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European Journal of Operational Research 209 (2011) 273–284 Contents lists available at ScienceDirect European Journal of Operational Research Production, Manufacturing and Logistics On a multi-period supply chain system with supplementary order opportunity Xiang Li a, Yongjian Li b,⇑, Xiaoqiang Cai c a College of Economic and Social Development, Nankai University, Tianjin 300071, PR Chinab Business School, Nankai University, Tianjin 300071, PR Chinac Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong This paper considers a supplementary supply–order system in a multi-period situation. In each period, Received 27 December 2008 the buyer first places an initial order based on the demand prediction; he has the opportunity to place Accepted 19 August 2010 a supplementary order with the supplier after the demand of that period is realized. The supplier main- Available online 24 August 2010 tains an inventory, and decides the quantity to be produced and the quantity to be provided for the sup-plementary order in each time period. We formulate the problem as a multi-period inventory game, and derive the optimal production and order policies for the supplier and buyer, respectively. The existence Supply chain management and uniqueness of Nash equilibrium is proved in the generalized multi-period setting, and the closed- form Nash equilibrium solution is obtained when the parameters are stationary. Numerical study is per- Game theorySupplementary order formed to reveal more managerial insights. We find that the supplementary supply–order mechanism, ifdesigned properly, can effectively improve the multi-period supply chain performance.
Ó 2010 Elsevier B.V. All rights reserved.
to a level higher than the normal order quantity, to prepare a sup-plementary supply for the buyer. However, if the supplier produces Consider a supply chain consisting of a supplier and a buyer, in an exceeded amount of product, she has to pay the holding cost for which the buyer faces a stochastic demand of product. The buyer her own leftover inventory after the buyer's supplementary order.
commits an initial order, referred to as ‘‘normal order" in this pa- To summarize, the buyer decides his normal order quantity at the per, before the demand realization, and then has an opportunity beginning of each period, to balance the tradeoff between the ben- to place a supplementary order with the supplier after the demand efit of lower inventory and the loss of shortage, while the supplier is realized. The stochastic nature of the demand usually causes decides her supplementary supply quantity, to balance the tradeoff insufficient/excess inventory, resulting in substantial revenue between the profitability of the higher wholesale price of the sup- loss/holding costs for the supply chain members. In a decentralized plementary order and the risk of holding excess inventory.
system the performance of the supply chain members could be In what follows, we briefly relate our paper to the literature, even worse because of their different and usually, conflicting including study on the emergency replenishment policy, on the objectives they attempt to optimize.
quantity flexibility contract, and on the supply chain with dual Some studies, e.g., Cachon (2004), Dong and Zhu (2007), have discussed the supplementary supply–order problem in a single- Numerous researchers have studied the use of emergency period situation. In this paper we will consider the problem in a replenishment to raise service level and to mitigate the risk of de- multi-period situation, which arises from the supply chain man- mand randomness, such as Chiang (2003, 2010), Chiang and Gut- agement of many non-seasonal products, like raw materials, com- ierrez (1998), and Sethi et al. (2001, 2003), among others. These papers study the optimal policies of periodic or continuous review commodities. In each period, the buyer first submits a normal or- inventory system under consideration of two replenishment der to the supplier based on the forecast of demand, and then modes: fast and slow, where the fast one is more expensive than places a supplementary order if the normal order cannot cover the slow one. Papers such as Fisher and Raman (1996, 2001), Eppen the realized demand. The supplementary order is submitted after and Iyer (1997), Brown and Lee (2003), Yan et al. (2003), and Choi the realization of the demand and charged at a higher wholesale (2007) have developed models for fashion product in which the price, if the supplier has available units to satisfy the order. To in- stochastic demand only occurs once but more than one replenish- crease profit, the supplier might rationally increase her inventory ment opportunity can be utilized. These papers study the effect ofadvance ordering/production and demand information updating where the interaction between supply chain members is not an Corresponding author. Tel.: +86 22 23505341.
E-mail address: [email protected] (Y. Li).
0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2010.08.019 Author's personal copy
X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 There also has been much interest in exploring the impact of related to the quantity decision in the channel (the buyer two or more order chances from a supply chain perspective. A moves first and then the supplier follows). This fundamental mainstream research is on the quantity flexibility contracts based difference leads to some different results (like the wholesale on option. These contracts allow the buyer to adjust his order price boundary) in our paper.
quantities after the normal order, and an option price is paid tothe supplier for this flexibility. For example, Barnes-Schuster In the paper, we investigate how the supplementary supply–or- et al. (2002) explores the contract with option in a two-period sup- der system works in a generalized multi-period setting by the ply chain, in which the supplier has two production opportunities game theoretic approach. We derive optimal normal order policy and the buyer is allowed to increase delivery quantity at the sec- for the buyer and optimal supplementary supply policy for the ond period by utilizing the options. Wang et al. (2006) investigates supplier, respectively, and prove the uniqueness of Nash equilib- a similar flexible contract strategy, except that the supplier has rium. When system parameters are stationary over time, the ana- only one production opportunity. Van Delft and Vial (2001) lytic solution of Nash equilibrium is obtained, and the condition presents a stochastic programming approach to analyze the mul- under which this supplementary supply–order mechanism takes ti-periodic supply chain contracts with options. Other relevant re- effect is given. We find that the supplementary supply–order searches include Bassok and Anupindi (1997, 1999), Tsay (1999), mechanism is a simple and applicable method to allocate multi- Deshmukh and Lian (2009), Zhao et al. (2010), etc.
period demand risk between the buyer and the supplier, and to Moreover, Deng and Yano (2002a) models a two-echelon supply achieve substantial improvement in the supply chain performance chain where a second purchase opportunity at an anticipated high- if properly designed.
er price induces the supplier to bear uncertainty risk. Deng and The rest of paper is organized as follows. In Section 2 we present Yano (2002b) studies the impact of a late ordering opportunity the description and assumptions of the model. In Section 3 we for- on a push system, assuming that the price of the late order is ran- mulate and analyze the model. We establish an inventory game dom. Ozer et al. (2007) considers a supply chain with a demand and prove the uniqueness of its Nash equilibrium. The analytic forecast update under a wholesale price contract and a dual pur- equilibrium solution and some properties are explored in the case chase contract, through which the manufacturer provides a dis- of stationary parameters. Section 4 presents a numerical study to count for orders placed before the forecast updates. They show provide more insights and Section 5 concludes the paper.
that the supply chain performance is better off in the dual purchasecontract than the one in the wholesale price contract under certaincircumstances.
2. Model description Cachon (2004), and Dong and Zhu (2007) are the literature that most closely relate to our work. Both papers explore the single- 2.1. Problem description period case in which the supplier has only one production oppor-tunity and the buyer has two order opportunities. Cachon (2004) We consider the following problem. In each time period of a analyzes scenarios where a buyer can order from a supplier before multi-period horizon, a buyer orders a product from a supplier the demand realization (push contract) or after realization (pull and then sells it in a market. The market demand of period t is a contract), or make a combination of these two (advance-pur- stochastic variable D(t) with known distribution information. The chase-discount contract), which is very similar to the supplemen- buyer has two order opportunities in each period. He places a ‘‘nor- tary supply–order system in our paper. The main focus of Cachon mal" order q(t) at the beginning of each period t, to raise his inven- (2004) is on investigating the Pareto set with three contracts and tory position from x(t) to y(t). Before the end of period t, he receives showing that the combining consideration of push and pull con- the normal order q(t) and demand D(t) is realized. Then, if he finds tracts yields higher supply chain efficiency. Dong and Zhu (2007) that there is a shortage to meet the demand, he can submit a ‘‘sup- studies push, pull, and advanced-purchase discount contracts in plementary" order, which would be delivered to him immediately an unified two-wholesale-price contract framework and shows to satisfy the demand of that period if the supplier has available how the channel decision changes along with the two wholesale inventory. The supplementary order is more costly than the normal prices. In the above two papers, only the single-period problem order for the buyer, and it is not obliged to be fulfilled by the sup- has been discussed.
plier. The demand shortage after the supplementary order, if any, Different from the above studies, our work makes the following becomes lost sale. However, if the normal order exceeds the real- ized demand in that period, the buyer carries the unsold units inhis inventory to the next period to meet future demands and pays (1) We study the supply chain system in a multi-period setting, the holding cost correspondingly. Therefore, when making the nor- derive the optimal production and order policy for both par- mal order decision, the buyer should balance between the cost for ties, and explore Nash equilibrium of the multi-period excess inventory and the cost for demand shortage.
inventory game. We show that the supplementary supply– On the other hand, the supplier has one production opportunity order mechanism is beneficial for a multi-period system if in each period and so needs to keep an inventory to meet the pos- the system parameters are properly designed; sible supplementary order of the buyer. At the beginning of period (2) When the parameters are stationary we characterize the t, the supplier raises her inventory position from xs(t) to a level analytic solution of the Nash equilibrium, and provide an equal or higher than the buyer's normal order q(t). The supplier might produce to a higher level since she can charge a higher price which the supplementary supply–order option will be for the fulfilled supplementary order. However, if the supplier pro- duces excessively, she needs to bear the holding cost for her own (3) The decision interaction in our supplementary supply–order inventory after the buyer's supplementary order. Therefore, when system is different from that under the two-wholesale-price making the production decision, the supplier should balance be- contract proposed in the literature. In our model we estab- tween the profitability of the supplementary supply and the risk lish a Nash game, in which the supplier decides the supply of excess inventory.
quantity and the buyer decides the order quantity in a Note that in our problem, the buyer is to raise, at the beginning simultaneously interactive way, while in the two-whole- of each period t, his inventory position to y(t) by the normal order, sale-price contract the buyer is virtually a Stackelberg leader given the initial inventory x(t). The normal order quantity q(t) is Author's personal copy
X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 (4) At the same time of period t, the supplier raises the product Notation used in the model.
quantity to the level q(t) + K(t) by producing q(t) + K(t)  xs(t) units, given the initial inventory is xs(t).2 Production isfinished before demand is realized and q(t) units are delivered Stochastic demand in period t Selling price in period t to the buyer. The remaining are to meet the buyer's supple- Wholesale price in a normal order in period t mentary order of that period. After the supplementary order Wholesale price in a supplementary order in period t is fulfilled, the leftover product, if any, is held in the supplier's Production cost in period t inventory for the use of next period and incurs holding cost to The buyer's inventory holding cost for leftovers by the end of the supplier.
The supplier's inventory holding cost for leftovers by the end of (5) In period t, the production cost is c(t). The supplier charges the wholesale price w1(t) for the normal order Salvage value of the leftover for the buyer by the end of the last and w2(t) for the supplementary order. The selling price is r(t). As it is not the objective to investigate pricing deci- Salvage value for the leftover for the supplier by the end of the lastperiod sions in this problem, we suppose that the buyer is in a The buyer's inventory position after the normal order in period t; competitive marketplace thus r(t) is exogenous. The decision variable specification of wholesale prices is prior to any order or The supplementary supply quantity; decision variable production quantity decision, e.g., they are determined in The buyer's inventory position at the beginning of period t advance according to the bargaining power between sup- The supplier's inventory position at the beginning of period t The system's inventory position at the beginning of period t, equal prices are given parameters and satisfy r(t) > w2(t) > Discount factor, 0 < c 6 1 w1(t) > c(t).
(6) For the buyer, unit leftover product at the end of period t = 0, 1, . . , T  2 incurs inventory holding cost h(t + 1). Weassume that h(t + 1) + w1(t) > cw1(t + 1) for t = 0, 1, . . , T  2; hence equal to y(t)  x(t). Meanwhile, the supplier is to raise her otherwise, the buyer may deliberately order more units than inventory position to q(t) + K(t) at the beginning of each period t, necessary in period t in order to satisfy the demand of the given the initial inventory x next period, because the procurement cost in period t has s(t). Thus the production quantity of the supplier in period t is q(t) + K(t)  x an economic advantage over period t + 1. At the end of the s(t). We refer to K(t) as the possible supplementary supply quantity in period t, which rep- last period t = T  1, leftover unit has a terminal value sT, in resents the maximum amount of units to be provided to the buyer which h(T) + w1(T  1) > csT. The inequation holds as other- in addition to the normal order. Therefore, K(t) + q(t) is the maxi- wise the buyer may deliberately order infinite amount in mum that could be used to satisfy the demand in period t.
the last period T  1 just to obtain the terminal value rather We list our notation in Table 1 and present in details the events than to fulfill the demand.
and assumptions in the next subsection.
(7) For the supplier, unit leftover product after supplementary order at the end of period t = 0, 1, . . , T  2 incurs inventory 2.2. Events and assumptions c(t) > cc(t + 1) for t = 0, 1, . . , T  2; otherwise, the suppliermay deliberately produce more units than necessary in per- (1) There is a sequence of time points t = 0, 1, . . , T. Period t is the iod t in order to satisfy the order of the next period, because time interval from the point t until just before the point t + 1.
the production cost in period t has an economic advantage The inventories of the buyer and supplier are both operated over period t + 1. At the end of the last period t = T  1, unit under a periodic review over the planning horizon.
leftover product has a terminal value ST, in which (2) Demand D(t) is a continuous random variable with the dis- hs(T) + c(T  1) > cST. The inequation holds as otherwise the tribution function F supplier may deliberately order infinite amount in the last D(t) and finite mean. The demands are independent of each other. Both Cachon (2004) and Dong period T  1 just to obtain the terminal value rather than and Zhu (2007) suppose that the demand has increasing to fulfill the order.
generalized failure rate (IGFR) property, while in our paper (8) The information of distribution FD(t) and all other parameters there is no such restrictive assumption on the demand are shared by the buyer and supplier. Both parties seek to maximize their own total expected profits over all periods, (3) At the beginning of period t, the buyer raises the inventory position to y(t) by submitting a normal order q(t) = y(t) x(t), given the initial inventory is x(t). The normal orderq(t) is delivered to the buyer and D(t) is realized by the 3. Model formulation, analysis, and results end of period t. After the demand is fulfilled, the leftoverproduct, if any, is held in the buyer's inventory for the use 3.1. The buyer's problem of next period and incurs holding cost to the buyer. If anyshortage occurs for the demand, the buyer submits a supple- First, we formulate a multi-period inventory model for the mentary order to the supplier, and if the supplier has no suf- buyer with the supplementary supply quantities given.
ficient units to cover the shortage, the stockout partbecomes lost sale. Without any loss of generality we assume 2 One might argue that due to the simultaneousness of the decisions, the number of the stockout penalty is 0.1 units at the supplier after production could be less than the normal order, i.e., K(t)could be negative. In fact, we can easily show that the supplier will optimally makethe product quantity equal or larger than the order. The only modification we need isto add an emergency procurement option for the supplier, i.e., the supplier will 1 Note that the stockout penalty can be incorporated into our model without any activate a more expensive procurement if there is any shortage for the normal order.
difficulty. Let b(t) be the penalty for unmet demand of each period t. Then all results This emergency procurement is, however, never activated as in the equilibrium the in this paper are still valid except that we need to change ‘‘r(t)" to ‘‘r(t) + b(t)".
production is always sufficient to cover the normal order.
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X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 Based on the above description, if the buyer raises the inventory (3) V+(t, x) (hence, V(t, x)) is a concave and nonincreasing function position to y in period t = 0, 1, . . , T  1, his profit in that particular (4) s*(t) is decreasing in w1(t) while increasing in w2(t); s*(t) decreasing in K(t).
Pðt; y; KÞ ¼ w1ðtÞðy  xðtÞÞ þ rðtÞE minðy; DðtÞÞ þ ðrðtÞ  w2ðtÞÞE min½KðtÞ; ðDðtÞ  yÞþ  hðt þ 1ÞE½ðy  DðtÞÞþ: Proof. Before the proof of this lemma, we note the following two The first term is the purchasing cost of a normal order and the sec- facts. Fact 1: if f(y) is a concave function of y, then g(x) = max{f(y): ond one is the revenue gained by the normal order. The third term y P x} is a concave and nonincreasing function of x. This fact can be represents the profit gained by the supplementary order and the seen directly from the geometry of concave function. Fact 2: if g(x) last one corresponds to the inventory holding cost by the end of is convex and f(x) is concave and nonincreasing, then f(g(x)) is con- cave too. This fact can be proved directly from the definition of Define V(t, x) as the maximized expectation of the buyer's profit concave function. It is also easy to see from (6) that P+(t, y, K) is con- over periods t, t + 1, . . , T  1, providing x(t) = x in period t. The ter- minal value V(T, x) = s The proof of part (1)–(3) is based on the inductive approach on Tx is a linear function of the leftover product x by the end of the last period T  1, in which h(T) + w t. First, consider the case t = T  1. As P+(T  1, y, K) is concave in y 1(T  1) > csT.
The recursion x(t + 1) = (y(t)  D(t))+ holds for t = 0, 1, . . , T  1 due and V+(T, x) = 0, H(T  1, y, K) is a concave function of y and the to the lost-sale assumption. Hence, we develop the following dy- optimality of base-stock policy is proved for t = T  1. According to namic formulation.
Fact 1, V+(T  1, x) is a concave and nonincreasing function of x.
Therefore, (1)–(3) is true for t = T  1.
VðT; xÞ ¼ sTx; Second, provided that the lemma is true for t = n + 1, we show that it is also true for n. Now that V+(n + 1, x) is a concave function Hðt; y; KÞ ¼ w1ðtÞy þ rðtÞE minðy; DðtÞÞ þ ðrðtÞ by the induction hypothesis and according to Fact 2, E[V+(n + 1  w2ðtÞÞE min½KðtÞ; ðDðtÞ  yÞþ  hðt þ 1ÞE½ðy  DðtÞÞþ (y  D(t))+] is a concave function of y. Remember that P+(t, y, K) is þ cE½Vðt þ 1; ðy  DðtÞÞþÞ; concave for all t, thus H(n, y, K) is concave and the optimality ofbase-stock policy is proved for period n. Once again according to Fact 1, V+(n, x) is a concave and nonincreasing function of x.
1ðtÞx þ maxfHðt; y; K Þ : y P xg: Therefore, (1)–(3) is true for t = n.
Here, H(t, y, K) represents the expectation of total discounted profit Finally, on part (4), it is easy to see that @2Hðt;y;KÞ ¼ 1 < 0, thus over periods t, t + 1, . . , T  1, when the inventory position is raised H(t, y, Y) is submodular with respect to w to y in period t. Eqs. (3) and (4) hold for all t = 0, 1, 1(t) and y. Therefore s*(t), which maximizes H(t, y, K(t)), is decreasing in w that {K(t), t = 0, 1, 1(t). Similarly we . . , T  1} are given, then the buyer's problem have @2Hðt;y;KðtÞÞ ¼ F is to determine the inventory position y(0) P x(0) to maximize ðtÞðy þ K Þ  FDðtÞðyÞ P 0 and @2Hðt;y;KðtÞÞ the expected total profit V(0, x(0)) in period 0, and to determine 2ðtÞÞfDðy þ K ðtÞÞ < 0, which yield the corresponding results, the optimal inventory position y(t) P x(t) thereafter in everyperiod t = 1, Lemma 1 states that the optimal normal order policy is up to le- . . , T  1 to maximize the profit V(t, x(t)) (this prop- erty is referred to as Bellman's principle of optimality in dynamic vel s*(t) for the buyer. It also indicates that this level decreases as theory, see Puterman (1994) for the wholesale price for normal order is higher, the wholesale price for supplementary order is lower, and more supplementary supply is provided. Now let s+(t) be the smallest value of y that maximizesP+(t, y, K). We call up-to-level s+(t) as a myopic policy for the buyer's normal order as it optimizes the current profit while ignoring thefuture. We show that the optimal solution s*(t) is bounded by Pþðt; y; KÞ ¼ ½hðt þ 1Þ þ w1ðtÞ  cw1ðt þ 1Þy þ ½hðt þ 1Þ the myopic solution s+(t).
þ w2ðtÞ  cw1ðt þ 1ÞED minðy; DÞþ ½rðtÞ  w Lemma 2. For all t: 2ðtÞED minðy þ K ðtÞ; DÞ; Vþðt; xÞ ¼ Vðt; xÞ  w (1) s+(t) equals y, which satisfies Then, the following recursion is equivalent to (2)–(4): ½hðt þ 1Þ þ w2ðtÞ  cw1ðt þ 1ÞFDðtÞðyÞ þ ½rðtÞ  w2ðtÞ FDðtÞðy þ KðtÞÞ ¼ rðtÞ  w1ðtÞ; VþðT; xÞ ¼ 0; (2) s*(t) 6 s+(t).
; y; KÞ ¼ Pþðt; y; KÞ þ cE½Vþðt þ 1; ðy  DðtÞÞþ; Vþðt; xÞ ¼ maxfHðt; y; KÞ : y P xg: Proof. Maximizing P+(t, y, K) and solving the first order optimalcondition immediately yield result (1). As V+(t, x) is nonincreasing Eqs. (9) and (10) hold for t = 0, 1, . . , T  1. And after the definition of in x, cE[V+(t + 1, (y  D(t))+] is nonincreasing in y. So H0(t, y, K) 6 1(T), we have h(t + 1) + w1(t) > cw1(t + 1) for all t = 0, 1, . . , T  1, P+0(t, y, K). In addition, H0(t, s+(t), K) 6 P+0(t, s+, K) = 0 6 H0(t, s*(t), K).
according to the assumption h(t + 1) + w1(t) > cw1(t + 1) for t = Due to the concavity of H(t, y, K) in y, we instantly obtain 0, 1, . . , T  2 and the terminal value assumption.
s*(t) 6 s+(t).
3.2. The supplier's problem (1) H(t, y, K) is a concave function of y.
For given normal order base-stock level {y(t): t = 0, 1, . . , T  1}, (2) A base-stock policy is optimal for the buyer's normal order in the supplier's decision is the supplementary supply quantities period t. The optimal base-stock level s*(t) is the smallest value {K(t): t = 0, 1, . . , T  1} to maximize her own total profit over all of y maximizing H(t, y, K).
periods. Note the production quantity for the supplier is Author's personal copy
X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 s(t) = y(t) + K(t)  x(t)  xs(t) P 0, maxfXðtÞ; yðtÞg < SðtÞ; max{0, xs(t) + x(t)  y(t)}.
ðXðtÞ  yðtÞÞþ maxfXðtÞ; yðtÞg P SðtÞ: Denote Ps(t, y, K) as the supplier's expected profit in period t with the decision variable K(t). We have (2) S*(t) is increasing in w2(t), but irrelevant with w1(t); S*(t) is increasing in y(t).
Psðt; y; KÞ ¼ w1ðtÞ½yðtÞ  xðtÞ  cðtÞ½yðtÞ þ KðtÞ  xðtÞ  xsðtÞ þ w2ðtÞE min½KðtÞ; ðDðtÞ  yðtÞÞþ sðt þ 1ÞE½K ðtÞ  ðDðtÞ  yðtÞÞþþ; where the first two terms represent the revenues of the normal (1) Rearranging the terms of (17) yields supply and supplementary supply, respectively; the third term isthe production cost; and the last term is the inventory holding Pþðt; y; KÞ ¼ ½w 1ðtÞ  cðtÞ½yðtÞ  xðtÞ þ ½w2ðtÞ  vðtÞE½yðtÞ  DðtÞþ  ½w2ðtÞ  cðtÞyðtÞ Define Vs(t, xs) as the maximized expectation of the supplier's þ ½w2ðtÞ  cðtÞ½KðtÞ þ yðtÞ profit over periods t, t + 1, . . , T  1, providing xs(t) = xs in period t.
The terminal value V  ½w2ðtÞ  vðtÞE½KðtÞ þ yðtÞ  DðtÞþ: s(T, xs) = STxs is a linear function of the leftover product x by the end of the last period T  1, in which In the following we use variable substitution to solve the problem.
hs(T) + c(T  1) > cST. The recursion xs(t + 1) = [K(t)  (y(t)  D(t))+]+ Denote Y(t) = K(t) + y(t) as the new decision variable representing holds for t = 0, 1, . . , T  1. Hence, we develop the following dy- the system inventory position raised to in period t, and xe = xs + x(t) namic formulation.
as the new state variable representing the initial system inventory.
The dynamic programming (19)–(21) is equivalent to sðT ; xsÞ ¼ ST xs; UsðT; xeÞ ¼ 0; Hsðt; y; KÞ ¼ w1ðtÞ½yðtÞ  xðtÞ  cðtÞ½yðtÞ þ K  xðtÞ þ w2ðtÞE min½K; ðDðtÞ  yðtÞÞþ  hsðt þ 1ÞEðK Gsðt; y; YÞ ¼ Mðt; YÞ þ cE½Usðt þ 1; ðY  DðtÞÞþ  ðyðtÞ  ðyðtÞ  DðtÞÞþÞþ þ cE½Vsðt þ 1; ðK  ðDðtÞ  yðtÞÞþÞþ; Usðt; xeÞ ¼ maxfGsðt; y; YÞ : Y P max½yðtÞ; xeg: Vsðt; xsÞ ¼ cðtÞxs þ maxfHsðt; y; KÞ : K P max½0; xs þ xðtÞ  yðtÞg: Mðt; YÞ ¼ ½w1ðtÞ  cðtÞ½yðtÞ  xðtÞ Here, Hs(t, y, K) represents the expectation of total discounted profitover periods t, t + 1, . . , T  1, when the supplementary supply is K þ ½w2ðtÞ  vðtÞE½ðyðtÞ  DðtÞÞþ in period t. Eqs. (13) and (14) hold for all t = 0, 1, . . , T  1. The sup-  ½w2ðtÞ  cðtÞyðtÞ þ ½w2ðtÞ  cðtÞY plier's problem is to determine the supplementary supply quantity  ½w2ðtÞ  vðtÞE½ðY  DðtÞÞþ; K(t) P max[0, xs(t) + x(t)  y(t)], t = 0, 1, . . , T  1 sequentially, for gi-ven inventory states of the buyer and supplier and the buyer's deci- is a concave function of Y. Now we declare for all t = 0, 1, . . , T  1, sion {y(t), t = 0, 1, . . , T  1}.
(a) Gs(t, y, Y) is a concave function of Y.
(b) The optimal decision Y*(t) is characterized by maxfXðtÞ; yðtÞg < SðtÞ; vðtÞ ¼ ccðt þ 1Þ  h maxfXðtÞ; yðtÞg maxfXðtÞ; yðtÞg P SðtÞ; Pþðt; y; KÞ ¼ w 1ðtÞ½yðtÞ  xðtÞ  cðtÞ½yðtÞ þ K ðtÞ  xðtÞ where S*(t) is the smallest value of Y maximizing Gs(t, y, Y).
2ðtÞE min½K ðtÞ; ðDðtÞ  yðtÞÞþ s(t, xs) is a concave and nonincreasing function of xs.
Note now y(t), t = 0, 1, . . , T  1 are regarded as constants, the þ vðtÞE½KðtÞ  ðDðtÞ  yðtÞÞþþ; proof of the above fact is very similar with Lemma 1 thus omitted.
According to (28), it is now easy to see the optimal decision K*(t) sÞ ¼ V sðt; xsÞ  cðtÞxs: can be expressed as (22).
Then, the following recursion is equivalent to (12)–(14): (2) It is easy to see that @2Gsðt;y;YÞ ¼ 1  F DðtÞðY Þ P 0, so Gs(t, y, Y) is supermodular with respect to w2(t) and Y. So S*(t), which maximizes Gs(t, y, Y), is increasing in w2(t). The irrelevance Hsðt; y; KÞ ¼ Pþðt; y; KÞ þ cE½Vþðt þ 1; ½K  ðDðtÞ  yðtÞÞþþ; 1(t) can be shown as w1(t) only emerges in the terms which are regarded to be constant value in (23). We also sÞ ¼ maxfHsðt; y; K Þ : K P max½0; xs þ xðtÞ  yðtÞg: have @2Gsðt;yðtÞ;YÞ ¼ @2Usðt;yðtÞ;YÞ þ @2Mðt;YÞ P 0. Therefore, S*(t) is Eqs. (20) and (21) hold for t = 0, 1, . . , T  1. And after the definition increasing in y(t).
of c(T) and v(t), we have v(t) < c(t) for all t = 0, 1, . . , T  1, accordingto the assumption hs(t + 1) + c(t) > cc(t + 1) for t = 0, 1, . . , T  2 and Lemma 3 shows that the supplier's production follows a ‘‘base- the terminal value assumption.
system-stock" S*(t) policy. That is, if the system inventory position Denote X(t) = x(t) + xs(t), which represents the initial inventory is higher than S*(t) after the normal order at the beginning of per- of the supply chain system at the beginning of period t. We have: iod t, then the supplier will use the leftover inventory to meet thesupplementary order if necessary. Otherwise the supplier will ac- tively produce an extra amount to be the supplementary supply,raising the system inventory position to S*(t). In other words, the (1) There exist constants S*(t), t = 0, 1, . . , T  1, such that the opti- supplier guarantees the system inventory position to be not less mal decision K*(t) for the supplier satisfies than S*(t) in period t.
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X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 It is also interesting to note that the benchmark S*(t) is increas- that. In contrast, in the system without the supplementary supply– ing in w2(t) and has no relation with w1(t), implying that the sup- order, the system inventory level is raised up to the classic base- plementary wholesale price w2(t) is a crucial factor to motivate a stock level by regarding {w1(t), t = 0, 1, . . , T  1} as the purchase high service level of the supply chain. This result is due to the fact costs. The difference between the two sets of levels designates that w2(t) is the wholesale price for the supplementary order and the effect of supplementary supply–order mechanism for the sup- w2(t) > w1(t). In fact, S*(t) indicates the system inventory position ply chain system.
to be raised when the supplementary supply–order takes effect, In the equilibrium, we consider the optimal dynamic policy for and in this situation it is intuitively reasonable that the price of each player while regarding the decision of his contender as a mul- supplementary order w2(t) would play a core role from the sys- ti-dimensional vector, i.e., the player is ignorant about the evolu- tem's perspective. Furthermore, S*(t) increases as the normal order tion of the decision process of his contender. Exploring the up-to-level y(t) is raised. This result, combining with (4) in Lemma closed-loop equilibrium of this model is an interesting problem 1, shows the sensitivity of one player's decision on the other's, and for the future study.
forms the basis of the game analysis.
Now let S+(t) be the smallest value of Y that maximizes M(t, Y) in 3.4. The stationary parameter case: The stationary Nash equilibrium (27). Thus the base-system-stock S+(t) is a myopic policy for the supplier since it optimizes the current profit while ignoring thefuture.
Suppose the parameters of the problem (all the costs, prices, and demands) are stationary across the entire planning horizon.
Lemma 4. For all t, In other words, r(t) = r, c(t) = c, h(t) = h, w1(t) = w1, etc., fort = 0, 1, . . , T, and all D(t),t = 0, 1, . . , T  1 share the same distribu- (1) SþðtÞ ¼ F1 ðw2ðtÞcðtÞÞ, tion. Let D denote the generic demand. The boundary condition DðtÞ w2ðtÞvðtÞ for the last period is V(T, x) = w 1x for the buyer and Vs(T, x) = cx for the supplier. This indicates that at the end of last period T  1, 6 y(t) for all t = 0, 1, . . , T  1, the myopic policy is opti- mal for the supplier, i.e., for all t, the optimal decision K*(t) the supplier refunds to the buyer at unit price w1 for the buyer's for the supplier satisfies leftover inventory, and salvage them with her own inventory at unit price c.
SþðtÞ  yðtÞ yðtÞ < SþðtÞ; yðtÞ P SþðtÞ: (1) Suppose that the supplier's decision is stationary over all peri- Proof. Maximizing M(t, Y) and solving the first order optimal con- ods as K(t) = K. The myopic base-stock s+ policy is optimal for dition immediately yield result (1). S*(t) 6 S+(t) can be proved in the buyer's order problem, where s+ is characterized by [h + the similar way with result (2) in Lemma 2. Now we prove result w2  cw1]FD(s+) + [r  w2]FD(s+ + K) = r  w1. In other words, (3). If X(t) 6 y(t), Eq. (26) becomes U the optimal normal order raises the buyer's inventory position s(t, xe) = max{Gs(t, y, Y):Y P y(t)} thus the lower bound constraint of decision Y(t) becomes a con- stant y(t) and no longer relevant to decision Y(t  1). In this case, the dynamic programming has a separative property and can be split to the summation of T disconnected items M(t, Y),t = 0, 1, . . ,T  1, each of which can be optimized independently. Maximizing (2) Suppose that the buyer's decision is stationary over all periods each M(t, Y) leads to the optimal policy which is base-system-stock as y(t) = y. The myopic base-system-stock Sþ ¼ F1 icy is optimal for the supplier's production problem. In otherwords, the optimal supplementary supply quantity is 3.3. The supplementary supply–order system: An inventory game ( Sþ  y y < Sþ; In the above subsections, Lemma 3 characterizes the supplier's optimal decisions {K(t), t = 0, 1, . . , T  1} given the buyer's deci-sions {y(t), t = 0, 1, . . , T  1}. On the other hand, Lemma 1 indicatesthat the buyer can determine his optimal policy {y(t), t = Proof. The optimality of myopic policy has been well studied in 0, 1, . . , T  1} for specified {K(t), t = 0, 1, . . , T  1}. Assume that the literature on stochastic inventory models when the parame- the buyer and the supplier are able to commit to their ordering ters are stationary and the terminal salvage value is linear with and production decisions, respectively. Therefore we establish a the slope equals to the procurement cost. For example, see supply–order game between them, in which each one chooses Zipkin (2000). The proof of our results is very similar and thus his/her {y(t)}, {K(t)}, respectively, to maximize his/her own objec- tive, given the decision of the other one. We call this a y  K inven- Lemma 5 states that in the stationary y  K inventory game, each player's optimal strategy is myopic, and also stationary, ifthe decision of the other player is stationary. Result (2) provides Theorem 1. For any cost and price parameters r(t), w1(t), w2(t), h(t), the analytic solution for the optimal system inventory position to r(t) > w2(t) > w1(t) > be raised in this stationary parameter case, which is consistent c(t) > v(t) = cc(t + 1)  hs(t + 1), h(t + 1) + w2(t) > cw1(t + 1), with Lemma 3 in the generalized case. An interpretation for this re- 1, . . , T  1, there exists an unique Nash Equilibrium {y*(t), K*(t), t = sult is that in this case the supplementary supply–order mecha- 0, 1, . . , T  1} for the y  K inventory game.
nism provides the supplier an opportunity to determine the Theorem 1 shows the uniqueness of Nash equilibrium in the inventory position for the entire supply chain by taking on the multi-period y  K inventory game. The proof can be found in the inventory risk at unit overage cost c  v and unit underage cost supplementary material online. It can be observed that in the sup- SðtÞ ¼ Sþ ¼ plementary supply–order system, the system inventory position at is obtained as the threshold for the system period t is raised up to S*(t) by the supplier whenever it falls below inventory position.
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X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 Now define GðwÞ ¼ chþcrvrðccvÞw in [c, r].
ðhþhsÞ2þ4ð1cÞðhcþð1cÞcrþhsrÞ 2 ðc; rÞ solving G(w) = w.
GðwÞ ¼ w () ð1  cÞw2 þ ðh þ cc  vÞw  ðhc þ cr vrÞ ¼ 0. Denote H(w) = (1  c)w2 + (h + c c  v)w  (hc + cr  vr).
According to the definition of v(t) and the stationary assumption, we have v = cc  hs < cc. Also c 6 1, so H(w) is an increasing func- G(w ) tion in w. It is easy to verify that H(c) = (c  v)(c  r) < 0 andH(r) = (r  c)[h + (1  c)r] > 0, thus we immediately have thatH(w) = 0 has a unique solution in (c, r). Solving H(w) = 0 yields w ¼ ðhþhsÞþ ðhþhsÞ2þ4ð1cÞðhcþð1cÞcrþhsrÞ.
In the following we suppose x Region 1 and Region 2: Supplementary Supply-Order Is Activated 0(0) = x(0) = 0, i.e., the initial inventory at the beginning of the planning horizon is 0 for both Region 3: Supplementary Supply-Order Is Not Activated the supplier and buyer. The following result is obtained immedi- ately from Lemma 5.
1, w2} map under supplementary-order system.
supply and the problem becomes a multi-period ‘‘selling-to-the-newsvendor" problem under single wholesale price w1. Region 3in Fig. 1 represents this case.
w and w2 > w1, or w1 6  w and w2 > G(w1), then We note that Dong and Zhu (2007) state that in their single-per- iod scenario, for any given w 2 there is a threshold for w1 above/be- low which the buyer will choose to use/not to use the 2 v Þðhþw2 cw1 Þ : KðtÞ ¼ F1ðw2c supplementary order (in their words, the system runs in the PAB/ w2vÞ  y > 0; Push regime). A counterpart observation for our problem is that, is the unique Nash equilibrium for the stationary y  K inventory game.
for any given w1 there is a threshold for w2 above/below which the supplier will choose to provide/not to provide the supplemen- w and w2 6 G(w1), then tary supply. So our result provides a different dividing line in the wholesale price space {w 1, w2} between the regimes that the sup- plementary supply–order will and will not be activated, for a moregeneralized multi-period setting. Moreover, Dong and Zhu (2007) is the unique Nash equilibrium for the stationary y  K inventory game.
doesn't offer any implicit or explicit solution of the threshold theyannounce for w1. In contrast we derive the analytic expression forthis important boundary line in {w1, w2} map. Finally, in Dong and Zhu (2007), the supply chain operating regimes depend on the de- mand distribution, while in our paper Region 2 and Region 3 are w2vÞ. If w1 >  w and w2 P w1, we divided by w2 = G(w1), which do not depend on the specific de- 2 P w1 > G(w1) thus F1 mand distribution at all.
is also obtained. In this case, it is easy to verify that {y*(t), K*(t)} The differences of the results stem from the fundamental differ- expressed by (32) is a Nash equilibrium according to Lemma 5.
ence of decision interaction: they assume the buyer is a Stackel- The initial inventory condition x0(0) = x(0) = 0 ensures that berg leader in the supply–order game, so when w1 is small xðtÞ < F1 ðw2vÞðrw1Þðw2cÞðrw2Þ and XðtÞ < Sþ ¼ F1 w2c enough the buyer will always choose to take full inventory risk; for all t = 0, 1, . . , T  1, thus the above stationary base-stock/ our paper is based on the equal status setting between the supplier base-system-stock policy can be proceeded and remains optimal and the buyer, so even if w1 is very small, the supplier has the op- for the buyer/supplier as period goes by. According to Theorem 1 tion to provide the supplementary supply for a large w2. Moreover, this is the unique Nash equilibrium. Part (2) can be proved in a most existent results in this paper are independent of the demand similar manner and thus omitted.
distribution, which is also the direct result of the action order and Theorem 2 states that the Nash equilibrium is also stationary information set of the two players in our model. From a game the- for the multi-period y  K inventory game with stationary param- oretic perspective, our paper is also an illustration where a slightly eters over periods. Moreover, it provides a specific condition of different interaction setting would lead to some very different {w1, w2} under which the supplementary supply–order mechanism takes effect. There exists a threshold value  w for the wholesale Finally, we compare two special cases with the supplementary price of normal order w1. If w1 is higher than this threshold, then supply–order system. One is the multi-period selling-to-the-news- any wholesale price of the supplementary order w2, as long as it vendor system where no supplementary order is allowed. In this is higher than w1, will motivate the supplier to actively offer a sup- case, the buyer bears all the inventory risk and orders only once plementary supply. See Region 1 in Fig. 1 for this case. If w1 is lower in each period with wholesale price w1 before the demand is real- than this threshold, the supplier would be induced to provide the ized. The supplier produces exactly the order quantity and holds no supplementary supply only at the wholesale price higher than inventory. Since the parameters are stationary over the periods, the 1) (now just w2 > w1 is no longer enough to stimulate the sup- buyer's optimal policy is order up to level y ¼ F1 plier). Note that G(w1) is decreasing in w1, which indicates that the other case is the centralized system, i.e., the entire supply chain lower w1 is, the higher w2 should be set to stimulate the supplier.
is controlled by a single manager and the total profit is maximized.
See Region 2 in Fig. 1 for this case. In contrast, if both w1 and w2 are In this case, w1 and w2 become trivial and the optimal policy for the small, the supplier is not motivated to provide any supplementary system is order up to level y ¼ F1 rc .
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X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 c 6 w1 6 w2 under a single-period setting. Theorem 3 extends this inventory position result into the multi-period situation with different decision inter- action between the supply chain members. 3 the system optimum 4. Numerical studies Service improvement In this section we present several groups of numerical examples to obtain more insights into the supplementary supply–order sys- Inventory benefit tem. To investigate the benefit of the supplementary supply–order v)(r w ) (w c)(r w ) opportunity, we mainly focus on the multi-period problem with v)(h w stationary parameters. We suppose that the demands are normallydistributed over the periods and the supplementary wholesaleprice w2 is large enough to stimulate the supplier (thus case 1 inTheorem 2 occurs).
For the supplementary in the supplementary in the selling to supply-order system centralized system supply-order system newsvendor system c = 0.95, and assume i.i.d variables D(t) are generated sto-chastically by a Normal distribution N(100, 30).
Fig. 2. Comparison on the base-stock inventory position among three systems.
The above two cases serve as benchmarks of the supplementary Fig. 3 shows the relationship between the demands and order supply–order system. We denote the supply chain profit of the quantities of the buyer with and without the supplementary order multi-period selling-to-the-newsvendor system, supplementary option. It indicates that the normal order quantity in the supple- supply–order system, and centralized system as P , P, andP, mentary supply–order system is smaller than that without the respectively. Theorem 2 has shown that when w supplementary order option, as the buyer has another opportunity to replenish her inventory after the demand is realized.
2 6 G(w1), the supplementary supply–order system reduces to the multi-period selling-to-the-newsvendor one. The opposite oc- Fig. 4 shows the buyer's inventories in each period with and casion, in which the supplementary mechanism takes effect, is without the supplementary order option. It indicates that the sup- compared with two benchmark cases as follows.
plementary order option reduces the buyer's inventory.
Fig. 5 shows the supplier's inventories in each period with and Theorem 3. When w without the supplementary order option. In the ‘‘selling to the w and w2 > w1, or w1 6  w and w2 > G(w1), newsvendor" case, the supplier does not hold any inventory. How-ever, in the supplementary supply–order system the supplier (1) y > yðtÞ þ KðtÞ > y > yðtÞ, keeps inventory and thus bears part of the demand risk. This risk (2) P > P > P .
sharing between the two members in the supply chain is the basis of the supplementary supply–order mechanism.
Figs. 6 and 7 show the improvement in performance when the (3) P* increases in w2. When w2 ? r, we have yðtÞ þ KðtÞ ! y, supplementary supply–order mechanism is utilized. It is clear that and P ! P.
both the buyer and the supplier are better off as profit increases astime passes.
Proof. When w1 >  w and w2 > w1, or w1 6  w and w2 6 G(w1), we (2) Then we show the influence of price and cost parameters on > yðtÞ. Fur- w2v ; which implies yðtÞ þ KðtÞ > yn the supplementary supply–order system. Normally distrib- thermore, y > yðtÞ due to w 2 < r. Result (2) immediately follows uted demand is again adopted, and the planning horizon T Result (1). Finally, yðtÞ þ KðtÞ ¼ F1 w2c is extended to 300. Each time we change one particular to the result that P* is also increasing in w2.
parameter and fixed the others, which have the same values Theorem 3 shows that when the supplementary supply–order as in the first experiments.
mechanism takes effect, the base-stock inventory position andthe performance of the supply chain system are between the ones More specifically, we investigate the parameter's sensitivity to in the multi-period selling-to-the-newsvendor case (the worst the profit increment ratio. The profit increment ratio is defined case) and the centralized case (the best case). More specifically, as the profit increment after allowing the supplementary supply– the buyer's base-stock inventory position is lower than that in order opportunity divided by the original profit without the oppor- the multi-period selling-to-the-newsvendor case, indicating that tunity, which indicates the effect of the supplementary supply–or- the supplementary mechanism helps the buyer to mitigate the de- der mechanism.
mand variance and lower his inventory cost. However, the system Table 2 summarizes Figs. 8–14 on the relationship between the base-stock inventory position (i.e., the sum of the buyer's and sup- parameters and the profit increment ratio of the buyer, supplier plier's) in the supplementary supply–order case is higher than that and supply chain. In the table, ‘+'/‘' means the ratio is increas- in the multi-period selling-to-the-newsvendor case, indicating that ing/decreasing in the parameter, while ‘+' means first increasing the supplementary mechanism raises the service level for the sup- then decreasing. We find that from supply chain perspective, the ply chain system. See Fig. 2 for a detailed illustration.
increase of cost c, price r and the supplier's holding cost h On the other hand, both the system base-stock level and the undermine the benefit of supplementary supply–order mechanism system profit move closer to the centralized case as the supple-mentary wholesale price w2 increases towards its upper bound r, the selling price. When w Note that the single-period situation of our problem is a special case of our 2 arrives at r, the supply chain efficiency stationary parameter situation, therefore all the results in Section 3.4, especially, reaches 100%, i.e., the supplementary supply–order mechanism Theorems 2 and 3, are still valid for the single-period supplementary supply–order helps the supply chain achieve system optimum. Cachon (2004) system. The only modification we need is to change all cw1  h into v, which states that the supply chain efficiency is 100% with w2 = r and represents the salvage value of leftover product after that single period.
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X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 supplier's inventory with supplementory supply supplier's inventory without supplementary supply supplementory supply (always zero) Fig. 3. Demands and buyer's orders.
Fig. 5. Supplier's inventory positions.
buyer's inventory with supplementary order buyer's inventory without buyer's profit with supplymentary order supplementary order buyer's profit withoutsupplementary order buyer's profit improvement: 4.22% Fig. 4. Buyer's inventory positions.
Fig. 6. Buyer's total profit over multiple periods.
as the supplier's motivation of provide supplementary supply isweakened by the increase in these parameters. On the other hand,the increase of the buyer's holding cost h, wholesale prices w1, w2, increase in the buyer's revenue but weakens the effect of the sup- and the demand's uncertainty will enhance the effect of the sup- plementary supply–order mechanism. The buyer's profit incre- plementary supply–order mechanism, since in these cases either ment ratio is first increasing and then decreasing in price r, the supplier is more induced to provide a supplementary supply which implies that the revenue increase compensates the weak- (when w2 increases) or the buyer is more needful for the supple- ness at first, but finally the loss of effectiveness of the supplemen- mentary order (when h, w1, r increases). In sum, given the same tary supply–order mechanism becomes the dominant factor.
supply–order contract price, the supplementary supply–order sys- Another interesting observation is that as w2 increases, the tem is more beneficial for a product with lower cost, lower value, buyer's profit increment ratio first increases and then decreases.
lower inventory cost of supplier, higher inventory cost of buyer This is because a higher w2 increases the buyer's cost since the sup- and higher demand variation.
plementary order is more expensive than the normal order, mean- We also notice that there are several cases in which the varia- while it provides an incentive for the supplier to produce more tion tendency of individual supply chain member is not consistent units for the supplementary supply. When w2 is relatively small with the system. For example, raising the selling price r incurs an and increases, the benefit of more supplementary supply over- Author's personal copy
X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 suppler's profit with supplementory supply suppler's profit without supplementory supply supplier's profit increment: 5.54% profit increment ratio Fig. 7. Supplier's total profit over multiple periods.
Fig. 9. Influence of hs on the profit increment ratio.
Table 2Relationship between parameters and profit increment ratio.
Profit increment ratio profit increment ratio profit increment ratio Fig. 10. Influence of c on the profit increment ratio.
supplier. From Fig. 14 we can see that the supplier's ratio first in-creases but then zigzags when the variation in demand is larger.
It is interesting that in the above figures (Figs. 8–13), three trajec-tories always intersect at one point where the supply chain's curve lies between the other two curves.
Fig. 8. Influence of h on the profit increment ratio.
This paper presents a decentralized supply chain system with a weights the side-effect of the increasing cost. However, when w2 supplementary supply–order opportunity under the multi-period continues to increase and after a certain point, the increasing cost situation. During each period the buyer places an order before eventually becomes the dominant factor.
the demand is realized, and the supplier decides the production Finally, the increase of w1 weakens the supplier's motivation to quantity. The buyer might place a more costly supplementary or- provide the supplementary supply thus reduces the benefit for the der after the uncertainty of the demand is resolved, and the order Author's personal copy
X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 profit increment ratio profit increment ratio Fig. 11. Influence of r on the profit increment ratio.
Fig. 13. Influence of w2 on the profit increment ratio.
the supplier chain profit increment ratio 10% profit increment ratio standard deviation of demand: σ Fig. 12. Influence of w1 on the profit increment ratio.
Fig. 14. Influence of r on the profit increment ratio.
can be satisfied by the supplier's supplementary supply. We adopt loop Nash equilibrium is an important future research direction.
a game theoretic approach to analyze the model, propose the opti- Moreover, in our study we assume that the selling price is exoge- mal supply and order policies for the supplier and the buyer, nous. Incorporating the pricing decisions into the model would respectively, and show the existence and uniqueness of Nash equi- make the problem more complicated but more interesting.
librium. Moreover, we obtain the analytical Nash equilibrium solu-tion when the parameters are stationary over the planning horizon.
It is shown that the mechanism, if properly designed, improves the supply chain performance in the multi-period setting and benefitsboth the supplier and the buyer.
The authors wish to express their sincerest thanks to the editors Our paper is among the first effort to study the multi-period and anonymous referees for their constructive comments and sug- supplementary supply–order game and the inventory decision gestions on the earlier versions of the paper. We gratefully interaction between the supply chain members. The equilibrium acknowledge the support of (i) National Natural Science Founda- concept is based on the information structure that each player con- tion of China (NSFC), Research Fund Nos. 71002077 and siders the strategy of his contender as static. Exploring the closed- 71002106, and 2009 Humanities and Social Science Youth Founda- Author's personal copy
X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 tion of Nankai University NKQ09027, for X. Li; (ii) NSFC Research Choi, Tsan-Ming., 2007. Pre-season stocking and pricing decisions for fashion Fund Nos. 70971069 and 71001106, and the Fok Ying-Tong Educa- retailers with multiple information updating. International Journal ofProduction Economics 106 (1), 146–170.
tion Foundation of China No. 121078, for Y.J. Li; and (iii) Research Deng, S., Yano, C., 2002a. On the Role of Second Purchase Opportunity in a Two- Grants Council of Hong Kong, General Research Fund No. 410509, Echelon Supply Chain. Working Paper, University of California, Berkeley.
and NSFC Key Program Grant No. 70932005, for X.Q. Cai.
Deng, S., Yano, C., 2002b. Combining Spot Purchases with Contracts in a Two- Echelon Supply Chain. Working Paper, University of California, Berkeley.
Deshmukh, A., Lian, Z., 2009. Analysis of supply contracts with quantity flexibility.
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