European Journal of Operational Research 209 (2011) 273–284

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

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

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 December 2008Accepted 19 August 2010Available online 24 August 2010

Keywords:Supply chain managementInventoryGame theorySupplementary order

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.08.019

⇑ Corresponding author. Tel.: +86 22 23505341.E-mail address: liyongjian@nankai.edu.cn (Y. Li).

This paper considers a supplementary supply–order system in a multi-period situation. In each period,the buyer first places an initial order based on the demand prediction; he has the opportunity to placea supplementary order with the supplier after the demand of that period is realized. The supplier main-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, andderive the optimal production and order policies for the supplier and buyer, respectively. The existenceand 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-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.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Consider a supply chain consisting of a supplier and a buyer, inwhich the buyer faces a stochastic demand of product. The buyercommits an initial order, referred to as ‘‘normal order” in this pa-per, before the demand realization, and then has an opportunityto place a supplementary order with the supplier after the demandis realized. The stochastic nature of the demand usually causesinsufficient/excess inventory, resulting in substantial revenueloss/holding costs for the supply chain members. In a decentralizedsystem the performance of the supply chain members could beeven worse because of their different and usually, conflictingobjectives they attempt to optimize.

Some studies, e.g., Cachon (2004), Dong and Zhu (2007), havediscussed the supplementary supply–order problem in a single-period situation. In this paper we will consider the problem in amulti-period situation, which arises from the supply chain man-agement of many non-seasonal products, like raw materials, com-ponents, recycled cores, and other continually demandedcommodities. In each period, the buyer first submits a normal or-der to the supplier based on the forecast of demand, and thenplaces a supplementary order if the normal order cannot coverthe realized demand. The supplementary order is submitted afterthe realization of the demand and charged at a higher wholesaleprice, if the supplier has available units to satisfy the order. To in-crease profit, the supplier might rationally increase her inventory

ll rights reserved.

to a level higher than the normal order quantity, to prepare a sup-plementary supply for the buyer. However, if the supplier producesan exceeded amount of product, she has to pay the holding cost forher own leftover inventory after the buyer’s supplementary order.To summarize, the buyer decides his normal order quantity at thebeginning of each period, to balance the tradeoff between the ben-efit of lower inventory and the loss of shortage, while the supplierdecides her supplementary supply quantity, to balance the tradeoffbetween the profitability of the higher wholesale price of the sup-plementary order and the risk of holding excess inventory.

In what follows, we briefly relate our paper to the literature,including study on the emergency replenishment policy, on thequantity flexibility contract, and on the supply chain with dualpurchase opportunity.

Numerous researchers have studied the use of emergencyreplenishment to raise service level and to mitigate the risk of de-mand randomness, such as Chiang (2003, 2010), Chiang and Gut-ierrez (1998), and Sethi et al. (2001, 2003), among others. Thesepapers study the optimal policies of periodic or continuous reviewinventory system under consideration of two replenishmentmodes: fast and slow, where the fast one is more expensive thanthe slow one. Papers such as Fisher and Raman (1996, 2001), Eppenand Iyer (1997), Brown and Lee (2003), Yan et al. (2003), and Choi(2007) have developed models for fashion product in which thestochastic demand only occurs once but more than one replenish-ment opportunity can be utilized. These papers study the effect ofadvance ordering/production and demand information updatingwhere the interaction between supply chain members is not anissue.

274 X. Li et al. / European Journal of Operational Research 209 (2011) 273–284

There also has been much interest in exploring the impact oftwo or more order chances from a supply chain perspective. Amainstream research is on the quantity flexibility contracts basedon option. These contracts allow the buyer to adjust his orderquantities after the normal order, and an option price is paid tothe supplier for this flexibility. For example, Barnes-Schusteret al. (2002) explores the contract with option in a two-period sup-ply chain, in which the supplier has two production opportunitiesand the buyer is allowed to increase delivery quantity at the sec-ond period by utilizing the options. Wang et al. (2006) investigatesa similar flexible contract strategy, except that the supplier hasonly one production opportunity. Van Delft and Vial (2001)presents a stochastic programming approach to analyze the mul-ti-periodic supply chain contracts with options. Other relevant re-searches include Bassok and Anupindi (1997, 1999), Tsay (1999),Deshmukh and Lian (2009), Zhao et al. (2010), etc.

Moreover, Deng and Yano (2002a) models a two-echelon supplychain where a second purchase opportunity at an anticipated high-er price induces the supplier to bear uncertainty risk. Deng andYano (2002b) studies the impact of a late ordering opportunityon a push system, assuming that the price of the late order is ran-dom. Ozer et al. (2007) considers a supply chain with a demandforecast update under a wholesale price contract and a dual pur-chase contract, through which the manufacturer provides a dis-count for orders placed before the forecast updates. They showthat the supply chain performance is better off in the dual purchasecontract than the one in the wholesale price contract under certaincircumstances.

Cachon (2004), and Dong and Zhu (2007) are the literature thatmost closely relate to our work. Both papers explore the single-period case in which the supplier has only one production oppor-tunity and the buyer has two order opportunities. Cachon (2004)analyzes scenarios where a buyer can order from a supplier beforethe demand realization (push contract) or after realization (pullcontract), or make a combination of these two (advance-pur-chase-discount contract), which is very similar to the supplemen-tary supply–order system in our paper. The main focus of Cachon(2004) is on investigating the Pareto set with three contracts andshowing that the combining consideration of push and pull con-tracts yields higher supply chain efficiency. Dong and Zhu (2007)studies push, pull, and advanced-purchase discount contracts inan unified two-wholesale-price contract framework and showshow the channel decision changes along with the two wholesaleprices. In the above two papers, only the single-period problemhas been discussed.

Different from the above studies, our work makes the followingcontributions.

(1) We study the supply chain system in a multi-period setting,derive the optimal production and order policy for both par-ties, and explore Nash equilibrium of the multi-periodinventory game. We show that the supplementary supply–order mechanism is beneficial for a multi-period system ifthe system parameters are properly designed;

(2) When the parameters are stationary we characterize theanalytic solution of the Nash equilibrium, and provide anexplicit condition of the wholesale prices underwhich the supplementary supply–order option will beactivated;

(3) The decision interaction in our supplementary supply–ordersystem is different from that under the two-wholesale-pricecontract proposed in the literature. In our model we estab-lish a Nash game, in which the supplier decides the supplyquantity and the buyer decides the order quantity in asimultaneously interactive way, while in the two-whole-sale-price contract the buyer is virtually a Stackelberg leader

related to the quantity decision in the channel (the buyermoves first and then the supplier follows). This fundamentaldifference leads to some different results (like the wholesaleprice boundary) in our paper.

In the paper, we investigate how the supplementary supply–or-der system works in a generalized multi-period setting by thegame theoretic approach. We derive optimal normal order policyfor the buyer and optimal supplementary supply policy for thesupplier, respectively, and prove the uniqueness of Nash equilib-rium. When system parameters are stationary over time, the ana-lytic solution of Nash equilibrium is obtained, and the conditionunder which this supplementary supply–order mechanism takeseffect is given. We find that the supplementary supply–ordermechanism is a simple and applicable method to allocate multi-period demand risk between the buyer and the supplier, and toachieve substantial improvement in the supply chain performanceif properly designed.

The rest of paper is organized as follows. In Section 2 we presentthe description and assumptions of the model. In Section 3 we for-mulate and analyze the model. We establish an inventory gameand prove the uniqueness of its Nash equilibrium. The analyticequilibrium solution and some properties are explored in the caseof stationary parameters. Section 4 presents a numerical study toprovide more insights and Section 5 concludes the paper.

2. Model description

2.1. Problem description

We consider the following problem. In each time period of amulti-period horizon, a buyer orders a product from a supplierand then sells it in a market. The market demand of period t is astochastic variable D(t) with known distribution information. Thebuyer has two order opportunities in each period. He places a ‘‘nor-mal” order q(t) at the beginning of each period t, to raise his inven-tory position from x(t) to y(t). Before the end of period t, he receivesthe normal order q(t) and demand D(t) is realized. Then, if he findsthat there is a shortage to meet the demand, he can submit a ‘‘sup-plementary” order, which would be delivered to him immediatelyto satisfy the demand of that period if the supplier has availableinventory. The supplementary order is more costly than the normalorder for the buyer, and it is not obliged to be fulfilled by the sup-plier. The demand shortage after the supplementary order, if any,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 paysthe holding cost correspondingly. Therefore, when making the nor-mal order decision, the buyer should balance between the cost forexcess inventory and the cost for demand shortage.

On the other hand, the supplier has one production opportunityin each period and so needs to keep an inventory to meet the pos-sible supplementary order of the buyer. At the beginning of periodt, the supplier raises her inventory position from xs(t) to a levelequal or higher than the buyer’s normal order q(t). The suppliermight produce to a higher level since she can charge a higher pricefor the fulfilled supplementary order. However, if the supplier pro-duces excessively, she needs to bear the holding cost for her owninventory after the buyer’s supplementary order. Therefore, whenmaking the production decision, the supplier should balance be-tween the profitability of the supplementary supply and the riskof excess inventory.

Note that in our problem, the buyer is to raise, at the beginningof each period t, his inventory position to y(t) by the normal order,given the initial inventory x(t). The normal order quantity q(t) is

Table 1Notation used in the model.

Symbol Description

D(t) Stochastic demand in period tr(t) Selling price in period tw1(t) Wholesale price in a normal order in period tw2(t) Wholesale price in a supplementary order in period tc(t) Production cost in period th(t + 1) The buyer’s inventory holding cost for leftovers by the end of

period ths(t + 1) The supplier’s inventory holding cost for leftovers by the end of

period tsT Salvage value of the leftover for the buyer by the end of the last

periodST Salvage value for the leftover for the supplier by the end of the last

periody(t) The buyer’s inventory position after the normal order in period t;

decision variableK(t) The supplementary supply quantity; decision variablex(t) The buyer’s inventory position at the beginning of period txs(t) The supplier’s inventory position at the beginning of period tX(t) The system’s inventory position at the beginning of period t, equal

to xs(t) + x(t)c Discount factor, 0 < c 6 1

2 One might argue that due to the simultaneousness of the decisions, the number ofunits at the supplier after production could be less than the normal order, i.e., K(t)

X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 275

hence equal to y(t) � x(t). Meanwhile, the supplier is to raise herinventory position to q(t) + K(t) at the beginning of each period t,given the initial inventory xs(t). Thus the production quantity ofthe supplier in period t is q(t) + K(t) � xs(t). We refer to K(t) asthe possible supplementary supply quantity in period t, which rep-resents the maximum amount of units to be provided to the buyerin addition to the normal order. Therefore, K(t) + q(t) is the maxi-mum that could be used to satisfy the demand in period t.

We list our notation in Table 1 and present in details the eventsand assumptions in the next subsection.

2.2. Events and assumptions

(1) There is a sequence of time points t = 0,1, . . . ,T. Period t is thetime interval from the point t until just before the point t + 1.The inventories of the buyer and supplier are both operatedunder a periodic review over the planning horizon.

(2) Demand D(t) is a continuous random variable with the dis-tribution function FD(t) and finite mean. The demands areindependent of each other. Both Cachon (2004) and Dongand Zhu (2007) suppose that the demand has increasinggeneralized failure rate (IGFR) property, while in our paperthere is no such restrictive assumption on the demanddistribution.

(3) At the beginning of period t, the buyer raises the inventoryposition 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 theend of period t. After the demand is fulfilled, the leftoverproduct, if any, is held in the buyer’s inventory for the useof next period and incurs holding cost to the buyer. If anyshortage occurs for the demand, the buyer submits a supple-mentary order to the supplier, and if the supplier has no suf-ficient units to cover the shortage, the stockout partbecomes lost sale. Without any loss of generality we assumethe stockout penalty is 0.1

1 Note that the stockout penalty can be incorporated into our model without anydifficulty. Let b(t) be the penalty for unmet demand of each period t. Then all resultsin this paper are still valid except that we need to change ‘‘r(t)” to ‘‘r(t) + b(t)”.

(4) At the same time of period t, the supplier raises the productquantity 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 deliveredto the buyer. The remaining are to meet the buyer’s supple-mentary order of that period. After the supplementary orderis fulfilled, the leftover product, if any, is held in the supplier’sinventory for the use of next period and incurs holding cost tothe supplier.

(5) In period t, the production cost is c(t). The suppliercharges the wholesale price w1(t) for the normal orderand w2(t) for the supplementary order. The selling priceis r(t). As it is not the objective to investigate pricing deci-sions in this problem, we suppose that the buyer is in acompetitive marketplace thus r(t) is exogenous. Thespecification of wholesale prices is prior to any order orproduction quantity decision, e.g., they are determined inadvance according to the bargaining power between sup-ply chain members. Therefore, in our model allprices are given parameters and satisfy r(t) > w2(t) >w1(t) > c(t).

(6) For the buyer, unit leftover product at the end of periodt = 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;otherwise, the buyer may deliberately order more units thannecessary in period t in order to satisfy the demand of thenext period, because the procurement cost in period t hasan economic advantage over period t + 1. At the end of thelast period t = T � 1, leftover unit has a terminal value sT, inwhich h(T) + w1(T � 1) > csT. The inequation holds as other-wise the buyer may deliberately order infinite amount inthe last period T � 1 just to obtain the terminal value ratherthan to fulfill the demand.

(7) For the supplier, unit leftover product after supplementaryorder at the end of period t = 0,1, . . . ,T � 2 incurs inventoryholding cost hs(t + 1). We assume that hs(t + 1) +c(t) > cc(t + 1) for t = 0,1, . . . ,T � 2; otherwise, the suppliermay deliberately produce more units than necessary in per-iod t in order to satisfy the order of the next period, becausethe production cost in period t has an economic advantageover period t + 1. At the end of the last period t = T � 1, unitleftover product has a terminal value ST, in whichhs(T) + c(T � 1) > cST. The inequation holds as otherwise thesupplier may deliberately order infinite amount in the lastperiod T � 1 just to obtain the terminal value rather thanto fulfill the order.

(8) The information of distribution FD(t) and all other parametersare shared by the buyer and supplier. Both parties seek tomaximize their own total expected profits over all periods,respectively.

3. Model formulation, analysis, and results

3.1. The buyer’s problem

First, we formulate a multi-period inventory model for thebuyer with the supplementary supply quantities given.

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 willactivate a more expensive procurement if there is any shortage for the normal order.This emergency procurement is, however, never activated as in the equilibrium theproduction is always sufficient to cover the normal order.

276 X. Li et al. / European Journal of Operational Research 209 (2011) 273–284

Based on the above description, if the buyer raises the inventoryposition to y in period t = 0,1, . . . ,T � 1, his profit in that particularperiod is

Pðt; y;KÞ ¼ �w1ðtÞðy� xðtÞÞ þ rðtÞE minðy;DðtÞÞ þ ðrðtÞ�w2ðtÞÞE min½KðtÞ; ðDðtÞ � yÞþ� � hðt þ 1ÞE½ðy� DðtÞÞþ�:

ð1Þ

The first term is the purchasing cost of a normal order and the sec-ond one is the revenue gained by the normal order. The third termrepresents the profit gained by the supplementary order and thelast one corresponds to the inventory holding cost by the end ofperiod t.

Define V(t,x) as the maximized expectation of the buyer’s profitover periods t, t + 1, . . . ,T � 1, providing x(t) = x in period t. The ter-minal value V(T,x) = sTx is a linear function of the leftover product xby the end of the last period T � 1, in which h(T) + w1(T � 1) > csT.The recursion x(t + 1) = (y(t) � D(t))+ holds for t = 0,1, . . . ,T � 1 dueto the lost-sale assumption. Hence, we develop the following dy-namic formulation.

VðT; xÞ ¼ sT x; ð2Þ

Hðt; y;KÞ ¼ �w1ðtÞyþ rðtÞE minðy;DðtÞÞ þ ðrðtÞ�w2ðtÞÞE min½KðtÞ; ðDðtÞ � yÞþ� � hðt þ 1ÞE½ðy� DðtÞÞþ�þ cE½Vðt þ 1; ðy� DðtÞÞþÞ�; ð3Þ

Vðt; xÞ ¼ w1ðtÞxþmaxfHðt; y;KÞ : y P xg: ð4Þ

Here, H(t,y,K) represents the expectation of total discounted profitover periods t, t + 1, . . . ,T � 1, when the inventory position is raisedto y in period t. Eqs. (3) and (4) hold for all t = 0,1, . . . ,T � 1. Supposethat {K(t), t = 0,1, . . . ,T � 1} are given, then the buyer’s problemis to determine the inventory position y(0) P x(0) to maximizethe expected total profit V(0,x(0)) in period 0, and to determinethe optimal inventory position y(t) P x(t) thereafter in everyperiod t = 1, . . . ,T � 1 to maximize the profit V(t,x(t)) (this prop-erty is referred to as Bellman’s principle of optimality in dynamicprogramming theory, see Puterman (1994) for a detaileddiscussion).

Define

w1ðTÞ ¼ sT ; ð5Þ

Pþðt; y;KÞ ¼ �½hðt þ 1Þ þw1ðtÞ � cw1ðt þ 1Þ�yþ ½hðt þ 1Þþw2ðtÞ � cw1ðt þ 1Þ�ED minðy;DÞþ ½rðtÞ �w2ðtÞ�ED minðyþ KðtÞ;DÞ; ð6Þ

Vþðt; xÞ ¼ Vðt; xÞ �w1ðtÞx: ð7Þ

Then, the following recursion is equivalent to (2)–(4):

VþðT; xÞ ¼ 0; ð8ÞHðt; y;KÞ ¼ Pþðt; y;KÞ þ cE½Vþðt þ 1; ðy� DðtÞÞþ�; ð9ÞVþðt; xÞ ¼maxfHðt; y;KÞ : y P xg: ð10Þ

Eqs. (9) and (10) hold for t = 0,1, . . . ,T � 1. And after the definition ofw1(T), we have h(t + 1) + w1(t) > cw1(t + 1) for all t = 0,1, . . . ,T � 1,according to the assumption h(t + 1) + w1(t) > cw1(t + 1) for t =0,1, . . . ,T � 2 and the terminal value assumption.

Lemma 1. For all t = 0,1, . . . , T � 1:

(1) H(t,y,K) is a concave function of y.(2) A base-stock policy is optimal for the buyer’s normal order in

period t. The optimal base-stock level s*(t) is the smallest valueof y maximizing H(t,y,K).

(3) V+(t,x) (hence, V(t,x)) is a concave and nonincreasing functionof x.

(4) s*(t) is decreasing in w1(t) while increasing in w2(t); s*(t)decreasing in K(t).

Proof. Before the proof of this lemma, we note the following twofacts. Fact 1: if f(y) is a concave function of y, then g(x) = max{f(y):y P x} is a concave and nonincreasing function of x. This fact can beseen directly from the geometry of concave function. Fact 2: if g(x)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 ofconcave function. It is also easy to see from (6) that P+(t,y,K) is con-cave in y for all t = 0,1, . . . ,T � 1.

The proof of part (1)–(3) is based on the inductive approach ont. First, consider the case t = T � 1. As P+(T � 1,y,K) is concave in yand V+(T,x) = 0, H(T � 1,y,K) is a concave function of y and theoptimality of base-stock policy is proved for t = T � 1. According toFact 1, V+(T � 1,x) is a concave and nonincreasing function of x.Therefore, (1)–(3) is true for t = T � 1.

Second, provided that the lemma is true for t = n + 1, we showthat it is also true for n. Now that V+(n + 1,x) is a concave functionby the induction hypothesis and according to Fact 2, E[V+(n + 1(y � D(t))+] is a concave function of y. Remember that P+(t,y,K) isconcave for all t, thus H(n,y,K) is concave and the optimality ofbase-stock policy is proved for period n. Once again according toFact 1, V+(n,x) is a concave and nonincreasing function of x.Therefore, (1)–(3) is true for t = n.

Finally, on part (4), it is easy to see that @2Hðt;y;KÞ@y@w1ðtÞ ¼ �1 < 0, thus

H(t,y,Y) is submodular with respect to w1(t) and y. Therefore s*(t),which maximizes H(t,y,K(t)), is decreasing in w1(t). Similarly wehave @2Hðt;y;KðtÞÞ

@y@w2ðtÞ ¼ FDðtÞðyþ KÞ � FDðtÞðyÞP 0 and @2Hðt;y;KðtÞÞ@y@KðtÞ ¼ �ðrðtÞ�

w2ðtÞÞfDðyþ KðtÞÞ < 0, which yield the corresponding results,respectively. h

Lemma 1 states that the optimal normal order policy is up to le-vel s*(t) for the buyer. It also indicates that this level decreases asthe wholesale price for normal order is higher, the wholesale pricefor supplementary order is lower, and more supplementary supplyis 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’snormal order as it optimizes the current profit while ignoring thefuture. We show that the optimal solution s*(t) is bounded bythe myopic solution s+(t).

Lemma 2. For all t:

(1) s+(t) equals y, which satisfies

½hðt þ 1Þ þw2ðtÞ � cw1ðt þ 1Þ�FDðtÞðyÞ þ ½rðtÞ �w2ðtÞ�FDðtÞðyþ KðtÞÞ ¼ rðtÞ �w1ðtÞ;

(2) s*(t) 6 s+(t).

Proof. Maximizing P+(t,y,K) and solving the first order optimalcondition immediately yield result (1). As V+(t,x) is nonincreasingin x, cE[V+(t + 1, (y � D(t))+] is nonincreasing in y. So H0(t,y,K) 6P+0(t,y,K). In addition, H0(t,s+(t),K) 6 P+0(t,s+,K) = 0 6 H0(t,s*(t),K).Due to the concavity of H(t,y,K) in y, we instantly obtains*(t) 6 s+(t). h

3.2. The supplier’s problem

For given normal order base-stock level {y(t): t = 0,1, . . . ,T � 1},the supplier’s decision is the supplementary supply quantities{K(t): t = 0,1, . . . ,T � 1} to maximize her own total profit over allperiods. Note the production quantity for the supplier is

X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 277

q(t) + K(t) � xs(t) = y(t) + K(t) � x(t) � xs(t) P 0, we have K(t) Pmax{0,xs(t) + x(t) � y(t)}.

Denote Ps(t,y,K) as the supplier’s expected profit in period twith the decision variable K(t). We have

Psðt; y;KÞ ¼ w1ðtÞ½yðtÞ � xðtÞ� � cðtÞ½yðtÞ þ KðtÞ � xðtÞ � xsðtÞ�þw2ðtÞE min½KðtÞ; ðDðtÞ � yðtÞÞþ�� hsðt þ 1ÞE½KðtÞ � ðDðtÞ � yðtÞÞþ�þ; ð11Þ

where the first two terms represent the revenues of the normalsupply and supplementary supply, respectively; the third term isthe production cost; and the last term is the inventory holdingcost.

Define Vs(t,xs) as the maximized expectation of the supplier’sprofit over periods t, t + 1, . . . ,T � 1, providing xs(t) = xs in period t.The terminal value Vs(T,xs) = STxs is a linear function of the leftoverproduct x by the end of the last period T � 1, in whichhs(T) + c(T � 1) > cST. The recursion xs(t + 1) = [K(t) � (y(t) � D(t))+]+

holds for t = 0,1, . . . ,T � 1. Hence, we develop the following dy-namic formulation.

VsðT; xsÞ ¼ ST xs; ð12Þ

Hsðt; y;KÞ ¼ w1ðtÞ½yðtÞ � xðtÞ� � cðtÞ½yðtÞ þ K � xðtÞ�þw2ðtÞE min½K; ðDðtÞ � yðtÞÞþ� � hsðt þ 1ÞEðK� ðyðtÞ � DðtÞÞþÞþ þ cE½Vsðt þ 1; ðK � ðDðtÞ � yðtÞÞþÞþ�;

ð13Þ

Vsðt; xsÞ ¼ cðtÞxs þmaxfHsðt; y;KÞ : K P max½0; xs þ xðtÞ � yðtÞ�g:ð14Þ

Here, Hs(t,y,K) represents the expectation of total discounted profitover periods t, t + 1, . . . ,T � 1, when the supplementary supply is Kin period t. Eqs. (13) and (14) hold for all t = 0,1, . . . ,T � 1. The sup-plier’s problem is to determine the supplementary supply quantityK(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-sion {y(t), t = 0,1, . . . ,T � 1}.

Define

cðTÞ ¼ ST ; ð15ÞvðtÞ ¼ ccðt þ 1Þ � hsðt þ 1Þ; ð16Þ

Pþs ðt; y;KÞ ¼ w1ðtÞ½yðtÞ � xðtÞ� � cðtÞ½yðtÞ þ KðtÞ � xðtÞ�þw2ðtÞE min½KðtÞ; ðDðtÞ � yðtÞÞþ�þ vðtÞE½KðtÞ � ðDðtÞ � yðtÞÞþ�þ; ð17Þ

Vþs ðt; xsÞ ¼ Vsðt; xsÞ � cðtÞxs: ð18Þ

Then, the following recursion is equivalent to (12)–(14):

Vþs ðT; xsÞ ¼ 0; ð19ÞHsðt; y;KÞ ¼ Pþs ðt; y;KÞ þ cE½Vþs ðt þ 1; ½K � ðDðtÞ � yðtÞÞþ�þ�; ð20ÞVþs ðt; xsÞ ¼maxfHsðt; y;KÞ : K P max½0; xs þ xðtÞ � yðtÞ�g: ð21Þ

Eqs. (20) and (21) hold for t = 0,1, . . . ,T � 1. And after the definitionof 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 andthe terminal value assumption.

Denote X(t) = x(t) + xs(t), which represents the initial inventoryof the supply chain system at the beginning of period t. We have:

Lemma 3.

(1) There exist constants S*(t), t = 0,1, . . . , T � 1, such that the opti-mal decision K*(t) for the supplier satisfies

K�ðtÞ ¼S�ðtÞ � yðtÞ maxfXðtÞ; yðtÞg < S�ðtÞ;ðXðtÞ � yðtÞÞþ maxfXðtÞ; yðtÞgP S�ðtÞ:

�ð22Þ

(2) S*(t) is increasing in w2(t), but irrelevant with w1(t); S*(t) isincreasing in y(t).

Proof.

(1) Rearranging the terms of (17) yields

Pþs ðt; y;KÞ ¼ ½w1ðtÞ � cðtÞ�½yðtÞ � xðtÞ� þ ½w2ðtÞ� vðtÞ�E½yðtÞ � DðtÞ�þ � ½w2ðtÞ � cðtÞ�yðtÞþ ½w2ðtÞ � cðtÞ�½KðtÞ þ yðtÞ�� ½w2ðtÞ � vðtÞ�E½KðtÞ þ yðtÞ � DðtÞ�þ: ð23Þ

In the following we use variable substitution to solve the problem.Denote Y(t) = K(t) + y(t) as the new decision variable representingthe system inventory position raised to in period t, and xe = xs + x(t)as the new state variable representing the initial system inventory.The dynamic programming (19)–(21) is equivalent to

UsðT; xeÞ ¼ 0; ð24Þ

Gsðt; y;YÞ ¼ Mðt;YÞ þ cE½Usðt þ 1; ðY � DðtÞÞþ � ðyðtÞ� DðtÞÞþÞ�; ð25Þ

Usðt; xeÞ ¼maxfGsðt; y;YÞ : Y P max½yðtÞ; xe�g: ð26Þ

where

Mðt;YÞ ¼ ½w1ðtÞ � cðtÞ�½yðtÞ � xðtÞ�þ ½w2ðtÞ � vðtÞ�E½ðyðtÞ � DðtÞÞþ�� ½w2ðtÞ � cðtÞ�yðtÞ þ ½w2ðtÞ � cðtÞ�Y� ½w2ðtÞ � vðtÞ�E½ðY � DðtÞÞþ�; ð27Þ

is a concave function of Y. Now we declare for all 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

Y�ðtÞ ¼S�ðtÞ maxfXðtÞ; yðtÞg < S�ðtÞ;maxfXðtÞ; yðtÞg maxfXðtÞ; yðtÞgP S�ðtÞ;

�ð28Þ

where S*(t) is the smallest value of Y maximizing Gs(t,y,Y).

(c) Us(t,xs) is a concave and nonincreasing function of xs.

Note now y(t), t = 0,1, . . . ,T � 1 are regarded as constants, theproof 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)can be expressed as (22).

(2) It is easy to see that @2Gsðt;y;YÞ@w2ðtÞ@Y ¼ 1� FDðtÞðYÞP 0, so Gs(t,y,Y) is

supermodular with respect to w2(t) and Y. So S*(t), whichmaximizes Gs(t,y,Y), is increasing in w2(t). The irrelevancewith w1(t) can be shown as w1(t) only emerges in the termswhich are regarded to be constant value in (23). We alsohave @2Gsðt;yðtÞ;YÞ

@yðtÞ@Y ¼ @2Usðt;yðtÞ;YÞ@yðtÞ@Y þ @2Mðt;YÞ

@yðtÞ@Y P 0. Therefore, S*(t) isincreasing in y(t). h

Lemma 3 shows that the supplier’s production follows a ‘‘base-system-stock” S*(t) policy. That is, if the system inventory positionis higher than S*(t) after the normal order at the beginning of per-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, thesupplier guarantees the system inventory position to be not lessthan S*(t) in period t.

278 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-ing in w2(t) and has no relation with w1(t), implying that the sup-plementary wholesale price w2(t) is a crucial factor to motivate ahigh service level of the supply chain. This result is due to the factthat w2(t) is the wholesale price for the supplementary order andw2(t) > w1(t). In fact, S*(t) indicates the system inventory positionto be raised when the supplementary supply–order takes effect,and in this situation it is intuitively reasonable that the price ofsupplementary order w2(t) would play a core role from the sys-tem’s perspective. Furthermore, S*(t) increases as the normal orderup-to-level y(t) is raised. This result, combining with (4) in Lemma1, shows the sensitivity of one player’s decision on the other’s, andforms the basis of the game analysis.

Now let S+(t) be the smallest value of Y that maximizes M(t,Y) in(27). Thus the base-system-stock S+(t) is a myopic policy for thesupplier since it optimizes the current profit while ignoring thefuture.

Lemma 4. For all t,

(1) SþðtÞ ¼ F�1DðtÞð

w2ðtÞ�cðtÞw2ðtÞ�vðtÞÞ,

(2) S*(t) 6 S+(t),(3) if X(t) 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)for the supplier satisfies

K�ðtÞ ¼ SþðtÞ � yðtÞ yðtÞ < SþðtÞ;0 yðtÞP SþðtÞ:

(ð29Þ

Proof. Maximizing M(t,Y) and solving the first order optimal con-dition immediately yield result (1). S*(t) 6 S+(t) can be proved inthe similar way with result (2) in Lemma 2. Now we prove result(3). If X(t) 6 y(t), Eq. (26) becomes Us(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 besplit to the summation of T disconnected items M(t,Y),t = 0,1, . . . ,T � 1, each of which can be optimized independently. Maximizingeach M(t,Y) leads to the optimal policy which is base-system-stockS+(t). h

3.3. The supplementary supply–order system: An inventory game

In the above subsections, Lemma 3 characterizes the supplier’soptimal 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 =0,1, . . . ,T � 1} for specified {K(t), t = 0,1, . . . ,T � 1}. Assume thatthe buyer and the supplier are able to commit to their orderingand production decisions, respectively. Therefore we establish asupply–order game between them, in which each one chooseshis/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-tory game.

Theorem 1. For any cost and price parameters r(t), w1(t), w2(t), h(t),c(t), v(t), c, satisfying the relationship r(t) > w2(t) > w1(t) >c(t) > v(t) = cc(t + 1) � hs(t + 1), h(t + 1) + w2(t) > cw1(t + 1), t = 0,1, . . . , T � 1, there exists an unique Nash Equilibrium {y*(t),K*(t), t =0,1, . . . , T � 1} for the y � K inventory game.

Theorem 1 shows the uniqueness of Nash equilibrium in themulti-period y � K inventory game. The proof can be found in thesupplementary material online. It can be observed that in the sup-plementary supply–order system, the system inventory position atperiod t is raised up to S*(t) by the supplier whenever it falls below

that. In contrast, in the system without the supplementary supply–order, the system inventory level is raised up to the classic base-stock level by regarding {w1(t), t = 0,1, . . . ,T � 1} as the purchasecosts. The difference between the two sets of levels designatesthe effect of supplementary supply–order mechanism for the sup-ply chain system.

In the equilibrium, we consider the optimal dynamic policy foreach player while regarding the decision of his contender as a mul-ti-dimensional vector, i.e., the player is ignorant about the evolu-tion of the decision process of his contender. Exploring theclosed-loop equilibrium of this model is an interesting problemfor the future study.

3.4. The stationary parameter case: The stationary Nash equilibriumsolution

Suppose the parameters of the problem (all the costs, prices,and demands) are stationary across the entire planning horizon.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-tion. Let D denote the generic demand. The boundary conditionfor the last period is V(T,x) = w1x for the buyer and Vs(T,x) = cx forthe supplier. This indicates that at the end of last period T � 1,the supplier refunds to the buyer at unit price w1 for the buyer’sleftover inventory, and salvage them with her own inventory atunit price c.

Lemma 5.

(1) Suppose that the supplier’s decision is stationary over all peri-ods as K(t) = K. The myopic base-stock s+ policy is optimal forthe buyer’s order problem, where s+ is characterized by [h +w2 � cw1]FD(s+) + [r � w2]FD(s+ + K) = r � w1. In other words,the optimal normal order raises the buyer’s inventory positionto

y�ðtÞ ¼sþ xðtÞ < sþ;

x xðtÞP sþ:

�ð30Þ

(2) Suppose that the buyer’s decision is stationary over all periodsas y(t) = y. The myopic base-system-stock Sþ ¼ F�1

Dw2�cw2�v

� �pol-

icy is optimal for the supplier’s production problem. In otherwords, the optimal supplementary supply quantity is

K�ðtÞ ¼ Sþ � y y < Sþ;

0 y P Sþ:

(ð31Þ

Proof. The optimality of myopic policy has been well studied inthe literature on stochastic inventory models when the parame-ters are stationary and the terminal salvage value is linear withthe slope equals to the procurement cost. For example, seeZipkin (2000). The proof of our results is very similar and thusomitted. h

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) providesthe analytic solution for the optimal system inventory position tobe raised in this stationary parameter case, which is consistentwith Lemma 3 in the generalized case. An interpretation for this re-sult is that in this case the supplementary supply–order mecha-nism provides the supplier an opportunity to determine theinventory position for the entire supply chain by taking on theinventory risk at unit overage cost c � v and unit underage costw2 � c. So the famous newsvendor solution S�ðtÞ ¼ Sþ ¼F�1

Dunderagecost

overagecostþunderagecost

� �is obtained as the threshold for the system

inventory position.

Region 2

2 1w w

2w

r

Region 1

X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 279

Now define GðwÞ ¼ chþcr�vr�ðcc�vÞwhþð1�cÞw in [c,r].

Lemma 6. There is an unique solution �w ¼�ðhþhsÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðhþhsÞ2þ4ð1�cÞðhcþð1�cÞcrþhsrÞp

2ð1�cÞ 2 ðc; rÞ solving G(w) = w.

w

Region 3w

1( )G w

1wr

Region 1 and Region 2: Supplementary Supply-Order Is Activated

Region 3: Supplementary Supply-Order Is Not Activated

c

Fig. 1. The {w1,w2} map under supplementary-order system.

Proof. Note GðwÞ ¼ w() ð1� cÞw2 þ ðhþ 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-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 �ðhþhsÞþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðhþhsÞ2þ4ð1�cÞðhcþð1�cÞcrþhsrÞp

2ð1�cÞ . h

In the following we suppose x0(0) = x(0) = 0, i.e., the initialinventory at the beginning of the planning horizon is 0 for boththe supplier and buyer. The following result is obtained immedi-ately from Lemma 5.

Theorem 2.

(1) If w1 > �w and w2 > w1, or w1 6 �w and w2 > G(w1), then

y�ðtÞ ¼ F�1D ð

ðw2�vÞðr�w1Þ�ðw2�cÞðr�w2Þðw2�vÞðhþw2�cw1Þ

Þ;

K�ðtÞ ¼ F�1D ð

w2�cw2�vÞ � y� > 0;

8<: ð32Þ

is the unique Nash equilibrium for the stationary y � K inventory game.(2) If w1 6 �w and w2 6 G(w1), then

y�ðtÞ ¼ F�1D

r�w1rþh�cw1

� �;

K�ðtÞ ¼ 0;

(ð33Þ

is the unique Nash equilibrium for the stationary y � K inventory game.

Proof. If w1 6 �w and w2 > G(w1), we haveF�1

D ððw2�vÞðr�w1Þ�ðw2�cÞðr�w2Þ

ðw2�vÞðhþw2�cw1ÞÞ < F�1

D ðw2�cw2�vÞ. If w1 > �w and w2 P w1, we

have w2 P w1 > G(w1) thus F�1D

ðw2�vÞðr�w1Þ�ðw2�cÞðr�w2Þðw2�vÞðhþw2�cw1Þ

� �< F�1

Dw2�cw2�v

� �is also obtained. In this case, it is easy to verify that {y*(t),K*(t)}expressed by (32) is a Nash equilibrium according to Lemma 5.The initial inventory condition x0(0) = x(0) = 0 ensures thatxðtÞ < F�1

Dðw2�vÞðr�w1Þ�ðw2�cÞðr�w2Þ

ðw2�vÞðhþw2�cw1Þ

� �and XðtÞ < Sþ ¼ F�1

Dw2�cw2�v

� �hold

for all t = 0,1, . . . ,T � 1, thus the above stationary base-stock/base-system-stock policy can be proceeded and remains optimalfor the buyer/supplier as period goes by. According to Theorem 1this is the unique Nash equilibrium. Part (2) can be proved in asimilar manner and thus omitted. h

Theorem 2 states that the Nash equilibrium is also stationaryfor the multi-period y � K inventory game with stationary param-eters over periods. Moreover, it provides a specific condition of{w1,w2} under which the supplementary supply–order mechanismtakes effect. There exists a threshold value �w for the wholesaleprice of normal order w1. If w1 is higher than this threshold, thenany wholesale price of the supplementary order w2, as long as itis higher than w1, will motivate the supplier to actively offer a sup-plementary supply. See Region 1 in Fig. 1 for this case. If w1 is lowerthan this threshold, the supplier would be induced to provide thesupplementary supply only at the wholesale price higher thanG(w1) (now just w2 > w1 is no longer enough to stimulate the sup-plier). Note that G(w1) is decreasing in w1, which indicates that thelower w1 is, the higher w2 should be set to stimulate the supplier.See Region 2 in Fig. 1 for this case. In contrast, if both w1 and w2 aresmall, the supplier is not motivated to provide any supplementary

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.

We note that Dong and Zhu (2007) state that in their single-per-iod scenario, for any given w2 there is a threshold for w1 above/be-low which the buyer will choose to use/not to use thesupplementary order (in their words, the system runs in the PAB/Push regime). A counterpart observation for our problem is that,for any given w1 there is a threshold for w2 above/below whichthe supplier will choose to provide/not to provide the supplemen-tary supply. So our result provides a different dividing line in thewholesale price space {w1,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)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 andZhu (2007), the supply chain operating regimes depend on the de-mand distribution, while in our paper Region 2 and Region 3 aredivided by w2 = G(w1), which do not depend on the specific de-mand distribution at all.

The differences of the results stem from the fundamental differ-ence of decision interaction: they assume the buyer is a Stackel-berg leader in the supply–order game, so when w1 is smallenough the buyer will always choose to take full inventory risk;our paper is based on the equal status setting between the supplierand the buyer, so even if w1 is very small, the supplier has the op-tion to provide the supplementary supply for a large w2. Moreover,most existent results in this paper are independent of the demanddistribution, which is also the direct result of the action order andinformation set of the two players in our model. From a game the-oretic perspective, our paper is also an illustration where a slightlydifferent interaction setting would lead to some very differentequilibrium results.

Finally, we compare two special cases with the supplementarysupply–order system. One is the multi-period selling-to-the-news-vendor system where no supplementary order is allowed. In thiscase, the buyer bears all the inventory risk and orders only oncein each period with wholesale price w1 before the demand is real-ized. The supplier produces exactly the order quantity and holds noinventory. Since the parameters are stationary over the periods, thebuyer’s optimal policy is order up to level y�n ¼ F�1

Dr�w1

r�cw1þh

� �. The

other case is the centralized system, i.e., the entire supply chainis controlled by a single manager and the total profit is maximized.In this case, w1 and w2 become trivial and the optimal policy for thesystem is order up to level y�c ¼ F�1

Dr�cr�v

� �.

1 2 1 2 2

2 2 1

( )( ) ( )( )( )

( )( )D

w v r w w c r wF

w v h w w

For the buyerin the supplementary supply-order system

For the buyer in the selling to

newsvendor system

For the supplementary supply-order system

For the centralized system

Inventory benefit for the buyer

Service improvement for the system

1 1

1

( )( )( )D

r wF

h r w

1 2

2

( )( )( )D

w cF

w v

1 ( )( )( )D

r cF

r vDistance from

the system optimum

Base-stockinventory position

Fig. 2. Comparison on the base-stock inventory position among three systems.

280 X. Li et al. / European Journal of Operational Research 209 (2011) 273–284

The above two cases serve as benchmarks of the supplementarysupply–order system. We denote the supply chain profit of themulti-period selling-to-the-newsvendor system, supplementarysupply–order system, and centralized system as P�n, P�s , andP�c ,respectively. Theorem 2 has shown that when w1 6 �w andw2 6 G(w1), the supplementary supply–order system reduces tothe multi-period selling-to-the-newsvendor one. The opposite oc-casion, in which the supplementary mechanism takes effect, iscompared with two benchmark cases as follows.

Theorem 3. When w1 > �w and w2 > w1, or w1 6 �w and w2 > G(w1),we have:

(1) y�c > y�ðtÞ þ K�ðtÞ > y�n > y�ðtÞ,(2) P�c > P�s > P�n.

(3) P* increases in w2. When w2 ? r, we have y�ðtÞ þ K�ðtÞ ! y�c ,and P�s ! P�c .

3 Note that the single-period situation of our problem is a special case of ourstationary parameter situation, therefore all the results in Section 3.4, especially,Theorems 2 and 3, are still valid for the single-period supplementary supply–ordersystem. The only modification we need is to change all cw1 � h into v, whichrepresents the salvage value of leftover product after that single period.

Proof. When w1 > �w and w2 > w1, or w1 6 �w and w2 6 G(w1), wehave r�w1

rþh�cw1< w2�c

w2�v ; which implies y�ðtÞ þ K�ðtÞ > y�n > y�ðtÞ. Fur-thermore, y�c > y�ðtÞ due to w2 < r. Result (2) immediately followsResult (1). Finally, y�ðtÞ þ K�ðtÞ ¼ F�1

Dw2�cw2�v

� �increases in w2, leads

to the result that P* is also increasing in w2. h

Theorem 3 shows that when the supplementary supply–ordermechanism takes effect, the base-stock inventory position andthe performance of the supply chain system are between the onesin the multi-period selling-to-the-newsvendor case (the worstcase) and the centralized case (the best case). More specifically,the buyer’s base-stock inventory position is lower than that inthe multi-period selling-to-the-newsvendor case, indicating thatthe supplementary mechanism helps the buyer to mitigate the de-mand variance and lower his inventory cost. However, the systembase-stock inventory position (i.e., the sum of the buyer’s and sup-plier’s) in the supplementary supply–order case is higher than thatin the multi-period selling-to-the-newsvendor case, indicating thatthe supplementary mechanism raises the service level for the sup-ply chain system. See Fig. 2 for a detailed illustration.

On the other hand, both the system base-stock level and thesystem profit move closer to the centralized case as the supple-mentary wholesale price w2 increases towards its upper bound r,the selling price. When w2 arrives at r, the supply chain efficiencyreaches 100%, i.e., the supplementary supply–order mechanismhelps the supply chain achieve system optimum. Cachon (2004)states that the supply chain efficiency is 100% with w2 = r and

c 6 w1 6 w2 under a single-period setting. Theorem 3 extends thisresult into the multi-period situation with different decision inter-action between the supply chain members. 3

4. Numerical studies

In this section we present several groups of numerical examplesto obtain more insights into the supplementary supply–order sys-tem. To investigate the benefit of the supplementary supply–orderopportunity, we mainly focus on the multi-period problem withstationary 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).

(1) We set T = 50; r = 13; w2 = 8; w1 = 5; c = 3; h = 3; hs = 1; andc = 0.95, and assume i.i.d variables D(t) are generated sto-chastically by a Normal distribution N(100,30).

Fig. 3 shows the relationship between the demands and orderquantities of the buyer with and without the supplementary orderoption. It indicates that the normal order quantity in the supple-mentary supply–order system is smaller than that without thesupplementary order option, as the buyer has another opportunityto replenish her inventory after the demand is realized.

Fig. 4 shows the buyer’s inventories in each period with andwithout the supplementary order option. It indicates that the sup-plementary order option reduces the buyer’s inventory.

Fig. 5 shows the supplier’s inventories in each period with andwithout the supplementary order option. In the ‘‘selling to thenewsvendor” case, the supplier does not hold any inventory. How-ever, in the supplementary supply–order system the supplierkeeps inventory and thus bears part of the demand risk. This risksharing between the two members in the supply chain is the basisof the supplementary supply–order mechanism.

Figs. 6 and 7 show the improvement in performance when thesupplementary supply–order mechanism is utilized. It is clear thatboth the buyer and the supplier are better off as profit increases astime passes.

(2) Then we show the influence of price and cost parameters onthe supplementary supply–order system. Normally distrib-uted demand is again adopted, and the planning horizon Tis extended to 300. Each time we change one particularparameter and fixed the others, which have the same valuesas in the first experiments.

More specifically, we investigate the parameter’s sensitivity tothe profit increment ratio. The profit increment ratio is definedas the profit increment after allowing the supplementary supply–order opportunity divided by the original profit without the oppor-tunity, which indicates the effect of the supplementary supply–or-der mechanism.

Table 2 summarizes Figs. 8–14 on the relationship between theparameters and the profit increment ratio of the buyer, supplierand supply chain. In the table, ‘+’/‘�’ means the ratio is increas-ing/decreasing in the parameter, while ‘+�’ means first increasingthen decreasing. We find that from supply chain perspective, theincrease of cost c, price r and the supplier’s holding cost hs willundermine the benefit of supplementary supply–order mechanism

0 10 20 30 40 500

20

40

60

80

100

120

140

160

180

200

Period

Uni

ts (P

rodu

ct)

demandnormal order

order withoutsupplementary supply

Fig. 3. Demands and buyer’s orders.

0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

110

Period

Uni

ts (P

rodu

ct)

buyer’s inventory withsupplementary orderbuyer’s inventory withoutsupplymentary order

Fig. 4. Buyer’s inventory positions.

0 10 20 30 40 500

5

10

15

20

25

30

Period

Uni

ts (P

rodu

ct)

supplier’s inventory withsupplementory supplysupplier’s inventory withoutsupplementory supply (always zero)

Fig. 5. Supplier’s inventory positions.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4

Period

Prof

it

buyer’s profit withsupplementary order

buyer’s profit withoutsupplementary order

buyer’s profit improvement: 4.22%

Fig. 6. Buyer’s total profit over multiple periods.

X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 281

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,and the demand’s uncertainty will enhance the effect of the sup-plementary supply–order mechanism, since in these cases eitherthe supplier is more induced to provide a supplementary supply(when w2 increases) or the buyer is more needful for the supple-mentary order (when h,w1,r increases). In sum, given the samesupply–order contract price, the supplementary supply–order sys-tem is more beneficial for a product with lower cost, lower value,lower inventory cost of supplier, higher inventory cost of buyerand higher demand variation.

We also notice that there are several cases in which the varia-tion tendency of individual supply chain member is not consistentwith the system. For example, raising the selling price r incurs an

increase in the buyer’s revenue but weakens the effect of the sup-plementary supply–order mechanism. The buyer’s profit incre-ment ratio is first increasing and then decreasing in price r,which implies that the revenue increase compensates the weak-ness at first, but finally the loss of effectiveness of the supplemen-tary supply–order mechanism becomes the dominant factor.

Another interesting observation is that as w2 increases, thebuyer’s profit increment ratio first increases and then decreases.This is because a higher w2 increases the buyer’s cost since the sup-plementary order is more expensive than the normal order, mean-while it provides an incentive for the supplier to produce moreunits for the supplementary supply. When w2 is relatively smalland increases, the benefit of more supplementary supply over-

0 10 20 30 40 500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Period

Prof

it

suppler’s profit withsupplementory supply

suppler’s profit withoutsupplementory supply

supplier’s profit increment: 5.54%

Fig. 7. Supplier’s total profit over multiple periods.

Table 2Relationship between parameters and profit increment ratio.

Profit increment ratio Parameters

c r h hs w1 w2 r

Buyer � ± + � + ± +Supplier � � + � � + N/ASupply chain � � + � + + +

1 2 3 4 5 6 7 80

5%

10%

15%

20%

25%

h

prof

it in

crem

ent r

atio

buyer

supplier

supply chain

Fig. 8. Influence of h on the profit increment ratio.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−5%

0

5%

10%

15%

20%

25%

30%

Hs

prof

it in

crem

ent r

atio

buyersuppliersupply chain

Fig. 9. Influence of hs on the profit increment ratio.

0.5 1 1.5 2 2.5 3 3.5 4 4.50.5%

1%

1.5%

2%

2.5%

3%

3.5%

4%

4.5%

c

prof

it in

crem

ent r

atio

buyer

supplier

supply chain

Fig. 10. Influence of c on the profit increment ratio.

282 X. Li et al. / European Journal of Operational Research 209 (2011) 273–284

weights the side-effect of the increasing cost. However, when w2

continues to increase and after a certain point, the increasing costeventually becomes the dominant factor.

Finally, the increase of w1 weakens the supplier’s motivation toprovide the supplementary supply thus reduces the benefit for the

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 curvelies between the other two curves.

5. Conclusions

This paper presents a decentralized supply chain system with asupplementary supply–order opportunity under the multi-periodsituation. During each period the buyer places an order beforethe demand is realized, and the supplier decides the productionquantity. The buyer might place a more costly supplementary or-der after the uncertainty of the demand is resolved, and the order

10 11 12 13 14 15 16 17 18 190 %

2%

4%

6%

8%

10%

12%

r

prof

it in

crem

ent r

atio

buyersuppliersupply chain

Fig. 11. Influence of r on the profit increment ratio.

3.5 4 4.5 5 5.5 6 6.5 7 7.5−5%

0

5%

10%

15%

20%

25%

30%

w1

prof

it in

crem

ent r

atio

buyer

supplier

the supplier chain

Fig. 12. Influence of w1 on the profit increment ratio.

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10−2%

0

2%

4%

6%

8%

10%

w2

prof

it in

crem

ent r

atio

retailer

supplier

supply chain

Fig. 13. Influence of w2 on the profit increment ratio.

5 10 15 20 25 30 35 40 45 500

2%

4%

6%

8%

10%

12%

standard deviation of demand: σ

prof

it in

crem

ent r

atio

buyer

supplier

supply chain

Fig. 14. Influence of r on the profit increment ratio.

X. Li et al. / European Journal of Operational Research 209 (2011) 273–284 283

can be satisfied by the supplier’s supplementary supply. We adopta game theoretic approach to analyze the model, propose the opti-mal supply and order policies for the supplier and the buyer,respectively, and show the existence and uniqueness of Nash equi-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 thesupply chain performance in the multi-period setting and benefitsboth the supplier and the buyer.

Our paper is among the first effort to study the multi-periodsupplementary supply–order game and the inventory decisioninteraction between the supply chain members. The equilibriumconcept is based on the information structure that each player con-siders the strategy of his contender as static. Exploring the closed-

loop Nash equilibrium is an important future research direction.Moreover, in our study we assume that the selling price is exoge-nous. Incorporating the pricing decisions into the model wouldmake the problem more complicated but more interesting.

Acknowledgments

The authors wish to express their sincerest thanks to the editorsand anonymous referees for their constructive comments and sug-gestions on the earlier versions of the paper. We gratefullyacknowledge the support of (i) National Natural Science Founda-tion of China (NSFC), Research Fund Nos. 71002077 and71002106, and 2009 Humanities and Social Science Youth Founda-

284 X. Li et al. / European Journal of Operational Research 209 (2011) 273–284

tion of Nankai University NKQ09027, for X. Li; (ii) NSFC ResearchFund Nos. 70971069 and 71001106, and the Fok Ying-Tong Educa-tion Foundation of China No. 121078, for Y.J. Li; and (iii) ResearchGrants Council of Hong Kong, General Research Fund No. 410509,and NSFC Key Program Grant No. 70932005, for X.Q. Cai.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ejor.2010.08.019.

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