We study the location-inventory model as introduced by Teo et al. (2001) to analyze the impact of consolidation of distribution centers on facility and inventory costs. We extend their result on profitability of consolidation. We associate a cooperative game with each location-inventory situation and prove that this game has a non-empty core for identical and independent demand processes. This illustrates that consolidation does not only lower joint costs (which was shown by Teo et al. (2001)), but it allows for a stable division of the minimal costs as well

Institut d'Anàlisi Econòmica

Klijn, Flip

Slikker, Marco

Universitat Autònoma de Barcelona. Unitat de Fonaments de l'Anàlisi Econòmica

Diposit Digital de Documents de la UAB

Distribution Center Consolidation GamesFlip Klijn∗ Marco Slikker†February 6, 2004AbstractWe study the location-inventory model as introduced by Teo et al. (2001) to analyzethe impact of consolidation of distribution centers on facility and inventory costs.We extend their result on profitability of consolidation. We associate a cooperativegame with each location-inventory situation and prove that this game has a non-empty core for identical and independent demand processes. This illustrates thatconsolidation does not only lower joint costs (which was shown by Teo et al. (2001)),but it allows for a stable division of the minimal costs as well.1 IntroductionDuring the last decade, consolidation of distribution centers (DCs) has turned out to bean excellent strategy in global logistics management. One of the explanations that is oftencited to rationalize its rate of success is the reduction of facility and inventory costs thatcomes with the consolidation and centralization of distribution operations.Facility location problems as well as inventory control have each been studied exten-sively, but separately, in a large literature. In fact, in several studies on facility locationproblems the demands processes are often (quite unrealistically) oversimplified for thesake of tractability. On the other hand, in the literature on inventory control there hasnot been much integration of other supply chain activities until recently. We refer thereader to Teo et al. (2001) for an overview on papers in which both lines of research domeet.As a matter of fact, in the paper mentioned above both aspects are dealt with simul-taneously. The paper can thus be viewed as a study of a facility location model that∗Corresponding author. Institut d’Ana`lisi Econo`mica (CSIC), Campus UAB, 08193 Bellaterra(Barcelona), Spain; e-mail: fklijn@pareto.uab.es. F. Klijn’s research is supported by a Ramo´n yCajal contract of the Spanish Ministerio de Ciencia y Tecnolog´ıa. His work is also partially supportedby Research Grant BEC2002-02130 from the Spanish Ministerio de Ciencia y Tecnolog´ıa and by theBarcelona Economics Program of CREA.†Department of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MBEindhoven, The Netherlands; e-mail: M.Slikker@tm.tue.nl.1captures the impact of the inventory related costs with more sophistication. In fact, Teoet al. (2001) allow for stochastic demands at the customer locations, which brings alongthe analytical problem of not having at hand closed form expressions for the optimalinventory costs for most demand distributions. As they pointed out, their analysis wasfacilitated in a great deal by recent advances in the area of stochastic inventory theory,mainly due to the papers by Zheng (1992) and Gallego (1998).In the location-inventory problem that Teo et al. (2001) study there is a firm with aproduction plant producing a single product for a collection of aggregated demand points(customers). There are a number of possible locations to set up regional distributioncenters, from which the customers are to be served. The main problem the firm facesis to choose the locations so as to minimize total facility and inventory costs. Someassumptions (indifference of the customers with respect to the DCs and constant outboundtransportation costs) are made to isolate the effect of ordering costs and replenishmentleadtimes on the consolidation strategy. One of their main results is that consolidationleads to lower facility investment and inventory costs if the demands are identically andindependently distributed.In this paper, we study the location-inventory problem described above and build onseveral results provided by Zheng (1992) and Teo et al. (2001) to extend the profitabilityresult in the latter work. Our main result says that the minimal total costs (whichcorrespond to the consolidation into a central DC) can also be divided in a stable wayamong the DC locations. There are (at least) two possible interpretations of the existenceof a stable allocation of the minimal total costs. In the first place, it means that there is away to divide the costs so that no set of DC locations can protest against the consolidationstrategy arguing that they could take care of their own customers at less costs. Anotherinterpretation is possible if we assume that the DCs have already been set up. In thiscase the set up costs, which are amortized over time, can be reinterpreted as being thecosts involved in running a DC. The existence of a stable cost allocation then implies thatcosts can be divided in such a way that no set of DC locations can object against the firmcentralizing its DCs into one central DC.The allocation of costs is one of the main issues that are addressed in cooperative gametheory. Cooperative game theory deals with situations involving several agents who canbenefit from cooperation. Establishing a relation between location-inventory problems andcooperative games then enables us to apply cooperative game theory to study the existenceof core allocations, i.e., stable cost allocations. Not surprisingly, cooperative game theoryhas been invoked before for the same reason in several other studies on inventory problems.Cooperative games associated with news-vendor situations were studied in Hartman etal. (2000), Slikker et al. (2001), and Mu¨ller et al. (2002). The last two of these papersindependently established the existence of stable allocations in news-vendor situations.Moreover, they provided conditions under which news-vendor games are convex. Thusthey neatly complement the results of Gerchak and Gupta (1991), Robinson (1993), andHartman and Dror (1996) on allocation rules in continuous review single period inventorysystems with complete back-ordering. Meca et al. (1999) studied cooperative gamesassociated with inventory situations with deterministic demand processes. They focus2on the information that is revealed by the companies and proportional cost allocationmechanisms. Finally, Borm et al. (2001) recently provided a survey of cooperative gamesassociated with operations research situations. They focus on five different natures of theunderlying optimization problem, one of which being ‘inventory’.The rest of the paper is organized as follows. We start in Section 2 with some pre-liminaries on the stochastic (Q, r)-model and cooperative game theory. Then, in Section3, we recall the location-inventory model of Teo et al. (2001), and introduce the class ofassociated DC consolidation games. In Section 4 we present our result on the balanced-ness of DC consolidation games in case the demands are identically and independentlydistributed.2 PreliminariesIn this section we recall some concepts and definitions of (Q, r)-policies and cooperativegame theory.The location-inventory model that we consider includes an order quantity/reorderpoint system, i.e., a (Q, r)-system. More precisely, we study a single-item continuous-review inventory system, where stochastic demands arrive at mean rate µ per unit time.Replenishment orders are delivered after a positive fixed leadtime L. All stockouts arebackordered. The inventory costs consist of an ordering cost k(Q) per order, Q beingthe ordering size, and proportional inventory holding (penalty) costs accumulating at aconstant rate h (p) per unit stock (backorder) per unit time. The ordering cost k(Q)is modeled as a piece-wise linear concave increasing function of the ordering size Q toaccount for the economies of scale in purchasing and transportation. We concentrate onminimizing the long run average total costs per unit time.Under mild conditions on the stochastic demand (see Serfozo and Stidham (1982) andBrowne and Zipkin (1991) for detailed discussions) the long run average total costs perunit time takes the following form:C(Q, r) = C(Q, r, µ) =µk(Q) +∫ r+QrG(y)dyQfor all Q, r > 0, (1)where G(y) is the rate at which the expected inventory costs accumulate at time t + Lwhen the inventory position at time t equals y ∈ IR, i.e.,1G(y) = E(h(y −D)+ + p(D − y)+),D being the sum of demands that occur during the time interval (t, t + L]. The termλk(Q)Qin (1) reflects the average ordering cost. The other term,∫ r+Qr G(y,µ)dyQ, reflects theaverage holding and stockout cost of the system, as the inventory level can be shown tobe uniformly distributed between r and r +Q (cf. Zheng (1992)).1For x ∈ IR we write x+ = max{0, x}.3The following (technical) results are due to Zheng (1992) and facilitate much of ouranalysis later on. Definer(Q) := argminr∫ r+QrG(y)dy for all Q > 0.Then, r(Q) is the optimal reorder point when the order quantity is fixed at Q. Zheng(1992) showed thatG(r(Q)) = G(r(Q) +Q) for all Q > 0.Moreover, he proved that the functions H and S defined byH(Q) :={minyG(y) if Q = 0;G(r(Q)) if Q > 0,andS(Q) :={0 if Q = 0;∫ r(Q)+Qr(Q)G(y)dy =∫ Q0H(y)dy if Q > 0,are both convex in Q. Hence, using S(0) = 0, we have thatS(αQ) ≤ αS(Q) for all Q > 0 and 0 ≤ α ≤ 1. (2)The other ingredient of this paper is cooperative game theory of which we recall thenecessary basic definitions below. A cooperative (cost) game with transferable utility(game, for short) is a pair (N, c), where N = {1, . . . , n} is the player set and c thecharacteristic function, which assigns to every subset S ⊆ N a value c(S), with c(∅) = 0.The number c(S) represents the minimal costs for coalition S when all its members decideto cooperate. The subgames of (N, c) are the games (S, c|S) where S ⊆ N and c|S : S → IRis the restriction of the map c to the coalition S.Cooperative game theory focuses on fair division rules for the value of c(N) of thegrand coalition. A core allocation x = (xi)i∈N ∈ IRN divides the value c(N) among theplayers in such a way that no coalition has an incentive to split off, i.e.,x(N) = c(N) and x(S) ≤ c(S) for all S ⊆ N,where x(S) =∑i∈S xi for all S ⊆ N . The core Core(N, c) is the (possibly empty) set ofcore allocations.A map λ : 2N\{∅} → [0, 1] is called balanced if for all i ∈ N it holds that∑S⊆N :i∈S λ(S) =1. A cost game (N, c) is called balanced if for each balanced map λ we have∑S⊆Nλ(S)c(S) ≥ c(N).4Bondareva (1963) and Shapley (1967) proved2 independently that non-emptiness of thecore and balancedness of a game are equivalent, i.e., for a game (N, c) we haveCore(N, c) 6= ∅ ⇔ (N, c) is balanced.A game is called totally balanced if all its subgames are balanced.3 The modelIn this section we recall the location-inventory problem as defined by Teo et al. (2001).Moreover, we introduce the class of consolidation games which are cost games associatedwith the location-inventory problems.In the location-inventory problem, there is a set M = {1, 2, . . . ,m} of demand points.We assume that the demand for the product at demand point j can be modeled by acontinuous stochastic process {dj(t) : t ≥ 0}, with mean rate µj per unit time.The demands are to be served by n possible distribution centers, which are indexedby 1, . . . , n. Let N = {1, . . . , n} be the set of possible DC locations. We assume thatthe demand points (i.e., customers) are indifferent about where their orders are shippedfrom, and also that the outbound transportation costs do not depend on where the ordersare shipped from. We assume that initially each DC i is to serve aggregate demand∑j∈M x0ijdj(t), where the (x0ij)i∈N,j∈M satisfy x0ij ∈ {0, 1} (i ∈ N, j ∈M) and∑ni=1 x0ij = 1(j ∈M). So, the demands at each demand point are exclusively served from a single DC.A group (coalition) S ⊆ N of distribution centers may decide to cooperate and re-assign the initial demands within the coalition. A demand re-assignment for S is a matrixXS = (xSij)i∈S,j∈M that satisfies xSij ∈ {0, 1} (i ∈ S, j ∈ M) and∑i∈S∑j∈M xSij =∑i∈S∑j∈M x0ij. Let ΠS be the set of demand re-assignments for S. If for a given demandre-assignment XS it holds that XSi :=∑mj=1 xij > 0, then DC i is set up for which itincurs a fixed setup cost fi. We assume that fi is a per unit time charge, which thenalso could be treated as and/or include the direct variable costs in running a DC. Let Libe the leadtime to supply from the plant to DC i. The inventory costs at DC i includean ordering cost ki(Q) per order, Q being the ordering size, and proportional inventoryholding (penalty) costs accumulating at a constant rate hi (pi) per unit stock (backorder)per unit time. The ordering cost ki(Q) is modeled as a piece-wise linear concave increasingfunction of the ordering size Q to account for the economies of scale in purchasing andtransportation.If a coalition S ⊆ N of distribution centers decides to cooperate, i.e., re-assign theinitial demand within S to minimize costs, then it follows readily that the optimal long2In fact, Bondareva (1963) and Shapley (1967) proved that a cost savings game (N, v) is balanced(i.e., there is an allocation x = (xi)i∈N ∈ IRN such that x(S) ≥ v(S) for all S ⊆ N with equality forS = N) if and only if for each balanced map λ we have∑S⊆N λ(S)v(S) ≤ v(N). This is easily shown tobe equivalent with the above statement.5run average costs for S equalc(S) := minXS∈ΠS{∑i∈S[fiχ(xSi > 0) + minr,QC(r,Q,∑j∈MxSijµj)]},where χ is the indicator function and C the cost function as defined in (1).4 BalancednessIn this section we restrict ourselves to identically and independently distributed (i.i.d.)demand processes. We prove that the associated consolidation games are balanced, i.e.,allow for stable divisions of the joint costs.Since demands are i.i.d. we can assume without loss of generality that µj = 1 forall demand points j ∈ M . Let ni := x0i =∑j∈M x0ij be the aggregate demand thatoriginally would be served by DC i ∈ N . Define nS :=∑i∈S ni for all S ⊆ N . Recall thatx0ij ∈ {0, 1} for all i ∈ N, j ∈M , so that nS ∈ IN for all S ⊆ N . Note that nN = m.Let us now consider the (Q, r)-system that corresponds to a distribution center thatserves a set M ′ ⊆M of demand points. Since the demand points are i.i.d. we can defineG(y,m′) as the rate at which the expected inventory costs accumulate at time t+L whenthe inventory position at time t. The next result is due to Teo et al. (2001) (page 102).Lemma 4.1 (Teo et al. (2001)) For all y ∈ IR and 0 ≤ m′ ≤ m we have m′mG(y,m) ≤G(m′my,m′).Theorem 4.2 Consolidation games have a non-empty core when the demand processesare i.i.d..Proof. Let λ : 2N\{∅} → [0, 1] be a balanced map.Let S ⊆ N . From Theorem 1 in Teo et al. (2001) we know that an optimal strategyfor coalition S is consolidation at some DC in S, say iS.Consider the strategy for the grand coalition in which with probability λ(S) nSnN(S ⊆N) consolidation at iS takes place. Equivalently, a DC i ∈ N is picked with probability∑S⊆N :i=iS λ(S)nSnN. Note that∑S⊆N λ(S)nSnN= 1nN∑S⊆N λ(S)nS =1nN∑S⊆N∑i∈S λ(S)ni =1nN∑i∈N∑S:i∈S λ(S)ni =1nN∑i∈N ni∑S:i∈S λ(S) =1nN∑i∈N ni = 1.Since this strategy is a weighted average of (pure) consolidation strategies, the asso-6ciated costs majorize the optimal costs of the grand coalition. Hence,c(N) ≤∑S⊆Nλ(S)nSnN[fiS +minr,QkiS(Q)nN +∫ r+QrGiS(y, nN)dyQ]=∑S⊆Nλ(S)[nSnNfiS +minr,QnSnNkiS(Q)nN +∫ r+QrGiS(y, nN)dyQ]≤∑S⊆Nλ(S)[fiS +minr,QnSnNkiS(Q)nN +∫ r+QrGiS(y, nN)dyQ]=∑S⊆Nλ(S)[fiS +minr,QkiS(Q)nS +∫ r+QrnSnNGiS(y, nN)dyQ]≤∑S⊆Nλ(S)[fiS +minr,QkiS(Q)nS +∫ r+QrGiS(nSnNy, nS)dyQ]=∑S⊆Nλ(S)fiS +minr,QkiS(Q)nS +nNnS∫ nS(r+Q)/nNnSr/nNGiS(nSnNy, nS)dyQ=∑S⊆Nλ(S)[fiS +minr,QkiS(Q)nS +nNnS∫ r+nSQ/nNrGiS(nSnNy, nS)dyQ]≤∑S⊆Nλ(S)[fiS +minr,QkiS(Q)nS +∫ r+QrGiS(y, nS)dyQ]=∑S⊆Nλ(S)c(S).The third inequality follows from Lemma 4.1 with m′ = nS. The last inequality followsfrom (2) with α = nSnN. The last equality follows from the optimality for S to consolidateinto DC iS.We conclude that the consolidation game is balanced and hence, using the result byBondareva (1963) and Shapley (1967), that it has a non-empty core. 2There are (at least) two possible interpretations of Theorem 4.2. In the first place, itmeans that there is a way to divide the minimal total costs so that no set of DC locationscan protest against the consolidation strategy arguing that they could take care of theirown customers at less costs. Another interpretation is possible if we assume that the DCshave already been set up. In this case the set up costs, which are amortized over time, canbe reinterpreted as being the costs involved in running a DC. The existence of a stablecost allocation then implies that costs can be divided in such a way that no set of DClocations can object against the firm centralizing its DCs into one central DC.7The following result follows immediately by noting that a subgame (S, c|S) of a consol-idation game (N, c) is again a consolidation game, associated with the original location-inventory problem restricted to S.Corollary 4.3 Every consolidation game is totally balanced when the demand processesare i.i.d..ReferencesBondareva, O.N. (1963). “Some Applications of the Methods of Linear Programming tothe Theory of Cooperative Games,” (in Russian) Problemy Kibernetiki, 10, 119-139.Borm, P., Hamers, H., and Hendrickx, R. (2001). “Operations Research Games: ASurvey,” TOP, 9, 139-198.Browne, S. and Zipkin, P. (1991). “Inventory Models with Continuous, Stochastic De-mands,” Annals of Applied Probability, 1, 419-435.Gallego G. (1998). “New Bounds and Heuristics for (Q, r) Policies,” Management Sci-ence, 44, 219-233.Gerchak, Y. and Gupta, D. (1991). “Technical Note: On Apportioning Costs to Cus-tomers in Centralized Continuous Review Inventory Systems,” Journal of OperationsManagement, 10, 546-551.Hartman, B.C. and Dror, M. (1996). “Cost Allocation in Continuous Review Inven-tory Models,” Naval Research Logistics Quarterly, 43, 549-561.Hartman, B.C., Dror, M., and Shaked, M. (2000). “Cores of Inventory CentralizationGames,” Games and Economic Behavior, 31, 26-49.Meca, A., Timmer, J., Garc´ıa-Jurado, I., and Borm, P. (1999). “Inventory Games,”CentER Discussion Paper 9953, Tilburg University, Tilburg, The Netherlands. To appearin European Journal of Operational Research.Mu¨ller, A., Scarsini M., and Shaked M. (2002). “The Newsvendor Game has a non-empty Core,” Games and Economic Behavior, 38, 118-126.Robinson, L. (1993). “A Comment on Gerchak and Gupta’s: On Apportioning Coststo Customers in Centralized Continuous Review Inventory Systems,” Journal of Opera-tions Management, 11, 99-102.Serfozo, R. and Stidham, S. (1982). “Semi-Stationary Clearing Processes,” Stochastic8Process and Applications, 6, 165-178.Shapley, L.S. (1967). “On Balanced Sets and Cores,” Naval Research Logistics Quar-terly, 14, 453-460.Slikker, M., Fransoo, J., and Wouters, M. (2001). “Joint Ordering in Multiple News-Vendor Problems: A Game-Theoretical Approach,” Beta Working Paper WP-64, Depart-ment of Technology Management, Eindhoven University of Technology, Eindhoven, TheNetherlands.Teo, C.P., Ou J., and Goh, M. (2001). “Impact on Inventory Costs with Consolida-tion of Distribution Centers,” IIE Transactions, 33, 99-110.Zheng Y.S. (1992). “On Properties of Stochastic Inventory Systems,” Management Sci-ence, 38, 87-103.9

Distribution center consolidation games

http://ddd.uab.cat/pub/worpap/2006/hdl_2072_1847/60204.pdf

Distribution center consolidation games

Abstract

Similar works

Full text

Available Versions

Diposit Digital de Documents de la UAB