Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption

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Competition and cooperation in a single-retailer two-supplier supply chainwith supply disruption

Jian Li a,�, Shouyang Wang b, T.C.E. Cheng c

a School of Economics and Management, Beijing University of Chemical Technology, Beijing 100029, Chinab Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, Chinac Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China

a r t i c l e i n f o

Article history:

Received 4 September 2008

Accepted 23 September 2009Available online 28 October 2009

Keywords:

Supply disruption

Non-cooperation game

Cooperation game

Supply chain coordination

a b s t r a c t

Many retailers diversify their supply disruption risk by sourcing from multiple suppliers. While a

retailer’s sourcing strategy impacts the profit of the supply chain, the pricing strategies of suppliers

influence all aspects of the supply chain. In this paper we investigate the sourcing strategy of a retailer

and the pricing strategies of two suppliers in a supply chain under an environment of supply disruption.

We characterize the sourcing strategies of the retailer in a centralized and a decentralized system. We

derive a sufficient condition for the existence of an equilibrium price in the decentralized system when

the suppliers are competitive. Based on the assumption of a uniform demand distribution, we obtain an

explicit form of the solutions when the suppliers are competitive. Finally we devise a coordination

mechanism to maximize the profits of both suppliers.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Over the past few years many types of unpredictable disastersincluding non-terrorist intentional acts, terrorist acts, accidents,natural calamities etc., have occurred, which indicate that ourworld is increasingly more uncertain and vulnerable. Moreover,supply chains seem to be more fragile today than in the past dueto the popularity of such industry trends and business practices asglobal sourcing, decentralized production, increased reliance onoutsourcing, reduced number of suppliers, and focusing onreducing inventory etc. Although these trends and practices havereduced the normal costs in supply chains, they have also madesupply chains more susceptible to disruptions and have createdlonger and more complex supply chains in which the dominoeffects of disruptions have been exacerbated (Christopher and Lee,2004).

Supply chain disruptions are unplanned and unanticipatedevents that disrupt the normal flow of goods and materials withina supply chain (Hendricks and Singhal, 2003; Kleindorfer andSaad, 2005) and, as a consequence, expose firms within the supplychain to operational and financial risks (Stauffer, 2003). Forexample, the one-month-long brutal winter weather caused byheavy snowfalls that occurred in large tracts of China in January2008 caused transport chaos and disrupted supplies of energy andfood. The delivery dates of the goods on most delivery trucks were

way overdue. It can be catastrophic for a short-life product if thedisruption coincides with the selling season.

Generally speaking, most supply chain disruptions can bebroadly classified into three categories, namely supply-related,demand-related, and miscellaneous risks (Oke and Gopalakrish-nana, 2009). Supply disruption occurs when suppliers are unableto fill the orders placed with them. These risks could potentiallyaffect or disrupt the supply of products or services that the supplychain offers its customers. Demand disruption may be due to asudden drop or a sudden rise in customer orders. Demand-relatedrisks could potentially affect or disrupt the operations of theretailer and affect its ability to make products available to itscustomers. Miscellaneous risks are risks that could potentiallyaffect the costs of doing business, such as unexpected changes topurchasing costs, interest rates, currency exchange rates, safetyregulations by government agencies etc. In this paper we mainlyfocus on supply disruption.

A supplier may be unable to fill an order for a variety ofreasons, including equipment failures, damaged facilities, pro-blems in procuring the necessary raw materials, or rationing itssupply among its customers. Hendricks and Singhal (2003)estimated the short-term effects of supply disruption such asproduction or shipment delays on shareholder value. They foundthat supply chain disruption announcements are associated withan abnormal decrease in shareholder value by 10.28%. In a followup study, Hendricks and Singhal (2005) investigated the long-term negative effects of supply disruption on the financialperformance of firms. They found that over a three-year timeperiod, the mean abnormal return of the sample firms in their

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Int. J. Production Economics 124 (2010) 137–150

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study was nearly �40%. Furthermore, they found that firmscannot quickly recover from the negative effects of a disruption.The significant negative economic consequences of disruptionsand the lack of evidence of a quick recovery underscore the needto pay close attention to the risk of supply disruption.

Supply disruption management has received increasing atten-tion from both industry and academia. Firms are starting torealize that supply disruption severely affects their ability tosuccessfully manage their supply chains. The academia hasdevoted much research effort to studying this issue. Many papershave been published that advise firms on how to manage theirsupply chains in the presence of supply disruption. For example,Oke and Gopalakrishnana (2009) suggested some kinds ofmeasures to mitigate supply risks, such as better planning andco-ordination of supply and demand, flexible capacity, identifyingsupply chain vulnerability points and having contingency plans,and multiple sourcing strategy etc.

While the literature on supply disruption management isgrowing, the vast majority of these studies only investigatedstrategies of retailers or strategies of supplier. Different from theexisting literature on supply disruption management, we inves-tigate not only the sourcing strategies of the retailer but also thepricing game played between suppliers in a single-retailer two-supplier supply chain in the presence of supply disruption. Weexamine the pricing game under two scenarios, namely onebetween non-cooperative suppliers and the other betweencooperative suppliers. As such, the literature on supply disruptionmanagement, the wholesale price setting problem, and non-cooperative and cooperative games in supply chains is all relevantto our study.

There is a large body of literature on the broad topic of supplydisruption management. However, most of these studies assumeda single supplier. With no alternative sources available in thesingle-supplier system, inventory mitigation was the onlydisruption management strategy under consideration in thesestudies. In the following we limit our discussion to studies thatconsidered multiple suppliers. Recent papers dealing with multi-ple suppliers include Parlar and Perry (1996), Gurler and Parlar(1997), and Tomlin (2005, 2006). Parlar and Perry (1996)considered a firm that faces a constant demand and sources fromtwo identical-cost, infinite-capacity suppliers. The inter-failuretime and the repair time are exponentially distributed for bothsuppliers. The authors proposed a suboptimal ordering policy,which was solved numerically. Gurler and Parlar (1997) extendedthe work of Parlar and Perry (1996) by considering the case ofErlang-k inter-failure times and general repair times. However,the identical-cost or/and infinite-capacity assumptions shift anydownside to sourcing mitigation or/and contingent rerouting.When the finite-capacity suppliers differ in cost, there are otherdisruption management strategies, e.g., dual sourcing, emergencysourcing etc. Tomlin (2005) went beyond the existing literatureby explicitly modelling the trade-offs and limitations inherent inmitigation and contingency strategies. Tomlin (2006) considereda model in which a firm may order from a cheap but unreliablesupplier and/or an expensive but reliable supplier. He examinedthe conditions under which the firm’s optimal strategy is tomanage supply disruption. He also investigated the influence ofthe firm’s attitude towards risk on mitigation and contingencystrategies for managing supply disruption risk.

The supply disruption papers cited above only investigated thestrategies of retailers under the assumption that suppliers areexogenous. However, suppliers’ responses, e.g., their pricingstrategies, are also crucial factors that impact the supply chain.The wholesale price setting problem has been extensively studiedin the literature. Recent literature includes the papers by Lariviereand Porteus (2001), Wang and Gerchak (2003), Tomlin (2003),

Bernstein and DeCroix (2004), and Cachon and Lariviere (2001),among others. They gave the optimal pricing strategies of thesuppliers under different scenarios. However, most of the abovepapers assumed perfectly reliable supply. The price settingproblem with unreliable supply has received much less attention.

We now turn our attention to the literature on game analysisof supply chains. Game theory can be divided broadly into twoapproaches, namely the non-cooperative and the cooperativeapproaches. In the last several years, it has been recognizedthat game theory is an effective tool for the analysis of supplychains with multiple agents. In recent years, there is awide variety of research papers that apply non-cooperativegame theory to the field of supply chain management. For thesake of conciseness, we do not provide a comprehensive reviewof the literature in this area. For an excellent survey, we referthe reader to Cachon and Netessine (2004). Papers employingcooperative game theory to study supply chain managementare much less prevalent, but are becoming more popular. Thistrend is probably due to the prevalence of bargaining andnegotiation in supply chain relationships. We refer the reader toNagarajan and Sosic (2008) for a detailed survey of the existingliterature on applications of cooperative games to supply chainmanagement.

Motivated by the above observations, we set out to study asupply chain consisting of one retailer and two suppliers andconsider the price setting problem in the presence of supplydisruption. In this paper we investigate both a centralized supplychain and a decentralized supply chain. Furthermore, we considertwo scenarios for the decentralized supply chain, i.e., the twosuppliers are competitive and cooperative. We seek to find theoptimal order quantities and the optimal wholesale prices in bothscenarios. Babich (2006) and Babich et al. (2007) developedsimilar models to investigate supplier pricing decisions withsupply disruption. Babich (2006) investigated how the supplierdefault risk and default co-dependence affect the procurementand production decisions of the manufacturer, supplier pricingdecisions, and the value of the supplier’s option to postpone itspricing decisions. Babich et al. (2007) examined the effects of co-dependence among supplier defaults on the performance of firmsand the consequences of the suppliers offering different paymentpolicies. One major difference between their papers and our studyis about the treatment of unfilled demand due to supplydisruption. In our paper, the unfilled demand is filled from a spotmarket rather than is lost. This is always true because the demandcan be filled by emergency sourcing or global sourcing, which is acommon business practice with advances in transportation andinformation technology. The incorporation of the spot market inthe model alters supplier competition. Moreover, we consider asituation in which the suppliers are cooperative. The study of thepricing decisions of cooperative suppliers in this setting is not far-fetched. First, examples of real-world supplier alliances in supplychains abound. For example, Greene (2002) presented severalinstances of alliances between component manufacturers in thesemiconductor industry. Second, retailers encourage cooperationbetween suppliers in hopes of converting difficult suppliers intosupportive suppliers through cooperation, which provides oppor-tunities for the sharing of good practices and experiences betweensuppliers. The pricing decisions of cooperative suppliers are ofinterest to our study.

We intend to contribute to knowledge in this area by addresstwo key questions: How does the supply disruption affect thesuppliers’ pricing and the retailer’s ordering behaviours? Howshall we coordinate the behaviours of cooperative suppliers in thepresence of supply disruption? This paper provides valuablemanagerial guidance for retailers to allocate their orders betweendifferent suppliers and for suppliers to price their supplies when

J. Li et al. / Int. J. Production Economics 124 (2010) 137–150138


facing supply disruption. Specially, we make four major contribu-tions:

1. We show the existence of an equilibrium price in thecompetitive scenario for two typical customer demanddistributions, namely the uniform distribution and the ex-ponential distribution.

2. Based on the uniform demand distribution, we obtain anexplicit form of the unique equilibrium price.

3. We investigate the impacts of supply disruption on theretailer’s sourcing strategy and the suppliers’ pricing strategyby both theoretical and computational analyses.

4. We devise a coordination mechanism to maximize the profitsof cooperative suppliers.

The rest of this paper is organized as follows. The problemunder consideration is introduced and formulated in Section 2.Section 3 analyses a benchmark scenario in which the wholechannel (i.e., the supply chain) is centrally controlled. Section 4investigates a decentralized supply chain under two scenarios,one with non-cooperative suppliers and the other with coopera-tive suppliers. In Section 5 numerical results are presented toillustrate the theoretical results. Conclusions and suggestions forfuture research are given in Section 6. All the proofs of thetheoretical results are given in Appendix A.

2. The problem

In this paper we study a supply chain consisting of one retailerand two suppliers with unreliable supply. All three firms areassumed to be risk neutral and pursue expected profit maximiza-tion. In addition, we assume that there is a spot market as acontingent supplier that is perfectly reliable.

The retailer buys a short-life product from the two suppliersand from the spot market, and sells the product to its customersin a single selling season. The uncertain source of supply is a stateof the suppliers, which are subject to random failures. If a supplieris in the success state, the orders placed with it will be deliveredon time. However, if a supplier is in the failure state, no orders canbe supplied. We assume that there are two types of failure:common-cause and supplier-specific failures. A common-causefailure affects both suppliers. For example, an earthquake mayaffect all the suppliers in a region. A supplier may still fail forsome supplier-specific reason even if there is no common-causefailure. For example, equipment failure might affect one supplierbut not the other supplier. We assume that supplier 2 is affectedonly by the common-cause failure but supplier 1 is affected byboth types of failure. First, supplier 1 and supplier 2 decide theirindividual wholesale prices. Then the retailer allocates its ordersbetween the two suppliers before the states of the suppliers arerealized. After the states have been realized, the retailer has achance to make an emergency order from the spot market. Weassume that the replenishment rate is infinite and the lead time iszero.

The following notation is used in the model:

i¼ 1;2;3 supplier 1, supplier 2 and the spot market, respectively;b the goodwill cost of a unit of unmet demand;ci the delivery cost of a unit of the product of supplier i,

i¼ 1;2;D the positive stochastic customer demand;f the positive probability density function of D;F the differentiable and strictly increasing cumulative

distribution function of D;

p the fixed selling price of a unit of the product;Qi the order quantity placed with supplier i, i¼ 1;2;Q3 the inventory level after making an emergency order

from the spot market;s the salvage value of a unit of the residual product;wi the wholesale price of a unit of the product offered by

supplier i, i¼ 1;2;w3 the fixed wholesale price of a unit of the product offered

by the spot market;a the probability of a common-cause failure not occurring,

where 0oao1;b the probability that supplier 1 does not fail conditional

on a common-cause failure not occurring, where0obo1;

g the total proportion of the marginal delivery cost in theevent of a failure, where 0ogo1;

Z the proportion of the cost incurred by the supplier whofails in the event of a failure, where 0rZr1.

Among the above variables, w1, w2, Q1, Q2, and Q3 are decisionvariables and the others are exogenous variables, which areknown to all the members of the supply chain. In this paper therevenues of supplier 1, supplier 2 and the retailer are our focus.We do not care about the revenue of the spot market and have noregard for its delivery cost. The spot market is not a decision-maker in the supply chain.

It should be noted that a marginal cost gci is incurred in theevent of a failure. We suppose that the failing supplier and theretailer assume this cost jointly. The marginal cost assumed bythe failing supplier is Zgci and the marginal cost assumed by theretailer is ð1� ZÞgci. This cost structure is different from that usedin most of the literature in which only the retailer assumes thecost in the event of a failure. But this is not always true. In fact,before supply failures are realized, both the retailer and suppliersusually have incurred some costs, which may include fixed set-upcosts and variable costs. For simplicity of analysis, we assume thatall the setup costs are zero and all the variable costs in the eventof a supply failure are proportional to the delivery cost and to theorder quantity.

Based on the reliability of the suppliers, it is reasonable toassume that c1oc2ow3. In addition, we assume that0rsoc1oc2ow3op. These inequalities ensure that each firmmakes a positive profit and the chain will not produce infinitequantities of the product.

In the following section we first consider a centralized systemin which all the decisions are centralized to maximize theperformance of the entire supply chain (including the retailer,suppliers 1 and 2). We give the conditions for both suppliers beingplaced with positive orders and the corresponding optimal orderquantities. The centralized system solution serves as a benchmarkfor the decentralized setting. Then we consider a decentralizedsupply chain under two different scenarios in which the suppliersare competitive or cooperative. For the two decentralizedproblems, information on each player’s demand function, coststructure, and decision rules is common knowledge to all theparties concerned. The decentralized supply chain with compe-titive suppliers, in which the players act independently and makedecisions that maximize their individual profits, can be viewed astwo static nested games. The first is a static non-cooperative gamebetween suppliers 1 and 2. They choose their wholesale pricessimultaneously and do not collude. The second is a Stackelberggame, which is nested within the static noncooperative game. Inthe Stackelberg game, the leaders (suppliers 1 and 2) select thewholesale prices, and the followers (the retailer facing randomyields) respond by selecting their order quantities. For thedecentralized supply chain with competitive suppliers, we give

J. Li et al. / Int. J. Production Economics 124 (2010) 137–150 139


the equilibrium wholesale price of the two suppliers and theoptimal order quantities of the retailer. Finally we investigate thedecentralized supply chain with cooperative suppliers in whichsuppliers 1 and 2 choose their individual wholesale prices tomaximize their total profits. To ensure stability and robustness ofthe cooperation, the Nash bargaining game in cooperative gametheory is used to divide the profit pie created through coopera-tion.

We use the following mathematical notation in the paper: aþ

represents the positive part of a, i.e., maxf0; ag. a3b represents thelarger value between a and b, i.e., maxfa; bg. a4b represents thesmaller value between a and b, i.e., minfa;bg. More notation willbe introduced and defined when needed.

3. The centralized supply chain

It is obvious that the supply chain will perform best if thechannel is centrally controlled. Since the wholesale price is onlyused to divide the profit between the retailer and the suppliers,w1 and w2 are no longer decision variables in the centralizedsupply chain. The decision variables are only Q1, Q2 and Q3. Weseek to determine the channel’s optimal order allocation decisionswhen there is supply uncertainty for a seasonal product. Thesequence of events in the centralized supply chain is as follows:

1. Orders are placed with suppliers 1 and 2, respectively, inanticipation of supply disruption and demand (stage 1).

2. An emergency order is placed with the spot market after asupply disruption has occurred but before demand occurs(stage 2).

3. When the selling season arrives, the product is sold at a fixedprice in the market. Any unmet demand incurs a goodwill costto the whole channel. After the selling season, the residualproduct will be salvaged (stage 3).

Denote z and Q3c as the inventory level of the supplychain before and after the emergency order is placed, respectively.Let €pcðQ3cjzÞ be the channel’s random profit in stage 2, i.e., therandom profit after the emergency order Q3c � z is placed(hereafter the subscript ‘c’ stands for the centralizedsupply chain and the superscript ‘€’ stands for stage 2). We have

€pcðQ3cjzÞ ¼ pðQ3c4DÞ �w3ðQ3c � zÞþ þsðQ3c � DÞþ

�bðD� Q3cÞþ : ð3:1Þ

Then we can deduce the channel’s expected profit in stage 2,denoted as €PcðQ3cjzÞ, which is given by

€PcðQ3cjzÞ ¼ðpþb�w3ÞQ3c � ðpþb� sÞ

R Q3c

0 FðxÞdxþw3z� bE½X�; Q3c Zz;

ðpþbÞz� ðpþb� sÞR z

0 FðxÞdx� bE½X�; Q3c ¼ z;

8<:

ð3:2Þ

where E½X� is the mean of the random demand D.The channel’s order problem in stage 2 is to choose the

emergency order quantity Q3c � z to maximize its expected profitfor any given initial inventory level z. This is the classicalnewsvendor problem. By using the first- and second-orderoptimality conditions, we can obtain that the order-up-to-level(OUL) policy is optimal for the channel and the threshold value ofthe inventory level is Q 3c ¼ F�1ððpþb�w3Þ=ðpþb� sÞÞ. Hence, theoptimal inventory level after the retailer placing an emergencyorder is as follows:

Q�3c ¼ Q 3c3z: ð3:3Þ

Then the maximum expected revenue of the channel instage 2 for any given initial inventory level z is deduced

as follows:

€P�

c ðzÞ ¼ðpþb�w3ÞQ 3c � ðpþb� sÞ

R Q 3c

0 FðxÞdxþw3z� bE½X�; zrQ 3c ;

ðpþbÞz� ðpþb� sÞR z

0 FðxÞdx� bE½X�; zZQ 3c :

8<:

ð3:4Þ

It is obvious that the overall probability of supplier 1 notfailing is ab and the overall probability of supplier 2 not failing isa. Hence the channel’s expected profit in stage 1 after the retailerplacing orders with suppliers 1 and 2, denoted as _PcðQ1c ;Q2cÞ, isgiven by (hereafter the superscript ‘_’ stands for stage 1)

_PcðQ1c ;Q2cÞ ¼ ab½ €P�

c ðQ1cþQ2cÞ � c1Q1c � c2Q2c�það1� bÞ½ €P�

c ðQ2cÞ

�gc1Q1c � c2Q2c�þð1� aÞ½ €P�

c ð0Þ � gc1Q1c � gc2Q2c�:

ð3:5Þ

The channel’s order problem in stage 1 is to choose orderquantities Q1c and Q2c to maximize its expected profit. We havethe following conclusions about the optimal sourcing strategy ofthe centralized supply chain.

Theorem 3.1. After a supply disruption has occurred, the optimal

ordering strategy of the centralized supply chain from the spot

market is the OUL policy and the threshold value of the inventory

level is Q 3c ¼ F�1ðpþb�w3=pþb� sÞ. The optimal sourcing strate-

gies from suppliers 1 and 2 are as follows:

1. If abðw3 � c1þgc1Þ � gc1o0 and aðw3 � c2þgc2Þ � gc2o0,both supplier 1 and supplier 2 are placed with zero order quantity

and the centralized supply chain only sources from the spot

market. The emergency order quantity is Q 3c.2. If abðw3 � c1þgc1Þ � gc1o0 and aðw3 � c2þgc2Þ � gc2Z0, the

optimal quantity ordered from supplier 1 is zero and the optimal

quantity ordered from supplier 2 is F�1ððaðpþb� c2þgc2Þ �

gc2Þ= aðpþb� sÞÞ.3. If 0rabðw3 � c1þgc1Þ � gc1rb ½aðw3 � c2þgc2Þ � gc2�, the

optimal quantity ordered from supplier 1 is zero and the optimal

quantity ordered from supplier 2 is F�1ððaðpþb� c2þgc2Þ�

gc2Þ=aðpþb� sÞÞ.4. If 0rb½aðw3 � c2þgc2Þ � gc2�rabðw3 � c1þgc1Þ � gc1r aðw3

�c2þgc2Þ � gc2, both supplier 1 and supplier 2 are selected to be

placed with positive orders, which are given by

Q�1c ¼ F�1 abðpþb� c1þgc1Þ � gc1

abðpþb� sÞ

� �

�F�1 aðpþb� c2þgc2Þ � gc2 � abðpþb� c1þgc1Þ � gc1

� �að1� bÞðpþb� sÞ

� �;

ð3:6Þ

Q�2c ¼ F�1 aðpþb� c2þgc2Þ � gc2 � ðabðpþb� c1þgc1Þ � gc1Þ

að1� bÞðpþb� sÞ

� �:

ð3:7Þ

5. If abðw3 � c1þgc1Þ � gc1Z0 and abðw3 � c1þgc1Þ � gc1Z

aðw3 � c2þgc2Þ � gc2, the optimal quantity ordered from

supplier 2 is zero and the optimal quantity ordered from supplier

1 is F�1ððabðpþb� c1þgc1Þ � gc1Þ=abðpþb� sÞÞ.

We obtain the conditions for both suppliers being placed withpositive order quantities as the following corollary.



Corollary 3.2. In the centralized supply chain, both suppliers are

placed with positive order quantities if and only if the following

conditions hold:

C 1: abðw3 � c1þgc1Þ � gc1Z0;C 2: aðw3 � c2þgc2Þ � gc2Zabðw3 � c1þgc1Þ � gc1;C 3: abðw3 � c1þgc1Þ � gc1Zbðaðw3 � c2þgc2Þ � gc2Þ.

For the centralized supply chain, it is possible that bothsuppliers 1 and 2 are not selected, i.e., the channel only sourcesfrom the spot market if abðw3 � c1þgc1Þ � gc1o0 andaðw3 � c2þgc2Þ � gc2o0. However, if Q�1cþQ�2c 40, thenQ�1cþQ�2c 4Q

�

3c , i.e., the total order quantity always exceeds thethreshold value Q

�

3c if any supplier is placed with a positive orderquantity. This indicates that once the channel selects one supplieror two suppliers, it prefers the supplier(s) to the spot market.

From Theorem 3.1, we see that the sourcing strategy of theretailer in the centralized supply chain is affected mainly by twokey factors, i.e., abðw3 � c1þgc1Þ � gc1 and aðw3 � c2þgc2Þ � gc2.These two factors can be regarded as the competitiveness of thetwo suppliers in the centralized supply chain. The larger the valueof a factor is, the more powerful is the corresponding supplier, i.e.,a higher probability that the supplier will be placed with apositive order quantity. Furthermore, the other factors that affectsupplier competitiveness include the fixed wholesale price of thespot market, the delivery cost of a unit of the product of thesupplier, the probability of delivering orders on time, and the totalproportion of the marginal delivery cost in the event of a failure.The supplier can improve his competitiveness by decreasing hisdelivery cost or improving his probability of delivering orders ontime. However, stable delivery usually increases the marginaldelivery cost. Thus, a tradeoff exists between the probability ofon-time delivery and the marginal cost of delivery.

4. The decentralized supply chain

Consider the decentralized supply chain in which the firmsmake their decisions independently. The sequence of events in thedecentralized supply chain is as follows:

1. The suppliers decide their individual wholesale prices eitherwithout cooperation or with cooperation (stage 0).

2. The retailer places orders of Q1 units and Q2 units withsuppliers 1 and 2, respectively, in anticipation of supplydisruption and demand (stage 1).

3. The retailer makes an emergency order from the spot marketafter a supply disruption has occurred but before demandoccurs (stage 2).

4. When the selling season arrives, the retailer sells the productat a fixed price in the market. Any unmet demand incurs agoodwill cost to the retailer. After the selling season, theresidual product will be salvaged (stage 3).

As mentioned in Section 2, the situation with competitivesuppliers can be viewed as two static nested games. In thefollowing we first investigate the response function of the retailer

for any given wholesale price. Then based on the optimal responsefunction, we derive the optimal wholesale price decisions for thesuppliers without cooperation. A sufficient condition for theexistence of an equilibrium is provided. Finally we devise acoordination mechanism to maximize the profits of both supplierswhen they are cooperative.

4.1. The optimal strategy of the retailer

This subsection aims to determine the retailer’s optimal orderallocation decisions to maximize its expected profit in stage 1 forany given wholesale price when the supply chain is decentralized.

It is straightforward to deduce that the retailer adopts thesame optimal strategies as those in the centralized channel instage 2 after a supply disruption has occurred. Both of them applythe OUL policy with the same threshold Q 3d ¼ Q 3c ¼ F�1ððpþb�

w3Þ=ðpþb� sÞÞ (hereafter the subscript ‘d’ stands for the decen-tralized supply chain). Hence the maximum expected profit of theretailer in stage 2, denoted by €P

�

r ðzÞ, is also the same as themaximum expected profit of the centralized supply chain in stage2 with the same initial inventory level z defined by Eq. (3.4).

Denote _PrðQ1;Q2Þ as the retailer’s expected profit in stage 1 forgiven wholesale prices w1 and w2, which is given by_PrðQ1;Q2Þ ¼ ab½ €P

�

r ðQ1þQ2Þ �w1Q1 �w2Q2�það1� bÞ½ €P�

r ðQ2Þ

�ð1� ZÞgc1Q1 �w2Q2�þð1� aÞ½ €P�

r ð0Þ

�ð1� ZÞgc1Q1 � ð1� ZÞgc2Q2�: ð4:1Þ

Derivation of the retailer’s optimal strategy is similar to that ofthe centralized supply chain. We only state the main conclusionsin the following theorem.

Theorem 4.1. After a supply disruption has occurred, the optimal

order strategy of the retailer from the spot market is the OUL policy

and the threshold value of the inventory level is

Q 3d ¼ F�1ððpþb�w3Þ=ðpþb� sÞÞ. The optimal sourcing strategies

from supplier 1 and supplier 2 of the retailer are given as follows:

1. If abðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1o0 and aðw3 �w2þ

ð1� ZÞgc2Þ � ð1� ZÞgc2o0, then both supplier 1 and supplier 2are placed with zero order quantity and the retailer only sources

from the spot market. The emergency order quantity is Q 3d.2. If abðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1o0 and aðw3� w2þ

ð1� ZÞgc2Þ � ð1� ZÞgc2Z0, then the optimal quantity ordered

from supplier 1 is zero and the optimal quantity ordered from

supplier 2 is F�1ððaðpþb�w2þð1� ZÞgc2Þ � ð1� ZÞgc2Þ= ðaðpþb� sÞÞÞ.

3. If 0rabðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1r b½aðw3�w2þð1�ZÞgc2Þ � ð1� ZÞgc2�, then the optimal quantity ordered from

supplier 1 is zero and the optimal quantity ordered from supplier

2 is F�1ððaðpþb� w2þð1� ZÞgc2Þ� ð1� ZÞgc2Þ= ðaðpþb� sÞÞÞ.4. If 0rb½aðw3 �w2þð1� ZÞgc2Þ � ð1� ZÞgc2�rabðw3 �w1þ

ð1� ZÞgc1Þ � ð1� ZÞgc1raðw3 �w2þ ð1� ZÞgc2Þ� ð1� ZÞgc2,both supplier 1 and supplier 2 are selected to be placed with

positive order quantities, which are given by

5. If abðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1Z0 and abðw3 �w1þ

ð1� ZÞgc1Þ � ð1� ZÞgc14 aðw3 �w2þð1� ZÞgc2Þ � ð1� ZÞgc2,then the optimal quantity ordered from supplier 2 is zero and

Q�1d ¼ F�1 abðpþb�w1þð1� ZÞgc1Þ � ð1� ZÞgc1

abðpþb� sÞ

� �� F�1 að1� bÞðpþbÞ � aw2þabw1 � ð1� aÞð1� ZÞgc2þð1� abÞð1� ZÞgc1


� �; ð4:2Þ

and

Q�2d ¼ F�1 að1� bÞðpþbÞ � aw2þabw1 � ð1� aÞð1� ZÞgc2þð1� abÞð1� ZÞgc1


� �: ð4:3Þ



the optimal quantity ordered from supplier 1 is F�1ððabðpþb�

w1þ ð1� ZÞgc1Þ � ð1� ZÞgc1Þ=ðabðpþb� sÞÞÞ.

Similar to the conclusions reached for the centralized supplychain, we have the following conditions for both suppliers beingplaced with positive order quantities.

Corollary 4.2. In the decentralized supply chain, both suppliers are

placed with positive order quantities if and only if the following

conditions hold:

C 4: abðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1Z0;C 5: aðw3 �w2þð1� ZÞgc2Þ � ð1� ZÞgc2Zabðw3 �w1þð1� ZÞ

gc1Þ � ð1� ZÞgc1;C 6: abðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1Zb½aðw3 �w2þð1� ZÞ

gc2Þ � ð1� ZÞgc2�.

For the decentralized supply chain, we also find that thesourcing strategy of the retailer is affected mainly by two keyfactors, i.e., abðw3 �w1þð1� ZÞgc1Þ � ð1� ZÞgc1 and aðw3 �w2

þð1� ZÞgc2Þ � ð1� ZÞgc2. These two factors can be regarded asthe competitiveness of the two suppliers in the decentralizedsupply chain. The larger the value of a factor is, the more powerfulis the corresponding supplier. Furthermore, the other factors thataffect supplier competitiveness include the wholesale priceoffered by the supplier and the proportion of the cost incurredby the supplier in addition to all the factors in the centralizedsupply chain. The supplier in the centralized supply chain canimprove his competitiveness by decreasing his wholesale price inaddition to decreasing his delivery cost or improving hisprobability of delivering orders on time. However, there is atradeoff between the order quantity and the wholesale price.

4.2. The optimal strategies of competitive suppliers

In this section suppliers 1 and 2 are assumed to becompetitive, i.e., suppliers 1 and 2 set their individual wholesaleprices simultaneously to maximize their respective expectedprofits before the retailer places its orders and the suppliers donot collude. As mentioned in Section 2, this is a static non-cooperative game between suppliers 1 and 2. We first derive thefeasible strategy space of both suppliers. Then we derive asufficient condition for the existence of an equilibrium pricestrategy in this game. Based on the assumption of a uniformdemand distribution, we further obtain an explicit form of theequilibrium strategy.

We have obtained the conditions in Corollary 4.2 in whichboth supplier are placed with positive order quantities. Based onthese conditions in Corollary 4.2, we obtain the feasible strategyspaces of both suppliers to make a positive expected profit andthe conditions for the existence of the spaces as follows:

Theorem 4.3. If bða� agþgÞc24 ðab� abgþgÞc1, then the feasible

strategy space of supplier 1 is ½ððab� abZgþZgÞ=abÞc1; ðða�agþgÞ=aÞc2� ðð1� abÞð1� ZÞg=abÞc1�. When supplier 1 sets its

wholesale price in the interval, it will obtain a positive profit.Similarly, if aw3 � ða� agþgÞc24abw3 � ðab� abgþgÞc1, then

the feasible strategy space of supplier 2 is ½ða� agZþgZ=aÞc2; ð1�bÞw3 � ðð1� aÞð1� ZÞg=aÞc2þðab� a bgþg=aÞc1�. When supplier 2sets its wholesale price in the interval, it will obtain a positive profit.

Note that the two conditions in Theorem 4.3 are C2 and C3 inCorollary 3.2. This means that if both suppliers are placed withpositive order quantities in the centralized supply chain, thenthere exist feasible strategy spaces for the two competitive

suppliers to obtain a positive profit in the decentralized supplychain. Hence, we have the following results.

Proposition 4.4. The two competitive suppliers can obtain a positive

expected profit in the decentralized supply chain if both of them are

placed with positive order quantities in the centralized supply chain.

From Theorem 4.3, we obtain the feasible strategy spaces forboth suppliers. Consequently, we discuss the existence of anequilibrium solution for the game. For any wholesale price,suppliers 1 and 2 can correctly anticipate the retailer’s demandcurves, i.e., Q1ðw1;w2Þ and Q2ðw1;w2Þ, which are given byEqs. (4.2) and (4.3). Hence, the suppliers face the inverse demandcurves

w1ðQ1;Q2Þ ¼ pþb� ðpþb� sÞFðQ1þQ2Þ �ð1� abÞð1� ZÞg

ab c1; ð4:4Þ

and

w2ðQ2;Q1Þ ¼ pþb� bðpþb� sÞFðQ1þQ2Þ

�ð1� bÞðpþb� sÞFðQ2Þ �ð1� aÞð1� ZÞg

a c2: ð4:5Þ

Because FðxÞ is continuous and strictly increasing, it is easy toverify that the corresponding feasible spaces for Q1 and Q2 arealso closed intervals if the feasible spaces for w1 and w2 are closedintervals. Moreover, the revenue functions are equivalent to

Ps1ðQ1;Q2Þ ¼ ½abw1ðQ1;Q2Þ � ðab� abZgþZgÞc1�Q1

¼ ½abðpþbÞ � abðpþb� sÞFðQ1þQ2Þ

�ðab� abgþgÞc1�Q1; ð4:6Þ

and

Ps2ðQ2;Q1Þ ¼ ½aw2ðQ2;Q1Þ � ða� agZþgZÞc2�Q2

¼ ½aðpþbÞ � abðpþb� sÞFðQ1þQ2Þ � að1� bÞ�ðpþb� sÞFðQ2Þ � ða� agþgÞc2�Q2: ð4:7Þ

The problems of suppliers 1 and 2 are equivalent to settingquantities Q1 and Q2 to maximize Ps1

ðQ1;Q2Þ and Ps2ðQ2;Q1Þ

simultaneously. Maximizing Ps1ðQ1;Q2Þ and Ps2

ðQ2;Q1Þ arestraightforward if they are unimodal. However the objectivefunctions Ps1

ðQ1;Q2Þ and Ps2ðQ2;Q1Þ are dependent on the

demand distribution. It should be noted that not all demanddistributions result in a unimodal objective function. In thefollowing, we derive a sufficient and less restrictive condition toensure the objective functions are unimodal.

Define a new function gðQ2jQ1;bÞ as follows:

gðQ2jQ1;bÞ9Q2½bðpþb� sÞf ðQ2þQ1Þþð1� bÞðpþb� sÞf ðQ2Þ�

b½pþb� ðpþb� sÞFðQ2þQ1Þ�þð1� bÞ½pþb� ðpþb� sÞFðQ2Þ�:

ð4:8Þ

We have the following conclusions about the objective functions.

Lemma 4.5. Suppose FðxÞ has a support ½a; bÞ. If gðQ2jQ1;bÞ is

weakly increasing for Q2, then supplier 2’s revenue function is

unimodal for Q2A ½0; þ1Þ. Moreover, supplier 1’s revenue function

Ps1ðQ1;Q2Þ is also unimodal for Q1A ½0; þ1Þ.

As pointed out in the above analysis, the strategy space for eachsupplier’s decision is a closed interval; hence, it is a nonemptycompact convex set of the Euclidean space. Along with the resultsin Lemma 4.5, we have the following theorem about the existenceof a Nash equilibrium of this game.

Theorem 4.6. If gðQ2jQ1;bÞ is weakly increasing for Q2, then a pure

strategy Nash equilibrium exists.

In the proof of Lemma 4.5, a new function gðQ2jQ1;bÞ wasdefined. If s¼ 0, Q1 ¼ 0 and b¼ 1, gðQ2jQ1;bÞ is the so-called



generalized failure rate defined by Lariviere and Porteus (2001).They proved that if the demand follows an increasing generalizedfailure rate (IGFR), the objective is unimodal. They also pointedout that most demand distributions follow an IGFR. In this paperwe obtain a similar condition, i.e., gðQ2jQ1;bÞ is weakly increasing.It should be noted that it is difficult to verify that all the demanddistributions meet this condition. Fortunately, it is straightfor-ward to verify that both the uniform and exponential distribu-tions possess this important property.

Corollary 4.7. If the demand follows a uniform distribution or an

exponential distribution, then a pure strategy Nash equilibrium exists.

Our analysis so far has not imposed any restrictions on thedemand distribution. Further analysis (e.g., the uniqueness of theequilibrium and the explicit expression of the equilibrium) for ageneral demand distribution is difficult. In order to gain furtherinsights, we suppose that the demand D is uniformly distributedin some interval, which without loss of generality can be taken asthe interval ½0;1�. Here the game between suppliers 1 and 2 is anon-cooperative static game. We have the following theorem.

Theorem 4.8. Suppose that the demand D is uniformly distributed in

the interval ½0;1�. If bða� agþgÞc24 ðab� abgþgÞc1 and

aw3 � ða� agþgÞc24abw3 � ðab� abgþgÞc1, then the unique

Nash equilibrium strategy of the game between the suppliers is given

by:

ðw�1n;w�2nÞ ¼

ðw11;w21Þ; w11rw1V

w1 and w21rw2V

w2;

ðw12;w23Þ; w1Zw124w1 and w23rw2V

w2;


w1 and w2Zw224w2;

ðw13;w23Þ; w1Zw134w1 and w2Zw234w2:

8>>>><>>>>:

ð4:9Þ

(All the variables appearing in this theorem are defined in the proof.Hereafter the subscript ‘n’ stands for the decentralized supply chain

with non-cooperation suppliers.)

4.3. The optimal strategies of cooperative suppliers

In this subsection, suppliers 1 and 2 are assumed to becooperative, i.e., the two suppliers set their individual wholesaleprices in order to maximize their total expected profits before theretailer places its orders. Obviously, a necessary precondition fortheir cooperation is that none of them is intended to set itswholesale price low enough to monopolize the market. It seemsthat the wholesale prices of both suppliers are set by onedecision-maker to ensure both suppliers are placed with non-negative order quantities and to maximize the total profit of thetwo suppliers. We first derive the optimal wholesale price ofthe two cooperative suppliers. Then we discuss how to divide theprofit and how to pool the cost between the two suppliers toexecute the cooperative wholesale price successfully. Finally, wediscuss the coordination of the whole channel.

From the analysis in subsection 4.1, if and only if theconditions in Corollary 4.2 hold, both suppliers will be placedwith positive order quantities, which are given by Eqs. (4.2) and(4.3). Then the total revenue function of the two suppliers is givenby

Pdcðw1;w2Þ ¼ ½abw1 � ðab� abZgþZgÞc1�Q1

þ½aw2 � ða� agZþgZÞc2�Q2: ð4:10Þ

(Hereafter the subscript ‘dc’ stands for the decentralized supplychain with cooperative suppliers).

The problem of cooperative suppliers is as follows:

max Pdcðw1;w2Þ

s:t: 0rw1rw3 �ð1� abÞð1� ZÞgc1

ab;

w2Zw1þð1� abÞð1� ZÞgc1

ab �ð1� aÞð1� ZÞgc2

a ;

w2rw3 �ð1� aÞð1� ZÞgc2

a

�abðw3 �w1Þ � ð1� abÞð1� ZÞgc1

a: ð4:11Þ

It is easy to verify that the Hessian matrix of Pdcðw1;w2Þ isnegative definite. Hence, Pdcðw1;w2Þ is jointly concave withrespect to w1 and w2. Moreover, the feasible spaces for w1 and w2

are convex. Problem (4.11) is a convex quadratic programmingproblem, which can be solved by some popular mathematicalsoftware such as MatLab or Mathematica.

It is obvious that if suppliers 1 and 2 are cooperative, they canobtain more total expected revenue than that under the non-cooperative scenario. How to divide the profit pie created throughcooperation is crucial for stability and robustness of the coopera-tion. The Nash bargaining game in cooperative game theory canbe used to ascertain the allocation ratio of the expected profit pieto ensure that both suppliers earn a rational expected revenue. Itshould be noted that the bargaining solution of the game isnot a randomized outcome. However, all the revenues of thesuppliers are random. Thus, only to ascertain the allocation ratioof the expected profit pie is not enough to guarantee suppliercooperation. An effective way to allocate the randomized revenueshould be developed to the effect that the total expectedprofit can be allocated according to the bargaining solution. Inthe following we primarily focus on two issues, i.e., allocationof the expected revenue and allocation of the randomizedrevenue.

We start by building a basic bargaining model initiated by Nash(1951). Recall that the Nash bargaining game requires us to identify afeasible set of payoffs F and a disagreement point d that are pre-determined and are independent of the negotiations. To do so, let usfirst suppose that the two suppliers negotiate on individual expectedrevenues denoted by (Pdcs1

;Pdcs2). Obviously, this negotiation is

conducted over the sharing of some fixed profit pie. Denote theoptimal value of Problem (4.11), i.e., the pie to be allocated betweenthe two suppliers, as P�dc . Thus, the feasible set of the bargaining isF ¼ ðPdcs1

;Pdcs2jPdcs1

þPdcs2¼P�dcÞ. Furthermore, according to the

rule of negotiation, the disagreement point is defined as the twosuppliers’ equilibrium expected revenues under the non-cooperativescenario, i.e., d¼ ðP�s1;P

�s2Þ. Hence the Nash bargaining solution

between the two suppliers is obtained by solving the followingoptimization problem:

arg maxðPdcs1

;Pdcs2ÞAF;ðPdcs1

;Pdcs2ÞZdðPdcs1

�P�s1ÞðPdcs2�P�s2Þ: ð4:12Þ

It is straightforward to obtain the solution of Problem (4.12) asfollows:

ðP�dcs1;P�dcs2

Þ ¼P�dcþP

�s1 �P�s2

2;P�dc �P�s1þP

�s2

2

� �: ð4:13Þ

Since all revenue uncertainty due to random demand falls onthe retailer, the uncertainty of the suppliers’ revenues is causedonly by their reliability. Suppliers 1 and 2 can allocate therandomized profit according to ratios y1, y2, and y3 as follows:

1. If both suppliers are in the success state, the total profit ofthem is ðw1 � c1ÞQ1þðw2 � c2ÞQ2. The profits of suppliers 1 and2 are y1½ðw1 � c1ÞQ1þðw2 � c2ÞQ2� and ð1� y1Þ½ðw1 � c1ÞQ1þ

ðw2 � c2Þ Q2�, respectively. The probability of this caseoccurring is ab.



2. If supplier 1 is in the failure state and supplier 2 is in thesuccess state, the total profit of them is ðw2 � c2ÞQ2 � Zgc1Q1.The profits of suppliers 1 and 2 are y2½ðw2 � c2ÞQ2 � Zgc1Q1�

and ð1� y2Þ½ðw2 � c2ÞQ2 � Zgc1Q1�, respectively. The probabil-ity of this case occurring is að1� bÞ.

3. If both suppliers are in the failure state, the total profit of themis �Zgðc1Q1þc2Q2Þ. The profits of suppliers 1 and 2 arey3½�Zgðc1Q1þc2Q2Þ� and ð1� y3Þ½�Zgðc1Q1þc2Q2Þ�, respec-tively. The probability of this case occurring is 1� a.

Obviously, the expected profit of supplier 1 can be writtenequivalently as

Ps1ðy1; y2; y3Þ ¼ aby1½ðw1 � c1ÞQ1þðw2 � c2ÞQ2�

�ð1� aÞy3Zgðc1Q1þc2Q2Þþað1� bÞ�y2½ðw2 � c2ÞQ2 � Zgc1Q1�:

Similarly, the expected profit of supplier 2 can be written as

Ps2ðy1; y2; y3Þ ¼ abð1� y1Þ½ðw1 � c1ÞQ1þðw2 � c2ÞQ2�

�ð1� aÞð1� y3ÞZgðc1Q1þc2Q2Þþað1� bÞ�ð1� y2Þ½ðw2 � c2ÞQ2 � Zgc1Q1�:

To obtain the bargaining solution, suppliers 1 and 2 cannegotiate parameters y1; y2 and y3, which are subject to thefollowing equations:

Ps1ðy1;y2; y3Þ ¼P�dcs1

;

Ps2ðy1;y2; y3Þ ¼P�dcs2

:

(ð4:14Þ

Eq. (4.14) is a system of two linear equations in three unknowns.This system of linear equations has infinitely many solutions.Suppliers 1 and 2 can select a combination of ðy1; y2; y3Þ subject to(4.14) to allocate their randomized profits.

In sum, the cooperative suppliers can obtain more profits bythe following mechanism:

1. Suppliers 1 and 2 decide their individual wholesale prices withcooperation to maximize their total expected profits.

2. Adopt the Nash bargaining framework to examine theexpected total profit allocations.

3. All the parameters (y1, y2, y3) subject to (4.14) are negotiatedto allocate their randomized profits.

We now turn our attention to coordination of the wholechannel. Note that the retailer in the decentralized supply chainadopts the same OUL policy as the centralized channel after asupply disruption has occurred. Thus, the whole channel iscoordinated if and only if the retailer in the decentralized systemchooses the same inventory vector as in the centralized system.Letting Q�1n ¼ Q�1c and Q�2n ¼ Q�2c , we have

w�1c ¼ðab� abgþgÞ � ð1� abÞð1� ZÞg

ab c1; ð4:15Þ

w�2c ¼ða� agþgÞ � ð1� aÞð1� ZÞg

ac2: ð4:16Þ

Table 1All the problems and computational results of the centralized supply chain.

Prob. g Z a b w�11 w�2 Q �1 Q �2 P�t

1 0.2 0.2 0.7 0.7 11.1557 12.3086 22.8571 483.4286 1068.9

2 0.2 0.2 0.7 0.9 j2

j j j j

3 0.2 0.2 0.9 0.7 10.8700 12.0800 35.5556 489.7778 1699.8

4 0.2 0.2 0.9 0.9 j j j j j

5 0.2 0.5 0.7 0.7 11.5929 12.5143 22.8571 483.4286 1068.9

6 0.2 0.5 0.7 0.9 j j j j j

7 0.2 0.5 0.9 0.7 11.1167 12.1333 35.5556 489.7778 1699.8

8 0.2 0.5 0.9 0.9 j j j j j

9 0.2 0.8 0.7 0.7 12.2486 12.8229 22.8571 483.4286 1068.9

10 0.2 0.8 0.7 0.9 j j j j j

11 0.2 0.8 0.9 0.7 11.4867 12.2133 35.5556 489.7778 1699.8

12 0.2 0.8 0.9 0.9 j j j j j

13 0.3 0.2 0.7 0.7 J3 J J J J

14 0.3 0.2 0.7 0.9 j j j j j

15 0.3 0.2 0.9 0.7 10.8700 12.0800 3.3333 509.6667 1632.3

16 0.3 0.2 0.9 0.9 j j j j j

17 0.3 0.5 0.7 0.7 J J J J J

18 0.3 0.5 0.7 0.9 j j j j j

19 0.3 0.5 0.9 0.7 11.4250 12.2000 3.3333 509.6667 1632.3

20 0.3 0.5 0.9 0.9 j j j j j

21 0.3 0.8 0.7 0.7 J J J J J

22 0.3 0.8 0.7 0.9 j j j j j

23 0.3 0.8 0.9 0.7 11.9800 12.3200 3.3333 509.6667 1632.3

24 0.3 0.8 0.9 0.9 j j j j j

25 0.4 0.2 0.7 0.7 J J J J J

26 0.4 0.2 0.7 0.9 j j j j j

27 0.4 0.2 0.9 0.7 J J J J J

28 0.4 0.2 0.9 0.9 j j j j j

29 0.4 0.5 0.7 0.7 J J J J J

30 0.4 0.5 0.7 0.9 j j j j j

31 0.4 0.5 0.9 0.7 J J J J J

32 0.4 0.5 0.9 0.9 j j j j j

33 0.4 0.8 0.7 0.7 J J J J J

34 0.4 0.8 0.7 0.9 j j j j j

35 0.4 0.8 0.9 0.7 J J J J J

36 0.4 0.8 0.9 0.9 j j j j j

1 For the centralized supply chain, w�1 and w�2 denote the wholesale prices at which the best performance of the whole supply chain can be achieved.2 Hereafter ‘j’ indicates that the retailer only orders from supplier 1.3 Hereafter ‘J’ indicates that the retailer only orders from supplier 2.



Hence, the maximum expected revenue of the whole supply chaincan be achieved if the corresponding wholesale prices areidentical to Eqs. (4.15) and (4.16). However, even if thecorresponding wholesale prices are identical to Eqs. (4.15) and(4.16), it is difficult to achieve full coordination only by thewholesale prices. The reason is that it is difficult to allocate thetotal revenue only by the wholesale prices between the retailerand the suppliers since the revenue of the retailer is stochasticdue to the random demand.

5. Computational analysis

In this section we present numerical examples to illustrate thetheoretical results and explore the differences between thecentralized supply chain and the decentralized supply chain. Westudied all 36 problems created by all possible combinations ofthe following parameters: w3 ¼ f16g; c1 ¼ f10:5g; c2 ¼ f12g;

p¼ f18g; b¼ f5g; s¼ f3g; g¼ f0:2;0:3;0:4g; Z¼ f0:2;0:5;0:8g;a¼ f0:7;0:9g; and b¼ f0:7;0:9g. g¼ f0:2;0:3;0:4g denotes thatthe marginal delivering cost in the event of a failure is low,moderate and high, respectively. The meanings ofZ¼ f0:2;0:5;0:8g are similar. a¼ f0:7;0:9g denotes a low and highprobability of a common-cause failure not occurring, respectively.

The meanings of b¼ f0:7;0:9g are similar. In these problems,demand was uniformly distributed over [300,700] and thus thecorresponding mean demand was 500.

All the different problems and the computational results of thecentralized supply chain are listed in Table 1. The computationalresults of the decentralized supply chain with non-cooperative

are listed in Table 2. The computational results of thedecentralized supply chain with cooperative are listed in Table 3.

From the computational results in Table 1, the followingobservations can be made:

� The total profit of the supply chain decreases as g increases.� The order quantity from supplier 2 increases as g increases.

However, the order quantity from supplier 1 decreases. Thisindicates that the larger the total proportion of the marginaldelivery cost in the event of a failure, the more important issupply stability.� If the supply stability of supplier 1 is high, the advantage of his

low cost is obvious. But if supply stability is too low, theadvantage does not exist any more.

From the computational results in Tables 2 and 3, the followingobservations can be made:

� If the two suppliers are cooperative, supplier 1 will set itswholesale price high enough to compel the retailer to sourceonly from supplier 2, who sets its wholesale price substantiallyhigher than the equilibrium wholesale price. Hence the sum ofthe suppliers’ profits increases and the profit of the retailerdecreases at the same time.� When the two suppliers are cooperative, the total profit

of the whole supply chain is lower than the profit whenthey are competitive. This indicates that cooperationof the suppliers does not necessarily lead to supply chainefficiency.

Table 2Computational results of the decentralized supply chain with competitive suppliers.

Prob. w�1 w�2 Q�1 Q�2 P�r P�s1 P�s2 P�t

1 11.4986 12.9600 43.4286 456.0000 849.1680 7.2960 207.9360 1064.4

2 j j j j j j j j

3 11.4033 12.8267 49.7778 464.8889 1364.7 16.7253 312.4053 1693.8

4 j j j j j j j j

5 11.9357 13.1657 43.4286 456 849.1680 7.2960 207.9360 1064.4

6 j j j j j j j j

7 11.6500 12.8800 49.7778 464.8889 1364.7 16.7253 312.4053 1693.8

8 j j j j j j j j

9 12.5914 13.4743 43.4286 456.0000 849.1680 7.2960 207.9360 1064.4

10 j j j j j j j j

11 12.0200 12.9600 49.7778 464.8889 1364.7 16.7253 312.4053 1693.8

12 j j j j j j j j

13 J J J J J J J J

14 j j j j j j j j

15 10.9200 13.1250 69.6667 442.3333 1183.5 2.1945 416.0145 1601.7

16 j j j j j j j j

17 J J J J J J J J

18 j j j j j j j j

19 11.4750 13.2450 69.6667 442.3333 1183.5 2.1945 416.0145 1601.7

20 j j j j j j j j

21 J J J J J J J J

22 j j j j j j j j

23 12.0300 13.3650 69.6667 442.3333 1183.5 2.1945 416.0145 1601.7

24 j j j j j j j j

25 J J J J J J J J

26 j j j j j j j j

27 J J J J J J J J

28 j j j j j j j j

29 J J J J J J J J

30 j j j j j j j j

31 J J J J J J J J

32 j j j j j j j j

33 J J J J J J J J

34 j j j j j j j j

35 J J J J J J J J

36 j j j j j j j j



Comparing the computational results in Tables 1–3, thefollowing observations can be made:

� If both suppliers are selected in the centralized supplychain, they will also be placed with positive order quan-tities when they are competitive in the decentralized supplychain.

� In the decentralized supply chain with non-cooperative orcooperative suppliers, the total profit of the supply chain is lessthan that of the centralized supply chain. Moreover, thewholesale prices are higher than the corresponding wholesaleprices at which the best performance of the whole supplychain can be achieved.� Under the three scenarios, the impact of Z on the supply chain

is not as significant as the impact of g. Only the impact on thewholesale prices is obvious.

6. Conclusions and further research

In this paper we considered both upstream and downstreamuncertainty in determining appropriate sourcing strategies forretailers and pricing strategies for suppliers. By assuming thatdemand is uniformly distributed, we derived the optimal orderquantities from different suppliers. This allows the retailer toexamine the critical trade-off between the low-cost supplier’sreliability versus its cost advantage relative to the other suppliers.We derived a sufficient condition for the existence of anequilibrium price in a decentralized system when the suppliersare competitive. Based on the assumption of a uniform demanddistribution, we obtained an explicit form of the solutions whenthe suppliers are competitive. We also constructed a coordinationmechanism to maximize the profits of the suppliers. Thesefindings can guide suppliers to find a trade-off between orderquantity and wholesale price and a tradeoff between theprobability of on-time delivery and the marginal cost of delivery.Comparing with the benchmark scenario, i.e., a centralized supplychain, we found that it is difficult to achieve full coordination bywholesale-price-only contracts. How to devise a mechanism tocoordinate the whole channel is a potential topic for futureresearch.

Acknowledgements

This research was supported in part by MADIS and theNational Natural Science Foundation of China under Grant nos.70731160635, 70731003, and 70801003. The authors are gratefulto two anonymous referees for their valuable comments andsuggestions that help improve early versions of the paper.

Table 3Computational results of the decentralized supply chain with cooperative

suppliers.

Prob. w�1 w�2 Q�1 Q�2 P�r P�s1 P�s2 P�t

1 14.4700 15.2800 0 440.0000 90.0000 0 915.2000 1005.2

2 j j j j j j j j

3 15.1367 15.8133 0 440.0000 90.0000 0 1478.4 1568.4

4 j j j j j j j j

5 14.9071 15.4857 0 440.0000 90.0000 0 915.2000 1005.2

6 j j j j j j j j

7 15.3833 15.8667 0 440.0000 90.0000 0 1478.4 1568.4

8 j j j j j j j j

9 15.5629 15.7943 0 440.0000 90.0000 0 915.2000 1005.2

10 j j j j j j j j

11 15.7533 15.9467 0 440.0000 90.0000 0 1478.4 1568.4

12 j j j j j j j j

13 J J J J J J J J

14 j j j j j j j j

15 14.5200 15.6800 0 440.0000 90.0000 0 1425.6 1515.6

16 j j j j j j j j

17 J J J J J J J J

18 j j j j j j j j

19 15.0750 15.8000 0 440.0000 90.0000 0 1425.6 1515.6

20 j j j j j j j j

21 J J J J J J J J

22 j j j j j j j j

23 15.6300 15.9200 0 440.0000 90.0000 0 1425.6 1515.6

24 j j j j j j j j

25 J J J J J J J J

26 j j j j j j j j

27 J J J J J J J J

28 j j j j j j j j

29 J J J J J J J J

30 j j j j j j j j

31 J J J J J J J J

32 j j j j j j j j

33 J J J J J J J J

34 j j j j j j j j

35 J J J J J J J J

36 j j j j j j j j

Appendix A

Proof of Theorem 3.1. We discuss the optimal strategies Q1c and Q2c of the channel based on the following different cases.

Case 1: Q2c rQ1cþQ2c rQ 3c.

Since the first-order derivative of the function in Eq. (3.4) with respect to z is given by

d €P�

c ðzÞ

dz¼

w3; zrQ 3c;

ðpþbÞ � ðpþb� sÞFðzÞ; zZQ 3c;

(ðA:1Þ

we obtain the first-order partial derivatives of _PcðQ1c ;Q2cÞ with respect to Q1c and Q2c in this case as follows:

@ _PcðQ1c ;Q2cÞ

@Q1c¼ abw3 � ðab� abgþgÞc1; ðA:2Þ

@ _PcðQ1c ;Q2cÞ

@Q2c¼ aw3 � ða� agþgÞc2: ðA:3Þ

From Eqs. (A.2) and (A.3), we reach the following conclusions:

1. If abw3 � ðab� abgþgÞc1Z0 and aw3 � ða� agþgÞc2Z0, then the expected revenue will increase as the order quantities Q1 and Q2

increase. Hence, Q�1cþQ�2c ZQ 3c.



2. If abw3 � ðab� abgþgÞc1o0 and aw3 � ða� agþgÞc2o0, then the expected revenue will increase as the order quantities Q1 and Q2

decrease. Hence, Q�1c ¼ Q�2c ¼ 0.3. If abw3 � ðab� abgþgÞc1o0 and aw3 � ða� agþgÞc2Z0, then the expected revenue will increase as Q1 decreases or as Q2 increases.

Hence, Q�1c ¼ 0 and Q�2c Z Q 3c .4. If abw3 � ðab� abgþgÞc1Z0 and aw3 � ða� agþgÞc2o0, then the expected revenue will increase as Q1 increases or as Q2 decreases.

Hence, Q�1c ZQ 3c and Q�2c ¼ 0.

From conclusions 2–4, it can be observed that there is at least one supplier that is not selected to receive orders when abw3 � ðab�abgþgÞc1o0 and/or aw3 � ða� agþgÞc2o0. The reason is that the supplier’s supply reliability is too low or its delivery cost is too high.

Case 2: Q2c r Q 3c rQ1cþQ2c .

The first-order partial derivatives of _PcðQ1c ;Q2cÞ with respect to Q1c and Q2c in this case are given by

@ _PcðQ1c ;Q2cÞ

@Q1c¼ ab½ðpþbÞ � ðpþb� sÞFðQ1cþQ2cÞ� � ðab� abgþgÞc1; ðA:4Þ

and

@ _PcðQ1c ;Q2cÞ

@Q2c¼ ab½ðpþbÞ � ðpþb� sÞFðQ1cþQ2cÞ�það1� bÞw3 � ða� agþgÞc2: ðA:5Þ

It is straightforward to verify that the Hessian matrix of _PcðQ1c ;Q2cÞ is negative definite. Hence, _PcðQ1c ;Q2cÞ is jointly concave with

respect to Q1c and Q2c. The optimal order quantity can be easily deduced via the first-order optimality condition.

From the assumption Q2c rQ 3c rQ1cþQ2c and the analysis in Case 1, it is straightforward to deduce that abw3 � ðab� abgþgÞc1Z0.

So we have

@ _PcðQ1c ;Q2cÞ

@Q1c

��Q1c þQ2c ¼ Q 3c

¼ abw3 � ðab� abgþgÞc1Z0:

Moreover, we have

@ _PcðQ1c ;Q2cÞ

@Q1c

��Q1c þQ2c ¼ þ1

¼ abs� ðab� abgþgÞc1o0;

and FðxÞ is strictly increasing. Hence @ _PcðQ1c;Q2cÞ=@Q1c has an unique zero point as follows:

ðQ1cþQ2cÞ�¼ F�1 abðpþbÞ � ðab� abgþgÞc1

abðpþb� sÞ

� �: ðA:6Þ

Substituting Eq. (A.6) into Eq. (A.5), we deduce that

@ _PcðQ1c ;Q2cÞ

@Q2c¼ aw3 � ða� agþgÞc2 � abw3þðab� abgþgÞc1: ðA:7Þ

So we have the following conclusions:

1. If abw3 � ðab� abgþgÞc14aw3 � ða� agþgÞc2, i.e., @ _PcðQ1c;Q2cÞ=@Q2c o0, then the expected revenue will increase as Q2 decreases.Hence, Q�1c ¼ ðQ1cþQ2cÞ

� and Q�2c ¼ 0.2. If abw3 � ðab� abgþgÞc1raw3 � ða� agþgÞc2, i.e., @ _PcðQ1c ;Q2cÞ=@Q2c Z0, then the expected revenue will increase as Q2 increases.

Hence, Q�1cþQ�2c ¼ ðQ1cþQ2cÞ� and Q�2c Z Q 3c .

Case 3: Q 3c rQ2c rQ1cþQ2c .

The first-order partial derivatives of _PcðQ1c ;Q2cÞ with respect to Q1c and Q2c in this case are given by

@ _PcðQ1c ;Q2cÞ

@Q1c¼ ab½ðpþbÞ � ðpþb� sÞFðQ1cþQ2cÞ� � ðab� abgþgÞc1; ðA:8Þ

@ _PcðQ1c ;Q2cÞ

@Q2c¼ aðpþbÞ � ðpþb� sÞ½abFðQ1cþQ2cÞþað1� bÞFðQ2cÞ� � ða� agþgÞc2: ðA:9Þ

It can be deduced that the Hessian matrix of _PcðQ1c ;Q2cÞ is negative definite. The optimal order quantity can be easily deduced via the

first-order derivatives.

The unique optimal total order quantity is deduced as follows:

ðQ1cþQ2cÞ�¼ F�1 abðpþbÞ � ðab� abgþgÞc1

abðpþb� sÞ

� �: ðA:10Þ

Substituting Eq. (A.10) into Eq. (A.9), it is straightforward to deduce the unique optimal order quantity from supplier 2 as follows:

Q�2c ¼ F�1 að1� bÞðpþbÞþðab� abgþgÞc1 � ða� agþgÞc2


� �: ðA:11Þ



Furthermore, we have

Q�1c Z03Q�2c r ðQ1cþQ2cÞ�3ðab� abgþgÞc1rbða� agþgÞc2: ðA:12Þ

From the above analysis of the three different cases, we reach the conclusions about the optimal sourcing strategy of the centralized

supply chain. &

Proof of Theorem 4.3. Obviously, only if both suppliers are placed with positive order quantities is it possible for them to obtain apositive expected profit. If both suppliers are placed with positive order quantities, the expected revenue function of supplier 1 in stage 0,denoted by Ps1

ðw1;w2Þ, is given by

Ps1ðw1;w2Þ ¼ ½abw1 � ðab� abZgþZgÞc1�Q

�1d: ðA:13Þ

Similarly, the expected revenue function of supplier 2 in stage 0, denoted by Ps2ðw2;w1Þ, is given by

Ps2ðw2;w1Þ ¼ ½aw2 � ða� agZþgZÞc2�Q

�2d: ðA:14Þ

From Eqs. (A.13) and (A.14), we observe that w1 and w2 should be subject to w14 ððab� abZgþZgÞ=abÞc1 and w24 ðða� agZþgZÞ=aÞc2 ifboth suppliers seek to obtain a positive expected profit.

Along with the results in Theorem 4.1, we obtain the following conclusions when supplier 1 adopts a different wholesale price for any

given w2A ½ðða� agZþgZÞ=aÞc2;w3 � ðð1� aÞð1� ZÞgc2=aÞ�.

1. If ððab� abZgþZgÞ=abÞc1ow1ow3 � ðð1� abÞð1� ZÞgc1=abÞ � ðaðw3 �w2Þ � ð1� aÞð1� ZÞgc2=abÞ, then Q�1d40 and Q�2d ¼ 0.2. If w3 � ðð1� abÞð1� ZÞgc1=abÞ � ðaðw3 �w2Þ � ð1� aÞð1� ZÞgc2=abÞow1ow2þðð1� aÞð1� ZÞgc2=aÞ � ðð1� abÞð1� ZÞgc1=abÞ, then

Q�1d40 and Q�2d40.3. If w2þðð1� aÞð1� ZÞgc2=aÞ � ðð1� abÞð1� ZÞgc1=abÞow1rw3 � ðð1� abÞð1� ZÞg=abÞc1, then Q�1d ¼ 0 and Q�2d40.

Similarly, there are the following conclusions when supplier 2 adopts a different wholesale price for any given: w1A ½ððab�abZgþZgÞ=abÞc1;w3 � ðð1� abÞð1� ZÞg=abÞc1�.

4. If ðða� agZþgZÞ=aÞc2ow2ow1þðð1� abÞð1� ZÞgc1=abÞ � ðð1� aÞð1� ZÞgc2=aÞ, then Q�1d ¼ 0 and Q�2d40.5. If w1þðð1� abÞð1� ZÞgc1=abÞ � ðð1� aÞð1� ZÞgc2=aÞow2ow3 � ðð1� aÞð1� ZÞgc2=aÞ � ðabðw3 �w1Þ � ð1� abÞð1� ZÞgc1=aÞ, then

Q�1d40 and Q�2d40.6. If w3 � ðð1� aÞð1� ZÞgc2=aÞ � ðabðw3 �w1Þ � ð1� abÞð1� ZÞgc1=aÞow2ow3 � ðð1� aÞð1� ZÞgc2=aÞ, then Q�1d40 and Q�2d ¼ 0.

We observe that if ððab� abZgþZgÞ=abÞc1ow3 � ðð1� abÞð1� ZÞgc1=abÞ � ðaðw3 �w2Þ � ð1� aÞð1� ZÞgc2=abÞ, then it is possible that

supplier 1 sets its wholesale price w1 in the interval ½ab� abZgþZg=abc1;w3 � ð1� abÞð1� ZÞgc1=ab� aðw3 �w2Þ � ð1� aÞð1�ZÞgc2=ab� in order to monopolize the market. To avoid this scenario, the countermeasure of supplier 2 is to set its wholesale price w2 such

that ððab� abZgþZgÞ=abÞc14w3 � ð1� abÞð1� ZÞgc1=ab� aðw3 �w2Þ � ð1� aÞð1� ZÞgc2=ab, i.e., w2oð1� bÞw3 � ðð1� aÞð1� ZÞg=aÞc2

þððab� abgþgÞ=aÞc1. Moreover, it is straightforward to verify that ð1� bÞw3 � ðð1� aÞð1� ZÞg=aÞc2þððab� abgþgÞ=aÞc1ow3 � ð1�

aÞð1� ZÞgc2=a� abðw3 �w1Þ � ð1� abÞð1� ZÞgc1=a when w14 ððab� abZgþZgÞ=abÞc1. Hence, the strategy space for supplier 2 is

½ða� agZþgZ=aÞc2; ð1� bÞw3 � ðð1� aÞð1� ZÞg=aÞc2þððab� abgþgÞ=aÞc1�. Obviously, to ensure the existence of this strategy space,

ðða� agZþgZÞ=aÞc2oð1� bÞw3 � ðð1� aÞð1� ZÞg=aÞc2þððab� abgþgÞ=aÞc1, i.e., aw3 � ða� agþgÞc24abw3 � ðab� abgþgÞc1.

Similarly, the feasible strategy space for supplier 1 is ½ððab� abZgþZgÞ=abÞc1; ðða� agþgÞ=aÞc2 � ðð1� abÞð1� ZÞg=abÞc1� if

bða� agþgÞc24ðab� abgþgÞc1. &

Proof of Lemma 4.5. Denote ðpþbÞ � bðpþb� sÞFðQ1þQ2Þ � ð1� bÞðpþb� sÞFðQ2Þ as U2ðQ2;Q1Þ and U2ðQ2;Q1ÞQ2 as R2ðQ2;Q1Þ.

The first-order derivative of R2ðQ2;Q1Þ is given by

dR2ðQ2;Q1Þ

dQ2¼U2ðQ2;Q1ÞþQ2

dU2ðQ2;Q1Þ

dQ2¼U2ðQ2;Q1Þð1� gðQ2jQ1;bÞÞ: ðA:15Þ

The second-order derivative of R2ðQ2;Q1Þ is given by

d2R2ðQ2;Q1Þ

dQ22

¼dU2ðQ2;Q1Þ

dQ2ð1� gðQ2jQ1;bÞÞ � U2ðQ2;Q1Þ

dgðQ2jQ1;bÞdQ2

: ðA:16Þ

Assume that the support of FðxÞ is the interval ½a; bÞ, i.e., 0oFðxÞo1 for xA ½a; bÞ and FðxÞ ¼ 0 for x=2½a; bÞ. Then gðQ2jQ1;bÞ ¼ 0 and

R2ðQ2;Q1Þ ¼ aðpþbÞQ2 for Q2A ½0; aÞ. Define Q 2 as the least upper bound on the set of points such that gðQ2jQ1;bÞr1. Since gðQ2jQ1;bÞ ¼ 0

for Q2A ½0; aÞ, Q 2Za.

If dgðQ2jQ1;bÞ=dQ2Z0, i.e., gðQ2jQ1;bÞ is weakly increasing, then gðQ2jQ1;bÞZ1 for Q2A ½Q 2;1Þ and dR2ðQ2;Q1Þ=dQ2r0 for

Q2A ½Q 2;1Þ. It can be deduced that R2ðQ2;Q1Þ is decreasing for Q2A ½Q 2; þ1Þ. Hence, Ps2ðQ2;Q1Þ ¼ aR2ðQ2;Q1Þ � ða� agþgÞc2Q2 is

decreasing for Q2A ½Q 2; þ1Þ, too.

Note that gðQ2jQ1;bÞr1 for Q2A ½a;Q 2Þ and w2ðQ2;Q1Þ is decreasing in Q2. Hence d2R2ðQ2;Q1Þ=dQ22r0 for Q2A ½a;Q 2Þ. Then R2ðQ2;Q1Þ is

concave for arQ2rQ 2. Hence it is straightforward to deduce that Ps2ðQ2;Q1Þ ¼ aR2ðQ2;Q1Þ � ða� agþgÞc2Q2 is concave for Q2A ½a;Q 2Þ,

too.



In addition, it is obvious that Ps2ðQ2;Q1Þ is linear and strictly increasing on ½0; aÞ.

From the above analysis, it is straightforward to deduce that Ps2ðQ2;Q1Þ is unimodal for Q2A ½0; þ1Þ.

The unimodality of Ps1ðQ1;Q2Þ can be proved by letting Q2 ¼Q1, Q1 ¼ 0 and b¼ 1 in Eq. (4.8). &

Proof of Theorem 4.8. We first assume that aðw3 �w2Þ � ð1� aÞð1� ZÞgc24abðw3 �w1Þ � ð1� abÞð1� ZÞgc140 and b½aw2þð1�aÞð1� ZÞgc2�4 abw1þð1� abÞð1� ZÞgc1. Under these conditions, the optimal order quantities from suppliers 1 and 2 are given by

Q1 ¼b½aw2þð1� aÞð1� ZÞgc2� � ½abw1þð1� abÞð1� ZÞgc1�

abð1� bÞðpþb� sÞ; ðA:17Þ

and

Q2 ¼aðpþb�w2Þ � ð1� aÞð1� ZÞgc2 � ½abðpþb�w1Þ � ð1� abÞð1� ZÞgc1�

að1� bÞðpþb� sÞ: ðA:18Þ

Substituting Eqs. (A.17) and (A.18) into the revenue functions Eqs. (A.13) and (A.14), we have

Ps1ðw1;w2Þ ¼

b½aw2þð1� aÞð1� ZÞgc2� � ½abw1þð1� abÞð1� ZÞgc1�

abð1� bÞðpþb� sÞ½abw1 � ðab� abZgþZgÞc1�; ðA:19Þ

and

Ps2ðw2;w1Þ ¼

aðpþb�w2Þ � ð1� aÞð1� ZÞgc2 � ½abðpþb�w1Þ � ð1� abÞð1� ZÞgc1�

að1� bÞðpþb� sÞ½aw2 � ða� agZþgZÞc2�: ðA:20Þ

It is straightforward to verify that Eqs. (A.19) and (A.20) are concave with respect to their own decision variables. By setting the first

partial derivative of each player’s revenue function with respect to its own decision variable equal to zero, we obtain the unconstrained

best response function as follows:

w1ðw2Þ ¼abw2þðab� 2abZgþabgþ2Zg� gÞc1þð1� aÞð1� ZÞgbc2

2ab ; ðA:21Þ

and

w2ðw1Þ ¼að1� bÞðpþbÞþabw1þð1� abÞð1� ZÞgc1þða� 2agZþagþ2gZ� gÞc2

2a : ðA:22Þ

Denote the feasible strategy interval of supplier 1 as ½w1;w1�, i.e., w

1¼ ððab� abZgþZgÞ=abÞc1 and w1 ¼ ðða� agþgÞ=

aÞc2 � ðð1� abÞð1� ZÞg=abÞc1. Similarly, denote the feasible strategy interval of supplier 2 as ½w2;w2�. It is straightforward to verify

that w1ðw2ÞZw1

for w2A ½w2;w2�. Hence, Ps1

ðw1;w2Þ is an increasing function of w1 in the interval ½w1;w1ðw2Þ� for any given

w2A ½w2;w2�. Similarly, w2ðw1ÞZw

2and Ps2

ðw2;w1Þ is an increasing function of w2 in the interval ½w2;w2ðw1Þ� for any given

w1A ½w1;w1�. Moreover,

w1ðw2Þrw13w2rw2; ðA:23Þ

and

w2ðw1Þrw23w1rw1; ðA:24Þ

where w29ðg� agþaþgZ� agZþa=aÞc2 � ðab� abgþg=abÞc1 and w19ð1� bÞð2w3 � p� bÞ=bþð2ab� abgþg� abZgþZgÞc1=ab�ðg� agþaÞc2=ab. Hence, we obtain supplier 1’s best response function for any given feasible w2 as follows:

w�1ðw2Þ ¼

abw2þðab� 2abZgþabgþ2Zg� gÞc1þð1� aÞð1� ZÞgbc2

2ab ; w2rw2;

a� agþga c2 �

ð1� abÞð1� ZÞgab c1; w24w2:

8>>><>>>:

ðA:25Þ

Similarly, we obtain the best response function for any given feasible w1 of supplier 2 as follows:

w�2ðw1Þ ¼

að1� bÞðpþbÞþabw1þð1� abÞð1� ZÞgc1þða� 2agZþagþ2gZ� gÞc2

2a; w1rw1;

ð1� bÞw3 �ð1� aÞð1� ZÞg

a c2þab� abgþg

a c1; w14w1:

8>><>>: ðA:26Þ

Since the absolute value of the first derivative of each best response with respect to its own decision variable is less than 1, the best

response mapping is a contraction. Hence the Nash equilibrium is unique.

By letting w1 ¼w1ðw2ðw1ÞÞ and w2 ¼w2ðw1ðw2ÞÞ, all the possible wholesale prices of suppliers 1 and 2 are derived as follows:

w119ðgb� gbZþab2gZ� ab2gþ2ab� 4agZbþ2agbþ4gZ� 2gÞc1þbða� agþgÞc2þabð1� bÞðpþbÞ

abð4� bÞ;

w129abð1� bÞw3þðab� 2abZgþabgþ2Zg� gþab2

� ab2gþbgÞc1

2ab;

w139a� agþg

a c2 �ð1� abÞð1� ZÞg

ab c1;



and

w219ðgb� gbZþabZg� abgþ2a� 4agZþ2agþ4gZ� 2gÞc2þðab� abgþgÞc1þ2að1� bÞðpþbÞ

að4� bÞ;

w229að1� bÞðpþbÞþða� 2agZþagþ2gZ� gþab� abgþbgÞc2

2a ;

w239ð1� bÞw3 �ð1� aÞð1� ZÞg

ac2þ

ab� abgþga

c1:

It is straightforward to verify that all the possible values of supplier 1 are larger than w1

and all the possible values of supplier 2 are larger

than w2. Jointly considering the constraints, we obtain the unique Nash equilibrium strategy of the game between the suppliers as

follows:

ðw�1n;w�2nÞ ¼


w1 and w21rw2V

w2;

ðw12;w23Þ; w1Zw124w1 and w23rw2V

w2;


w1 and w2Zw224w2;

ðw13;w23Þ; w1Zw134w1 and w2Zw234w2:

8>>>><>>>>:

&

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Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption

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