which illustrates some of the concepts introduced above. For notational and computational purposes the specifications have been kept very simple. This does not hamper the illustration of the concepts. Example 3.1. Consider the 2-person news-vendor situation ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ with N ¼ f1; 2g, c1 ¼ c2 ¼ 1, p1 ¼ p2 ¼ 16, free transshipments, and a stochastic demand X1 for company 1 described by probability mass function 8 > < 0 if x 62 f0; 1g; ð6Þ p1 ðxÞ ¼ 78 if x ¼ 0; > :1 if x ¼ 1; 8 while company 2 faces a demand with the same distribution that is independent of the demand of company 1. It is easily verified that ð1 pc11 Þ-quantile of the distribution of X1 equals 1, implying that the optimal order size of coalition {1} equals 1, resulting in expected profit f1g ð1; X1 Þ ¼ 1 1 þ p
1 1 16 ¼ 1 þ 2 ¼ 1: 8
Obviously, similar calculations result in the optimal expected profit for coalition 2. The optimal order for coalition f1; 2g equals 1 as well. Since transshipments are free, any vector q 2 RNþ with q1 þ q2 ¼ 1 results in an optimal profit for the grand coalition, equal to f1;2g ðq; X f1;2g Þ ¼ 1 1 þ p
15 3 1 16 ¼ 2 : 64 4
By cooperating, the companies can increase their joint profit from 2 to 2 34. This is due to the fact that they can reduce the total amount they order significantly, while this results in a small increase in expected lost sales only. The associated news-vendor game ðN ; vC Þ is described by
8 > <0 vC ðSÞ ¼ 1 > : 3 24
377
if S ¼ ;; if jSj ¼ 1; if S ¼ N :
Note that this news-vendor game is balanced, i.e., it has a non-empty core, since any x 2 R2 with x1 P 1, x2 P 1 and x1 þ x2 ¼ 2 34 belongs to the core, for example x ¼ ð118 ; 118Þ. Moreover, ðN ; vC Þ is convex. In the following section we will focus on balancedness of general news-vendor games. M€ uller et al. (2002) proved balancedness for anonymous wholesale and customer prices and free transshipments. Furthermore, Slikker et al. (2001) proved the same results independently, and showed additionally that if, on top of the restrictions of M€ uller et al. (2002) we assume that the demand distributions have a normal distribution and that the demands are independent then the games are convex. If anonymity or independence is dropped or if discrete rather than normal demand is considered, convexity is no longer guaranteed.
4. Balancedness In this section we show that any general newsvendor game has a non-empty core. In the process of proving that general newsvendor games are balanced, we will use the following lemma. Lemma 4.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. Let j be an associated balanced map. Denote an optimal order vector of coalition S N by qS; . Then X
N
p
jðSÞq ;
SN :S6¼;
P
S;
X
X
! jðSÞX
S
SN :S6¼;
S jðSÞqS; ; jðSÞX S : p
SN :S6¼;
Proof. First, note that total wholesale costs are the same. Hence, it suffices to show that
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R
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X
N
S;
jðSÞq ;
SN :S6¼;
P
X
X
! jðSÞX
Theorem 4.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. The associated news-vendor game is balanced.
S
SN :S6¼;
R jðSÞqS; ; jðSÞX S : S
SN :S6¼;
Let xN be a realization of demand for the grand coalition and let xS be the associated demand of any coalition S N . As before, denote a reallocation of qS; that maximizes the profit of coalition S; S by AS; . Denote the part is actually Pof A that S; S; N sold by A . Then B ¼ SN jðSÞA denotes a possible selling part of a reallocation of the grand coalition associated with order vector P S; SN :S6¼; jðSÞq . Denote the sold part of an optimal reallocation of N associated with this order vector by C N . Then ! X X N S; S r jðSÞq ; jðSÞx SN :S6¼;
XX
¼
i2N
P
SN :S6¼;
CijN ðpj
XX i2N
BNij ðpj tij Þ
j2N
XXX
¼
i2N
SN i2N
X
¼
S;
jðSÞA ðpj tij Þ
N ðqN ; X N Þ P p N ðzN ; X N Þ vðN Þ ¼ p ! X X N S S ¼p jðSÞq ; jðSÞX
j2N SN
XXX
¼
tij Þ
j2N
S;
jðSÞAij ðpj tij Þ
j2N
S
r ðjðSÞqS; ; jðSÞxS Þ;
ð7Þ
SN S;
where the last equality holds since jðSÞA is obviously the sold part of an optimal reallocation for coalition S and demand realization jðSÞxS . Combining equal wholesale costs with (7) gives ! X X jðSÞqS; ; jðSÞxS pN SN :S6¼;
P
X
SN :S6¼; XSN :S6¼; S S ðjðSÞq ; jðSÞX S Þ P p SN :S6¼;
¼
X
SN :S6¼;
jðSÞ pS ðqS ; X S Þ ¼
X
jðSÞvðSÞ:
SN :S6¼;
The second inequality holds by Lemma 4.1. Hence, the cooperative game associated with the general news-vendor situation is balanced, i.e., it has a non-empty core.
SN :S6¼;
p jðSÞqS; ; jðSÞxS : S
Proof. Let j : 2N n f;g ! ½0; 1 be a balanced map. Recall that X N ¼ ðXi Þi2N Pand, for all i 2 N , note that P S jðSÞX ¼ X where i i SN :S6¼; SN :i2S jðSÞ ¼ Xi , the last equality follows since j is a balanced map. Let ðqS ÞSN :S6¼; be optimal ordervectors for the different coalitions. Then zN defined by zNi ¼ P jðSÞ qSi for all i 2 N denotes a possible SN :i2S ordervector for coalition N , while qN is the optimal N ðqN ; X N Þ P ordervector. Hence, we know that p N N N ðz ; X Þ. Furthermore, we know that for all p S N with S 6¼ ; it holds that jðSÞqS maximizes S ð:; jðSÞX S Þ since p S ðkq; kX Þ ¼ k profit p pS ðq; X Þ for any stochastic demand vector X of coalition S and any ordervector q of coalition S. This implies that jðSÞqS is an optimal ordervector for stochastic demand vector jðSÞX S if and only if qS is an optimal ordervector for stochastic demand X S . Furthermore, it implies that if the stochastic demand vector changes by some factor then the optimal expected profit changes by the same factor. Using this, we find
SN :S6¼;
Since this inequality holds for any realization xN of X N , taking expectations over the distribution of X N completes the proof. The following theorem states that general newsvendor games are balanced.
The result in Theorem 4.1 can be strengthened by noting that every subgame of a general newsvendor game is a general news-vendor game itself. Hence, the following corollary follows immediately from Theorem 4.1. Corollary 4.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. Then the associated news-vendor game is totally balanced.
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5. Remarks and discussion This work provides a further contribution to the game-theoretical analysis of situations in which retailers can coordinate their orders. In Section 1 several extensions to the work of M€ uller et al. (2002) and Slikker et al. (2001) were proposed. Out of these, the inclusion of transshipments into the model seemed to be the most interesting generalization. We showed that stable divisions of expected profits also exist if transshipment cost are positive. Furthermore, we showed that in case of different retail and wholesale prices rather than anonymous prices, stable divisions of expected profits exist as well. These two extensions of the earlier results by M€ uller et al. (2002) and Slikker et al. (2001) provide significant progress in the analysis of multiple news-vendor games. The next step would be to cover the issues that are mentioned in the introduction, but are not covered in this paper. Firstly, one could relax the assumption that actual demand is known at the moment the joint order is divided or at the moment at which transshipment decisions have to be made. Other research questions deal with a cooperative analysis from a stochastic game-theoretical point of view and with a non-cooperative analysis of the process of allocation and reallocation, following the works of Suijs (1999) and Klaus (1998) respectively.
Acknowledgements The authors thank Henk Norde and others for useful suggestions and comments.
References Anupindi, R., Bassok, Y., Zemel, E., 2001. A general framework for the study of decentralized distribution systems. Manufacturing and Service Operations Management 3, 349–368. Bondareva, O., 1963. Certain applications of the methods of linear programming to the theory of cooperative games. Problemy Kibernetiki 10, 119–139 (In Russian). Borm, P., Hamers, H., Hendrickx, R., 2001. Operations research games: A survey. TOP 9, 139–198.
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Cachon, G., 1999. Competitive supply chain inventory management. In: Tayur, S., Ganeshan, R., Magazine, M. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston. Cachon, G., Lariviere, M., 1999. Capacity choice and allocation: Strategic behavior and supply chain performance. Management Science 45, 1091–1108. Fisher, M., Raman, A., 1996. Reducing the cost of demand uncertainty through accurate response to early sales. Operations Research 44, 87–99. Fransoo, J., Wouters, M., de Kok, T., 2001. Multi-echelon multi-company inventory planning with limited information exchange. Journal of the Operational Research Society 130, 830–838. Gerchak, Y., Gupta, D., 1991. Technical note: On apportioning costs to customers in centralized continuous review inventory systems. Journal of Operations Management 10, 546– 551. Granot, D., Sosic, G., 2003. A three stage model for a decentralized distribution system of retailers. Operations Research 51 (5), 771–784. Hartman, B., Dror, M., 1996. Cost allocation in continuous review inventory models. Naval Research Logistics Quarterly 43, 549–561. Hartman, B., Dror, M. 1997. Inventory centralization costs with varying cost parameters. Working paper, MIS Department, University of Arizona. Hartman, B., Dror, M., Shaked, M., 2000. Cores of inventory centralization games. Games and Economic Behavior 31, 26–49. Khouja, M., 1999. The single-period (news-vendor) problem: Literature review and suggestions for future research. Omega 27, 537–553. Klaus, B. 1998. Fair allocation and reallocation: An axiomatic study. PhD thesis, Maastricht University, Maastricht, The Netherlands. Lee, H., Padmanabhan, P., Wang, S., 1997. Information distortion in a supply chain: The bullwhip effect. Management Science 43, 546–558. Meca-Martınez, A., Timmer, J., Garcıa-Jurado, I., Borm, P. 1999. Inventory games. CentER Discussion Paper 9953, Tilburg University, Tilburg, The Netherlands. M€ uller, A., Scarsini, M., Shaked, M., 2002. The newsvendor game has a non-empty core. Games and Economic Behavior 38, 118–126. Robinson, L., 1993. A comment on Gerchak and Gupta’s: On apportioning costs to customers in centralized continuous review inventory systems. Journal of Operations Management 11, 99–102. Rudi, N., Kapur, S., Pyke, D., 2001. A two-location inventory model with transshipment and local decision making. Management Science 47, 1668–1680. Shapley, L., 1953. A value for n-person games. In: Tucker, A., Kuhn, H. (Eds.), Contributions to the Theory of Games II. Princeton University Press, Princeton, NJ, pp. 307–317. Shapley, L., 1967. On balanced sets and cores. Naval Research Logistics Quarterly 14, 453–460.
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Silver, E., Pyke, D., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling. Wiley, New York. Slikker, M., van den Nouweland, A., 2001. Social and Economic Networks in Cooperative Game Theory. Kluwer Academic Publishers, Boston. Slikker, M., Fransoo, J., Wouters, M. 2001. Joint ordering in multiple news-vendor situations: A game-theoretical approach. Working paper, Eindhoven University of Technology, Eindhoven, The Netherlands.
Suijs, J., 1999. Cooperative Decision-Making under Risk. Kluwer Academic Publishers, Boston. Tayur, S., Ganeshan, R., Magazine, M. (Eds.), 1999. Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston. Tsay, A., Nahmias, S., Agrawal, N., 1999. Modeling supply chain contracts: A review. In: Tayur, S., Ganeshan, R., Magazine, M. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston.
Production, Manufacturing and Logistics
Cooperation between multiple news-vendors with transshipments Marco Slikker
a,*
, Jan Fransoo a, Marc Wouters
b
a
b
Department of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Department of Technology Management, University of Twente, P.O. Box 217, 7522 NB Enschede, The Netherlands Received 27 February 2002; accepted 17 March 2004 Available online 17 June 2004
Abstract We study a situation with n retailers, each of them facing a news-vendor problem, i.e., selling to customers over a finite period of time (product with a short life cycle, such as fashion). Groups of retailers might improve their expected joint profit by coordinating their orders, followed by transshipments after demand realization is known. We analyze these situations by defining a cooperative game, called a general news-vendor game, for such a situation with n retailers. We concentrate on whether it makes sense to cooperate by studying properties of general news-vendor games. Our main result states that general news-vendor games have non-empty cores. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Game theory; News-vendor; Transshipments; Balancedness
1. Introduction Consider a group of retailers who buy from the same supplier and who have to place their purchase orders well in advance of receiving customer orders. For example, imagine a supplier located in Asia and retailers located in Europe who have to place orders for a seasonal product. They have to
*
Corresponding author. Tel.: +31-40-247-3940; fax: +31-40246-5949. E-mail addresses: [email protected] (M. Slikker), J.C. [email protected] (J. Fransoo), M.J.F. Wouters@sms. utwente.nl (M. Wouters).
place their orders in advance of knowing actual demand to cover the manufacturing and transportation lead time. After all orders have been received, the supplier has to make a release decision on how much to produce in total (i.e., for all retailers). The retailers might improve their joint expected profit, for example when it is possible to postpone the allocation decision (i.e., which portion of the quantity manufactured to allocate to each of the retailers). This raises the question of whether in such a setting it is always beneficial for companies to cooperate and to order jointly. In this paper we use cooperative game theory to analyze this question. We look at the benefits for the total supply chain and at benefits for the
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.03.014
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individual companies, assuming that under cooperation demand information is shared and orders are based on combined demand. Companies may benefit even more from cooperation by not sharing all information. For example, they may have private information about actual demand as it materializes after placing the order and they may use this information to their own benefit. In case of shortages, they order more than they really need while expecting to actually receive what they anticipate to need (shortage gaming, see Lee et al. (1997)). See, for example, Cachon and Lariviere (1999) for a model in a news-vendor setting where retailers have private information about demand and retailers can influence their allocation through their orders. Cachon and Lariviere (1999) investigate allocation rules that maximize expected total profit and are attractive for the individual companies. They show that supply chains may not benefit from allocation rules that lead retailers to tell the truth (i.e., order exactly their needs). We refer to Cachon (1999) for a review and analysis of non-cooperative game theory in supply chain settings. Further, single period supply chain ordering (usually referred to as contracts), possibly consisting of two consecutive decisions (initial order-reallocation) is reviewed by Tsay et al. (1999). Yet, strategic behavior with non-cooperative gaming is outside the scope of this paper. This paper contributes to the literature that applies cooperative game theory to Operations Research problems. Borm et al. (2001) provide a recent survey of cooperative games associated with operations research games, in which five types of underlying OR-problems are distinguished, one of them being ‘inventory’. We consider a general news-vendor situation in a supply chain consisting of a single supplier (wholesaler) and n retailers. The retailers order the same product from a single supplier and resell the product to consumers. Each retailer i orders a quantity qi at the supplier, who in its turn orders a quantity q at the manufacturer of the goods. Every retailer experiences stochastic demand and realization of demand is not known at the moment of ordering. News-vendor models are single period models, which means that inventory is not carried
371
over to another period. Furthermore, any remaining products at the end of the period can be disposed of at a certain expense, or can be sold at a lower price than the market price. Initially, this type of modeling was applied to products with very high perishability, such as newspapers. Later, especially in the fashion industry, news-vendor models were proven to be of use for short life cycle production (see Fisher and Raman (1996) who study the single period setting in the fashion industry), and following the decrease of product life cycles in the high-tech industry, such as personal computers and mobile phones, news-vendor models are now well-accepted to model ordering decisions in these environments (see, e.g., Tayur et al. (1999), for a series of papers using this setting). This means that the news-vendor setting studied in this paper has been widely accepted as one of practical relevance. In the news-vendor situation discussed above, retailers can increase total expected profits (the sum of expected profits of all parties in the supply chain) by combining their orders. Besides price effects, this is because if some companies have ordered more and others have ordered less than they can sell, products are transferred between these companies. This builds on the traditional news-vendor problems (see, e.g., Silver et al. (1998) and many other textbooks; see also Khouja (1999) for a review). Although ordering jointly is collectively rational, the feasibility of such an arrangement depends on whether the expected profits of individual companies increase as a result of cooperation. It has been investigated in several studies whether it makes sense for companies to cooperate. Gerchak and Gupta (1991) compare four simple allocation rules in a continuous review single period inventory model with complete backordering. They show that individual stores may be unhappy. Robinson (1993) reexamines their results in terms of the core and subsequently studies the Shapley value (cf. Shapley, 1953) for these games and an alternative allocation rule for games with a large number of retailers. Hartman and Dror (1996) formulate three criteria for allocation rules in this setting: non-emptiness of the core, computational ease, and justifiability. This last criterion demands the existence of an appealing dual
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allocation rule in case the situation is analyzed in terms of profits rather than in terms of costs. Their results mainly deal with the existence of justifiable pairs, i.e., the analysis of an allocation rule and its dual. Recently, an increasing amount of attention is paid to coopetitive models in which ordering decisions of different retailers are assumed to be made independently/competitively, while allocation or transshipment decisions are assumed to take place cooperatively. We mention Rudi et al. (2001), Anupindi et al. (2001), and Granot and Sosic (2003). They mainly focus on Nash equilibria for the ordering decisions and investigate whether joint optimal solutions can be obtained in equilibrium. The cooperative behavior in the last stage is modeled by assuming or requiring that a coreelement is attained (for this stage only). Finally, we mention Meca-Martınez et al. (1999) who consider cooperation within the field of inventory management. They analyze cooperative games that depend on the information that is revealed by the companies and then focus on proportional costallocation mechanisms. Of specific interest is the work of Hartman et al. (2000) and M€ uller et al. (2002). They consider newsvendor games in which the value of a group of retailers is assumed to be their optimal profit if they coordinate their orders. Hartman et al. (2000) remark that Hartman and Dror (1997) show by means of a simple 3-person example that news-vendor games are not necessarily convex. Furthermore, Hartman et al. (2000) prove that news-vendor games have a non-empty core for specific demand distributions of the retailers. A non-empty core means that no group of players has an incentive not to cooperate, because with cooperation there exists a division of the joint profit that gives each group of players at least as much as they can obtain for themselves. Hartman et al. (2000) end with an open question, which can be reformulated as follows: ‘Do news-vendor games have a nonempty core?’. M€ uller et al. (2002) answer this question affirmatively, restricted to anonymous wholesale and customer prices. 1
1 We remark that the same result was independently derived by Slikker et al. (2001).
Several generalizations, starting from the work of Hartman et al. (2000) and M€ uller et al. (2002), seem relevant. First, the prices that companies pay to the producer and the customers pay to the companies do not need to be identical. Even for identical products, some companies may have more power to negotiate lower prices, for example because they buy many other products from the same supplier, and some companies may be able to charge higher prices to customers, for example because they sell through outlets that have a wider product assortment, better service, nicer stores, and a more expensive image. Furthermore, a reason for different prices may be that the retailers do not order exactly the same product, but products with slight differences, for example regarding packaging and labels. This requires having separate inventories per retailer. It also means that if the product of one retailer is going to be sold by another retailer, some modification might be involved. This leads to a second generalization that is relevant, namely that transshipment costs might be involved when there are separate inventories per retailer and products destined for one retailer are sold by another retailer. Transshipment costs might involve handling and transportation activities if products are identical, but they could also involve costs related to modifying products that have some characteristics that are specific to one retailer, for example repackaging and relabeling. Thirdly, the existing models could be made more general by relaxing the assumption that actual demand is known with certainty at the point in time at which the joint order has to be allocated among the retailers. For each retailer, demand might at that point in time better be modeled by a stochastic variable with less uncertainty (e.g., lower standard deviation) than at the first point in time, instead with no uncertainty. A more fundamental generalization concerns relaxing the assumption that retailers are risk-neutral. This demands an application of stochastic cooperative game theory. For a survey of research in this area we refer to Suijs (1999). Furthermore, strategic behavior of the players could be taken into account. In the models used in the literature, at two moments information is needed to make optimal decisions: At the point in time where the quantity
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
of the joint order is determined and at the moment that retailers have to reveal their actual demand, i.e., the point in time at which their actual demand is known. The events at the second point in time can be modeled as a process of allocation or reallocation. Here, the work of Klaus (1998) can be a guidance. In this paper, we formulate a coalitional game associated with a general news-vendor situation. This coalitional game corresponds essentially to the inventory centralization game introduced by Hartman et al. (2000) and M€ uller et al. (2002). The main difference being that they concentrate on costs of a coalition, whereas we concentrate on profits. We include the first two generalizations mentioned above. Extending the model of Hartman et al. (2000) and M€ uller et al. (2002), we allow that different retailers may experience different wholesale and/or customer prices. We also introduce the possibility of transshipments and make assumptions about costs associated with these. After some remarks about convexity we show that any general news-vendor game has a non-empty core. The plan of this paper is as follows. We start in Section 2 with some preliminaries on cooperative game theory to make this paper self-contained. Then, in Section 3, we introduce general newsvendor situations and associated cooperative games with these situations. These so-called general news-vendor games are studied in Section 4, which concentrates on balancedness. We conclude in Section 5 with some final remarks and discussion.
2. Preliminaries cooperative game theory For reasons of self-containedness, we here give a brief introduction to cooperative game theory, based on the work by Slikker and van den Nouweland (2001). Cooperative game theory covers the problem setting where different parties cooperate to reach a common goal. Let us assume we have n different companies pursuing similar objectives. In the setting studied in this paper, these companies are retailers possibly ordering jointly in order to reach a better overall service level to serve a set of independent markets. In game theory, these companies are referred to as
373
players, with N the set of players. For convenience, we number the players such that the player set is N ¼ f1; 2; . . . ; ng. A subset of N is called a coalition and is denoted by S. We are now interested in various coalitions S, and particularly to what extent a specific coalition can reach the common objective without the players who are not part of the coalition. We assume that the benefits of the cooperation within the coalition S are transferable between the players of S, and denote the benefits of the cooperation as vðSÞ. In the example of the retailers described above, this can be defined as the reduced cost of ordering jointly, or as the increased profits of reaching a higher service level. The function v that assigns to every coalition S N its value or worth vðSÞ, with vð;Þ ¼ 0, is commonly referred to as the characteristic function. A pair ðN ; vÞ consisting of a player set N and a characteristic function v constitutes a cooperative game or coalitional game. For notational convenience we will sometimes write simply game rather than coalitional game or cooperative game. In our analysis, we sometimes focus on only a few of the players involved in a game ðN ; vÞ. For a coalition S N , vjS denotes the restriction of the characteristic function v to the player set S, i.e., vjS ðT Þ ¼ vðT Þ for each coalition T S. The pair ðS; vjS Þ is a game with player set S that is then closely related to the game ðN ; vÞ. Two interesting properties that a game might satisfy are superadditivity and monotonicity. A game (or its characteristic function) is called superadditive if for any two disjoint coalitions S and T of players it holds that vðSÞ þ vðT Þ 6 vðS [ T Þ. An important consequence of a superadditive characteristic function, is that it is always attractive for these two disjoint coalitions to form one big coalition S [ T rather than operating separately. A game is called monotonic if the addition of more players will increase the value obtainable. Formally, ðN ; vÞ is called monotonic if vðSÞ 6 vðT Þ for all S T . Finally, we remark that a game is called additive if for any two disjoint coalitions S and T of players it holds that vðSÞ þ vðT Þ ¼ vðS [ T Þ. In line with any reality involving individual companies and increasingly with papers in the
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Operations Management literature (e.g., Cachon, 1999; Fransoo et al., 2001), players are not primarily interested in the benefits of a coalition, but in the individual benefits they are getting out of the coalition. The allocation is represented as a payoff vector x ¼ ðxi Þi2N 2 RN , specifying for each player i 2 N the benefit (e.g., (extra) profit) xi that this player can expect if he cooperates with the other players. An allocation is called efficient if the payoffs to the various players add up to exactly vðN Þ. The set consisting of all efficient allocations P is the set fx 2 RN j i2N xi ¼ vðN Þg. Note that not all these allocations will be acceptable to the players, as each player will require that he gets at least as much as what he can obtain when staying alone. An allocation x 2 RN with the property that xi P vðiÞ for all i 2 N is called individually rational. The set of all individually rational and efficient allocations is the imputation set IðN ; vÞ ¼ fx 2 RN j P i2N xi ¼ vðN Þ and xi P vðiÞ for each i 2 N g. This type of rationality requirement can be extended to all coalitions, not just individual players, to N P obtain the core CðN ; vÞ ¼ fx 2 R j i2N xi ¼ vðN Þ P and i2S xi P vðSÞ for all S N g. 2 The interpretation of the core is that it consists of all imputations that are such that no group of players has an incentive to split off from the grand coalition N and form a smaller coalition S because they collectively receive at least as much as what they can obtain for themselves as a coalition. A second interpretation of the core is that no group of players gets more than what they collectively add to the value obtainable by the grand coalition N . This follows since forPeach x 2PCðN ; vÞ P and S N it holds that x ¼ x i2S i i2N i i2N nS xi 6 vðN Þ vðN n SÞ. Bondareva (1963) and Shapley (1967) independently identified the class of games that have non-empty cores as the class of balanced games. To describe this class, we define for all S N the vector eS by eSi ¼ 1 for all i 2 S and eSi ¼ 0 for all i 2 N n S. A map jP: 2N n f;g ! ½0; 1 is S N called a balanced map if S22N nf;g jðSÞe ¼ e . Further, a game ðN ; vÞ is called balanced if for every balanced map j : 2N n f;g ! ½0; 1. it holds
2
We define the empty sum to be equal to 0.
P The following that S22N nf;g jðSÞvðSÞ 6 vðN Þ. theorem is due to Bondareva (1963) and Shapley (1967). Theorem 2.1. Let ðN ; vÞ be a coalitional game. Then CðN ; vÞ 6¼ ; if and only if ðN ; vÞ is balanced. A coalitional game ðN ; vÞ is called totally balanced if it is balanced and each of its subgames is balanced as well. The last notion we wish to introduce is the notion of convexity. A coalitional game is convex if a player’s marginal contribution increases if he joins a larger coalition. Formally, coalitional game ðN ; vÞ is convex if for each i 2 N and for all S T N n fig it holds that vðS [ iÞ vðSÞ 6 vðT [ iÞ vðT Þ. We remark that a convex coalitional game has a non-empty core.
3. The model In this section we introduce general news-vendor situations. These are situations where several companies order the same good from a producer. This good can be bought by company i at (wholesale) price ci and subsequently sold to customers at customer price pi . Every company experiences a stochastic demand. As in the standard news-vendor problem it is assumed that the realization of this stochastic demand is not known at the moment of ordering. If several companies cooperate they can, after realizations of demand are known, transship goods. Unit costs for transshipping goods from i to j are denoted by tij . A general news-vendor situation is a tuple ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ, where N denotes a set of companies and Xi the stochastic demand for the good at company i 2 N . Furthermore, ci and pi denote the prices that companies pay to the producer and the customers pay to the companies, respectively, and tij denotes transshipment costs. Throughout this work we assume that pi > ci > 0 for all i 2 N , that tij P 0 for all i, j 2 N , and that each company has a finite expected demand. A general news-vendor situation with one company only corresponds to the standard newsvendor problem.
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This model extends the model of Hartman et al. (2000), who restrict themselves to anonymous wholesale prices (ci ¼ c for all i 2 N ), anonymous customer prices (pi ¼ p for all i 2 N ), and free transshipments (tij ¼ 0 for all i, j 2 N ). Consider a general news-vendor situation and a collection of retailers S who will jointly determine an order vector qS 2 QS :¼ fq 2 RN jqSi ¼ 0 for all i 2 N n S and qSi P 0 for all i 2 Sg: Suppose coalition S has ordered vector qS 2 QS and subsequently, they face demand vector xS 2 RN with xSi ¼ 0 for all i 2 N n S. A reallocation of qS is a matrix S
S
S
S
RNþN jASij
¼ 0 if A 2 M ðq Þ :¼ fA 2 X S S Aij ¼ qi for all i 2 Sg: i 62 S or j 62 S; j2S
ASij
denotes the amount of products that are Here, transshipped from retailer i to retailer j. For j ¼ i this denotes the amount that is not transshipped. Hence, the total order of a retailer has to be equal to the sum of the transshipments from this retailer. Of course no transshipments with retailers outside S are allowed, i.e., they are equal to 0. The following lemma states that an optimal reallocation exists. Lemma 3.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation, let S N , let qS 2 QS , and let xS be a demand realization vector. Then there exist a reallocation AS 2 M S ðqS Þ that maximizes the expected profit of coalition S. Proof. Let hS ðAS ; qS ; xS Þ be the total revenue minus total transshipment costs if qS is rescheduled according to AS 2 M SP ðqS Þ. Since,PhS ðAS ; qS ; xS Þ P P S S S ¼ i2N j2N Aij tij þ j2S pj minf i2S Aij ; xj g, S S S we conclude that h ð:; q ; x Þ is a continuous function. Since it is defined on M S ðqS Þ, which is compact, we conclude that hS ð:; qS ; xS Þ attains its maximum. If we denote an optimal reallocation of this lemma by AS; then the optimal profit of coalition
375
S with order qS and demand realization xS is described by X pS ðqS ; xS Þ ¼ qSi ci þ hS ðAS; ; qS ; xS Þ: i2S
From now on, we will refer to hS ðAS; ; qS ; xS Þ as rS ðqS ; xS Þ. The expected profit of coalition S depends on their order vector qS and the demand faced by each retailer. We introduce for any vector of demands ðXi Þi2N and any S N the vector of demands X S , where ðX S Þi ¼ Xi for all i 2 S and ðX S Þi ¼ 0 for all i 2 N n S. We will write XiS instead of ðX S Þi . The expected profit of a coalition S that ordered qS and faces stochastic demand vector X S is then given by S ðqS ; X S Þ ¼ E½pS ðqS ; X S Þ: p
ð1Þ
Furthermore, we denote the expected profit that abstains from the initial costs by RS ðqS ; X S Þ ¼ E½rS ðqS ; X S Þ: ð2Þ P S S S S S S S ðq ; X Þ ¼ i2S qi ci þ R ðq ; X Þ. Hence, p A coalition may achieve higher expected profits than the sum of the expected profits of the individual companies. Hence, retailers may have an incentive to cooperate. We will construct a coalitional game with each general news-vendor situation. In this game, the value of a coalition is the maximum expected profit this coalition can obtain if they order jointly. Formally, let C ¼ ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. The associated general news-vendor game ðN ; vC Þ is defined by S ðq; X S Þ for all S N ; vC ðSÞ ¼ max p q2QS
ð3Þ
S are as defined above. 3 In the where X S and p following theorem we will show that each coalition
3 Hartman et al. (2000) and M€ uller et al. (2002) consider for the setting of anonymous wholesale prices ðcÞ, anonymous customer prices ðpÞ and free transshipments the cost game ðN ; cC Þ with cC ðSÞ ¼ ðp cÞE½XS vC ðSÞ for all S N . Additionally, they have a slightly different notation by focusing on oversupplying costs ðcÞ and undersupplying costs ðp cÞ. Since P ðp cÞE½XS ¼ i2S ðp cÞE½Xi , i.e., to get cC , an additive game is added to vC , we have that non-emptiness of the core in our general news-vendor game corresponds to non-emptiness of the (anti-)core in their cost game.
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has an optimal order vector, implying that the general news-vendor game is well-defined. Theorem 3.1. Let C ¼ ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation and let S N . Then there exists an order vector qS; that maximizes the expected profit of coalition S. Proof. We will show that the expected profit S ð:; XS Þ is a continuous function of the order p vector. Additionally we will show that any order outside a specific compact set results in lower expected profits than an order equal to zero. Hence, coalition S optimizes a continuous function over a compact set, which implies that an optimal order vector exists. First, we prove that the optimal profit is a continuous function of the order vector. Therefore, let q 2 QS denote an order vector and let > 0. Define cS ¼ maxi2S ðci ; pS Þ ¼ maxi2S pi , d ¼ jSjcSþpS . Let q0 2 QS with jq q0 j < d, where j j denotes the Euclidean norm. Let qmax and qmin be defined by ¼ maxfqi ; q0i g for all i 2 N ; qmax i min qi ¼ minfqi ; q0i g for all i 2 N : Using the triangle inequality we have for any realization xS , jpS ðq; xS Þ pS ðq0 ; xS Þj X jqi q0i j þ jrS ðq; xS Þ rS ðq0 ; xS Þj: 6 cS
ð4Þ
i2S
Hence, for the expected profit functions we have S ðq0 ; X S Þj j pS ðq; X S Þ p X jqi q0i j þ jRS ðq; X S Þ RS ðq0 ; X S Þj 6 cS i2S
6 cS
X
jqi q0i j þ jRS ðqmax ; X S Þ RS ðqmin ; X S Þj
i2S
6
X
! jqi
q0i j
ðcS þ pS Þ
i2S
6 jq q0 jjSjðcS þ pS Þ < d jSjðcS þ pS Þ ¼ : The first inequality follows by taking expectations over the inequality in (4). The second holds since RS ðqmax ; X S Þ P RS ðq ; X S Þ for q 2 fq; q0 g and re-
verse inequalities for RS ðqmin ; X S Þ. The third inequality holds since RS ðqmin ; XPS Þ 6 RS ðqmax ; X S Þ P max max pP qmin and qmin S i j i j ¼ i2S jqi i2S jqi 0 jq q j. The fourth inequality holds since i i i2S is jq q0 j P jqi q0i j for i 2 S. We conclude that p a continuous function of the order vector. It remains to show that any order vector outside a specific compact set results in lower expected profits than an order equal to zero. Let " # X maxi2S pi X S qm ¼ E½Xi þ E½Xi ; ð5Þ mini2S ci i2S i2S where Xi ¼ maxfXi ; 0g. 4 Then for all q 2 QS with qi > qSm for some i 2 S, say j, we have X ðq; X S Þ 6 qj min ci þ max ðpi Þ p E½Xi i2S
< max ðpi Þ i2S
i2S
X
i2S
S ð0; X S Þ: E½Xi 6 p
i2S
The first inequality follows by taking only part of the wholesale costs and an upperbound of the remaining revenues into consideration. The second inequality follows by definition of qSm . The last inequality follows immediately. Hence, we can restrict attention to order vectors in the compact set fqj0 6 qi 6 qSm for all i 2 Sg. This completes the proof. If S consists of a single retailer or if all retailers in S experience the same wholesale price, the same customer price, and free transshipments, then the determination of an optimal order vector boils down to the determination of a total order (distribution over i 2 S is irrelevant). This, in turn, comes down to a standard P news-vendor problem with stochastic demand i2S Xi , wholesale price c ð¼ ci for all i 2 SÞ and customer price p ð¼ pi for all i 2 SÞ. It is well known that an optimal order size is then given by the ð1 pcÞ-quantile, which need not be a unique order size. For a normal distribution Normðl; rÞ this implies that the optimal order quantity is given by l þ rzc=p ,
4 The introduction of Xi is superfluous if only non-negative demands are possible.
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
where za is defined as the unique real number such that P ðX P za Þ ¼ a if X is a stochastic variable with the standard normal distribution Normð0; 1Þ. A general news-vendor situation with anonymous wholesale and customer prices (ci ¼ c and pi ¼ p for all i 2 N ) and free transshipments is called a news-vendor situation with associated news-vendor game. Consider the following example, which illustrates some of the concepts introduced above. For notational and computational purposes the specifications have been kept very simple. This does not hamper the illustration of the concepts. Example 3.1. Consider the 2-person news-vendor situation ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ with N ¼ f1; 2g, c1 ¼ c2 ¼ 1, p1 ¼ p2 ¼ 16, free transshipments, and a stochastic demand X1 for company 1 described by probability mass function 8 > < 0 if x 62 f0; 1g; ð6Þ p1 ðxÞ ¼ 78 if x ¼ 0; > :1 if x ¼ 1; 8 while company 2 faces a demand with the same distribution that is independent of the demand of company 1. It is easily verified that ð1 pc11 Þ-quantile of the distribution of X1 equals 1, implying that the optimal order size of coalition {1} equals 1, resulting in expected profit f1g ð1; X1 Þ ¼ 1 1 þ p
1 1 16 ¼ 1 þ 2 ¼ 1: 8
Obviously, similar calculations result in the optimal expected profit for coalition 2. The optimal order for coalition f1; 2g equals 1 as well. Since transshipments are free, any vector q 2 RNþ with q1 þ q2 ¼ 1 results in an optimal profit for the grand coalition, equal to f1;2g ðq; X f1;2g Þ ¼ 1 1 þ p
15 3 1 16 ¼ 2 : 64 4
By cooperating, the companies can increase their joint profit from 2 to 2 34. This is due to the fact that they can reduce the total amount they order significantly, while this results in a small increase in expected lost sales only. The associated news-vendor game ðN ; vC Þ is described by
8 > <0 vC ðSÞ ¼ 1 > : 3 24
377
if S ¼ ;; if jSj ¼ 1; if S ¼ N :
Note that this news-vendor game is balanced, i.e., it has a non-empty core, since any x 2 R2 with x1 P 1, x2 P 1 and x1 þ x2 ¼ 2 34 belongs to the core, for example x ¼ ð118 ; 118Þ. Moreover, ðN ; vC Þ is convex. In the following section we will focus on balancedness of general news-vendor games. M€ uller et al. (2002) proved balancedness for anonymous wholesale and customer prices and free transshipments. Furthermore, Slikker et al. (2001) proved the same results independently, and showed additionally that if, on top of the restrictions of M€ uller et al. (2002) we assume that the demand distributions have a normal distribution and that the demands are independent then the games are convex. If anonymity or independence is dropped or if discrete rather than normal demand is considered, convexity is no longer guaranteed.
4. Balancedness In this section we show that any general newsvendor game has a non-empty core. In the process of proving that general newsvendor games are balanced, we will use the following lemma. Lemma 4.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. Let j be an associated balanced map. Denote an optimal order vector of coalition S N by qS; . Then X
N
p
jðSÞq ;
SN :S6¼;
P
S;
X
X
! jðSÞX
S
SN :S6¼;
S jðSÞqS; ; jðSÞX S : p
SN :S6¼;
Proof. First, note that total wholesale costs are the same. Hence, it suffices to show that
378
R
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
X
N
S;
jðSÞq ;
SN :S6¼;
P
X
X
! jðSÞX
Theorem 4.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. The associated news-vendor game is balanced.
S
SN :S6¼;
R jðSÞqS; ; jðSÞX S : S
SN :S6¼;
Let xN be a realization of demand for the grand coalition and let xS be the associated demand of any coalition S N . As before, denote a reallocation of qS; that maximizes the profit of coalition S; S by AS; . Denote the part is actually Pof A that S; S; N sold by A . Then B ¼ SN jðSÞA denotes a possible selling part of a reallocation of the grand coalition associated with order vector P S; SN :S6¼; jðSÞq . Denote the sold part of an optimal reallocation of N associated with this order vector by C N . Then ! X X N S; S r jðSÞq ; jðSÞx SN :S6¼;
XX
¼
i2N
P
SN :S6¼;
CijN ðpj
XX i2N
BNij ðpj tij Þ
j2N
XXX
¼
i2N
SN i2N
X
¼
S;
jðSÞA ðpj tij Þ
N ðqN ; X N Þ P p N ðzN ; X N Þ vðN Þ ¼ p ! X X N S S ¼p jðSÞq ; jðSÞX
j2N SN
XXX
¼
tij Þ
j2N
S;
jðSÞAij ðpj tij Þ
j2N
S
r ðjðSÞqS; ; jðSÞxS Þ;
ð7Þ
SN S;
where the last equality holds since jðSÞA is obviously the sold part of an optimal reallocation for coalition S and demand realization jðSÞxS . Combining equal wholesale costs with (7) gives ! X X jðSÞqS; ; jðSÞxS pN SN :S6¼;
P
X
SN :S6¼; XSN :S6¼; S S ðjðSÞq ; jðSÞX S Þ P p SN :S6¼;
¼
X
SN :S6¼;
jðSÞ pS ðqS ; X S Þ ¼
X
jðSÞvðSÞ:
SN :S6¼;
The second inequality holds by Lemma 4.1. Hence, the cooperative game associated with the general news-vendor situation is balanced, i.e., it has a non-empty core.
SN :S6¼;
p jðSÞqS; ; jðSÞxS : S
Proof. Let j : 2N n f;g ! ½0; 1 be a balanced map. Recall that X N ¼ ðXi Þi2N Pand, for all i 2 N , note that P S jðSÞX ¼ X where i i SN :S6¼; SN :i2S jðSÞ ¼ Xi , the last equality follows since j is a balanced map. Let ðqS ÞSN :S6¼; be optimal ordervectors for the different coalitions. Then zN defined by zNi ¼ P jðSÞ qSi for all i 2 N denotes a possible SN :i2S ordervector for coalition N , while qN is the optimal N ðqN ; X N Þ P ordervector. Hence, we know that p N N N ðz ; X Þ. Furthermore, we know that for all p S N with S 6¼ ; it holds that jðSÞqS maximizes S ð:; jðSÞX S Þ since p S ðkq; kX Þ ¼ k profit p pS ðq; X Þ for any stochastic demand vector X of coalition S and any ordervector q of coalition S. This implies that jðSÞqS is an optimal ordervector for stochastic demand vector jðSÞX S if and only if qS is an optimal ordervector for stochastic demand X S . Furthermore, it implies that if the stochastic demand vector changes by some factor then the optimal expected profit changes by the same factor. Using this, we find
SN :S6¼;
Since this inequality holds for any realization xN of X N , taking expectations over the distribution of X N completes the proof. The following theorem states that general newsvendor games are balanced.
The result in Theorem 4.1 can be strengthened by noting that every subgame of a general newsvendor game is a general news-vendor game itself. Hence, the following corollary follows immediately from Theorem 4.1. Corollary 4.1. Let ðN ; ðXi Þi2N ; ðci Þi2N ; ðpi Þi2N ; ðtij Þi2N ;j2N Þ be a general news-vendor situation. Then the associated news-vendor game is totally balanced.
M. Slikker et al. / European Journal of Operational Research 167 (2005) 370–380
5. Remarks and discussion This work provides a further contribution to the game-theoretical analysis of situations in which retailers can coordinate their orders. In Section 1 several extensions to the work of M€ uller et al. (2002) and Slikker et al. (2001) were proposed. Out of these, the inclusion of transshipments into the model seemed to be the most interesting generalization. We showed that stable divisions of expected profits also exist if transshipment cost are positive. Furthermore, we showed that in case of different retail and wholesale prices rather than anonymous prices, stable divisions of expected profits exist as well. These two extensions of the earlier results by M€ uller et al. (2002) and Slikker et al. (2001) provide significant progress in the analysis of multiple news-vendor games. The next step would be to cover the issues that are mentioned in the introduction, but are not covered in this paper. Firstly, one could relax the assumption that actual demand is known at the moment the joint order is divided or at the moment at which transshipment decisions have to be made. Other research questions deal with a cooperative analysis from a stochastic game-theoretical point of view and with a non-cooperative analysis of the process of allocation and reallocation, following the works of Suijs (1999) and Klaus (1998) respectively.
Acknowledgements The authors thank Henk Norde and others for useful suggestions and comments.
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