Int. J. Production Economics 138 (2012) 68–75

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Int. J. Production Economics

0925-52

doi:10.1

n Corr

South C

Tel.: þ8

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journal homepage: www.elsevier.com/locate/ijpe

Coordination of supply chain with a revenue-sharing contractunder demand disruptions when retailers compete

Wei-Guo Zhang a, Junhui Fu a,n, Hongyi Li b, Weijun Xu a

a School of Business Administration, South China University of Technology, Guangzhou 510640, Chinab Department of Decision Sciences and Managerial Economics, Faculty of Business Administration, Chinese University of Hong Kong, Shatin, NT, Hong Kong

a r t i c l e i n f o

Article history:

Received 9 September 2010

Accepted 20 February 2012Available online 8 March 2012

Keywords:

Supply chain management

Revenue-sharing contract

Demand disruptions

Game theory

73/$ - see front matter & 2012 Elsevier B.V. A

016/j.ijpe.2012.03.001

esponding author. Postal address: School

hina University of Technology, Guangzhou, 5

6 20 87114121; fax: þ86 20 22236282.

ail address: fujh02@gmail.com (J. Fu).

a b s t r a c t

This paper investigates how to coordinate a one-manufacturer–two-retailers supply chain with

demand disruptions by revenue-sharing contracts. Firstly, we study the coordination of the supply

chain without demand disruptions and give the feasible revenue-sharing contracts, which assure the

desirability of the chain partners and the legality of selling. Next, we discuss how the supply chain is

coordinated under one demand disruption. In the case, we analyze the effects of demand disruptions on

the centralized supply chain and derive the coordinating revenue-sharing contracts. We also extend the

theoretical results to the case of two demand disruptions. We find that it is harmful for the chain

partners to keep the original revenue-sharing contracts without demand disruptions when there are

demand disruptions. It is necessary to adjust the original revenue-sharing contracts to demand

disruptions. Finally, some numerical examples are given to illustrate the theoretical results.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

There are many unexpected changes of the market demand inthe real world. For example, the outbreak of swine flu can cause alarge sudden demand for disinfectors and vaccines and harmobviously the demand of pork consumption. Similarly, a series ofaccidents caused by the Toyota car break systems will lower thedemand for Toyota vehicles. All the retailers, wholesalers andmanufacturers in the entire supply chain will be severely affectedby these demand disruptions, which could make the originalcoordination scheme designed for situations with no demanddisruptions invalid. Therefore, the issue of the coordination ofsupply chain management under demand disruptions has been anactive research topic in recent years.

A large amount of research has been conducted on supply chaindisruption management. Yu (1998) pioneered the topic and offeredsome customized solutions to certain scheduling disruptions.Thengvall et al. (2000) and Yu et al. (2003) successfully applieddisruption management to the airline industry. Yu and Qi (2004)gave an overview of general applications for disruption manage-ment in the supply chain. Golany et al. (2002) proposed aninteractive Goal Programming to address various types of disrup-tions. Xu et al. (2003) showed how to effectively handle demand

ll rights reserved.

of Business Administration,

10641, P.R. China.

variations in a one-supplier–one-retailer supply chain system withnonlinear demand functions. Qi et al. (2004) studied a one-supplier–one-retailer supply chain coordination problem withdemand disruptions during the planning horizon. Xia et al. (2004)proposed a general disruption management approach for a two-stage production and inventory system. Xu et al. (2006) studiedhow to model the production cost disruptions and their impacts onthe supply chain, and discussed how to design coordinationschemes under disruptions. Yang et al. (2005) considered theproblem of recovering a production plan for the general-cost caseand the convex-cost case after the demand and cost are disrupted.Huang et al. (2006) adopted a novel exponential market demandfunction and investigated the coordination of a supply chaincomprised of one supplier and one retailer under market demanddisruptions with appropriate price discount policies. Lei et al.(2011) studied the risk management strategies in supplier chainwhen the disruptions of demand and cost are the private informa-tion and used linear contract menus to analyze the supply chainunder demand and cost disruptions with asymmetric information.

The above papers mainly focused on the coordination mechan-isms of a supply chain with one supplier and one retailer. Theseresearches did not consider how to coordinate the disruptedsupply chains with multiple retailers. Chen et al. (2001) investi-gated the coordination mechanisms for a distribution system withone supplier and multiple independent retailers. As an extensionof Chen et al. (2001), Bernstein and Federgruen (2003) discussedpricing and replenishment strategies in a distribution systemwith competing retailers. Ingene and Parry (1995) studied the

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–75 69

coordination of a supply chain with one manufacturer and tworetailers competing in price and examined the implications of alinear quantity discount schedule and a two-part tariff. Iyer(1998) discussed the case how manufacturers coordinate thedistribution channels when two retailers compete in price andat the same time in other non-price factors, for example, theprovision of product information, faster check-out, and free repairor after-sales services. Xiao et al. (2005) studied the coordinationof a supply chain system with one manufacturer and twocompeting retailers when there are demand disruptions.A price-subsidy rate contract is considered to coordinate theinvestments of the competing retailers with sales promotionopportunities and demand disruptions. Xiao and Yu (2006)proposed an indirect evolutionary game model with two-verti-cally integrated channels to study the retailers’ evolutionarilystable strategies (ESS) in the quantity-setting duopoly competi-tion with homogeneous products. Xiao et al. (2007) studied thecoordination of the supply chain with demand disruptions in thecases of a linear quantity discount schedule or an all-unit quantitydiscount schedule. They also considered different scenarios of theproblem: the production deviation cost was borne by the retailersor the manufacturer. Xiao and Qi (2008) investigated how tocoordinate a supply chain with one manufacturer and twocompeting retailers using an all-unit quantity discount or anincremental quantity discount after the production cost of themanufacturer was disrupted. Chen and Xiao (2009) consideredtwo coordination schedules, a linear quantity discount scheduleand a Groves wholesale price schedule, to coordinate a supplychain consisting of one manufacturer, one dominant retailer andmultiple fringe retailers after demand disruptions.

However, the revenue-sharing contract has not been used inthe study of the coordination problem of a disrupted supply chainwith one supplier and multiple competing retailers in the existingliterature. In this paper, we analyze the effects of demanddisruptions on the supply chain with one manufacturer and twocompeting retailers and investigate how to coordinate the supplychain with demand disruptions by revenue-sharing contracts.

The rest of the paper is organized as follows. Section 2introduces the basic coordination model when two retailerscompete in a Bertrand market, and discusses how the supplychain is coordinated by revenue-sharing contracts. Section 3studies the coordination mechanism of a supply chain withrevenue-sharing contracts after one demand is disrupted.Section 4 investigates the coordination mechanism for a supplychain with a revenue-sharing contract when there are twodemand disruptions. Some numerical examples are given inSection 5 to elaborate our results. Finally, Section 6 providesconclusions.

2. The coordination of a general model

Ingene and Parry (1995) considered a supply chain model withone manufacturer and two competing retailers. The commoditiesare produced by the manufacturer with a unit cost c0 and are soldto retailer i at a unit price wi. After purchasing the commoditiesfrom the manufacturer, retailer i adds some values to thecommodities with a unit cost ci(i¼1, 2). Then retailer i offersthe final commodities to consumers with the retail price pi.In particular, we assume that the market demand for retailer i

is as the following:

qi ¼ ai�piþdpj, 0odo1, i,j¼ 1,2, ja i, ð1Þ

where ai represents the market scale for retailer i (a1aa2) andd represents the degree of substitutability between retailers.The parameter ai measures the maximum possible market demand

for retailer i while the cross-price parameter d is a measure of thesensitivity of the sales of retailer i to the change of the price ofretailer j. For more details, see Ingene and Parry (1995).

Revenue-sharing contracts have been proven to be effective inimproving supply chain performance. Mortimer (2000) estimatedin a rigorous empirical analysis that the adoption of revenue-sharing contracts has increased the industry’s total industryprofits by 7%. Cachon and Lariviere (2005) compared revenue-sharing contracts to several other contracts that enhance channelcoordination, e.g., buy-back contracts, quantity discounts con-tracts and sales-rebate contracts. None of these contracts matchesrevenue-sharing contracts’ ability to coordinate a wide range ofsupply chains. Therefore, we will investigate how to coordinate aone-manufacturer–two-retailers supply chain by revenue-sharingcontracts in this paper.

The revenue-sharing contract is effective (first objective) if itassures supply chain coordination, while it is desirable by thechain partners (second objective) if they each attain profit that isas least as great as they would earn in the model withoutrevenue-sharing contract (Giannoccaro and Pontrandolfo, 2004).

First, we will consider how to achieve the first objective.A revenue-sharing contract usually includes two parameters.The first is the wholesale unit price wi that retailer i pay. The secondis the revenue share of retailer i represented by ji (0ojio1).

According to the above descriptions, we know that the profitfunction of retailer i is

pi ¼ ½jipi�ðwiþciÞ�qi, ð2Þ

where jipi4wiþci. In other words, the profit of a unit productshould be positive. The profit function of the manufacturer is

p0 ¼X2

i ¼ 1

ðwi�c0þð1�jiÞpiÞqi: ð3Þ

The total profit of the centralized supply chain with revenue-sharing contracts is

Tðp1,p2Þ ¼X2

i ¼ 1

ðpi�ci�c0Þqi: ð4Þ

From (4), we can obtain the optimal retail price of retailer i inthe centralized supply chain

pn

i ¼aiþdajþð1�d2

Þðciþc0Þ

2ð1�d2Þ

:

The corresponding optimal order quantity of retailer i is

qn

i ¼ ½ai�ciþdcj�ð1�dÞc0�=2:

In the decentralized supply chain, the retailers determine theirretail prices to maximize their own profits simultaneously.Solving the first order conditions of (2) with respect to pi, wecan obtain

ji½2pDi �ðaiþdpD

j Þ� ¼ ciþwi, i,j¼ 1,2, ja i: ð5Þ

We know that the decentralized supply chain is coordinatedwhen the Nash equilibrium retail prices are equivalent to theircorresponding optimal retail prices of the centralized supplychain. Let pD

i ¼ pn

i be given, we have

wn

i ¼jn

i ½2pn

i �ðaiþdpn

j Þ��ci, i,j¼ 1,2, ja i:

With respect to the second objective, a market-like setting with-out the revenue-sharing contracts is adopted. In the market-likesetting, the system would work as follows. The manufacturerproduces the commodities at a unit cost c0 and sell them toretailer i at a unit price wim (wim4c0). Based on the marketdemand and the retail price pim, retailer i orders the quantity qim

(i¼1, 2)and his unit cost is ci.

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–7570

In the market-like setting, the profit function of retailer i is

pim ¼ ½pim�ðwimþciÞ�qim:

The profit function of the manufacturer is

p0m ¼X2

i ¼ 1

ðwim�c0Þqim, ð6Þ

where qim¼ai�pimþpjm, i,j¼1,2, ja i.The total profit of the supply chain in the market-like setting is

Tm ¼X2

i ¼ 1

ðpim�ci�c0Þqim:

By solving the first-order condition qpim/qpim¼0, we canobtain the optimal retail price of retailer i in the market-likesetting:

pim ¼ ½2ðwimþaiþciÞþdðwjmþajþcjÞ�=ð4�d2Þ, i, j¼ 1,2, ja i:

ð7Þ

Substitute (7) into (6) and solve the first order conditions of (6)with respect to wim, we can obtain the optimal wholesale unitprice wn

im:

wn

im ¼aiþdajþðc0�ciÞð1�d2

Þ

2ð1�d2Þ

, i, j¼ 1,2, ja i:

The optimal retail prices pn

im and the corresponding optimalorder quantities qn

im in the market-like setting are

pn

im ¼ ½2ðwn

imþaiþciÞþdðwn

jmþajþcjÞ�=ð4�d2Þ,

qn

im ¼ ðai�pn

imþdpn

jmÞ, i, j¼ 1,2, ja i:

Based on the above discussions, we can derive Theorem 1.

Theorem 1. The following revenue-sharing contracts can coordinate

the supply chain with competing retailers:

wn

i ¼jn

i ½2pn

i �ðaiþdpn

j Þ��ci, ð8Þ

maxciþc0

2pn

i �ðaiþdpn

j Þ,½pn

im�ðwn

imþciÞ�qn

im

ðaiþdpn

j �pn

i Þqn

i

!ojn

i o1, ð9Þ

X2

i¼ 1

ja i

jn

i ðaiþdpn

j�pn

i Þqn

i oX2

i ¼ 1

ðpn

i �ci�c0Þqn

i �X2

i ¼ 1

ðwn

im�c0Þqn

im, ð10Þ

Proof. With respect to the first objective, the revenue-sharingcontracts have to be designed such that pD

1 ¼ pn

1 and pD2 ¼ pn

2.Substitute them into (5), we can get (8).

Because 0ojn

i o1 and it is illegal to sell below cost in practice(i.e.wn

i 4c0), the retailer’s revenue share jn

i must satisfy

ciþc0

2pn

i �ðaiþdpn

j Þojn

i o1: ð11Þ

The second objective is to analyze how the contract can bedesigned to satisfy win–win condition for the chain partners.Assume that pn

k is the optimal profit of the actor k (k¼0, 1, 2) inthe supply chain with the revenue-sharing contracts and pn

km isthe optimal profit of the actor k (k¼0, 1, 2) in the market-likesetting. We know that the revenue-sharing contracts are accep-table to the chain partners only if pn

k 4pn

km. Then we have

½jn

i pn

i �ðwn

i þciÞ�qn

i 4 ½pn

im�ðwn

imþciÞ�qn

im, i,j¼ 1,2, ja i, ð12Þ

X2

i ¼ 1

½wn

i �c0þð1�jn

i Þpn

i �qn

i 4X2

i ¼ 1

ðwn

im�c0Þqn

im: ð13Þ

Substitute (8) into (12), we can obtain

jn

i 4½pn

im�ðwn

imþciÞ�qn

im

ðaiþdpn

j�pn

i Þqn

i

: ð14Þ

According to (11) and (14), we can get (9). Similarly, (10) canbe derived. &

Based on the above discussions, we find that the supply chaincan be coordinated under the revenue-sharing contracts whenthere are no demand disruptions.

In the following, we will determine whether the supply chaincan be coordinated by the revenue-sharing contracts when thedemands are disrupted. The demand disruptions usually lead tothe production cost disruption. We assume that the productioncost may be disrupted to c0þDc40.

3. Coordination of supply chain with one demand disruption

When there is only one demand disruption, the originalcoordination scheme under no demand disruptions may becomeinvalid. Therefore, the original coordination mechanism needs tobe adjusted. In this section, we will study how to coordinate thesupply chain after one demand is disrupted. We assume thatthere is a shock Da to the market demand for retailer 1 and thereal market scale a1 is now a1þDa40. Note that a positiveDa represents an increased market demand, and a negativeDa represents a decreased market demand. The shock to thedemand will generate a deviation cost which is new to the supplychain and can be borne by either the retailers or the manufac-turer. However, a manufacturer often employs a return policy inwhich the retailers will return the unsold goods. This means thatthe manufacturer has to bear the deviation cost fully.

When one demand disruption occurs, the real market demandsfor retailers are

q1 ¼ a1þDa�p1þdp2, q2 ¼ a2�p2þdp1:

The total production deviation quantity is

DQ ¼ q1þq2�qn

1�qn

2 ¼ a1þDaþa2�ð1�dÞðp1�p2Þ�qn

1�qn

2:

Demand disruptions usually cause a deviation cost that isassociated with the total production deviation quantity (Xiaoet al., 2007; Xiao and Qi, 2008). When DQ40, more productsneed to be produced to meet the unplanned increased marketdemand. The producer will spend more in machines, labor inputand raw materials, which will cause an extra cost. When DQo0,an excess supply which generate some leftover inventory thatresult in some extra holding costs. Therefore, we define a unitpenalty cost cu40 for the increased production and similarly aunit penalty cost cs40 for the decreased production. The totalprofit of the centralized supply chain is

T ¼X2

i ¼ 1

ðpi�ci�c0�DcÞqi�cuðDQ Þþ�csð�DQ Þþ : ð15Þ

By solving the Kuhn–Tucker condition of (15), we obtain thefollowing results.

Corollary 1. When one demand disruption occurs, the optimal retail

prices and the corresponding optimal order quantities in the cen-

tralized supply chain are

(1)

if Da=2ð1�dÞ�DcZcu, then

pn

1 ¼ pn

1þDa

2ð1�d2Þþ

cuþDc

2, pn

2 ¼ pn

2þdDa

2ð1�d2Þþ

cuþDc

2,

ð16Þ

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–75 71

qn

1 ¼ qn

1þDa

2�ð1�dÞðcuþDcÞ

2, qn

2 ¼ qn

2�ð1�dÞðcuþDcÞ

2

(2)

if �csoDa=2ð1�dÞ�Dcocu, then

pn

1 ¼ pn

1þð3þdÞDa

4ð1�d2Þ

, pn

2 ¼ pn

2þð1þ3dÞDa

4ð1�d2Þ

, ð17Þ

qn

1 ¼ qn

1þDa

4, qn

2 ¼ qn

2�Da

4

(3)

if Da=2ð1�dÞ�Dcr�cs, then

pn

1 ¼ pn

1þDa

2ð1�d2ÞþDc�cs

2, pn

2 ¼ pn

2þdDa

2ð1�d2ÞþDc�cs

2,

ð18Þ

qn

1 ¼ qn

1þDa

2�ð1�dÞðDc�csÞ

2, qn

2 ¼ qn

2�ð1�dÞðDc�csÞ

2

From the above, we can see that one demand disruption hasthe following impact on the centralized supply chain.

If �csrDa/2(1�d)�Dcrcu, the supply chain should hold theline on production quantity (qn

1þqn

2 ¼ qn

1þqn

2) to avoid the pro-duction deviation cost and change the retail prices to offset theeffect of the demand disruption. Otherwise, the productionquantity is adjusted to the demand disruption. For instance, themanufacturer should increase production quantity to satisfy theincreased market demand if Da/2(1�d)�DcZcu. In addition,retailer 1 should increase the order quantity (retail price) withan increased demand and decrease the order quantity (retailprice) with a decreased demand.

In the decentralized supply chain, retailer i will select anoptimal price to maximize his profit. It seems that the retailersplay a static game with each other in a Bertrand market. Whenthe manufacturer bears the deviation cost, the profit function ofretailer i is

piðp1,p2Þ ¼ ½jipi�ðwiþciÞ�qi:

Solving the first order conditions of the above equations, theNash equilibrium retail prices satisfy

j1½2pD1�ða1þDaþdpD

2 Þ� ¼w1þc1, j2½2pD2�ða2þdpD

1 Þ� ¼w2þc2:

ð19Þ

With respect to the desirability of the new revenue-sharingcontracts, Revenue-sharing setting without demand disruptionsis adopted. When one demand disruption occurs, the chainpartners keep the original mechanism: retailer i keeps the optimalretail price pn

i and the optimal quantity qn

i , and his unit cost is ci;the manufacturer keep the contract (wn

i ,jn

i ) without demanddisruptions and his production cost is still c0. Then the realmarket demands for retailers 1 and 2 are

qn

1r ¼ a1þDa�pn

1þdpn

2, qn

2r ¼ a2�pn

2þdpn

1:

The profit functions of the retailers and the manufacturer inthe setting are

pir ¼ ½jn

i pn

i �ðwn

i þciÞ�qn

ir ,

p0r ¼X2

i ¼ 1

½ð1�jn

i Þpn

i þwn

i �c0�qn

ir�cuðDQrÞþ�csð�DQrÞ

þ ,

where DQr ¼ qn

1rþqn

2r�qn

1�qn

2.The total profit of the supply chain in the setting is

T r ¼X2

i ¼ 1

ðpn

i �ci�c0�qn

ir�cuðDQrÞþ�csð�DQrÞ

þ :

According to the above descriptions, we can derive Theorem 2.

Theorem 2. If one demand disruption occurs, then the supply chain

is coordinated by the following revenue-sharing contracts with

wn

1 ¼jn

1½2pn

1�ða1þDaþdpn

2��c1, wn

2 ¼jn

2½2pn

2�ða2þdpn

1��c2,

ð20Þ

ðc1þc0þDcÞ

2pn

1�ða1þDaþdpn

2Þ,½jn

1pn

1�ðwn

1þc1Þ�qn

1r

ða1þDaþdpn

2�pn

1Þqn

1

!ojn

1o1, ð21Þ

ðc2þc0þDcÞ

2pn

2�ða2þdpn

1Þ,½jn

2pn

2�ðwn

2þc2Þ�qn

2r

ða2þdpn

1�pn

2Þqn

2

!ojn

2o1, ð22Þ

jn

1ða1þDaþdpn

2�pn

1Þqn

1þjn

2ða2þdpn

1�pn

2Þqn

2oTn

0�p0r , ð23Þ

where

qn

1 ¼ a1þDa�pn

1þdpn

2, qn

2 ¼ a2�pn

2þdpn

1; pn

1 and pn

2 is given

by (16) if Da/2(1�d)�DcZcu, by (17) if �csrDa=2ð1�dÞ�

Dcrcu, or by (18) if Da=2ð1�dÞ�Dcr�cs.

Proof. When the decentralized supply chain is coordinated, we havepD

1 ¼ pn

1 and pD2 ¼ pn

2. Substitute them into (19), we can derive (20).

Because 0ojn

i o1 and wn

i 4c0þDc, jn

1 and jn

2 must satisfy:

c1þc0þDc

2pn

1�ða1þDaþdpn

2Þojn

1o1,c2þc0þDc

2pn

2�ða2þdpn

1Þojn

2o1: ð24Þ

Assume that pn

k and pkr is the optimal profit of the actor k

(k¼0, 1, 2) in the supply chain with the new revenue-sharingcontracts and the original contracts. In order to assure thedesirability of the adopted revenue-sharing contracts by the chainpartners, pn

k should be higher than pkr. Let pn

k 4pkr be given, wehave

½jn

i pn

i �ðwn

i þciÞ�qn

i 4 ½jn

i pn

i �ðwn

i þciÞ�qn

ir , ð25Þ

X2

i ¼ 1

½ð1�jn

i Þpn

i þwn

i �c0�qn

i �cuðDQ Þþ�csð�DQ Þþ4p0r : ð26Þ

Substitute (20) into (25), we can obtain

jn

14½jn

1pn

1�ðwn

1þc1Þ�qn

1r

ða1þDaþdpn

2�pn

1Þqn

1

, jn

24½jn

2pn

2�ðwn

2þc2Þ�qn

2r

ða2þdpn

1�pn

2Þqn

2

: ð27Þ

According to (24) and (27), we can get (21) and (22). Similarly,(23) can be obtained. &

Theorem 2 investigates how to coordinate the supply chain byrevenue-sharing contracts after one demand is disrupted. Com-paring Theorem 2 with Theorem 1, we notice that it is necessaryto adjust the original revenue-sharing contracts in Theorem 1 tothe demand disruption.

4. Coordination of supply chain with two demand disruptions

In Section 3, we considered how to coordinate the supply chainwith a single demand disruption. It is also very likely that due tosome reasons, such as the emergence of new technology and newpolicy, there can be to two demand disruptions. In this section, wewill investigate how to use the revenue-sharing contracts tocoordinate the supply chain with two demand disruptions. Tosimplify the notations, we assume that ai ¼ aiþDaiði¼ 1, 2Þ. Aftertwo demand disruptions, the market demand for retailer i is

qi ¼ ai�piþdpj:

Then the total production deviation quantity is

DQ ¼ q1þq2�qn

1�qn

2 ¼ a1þa2�ð1�dÞðp1�p2Þ�qn

1�qn

2

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–7572

The total profit of the centralized supply chain is

T ¼X2

i ¼ 1

ðpi�ci�c0�DcÞqi�cuðDQ Þþ�csð�DQ Þþ : ð28Þ

From (28), we obtain the optimal retail prices and thecorresponding optimal order quantities in a decentralized supplychain.

Corollary 2. When two demand disruptions occur, the optimal retail

prices and the corresponding optimal order quantities in the

centralized channel are:

(1)

if ðDa1þDa2Þ=2ð1�dÞ�DcZcu, then

pn

i ¼ pn

i þðcuþDcÞ

2þDaiþdDaj

2ð1�d2Þ

, i,j¼ 1,2, ia j, ð29Þ

qn

i ¼ qn

i þDai

2�ð1�dÞðcuþDcÞ

2, i¼ 1,2

(2)

if �cso ðDa1þDa2Þ=2ð1�dÞ�Dcocu, then

pn

i ¼ pn

i þð3þdÞDaiþð1þ3dÞDaj

4ð1�d2Þ

, i,j¼ 1,2, ia j ð30Þ

qn

i ¼ qn

i þðDai�DajÞ

4i,j¼ 1,2, ia j,

(3)

if ðDa1þDa2Þ=2ð1�dÞ�Dcr�cs, Da14�a1, and Da24�a2,then

pn

i ¼ pn

i þðDc�csÞ

2þDaiþdDaj

2ð1�d2Þ

i,j¼ 1,2, ia j, ð31Þ

qn

i ¼ qn

i þDai

2�ð1�dÞðDc�csÞ

2, i¼ 1,2:

From Corollary 2, we know that it is optimal for the supplychain to keep the original quantities if the changed amount9ðDa1þDa2Þ=2ð1�dÞ�Dc9 is sufficiently small. Hence, the supplychain should keep the original quantities to avoid the productiondeviation cost and change the original retail prices to offset theeffect of the demand disruptions. If the changed amount9ðDa1þDa2Þ=2ð1�dÞ�Dc9 is large, the manufacturer should changethe production quantity to satisfy the new market demand.

In a decentralized supply chain, we know that the profitfunction of retailers i is

pi ¼ ½jipi�ðwiþciÞ�qi, ð32Þ

By solving the first-order conditions of (32), the Nash equili-brium retail price in the decentralized supply chain is

ji½2pDi �ðaiþdpD

j Þ� ¼wiþci, i,j¼ 1,2, ja i: ð33Þ

With respect to the desirability of the new revenue-sharingcontracts, Revenue-sharing setting without demand disruptionsis adopted. When two demand disruptions occur, the chainpartners will keep the original mechanism without demanddisruptions. Then the real market demand for retailer i isqn

is ¼ ai�pn

i þdpn

j , i,j¼ 1,2, ja i:

The profit functions of the retailers and the manufacturer inthe setting are

pis ¼ ½jn

i pn

i �ðwn

i þciÞ�qn

is,

p0s ¼X2

i ¼ 1

½ð1�jn

i Þpn

i þwn

i �c0�qn

is�cuðDQsÞþ�csð�DQsÞ

þ ,

where DQs ¼ qn

1sþqn

2s�qn

1�qn

2.

The total profit of the centralized supply chain in the setting is

T s ¼X2

i ¼ 1

½pn

i �ci�c0�qn

is�cuðDQsÞþ�csð�DQsÞ

þ :

According to the above discussions, we can derive Theorem 3.

Theorem 3. If two demand disruptions occur, then the supply chain

is coordinated by the following revenue-sharing contracts with:

wn

i ¼jn

i ½2pn

i �ðaiþdpn

j Þ��ci, ð34Þ

ðciþc0þDcÞ

2pn

i �ðaiþdpn

j Þ,½jn

i pn

i �ðwn

i þciÞ�qn

is

ðaiþdpn

j�pn

i Þqn

i

!ojn

i o1, ð35Þ

X2

i ¼ 1

jn

i ðaiþdpn

j�pn

i Þqn

i oTn�p0s, ð36Þ

where qn

1 ¼ a1þDa�pn

1þdpn

2, qn

2 ¼ a2�pn

2þdpn

1, pn

1 and pn

2 are

given by (29) if Da=2ð1�dÞ�DcZcu, by (30) if �csrDa=2ð1�dÞ�

Dcrcu, or by (31) if Da=2ð1�dÞ�Dcr�cs:

Proof. When the supply chain is coordinated, we have pD1 ¼ pn

1

and pD2 ¼ pn

2. Substitute them into (33), we easily obtain (34).

Because 0ojn

i o1 and wn

i 4c0þDc, jn

1 and jn

2 must satisfy:

ðciþc0þDcÞ

2pn

i �ðaiþdpn

j Þojn

i o1, ð37Þ

In order to assure that the revenue-sharing contracts areacceptable to all the chain partners, pn

k should be higher than pn

ks.Then we have

½jn

i pn

i �ðwn

i þciÞ�qn

i 4 ½jn

i pn

i �ðwn

i þciÞ�qn

is, ð38Þ

X2

i ¼ 1

½ð1�jn

i Þpn

i þwn

i �c0�qn

i �cuðDQ Þþ�csð�DQ Þþ4p0s: ð39Þ

Substitute (34) into (38), we can get

½jn

i pn

i �ðwn

i þciÞ�qn

is

ðaiþdpn

j�pn

i Þqn

i

ojn

i ð40Þ

From (37) and (40), we can derive (35). Likewise, (36) can beobtained. &

Theorem 3 gives the coordination mechanism of the decen-tralized supply chain with two demand disruptions. Similar toTheorem 2, Theorem 3 indicates that it is necessary to adjust theoriginal revenue-sharing contracts to the demand disruptions.Furthermore, Theorem 2 is a special case of Theorem 3 when themarket demand for retailer 2 is not disrupted (i.e. when Da2¼0).

5. Numerical examples

In the above sections, we discuss theoretically how to coordi-nate the supply chain in the different scenarios of demanddisruptions. To well illustrate the theoretical results, we givesome numerical examples in this section.

Example 1. Coordination of the basic model.

Firstly, we consider the revenue-sharing contracts withoutdemand disruptions. Consider the following example with para-meters: a1¼20, a2¼20, c1¼3, c2¼2, d¼0.5, c0¼5. According toSection 2, the optimal retail prices are pn

1¼24 and pn

2¼23.5, andthe optimal order quantities are qn

1 ¼ 7:75 and qn2 ¼ 8:5.

In the following, we will describe the revenue-sharing con-tracts for assuring the coordination of supply chain, which isdepicted in Figs. 1 and 2. From Fig. 1, we find that only the values

1∗

1w∗

( )21

2 0.47 1

K ∗

∗

=

< <

2∗

2w∗

( )12

1

min (88 60 ) 72, 1 0.47,

0.49 1

K ∗

∗

= − >

< <

�

�

�

�

��

>min (88–72 60, 1 0.49,)

0.2

0.18 0.13 0.470.49 K1K2

0

3

6

9

12

15

3

6

9

12

15

0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Fig. 1. The revenue-sharing contracts at the manufacturer/retailer i interface: wn

i vs. jn

i .

1∗

2∗

1 0.49∗ =

2 0.47∗ =

1 260 72 88∗ ∗+ <

�

�

�

�

� �

00 .2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Fig. 2. The feasible domains for both jn

1 and jn

2.

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–75 73

of wn

i and jn

i along the continuous line can make the supply chainbe coordinated.

From (9) and (10) in Theorem 1, we can obtain the feasibledomains for both jn

1 and jn

2, which is depicted in Fig. 2.The shaded area satisfies the legality issue, i.e., the commoditiesshould be sold above cost. The shaded area also satisfies thedesirability issue, namely let the chain partners earn more than inthe market-like setting. Therefore, the revenue-sharing contractsshould not only assure the coordination of supply chain, but alsoconsider the legality issue and the desirability issue.

Suppose that jn

1 ¼ 0:7 and jn

2 ¼ 0:6. Then we can obtain theassociated parameters under Revenue-sharing setting, which arereported in Table 1. In order to illustrate the efficiency of therevenue-sharing contracts, the market-like setting is adopted andthe associated parameters are depicted in Table 2. FromTables 1 and 2, the profit of the supply chain increases by 12.5%if the revenue-sharing contracts are employed to coordinate thesupply chain. Moreover, the chain partners each attain moreprofits than in the market-like setting, which guarantees thedesirability of them.

Example 2. Coordination of supply chain with one demanddisruption.

In Section 3, we discuss how to coordinate a one-manufac-turer–two-retailers supply chain by revenue-sharing contractswhen one demand is disrupted. Here we give a numericalexample to show the effects of demand disruption on the

coordination mechanism. Consider the example with parameters:a1¼20, a2¼20, c1¼3, c2¼2, d¼0.5, c0¼5, cu¼cs¼2.

Table 3 illustrates the effects of demand disruptions on thecentralized supply chain. When Da�DcZ2(Da�Dcr�2),the supply chain should increase (decrease) the productionquantity to satisfy the increased (poor) market demand. When�2oDa�Dco2, it is optimal to keep the original productionquantity for the supply chain, such as Cases 3 and 4. To keep theoriginal production quantity, the supply chain should change theoriginal retail prices to avoid the production deviation cost.

Let the fraction of the increase of the channel profit due todisruption management be a¼ ðT�TrÞ=Tr . From Table 3, thiseffect of disruption management is very remarkable when theoverall effect of the disruptions is large, i.e., 9Da/2(1�d)�Dc9Z2.

According to Theorem 2, we can get the feasible revenue-sharing contracts under different demand disruptions, which arereported in Table 4. In order to illustrate the desirability issue, wewill select one feasible revenue-sharing contract from each casein Table 4 and compute the profits of the chain partners. These aredepicted in Table 5.

From Table 5, we find it harmful for the chain partners to keepthe original revenue-sharing contracts without demand disruptions.If the new coordinating contracts are employed, the chain partnerswill earn more profits than in the original contracts. Therefore, it isnecessary to adjust the original revenue-sharing contracts.

Example 3. Coordination of supply chain with two demanddisruptions.

In Section 4, we study the coordination of supply chain withtwo-demand disruptions. Now we use an example to show howrevenue-sharing contracts can satisfy win–win condition for thechain partners. Assume the following values of the parameters:a1¼20, a2¼20, c1¼3, c2¼2, c0¼5, d¼0.5, cu¼cs¼2.

Based on Corollary 2, we can illustrate the effects of demanddisruptions on the centralized supply chain, which are depictedin Table 6. When 9Da1þDa2�Dc9o2, the supply chain shouldkeep the original production quantity and change the originalretail prices to offset the effect of the demand disruption.When 9Da1þDa2�Dc9Z2, the supply chain should change theoriginal production quantity and retail prices due to demanddisruptions. Table 6 shows that the total profit of the centralizedsupply chain with disruption management obviously increases if9Da1þDa2�Dc9Z2.

In order to illustrate the desirability issue, we present Table 7.Table 7 reveals that the chain partners benefit from the newcoordinating contracts. Therefore, it is necessary to adjust the originalrevenue-sharing contracts when two demand disruptions occur.

Table 1Parameters under Revenue-sharing setting.

pn

1 pn

2 qn

1 qn

2 jn

1 jn

2 wn

1 wn

2 pn

1 pn

2 pn

0 T*

24.00 23.50 7.75 8.50 0.70 0.60 8.38 7.00 42.04 43.35 178.86 264.25

Table 2Parameters under Market-like setting.

pn

1m pn

2m qn

1m qn

2m wn

1m wn

2m pn

1m pn

2m pn

0m Tn

m

29.27 29.07 5.27 5.57 21.00 21.50 27.74 30.99 176.12 234.84

Table 3The centralized solution under one demand disruption.

Dc Da pn

1 pn

2 qn

1 qn

2 T r T a (%)

Case 1 0.5 �7 18.58 20.42 4.63 8.88 138.25 155.77 12.67

Case 2 0.5 �4 20.58 21.42 6.13 8.88 192.25 195.02 1.44

Case 3 0 �1 22.83 22.67 7.50 8.75 246.25 248.33 0.84

Case 4 0 1 25.17 24.33 8.00 8.25 278.25 280.33 0.75

Case 5 �0.5 4 27.42 25.58 9.38 8.13 320.25 344.27 7.50

Case 6 �0.5 7 29.42 26.58 10.88 8.13 362.25 407.02 12.36

Table 4The revenue-sharing contracts at the feasible domains under one demand disruption.

Dc Da ðwn

1 , jn

1 ,wn

2 , jn

2Þ

0.5 �7 wn

1 ¼ 13:96jn

1�3, wn

2 ¼ 11:54jn

2�2, jn

1 40:61, jn

2 40:65, 21jn

1þ79jn

2 o68

0.5 �4 wn

1 ¼ 14:46jn

1�3, wn

2 ¼ 12:54jn

2�2, jn

1 40:59, jn

2 40:55, 38jn

1þ79jn

2 o68

0 �1 wn

1 ¼ 15:33jn

1�3, wn

2 ¼ 13:92jn

2�2, jn

1 40:65, jn

2 40:57, 56jn

1þ77jn

2 o82

0 1 wn

1 ¼ 17:17jn

1�3, wn

2 ¼ 16:09jn

2�2, jn

1 40:70, jn

2 40:64, 64jn

1þ68jn

2 o97

�0.5 4 wn

1 ¼ 18:04jn

1�3, wn

2 ¼ 17:46jn

2�2, jn

1 40:73, jn

2 40:66, 88jn

1þ66jn

2 o130

�0.5 7 wn

1 ¼ 18:54jn

1�3, wn

2 ¼ 18:46jn

2�2, jn

1 40:68, jn

2 40:66, 118jn

1þ66jn

2 o165

Table 5The profits of chain partners under one demand disruption.

Dc Da jn

1 wn

1 jn

2 wn

2 pn

1 pn

2 pn

0 p1r p2r p0r

0.5 �7 0.68 6.49 0.62 5.16 14.55 48.83 92.39 4.07 43.35 90.83

0.5 �4 0.60 5.68 0.54 4.77 22.51 42.53 129.98 20.34 43.35 128.56

0 �1 0.66 7.12 0.58 6.07 37.13 44.41 166.80 36.62 43.35 166.28

0 1 0.75 9.88 0.65 8.45 48.00 44.24 188.09 47.47 43.35 187.43

�0.5 4 0.78 11.07 0.86 13.01 68.55 56.77 218.94 63.74 43.35 213.16

�0.5 7 0.74 10.72 0.82 13.14 87.52 54.13 265.37 80.02 43.35 238.88

Table 6The centralized solution under two demand disruptions.

Dc Da1 Da2 pn

1 pn

2 qn

1 qn

2 Tn

r Tn a(%)

Case 1 0.5 �2 �2 22.13 21.63 7.13 7.88 191.25 205.81 7.61

Case 2 0.5 0 �2 22.88 22.00 8.13 7.88 227.25 230.48 1.42

Case 3 0.5 �2 0 22.50 22.38 7.13 8.88 228.25 231.27 1.32

Case 4 0 �2 2 24.67 22.83 8.75 7.50 263.25 264.58 0.51

Case 5 0 2 �2 23.33 24.17 6.75 9.50 265.25 266.58 0.50

Case 6 �0.5 2 0 26.08 24.92 8.38 8.13 292.25 304.77 4.28

Case 7 �0.5 0 2 25.42 25.58 7.38 9.13 293.25 305.77 4.27

Case 8 �0.5 2 2 26.75 26.25 8.38 9.13 321.25 338.94 5.51

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–7574

Table 7The profits of the chain partners under two-demand disruptions.

jn

1 wn

1 jn

2 wn

2 pn

1 pn

2 pn

0p1s p2s p0s

0.68 7.50 0.62 6.80 32.40 36.31 137.10 31.19 33.15 126.91

0.64 6.44 0.58 6.45 42.25 33.97 154.26 42.04 33.15 152.06

0.66 7.44 0.56 5.56 31.45 44.11 155.71 31.19 43.35 153.71

0.70 8.14 0.59 7.05 53.59 33.19 177.80 52.89 33.15 177.21

0.69 8.44 0.60 6.80 31.44 54.15 181.00 31.19 53.55 180.51

0.76 10.46 0.67 9.25 53.31 44.23 207.23 52.89 43.35 196.01

0.80 11.43 0.68 9.19 43.51 56.62 205.64 42.04 53.55 197.66

0.76 10.97 0.65 9.13 53.31 54.12 231.51 52.89 53.55 214.81

W.-G. Zhang et al. / Int. J. Production Economics 138 (2012) 68–75 75

6. Conclusions

In this paper, we study the coordination mechanism of asupply chain with one manufacturer and two competing retailers.We mainly focus on the coordination of the supply chain byrevenue-sharing contracts when the demands are disrupted. Twocases are considered: one demand disruption and two demanddisruptions. In each case, we obtain the optimal retail prices andthe corresponding optimal order quantities for the two retailers inthe centralized channel, and then derive revenue-sharing con-tracts that can coordinate the disrupted supply chain. We findthat it is harmful for the chain partners to keep the originalrevenue-sharing contracts without demand disruptions whenthere are demand disruptions. It is necessary to adjust the originalrevenue-sharing contracts to demand disruptions.

The demand disruption management in the supply chain isvery important and more research effect in this area is required.In our future study, we will consider how to coordinate the supplychain by revenue-sharing contracts when multiple retailers com-pete. We will also study the coordination mechanism of thesupply chain when one retailer has a priority to make his decisionfirst. Finally, we will extend our model to consider the case withtwo or more manufacturers.

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos. 70825005 and 71171086), Humanitiesand GDUPS (2010), and New Century Excellent Talents in Uni-versity (no. NCET-10-0401).

Authors would like to thank Qin Mei, Edwin Cheng (the editor),and an anonymous referee for their helpful comments andsuggestions.

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