Stochastic programming for decentralized newsvendor with transshipment

Int. J. Production Economics 137 (2012) 292–303

Contents lists available at SciVerse ScienceDirect

Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Stochastic programming for decentralized newsvendor with transshipment Nichalin S. Summerfield n, Moshe Dror MIS Department, The University of Arizona, Tucson, AZ 85721, United States

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 August 2011 Accepted 14 February 2012 Available online 23 February 2012

This paper discusses a family of two-stage decentralized inventory problems using a unifying framework (taxonomy) depicted as a multilevel graph. This framework allows us to model and link different problems of competing retailers who independently procure inventory in response to uncertain demand and anticipated inventory decisions of other retailers. In this family of problems, in the ex-post stage, the retailers exercise recourse actions in response to the realized demand and competitors’ chosen procurement levels. For example, retailers could coordinate inventory transshipment to satisfy shortage with overage based on profit sharing agreements. Our framework provides a unifying parsimonious view using a single methodological prism for a variety of problems. Equally importantly, as recourse options are laid out, our framework clarifies and contributes a modeling connection between problems in a clear taxonomy of models. This unifying perspective explicitly links work that appeared in isolation and offers future research directions. & 2012 Elsevier B.V. All rights reserved.

Keywords: Stochastic programming Inventory management Cooperative and non-cooperative game

1. Introduction The query in this paper pertains to a group of independent retailers, like independent car dealerships, aviation parts merchants, and others, who decide, each one independently, about their individual procurement strategy for a single product of uncertain demand. The retailers’ initial procurement decisions account for all the available options beyond the point of learning the actual demands. For instance, if a car dealership does not have a car requested by a customer, it might consider acquiring the car from a competing dealer. All parties have to be appropriately compensated. That is, the car dealership can anticipate ex post cooperation options with other dealerships by means of a transshipment mechanism, etc. We propose to view this setting and all its problems through the prism of the stochastic programming methodology. More to the point, we view the inventory procurement options of a retailer competing with similar retailers as a strategic decision of how much to order at the outset (ex ante) without knowing the exact nature of the demand. Nevertheless, it is presumed that the retailer has different options of communicating and collaborating with the competitor(s) at a later stage and responding ex post in a variety of options once the retailer learns the true demand. Without inferring any gender bias, we refer to an individual retailer in this setting by the generic ‘she.’

n

Corresponding author. E-mail addresses: [email protected] (N.S. Summerfield), [email protected] (M. Dror). 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2012.02.014

Stochastic programming is a mathematical methodology for solving optimization problems with time-dependent stochastic variables representing uncertainty of future values. The methodology was introduced by Dantzig (1955), Beale (1955), Walkup and Wets (1967), and among others. For basic exposition, definitions, concepts, and additional references, see Birge and Louveaux (1997). Stochastic programming model envisions two (or more) decision stages: given a probability space (O, X,P ) and certain known initial information, e.g., cost parameters and demand uncertainty parameters, the first-stage decisions are taken at some cost. Subsequently, the true value of demand is revealed. At that point, another set of decisions is taken at some cost in response to the realized demand and the first stage decision. The common objective is to minimize the expected cost or to maximize the expected profit. Stochastic programming methodology has also been used to analyze centralized inventory system with demand uncertainty (Chen and Zhang, 2009; Ozen et al., 2009). As many stochastic programming professionals know well, for a majority of real-life problems, after the realization of the demand, the second-stage decision domain can be very large and computationally ‘unmanageable’. To mitigate this problem, one has to consider restrictions in the recourse options by limiting the recourse policies in the second stage. We have to examine carefully what restrictions lead to computational resolution of the strategy domain and potential real-life implementation of effective solutions. In this exposition, we present a systematic exploration process of the different recourse restrictions that are applicable in the case of decentralized inventory with transshipment. The presumed aim of imposing recourse restrictions in the second stage of our stochastic

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

programming model is conceptual and mathematical ‘tractability’ that retains a meaningful modeling interpretation. A loose meaning of mathematical tractability is being able to prescribe an algorithmic approach that allows for the computation of solutions (either exact or approximate) in a reasonable time. Not every real-life decentralized inventory procurement problem with all possible decision options allows for computational solvability. There are important trade-offs between allowing a richer set of recourse options and being able to compute a solution. For instance, allowing retailer i to consider any subset of retailers for cooperation ex post brings forth from the outset the issue of real-life computability. To illustrate our concept of restricted recourse we point to stochastic demand problem in Dror et al. (1989), where a recourse restriction is imposed on vehicle routing options versus Dror (1993) where such recourse restriction is removed. Thus, one of the aims of this paper’s taxonomy is also to inform the reader of the modeling options set by the recourse assumption that allow to consider computational tractability in tandem with the expressiveness of the modeling representation. Our contribution can be summarized as follows: we propose a single, explicit, well understood, methodological prism of stochastic programming with recourse for an important family of inventory problems that in the past have been modeled using ad hoc methods or modeling points of view. The proposed taxonomy graph provides a framework for a systematic examination of operational options, their mathematical consequences, and available solution properties. At the same time we indicate modeling limitations and open research venues for a number of models examined in the past. In addition, we contribute results and questions related to the adequate pricing of the swapped items and the existence of Walrasian equilibria for the salvage goods market. Decentralized inventory system is applicable to many industries. For instance, the operation of independent lumber companies in Scandinavia that motivates the work of Sandsmark (2009) and the pooling of spare parts inventories at air carrier companies in Brussels (Wong et al., 2006) are all of similar flavor. It has been studied by a number of various authors under many specific assumptions. With apologies for being too ‘‘narrow’’ with the citations of related work, we only mention the more recent papers, since the vast body of literature does not allow a single paper to ‘‘do’’ justice to all pertinent contributions. For a start, Anupindi et al. (2001) analyze two-stage systems with an assumption that retailer’s local demand must be satisfied first before she can transship overage to other retailers. In addition, they assume that the retailers do not hold back shortage or overage when transshipment decisions are made. Retailers participation is non-binding, hence Anupindi et al. (2001) restrict their focus on profit allocation schemes that are stable. The allocation schemes they studied include allocation based on dual prices and modified fractional allocation (see Section 3.2.1). Granot and Soˇsic´ (2003) followed by removing the assumption that the retailers fully contribute their shortage or overage. They model their problem version as a ‘‘three-stage’’ problem. In terms of second stage transshipment, Stuart (2005) considers a duopoly model of inventory competition, but assumes stable price competition in the second stage. This differs from the ex ante transshipment price coordination as described by Rudi et al. (2001), that aims to set transshipment prices that maximize systems’ expected profit. This issue of transshipment prices maximizing systems’ expected profit is further taken up in Hu et al. (2007) who shows the sufficient and necessary conditions for existence of coordinating transshipment prices in a two-location inventory model. Hezarkhani and Kubiak (2010a) proposed transshipment prices for two-retailer system based on Nash bargaining solution that induces retailers to choose the first-best inventory quantities. In

293

terms of ex post transshipment price announcement, Hezarkhani and Kubiak (2010b) proposed pair-wise transshipment prices based on the allocation rule proposed by Anupindi et al. (2001). This requires an ex ante agreement on the price calculations. Our interest is in a market clearing prices (Walrasian equilibrium). To our knowledge, it has not been studied. Other papers on related topics include preventive lateral transshipment, emergency lateral transshipment, and product substitution. Preventive lateral transshipments are the transshipments that are done before shortage happens. This type of transshipments was studied by Tiacci and Saetta (2011a,b) and Banerjee et al. (2003). Emergency lateral transshipment, on the other hand, happens after shortage happens. This type of transshipments was studied by Olsson (2009), and Minner and Silver (2005). The focus of our paper could be seen as a subset of emergency lateral transshipment. The combination of two types was studied by Lee et al. (2007). For product substitution, it assume that only fraction of customers will accept transshipped/substituted goods. In this context, Lippman and McCardle (1997) analyze both a duopoly and an n -firm (oligopoly) model of inventory competition. In such setting, Netessine and Rudi (2003) show conditions for the uniqueness of Nash equilibrium. We exclude the topic of product substitution from our taxonomy. Since this paper puts forth a unifying framework for a variety of decentralized procurement models, we point to another paper that has proposed a more comprehensive framework for both procurement and capacity decisions under the heading of newsvendor networks (Van Mieghem and Rudi, 2002). Our goal is less ambitious than that of Van Mieghem and Rudi (2002) since we present a more detailed unifying view of basic, single period problems, without extending it to dynamic settings. We do not presume that the cited papers are fully inclusive but just a representative sample. For a more comprehensive review and listing of references for the broader area of inventory games and cooperation, see Nagarajan and Soˇsic´ (2008). Fig. 1 shows the mapping of various options and actions that occur in decentralized inventory with transshipment systems to a general two-stage games. Before retailers can decide on the quantity of inventory to be replenished, they should know the market’s rules and behaviors. We assume that ex ante communication is allowed. One of the options is for the retailers to behave freely without agreement and to assume that the ex post transshipment will settle at the market equilibrium. Another option is for the retailers to have an ex ante agreement on an allocation rule to be used ex post as done in Anupindi et al. (2001) and Granot and Soˇsic´ (2003). Another option is for the retailers to set the transshipment prices to maximize expected profits as done in Rudi et al. (2001) and Huang and Soˇsic´ (2010a). Other rules and behaviors that should be communicated among the retailers include an issue whether each retailer must satisfy her local demand first as assumed by most models in the past studies, and an issue whether each retailer is allowed to hold back shortage/overage as allowed in the model by Granot and Soˇsic´ (2003). The retailers may also communicate about their options related to disposal of overage after transshipments, whether it can be done remotely by transporting overage to get disposed at another retailers at a lower cost. After the rules and setting are understood by all, ex ante strategic planning and execution can be performed as a competitive game. That is, each retailer replenishes her inventory at the Nash equilibrium inventory level that maximizes individual expected profit, assuming that other retailers would do the same. After inventory replenishment, the retailers may, once again, set transshipment prices to maximize expected profits as assumed by Hezarkhani and Kubiak (2010a).

294

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

General Two-Stage Games

Ex Ante Communication

First Stage

Competitive game Ex Ante Strategic Planning and Execution

Second Stage

Uncertainty realization Ex Post Local Actions

Decentralized Inventory with Transshipment Option: -Retailers have no ex ante agreement on ex post cooperation. -Retailers have an ex ante agreement (either binding or non-binding). -Retailers set transshipment prices. Actions: -Retailers communicate whether each must satisfy her local demand first. -Retailers communicate whether each is allowed to hold back shortage/overage. -Retailers communicate about their options related to disposal. Actions: -Each retailer replenishes her inventory. Option: -Retailers set transshipment prices to maximize expected profits given inventory decisions of all players. Demand realization / replenishment information revealed Actions: -Each retailer satisfies her local demand at an optimal quantity (full response to local demand if so required).

Cooperative game

Ex Post Communication and Recourse Actions

Shortage/overage information revealed Option: -Retailers do not set an ex post agreement and settle at market equilibrium. -Retailers set a binding ex post agreement. -Retailers set transshipment prices based on an ex ante agreement. Actions: -Solve for an optimal transshipping pattern and profit sharing. -Execute transshipment (satisfying shortage with overage). -Execute transshipment for disposal.

Fig. 1. Mapping of decentralized inventory with transshipment problems.

We consider the problem up to this point as the first-stage decision problem in the two-stage stochastic programming with recourse model. After the demand realization, the second-stage decisions take place. First, for local actions, each retailer might need to satisfy her local demand at an optimal quantity (or fully respond to her local demand if so required). Most studies assume that it is more profitable to satisfy local demand than transshipment to other retailers. After satisfying local demands, part or all of the shortage/overage information is revealed. Ex post communication and recourse actions can be performed. In the case that the retailers have not set an ex post agreement, they may settle the shortage/overage transshipment at the market equilibrium price. Otherwise, they may set a non-binding ex post agreement using a certain profit sharing scheme and solve for an optimal transshipping pattern in the cooperative game fashion as assumed in Stuart (2005). In the case that the retailers already have an ex ante agreement, they may proceed with that agreement or consider breaching the agreement. In this paper, we discuss these various options via a graphical taxonomy tree. We show that it can be clearly modeled using stochastic programming with recourse model. The aim of this paper is to provide a unified approach for future research in the field. More details of the taxonomy are provided in Section 3.1 and the subsequent sections.

cost, and unit salvage value of a retailer i,ðiA N ), and t ij denote the transshipping cost from retailer ito retailer j . r i , ci , vi , and t ij are deterministic, and known to all retailers. The inventory transshipment (for residual demand) and salvage transshipment (for disposal) can take place at the same time. Retailers may designate the residual for disposal instead of fulfilling the demand when it is more profitable (or less costly) to do so. Hence, the model consists of two stages: prior and after realization of demand. In the first stage, each retailer makes an individual inventory procurement decision. Let xi A X i denote the inventory quantity ordered by retailer i in the first stage. Assume that X i is a given nonempty compact convex subset (say, an interval) of R þ and that Di D X i . From the perspective of retailer i, her objective is to choose an order quantity xi that maximizes her expected profit, assuming that she is aware that other retailers would do the same. Let Pi ðx1 , . . . ,xi1 ,xi ,xi þ 1 , . . . ,xn Þ denote the expected profit for retailer i when each retailer k A N, respectively, orders a nonnegative inventory quantity xk . Let Pni ðx1 , . . . ,xi1 ,x0i ,xi þ 1 , . . . ,xn Þ denote the optimized value of Pi ðÞ for retailer iin response to the inventory values x1 , . . . ,xi1 ,xi þ 1 , . . . ,xn selected by the other n1 retailers. That is ! !0

Pni ðx1 , . . . ,xi1 ,x0i ,xi þ 1 , . . . ,xn Þ ¼ maxci xi þ E!½Q i ð x , d Þ, xi A X i

2. The model Consider a decentralized inventory distribution system such that for each retailer the individual generic stochastic programming model corresponds to a two-stage stochastic program. The details of the decisions, random variables, and the process are as follows: let N ¼ f1, . . . ,ng be the index set of the n competing retailers (n Z2 ) in a decentralized inventory distribution system. Each of the retailers responds to an uncertain demand vector ! d ¼ ðd1 , . . . ,dn Þ that, in the second stage, takes on a realized value !0 d ¼ ðd01 , . . . ,d0n Þ A ðD1      Dn Þ, where each Di , i A N is a given nonempty subset of R þ (say, contiguous intervals of the nonnegative real line). Let r i , ci , vi , i A N, represent unit revenue, unit

d

ð1Þ

! !0 where E!½Q i ð x , d Þ is retailer i’s second-stage expected profit. d In terms of the chronology of events, the following unfolds: first, retailer iorders xi at a cost of ci per unit. Simultaneously, other retailers also individually order their inventories x1 , . . . ,xi1 , xi þ 1 , . . . ,xn . Next, after observing demand realization (denoted !0 as d ), retailer iresponds to her local demand and the realized demand of the other retailers and decides on recourse action. In an aside note, the default degenerate problem version, not considered here, is the case when excess demand is lost and excess inventory is simply salvaged—the classical newsvendor problem.

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

To relate the first-stage decision problem to Cournot oligopoly, we can think of (1) as retailer i’s utility for choosing inventory procurement strategy x0i , that is the optimal solution of the program in (1), given that other retailers choose ðx1 , . . . ,xi1 , xi þ 1 , . . . ,xn Þ. Formally, we can write retailer i ’s utility function !! as Pi ðx1 , . . . ,xn Þ ¼ ci xi þ E!½Q i ð x , d Þ. d A tuple ðX 1 , . . . ,X n ; P1 , . . . , Pn Þ denotes a competitive inventory game. The inventory procurement strategies ðxn1 , . . . xnn Þ are a pure strategy Nash equilibrium (PSNE) if, for each retailer i, xni is retailer i’s best response to the strategies ðxn1 , . . . ,xni1 ,xni þ 1 , . . . ,xnn Þ chosen by other retailers; that is, xni solves maxxi A X i Pi ðxn1 , . . . ,xni1 , xi ,xni þ 1 , . . . ,xnn Þ. Solving (1) for PSNE is not our main focus here. Note however that a system of n stochastic programming models must be solved simultaneously to obtain the PSNE(s)—the equilibrium inventory procurement strategy. Gaining an understanding of the different potential connections between the real-life second-stage expected profits options in (1) and their mathematical formulations is the main objective of this paper. To keep the second-stage model mathematically tenable, we systematically describe a taxonomy of recourse restrictions for the resulting ! !0 mathematical structures of Q i ð x , d Þ reflecting the different assumptions. Two-stage stochastic programming with recourse methodology provides a framework for such analysis with a plethora of options that represents different restrictions on the form of recourse. Not all recourse options (restriction) have to account for economically rational profit maximizing impetus. One might also incorporate restrictions that reflect an industry norm, legal business context, behavioral environment, etc.

3. Second-stage models In this section, we introduce the second-stage models using a ‘recourse tree’ representing a taxonomy for the different recourse restriction options. It is a synthesis of earlier works and new analysis/directions. At the outset we note that the second-stage profit function ! !0 ! ! !0 Q i ð x , d Þ has generally two parts: local revenue Li ð y i , x , d Þ and ! ! ! !0 ! a cooperative transshipment profit T i ð y i , y i , x , d Þ, where y i is ! retailer i’s recourse decision and y i represents all other retailers’ recourse decisions. These local revenue and transshipment profits ! ! are maximized with respect to recourse decision y i A Y i . That is ! !0 ! ! !0 ! ! ! !0 Q i ð x , d Þ ¼ max Li ð y i , x , d Þ þ T i ð y i , y i , x , d Þ: ! ! y iA Y i In decentralized inventory system, if there are no restrictions ! imposed on retailer’s recourse actions Y i , and the analytic form ! of the T i function, the recourse decision vector y i will be composed of a large number of recourse variables and the T i function may be ‘intractable’. Each restriction limits the number ! and the ‘range’ of components of y i and results in a functional (implicit or explicit) form for T i . The T i functions capture the different assumptions of recourse options. 3.1. Basic options for local demand We start the taxonomy with two basic assumptions on local demand (see Fig. 2). 3.1.1. Restriction on satisfying the local demand first Imposing a policy that requires each retailer to adhere to local demand before anything else represents a basic recourse

295

Decentralized inventory model

Required to Satisfy local demand first

…

Without necessarily satisfying local demand first

…

Fig. 2. Recourse tree structure (two levels).

restriction. In a number of cases, such policy is natural, especially if the profitability in a long run is heavily dependent on the loyalty of regular customers. In other cases adhering to local demand might be the accepted business culture. Let li denote the quantity of locally satisfied demand. When every retailer has to satisfy local demand before considering collaboration regarding the excess inventory (transshipping if any), with other retailers, the local revenue is modeled as ! !0 Li ð x , d Þ ¼ r i li þvi ðxi li Þ, where li ¼ minðxi ,d0i Þ. Thus, li is not a ! component of a second-stage decision vector y i . Note that vi ðxi li Þ is included in the local profit Li because retailer ican always guarantee for herself a salvage value vi per unit before transshipment. Hence, T i ðÞ is an amount, in addition to salvage value, earned by retailer ibased on her recourse action.

3.1.2. No commitment on satisfying local demand first Relaxing the adherence to local demand, retailer i will transship her inventory to any subset of the retailers that will result in the highest profit for herself. Since we assume that each retailer occupies an exclusive ‘close by’ market in which she normally operates and where the revenue parameters are endogenous, there could be situations where a retailer is not forced (contracted) to respond first to her local demand and has the option to sell the items she ordered in other markets as well. Say, a Christmas tree operation in mid December might not hesitate shipping a fraction of its trees to competitors in lieu of higher profits. Although this may create competition between the different retailers, we assume that cooperation is possible. In this case, the amount of locally satisfied demand li is a ! component of a second-stage recourse decision vector y i . In essence, the amount of locally satisfied demand is decided only after more profitable transshipments have been addressed. Obviously, the feasible region of li is in the interval ½0,minðxi ,d0i Þ. For instance, with an assumption that all other retailers’ recourse ! decisions in y i are similarly restricted, we can model the secondstage profit function as ! !0 Q i ð x , d Þ ¼ max li

! ! !0 r i li þ vi ðxi li Þ þ T i ðli , l i , x , d Þ

li A ½0,minðxi ,d0i Þ, ! where l i denotes all other retailers’ quantities of locally satisfied demand. In the literature, it is usually assumed that r i 4r j t ij for all pairs i and j guaranteing that local demand will be satisfied to the fullest extent and only residuals are pooled. In such a case, this recourse options can be dropped. We were unable to find any literature without this assumption (r i 4r j t ij ). It would be valuable to examine empirical data and comment on the managerial policies prevalent in such cases. s:t:

296

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

Decentralized inventory model Without necessarily satisfying local demand first

Required to Satisfy local demand first

A Non-binding collaboration

B

C

Binding collaboration

Partial sharing

D

…

Price competition

Fig. 3. Recourse ‘tree’ structure (three levels).

3.2. Restrictions related to transshipment This section discusses the restrictions related to the transshipment stage (see Fig. 3.) Note that the related third level nodes in Fig. 3 are connected to both nodes in the level above since imposing transshipment policy restriction is applicable to both second level nodes. Conceptually nodes C and D and nodes A and B are not on the same time horizon. The literature on partial sharing option tolerates non-binding second-stage collaboration, yet those involving price competition may not. 3.2.1. Non-binding (zero penalty) second-stage collaboration with complete sharing (node A) Examining Fig. 3, third level nodes from left to right we start with node A—the case where retailers have a non-binding ex ante agreement about a centralized optimal second-stage transshipment policy sharing all shortage and overage and an allocation rule a for sharing the resulting profits. Consider the following: after satisfying local demand, retailer i in collaboration with some proper subset of retailers can improve her profit over her share in the centralized transhipment profit based on the allocation rule a. In such case, since the ex ante collaborative agreement is not binding, a rational retailer will consider defecting in the second-stage and, subsequently, retai! ! ! !0 ler’s second-stage transshipment profit T i ð y i , y i , x , d Þ will depend on the subset of retailers who bind together for transshipment. The corresponding recourse model ought to capture the recourse decision space that includes all possible ‘stable’ subcoalitional outcomes. We define stable subcoalition as a subcoalition that no member, or a group of members will be better off by deviating from the subcoalition. By assuming that all other ! retailers’ recourse decisions in y i are similarly restricted, the second-stage profit function is ! !0 Q i ð x , d Þ ¼ max S3i

s:t:

S D N,

! !0 ! !0 Li ð x , d Þ þ T ai ðS, x , d Þ

S is stable,

Note that node A assumes complete sharing of shortage and surplus in the second stage transshipment. That is, retailers do not hold back shortage or overage in the second stage. The ! !0 local revenue function can be stated as Li ð x , d Þ ¼ r i minðxi ,d0i Þ þ vi maxðxi d0i ,0Þ. ! !0 A special case of the above is the case where T ai ðN, x , d Þ is a profit function of a game ðN,WÞ that is balanced (has a nonempty ! !0 ! !0 core) given x , d , with the characteristic function WðS, x , d Þ ¼ 0 P a !! i A S T i ðS, x , d Þ for S DN (for more recent work on stability of such games see Baiou and Balinski, 2002). To ensure that all (rational) retailers voluntarily collaborate in the centralized transshipment without actually imposing it with a contract, one must select ex ante an allocation rule a that is in the core of the balanced cooperative game ðN,WÞ. That is, no subcoalition is strictly better off on their own. We also assume that in the game ðN,WÞ any i A N prefers a larger coalition set for the same payoff. If an allocation rule a is in the core, the optimal solution to (2) is Sn ¼ N and the second-stage profit function can be reduced to ! !0 ! !0 ! !0 Q i ð x , d Þ ¼ Li ð x , d Þ þT ai ðN, x , d Þ ! !0 ¼ r i minðxi ,d0i Þ þvi maxðxi d0i ,0Þ þT ai ðN, x , d Þ:

Assuming that in the corresponding ðN,WÞ game any i A N prefers a larger coalition set for the same payoff imposes a recourse restriction that has computational implications. We do not have to examine all proper subsets S  N in our Q i ð,Þ functional. However, reversing this restriction might introduce considerable conceptual and computational difficulties and could lead to new interesting research findings. Some instances of the model corresponding to node A were analyzed in Anupindi et al. (2001) and Suakkaphong and Dror (2010). A short synopsis of their results exposes the generic issues that have to be addressed as a part of selecting a procurement strategy. We start with a allocation that satisfies the core conditions. ! !0 Since T ai ðN, x , d Þ corresponds to the profit function of a coopera! !0 tive game with a given x , d , the core conditions require that X a ! !0 ! !0 T i ðN, x , d Þ ¼ WðN, x , d Þ iAN

X a ! !0 X a ! !0 ! !0 T i ðN, x , d Þ Z WðS, x , d Þ ¼ T i ðS, x , d Þ iAS

ð2Þ

for all S D N,

ð4Þ

iAS

where X X ! !0 ðr j vi t ij Þyij WðS, x , d Þ ¼ max ! i A S j A S,j a i y X s:t: yij r max fxi d0i ,0g for all iA S, X

! !0 where T ai ðS 3 i, x , d Þ is an amount, according to the allocation rule a allocated to retailer i when applied to a subcoalition S 3 i. Clearly, the coalitions Swhose members extract their maximal !0 profits by forming Sare dependent on a realization d and it is not well understood how they would form. Note that in order for (2) to be a meaningful expression, the resulting subcoalition Shas to be a maximizing subset for all its members. Otherwise, (2) is ill defined. The existence of such a nontrivial Swould be an outcome of the expected behavior of rational agents. At this point we are not aware of studies on this topic.

ð3Þ

j A S,j a i

yij r max fd0j xj ,0g for all j A S,

i A S,i a j

for all yij Z 0:

ð5Þ

The optimal transshipping pattern ynij for (5) represents the number of units of product transshipped from retailer i to retailer j in the coalition S. Note that each coefficient term in the objective includes a negative salvage value (vi ), since, as mentioned before, a retailer with excess inventory is expected to receive at least the salvage value vi per unit from transshipment. Hence, there is no double counting of salvage. The following are a number of known results: Theorem 1 (Anupindi et al., 2001, Theorem 4.1). The core of the second-stage transshipment game is nonempty. In particular, the

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

!! allocation based on dual prices adi ð x , d Þ of transshipment problem ! !0 WðN, x , d Þ is an allocation in the core. This result is based on a classic work of Shapley and Shubik (1975) and Owen (1975) (expanded on by Samet and Zemel, 1984; Sa´nchez-Soriano et al., 2001). In terms of system performance, a decentralized solution is compared with the performance of the ‘first-best’ solution—the centralized inventory decision. Definition 1. The expected combined profit (first-best profit) is represented by (  n X n !c ! !0 Pc ð x Þ ¼ max  ci xi þ E! WðN, x , d Þ ! ! d iAN x AX #) X þ r i minðxi ,d0i Þ þvi maxðxi d0i ,0Þ : iAN

!cn is the solution that maximizes then The first-best solution x !c !c expected combined profit for the set N of retailers ( x and x stand for ‘centralized inventory decision’ and ‘optimal centralized n inventory decision’ or first-best respectively; Pc and Pc being the centralized inventory profit and optimal centralized inventory profit respectively). Generally, if the decentralized inventory decisions coincide with the first-best solution, the first-best profit can be achieved. However, allocations based on dual prices of transshipment do not necessarily result in a profit value equal to the first-best solution (Anupindi et al., 2001). Therefore, Anupindi et al. (2001) propose another core allocation rule (see also Suakkaphong and Dror, 2010). For notational expediency we omit reference to the set N. ! !0 Definition 2. Let the fractional allocation afi ð x , d Þ be defined as !0

!0

! afi ð! x , d Þ ¼ gi Pc ð x , d Þ½r i minðxi ,d0i Þ þ vi maxðxi d0i ,0Þci xi , ! !0 ! !0 P where Pc ð x , d Þ ¼ WðN, x , d Þ þ i A N ci xi þ r i min ðxi ,d0i Þ þ vi max ðxi d0i ,0Þ, and gi Z 0 is a fraction determined by the retailers P f !! such that i A N gi ¼ 1. Note that ai ð x , x Þ can be negative. If we allow to consider gi r 0 for some i’s, the interpretation of fractional allocation changes. The option of subsidizing a facility in order to raise the collective profit will be considered in future research. Theorem 2 (Anupindi et al., 2001, Theorems 5.1 and 5.2, Corollary 5.1). Consider a modified fractional allocation rule that allocates the residual profits to player iA N as follows: !0

!0

cn

!0

cn

!0

! ! f ! d ! am , d Þafi ð x , d Þ, i ð x , d Þ ¼ ai ð x , d Þ þ ai ð x !cn is the first-best solution. Then the pure strategy Nash where x ! !0 equilibrium based on core allocation rule am i ð x , d Þ corresponds to a first-best solution. Claim 1. At non-pure strategy Nash equilibrium inventory position, ! !0 ! !cn when x a x the allocation am i ð x , d Þ is not always in the core of the transshipment game. ! Claim 2. If the demand distribution of d is not of common knowledge, the decentralized inventory system that adopts ABZ allocation rules is not incentive compatible. Suakkaphong and Dror (2010) propose sufficient conditions for uniqueness of pure strategy Nash equilibrium (PSNE). The sufficient conditions include: (i) the expected centralized profit function has to have a unique global maximum, and (ii) there is no

297

‘ridge’ present in the expected centralized profit function as ! plotted on x . Hezarkhani and Kubiak (2010b) proposed pair-wise transshipment prices based on the allocation rule proposed by Anupindi et al. (2001). This price setting guarantees to be pair-wise stable and that the individual retailers will choose their transshipment partners optimally. 3.2.2. Binding second-stage collaborative ex ante transshipment agreement (node B) Binding ex ante second-stage collaborative transshipment agreement to include all overage and shortage represents a different recourse restriction in the two-stage stochastic programming model. This type of agreement forbids formation of subcoalitions. With a binding agreement, each retailer must participate in collaborative transshipment solution and share their overage/shortage. Retailers cannot consider proper subcoalitions. Hence, the second-stage profit function is modeled as (3). We can show the following: ! !0 ! !0 Theorem 3. If T ai ðN, x , d Þ for all iA N increases as Wð x , d Þ increases (monotonic in W ) and Y ij 3 yij is a closed set, then there exist a transshipment pattern yij that maximizes every individual !0 retailer’s profit for a realization d . Hence ! !0 xni ¼ argmaxfci xi þ E- ½r i minðxi ,d0i Þ þ vi maxðxi d0i ,0Þ þ Wð x , d Þg: d

xi A X i

ð6Þ

Since the above result is straight-forward, we omit the proof. Allocation rules that satisfy the above theorem include a propor! !0 ! !0 tional allocation api ð x , d Þ ¼ gi Wð x , d Þ where gi is a fraction reflecting retailer’s contribution or negotiation power such that P i A N gi ¼ 1 and for all i, gi A ð0; 1Þ. 3.2.3. Partial sharing option (node C) The case of retailers being free to hold back any part of shortage or overage assets in the second stage in order to increase their individual share of the transshipment profits. In this case, ! the second-stage recourse decision vector y i will include the amounts of excess individual demand ei and individual supply hi that i frees up for transshipment (note that ei  hi ¼ 0 for all i). ! ! ! !0 Given an ex ante allocation rule a, let T ai ðhi , h i ,ei , e i ,S, x , d Þ be the profit allocated to retailer i from transshipment (in addition to salvage value) when (i) retailer i contributes hi units of excess supply and ei units of excess demand, (ii) all other ! ! retailers contribute h i of excess supply and e i of excess demand, and (iii) retailer i is a member of subcoalition S. The second-stage profit function can be stated as ! !0 Q i ð x , d Þ ¼ max

hi ,ei ,S3i

! ! ! !0 r i li þ vi ðxi li Þ þ T ai ðhi , h i ,ei , e i , S, x , d Þ

s:t: S DN, hi r xi li , S is stable,

ei rdi li , hi ,

ei Z0 ð7Þ

where li ¼ minðxi ,d0i Þ. Note that, similar to node A, in order for (7) to be a meaningful expression, the resulting subcoalition S has to be a maximizing subset for all its members. Otherwise, (7) is ill defined. This problem was examined in Granot and Soˇsic´ (2003) under the heading of a three-stage model. First, retailers decide on their procurement levels. Then, after demand is realized and each retailer responds to her local demand, she determines how much shortage/overage to share with the other retailers. Finally, the collaborated transshipment decisions take place. An allocation rule is value preserving (Granot and Soˇsic´, 2003) if it induces all

298

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

retailers to share their shortage/overage in amounts that do not result in a decrease in the total additional profit, and it is complete sharing if it induces all retailers to share all of their (total) shortage/overage.

Decentralized inventory model

Theorem 4 (Granot and Soˇsic´, 2003, Theorem 9). There are no completely sharing or value-preserving allocation rules for transshipment games based on dual prices. Hence, transshipment dual prices are not an appropriate shortage/overage sharing mechanism in this case. Retailers may not necessarily choose hi ¼ xi li and ei ¼ di li , for all iA N in (7). In the case of two retailers, both retailers will actually choose hi ¼ 0 and ei ¼ 0. Consequently, an allocation rule based on dual prices does not lead to the first-best outcome. Granot and Soˇsic´ (2003, Theorem 11) observe (based on results from Young (1985) and Housman and Clark (1998) that there are no completely sharing core allocation rules for transshipment games with four or more retailers. This suggests that for any allocation function ! ! ! !0 T ai ðhi , h i ,ei , e i ,S, x , d Þ in (7), for S ¼ N, retailers may not necessarily choose hi ¼ xi li and ei ¼ di li , for i A N. Moreover, Granot and Soˇsic´ (2003, Theorem 12) observe that: there are no value-preserving core allocation rules for transshipment games with six or more retailers. To sidestep the functional and computational difficulties of this issue, the individual subset choice S 3 i is removed from the second-stage decision options. That is, all retailers are bind to cooperate in transshipment while still having the option for partial sharing of shortage/overage. The restricted second-stage profit function can now be stated as ! !0 Q i ð x , d Þ ¼ max hi ,ei

s:t:

hi r xi li ,

! ! ! !0 r i li þ vi ðxi li Þ þ T ai ðhi , h i ,ei , e i , x , d Þ ei rdi li ,

hi ,ei Z0,

where li ¼ minðxi ,d0i Þ. By manipulating a fractional allocation rule, Granot and Soˇsic´ ! ! ! !0 (2003) proposed another allocation, i.e., T ai ðhi , h i ,ei , e i , x , d Þ, that leads to the first-best outcome. This allocation is not necessarily in the core of the corresponding game ðN,WÞ where X a ! ! ! ! !0 ! ! !0 T i ðhi , h i ,ei , e i , S, x , d Þ for S D N: Wð h , e ,S, x , d Þ ¼

A Non-binding collaboration

!

!0

!0

ð8Þ

P with gi A ð0; 1Þ, j A N gj ¼ 1, is an efficient value-preserving allocation rule which induces the inventory levels in a first-best solution to be a Nash equilibrium profile. The individual rationality is satisfied if retailers choose gi s that are proportional to the expected individual profit. In an infinitely repeated game setting with a large enough discount factor, Huang and Soˇsic´ (2010b) show that the dual allocations result in a subgame perfect Nash equilibrium with all retailers fully sharing their overage and shortage. 3.2.4. Price competition (node D) In a less structured version of the transshipment, one does not assume that the individual retailers value their shortage and overage uniformly using the fixed transshipment matrix ðt ij Þ and their r i ,vi values and cooperate in transhipment. Essentially, the decentralized inventory system is a market of overage/shortage with price competition (Fig. 4). We examine two cases in terms of timing pricing and roles of retailers. For the pricing, the model assumes that the prices

C

Binding collaboration

Partial sharing

F

Binding transshipmentfor-disposal

D Price competition G

Non-binding transshipmentfor-disposal

…

H

Ex-ante price Ex-post price competition competition

Fig. 4. Recourse tree structure (four levels).

are either set ex ante, or set ex post. In terms of the roles of retailers, the prices can be set by either sellers or buyers or both. Fig. 4 has nodes G and H on the same level as E and F, but G and H are actually on the same decision epoch of D. Ex ante price competition is probably more common in the retail setting where retailers are already part of a network, e.g. they buy products from the same vendor or distributor. On the other hand, ex-post price competition is more suitable for settings where a market for selling excess supply already exists. Ex ante price competition (node G): Consider the case when the prices are set ex ante by the sellers (node G). Other cases of retailers’ roles can be modeled similarly. Price tij for unit overage is set by retailer i for each retailer j A N,j a i. A simpler version is the case with a policy tij ¼ ti , for all j A N,j ai. In ex ante price competition, the tij ’s are first-stage decision variables. That is, the first-stage expected profit of retailer i is expressed as

Pni ððx1 , ! t 1 Þ, . . . ,ðxi1 , ! t i1 Þ,ðxni , ! t i Þ,ðxi þ 1 , ! t i þ 1 Þ, . . . ,ðxn , ! t n ÞÞ n

! ! !0 max ci xi þ E!½Q i ð x , t , d Þ, ! ! d

¼

xi A X i ,

! ! agi ðhi , h i ,ei , ! e i , x , d Þ ¼ gi Pc ð x , d Þ½r i li þvi ðxi li Þci xi ,

B

E

iAS

Theorem 5 (Granot and Soˇsic´, 2003, Theorem 13). The allocation rule defined by

Without necessarily satisfying local demand first

Required to Satisfy local demand first

ð9Þ

t iATi

! ! ! ! ! where t i ¼ ðti1 , ti2 , . . . , tin Þ, t ¼ ð t 1 , t 2 , . . . , t n Þ. Because these ! prices t are set ex ante, the second stage decision model can be stated as ! ! !0 Q ið x , t , d Þ ¼ max r i li þ vi ðxi li Þ ðyi1 ,...,yin Þ,ðy1i ,...,yni Þ X X þ ðtij t ij vi Þyij þ ðr i tji Þyji jAN

s:t: X

X

jAN

yij ¼ xi li ,

jAN

yji ¼ di li ,

jAN

! ! y A Y,

ð10Þ li ¼ minðxi ,d0i Þ,

complete sharing of residuals is assumed, where and Y represents the set of transshipment acceptable to all retailers. Model (10) assumes that the retailers who have overage pay for the transshipping cost. In the two-retailer case, ! it can be shown that Y ¼ ðY12 , Y21 Þ ¼ ð½0,x1 l1 ,½0,x2 l2 Þ when t12 A ½t 12 þ v1 ,r2  and t21 A ½t 21 þ v2 ,r1  because the solution to (10) for retailer 1 is also the optimal solution for retailer 2. The transshipment price coordination for two-retailer system as described by Rudi et al. (2001) aims on setting a transshipment price that maximizes system expected profit. Hu et al. (2007) show that such transshipment price does not always exist.

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

Hezarkhani and Kubiak (2010a) recently proposed a new transshipment price scheme for a similar two-retailer system based on Nash bargaining solution that induces retailers to choose the firstbest inventory quantities. This price is set after the inventories are ordered but prior to demand realization. However, to define Y formally for n-retailer system, further assumptions are required because another retailer k may find that ! retailer i’s proposed transshipment pattern y is not maximizing her expected profit and may decide not to participate (cooperate) in the transshipment. To resolve this mathematical dilemma, Huang and Soˇsic´ (2010a) suggest an approach—a restriction that employs a central coordinator (depot) who unilaterally decides on the overall transshipment for all the retailers. That is, first, the transshipment prices tij are separated into two elements: ti0 which is the price retail i asks for transshipping to the depot and t0j which is the price retail j will pay for transshipping from the depot. The depot is responsible for the actual transshipment cost t ij ¼ t i0 þ t 0j . Huang and Soˇsic´ (2010a) omit the ‘managerial’ objective of the depot. They simply impose a constraint on non-negative operating profit for the depot. That is, t0j ti0 Zt ij ; they denote this condition by (ID). They denote seller i’s rational constraint ti0 vi Z 0 by (IRS), and buyer j’s rational constraint r j t0j Z0 by (IRB). Huang and Soˇsic´ (2010a) propose a pricing scheme such that (ID), (IRS), and (IRB) are satisfied, and the depot will be ‘neutral’, i.e., will end up with zero profit for any demand realizations and inventory positions. Theorem 6 (Huang and Soˇsic´, 2010a, Proposition 8). Let vm þ t m0 ¼ maxi fvi þ t i0 g and denote a transshipment price agreement by

ti0 ¼ vm þ tm0 ti0 8i, t0j ¼ vm þtm0 þt 0j 8j: This price agreement makes the central depot neutral and satisfies (ID), (IRS), and (IRB). Such pricing scheme assumes that the retailers do not consider sharing the depot’s operating profit. This pricing scheme is not well engineered because it does not have a clear objective; it does not maximize expected profit of an individual retailer, expected total profit of the system, transshipment profit of individual retailer, nor the realized total profit of the system. Moreover, the depot has no incentives for proposing an efficient transshipment patterns because its operating cost is zero independent of the transshipment solution. Hanany et al. (2010) proposed a similar transshipment mechanism which suffers from the same disadvantages as Huang and Soˇsic´ (2010a)—it does not have an incentive to maximize transshipping profit. They propose a new mechanism to be used in a fully noncooperative setting (including the secondstage transshipment). The transshipment fund computation in the second stage is based on the Vickrey–Clarke–Groves (VCG) mechanism which motivates truthful announcement of excess supply/demand. As an alternative to Huang and Soˇsic´ (2010a) and Hanany et al. (2010) assume that prices ti0 and t0i were set ex ante by retailer i and the central depot maximizes its operating surplus. That is !

Y ¼ arg max ! y X

s:t: X

X X

ðt0j t ij ti0 Þyij

i A Nj A N,j a i

yij r max

fxi d0i ,0g

yij r max fd0j xj ,0g

i A N,i a j

for all yij Z0:

! First, consider a case when Y is not a singleton. The depot is indifferent among the transshipment solutions, but each retailer is not indifferent among these solutions since these solutions impact the calculation of the first-stage individual expected profit. Hence, the depot is required to implement an algorithm to find the best transshipment solutions. One option for resolving this dilemma is to minimize the maximum difference between the best-off and the worst-off retailers. That is 8 8 9 <
jAN

!n

In the case that Y is still not a singleton, the depot may perturb each retailer’s price parameter with some small Ei to obtain a unique solution. The depot is ex ante required to communicate explicitly the perturbation method used. ! Now, consider the case with Y a singleton. All retailers will have to agree to this transshipment pattern because of the ex ante constraint. In this case, the optimal value of (11) can be greater than 0. The model shown in (9) assumes that the central depot’s operating profit is shared by retailers. If the central depot’s operating profit is shared, it is not clear how to incorporate this profit in the first-stage expected profit calculations. We are aware that this proposed solution may seem ad hoc but it has a clear interpretation. Detailed analysis of this approach is left for future studies. Adding the depot as a third-party profit center constitutes a deviation in this framework. Nevertheless, we include it in this paper for completeness. Ex-post price competition (node H): Now, consider the case when the prices are set ex post (node H). The price competition happens ex post with respect to market demand and supply after demand realization and after satisfying each local demand. That is, the first-stage expected profit is defined as in (1) while the second-stage profit is defined as ! !0 ! !0 !n !n Q i ð x , d Þ ¼ r i minðxi ,d0i Þ þ vi maxð0,xi d0i Þ þU ni ð x , d , p , q Þ, ð12Þ ! !0 !n !n where U ni ð x , d , p , q Þ represents retailer i’s utility (profit) given that the transshipment decision depends on the exchange !n !n market equilibrium price p and quantity q . The above model assumes that there exists a market equilibrium such that the market will be cleared. That is, shortage/overage that may not be traded/exchanged with profit will leave the market. Retailers are ! !0 !n !n price takers. Before we formally define U ni ð x , d , p , q Þ, we briefly discuss the setting of the exchange market. The equilibrium that we are referring to here is a ‘general equilibrium’, or sometimes, referred to as ‘Walrasian equilibrium.’ In our case, all n retailers are consumers who consume 2n types of commodities. These commodities are overage and shortage at n retailers. Because of varying cost parameters at each retailer, it is insufficient to model overage and shortage as two types of commodities. Let the amount of retailer i’s initial endowment be ! hn e1 en denoted by w i ¼ ðwh1 i , . . . ,wi ,wi , . . . ,wi Þ whi i ¼ maxðxi di ,0Þ,

whj ¼ 0, i

wei i ¼ maxðdi xi ,0Þ,

wej ¼0 i

for all i A N,

j A N,j a i

for all j A N,

299

for all j ai:

Let the amount of retailer i’s consumption bundle be denoted by ! ! ! ! hn e1 en q i ¼ ðqh1 i , . . . ,qi ,qi , . . . ,qi Þ. An allocation q ¼ ð q 1 , . . . , q n Þ is a collection of n consumption bundles describing the result of

300

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

exchange for n retailers. A feasible allocation is one that satisfies P ! P ! iAN q i r i A N w i . Each retailer i acts as if she was solving the following problem: ! maxU i ð q i Þ ! qi

k A N,k a j

X

!! !! s:t: p q i r p w i ,

k A N,k a j

!T

where p ¼ ðph1 , . . . ,phn ,pe1 , . . . ,pen Þ is transpose of a vector of market prices and the retailer i’s utility is defined as X X ! U i ð q i Þ ¼ max ðr j vk t kj Þykj ! k A Nj A N,j a k y X s:t: yjk r qhj for all j A N, i k A N,k a j

X k A N,k a j

ykj rqej i

for all j A N,

! That is, if retailer i is in control of consumption bundle q i , she will maximize the profit generated from that consumption bundle by transshipping overage to shortage in the most efficient way. !n !n The Walrasian equilibrium is a pair ð p , q Þ such that P P ! !n !n ! ! i A N q ið p , p w iÞ r i A N w i . Retailer i’s utility is written as ! !0 !n !n U i ð x , d , p , q Þ ¼ max ! qi n

!n ! !n ! p q i r p w i,

! Ui ð q i Þ

ð13Þ

!0 ! ! where w i is a function of x and d . The above decentralized inventory model as defined by (1), (12), and (13) requires that there is only one Walrasian equilibrium, or if there are multiple Walrasian equilibria, all equilibria must have the same profit value in order to compute expected profit. We know that all retailers have the same form of utility ! function U i ð q i Þ which is continuous, (weakly) concave, and ! ! non-decreasing in q i . Hence, given a certain price p , it is n 1 n k ! ! possible to have a set L ¼ f q , . . . , q g, 9L9 41, of consump!n1 tions that maximize individual retailer’s utility with U i ð q i Þ ¼ !nk . . . ¼ U i ð q i Þ for all i A N. Thus, uniqueness of Walrasian equilibrium cannot be guaranteed. It would be interesting to examine for this setting the conditions under which Walrasian equilibria have the same profit !n1 !n1 !nk !nk value. That is, if there is a set Z ¼ fð p , q Þ, . . . ,ð p , q Þg, ! !0 !n1 !n1 9Z9 41, of Walrasian equilibria, then U ni ð x , d , p , q Þ ¼ ! !0 !nk !nk    ¼ U ni ð x , d , p , q Þ for all i A N. This is yet another topic left for future studies. ! Because the utility function is weakly increasing in q i , it lacks several useful properties. For instance, Walrasian equilibrium allocation may not be a member of the core of the corresponding transshipment game. We note that when overall shortage P P ej hj j A N qi is less than overall overage j A N qi , an increase in hj overage qi does not increase the utility, and vice versa. To ! guarantee that the utility function is strictly increasing in q , we can add a small perturbation to the retailers’ preference (utility function) and a requirement that retailers not trade without profit. To accomplish this, we add an epsilon value to overage and shortage. This will make retailers prefer to hold on to overage and shortage rather than trade them with zero increase in profit

ykj r qej i

for all j A N,

for all ykj Z 0: ! The above utility0 function is strictly increasing in q i . !! Let WðS, x , d Þ represent the transshipment profit of coalition !0 ! S for a given x and d X X ! !0 WðS, x , d Þ ¼ max ðr j vi t ij Þyij ! i A S j A S,j a i y ( ) X þE ð max fxi d0i ,0g þ max fd0i xi ,0gÞ X

s:t:

for all ykj Z 0:

s:t:

(utility). The new utility function is written as 8 9
X

iAS

yij r max fxi d0i ,0g

for all iA S,

j A S,j a i

yij r max fd0j xj ,0g

for all j A S,

i A S,i a j

for all yij Z 0:

ð14Þ

Theorem 7. Assume that retailers are not trading without profit. All Walrasian equilibrium allocations that correspond to Walrasian equilibria in decentralized inventory system are members of the core of the second-stage transshipment game ðN,WÞ where W is defined in (14). The proof of this new result is based on Jehle and Reny (2000) in their Theorem 5.6. We note the similarities between the motivation of this model and that of Stuart (2005). However, Stuart (2005) assumes that their market consists of buyers and sellers, while in our market there is no distinction between buyer and seller (of overages). In addition, Stuart’s unit cost and salvage value are constant for all retailers (producer). That is, ci ¼ c,vi ¼ v. Moreover, their transshipping cost is zero, i.e., t ij ¼ 0. Finally, their demand is a function of price while our demand is exogenous. As a result, in Stuart (2005) the problem is to find the market clearing price tij ¼ t.

4. Restrictions related to transshipment for disposal Up to this point, we assumed that the overage without profitable demand is disposed at a local salvage value vi . We now modify our assumptions to allow retailers to transship their overage for disposal at other retailers in a manner that generates the greatest salvage net revenue. The corresponding analysis depends on the retailers’ agreements with respect to sharing the salvage value. 4.1. Binding transshipment for disposal agreement (node E) Related literature (Anupindi et al., 2001; Granot and Soˇsic´, 2003; Suakkaphong and Dror, 2010) generally assumes that the overage that is not transshipped (has no ‘profitable’ demand) is disposed at a local salvage value vi , vi Zvj t ij for all i,j A N. However, one may consider the case when retailers have options to transship their overage to dispose it (salvage) at a retailer that generates the greatest net revenue. Specifically, we examine some models related to transshipment pattern (solution) and profit sharing for salvage options. Say, we are dealing with agricultural

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

301

Retailer B

$4

Retailer A

$10 $4

$1 3 units of overage

$30

Retailer C

$17

$1

$1 6 units of shortage

$1

$15 4 units of overage

Fig. 5. Example.

products and an option of processing unsold volumes for other useful purposes (compost, biofuel, etc.). Consider an example shown in Fig. 5. This three-retailer system has the following parameters: ððr A ,r B ,r C Þ, ðt AB ,t BA , t AC ,t CA ,t BC ,t CB Þ,ðvA ,vB ,vC ÞÞ ¼ ðð30; 30,30Þ,ð4; 4,1; 1,1; 1Þ,ð17; 10,15ÞÞ. Suppose a demand realization instance with retailer A having six units of shortage, and retailers B and C having three and four units of overage, respectively. If the overage cannot be transshipped for disposal, with the three-retailers cooperating ex post we solve the following centralized transshipment problem: X X ! !0 WðN, x , d Þ ¼ max ðr j vi t ij Þyij ! i A N j A N,j a i y X s:t: yij r max fxi d0i ,0g for all i A N, X

j A N,j a i

yij r max fd0j xj ,0g

for all j A N,

i A N,i a j

for all yij Z0,

ð15Þ

where N¼ {A,B,C} and yij is the amount transshipped from retailer i to j. The optimal solution requires the transshipment of three units from B to A, three units from C to A and the disposal of one unit of overage at C. The corresponding total revenue (optimal value of the above problem) is $90. Because this model computes the revenue in addition to the salvage value, the actual revenue from transshipment, including the local salvage revenue, is $90 þ(3  $10)þ(4  $15)¼$180. Now we examine how this revenue is shared. Consider two allocation rules: one based on dual prices and the other based on an equal surplus sharing rule. That is: Rule I: Retailers agree ex ante to share transshipment profit based on dual prices of the transshipment problem. The allocation based on dual prices of transshipment is defined as !0

adi ð! x , d Þ ¼ li max fxi d0i ,0g þ di max fd0i xi ,0g,

ð16Þ

! !0 where li and di represent dual prices of WðN, x , d Þ. The dual prices are determined for each unit of shortage as well as each unit of overage. Assume also that they are not allowed to hold back shortage or overage. (A similar scenario was discussed in Section 3.2.3.) The amounts of revenue shared to each retailer are $84, $36, and $60 to A, B, and C, respectively. Note that the dual prices for overage at B and shortage at A are $2 and $14, respectively. Since sharing based on dual prices is in the core of this ‘cooperative game’, no retailer or a subset of retailers is

better-off by deviating from the centralized transshipment solution. The transshipment is stable. Rule II: Assume that these retailers have a binding ex ante agreement using a simple equal surplus sharing rule, that is, the profit between two retailers involved in transshipment are equally split. The corresponding revenue amounts allocated to each retailer are $45, $54, and $81 to A, B, and C, respectively. Note that the allocation rule based on an equal surplus sharing is not necessarily in the core of the corresponding cooperative transshipment game for all realizations and usually requires a binding agreement for retailers to cooperate. We compare the above two solution concepts in the case when retailers are allowed to transship for disposal at other locations. First, consider a case when an increase in salvage net revenue from transshipping for disposal does not have to be shared between participating retailers, implying that the owner of overage receives full amount of salvage value less the transshipping cost. The salvage net revenue at retailer i can be updated to a new value that reflects the highest possible salvage value less transshipping cost. That is, v0i ¼ maxðvi , maxj A N,j a i ðvj t ij ÞÞ. All other parts of the model stay the same. In that case we update the salvage values to ðv0A ,v0B ,v0C Þ ¼ ð17; 14,16Þ—the disposal locations do not receive any share from disposal revenues. The solution is to transship two units from B to A, and four units from C to A. B disposes one unit of overage. In terms of revenue, the total revenue (optimal value of the above problem) is $76. The actual revenue from transshipment, including the local salvage revenue, is $76 þ(3  $14) þ(4  $16)¼ $182. We examine how this revenue is shared using the two allocation rules: dual prices and an equal surplus sharing rule: (i) assume that the retailers agree ex ante to share transshipment profit based on dual prices of the transshipment problem. The amounts of revenue shared to each retailer are $72, $42, and $68 to A, B, and C, respectively; (ii) assume that these retailers have a binding ex ante agreement using an equal surplus sharing rule. The amounts of revenue shared to each retailer are $38, $54, and $90 to A, B, and C, respectively. 4.1.1. Non-binding transshipment for disposal agreement (node F) Now consider a case when an increase in salvage net revenue from transshipping for disposal is shared between participating retailers using dual prices. The salvage net revenue at retailer i can be updated to a new value that reflects the highest possible salvage value less transshipping cost. That is, v0i ¼ max ðvi , maxj A N,j a i ðvj t ij ÞÞ. All other parts of the model stay the same.

302

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

Table 1 Transshipment solutions. Description

Total revenue

A

B

C

No transshipment for disposal, dual allocation 3ðB-AÞ, 3ðC-AÞ,1ðC-C disposalÞ No transshipment for disposal, equal surplus allocation 3ðB-AÞ, 3ðC-AÞ, 1ðC-C disposalÞ Allow transshipment for disposal, dual allocation 2ðB-AÞ, 4ðC-AÞ, 1ðB-C disposalÞ Allow transshipment for disposal, equal surplus allocation, disposal locations do not receive disposal revenue 2ðB-AÞ, 4ðC-AÞ, 1ðB-C disposalÞ Allow transshipment for disposal, equal surplus allocation 2ðB-AÞ, 4ðC-AÞ, 1ðB-C disposalÞ

180

84 36 60

180

45 54 81

182

72 42 68

182

38 54 90

182

41 50 91

This case can be justified by the fact that zero dual price value is linked with an infinite quantity ( Z 0 non-negativity constraint). Because an infinite quantity can be disposed at any retailer, the disposal locations (retailers) will not receive share of disposal revenue when this rule based on dual prices is agreed upon. If there is no ex ante agreement on sharing disposal revenue, the transshipment and disposal of overage can be seen as a competitive market. Because of high competition between the disposal locations (retailers), the disposal locations will not receive share of disposal revenue when this rule based on dual prices is agreed upon. Thus, the salvage net revenue at retailer i can also be updated to a new value that reflects the highest possible salvage value less transshipping cost. Next consider the case when an increase in salvage net revenue from transshipping for disposal is shared between participating retailers using a rule of equal partition of surplus. The solution of a problem in (15) using updated salvage value v0i ¼ maxðvi , maxj A N,j a i ðvj t ij ÞÞ is also an optimal solution that generates highest transshipment profit. The solution is to transship two units from B to A, and four units from C to A. B disposes one unit of overage. The actual revenue from transshipment is $182. However, the solution of (15) does not immediately provide the allocation amounts for each retailers. The following model shows another equivalent setting for this case: XX XX !0 ^ ðN, ! W x , d Þ ¼ max ðr j vi t i,j Þyi,j þ ðvj vi t i,j Þzi,j ! i A Nj A N i A Nj A N y X s:t: ðyi,j þ zi,j Þ ¼ max fxi d0i ,0g for all iA N X

jAN

yi,j r max fd0j xj ,0g

for all j A N

iAN

for all yi,j ,zi,j Z 0: P With this setting, retailer k will receive j A N 0:5fðr j vk t k, P jÞyk,j þðvj vk t k,j Þzk,j g þ i A N 0:5fðr k vi t i,k Þyi,k þðvk vi t i,k Þzi,k g. The amounts of revenue shared to each retailer are $41, $50, and $91 to A, B, and C, respectively. Table 1 summarizes the transshipment solutions based on different allocation rules discussed in this section. In summary, allowing transshipment of salvage expands the recourse options and must be handled with some care in the second-stage model. Is it a third stage? It does not need to be considered a third stage if the expected values are rolled back to influence the first stage strategic procurement decisions. We note that the literature on this topic, if any, is very skimp. 5. Discussion and conclusion This paper presents the decentralized newsvendor with transshipment problem as a taxonomy graph that links the different

decisions as graph nodes representing assumptions and analytical resolutions. It is based on the premise that a newsvendor is maximizing her expected profit. Why should we, or anyone else, care? Because we want to understand as much as possible what is involved in doing this trade. It has a tremendous economic impact on everyone. What can we learn from the taxonomy graph and the analysis of the linked nodes? First of all it depicts a clear picture of interrelations of assumptions and decision options. Secondly, using the umbrella of stochastic programming with recourse methodology, we learn what answers to expect, and are these answers computable, offer a stable business configuration, present the best decision options, or that we not yet know how certain situations will play out. The analytic methodology is that of a two-stage stochastic programming with recourse. This does not mean it is a two-stage actions problem. It only means that under the second-stage heading, after learning the demand, we move all the implications in its expected value projection of the actual recourse operational costs and profits. In general, the variety of potential recourse options is too large for exact computational analysis. Thus, the modeling considers restrictions and approximations to real-life options. The aim of stating recourse restrictions in the second stage of the stochastic programming model is the communication of an ‘appropriate’ problem description and the mathematical ‘tractability’ that retains a meaningful modeling interpretation. We describe a progression of a number of recourse restrictions and the subsequent mathematical expressions of second-stage profit function reflecting different modeling assumptions. Past studies on decentralized inventory system are mapped onto our taxonomy. Anupindi et al. (2001) work (mapped to node A) analyzed two-stage systems under the assumption that each retailer’s local demand must be satisfied first before she can transship overage to other retailers. The contract between retailers is assumed to be non-binding, hence Anupindi et al. (2001) focus on profit allocation schemes that are stable. They also assume that all retailers do not hold back shortage or overage when transshipment decision is made. The allocation schemes that they studied include allocation based on dual prices and modified fractional allocation. When the contract is binding, the system may use the second-stage profit allocation scheme that is unstable such as fractional allocation and proportional allocation. This case is mapped to node B. Granot and Soˇsic´ (2003) (mapped to node C) removed the assumption that all retailers do not hold back shortage or overage. In this setting with four or more retailers, first-best profit cannot be guaranteed. Node D represents the system with price competition. The works of Rudi et al. (2001) and Huang and Soˇsic´ (2010a), ex ante price competition, were discussed and mapped to node G (a child node of node D). However, we noted that this setting is not computationally practical for more than two retailers. Another approach with price competition ex post is offered. It is mapped to node H. We use Walrasian equilibrium to explain the individual second-stage profit behavior. Fourth level nodes can be added under each of the third-level nodes to map the different assumptions on transshipment for disposal as we show at node E and F. The different assumptions will lead to different second-stage profit models. Moreover, when relaxing the assumption on satisfying local demand first, additional models can be generated and analyzed as represented by dotted lines connecting the second node in the second level to the third-level nodes in all taxonomy figures. Other problem instances can also be mapped into this taxonomy. For example, we may add a node in the third level for adversarial actions among retailers (node I in Fig. 6). Consider

N.S. Summerfield, M. Dror / Int. J. Production Economics 137 (2012) 292–303

Decentralized inventory model Without necessarily satisfying local Required to demand first Satisfy local demand first

A B C Non-binding Binding Partial collaboration collaboration sharing E

F

Binding Non-binding transshipment- transshipmentfor-disposal for-disposal

… D I Price Adversarial competition actions G

H

Ex-ante price Ex-post price competition competition

Fig. 6. Recourse tree structure (four levels).

a group of retailers a, b, and c. When they agree (ex ante or ex post) to transship their overage, their individual expected profits increase. However, retailer a might find that if she deviates from transshipping agreement, it will raise the cost for b and/or c by a certain dollar amounts. In the long run, one of them might be driven out of the market with profit implications for the remaining retailers. This behavior is called ‘‘Raising rivals’ costs’’ and has been around for a long time. It was formally introduced in economics literature by Salop and Scheffman (1983). Number of important extensions to the proposed taxonomy are ripe for future research. For node A, past studies focused on core allocations. We are not aware of analysis when allocation a is not in the core, except the one by Granot and Soˇsic´ (2003). If allocation a is not in the core, at some realization, a subset of retailers would have an incentive to deviate from the grand coalition. For node G (ex ante price competition), further study is required on how to model the operation of the depot (central coordinator for transshipments). If the depot is a profit maximizing entity, the profit at the central depot should be shared among retailers. The profit sharing scheme will effect the inventory decision in the first stage. For node H (ex post price competition), it would be interesting to examine the conditions for unique Walrasian equilibrium value and the affect of equilibrium on inventory decision. In addition, one may study the effect of ex post price competition under the case of partial sharing (node C). Other different extensions of taxonomy can be easily deduced from combinations of existing nodes. We claim that this taxonomy can serve as a useful and effective framework for analyzing decentralized inventory system. The algorithmic and computational issues related to solutions for the stochastic programming with recourse models deserve their own dedicated exposition. References Anupindi, R., Bassok, Y., Zemel, E., 2001. A general framework for the study of decentralized distribution systems. M & SOM 4 (3), 349–368. Baiou, M., Balinski, M., 2002. The stable allocation or ordinal transportation problem. Mathematics of Operations Research 27 (3), 485–503. Banerjee, A., Burton, J., Banerjee, S., 2003. A simulation study of lateral shipments in single supplier, multiple buyers supply chain networks. International Journal of Production Economics 81–82, 103–114. Beale, E.M.L., 1955. On minimizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society Series B 17 (2), 173–184. Birge, J.R., Louveaux, F., 1997. Introduction to Stochastic Programming. SpringerVerlag, New York, NY.

303

Chen, X., Zhang, J., 2009. A stochastic programming duality approach to inventory centralization games. Operation Research 57 (4), 840–851. Dantzig, G.W., 1955. Linear programming under uncertainty. Management Science 1 (3/4), 197–206. Dror, M., 1993. Modeling vehicle routing with uncertain demands as a stochastic program: properties of the corresponding solution. European Journal of Operational Research 64 (3), 432–441. Dror, M., Laporte, G., Trudeau, P., 1989. Vehicle routing with stochastic demands: properties and solution frameworks. Transportation Science 23 (3), 166–176. Granot, D., Soˇsic´, G., 2003. A three-stage model for a decentralized distribution system of retailers. Operation Research 51 (5), 771–784. Hanany, E., Tzur, M., Levran, A., 2010. The transshipment fund mechanism: coordinating the decentralized multilocation transshipment problem. Naval Research Logistics 57, 342–353. Hezarkhani, B., Kubiak, W., 2010a. A coordinating contract for transshipment in a two-company supply chain. European Journal of Operational Research 207, 232–237. Hezarkhani, B., Kubiak, W., 2010b. Transshipment prices and pair-wise stability in coordinating the decentralized transshipment problem. In: The Behavioral and Quantitative Game Theory: Conference on Future Directions. ACM, Newport Beach, CA. Housman, D., Clark, L., 1998. Core and monotonic allocation methods. International Journal of Game Theory 27, 611–616. Hu, X., Duenyas, I., Kapuscinski, R., 2007. Existence of coordinating transshipment prices in a two-location inventory model. Management Science 53, 1289–1302. Huang, X., Soˇsic´, G., 2010a. Transshipment of inventories: dual allocations vs. transshipment prices. M & SOM 12 (2), 299–312. Huang, X., Soˇsic´, G., 2010b. Repeated newsvendor game with transshipment under dual allocations. European Journal of Operational Research 204 (2), 274–284. Jehle, G.A., Reny, P.J., 2000. Advanced Microeconomic Theory. Addison-Wesley. Lee, Y.H., Jung, J.W., Jeon, Y.S., 2007. An effective lateral transshipment policy to improve service level in the supply chain. International Journal of Production Economics 106, 115–126. Lippman, S.A., McCardle, K.F., 1997. The competitive newsboy. Operation Research 45 (1), 54–65. Minner, S., Silver, E.A., 2005. Evaluation of two simple extreme transshipment strategies. International Journal of Production Economics 93–94, 1–11. Nagarajan, M., Soˇsic´, G., 2008. Game-theoretic analysis of cooperation among supply chain agents: review and extensions. European Journal of Operational Research 187, 719–745. Netessine, S., Rudi, N., 2003. Centralized and competitive inventory models with demand substitution. Operation Research 51 (2), 329–335. Olsson, F., 2009. Optimal policies for inventory systems with lateral transshipments. International Journal of Production Economics 118, 175–184. Owen, G., 1975. On the core of linear production games. Mathematical Programming 9 (1), 358–370. Ozen, U., Slikker, M., Norde, H., 2009. A general framework for cooperation under uncertainty. Operations Research Letters 37, 148–154. Rudi, N., Kapur, S., Pyke, D.F., 2001. A two-location inventory model with transshipment and local decision making. Management Science 47 (12), 1668–1680. Salop, S.C., Scheffman, D.T., 1983. Raising rivals’ costs. American Economic Review 73, 267–271. Samet, D., Zemel, E., 1984. On the core and dual set of linear production games. Mathematics of Operations Research 9 (2), 309–316. Sa´nchez-Soriano, J., Lo´pez, M.A., Garcı´a-Jurado, I., 2001. On the core of transportation games. Mathematical Social Sciences 41, 215–225. Sandsmark, M., 2009. Spatial oligopolies with cooperative distribution. International Game Theory Review 11 (1), 33–40. Shapley, L., Shubik, M., 1975. Competitive outcomes in the cores of market games. International Journal of Game Theory 4 (4), 229–237. Stuart Jr., H.W., 2005. Biform analysis of inventory competition. M & SOM 7 (4), 347–359. Suakkaphong, N., Dror, M., 2010. Competition and cooperation in decentralized distribution. TOP-Journal of the Spanish Society of Statistics and Operations Research 19 (2), 480–506. Tiacci, L., Saetta, S., 2011a. A heuristic for balancing the inventory level of different locations through lateral shipments. International Journal of Production Economics 131, 87–95. Tiacci, L., Saetta, S., 2011b. Reducing the mean supply delay of spare parts using lateral transshipments policies. International Journal of Production Economics 133, 182–191. Van Mieghem, J., Rudi, N., 2002. Newsvendor networks: inventory management and capacity investment with discretionary activities. M & SOM 4 (4), 313–335. Walkup, D.W., Wets, R.J.-B., 1967. Stochastic programs with recourse. SIAM Journal on Applied Mathematics 15 (5), 1299–1314. Wong, H., Van Houtum, G.J., Cattrysse, D., Van Oudheusden, D., 2006. Multi-item spare parts systems with lateral transshipments and waiting time constraints. European Journal of Operation Research 171 (3), 1071–1093. Young, H.P., 1985. Monotonic solutions of cooperative games. International Journal of Game Theory 14, 65–72.