Distribution Center Consolidation Games∗ Flip Klijn†

Marco Slikker‡ May 2004 Abstract

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 associate a cooperative game with each location-inventory situation and prove that when demand processes are i.i.d. this game has a non-empty core. Hence, consolidation does not only lower joint costs (Teo et al., 2001), but it allows for a stable division of the minimal costs as well. Keywords: inventory management, cooperative games

1

Background

In the location-inventory problem as defined by Teo et al. (2001) there is a set M = {1, 2, . . . , m} of demand points. The demand for the product at demand point j can be modeled by a continuous stochastic process {dj (t) : t ≥ 0} with mean rate µj per unit time. In this note we assume that the demand processes are identically and independently distributed (i.i.d.). We assume, without loss of generality, that µj = 1 for all demand points j ∈ M . The demands are to be served by n distribution centers (DCs), which are indexed by 1, . . . , n. Let N = {1, . . . , n}. We assume that the demand points (i.e., customers) are indifferent about where their orders are shipped from, and also that the outbound transportation costs do not depend on where the assume that initially each DC i is to serve P orders0 are shipped from. We 0 aggregate demand j∈M xij dj (t), where the (xij )i∈N,j∈M satisfy x0ij ∈ {0, 1} (i ∈ N, j ∈ M ) P and i∈N 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 P S is a matrix X S = P S S S (xij )i∈S,j∈M that satisfies xij ∈ {0, 1} (i ∈ S, j ∈ M ) and i∈S,j∈M xij = i∈S,j∈M x0ij . Let ΠS be the set re-assignments for S. If for a given demand re-assignment X S it holds that P of demand S S Xi := j∈M xij > 0, then DC i is set up for which it incurs a fixed setup cost fi . We assume that fi is a per unit time charge, which then also could be treated as and/or include the direct ∗

We thank Bettina Klaus and the referee for helpful comments. Corresponding author. Institut d’An` alisi Econ` omica (CSIC), Campus UAB, 08193 Bellaterra (Barcelona), Spain; e-mail: [email protected]. F. Klijn’s research is supported by a Ram´ on y Cajal contract of the Spanish Ministerio de Ciencia y Tecnolog´ıa. His work is also partially supported by Research Grant BEC200202130 from the Spanish Ministerio de Ciencia y Tecnolog´ıa and by the Barcelona Economics Program of CREA. ‡ Department of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands; e-mail: [email protected]. †

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variable costs in running a DC. Let Li be the positive fixed leadtime to supply from the plant to DC i. All stockouts are backordered. The inventory costs at DC i include an ordering cost ki (Q) per order, Q being the ordering size, and proportional inventory holding (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 increasing function of the ordering size Q to account for the economies of scale in and transportation. Ppurchasing S S Let S ⊆ N . For i ∈ S define µi = j∈M xij µj . If S decides to cooperate, i.e., re-assign the initial demand within S to minimize costs, then the optimal long run average costs for S equal ( ) X S S c(S) := min fi χ(Xi > 0) + min C(Q, r, µi ) , (1) X S ∈ΠS

Q,r

i∈S

where χ is the indicator function and C the cost function from the corresponding order quantity/reorder point system, i.e., (Q, r)-system, which we explain next. Given a demand re-assignment X S and under mild conditions on the stochastic demand (see Serfozo and Stidham (1982) and Browne and Zipkin (1991) for detailed discussions) the long run average total costs per unit time for a DC i ∈ S takes the following form: R S k(Q) + r+Q G(y)dy µ i r C(Q, r) = C(Q, r, µSi ) = for all Q, r > 0, (2) Q where G(y) is the rate at which the expected inventory costs accumulate at time t + L when the inventory position at time t equals y ∈ IR, i.e., G(y) = E(h(y − D)+ + p(D − y)+ ), D being the sum of demands that occur during the time interval (t, t + L]. (For x ∈ IR we write x+ = max{0, x}.) The term R r+Q

µS i k(Q) Q

in (2) reflects the average ordering cost. The other term,

G(y)dy , Q

reflects the average holding and stockout cost of the system, as the inventory level can be shown to be uniformly distributed between r and r + Q (cf. Zheng, 1992). Define r(Q) as the optimal reorder point when the order quantity is fixed at Q, i.e., Z r+Q G(y)dy for all Q > 0. r(Q) = argminr r

r

Zheng (1992) showed that ( S(Q) =

0 R r(Q)+Q r(Q)

if Q = 0; G(y)dy if Q > 0,

is convex in Q. Hence, using S(0) = 0, we have that S(αQ) ≤ αS(Q) for all Q > 0 and 0 ≤ α ≤ 1.

(3)

Finally, consider the (Q, r)-system that corresponds to a distribution center that serves a set M 0 ⊆ M of demand points. Since the demand points are i.i.d. we can define G(y, m0 ) as the associated rate at which the expected inventory costs accumulate at time t + L when the inventory position at time t equals y. The next result is due to Teo et al. (2001). Lemma 1.1 (Teo et al. (2001), p. 102) For all y ∈ IR and 0 ≤ m0 ≤ m we have 0 0 G( m m y, m ). 2

m0 m G(y, m)

≤

2

Result

One of the main results in Teo et al. (2001) is that consolidation leads to lower facility investment and inventory costs if the demands are identically and independently distributed. Our result states that these minimal total costs (which correspond to the consolidation into a central DC) can be divided in a stable way among the DC locations. If we reinterpret the set up costs, which are amortized over time, as the costs involved in running a DC, then the existence of a stable cost allocation implies that costs can be divided in such a way that no coalition of DCs can object against the firm centralizing its DCs into one central DC. To prove our result we show the existence of a core allocation of the cooperative game (N, c) where c is the characteristic function that assigns to every coalition S ⊆ N the optimal long run average costs c(S), as given by (1). The game (N, c) is called the consolidation game associated with the location-inventory situation described in the previous section. A core allocation x = (xi )i∈N ∈ IRN divides the value c(N ) among the DCs in such a way that no coalition P 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 set of core allocations. Our result can now be formulated as follows. Theorem 2.1 Consolidation games have a non-empty core. Proof. Let (N, c) be a consolidation game. Bondareva (1963) and Shapley (1967) proved independently that the existence of a core allocation is equivalent to balancedness. P The game (N, c) N is balanced if for each balanced map λ : 2 \{∅} → [0, 1] (i.e., for all i ∈ N , S⊆N :i∈S λ(S) = 1) P we have c(N ) ≤ S⊆N λ(S)c(S). 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 strategy for coalition S is P consolidation at some DC in S, say iS . Let ni := P j∈M x0ij be the aggregate demand that initially would be served by DC i ∈ N . Define nS := i∈S ni ∈ IN for all S ⊆ N . Note that nN = m. Consider the strategy for the grand coalition in which with probability λ(S) nnNS (S ⊆ N ) consolidation at iS takes place. a DC i ∈ N is picked with P Equivalently, P probability P nS nS 1 P 1 P . Note that λ(S) = λ(S)n = λ(S) S S⊆N S⊆N S⊆N i∈S λ(S)ni = S⊆N :i=iS nN nN nN P P nN 1 P 1 P 1 P i∈N S:i∈S λ(S)ni = nN i∈N ni S:i∈S λ(S) = nN i∈N ni = 1. nN Since this strategy is a weighted average of (pure) consolidation strategies, the associated costs majorize the optimal costs of the grand coalition. Hence, " # R r+Q X kiS (Q)nN + r GiS (y, nN )dy nS fiS + min c(N ) ≤ λ(S) r,Q nN Q S⊆N " # R r+Q X GiS (y, nN )dy nS nS kiS (Q)nN + r = λ(S) fi + min r,Q nN nN S Q S⊆N " # R r+Q X GiS (y, nN )dy nS kiS (Q)nN + r ≤ λ(S) fiS + min r,Q nN Q S⊆N " # R r+Q nS X kiS (Q)nS + r nN GiS (y, nN )dy = λ(S) fiS + min r,Q Q S⊆N

3

(4)

≤

" X

λ(S) fiS + min 

(5)

=

λ(S) fiS + min 

=

X

λ(S) fiS + min

≤

" X

λ(S) fiS + min

=

X

nN nS

nS r/nN

#

GiS (z, nS )dz

kiS (Q)nS +

nN nS

R r+nS Q/nN r

GiS (z, nS )dz

R r+Q r

GiS (z, nS )dz

 

Q kiS (Q)nS +

 

Q

#

Q

r,Q

S⊆N (8)

kiS (Q)nS +

GiS ( nnNS y, nS )dy

Q R nS (r+Q)/nN

r,Q

S⊆N (7)

r

r,Q

S⊆N (6)

R r+Q

r,Q

S⊆N

X

kiS (Q)nS +

λ(S)c(S).

S⊆N

Inequality (4) follows from Lemma 1.1 with m0 = nS . Equality (5) follows by substituting z = nnNS y. Equality (6) follows by redefining r (as nnNS r). Inequality (7) follows from (3) with α = nnNS . Equality (8) follows from the optimality for S to consolidate into DC iS . 2

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Discussion

In this note we have taken up the analysis of location-inventory models from a game-theoretical point of view. We have shown that instances of the location-inventory model of Teo et al. (2001) with i.d.d. demand processes result in balanced games, i.e., cooperation is not only profitable, but profits can be divided in a stable way as well. This result fits in the recent stream of literature on the interface of inventory control and game theory (see Borm et al. (2001) for a review of operations research games). The i.i.d. demand assumption in our model is certainly restrictive. However, as also noted by Teo et al. (2001), if demand processes are for example independent and Poisson with different rates that allow for some common base rate then our result holds as well. An approximation can be used in case there is no common base rate. Further research could focus on general demand distributions. It is well-known, however, that associated optimal inventory cost functions are very complex. Another direction for further research would be the study of more restrictive concepts like concavity to shed more light on the structure of the core.

References Bondareva, O.N. (1963). “Some Applications of the Methods of Linear Programming to the Theory of Cooperative Games,” (in Russian) Problemy Kibernetiki, 10, 119-139. Borm, P., Hamers, H., and Hendrickx, R. (2001). vey,” TOP, 9, 139-198.

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Browne, S. and Zipkin, P. (1991). “Inventory Models with Continuous, Stochastic Demands,” Annals of Applied Probability, 1, 419-435. Serfozo, R. and Stidham, S. (1982). Process and Applications, 6, 165-178. Shapley, L.S. (1967). terly, 14, 453-460.

“Semi-Stationary Clearing Processes,” Stochastic

“On Balanced Sets and Cores,” Naval Research Logistics Quar-

Teo, C.P., Ou J., and Goh, M. (2001). “Impact on Inventory Costs with Consolidation of Distribution Centers,” IIE Transactions, 33, 99-110. Zheng, Y.S. (1992). “On Properties of Stochastic Inventory Systems,” Management Science, 38, 87-103.

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