Cooperative Replenishment in the Presence of Intermediaries Behzad Hezarkhani, Marco Slikker, Tom Van Woensel School of Industrial Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands March 17, 2015 Abstract In complex supply chains, downstream agents often find it beneficial to replenish their required products indirectly from intermediaries rather than directly from the original manufacturers. By aligning their replenishment activities, however, downstream agents could reap the benefits of direct replenishments such as lower purchase prices. This paper constructs a general framework for multi-product cooperative procurement under the presence of supply chain intermediaries. We introduce a sufficient condition for the stability of cooperative games associated with these situations. To allocate the joint costs, we suggest Shapley value as it induces complete participation in a class of two-stage games associated with these situations. We show that in order to guarantee the optimal performance of the corresponding decentralized replenishment system, the options for indirect replenishments from the intermediaries must be explicitly considered in the cooperative organization which coordinates the joint replenishments.
1
Introduction
Intermediaries are economic entities who coordinate and arbitrate transactions in between upstream suppliers and downstream players in supply chains (Wu, 2004). According to the intermediation theory of the firm (Spulber, 1996) a firm is created when the gains from intermediated exchange exceed the gains from direct exchange. Thus intermediaries can alleviate the high transaction costs of direct trade. The gains created by intermediaries in supply chains stem from aggregating demand to achieve economy of scale, or consolidating supply to reduce order and delivery costs. The traditional way that enables the intermediaries to generate these benefits is through procuring products, holding inventories, and reselling them at a margin. Such activities add to the total cost of supply chains. This paper investigates the possibilities of achieving the benefits of intermediation without undergoing the additional costs due to double marginalization (Spengler, 1950) and excessive inventory holding costs – an objective which is possible via cooperation among the downstream players. 1
The role of supply chain intermediaries are more pronounced in industries with high degree of product differentiation, market fragmentation and sourcing globalization, e.g. fashion (Purvis et al., 2013), healthcare (), agro-food (), and construction (). This paper is in particular motivated by an initiative to promote cooperative replenishment organizations in the European automotive after-market. The after-market constitutes an integral part of the automotive industry – it generated about half of the profits of European automotive OEMs in 2007 (Capgemini, 2010). The automotive after-market supply chain deals with thousands of products and includes many echelons – e.g. manufacturers, importers, wholesalers, players, garages, and finally, car owners. Since quick response to demand for parts is one of the most important performance indicator in the after-market, little emphasis has been placed on efficiency and cost reduction in this sector. As a result, the automotive after-market supply chain is filled with inventories and redundancies at various echelons (AASA, 2012). However, an industry traditionally enjoying high margins, is forced to cope with new realities: in 2013 the sales of new vehicles reached its lowest in 20 years (Euronews, 2013). Thus cost savings in all aspects of the industry are becoming unavoidable. Cooperative replenishment is a strategy to reduce costs and improve the efficiency of the whole supply chain. The sustained presence of intermediaries in supply chains implies that downstream players find it worthwhile to replenish via intermediaries even though their prices are considerably higher than those of the manufacturers. This is due to the lack of economies of scale that renders direct replenishment from manufacturers unprofitable, or due to the option to bundle orders for different products at the intermediary instead of dealing with numerous manufacturers. However, there are real-life examples in both private and public sectors and across many industries, e.g. healthcare sector (Nollet and Beaulieu, 2003), and electronic markets (Huber et al., 2004), wherein a consortia of cooperating downstream players allow benefits to be accrued from direct transaction with manufacturers. However, very little quantitative research involves purchasing organizations or other contracting intermediaries (Hu et al., 2012). We introduce a general framework for cooperative replenishment of multiple products with direct and indirect sourcing options. The Cooperative Replenishment with the Intermediaries (CRI) situations formalize supply chains with several downstream players, multiple manufacturers and local intermediaries. The downstream players represent points of sale to the market. Each manufacturer is specialized in producing a single product. The intermediaries buys products from the manufacturers, keeps them in the stock, and offers them to the downstream players. Downstream players have the option to replenish a product either directly from the corresponding manufacturer or indirectly from the intermediary (Figure 1). The incentives for sourcing from either options are conflicting. Although the purchase price at a manufacturer is lower than that at the intermediary, it may not be economically justifiable for individual players to replenish directly. Firstly, direct replenishment often involves higher costs, because manufacturers are located at a farther distance, require orders of some minimum quantities, or charge additional fees for handling orders, e.g. the so-called order-picking costs. Secondly, while each product must be ordered separately from its corresponding manufacturer the downstream players could individually place orders for multiple products with the intermediary and receive them in 2
Figure 1: Replenishment sources – from intermediary (left), from manufacturers (right) one delivery – so that they pay the fixed order cost once for procuring multiple products. Therefore, replenishing more products by a downstream player from the intermediaries brings in more saving due to batching of orders, price discounts, etc. Although replenishments from manufacturers must be organized across individual products, the more players cooperating to replenish the same product from the manufacturer, the less total cost need to be incurred by the group. In order to find the optimal replenishment policy for a group of players one must balance the lower fixed order costs of the intermediary with the less expensive prices of the manufacturers. The optimal replenishment policies in CRI situations combine direct as well as indirect orders to achieve the minimum replenishment cost including the purchasing, inventory holding, and ordering costs. An optimal replenishment policy determines who replenishes which product directly from its manufacturer. The optimal replenishment policies might require players to obtain low-demand products from the intermediary and high-demand products from the manufacturers. The central message of this paper is that if the individual replenishment costs from intermediaries and manufacturers are submodular, then whenever downstream players can take advantage of cooperation, the corresponding cooperative organization would be stable. There have been several papers in the recent literature that elaborate on the submodularity of single-source-single-item replenishment models. We employ such results in the CRI situations and prove that if the individual replenishment functions for downstream players and manufacturers separately satisfy the submodularity condition, then the replenishment cost function in CRI situations – which incorporates multiple products and dual sources for procurements of products – is also submodular. An immediate outcome of the submodular structure of replenishment cost function in CRI situations is that the optimal replenishment policies can be calculated effectively. However, We provide some observations with regard to the structure of optimal replenishment policies. We exhibit that the cooperative replenishment of a product from its manufacturer has an spill-over effect on the rest of the products in the sense that the latter increases the possibility of beneficial joint replenishment of other products from their manufacturers. 3
Under the submodularity assumption of individual cost components, we observe a nested property with regard to that optimal replenishment policies in growing subset of downstream players. That is, if it is optimal for a groups of downstream players to replenish a given product directly via its manufacturer in a subset of players, it would remain optimal for the group to keep replenishing that product from its manufacturer in larger subsets. Thus, the optimal replenishment policies for larger groups can be built upon those of smaller groups. This observation sets the stage for obtaining the main result of the second part of the paper which addresses the cooperative games associated with CRI situations, i.e. CRI games. Drawing upon our structural results, we show that submodularity of individual cost components result in concavity of associated CRI games. This implies a nonincreasing effect in every downstream player’s contribution to the total costs of growing subsets of players. The concavity of CRI games guarantees the existence of stable allocations. By assuring that the total cost paid by any group of players is not more that the cost that they incur when replenishing apart from the rest, an stable allocation provides sufficient incentives for the downstream players to remain with the set of all players (grand coalition), i.e. the cooperative organization which includes all the players. To single out stable allocations among the infinitely many possible allocations, we suggest the Shapley value (Shapley, 1953) which in CRI games also has additional benefits. Strategic games ... The rest of this paper is organized as following. In Section 2, we briefly overview the literature relevant to cooperative purchasing, replenishment, and inventory management. Section 3 formally presents the CRI situations and their components. In Section 4 we discuss the replenishment policies and some of their properties. Section 5 focuses on the optimal replenishment policies for groups of downstream players. The cooperative cost games associated with CRI situations are introduced in Section 6 where the corresponding cost-sharing problem is also addressed. Section 9 concludes the paper.
2
Literature Review
4
3
Mathematical Model
Consider a supply chain with a set of downstream agents, hereafter the players, represented by the index set N = {1, ..., n}, replenishing a variety of different products to resell in their local markets. A reference set E contains all products that could be replenished by the players. Each product is produced and sold by a distinct manufacturer. Thus the set of products also represents the set of manufacturers. Each player may replenish only a subset of products in E. The set of products replenished by a player i ∈ N is denoted by Ei ⊆ E. The vector E = (Ei )i∈N denotes the player-specific product sets. In addition to the manufacturers, supply chain intermediaries–e.g. regional wholesalers or volume distributors– also sell some or all products in E. The intermediaries by themselves procure products from the manufacturers and keep them in stock. The players have the option to obtain each product either from its corresponding manufacturer or from an intermediary who sells it. Before describing the two replenishment options in more detail, we briefly overview set functions which will be used extensively in the rest of this paper. Given a set Ω, f ∶ 2Ω → R is a set function that gives real values to subsets of Ω. The following properties of set functions are of interest. Definition 1. Let f be a set function defined on a set Ω. • f is nondecreasing if for every A ⊂ B ⊆ Ω we have f (A) ≤ f (B). • f is subadditive if for every A, B ⊂ Ω, A ∩ B = ∅, we have f (A ∪ B) ≤ f (A) + f (B). • f is submodular if for every A ⊆ B ⊂ Ω and every element a ∈ Ω ∖ B it holds that f (B ∪ a) − f (B) ≤ f (A ∪ a) − f (A).1 The value of a nondecreasing set function never decreases as the result of including more elements. Subadditivity limits the amount of possible increase due to including more elements in such a way that the value of the union of two disjoint sets does not exceed the sum of the individual values of those sets. A submodular set function demonstrates a diminishing returns property which makes it comparable to concave continuous functions.
3.1
Replenishments from the intermediaries
Intermediaries constitute flexible replenishment sources which cater to individual players. When a player i ∈ N replenishes a subset of products P ⊆ Ei from the intermediaries, it will do so by choosing corresponding decision variables–batch sizes, ordering cycles, the choice of intermediaries etc.–optimally to attain the minimum possible per-period replenishment cost. We refrain from the operational details and instead introduce the indirect replenishment cost function Ciw ∶ 2E → R that gives the minimum per-period replenishment cost from intermediaries of player i for subsets of products. The vector C w = (Ciw )i∈N denotes indirect replenishment cost functions for all players. 1
For notational convenience we do not use braces for union and exclusion of single element sets. That is, we write A ∪ a instead of A ∪ {a} and A ∖ a instead of A ∖ {a}.
5
We require that for every player i ∈ N the indirect replenishment cost function Ciw is subadditive on the set of products E. This condition asserts that by combining the replenishments of multiple sets of products from the intermediaries, their total per-period replenishment cost does not increase—although in practice it is usually the case that combining the replenishments of multiple products from the same source provides opportunities to obtain additional savings by taking advantage of discounts or batch deliveries. It may seem natural to think of Ciw as a nondecreasing function for every i ∈ N , reflecting the fact that replenishing more products never results in a reduction in costs, yet, we do not impose this condition formally as the results obtained in this paper do not require indirect replenishment cost functions to be nondecreasing. We assume that replenishment costs from intermediaries are additive over the set of players, that is, no savings can be obtained by combining the indirect replenishments of different players. This justifies our expression of indirect replenishment cost functions in terms of individual players.
3.2
Replenishments from the manufacturers
Alternatively, every product can be replenished directly from its manufacturer. Replenishing directly from the manufacturers has some advantages. Above all, the purchase prices at the manufacturers are usually considerably lower compared to those at the intermediaries. However, factors such as high order costs may render direct replenishments unattractive for individual players. In spite of this, by aligning replenishment cycles and placing joint orders, groups of players could obtain savings when replenishing a product directly from its manufacturer. When a group of players jointly replenish a product from its manufacturer, corresponding decision variables would be chosen to minimize the per-period replenishment cost of that product for the group. With this perspective we can abstract away from the operational details of joint replenishments and introduce the direct replenishment cost function Clm ∶ 2N → R as the set function that obtains the minimum per-period replenishment cost from manufacturer l for different groups of players. Accordingly, the vector C m = (Clm )l∈E denotes direct replenishment cost functions for all products. We impose the following basic conditions on direct replenishment cost functions. For every l ∈ E, we require Clm to be nondecreasing and subadditive on the set of players N . That is, including more players in joint orders from a manufacturer never results in lower total costs although it can be cheaper to replenish a product directly as a single group instead of replenishing it in separate groups. Finally, in line with the assumption that each manufacturer produces a distinct product, we assume that direct replenishment costs from the manufacturers are additive over the set of products. Thus orders for multiple products from different manufacturers cannot be consolidated to make any savings. This explains our expression of direct replenishment cost functions in terms of distinct products.
6
3.3
CRI situations
We introduce CRI situations in order to formally encapsulate the relevant information necessary to analyze the setting described above. An instance of CRI situations can be described by the tuple Γ = (N, E, E, C w , C m ) with its elements defined as above. The vector of indirect and direct replenishment costs, C w and C m , are refereed to as the cost components of the situation. The set of all CRI situations is denoted by Γ.
3.4
An explanatory situation
We provide a more in-depth example of a CRI situation which fits into the motivating case of this paper in automotive aftermarket. The players of the situation are automotive part retailers who sell a large variety of products either to their customers (either repair garages or car owners). different retailers can be focused on certain groups of products (based on brand or functionality among others) so not all products are offered by all retailers. The demands for most products are fairly predictable at this level of supply chain—although demands show rather stochastic nature at a lower level of supply chain such as repair garages. Therefore, the assumption of deterministic demands at the retailers is rather a reasonable simplification for most products. Some groups of products also show a more or less constant demand rate over the year. Importers and wholesalers play the role of intermediaries in this chain offering products to the retailers. Although the unit purchase price of the same product at the intermediary is considerably higher than that at the corresponding manufacturer, the close proximity of intermediaries mean low transportation and ordering cost as well as short leadtimes. The retailers place their orders individually with the intermediaries. A retailer can replenish multiple products from the same intermediary and receive them in single deliveries. In this manner, the ordering and transportation costs can be broken down among the products ordered in one batch. Through a supply chain control center, retailers can also place joint replenishment orders for different products with their corresponding manufacturers. The delivery of the latter orders in most cases involve two types of costs: (1) major ordering and transportation costs which occurs irrespective of the volume and particular retailers, and (2) minor ordering and transportation costs which are specific to the particular retailers who are participating in a joint replenishment order. The above situation can be mathematically modelled by drawing upon the available purchasing and inventory management models in the literature. More specifically, the orders of an individual retailer for multiple products at an intermediary can be modelled by combining a linear purchasing component and the multi-product EOQ model in (Meca et al., 2004). The joint replenishments from the manufacturers can also be modelled by combining a linear purchasing component and the power-of-two joint replenishment model in (Anily and Haviv, 2007).
7
4
Replenishment Policies
Replenishment polices represent the various choices regarding the replenishment sources of different products for different players. Players are able to replenish any subsets of their required products from the manufacturers and their remaining requirements from the intermediaries so that they could possibility reap the benefits of both sources. Thus replenishment policies are the main decision variables in CRI situations. In this paper we assume that the choices of replenishment sources of all products and all players are binary, i.e. every single product is sourced entirely either from an intermediary or the its corresponding manufacturer. The remainder of this section formalizes and studies the notion of replenishment policies in CRI situations. Given the CRI situation Γ with player set N and a player i ∈ N , we define player i’s replenishment choice domain as X Γi = {(i, l)∣i ∈ N, l ∈ Ei } comprising player-product pairs specific to player i. The replenishment choice domains for subsets of players are obtained accordingly by concatenating their individual choice domains. For every S ⊆ N , we let X ΓS = ⋃ X Γi i∈S
denote the replenishment choice domain of S consisting of all its members’ individual choice domains. A direct replenishment policy X is a collection of player-product pairs that will be replenished directly via the manufacturers. Since the decisions regarding replenishment sources of all products for all players are binary, a direct replenishment policy also describes the indirect replenishment policies. Therefore, from this point on we refer to a direct replenishment policy simply as a replenishment policy. Based on the above definitions, a direct replenishment policy of a player (or a group of players) must be a subset of its (their) replenishment choice domain(s). Thus, we call a replenishment policy X feasible for players in S ⊆ N whenever X ⊆ X ΓS . Note that with this definition a feasible replenishment policy for a subset of players is also feasible for subsets of players which contain the former players. However, the reverse does not hold necessarily. In order to evaluate the cost of replenishment policies we define two auxiliary functions. These functions explicitly determine the replenishment actions of players and assist us in developing the total replenishment cost in CRI situations. Given a CRI situation Γ with player set N , a replenishment policy X, and a product l ∈ E, the direct replenishers of l in X are denoted by the set IlΓ [X] = {i ∈ N ∣(i, l) ∈ X}. (1) The direct replenishers of a product in a replenishment policy are individual players (hence the notation I) who obtain the product l from its corresponding manufacturer. A replenishment policy readily reveals the products that are replenished from the manufacturers by the players. The other input for calculating the total cost of a replenishment 8
policy is the products that players replenish indirectly from the intermediaries. This can be obtained by excluding the directly replenished products of a player from its specific product set. Given a situation Γ with player set N , a replenishment policy X, and a player i ∈ N , the indirectly replenished products of i in X is denoted by PiΓ [X] = {l ∈ Ei ∣(i, l) ∉ X}.
(2)
The indirectly replenished products in a replenishment policy are the products (hence the notation P) that are obtained from the intermediaries. We are now ready to calculate the total replenishment cost associated with a replenishment policy for a subset of players. For each group of players S ⊆ N , we define the replenishment cost function for S, CSΓ ∶ X ΓS → R, such that for every feasible replenishment policy for S, X ∈ X ΓS , we have CSΓ (X) = ∑ Ciw (PiΓ [X]) + ∑ Clm (IlΓ [X]). i∈S
(3)
l∈E
Hence the cost of a given replenishment policy X for S is the sum of replenishment costs from intermediaries of players in S for their indirectly replenished products in X plus the sum of replenishment costs from manufacturers of all products for their direct replenishers in X. The following lemma illustrates a relation between the costs of a feasible replenishment policy for two subsets of players. Lemma 1. Let Γ be a CRI situation with player set N and consider S ⊂ T ⊆ N . Let X be a feasible replenishment policy for S. We have CTΓ (X) = ∑i∈T ∖S Ciw (Ei ) + CSΓ (X). Proof. From definition of CSΓ (X) in (3) we have CSΓ (X) = ∑ Ciw (PiΓ [X]) + ∑ Clm (IlΓ [X]). i∈S
(4)
l∈E
Since X ⊆ X ΓS , for every player i ∈ T ∖ S there exists no l ∈ E such that (i, l) ∈ X. Therefore, for every i ∈ T ∖ S we have PiΓ [X] = Ei . Consequently, we have CTΓ (X) = ∑ Ciw (PiΓ [X]) + ∑ Clm (IlΓ [X]) i∈T
= =
l∈E w ∑ Ci (Ei ) + ∑ Ciw (PiΓ [X]) + ∑ Clm (IlΓ [X]) i∈T ∖S i∈S l∈E w Γ ∑ Ci (Ei ) + CS (X). i∈T ∖S
Lemma 1 allows one to evaluate the cost of a replenishment policy which is feasible for players in S for larger subsets of players. In doing so, it is sufficient to include the replenishment costs of additional players working individually with their intermediaries. An optimal replenishment policy for a subset of players has the lowest cost among all replenishment policies that are feasible for that subset of players. The cost of an optimal 9
replenishment policy for S ⊆ N is obtained by solving the following optimization problem: cΓ (S) = minΓ CSΓ (X)
(5)
X⊆X S
4.1
Sub-modular cost components
Our general framework allows one to draw upon the existing joint replenishment models to investigate the corresponding CRI situations. In this section, we focus our attention to CRI situations wherein the cost components are submodular. Operations Management literature highlights several classes of joint replenishment problems with submodular cost functions. Submodularity of cost functions in latter models has interesting consequences in terms of the underlying optimization problem and associated cooperative games. Examples include deterministic joint replenishment problems discussed in Meca et al. (2004), Anily and Haviv (2007), Zhang (2009), special cases in Van den Heuvel et al. (2007), as well as some stochastic models such as in Hartman et al. (2000).2 Consequently, we show that submodularity of cost components of CRI situations results in interesting properties for the entire situation as well. The first result in this paper provides a sufficient condition for the submodularity of the replenishment cost function. Theorem 1. Let Γ be a CRI situations with player set N , product set E, and vector of player-specific product sets (Ei )i∈N . Assume that Ciw is submodular on Ei for every i ∈ N and Clm is submodular on N for every l ∈ E. Then for every S ⊆ N the replenishment cost function CSΓ is submodular on X ΓS . Proof. Fix S ⊆ N . Let X, X ⊆ X ΓS be arbitrary feasible replenishment policies of S such ′ that X ⊆ X. Consider an arbitrary player-product pair (j, h) with j ∈ S, h ∈ Ej , and (j, h) ∈ X ΓS ∖ X. The replenishment cost function CSΓ is submodular if ′
CSΓ (X ∪ (j, h)) − CSΓ (X) ≤ CSΓ (X ∪ (j, h)) − CSΓ (X ). ′
′
(6)
We continue in two steps: (Step 1) By definition of PiΓ we have PiΓ [X ∪ (j, h)] = PiΓ [X] for every i ∈ S ∖ j, and ′ PjΓ [X ∪ (j, h)] = PjΓ [X] ∖ h. The assumption X ⊆ X implies that for every i ∈ S we have ′ PiΓ [X] ⊆ PiΓ [X ]. Submodularity of Cjw on Ei implies that Cjw (PjΓ [X ]) − Cjw (PjΓ [X ] ∖ h) ≤ Cjw (PjΓ [X]) − Cjw (PjΓ [X] ∖ h) ′
′
or equivalently Cjw (PjΓ [X] ∖ h) − Cjw (PjΓ [X]) ≤ Cjw (PjΓ [X ] ∖ h) − Cjw (PjΓ [X ]). ′
2
We elaborated on these papers in the literature review Section.
10
′
(7)
Adding ∑i∈S∖j Ciw (PiΓ [X∪(j, h)])−Ciw (PiΓ [X]) = 0 and ∑i∈S∖j Ciw (PiΓ [X ∪(j, h)])−Ciw (PiΓ [X ]) = 0 to the left and right sides of (7) respectively obtains ′
′
∑ Ciw (PiΓ [X ∪ (j, h)]) − Ciw (PiΓ [X]) ≤ ∑ Ciw (PiΓ [X ∪ (j, h)]) − Ciw (PiΓ [X ]). ′
i∈S
′
(8)
i∈S
(Step 2) By definition of IlΓ we have IlΓ [X ∪ (j, h)] = IlΓ [X] for every l ∈ E ∖ h, and ′ IhΓ [X ∪ (j, h)] = IhΓ [X] ∪ j. The assumption X ⊆ X implies that for every l ∈ E we have ′ IhΓ [X ] ⊆ IhΓ [X]. On the other hand submodularity of Chm on N yields Chm (IhΓ [X] ∪ j) − Chm (IhΓ [X]) ≤ Chm (IhΓ [X ] ∪ j) − Chm (IhΓ [X ]). ′
′
By adding ∑l∈E∖h Clm (IlΓ [X ∪ (j, h)]) − Clm (IlΓ [X]) = 0 and ∑l∈E∖h Clm (IlΓ [X ∪ (j, h)]) − ′ Clm (IlΓ [X ]) = 0 to the left and right sides of the above inequality respectively we get ′
∑ Clm (IlΓ [X ∪ (j, h)]) − Clm (IlΓ [X]) ≤ ∑ Clm (IlΓ [X ∪ (j, h)]) − Clm (IlΓ [X ]). ′
′
(9)
l∈E
l∈E
To conclude the proof, add (8) and (9) to get ∑ Ciw (PiΓ [X ∪ (i, h)]) + ∑ Clm (IlΓ [X ∪ (i, h)]) − (∑ Ciw (PiΓ [X]) + ∑ Clm (IlΓ [X])) i∈S
i∈S
l∈E
l∈E
≤ ∑ Ciw (PiΓ [X ∪ (i, h)]) + ∑ Clm (IlΓ [X ∪ (i, h)]) − (∑ Ciw (PiΓ [X ]) + ∑ Clm (IlΓ [X ])) . ′
′
i∈S
′
′
i∈S
l∈E
l∈E
which is equivalent to (6). Thus CSΓ is submodular on X ΓS . According to Theorem 1, submodularity of the cost components is a sufficient condition for a CRI situation to be submodular on the replenishment choice domain of the player set. In addition to outlining a sufficient condition for submodularity of replenishment cost function, Theorem 1 provides some immediate insights with regard to the benefits of joint replenishments from the manufacturers when the cost components are submodular. Assume ′ that for two replenishment policies X and X and an arbitrary player i ∈ N it holds that ′ ′ PiΓ [X] = PiΓ [X ] and IlΓ [X] ⊇ IlΓ [X ] for every l ∈ E. This means that while player i ′ does exactly the same in X as in X, the set of players who procure any product l jointly ′ from its manufacturer in X includes those players in X as well. Then it follows from the ′ submodularity of CNΓ that even if including i to IhΓ [X ], h ∈ Ei , does not obtain a less costly ′ replenishment policy than X , including i to IhΓ [X] could obtain a less costly replenishment ′ policy than X. When IhΓ [X] ⊃ IhΓ [X ] this means that, all other things held constant, the larger the set of direct replenishers of a product (h), the more likely that the inclusion of a new player (i), who has that product in its product set, to that subset be profitable. This demonstrates the economy of scale in direct replenishments from the manufacturers. Also, inclusion of i to IhΓ [X] could obtain a less costly replenishment policy than X (even if this ′ ′ ′ is not the case in X ) when IhΓ [X] = IhΓ [X ] and for some l ∈ E ∖ h we have IlΓ [X] ⊃ IlΓ [X ]. Thus, expansion of the direct replenishers of any product increases the chances for profitably 11
of direct replenishments in general. This reflects a spill-over effect with regard to the products replenished from the manufacturers. The submodularity of replenishment cost functions has important consequences with regard to the tractability of the optimization problem in (5). Gr¨otschel et al. (1988) show that for a submodular function, the Ellipsoid method can be used to construct a strongly polynomial algorithm for its minimization. Hence, submodularity of cost components implies that the optimal replenishment policies can be found efficiently in CRI situations. In the remainder of this section we show that whenever submodularity conditions of cost components are met, the optimal replenishment policies exhibit a nested property in growing subsets of players. That is, if it is optimal for a subset of players to collectively replenish certain products via their manufacturers, it would also be optimal that this subset of players keep on doing the same in any other subset that contains the former players. Theorem 2. Let Γ be a CRI situation with player set N and product set E, and assume that for every S ⊆ N the replenishment cost function CSΓ is submodular on X ΓS . Let XS∗ be an optimal replenishment policy for S ⊂ N . For every T ⊆ N , T ⊃ S, there exists an optimal replenishment policy XT∗ such that XT∗ ⊇ XS∗ . Proof. Fix T as above and assume that for an arbitrary optimal replenishment policy for T , i.e. XT∗ , we have XS∗ ∖ XT∗ ≠ ∅. Let XT = XT∗ ∪ XS∗ . Clearly XT is a feasible policy for T and furthermore XT ⊇ XS∗ . We show that XT has the lowest possible cost among all other feasible policies. We have CTΓ (XT ) − CTΓ (XT∗ ) = ≤ = = ≤
CTΓ (XT∗ ∪ [XS∗ ∖ XT∗ ]) − CTΓ (XT∗ ) CTΓ ([XS∗ ∩ XT∗ ] ∪ [XS∗ ∖ XT∗ ]) − CTΓ (XS∗ ∩ XT∗ ) CTΓ (XS∗ ) − CTΓ (XS∗ ∩ XT∗ ) CSΓ (XS∗ ) − CSΓ (XS∗ ∩ XT∗ ) 0.
(10) (11) (12) (13) (14)
The steps above are explained accordingly: equality (10) uses the fact that XT = XT∗ ∪ [XS∗ ∖ XT∗ ]. Inequality (11) follows from submodularity of CTΓ . Equality (12) follows since XS∗ = [XS∗ ∩ XT∗ ] ∪ [XS∗ ∖ XT∗ ]. Since XS∗ and XS∗ ∩ ST∗ both are feasible policies for S, equality (13) can be obtained by using Lemma 1. Finally, inequality (14) follows from the optimality of XS∗ for S. Therefore CTΓ (XT ) − CTΓ (XT∗ ) ≤ 0. By assumption, XT∗ is an optimal replenishment policy for T , thus it can only be the case that CTΓ (XT ) = CTΓ (XT∗ ) which implies that XT is also an optimal replenishment strategy for T . Theorem 2 can be interpreted in an alternative way: when submodularity conditions hold, direct replenishers of a product never shrink as the result of including more players to the cooperative organization. A direct consequence of Theorem 2 is that the optimal replenishment policies in CRI situations have a nested structure that allows one to built the optimal policies for larger subsets of players based on those of the smaller subsets.
12
5
Cooperative CRI Games
In this chapter we study the cooperation among players in CRI situations with the help of a class of cooperative cost games associated with these situations. A cooperative cost game is comprised of a set of players and a characteristics function that assigns a real value to every subset of players and the value zero to the empty set. The cooperative cost games associated with CRI situations, hereafter cooperative CRI games, can be constructed by considering the set of players N and defining the characteristics function to be the optimal replenishment cost function. Thus, for every CRI situation Γ ∈ Γ, one can define an associated cooperative cost game by (N, cΓ ) where for every S ⊆ N , cΓ (S) is defined as in equation (5). We denote the set of all CRI games with player set N by GN . A cooperative cost game is called subadditive if its characteristics function is subadditive on the set of players. If a game is subadditive, then the cost for the case where all players are collaborating is never higher than the sum of the costs of any other partitioning of players. Thus participation of all players in the game is always a reasonable option from the point of view of total costs. The next theorem exhibits this property in CRI games. Theorem 3. Cooperative CRI games are subadditive. Proof. Let Γ be a CRI situation with player set N and (N, cΓ ) be its associated cooperative CRI game. Assume S, T ⊆ N such that S∩T = ∅ and let XS∗ and XT∗ be optimal replenishment policies for S and T respectively. Let X = XS∗ ∪ XT∗ and observe that X is a feasible replenishment policy for S ∪ T . By definition of PiΓ it follows that for every i ∈ S we have PiΓ [X] = PiΓ [XS∗ ] and for every i ∈ T we have PiΓ [X] = PiΓ [XT∗ ]. By definition of IiΓ , on the other hand, it follows that for every l ∈ E we have IlΓ [X] = IlΓ [XS∗ ] ∪ IlΓ [XT∗ ]. Thus we have Γ (X) cΓ (S ∪ T ) ≤ CS∪T
= ∑ Ciw (PiΓ [XS∗ ]) + ∑ Ciw (PiΓ [XT∗ ]) + ∑ Clm (IlΓ [XS∗ ] ∪ IlΓ [XT∗ ]) i∈T l∈E m Γ ∗ m Γ ∗ ∗ w Γ ∗ w Γ ∑ Ci (Pi [XS ]) + ∑ Ci (Pi [XT ]) + ∑ Cl (Il [XS ]) + ∑ Cl (Il [XT ]) i∈S i∈T l∈E l∈E Γ ∗ Γ ∗ Γ Γ CS (XS ) + CT (XT ) = c (S) + c (T ) i∈S
≤ =
(15)
where inequality (15) follows from the subadditivity of Clm . It should be noted that the proof of Theorem 3 only uses the subadditivity of direct replenishment cost functions and does not require the subadditivity of indirect replenishment costs.
5.1
Concavity of CRI games
A cooperative cost game is concave if its characteristics function is submodular on the set of players. In a concave game the contributions of players to the cost of growing subsets of 13
players are nonincreasing. The concavity of a cooperative game has important implications with regard to its stability which will be discussed in the next section. Our main result in this section establishes a sufficient condition for the concavity of cooperative CRI games. Theorem 4. Let Γ be a CRI situations with player set N , product set E, and vector of player-specific product sets (Ei )i∈N . Assume that Ciw is submodular on Ei for every i ∈ N and Clm is submodular on N for every l ∈ E. The associated cooperative CRI game is concave. Proof. Consider a player i ∈ N and let S ⊂ T ⊂ N ∖ i. In the rest of the proof we use the shorthand notation S i = S ∪ i. Let XT∗ i , XT∗ , XS∗i , and XS∗ be the optimal replenishment policies for T i , T , S i , and S respectively in such a way that XT∗ i ⊇ XT∗ ⊇ XS∗ , and XT∗ i ⊇ XS∗i ⊇ XS∗ . From Theorem 2 we know that such optimal replenishment policies always can be found. We must show that CTΓi (XT∗ i ) − CTΓ (XT∗ ) ≤ CSΓi (XS∗i ) − CSΓ (XS∗ ).
(16)
By subtracting Ciw (Ei ) from both sides of (16) and noting (with the help of Lemma 1) that CTΓ (XT∗ ) + Ciw (Ei ) = CTΓi (XT∗ ) and CSΓ (XS∗ ) + Ciw (Ei ) = CSΓi (XS∗ ), it can be observed that in order to show (16) it is sufficient to show that CTΓi (XT∗ i ) − CTΓi (XT∗ ) ≤ CSΓi (XS∗i ) − CSΓi (XS∗ ).
(17)
To prove that the latter holds, note that XT∗ i is an optimal replenishment policy for T i so it is at most as costly as any other feasible replenishment policy for T i . Since XT∗ ∪[XS∗i ∖XT∗ ] ⊆ X ΓS∪i , it is the case that XT∗ ∪ [XS∗i ∖ XT∗ ] is a feasible replenishment policy for T i . Therefore: CTΓi (XT∗ i ) − CTΓi (XT∗ ) ≤ CTΓi (XT∗ ∪ [XS∗i ∖ XT∗ ]) − CTΓi (XT∗ )
≤ CTΓi ([XS∗i ∩ XT∗ ] ∪ [XS∗i ∖ XT∗ ]) − CTΓi (XS∗i ∩ XT∗ )
(18)
=
(19)
= ≤
CTΓi (XS∗i ) − CTΓi (XS∗i ∩ XT∗ ) CSΓi (XS∗i ) − CSΓi (XS∗i ∩ XT∗ ) CSΓi (XS∗i ) − CSΓi (XS∗ )
(20) (21)
Inequality (18) follows from the fact that XS∗i ∩XT∗ ⊆ XT∗ i and drawing upon the submodularity of CTΓi (in light of Theorem 1). Equality (19) follows since [XS∗i ∩ XT∗ ] ∪ [XS∗i ∖ XT∗ ] = XS∗i . Equality (20) holds as both XS∗i and XS∗i ∩ XT∗ are feasible policies for S i and by Lemma 1 we have CTΓi (XS∗i ) = CSΓi (XS∗i ) + ∑j∈T i ∖S i Cjw (Ej ) and CTΓi (XS∗i ∩ XT∗ ) = CSΓi (XS∗i ∩ XT∗ ) + ∑j∈T i ∖S i Cjw (Ej ). Finally, inequality (21) holds since XS∗i ∩ XT∗ is a feasible replenishment policy for S which means that CSΓ (XS∗i ∩ XT∗ ) ≥ CSΓ (XS∗ ) therefore CSΓ (XS∗i ∩ XT∗ )+Ciw (Ei ) ≥ CSΓ (XS∗ ) + Ciw (Ei ) and eventually CSΓi (XS∗i ∩ XT∗ ) ≥ CSΓi (XS∗ ). Thus (17) holds and the proof is complete.
14
5.2
Allocation rules
An important question in every cooperative situation concerns the division of joint costs among the participating members. The subadditivity of CRI games has already established that cooperation among players in their replenishment activities could reduce their total costs. However, the decisions with regard to joining the cooperative organization are made by rational and self-interested players who evaluate the outcomes of their participation on an individual basis. Hence, it is crucially important to ensure that, when dividing the total costs, every player is satisfied with its individual allocation. This is the main theme of the cost-sharing problem discussed in this section. An allocation for players in N is β = (βi )i∈N which determines the cost to be paid by each player. There are certain properties that a desirable allocation must satisfy. One of the most basic desirable properties of an allocation is the efficiency property which requires that the total cost of the set of all players (grand coalition) is entirely divided among the players. Considering (N, c) to be a generic cooperative cost game, an allocation β is efficient for (N, c) if ∑i∈N βi = c(N ). An allocation rule is individually rational for (N, c) if for every i ∈ N it holds that βi ≤ c(i). An allocation rule is stable in (N, c) if for any S ⊂ N it holds that ∑i∈S βi ≤ c(S). A stable allocations provides sufficient incentives for all players not to break apart from the grand coalition to collaborative in sub-coalitions. The core of a game contains all of its efficient and stable allocations. Due to the appealing features of allocations in the core, an important question with regard to every cooperative game is the nonemptiness of its core. In general the core of CRI games can be empty. However, Shapley (1971) prove that the core of a concave game is always nonempty. Our results in the previous section regarding the concavity of CRI games also establishes a sufficient condition for the nonemptiness of the core in these games. Corollary 1. Let Γ be a CRI situations with player set N , product set E, and vector of player-specific product sets (Ei )i∈N . Assume that Ciw is submodular on Ei for every i ∈ N and Clm is submodular on N for every l ∈ E. The core of the associated cooperative CRI game is nonempty. Next, we present an observation with regard to the lower bound of players’ allocations in any core allocation in CRI games. Theorem 5. Let Γ be a CRI situation with player set N such that the core of its associated game (N, cΓ ) is nonempty. For every allocation β in the core, every optimal replenishment policy for the grand coalition X ∗ , and every player i ∈ N it holds that βi ≥ Ciw (PiΓ [X ∗ ]). Proof. Let β be an allocation in the core of CRI game (N, cΓ ). By definition of core allocation it must be that ∑j∈N ∖i βj ≤ cΓ (N ∖ i). Also the efficiency of β requires that ∑j∈N βj = cΓ (N ). Thus we can write βi = cΓ (N ) − ∑ βj ≥ cΓ (N ) − cΓ (N ∖ i) (22) j∈N ∖i
To complete the proof it suffices to show that cΓ (N ) − cΓ (N ∖ i) ≥ Ciw (PiΓ [X ∗ ]). 15
Let X ∗ be an optimal replenishment policy for N and consider a player i ∈ N . Let X = that PjΓ [X] = PjΓ [X ∗ ]. Also we have IlΓ [X] = IlΓ [X ∗ ] i ∈ IlΓ [X ∗ ]. Since X is a feasible replenishment policy for N ∖ i, we can write: X ∗ ∩ X ΓN ∖i . For every j ∈ N ∖ i it holds if i ∉ IlΓ [X ∗ ] and IlΓ [X] = IlΓ [X ∗ ] ∖ i if CNΓ ∖i (X) = ∑ Cjw (PjΓ [X ∗ ]) + j∈N ∖i
∑
l∈E∶i∉IlΓ [X ∗ ]
Clm (IlΓ [X ∗ ]) +
∑
l∈E∶i∈IlΓ [X ∗ ]
Clm (IlΓ [X ∗ ] ∖ i)
(23)
Every optimal replenishment policy for N ∖ i is at most as costly as X, thus cΓ (N ∖ i) ≤ CNΓ ∖i (X). We have cΓ (N ) − cΓ (N ∖ i) ≥ cΓ (N ) − CNΓ ∖i (X) = ∑ Cjw (PjΓ [X ∗ ]) + j∈N
−
∑
l∈E∶i∉IlΓ [X ∗ ]
∑ Cjw (PjΓ [X ∗ ]) −
j∈N ∖i
= Ciw (PiΓ [X ∗ ]) +
∑
Clm (IlΓ [X ∗ ]) +
l∈E∶i∉IlΓ [X ∗ ]
∑
l∈E∶i∈IlΓ [X ∗ ]
∑
l∈E∶i∈IlΓ [X ∗ ]
Clm (IlΓ [X ∗ ]) −
∑
Clm (IlΓ [X ∗ ])
l∈E∶i∈IlΓ [X ∗ ]
Clm (IlΓ [X ∗ ] ∖ i)
Clm (IlΓ [X ∗ ]) − Clm (IlΓ [X ∗ ] ∖ i).
For every l ∈ E the function Clm is nondecreasing on N thus Clm (IlΓ [X ∗ ])−Clm (IlΓ [X ∗ ]∖i) > 0 and consequently we have cΓ (N ) − cΓ (N ∖ i) ≥ Ciw (PiΓ [X ∗ ]) and the proof is complete. Theorem 5 asserts that in every stable allocation, each player has to pay at least its indirect replenishment cost for the product it obtains from the intermediaries. Therefore, irrespective of the contribution of a player to the total cost savings in grand coalition, the indirect replenishment cost of no player would be subsidized in any core allocation. It is straightforward to observe that if a player replenishes a product directly from its manufacturer in an optimal replenishment policy, then in addition to its indirect replenishment cost, the player must pay a positive portion of its direct replenishment cost as well. Note that Theorem 5 does not require submodularity of cost components and holds whenever a CRI game has a nonempty core. In order for the cooperating organization to be able to repeatedly carry out joint replenishments without the need of renegotiating the appropriate allocations, a formal scheme for allocating the costs in different situations should be in place. This requirement is formalized with the notion of allocation rule. An allocation rule is a function which determines an allocation for every game in its domain of definition. For CRI games with player set N , an allocation rule is defined with the function B ∶ GN → RN . The desirability of an allocation rule can be evaluated by the desirable properties of the allocations it generates. For example, an allocation rule is called efficient if it always generates efficient allocations. The allocation to player i under allocation rule B is denoted with Bi . A well-known allocation rule in cooperative games literature is the celebrated Shapley
16
value (Shapley, 1953). Shapley value of a cost game3 (N, c), i.e. Φ(N, c), is calculated by the following formula: ∣S∣!(n − ∣S∣ − 1)! [c(S ∪ i) − c(S)] , n! S⊆N ∖i
Φi (N, c) = ∑
for every i ∈ N.
(24)
Shapley value divides the total cost of the grand coalition according to average contributions of the players in all subsets that they form. Several characterizations of Shapley value are given in the literature (see Peleg and Sudh¨olter (2007)). Although in general Shapley value might not belong to the core of a game, in concave games it always belongs to the core (Shapley, 1971). Therefore, in CRI situations with submodular cost components, players can always divide the costs among themselves in a stable way by implementing Shapley value as the allocation rule. In the next section we demonstrate another appealing property of Shapley value for CRI situations.
6
Strategic Participation in Two-stage CRI Games
An implicit assumption made in the cooperative CRI game studied in previous section was that once a player decides to join the cooperative organization, it puts forward its entire player-specific product set so that the cooperative organization decides the replenishment sources of all the products in that set in order to optimize the replenishment cost of the grand coalition. However, in reality the players’ decisions with regard to their participation in cooperative replenishment activities can be more detailed. One of the major dimensions of such decisions is the extent of the players’ participation in cooperative organization in terms of the products whose replenishment policies are delegated to the cooperative organization. In this section, the players are given with the option to strategically participate in the cooperative replenishment organization. Thus, by allowing players to withhold some of their required products from the cooperative organization, they are able to partially cooperate. The question we investigate is the conditions under which strategic participation of the players would not negatively affect the system as a whole. A crucial input in the players’ strategic decision making processes is the allocation rule that will be implemented in the cooperative organization to divide the joint costs. Hence, our analysis in this section enables us to comment on the appropriateness of different allocation rules for cooperative replenishment organizations. In order to achieve the latter, we construct a two-stage game comprising a noncooperative stage followed by a subsequent cooperative stage. The two-stage CRI game allows us to investigate the ability of allocation rules to induce full versus partial participation of the players. The sequence of events in our two-stage CRI game is as follows. First, the cooperative organization announces an allocation rule to be employed in the second (cooperative) stage for dividing the joint costs. With this knowledge, the players make their decisions regarding the extend of their participation in cooperative replenishment organization. That is, each 3
Note that the Shapley value is originally proposed for saving games.
17
Figure 2: Sequence of events in two-stage CRI game player strategically chooses the products it would replenish by itself outside the cooperative organization and announces the rest of its products to be replenished in the cooperative organization. The cooperative CRI game which is played in the second stage is associated with the modified version of the original CRI situation wherein only the announced product sets of players are included. The joint costs will be distributed according to the pre-fixed allocation rule. We assume that all information contained in the situation is known by all players. Figure (2) illustrates the sequence of events. We formulate the players’ participation strategies in terms of products they withhold from the cooperative organization and replenish individually from the intermediaries. Given a CRI situation Γ with player set N and a player i ∈ N , let E i ⊆ Ei be the set of withheld products of i. In this manner, E i is the participation strategy of player i. The vector of players’ strategies E = (E i )i∈N is refereed to as a participation strategy profile. One can use a participation strategy profile to construct a modified CRI situation wherein only the products intended by the players are present. We define the modified situation Γ∖E by excluding the products manifested in the players’ participation strategies in the following manner: Γ ∖ E = (N, E, E ∖ E, C w , C m ) (25) where E∖E = (Ei ∖E i )i∈N is the modified vector of player-specific product sets. Subsequently, the game associated with the modified situation, to be played in the second stage, is defined by (N, cΓ∖E ). The strategic participation decisions are made considering the outcome of cooperative replenishment in terms of costs allocated to each player. Suppose that a participation strategy profile E as well as an allocation rule B ∶ GN → RN are given. The individual replenishment cost of player i ∈ N in the associated two-stage CRI game is ziΓ (E; B) = Ciw (E i ) + Bi (N, cΓ∖E ).
(26)
The individual replenishment cost of player i is comprised of the indirect replenishment cost of player i for its withheld products and its allocation under B in the modified cooperative CRI game associated with the modified CRI situation Γ ∖ E. A measure for assessing the performance of the system as a whole is the sum of individual replenishment costs of all players. Among all strategies profiles, we highlight the complete
18
participation strategy profile,
o
o
E = (E i = ∅)i∈N , wherein all players announce their entire player-specific product sets to the cooperative organization. The importance of this strategy profile stems from the fact that under any efficient allocation rule, the sum of individual replenishment costs of all players is minimized with the choice of complete participation strategy profile. This is formalized in the following lemma. Lemma 2. Let Γ be a CRI situation with player set N and B ∶ GN → RN be an efficient o allocation rule for cooperative CRI games. Complete participation strategy profile E minimizes the sum of individual replenishment costs of all players in the associated two-stage CRI game. Proof. By efficiency of B, for every E we have ∑ ziΓ (E; B) = ∑ Ciw (E i ) + cΓ∖E (N ). i∈N
i∈N
Suppose E is the strategy that minimizes ∑i∈N ziΓ (E; B) and it is not the complete participation strategy profile. Let X ∗ be an optimal replenishment policy for N in the modified situation Γ ∖ E. We have ∑ ziΓ (E; B) = ∑ Ciw (E i ) + CNΓ∖E (X ∗ ) i∈N
i∈N
= ∑ Ciw (E i ) + ∑ Ciw (PiΓ∖E [X ∗ ]) + ∑ Clm (IlΓ∖E [X ∗ ]) i∈N
≥ = =
i∈N
l∈E
∑ ∑ Clm (IlΓ∖E [X ∗ ]) i∈N l∈E m w Γ ∗ ∑ Ci (Pi [X ]) + ∑ Cl (IlΓ [X ∗ ]) i∈N l∈E Γ ∗ Γ CN (X ) ≥ c (N ) Ciw (PiΓ∖E [X ∗ ] ∪ E i ) +
(27) (28) (29)
Inequality (27) draws upon the subadditivity of Ciw and equality (28) used the fact that (a) PiΓ∖E [X ∗ ] ∪ E i = PiΓ [X ∗ ] and (b) IlΓ∖E [X ∗ ] = IlΓ [X ∗ ]. Finally, inequality (in 29) holds since X ∗ ⊂ X ΓN and it is possible to find less costly replenishment policies than X ∗ in X ΓN . Despite the optimality of complete participation strategy profile for the performance of the aggregated system, players may choose other participation strategies if such strategies result in lower individual replenishment costs for them. Therefore, in order to assess the performance of the system in this case one needs to understand the individual decision making processes of the players in choosing their participation strategies. In two-stage CRI games introduced above, the individual decision making processes of the players are intertwined as the individual replenishment cost of players will be affected by the choices of strategies of other players as well. Thus the rational players choose their individual participation strategies in anticipation of the other players’ choices in order to minimize their individual replenishment costs. 19
Given the allocation rule, the best choice of strategy for a player is the strategy which always results in the lowest possible individual replenishment cost irrespective other players’ choices. Upon existence, such a strategy is called dominant. Accordingly, a strategy profile is dominant if the individual strategies of all players are dominant. The following definition formalizes the notion of dominant strategies and their relation to the allocation rules. We use the notation E −i in order to refer to strategy profile E without its i’s element. Definition 2. Let Γ be a CRI situation with player set N , product set E, and the vector of player-specific product sets (Ei )i∈N . The allocation rule B induces the participation strategy d profile E in dominant strategies if for every player i ∈ N and every participation strategy d profile E it holds that ziΓ (E; B) ≥ ziΓ (E −i , E i ; B). In the next step we present the main result of this section regarding the ability of Shapley value to induce complete participation of all players in two-stage CRI games. Theorem 6. Shapley value induces the complete participation strategy profile in dominant strategies in every two-stage CRI game. Proof. Let Γ be a CRI situation with player set N , product set E, and the vector of playerspecific product sets (Ei )i∈N . We must show that for every player i ∈ N and every E, it holds that ziΓ (E; Φ) ≥ ziΓ (E −i , ∅; Φ), or equivalently Ciw (E i ) + Φi (N, cΓ∖E ) ≥ Φi (N, cΓ∖E −i ).
(30)
The left hand side of (30) can be written as ∣S∣!(n − ∣S∣ − 1)! Γ∖E (S ∪ i) − cΓ∖E (S)] [c n! S⊆N ∖i ∣S∣!(n − ∣S∣ − 1)! w = ∑ [Ci (E i ) + cΓ∖E (S ∪ i) − cΓ∖E (S)] n! S⊆N ∖i Ciw (E i ) + ∑
(31)
where the equality (31) follows from the fact that ∑S⊆N ∖i ∣S∣!(n − ∣S∣ − 1)!/n! = 1. We focus
20
on the expression inside the braces in (31) for an arbitrary S ⊆ N ∖ i: Ciw (E i ) + cΓ∖E (S ∪ i) − cΓ∖E (S)
Γ∖E = Ciw (E i ) + CS∪i (X ∗ ) − cΓ∖E (S)
(32)
= Ciw (E i ) + ∑ Cjw (PjΓ∖E [X ∗ ]) + ∑ Clm (IlΓ∖E [X ∗ ]) − cΓ∖E (S) j∈S∪i
≥ Ciw (PiΓ∖E [X ∗ ] ∪ E i ) + ∑ Cjw (PjΓ∖E [X ∗ ]) + ∑ Clm (IlΓ∖E [X ∗ ]) − cΓ∖E (S) j∈S
=
(35)
l∈E
Γ∖E −i (X ∗ ) − cΓ∖E −i (S) = CS∪i
≥
(34)
l∈E
Γ∖E Γ∖E ∑ Cjw (Pj −i [X ∗ ]) + ∑ Clm (Il −i [X ∗ ]) − cΓ∖E −i (S) j∈S∪i
(33)
l∈E
min Γ∖E X⊆X S∪i −i
(36)
Γ∖E −i CS∪i (X) − cΓ∖E −i (S)
(37)
= cΓ∖E −i (S ∪ i) − cΓ∖E −i (S). In (32) we let X ∗ be an optimal replenishment policy for S ∪ i in the modified situation Γ∖E (X ∗ ). To obtain inequality (34) the subadditivity of Γ ∖ E and in (33) we expanded CS∪i Ciw is used. In (35) we use the following observations: (a) PiΓ∖E [X ∗ ] ∪ E i = PiΓ∖E −i [X ∗ ], (b) PjΓ∖E [X ∗ ] = PjΓ∖E −i [X ∗ ] for every j ∈ S, (c) IlΓ∖E [X ∗ ] = IlΓ∖E −i [X ∗ ], and (d) cΓ∖E (S) =
Γ∖E −i cΓ∖E −i (S). In (36) we combined the first two terms in (35). Finally, since X ∗ ⊂ X S∪i , there Γ∖E −i might exists better replenishment policies available when minimizing CS∪i thus inequality (37) follows. Therefore, for every S ⊆ N ∖ i we have
Ciw (E i ) + cΓ∖E (S ∪ i) − cΓ∖E (S) ≥ cΓ∖E −i (S ∪ i) − cΓ∖E −i (S).
(38)
Multiplying the both side by ∣S∣!(n − ∣S∣ − 1)!/n! and summing over all S ⊆ N ∖ i we get ∣S∣!(n − ∣S∣ − 1)! w [Ci (E i ) + cΓ∖E (S ∪ i) − cΓ∖E (S)] n! S⊆N ∖i ∣S∣!(n − ∣S∣ − 1)! Γ∖E −i [c (S ∪ i) − cΓ∖E −i (S)] . ≥ ∑ n! S⊆N ∖i ∑
Therefore, (30) holds and proof is complete. Theorem 6 exhibits a very appealing feature of Shapley value in CRI situations. That is, if Shapley value is set as the allocation rule, no player can obtain any benefit by withholding some of its products from the cooperative organization.
21
7
Disregarding Intermediaries in the Cooperative Stage
Drawing upon the logic of two-stage games elaborated previously, the purpose of this section is to answer the following question: is it really necessary to consider the replenishment options from intermediaries in the cooperative stage? Alternatively, what happens if the cooperation in second stage disregards the options to replenish from intermediaries? The motivation for this question is twofold. First, it has already been established that in every optimal replenishment policy, one can separate the player-product pairs that are replenished via the intermediaries from those that are replenished via the manufacturers. Thus, in a strategic game where players are free to withhold some of their products and replenish them via the intermediaries, there can be strategy profiles with partial cooperation which result in total costs equal to those in complete cooperation. Second, from the practical point of view it might be easier to set up the cooperative organization to only deal with direct replenishments from the manufacturers and the indirect replenishments be left out for players to manage individually. In this section we prove that for the optimal performance of the system it is vital that indirect replenishment options be considered in the cooperative stage. To carry out the analysis, we start by constructing an alternative cooperative game which disregards the options to replenish from the intermediaries. Given the CRI situation Γ, define the direct replenishment game (N, c˜Γ ) where for every S ⊆ N : c˜Γ (S) = CSΓ (X ΓS ) = ∑ Clm (IlΓ [X ΓS ])
(39)
l∈E
The cost to every coalition in the direct replenishment game is the direct replenishment cost of all products of every player. In this manner, direct replenishment games disregard the intermediaries. We denote the set of all direct replenishment games with player set N by ˜ N. G Subsequently, we define an alternative two-stage game associated with every CRI situation in the same spirit as in the previous section. In the first stage of this alternative two-stage game, each player decides its withheld product set which it replenishes individually from the intermediaries. In the second stage of the alternative two-stage game, the direct replenishment game is played and the costs will be divided according to a pre-specified allocation rule which is defined for direct replenishment games. We refer to this two-stage game as the alternative two-stage CRI game. Suppose the situation Γ with player set N , product set E, and vector of player-specific product sets (Ei )i∈N is given. For a participation strategy profile E and an allocation rule ˜ N → RN , the individual replenishment cost to player i in the alternative two-stage CRI B˜ ∶ G game is yiΓ (E; B) = Cim (E i ) + Bi (N, c˜Γ∖E ) (40) The following observation shows that for every alternative two-stage CRI games there exists participation strategies that make the total individual replenishment costs of the players to be as low as the minimum total replenishment cost of the corresponding (original) two-stage CRI game wherein the indirect replenishment options are also considered in the cooperative stage. 22
Lemma 3. Let Γ be a CRI situation with player set N , product set E, and the vector of player-specific product sets (Ei )i∈N . Let X ∗ be an optimal replenishment policy for N and B˜ ˜ N . The participation strategy profile be an arbitrary efficient allocation rule for G ∗
∗
E = (E i = PiΓ [X ∗ ])i∈N , minimizes the sum of individual replenishment costs of all players in the alternative two-stage ∗ ˜ = cΓ (N ). CRI game. Moreover, we have ∑i∈N yiΓ (E ; B) Proof. We show that every participation strategy profile in the alternative two-stage CRI game associated with situation Γ corresponds to a replenishment policy in this situation. Let E be an arbitrary participation strategy profile. By efficiency of B˜ we have ˜ = ∑ Ciw (E i ) + c˜Γ∖E (N ) = ∑ Ciw (E i ) + ∑ C m (I Γ∖E [X Γ∖E ∑ yiΓ (E; B) N ]). l l i∈N
i∈N
i∈N
l∈E
Γ∖E Let Xi = {(i, l)∣l ∈ Ei ∖ E i } and X = ⋃i∈N Xi . Note that we have IlΓ∖E [X N ] = IlΓ [X]. Also, PiΓ [X] = E i . Finally, it holds that IlΓ∖E [X] = IlΓ [X]. Therefore, we have
˜ = ∑ Ciw (X) + ∑ C m (I Γ [X]) = CNΓ (X). ∑ yiΓ (E; B) l l i∈N
i∈N
l∈E
Thus the sum of individual replenishment costs in the alternative two-stage CRI game under any efficient allocation rule equals the cost of a feasible replenishment policy for N in the corresponding CRI situation. Hence, the former is minimized when the latter is minimized. Let X ∗ be an optimal replenishment policy for N in Γ. It is straightforward to see that the ∗ ∗ participation strategy profile corresponding to X ∗ is E = (E i = PiΓ [X ∗ ])i∈N . Also, in this ∗ ˜ = cΓ (N ). case we have ∑i∈N yiΓ (E ; B) According to Lemma 3, in order to achieve the minimum cost for the entire system in the alternative two-stage CRI game, all players must choose to withhold the exact same set of products from the cooperative organization that they would have replenished individually in an optimal replenishment policy for the corresponding cooperative CRI game. But is there an allocation rule B˜ for the alternative two-stage CRI game that induces the latter participation strategies? As we show below, the answer to this question is negative if we require an allocation rule to induce an optimal participation strategy profile in dominant strategies. However, the answer remains negative even if we use a weaker condition than strategic dominance for induction of an strategy profile by an allocation rule. Definition 3. Let Γ be a CRI situation with player set N , product set E, and the vector of player-specific product sets (Ei )i∈N . The allocation rule B˜ induces the participation strategy e profile E in Nash equilibrium if for every player i ∈ N and every individual participation e ˜ ≥ y Γ (E e ; B). ˜ strategy E i it holds that yiΓ (E −i , E i ; B) i The deviation from a Nash equilibrium strategy profile is unprofitable for a player if all other players choose their corresponding Nash equilibrium participation strategies. The 23
requirement for an allocation rule to induce a strategy profile in Nash equilibrium is weaker than that in dominant strategies since if an allocation rule induces a strategy profile in dominant strategies, it also does so in Nash equilibrium. However, the reverse is not necessarily true. The last result in this paper shows that no allocation rule for direct replenishment games can be found that induces an optimal participation strategy profile in Nash equilibrium (and consequently dominant strategies) in alternative two-stage CRI games associated with every CRI situation. ˜ N → RN that induces an opTheorem 7. There exists no efficient allocation rule B˜ ∶ G timal participation strategy profile in Nash equilibrium in alternative two-stage CRI game associated with every CRI situation with player set N . Proof. Proof is done via a counterexample. Consider situation Γ with N = {1, 2} and E = {a, b}. While player 1 replenishes both products, player two only replenishes product a, i.e. E1 = {a, b} and E2 = {a}. The cost components in this situation are as following. For i ∈ N we have Ciw (∅) = 0, Ciw ({a}) = 9, Ciw ({b}) = 9, Ciw (E) = 16, and for l ∈ E we have Clm (∅) = 0,
Clm ({1}) = Clm ({2}) = 10,
Clm (N ) = 15.
It can be seen that cΓ ({1}) = C1w (E) = 16,
cΓ ({2}) = C2w ({a}) = 9,
cΓ (N ) = C1w ({b}) + Cam (N ) = 9 + 15 = 24.
The optimal replenishment policy for N requires that player 1 replenishes product b from the intermediary and both players replenish product a from its manufacturer. Consider the alternative two-stage game associated with this situation. Observe that the ∗ ∗ ∗ optimal participation strategy profile is E = (E 1 = {b}, E 2 = ∅). The modified situation associated with the latter participation strategy profile is ∗
Γ ∖ E = ({1, 2}, {a, b}, ({a}, {a}), C w , C m ). In the modified situation, players are identical in every way except their names. For the ∗ Γ∖E direct replenishment game associated with this situation, (N, c˜ ), we have ∗
c˜Γ∖E ({1}) = C1w ({a}) = 9,
∗
c˜Γ∖E ({2}) = C2w ({a}) = 9,
∗
c˜Γ∖E (N ) = Cam (E) = 15.
Suppose that an efficient allocation rule has been chosen for the system which divides ∗ ˜ the costs equally in this situation (Shapley values does the latter). We get B(N, c˜Γ∖E ) = (7.5, 7.5) and consequently ∗ ˜ = C1w ({b}) + B˜1 (N, c˜Γ∖E ) = 9 + 7.5 = 16.5. y1 (E ; B) ∗
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If player 2 does not change its strategy and player 1 chooses E 1 = {a, b}, the individual re∗ ˜ = 16 which is lower than its plenishment cost for player 1 in this case would be y1 (E 1 , E 2 ; B) individual replenishment cost under the choice of its optimal participation strategy. Therefore the equal cost allocation rule (Shapley value) does not induce the optimal participation in Nash equilibrium in this situation. To induce the optimal participation strategy profile in Nash equilibrium, an allocation rule must give player 1 an allocation smaller than that of player 2. Suppose we select another efficient allocation rule for the direct replenishment games which in this situation gives an allocation smaller than 7 to player 1. Although such an allocation rule induces the optimal participation strategy profile in Nash equilibrium for this situation, it would not be able to do so in the alternative situation which is identical to Γ in every aspect expect for players’ names which are now switched. In this alternative situation only an allocation rule would induce the optimal participation strategy profile which gives a lower allocation to the new player 2 than the new player 1. However, the direct replenishment game associated with the optimal participation strategy profile for the alternative situation would be identical to that for the original situation and the chosen allocation rule would give a lower allocation to the new player 1. We conclude that there exists no allocation rule for the direct replenishment game which induces the optimal strategy profiles in alternative two-stage CRI games. In light of Theorem 7, we conclude that the optimal participation of players in cooperative replenishment systems which disregard the presence of intermediaries cannot be guaranteed. Hence, to obtain the system wide optimal performance, it is crucial that the cooperative organizations do consider the players’ options for replenishments from the intermediaries.
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Final Remarks
In this paper, we discussed potential opportunities for cooperative replenishment among downstream players in the presence of intermediaries. The main insight obtained from our study is that under certain assumptions cooperation enables downstream player to bypass intermediaries and directly replenish from the original manufacturers. The downstream cooperation could increase the overall supply chain efficiency via eliminating double marginalization and excessive inventories. The submodularity of the corresponding cost function implies that the optimal replenishment strategies can be calculated efficiently. In lieu of the concave structure of CRI games, players can always find stable allocations. The model developed in this paper, along with its underlying assumptions, fit many practical situations. In subadditive games the larger the number of the players the higher the possibility of overall cost savings. In this regard, a desirable property of allocation rule is its support for the growth of subsets of players. The class of population monotone allocation rules (Sprumont, 1990) guarantees that the allocated payment to each player does not increase as the result of expanding the player set. As proved by Sprumont (1990), in concave games the Shapley value is a population monotone allocation rule. Thus, implementation of Shapley value in CRI games with submodular cost components also population monotonicity.
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There are several other perspectives to consider when the supply chain involves intermediaries. Joint replenishment activities are likely to affect the pricing schemes of manufacturers, intermediaries, and downstream players. James and Dana (2012) study the impact cooperative purchasing organization on price competition among the suppliers. Also, cooperative replenishment in the presence of intermediaries might be investigated in situations where the demand is nonuniform yet deterministic. Thus cooperative lot-sizing situations with the presence of intermediaries would be an immediate direction for further research. Finally, it worth mentioning that although cooperative purchasing results in lower purchasing prices for the downstream players, competition in price-setting may leave then worse off. An instance of this situation observed by Chen and Roma (2011) where they show that competitive pricing by two downstream players cooperatively purchasing from a supplier with known discount schedule could only make the less efficient player better off. Arya et al. (2015) shows that the intermediary can help coordinating the supply chain when the pricing efforts of the manufacturer and players are taken into consideration. In this respect, the benefits of cooperative replenishment must be also tested in the presence of price-setting downstream players as well as supply sources.
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