Coordination of a supply chain with one manufacturer and multiple competing retailers under simultaneous demand and cost disruptions

Int. J. Production Economics 141 (2013) 425–433

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Coordination of a supply chain with one manufacturer and multiple competing retailers under simultaneous demand and cost disruptions Erbao Cao a,b,n, Can Wan c, Mingyong Lai a,b a

College of Economics and Trade, Hunan University, Changsha 410079, China Hunan Province Key Laboratory of Logistics Information and Simulation Technology, Changsha 410079, China c Department of Electrical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong b

a r t i c l e i n f o

abstract

Article history: Received 14 September 2011 Accepted 13 September 2012 Available online 23 September 2012

This paper develops a coordination mechanism for a supply chain consisting of one manufacturer and n Cournot competing retailers when the production cost and demands are simultaneously disrupted. This differs from traditional supply chain coordination models under a static case and the case with only demand or cost disruption. The coordination mechanism with revenue sharing is considered, and the effects of production cost and demand disruptions on revenue sharing contract are discussed to investigate the optimal strategies of players with disruptions. The penalty cost is introduced explicitly to obtain the production deviation cost caused by the disruptions. In this study, it is obtained that the coordination contract considering the production deviation cost differs from that without disruption. Besides that, the disruptions may affect the order quantities, wholesale prices as well as revenue sharing contract. Then, the optimal strategies for different disruption levels under the centralized decision-making mode are proposed. Concerning the decentralized mode, the improved revenue sharing contract can be used to coordinate the decentralized decision-making supply chain effectively. Finally, the theoretical results are illustrated by conducting some numerical examples. & 2012 Elsevier B.V. All rights reserved.

Keywords: Supply chain management Disruption management Coordination mechanism Revenue sharing contract Game theory Optimization

1. Introduction With the development of technology and globalization of the economy, the ways that firms compete with each other have become competition in supply chains. Supply chain coordination has emerged as a focused opportunity in recent years. Generally, the conventional research on supply chain coordination focused on the decision making under normal environment, where the demand was deterministic and known, the manufacturer had perfect information about the market, and the operational costs could be excluded from the analysis. However, after the production and sell plan have been settled down, the environment is often disrupted by some unexpected haphazard events, such as terrorism, earthquake, SARS epidemic, financial crisis, labor strikes, raw material shortage, new tax or tariff policy, machine breakdown, and so on. The disruption caused by these unexpected haphazard events has made the members of supply chain aware of the need for active disruption management and focus on whether and how to promise the originally planned coordination scheme valid in the new disrupted environment. n Corresponding author at: Hunan University College of Economics and Trade, Changsha 410079, China. Tel.: þ 86 731 8868 4825; fax: þ 86 731 8868 4825. E-mail addresses: [email protected] (E. Cao), [email protected] (C. Wan), [email protected] (M. Lai).

0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.09.009

Disruption events can affect the performance of a supply chain significantly. These disruptions may destroy facilities and cause that the supply chain cannot be coordinated any more. Therefore, it is meaningful to know how the supply chain can still be coordinated under different levels of disruptions. Handling unexpected disruptions in an efficient and effective way is becoming increasingly important to the success of supply chain management. While the conventional study on supply chain coordination management has developed a static coordination mechanism under a deterministic environment with a known market demand and production cost (Cachon, 2003). In contrast to many traditional studies on the design of supply chain coordination scheme, this paper will address another two aspects of the problem: how disruptions affect the coordination scheme and how to coordinate the supply chain after disruptions. In particular, this paper focuses on both production cost and demands disruptions simultaneously. Generally speaking, production cost disruption is defined as the production cost changes from its estimated value used to design the original coordination scheme. Production cost disruption can occur in various forms at every stage of the production process with different consequences (Xu et al., 2006), for example, raw material prices and transportation costs change, certain equipments fail, and interest rates fluctuate. As such, different solutions are needed to deal with different situations. Practically,

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unexpected changes of the market demands are very common. For instance, the outbreak of earthquake may cause a large sudden demand for tent; the epidemic of mad cow disease reduces the demand for beef consumption; the execution of new environment legislations can affect the demand of some electrical and electronic products. These cost and demand disruptions have great influence on consumers, retailers, wholesalers, and manufacturers in the entire supply chain. Therefore, new models and effective coordination mechanisms for supply chain coordination are required to handle the disruptions. In this paper, the production cost and demand disruptions are considered in a supply chain with a single manufacturer and n retailers competing in a Cournot market. The particular purpose of the study is to coordinate the disrupted supply chain, especially regarding how the supply chain can be coordinated under production cost and demand disruptions for the manufacturer; and the effects of the production cost and demand disruptions on supply chain coordination with revenue sharing between the manufacturer and the retailers will be investigated. To the best of our knowledge, such a case is considered the first time in this paper. The decision process is expressed as follows. First, the manufacturer makes an initial production plan before the selling season based on a market forecasting. When the selling season arrives, the real market demands and production cost may be found to be different from the forecasted one. Thus, the manufacturer has to adjust the production plan, which will usually cause deviation cost. Consequently, the manufacturer must decide how to adjust the production plan and how to design a new revenue sharing contract, by means of which the manufacturer can induce the retailers to order the proper quantity of products and maximize the profit of the whole supply chain system. So far, there is no research on coordination mechanism for supply chain that consists of a manufacture and n competing retailers in terms of dealing with the simultaneous cost and demands disruptions. The rest of the paper is organized as follows. The related literature is reviewed in Section 2. Section 3 introduces the basic coordination model when n retailers competing in a Cournot market without any disruption. Section 4 studies the coordination of a centralized supply chain with revenue sharing contract in which the manufacturer bears the deviation cost when the demand for every retailer and production cost for manufacturer are disrupted simultaneously. In Section 5, the coordination of a decentralized supply chain with improved revenue sharing contract under both cost and demand disruptions is investigated in detail. The analytical results are illustrated by numerical examples in Section 6. Finally, this paper is summarized in Section 7.

2. State of the art in supply chain coordination and disruption management In this paper, the study stems from the intersection of supply chain coordination and disruption management, and also touches upon two elements in supply chain management, namely revenue sharing contract and supply chain disruption management. In recent years, the studies of supply chain management focus on how to design the coordination schemes. Since the revenue sharing contract is beneficial to a manufacture, revenue sharing contracts among supply chain participants have become popular (Cachon and Lariviere, 2005). Pasternack (1985) found that the right revenue sharing contract could coordinate two dyadic supply chains with stochastic demand. Dana and Spier (2001) adopted a revenue sharing contract to coordinate a supply chain in a basic supplier–retailer channel setting and to cope with the demand uncertainty and vertical control of competing firms.

Cachon and Lariviere (2005) generalized the research achievements of Dana and Spier (2001) by studying the revenue-sharing contract applied in a two-echelon distribution channel with competing retailers. Cachon and Lariviere (2005) found that only a revenue sharing contract could coordinate the channel in an extended setting with stochastic and price-dependent market demand. Giannoccaro and Pontrandolfo (2004) introduced a revenue sharing contract to coordinate a three-stage supply chain. Gerchak and Wang (2004) studied a revenue sharing contract in assembly systems with stochastic demand. Wang et al. (2004) examined the channel performance of supply chain under consignment contract with revenue sharing, and found that both the overall channel performance and the performance of individual firms depended on demand price elasticity and the retailer’s share of channel cost critically. Weng (2004) also considered the revenue sharing contract in a one-period supplier– buyer channel with uncertain and price-dependent market demand, and found that the loss of channel profit increased with demand price elasticity and decreased with the buyer’s share of cost. Zou et al. (2004) proposed an analytical model to associate different processing times through strategic placement of safety stocks at each player’s premise, and found that a revenue sharing contract could be used for channel coordination in a two-echelon decentralized system under uncertain market demand. Cachon (2003) summarized the supply chain coordination mechanisms and provided a detailed review. Disruption management is a new and fledging field in the study of supply chain management. The main difference between the coordination under disruptions and the coordination under normal circumstances is that the sudden change of demand will lead to certain deviation cost which does not exist before. The deviation cost may be caused by the over-time production and the expedited delivery for an increased demand, or the extra inventory holding and possible disposal for a decreased demand. The deviation cost may be incurred to either the manufacturer or the retailers. To achieve an effective supply chain management scheme, the deviation cost should be appropriately taken into account. The concept of disruption management was firstly introduced by Clausen (2001), and applied successfully in the airline operations. Xiao and Yu (2006) investigated the impacts of supply chain disruptions on the evolution of retailers’ behaviors in a certain supply chain, where retailers with bounded rationality addressed quantity completion in a duopoly market with homogeneous goods. All the above work assumed a centralized system without considering coordination schemes. The earliest work on supply chain coordination for a demand disruption was completed by Qi et al. (2004). They introduced a quantity discount contract to coordinate a two-stage supply chain with one manufacturer and one retailer. Xu et al. (2006) studied the case with production cost disruption, and proposed a quantity discount contract to coordinate a supplier–retailer supply chain. Huang (2006) considered a novel exponential demand disruption, and applied an all-quantity discount policy to the coordination of a supply chain. Hou (2010) developed a buy-back contract between a buyer and a backup supplier when the buyer’s main supplier encountered disruptions. Xiao and Qi (2008) applied a quantity discount contract to coordinating the supply chain with two Bertrand completing retailers when the production cost and market demand were both disrupted. Xiao et al. (2007) found that the linear quantity discount scheme could coordinate the supply chain with two competing retailers and the all-unit quantity discount scheme could coordinate the supply chain where the retailers were identical after the disruption of market demand. Lei et al. (2012) adopted linear wholesale price contract menus to analyze the supply chain under demand and cost disruptions with asymmetric information. Zhang (2012) found that the revenue

E. Cao et al. / Int. J. Production Economics 141 (2013) 425–433

sharing contract could coordinate a one-manufacturer–two-retailers supply chain with demand disruptions. While this paper aims at introducing the revenue sharing contract to coordinate n Cournot competing retailers when the demand and production cost are disrupted simultaneously. In this paper, we consider a supply chain with one manufacturer and n different retailers competing in quantity, and investigate how the production cost and demand disruptions affect the revenue sharing contract.

3. The basic model of supply chain coordination First, the coordination mechanism of supply chain is studied in the baseline case (without disruption or under normal circumstances). We begin with a manufacturer and n retailer model in which the price–demand relationship is deterministic and known. The manufacturer produces commodities purchased by the retailers who then sell them on the open terminal market. The situation can be considered as a Stackelberg game with the manufacturer acting as the role of the leader and the retailers acting as the followers. In this context, the manufacturer plays first by offering the retailers a menu of contracts {(wi, fi)}, in which the contract i is only a component contract, and each retailer i is free to choose its preferred component contract from the menu. The revenue sharing contract i includes the whole-sale price wi, and share of revenue fi generated from each unit. The retailers acting as the followers then react to the menu of contracts by choosing the most preferred contract. At the same time, we assume the system information is perfect, i.e., the individuals within the supply chain have the perfect information about the demand and cost. Let cs be the manufacturer’s unit production cost and cri be the unit production cost of retailer i, define ci ¼cs þcri as the total production cost of location i, and suppose that the retail price of retailer i is described by the relationship X pi ¼ ai qi d qj , 0 od o 1, i,j ¼ 1,2, . . ., n, ð1Þ jai

where ai is the market scale for retailer i (i.e., the maximum possible demand), qi (a decision variable) is the order number of retailer i, and d is the substitutability coefficient of the retailers’ real demand (i.e., the quantity that the manufacturer must provide) under retail price pi; the substitutability coefficient (substitution elasticity) is a measure of the sensitivity of the other retailer’s sales to the change of ith retailer’s price. In this paper, the demand function of retailers in Cournot competition is similar with Cachon and Lariviere (2005). However, the retailers with different market scales are considered in the study, and the disruptions will lead to different levels of disturbance to different market scales (Xiao et al. 2007). Let ! n X qi ¼ qqi ¼ qi qi ; ð2Þ i¼1

then the revenue of retailer i can be expressed as Ri ðqÞ ¼ qi Uðai qi dqi Þ

qRi(q)/qqj o0. In addition, inventory at retailers i and j are substitutes; i.e., q2Ri(q)/qqiqqj o0. Let Pi(q) be the supply chain’s profit earned at location i that consists of retailer i and supplier, Pi(q)¼Ri(q)  ciqi, and let PT(q) be the supply chain’s total profit

PT ðqÞ ¼

n X

½Ri ðqÞci qi ,

ð4Þ

i¼1

obviously @2 PT ðqÞ=@qi 2 o 0, therefore the supply chain’s total profit is convex about the production quantity. Denote a system optimal vector of quantities as q0 ¼ ðq01 ,q02 , . . ., q0n Þ, then q0 must satisfy the following system of first-order conditions Rii ðqÞ þ P i j a i Rj ðqÞ ¼ ci , i ¼ 1,2,. . .,n, accordingly, we have ai 2q0i 2d

n X

q0j ¼ ci :

ð5Þ

jai

The equilibrium order quantity of the n retailers can be obtained by solving simultaneously the n equations, in which the n retailers simultaneously choose their order quantities at the unique Nash-equilibrium point. ai ci dU

n P

ðai ci Þ=ðn1Þd þ1

i¼1

q0i ¼

,

2ð1dÞ

i ¼ 1,2,. . .,n:

ð6Þ

In a decentralized system, the manufacturer offers each retailer a revenue sharing contract (wi, fi), and the terms of those contracts can vary across retailers. Retailer i sets qi to maximize its own profit without decision coordinating among retailers in the contractual terms. Then the profit of the retailer i is

Pri ¼ fi Ri ðwi þ cri Þqi ,

ð7Þ

and manufacturer’s profit obtained from location i is

Psi ¼ ð1fi ÞRi þ ðwi cs Þqi :

ð8Þ

The following Lemma indicates that the manufacturer can use the revenue sharing contract to induce the retailer i to order quantity of q0i and set the retailer price at p0i . As a consequence, the manufacturer’s goal and maximum supply chain profit can be achieved. Lemma 1. The following revenue sharing contracts coordinate the supply chain with n Cournot competing retailers, and the contractual terms (wi, fi) satisfy the following equation 2 3 n X i 0 Rj ðq Þ5cri ð9Þ wi ¼ fi 4ci  jai

and

Pi ðq0 Þ

0 o fi o

Pi ðq0 Þ þq0i

n P jai

,

i ¼ 1,2,. . .,n;

Rij ðq0 Þ

The firms’ optimal profits are 2 3 n X i 0 0 0 0 Pri ðq Þ ¼ fi U4Pi ðq Þ þqi Rj ðq Þ5,

j ai:

ð10Þ

ð11Þ

jai

2

ð3Þ

For simplicity, we make a similar assumption as in Cachon and Lariviere (2005) that Rii ðqÞ o0,where Rii ðqÞ ¼

427

@Ri ðqÞ : @qi

Meanwhile retailer i’s marginal revenue is decreasing in qi, that is, to say @Rii ðqÞ=@qi o 0. Let Rij ðqÞ ¼ @Rj ðqÞ=@qi , and the revenue of retailer i is decreasing if retailer j increases order quantity, i.e.,

Psi ðq Þ ¼ ð1fi ÞU4Pi ðq 0

0

Þ þ q0i

n X

3 Rij ðq0 Þ5

jai

q0i

n X

Rij ðq0 Þ,

i ¼ 1,2,. . .,n;

j a i:

ð12Þ

jai

According to the study of Cachon and Lariviere (2005) and our earlier assumptions, the revenue sharing contract (wi, fi) can coordinate the supply chain with n Cournot competing retailers.

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4. The centralized solution with disruptions

n X

cu U In a decentralized system, there is a central decision maker in the supply chain who seeks to maximize the total profit of the supply chain after the uncertainty is resolved. Now we take into account the situation with demand and cost disruptions. Suppose that the demand market scale changes from ai to ai þ Dai, and the manufacturer’s production cost changes from cs to cs þ Dc. Then the demand-price function retailer i confronts is pi ¼ ai þ Dai qi d

n X

qj ,

ð13Þ

jai

further, we assume ai þ Dai 4 qi þ d

n P jai

qj , cs þ Dc 40. Let qi be the

Pn real demand of retailer i under the disruptions and Dq ¼ i¼1 0 ðqi qi Þ be the corresponding production deviation of the manuPn facturer incurred. When Dq ¼ i ¼ 1 ðqi qi 0 Þ 40, production must be increased in order to meet new market demand. Usually the unit production cost for the additional products will be higher than the normal unit production cost because it has to get some P extra production resources at a higher price. When Dq ¼ ni¼ 1 ðqi qi 0 Þ o 0, there will be excess inventory that has to be sold in a secondary market at a low price (Xiao et al., 2005; Xu et al., 2006). In both cases, the cost and demand disruptions will cause adjustment to the original production plan and finally affect the whole supply chain. We assume that the manufacturer bears the production deviation cost completely. For the sake of generality, we further assume that the manufacturer bears the location production deviation cost, which is determined by the proportion of deviation quantity of each location in the supply chain system. To put it another way, the fraction of the deviation cost of location i is   q q0  ð14Þ Di ¼ n i i  : P  qi q0  i

i¼1

qi 

i¼1

n X

!þ q0i

cs U

i¼1

jai

q0i 

i¼1

n X

!þ qi

ð17Þ

i¼1

where ðxÞ þ ¼ max0,x, cu 40, cs 40 are the marginal extra cost of increased and decreased production from the original plan, respectively. cu is the unit extra production cost when the actual demand exceeds the original plan, Dq40; and cs is the unit cost of handling the leftover inventory when the actual demand is inferior to the original plan, Dq o0 (Xiao et al., 2005; Xu et al., 2006). If there is a centralized decision maker who tries to maximize the total supply chain profit PT ðqÞ, the decision maker will find an optimal order quantity qn ¼ ðqn1 ,qn2 , . . ., qnn Þ that maximizes (17). Intuitively, when the production cost increases and the demand decreases, the optimal order quantity should decrease, and vice versa. This is stated in the following proposition. Proposition 1. Suppose qn ¼ ðqn1 ,qn2 , . . ., qnn Þis the optimal order quantity under the production cost and demand disruption described by (17).Then Dq40 if Dai 40 and Dco0; and Dqo0 if Dai o0 and Dc 40. Proof. Note that q0 ¼ ðq01 ,q02 ,. . .,q0n Þis the original optimal order quantity, then the inequality (18) holds for any q ¼ ðq1 ,q2 ,. . .,qn Þ. n  n X  X PT ðq0 Þ ¼ Ri ðq0 Þci qi 0 Z ½Ri ðqÞci qi  ¼ PT ðqÞ ð18Þ i¼1

i¼1

Suppose that Dq40 when Dai 40 and Dco0. Then we have ! n  n n X X X  PT ðqn Þ ¼ pi ci Dc qi n cs U q0i  qni i¼1

¼

n X

i¼1

0 @ai qi n d

i¼1

1

qj n ci Aqi n

jai

i¼1

þ

n X

n X

n X

ðDai DcÞqi n cs U

i¼1

q0i 

i¼1

Pn

n X

! qni

ð19Þ

i¼1

qi n o

Pn

qi 0 , we have ! n n n X X X n n n 0 n PT ðq Þ ¼ PT ðq Þ þ ðDai DcÞqi cs U qi  qi Since Dai 40 and Dco0,

Note that the above assumption is similar to some researchers’ work such as Qi et al. (2004), Xu et al. (2006) and Xiao et al. (2007). Therefore, the revenue and profit of location i after disruptions can be obtained 0 1 n X @ Ri ¼ ai þ Dai qi d qj Aqi , ð15Þ

n X

0

r PT ðq Þ þ

i¼1

i¼1

i¼1

i¼1

n X

n X

n X

0

ðDai DcÞqi cs U

i¼1

o PT ðq0 Þ þ

i¼1

n X

i¼1

ðDai DcÞq0i ¼ PT ðq0 Þ:

q0i 

! n

qi

i¼1

ð20Þ

i¼1

0

Pi ¼ @ai þ Dai qi d

n X

1 qj Aqi ðci þ c0 þ DcÞqi

jai

!þ !þ #   " n n n n q q0  X X X X  n i i  cu U qi  q0i þ cs U q0i  qi , P  i¼1 i¼1 i¼1 i¼1 qi q0i  i¼1

ð16Þ and the total supply chain profit is given by 8 > >   < n n > q q0  X X PT ¼ Pi ¼ pi ci DcÞqi  n i i  P  > i¼1 i¼1> > qi q0i  : "  cu U

¼

n X i¼1

20

n X i¼1

qi 

n X

!þ q0i

i¼1

i¼1

þ cs U

n X

q0i 

i¼1

13

X 4@ai þ Dai qi d qj ci DcA5qi jai

n X i¼1

! þ #) qi

This is a contradiction to the assumption that qn ¼ ðqn1 ,qn2 ,. . .,qnn Þmaximizes (17). Therefore, when Dai 40 and Dc o0, Dq40 must hold. Similarly, we must have Dqo0 when Dai o0 and Dc40. Proposition 1 indicates that when the demand scale is increasing and the production cost is decreasing, the manufacturer must increase the production quantity in order to meet the increasing market demand. On the other hand, when the demand scale is decreasing and the production cost is increasing, the manufacturer must decrease the production quantity in order to meet the decreasing market demand. Next we analyze how to determine P the optimal production quantity qi n . Based on (17), when ni¼ 1 Pn qi Z i ¼ 1 q0i , the problem of maximizing PT ðqÞ reduces to maximizing the following strict concave function. 0 1 n X X @ PT1 ðqÞ ¼ ai þ Dai qi d qj ci DcAqi jai

i¼1

cu U

n X i¼1

qi 

n X i¼1

! q0i

ð21Þ

E. Cao et al. / Int. J. Production Economics 141 (2013) 425–433

subject to n X

qi Z

i¼1

n X

q0i :

ð22Þ

i¼1

The Kuhn–Tucker condition of Eqs. (21) and (22) is that at the optimal order quantity qn ¼ ðqn1 ,qn2 ,    qnn Þ, there is a Lagrangian multiplier l Z0 such that the following equations hold. 8 @PT1 ðqn Þ > þl ¼ 0 > > @qni > > > > n n X X > > < qni  q0i Z 0 ð23Þ i ¼ 1 i ¼ 1 > ! > > n n > > > l X qn  X q0 ¼ 0 > > i i > : i¼1 i¼1 Solving the Kuhn–Tucker condition, we have two different cases with respect to the Lagrangian multiplier l. Case 1. If l ¼0, function (23) is equivalent to n X qnj ¼ ci þcu þ DcDai , ði ¼ 1,2,. . .,nÞ ai 2qni 2d

ð24Þ

P P and ni¼ 1 qni  ni¼ 1 q0i 40 holds simultaneously. We can obtain the total production quantity of the manufacturer by adding all the n equations together,

qn ¼

ðai ci þ Dai Þnðcu þ DcÞ

i¼1

:

2½ðn1Þd þ1

ð25Þ

Substituting the total production quantity into (24) we obtain the equilibrium order quantity of the n retailers by solving the n simultaneous equations. The optimal order quantity of retailer i is ai þ Dai ðcu þci þ DcÞdU

n P

ðai þ Dai ci Þnðcu þ DcÞ=ðn1Þd þ 1 2ð1dÞ

Dai ðcu þ DcÞdU

n P

Dai nðcu þ DcÞ=ðn1Þd þ 1

i¼1

¼ q0i þ

ocu :

ð28Þ

n 0 and i ¼ 1 qi  i ¼ 1 qi ¼ 0 holds simultaneously. Adding all the n equations in (28) together, we can obtain

:

ð29Þ

Substituting (29) into (23) the optimal order quantity of retailer i can be derived ! n n P P ai þ Dai  ci þ Dai =n dU ðDai ci Þ=ðn1Þd þ 1 qni ¼

ð31Þ

jai

i¼1

i¼1

i¼1

ð32Þ subject to n X

qi r

n X

q0i :

ð33Þ

i¼1

The Kuhn–Tucker condition of Eqs. (32) and (33) is that at the optimal order quantity qn ¼ ðqn1 ,qn2 ,    qnn Þ, there is a Lagrangian multiplier uZ0 such that the following equations hold 8 @PT2 ðqn Þ > > u ¼ 0 > @qni > > > > n n X > X > < q0i  qni Z 0 ð34Þ i¼1 i¼1 > ! > > n n > X X > > > q0i  qni ¼ 0 u > > : i¼1 i¼1 Solving the Kuhn–Tucker condition, we have two different cases with respect to the Lagrangian multiplier u.

ð35Þ

Pn Pn 0 n and i ¼ 1 qi  i ¼ 1 qi 4 0 holds simultaneously. Adding the equations in (35), we can obtain the product amount from the manufacturer, n P

ðai ci þ Dai ÞnðDccs Þ

i¼1

qn ¼

:

2½ðn1Þd þ 1

ð36Þ

n P

ðai þ Dai ci ÞnðDccs Þ=ðn1Þdþ 1

i¼1

2ð1dÞ n P

Dai ðDccs ÞdU

Dai nðDccs Þ=ðn1Þd þ 1

i¼1

¼ q0i þ

2ð1dÞ

ð37Þ

The constraint condition is given 2

Dai n

4 cu :

When i ¼ 1 qi r i ¼ 1 q0i , the problem of maximizing PT ðqÞ is simplified to maximize the following strict concave function 0 1 ! n n n X X X X @ai þ Dai qi d qj ci DcAqi cs U PT2 ðqÞ ¼ q0i  qi

qni ¼

Pn

l ¼ cu þ Dc i ¼ 1

n

Pn

ai þ Dai ðci þ Dccs ÞdU

jai

n P

n

Dai

Dc i ¼ 1

i:e:,

40,

ð27Þ

Case 2. If l Z0, the Kuhn–Tucker condition (23) equals n X qnj ¼ ci þcu þ DcDai l ai 2qni 2d Pn

l ¼ cu þ Dc i ¼ 1 Pn

n P

Dai

Substituting it into (34) we obtain the order quantity of the ith retailer.

Dai

i:e:, Dc i ¼ 1 n

n P

ð26Þ

and the constraint conditions are given as the following inequality 2 3 n X D a nðc þ D cÞ u i 6 7 n 6 X 7 6Dai ðcu þ DcÞdU i ¼ 1 7  40, 6 7 ðn1Þd þ 1 4 5 i¼1 n X

and the constraint conditions equal to the following inequality

jai

,

2ð1dÞ

ð30Þ

2ð1dÞ

Case 3. When u¼0, function (34) is equivalent to n X ai 2qni 2d qnj ¼ ci cs þ DcDai

i¼1

qni ¼

Dai =n

i¼1

¼ q0i þ

i¼1

jai

n P

n P

Dai 

429

i¼1

i¼1

2ð1dÞ

3 n X D a nð D cc Þ s i 6 7 n 6 X 7 6Dai ðDccs ÞdU i ¼ 1 7 o0, 6 7 ðn1Þd þ 1 5 i ¼ 14 n X

Dai

i:e:, Dc i ¼ 1 n

4 cs :

ð38Þ

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E. Cao et al. / Int. J. Production Economics 141 (2013) 425–433

Case 4. When u 40, the Kuhn–Tucker condition is equivalent to the following equation n X qnj ¼ ci cs þ DcEai þ u ð39Þ ai 2qni 2d jai

Pn

Pn n 0 and i ¼ 1 qi  i ¼ 1 qi ¼ 0 holds simultaneously. Adding the n equations in (39), then the Lagrangian multiplier u can be expressed as n P

manufacturer can use a revenue sharing contract to induce the retailer to order q0 ¼ ðq01 ,q02 ,. . .,q0n Þ when the market demand and production cost are disrupted and the decision making is decentralized. We demonstrate that a similar revenue sharing contract can also be used to coordinate the supply chain. Let Sðqn Þ ¼

!þ !þ #  n 0 " n n n n q q  X X X X i i : qni  q0i þ cs U q0i  qni  n  cu U q q0  i¼1 i¼1 i¼1 i¼1 i i

n P

i¼1

Dai ð40Þ

ð44Þ

We can obtain the total order quantity of all retailers in (41), 0 1 n n P P Dai ðDai ci Þ B C i ¼ 1 i ¼ 1 ai þ Dai @ci þ n AdU ðn1Þd þ 1

and fi denotes a given retailer’s share of revenue, the wholesale price charged by the manufacturer is 2 3 n X Sðqn Þ i Rj ðqn Þ þ n 5cri , ð45Þ wi ðqn Þ ¼ fi 4ci þ Dc qi jai

u ¼ cs Dc þ i ¼ 1 n

,

qni ¼

2ð1dÞ n P

¼ q0i þ

Dai 

and

Dai

i ¼ 1

n

2ð1dÞ

:

ð41Þ

Pri

0 o fi o

Pi þqi

n

The constraint condition is defined by the following inequality n P

n P

Dai

u ¼ cs Dc þ i ¼ 1 n

i

4 0,

i:e:,

Dai

Dc i ¼ 1

n

o cs :

ð42Þ

Summarizing the above results, we have the following proposition. Proposition 2. When the Cournot competing retailers’ demandP price relationship satisfies pi ¼ai  qi  d j a iqj , Dc denotes the production cost disruption and Dai denotes the market demand scale disruption, the total supply chain profit is maximized at the optimal production quantity qn ¼ ðqn1 ,qn2 ,    qnn Þ.

qni ¼

8 n X > > Dai nðcu þ DcÞ > > > > > Dai ðcu þ DcÞdUi ¼ 1 > ðn1Þd þ 1 > 0 > qi þ > 2ð1dÞ > > > > n X > > < Dai D a i ¼ 1

> > q0i þ i 2ð1dÞn > > > > > n > X > > > Dai nðDccs Þ > > > > i¼1 > > ðn1Þd þ 1 : q0 þ Dai ðDccs ÞdU i 2ð1dÞ

n X

Dai

if Dc i ¼ 1n

if

D ai

cu o Dc i ¼ 1n n X

,

where Rj ðq Þ ¼ @Rj =@qi . The manufacturer can use a revenue sharing contract to induce the retailer to order quantity qn ¼ ðqn1 ,qn2 ,. . .,qnn Þ so that the total profit of supply chain can be maximized when the production cost and demand are disrupted simultaneously. This is stated in the following proposition. n

Proposition 3. For the decentralized case with both production cost and demand disruptions, the retailers select their most preferred contracts accordingly from the menu of contracts {(wi, fi)} offered by the manufacturer. Then the supply chain can be coordinated and the arbitrary profit allocation of optimal supply chain coordination can be realized.

Pri ¼ fi Ri ðwi þ cri Þqni 2

o cs

¼ fi Ri 4ðfi ðci þ Dc

Dai

if Dc i ¼ 1n

jai

ð46Þ

i

R j ðqn Þ

Proof. Given the above contractual terms (wi, fi), the profit function of retailer i can be described as

r cu , n X

n P

2

Z cs ,

ð43Þ Proposition 2 indicates how to correctly respond to the demand and cost disruptions. When the market scale change of each retailer is small, the manufacturer will keep the original production plan qni ; this shows that the original production plan has certain robustness under the variable market scale. When the market scale changes largely enough, adjusting the production quantity becomes necessary.

For the decentralized decisions between the manufacturer and the retailers, with demand and cost disruptions, all members will adjust their respective decisions. The manufacturer will adjust the menu of revenue contracts which includes the wholesale price wi and the retailer’s share of revenue fi generated from each unit. Then the retailers will select their favorite contracts again. Therefore, the previously derived coordination scheme (menu of contracts) must be modified to achieve the maximum supply chain profit. Recall that the optimal order quantity qn ¼ ðqn1 ,qn2 ,. . .,qnn Þ is given in Proposition 2 for this case. In the baseline case, the

i

n

1 3 n Sðq Þ ¼ fi 4Ri @ci þ Dc Rj ðq Þ þ n Aqni 5 qi jai n X

2 ¼ fi 4Pi þ qi

n

n X

i

n

3 i

Rj ðq Þ5 n

ð47Þ

jai

From the optimal condition of the total supply chain profit maximization, the following equation holds, n X

5. Coordination of supply chain after disruptions

0

3 Sðqn Þ Rj ðq Þ þ n Þcri þcri 5qni qi jai

n X

i

i

Rj ðqn Þ ¼ ci þ DcRi :

ð48Þ

jai

Moreover, we have the following equation that holds as well,

Pi þqni

n X

i

i

i

Rj ðqn Þ ¼ Pi þðci þ DcRi Þqni ¼ Ri Ri qni :

ð49Þ

jai

Because  the marginal revenue of retailer i is decreasing in i qni Ri o 0 , the profit of retailer i obtained is positive, i.e.

Pri ¼ fi ðPi þqni

n X jai

i

i

Rj ðqn ÞÞ ¼ fi ðRi qni Ri Þ 40

ð50Þ

E. Cao et al. / Int. J. Production Economics 141 (2013) 425–433

6. Numerical experiment

The profit of manufacturer can be obtained

Ps ðqn Þ ¼

n X

Psi ðqn Þ

i¼1

¼

n X

ð1fi Þpi Uqni þ ðwi c0 DcÞqni

i¼1 n X

cu U

qni 

i¼1

n X

!þ q0i

cs U

i¼1

n X

q0i 

i¼1

n X

!þ qni

i¼1

8 9 2 3 n < n n = X X X i n i n n 4 5 ¼ ð1fi ÞU Pi qi Rj ðq Þ þqi Rj ðq Þ : : ; jai

i¼1

ð51Þ

jai

The retailers’ profit function is an affine function of supply chain total profit; accordingly, the optimal order quantity of retailer i is also the optimal order quantity qi n of the supply chain system, and each retailer can choose his own optimal contract parameter based on his own optimal order quantity. The bounds on fi ensure that the manufacturer and retailers earn a positive profit, and the profit of manufacturer (the retailers) obtained is decreasing (increasing) for a given retailer’s share of revenue fi. The arbitrary profit allocation of optimal supply chain profit can be achieved by adjusting the retailer’s share of revenue fi. So the improved revenue sharing contract (wi, fi) can implement the supply chain coordination when market demands and production cost are disrupted simultaneously. Furthermore, we have the following results hold: if Dai ¼0 and Dc ¼0, then cu ¼cs ¼ 0, and the total supply chain profit can be expressed as the following simplified equation

PT ðqÞ ¼ PT ðqÞ ¼

n X

431

½Ri ðqÞci qi ,

ð52Þ

i¼1

and the wholesale changes into 2 3 n X i 0 5 4 Rj ðq Þ cri wi ¼ fi ci 

ð53Þ

jai

Consequently, the revenue sharing contract (wi, fi) also can achieve supply chain coordination under normal circumstances. That is to say the improved revenue sharing contract has antidisruption ability. The above result is consistent with that obtained by Cachon and Lariviere (2005). From Proposition 2 and Proposition 3, we find that when the cost and demand disruptions happen simultaneously, there is no need to change the order and supply plans if the degree of disruptions is within a certain range, while only the revenue sharing fraction for the retailers should be changed; if the degree of disruptions exceeds the range, both the plans and the revenue sharing fraction should be changed for the possible coordination of the supply chain.

In this section, some numerical examples will be presented to illustrate the theoretical results obtained in the previous sections. We are particularly interested in the effects of the disruption management after/along with various demand and cost changes. Realizing the demand and cost changes, the manufacturer will adopt an appropriate improved revenue sharing contract to recoordinate the supply chain. On the contrary, if the manufacturer is not aware of the demand and cost changes, he will keep the pre-assumed revenue sharing contract, and the retailers will have the freedom to make orders according to the real market. For these two cases, the profits from the perspectives of the manufacturer, the retailers, and the whole supply chain will be compared and analyzed, respectively. The disruptions affect the optimal decisions with high degree and the profitability of the supply chain, the manufacturer, and the retailers. Then the effects of the cost and demand disruptions are illustrated by using the following numerical examples. Without loss of generality, we consider a supply chain system consisting of a manufacturer and two retailers, the demand-price function is pi ¼ai  qi  dqj, (i, j ¼1, 2; jai), considering this problem with following parameters: a1 ¼a2 ¼20, c1 ¼ 3, c2 ¼2, c0 ¼5, d ¼0.5, Da1 ¼{  1, 1}, Da2 ¼{  1, 1}, Dc ¼{  1.5, 1.5}, cu ¼cs ¼1. We assume that the manufacturer and retailers agree to share the total supply chain profit as 4:6 under the assumption of symmetric information, which is to say f1 ¼ f2 ¼0.6. Under normal circumstances, using Lemm1 expressed in Section 3, given the above values, we can easily obtain the order quantities of retailer 1 and retailer 2, q01 ¼ 11=3, q02 ¼ 14=3, respectively, and the corresponding retailer price is p01 ¼ 14 and p02 ¼ 13:5. The optimal production quantity of manufacturer is 25/3, the profits of location 1 and location 2 are P1 ¼22 and P2 ¼30.333, respectively. The wholesale price and profit of retailers are w1 ¼ 3.2 and w2 ¼3.3, Pr1 ¼ 8.064 and Pr2 ¼13.0668, respectively. The optimal profits of manufacturer and the supply chain are Ps ¼31.1983 and PT ¼52.33, respectively. In disruptions circumstances, from Proposition 2 described in Section 4, given the above values, there are eight cases in which demands and production cost are disrupted simultaneously, and six cases in which the original production quantity is changed. According to Proposition 3 given in Section 5, different disruptions’ effects on optimal parameters of revenue sharing contract and the profit of the total supply chain system are given in Table 1. In Table 1, from the first (the last) case, when the demands simultaneously decrease (increase) and the production cost P decreases (increases) and satisfies cu o Dc ni¼ 1 Dai =n o cs , according to Proposition 2, the original production plan (qni ¼ q0i ) is not affected. Additionally, the demand and production cost disruption mutually contend and restrict. This is to say the adverse effect of one disrupted event is eliminated by another

Table 1 The optimal parameters and the total profit under simultaneous demands and cost disruptions. Case

1 2 3 4 5 6 7 8

Da1

1 1 1 1 1 1 1 1

Da2

1 1 1 1 1 1 1 1

Dc

 1.5 1.5  1.5 1.5  1.5 1.5  1.5 1.5

qn1

3.667 3.1667 2.8333 2.5 4.8333 4.5 4.1667 3.667

qn2

4.667 4.1667 5.8333 5.5 3.8333 3.5 5.1667 4.667

pn1

13 13.75 13.25 13.75 14.25 14.75 14.25 15

pn2

12.5 13.25 13.75 14.25 12.75 13.25 13.75 14.5

wn1

2.3 4.0447 2.6794 4.3967 2.0741 3.7685 2.522 4.1

wn2

2.4 4.122 2.17 3.8652 2.7717 4.4833 2.6081 4.2

Total supply chain profit Original policy

New policy

53.1667 32.1667 65.8333 40.8333 63.8333 38.8333 73.8333 44.8333

56.5 32.2500 66.9167 41.9167 64.9167 39.9167 73.9167 48.1667

432

E. Cao et al. / Int. J. Production Economics 141 (2013) 425–433

Table 2 The optimal parameters and the total profit of supply chain under one demand and cost disruptions. Case

1 2 3 4 5 6 7 8

Da1

0 0 0 0 1 1 1 1

Da2

1 1 1 1 0 0 0 0

Dc

 1.5 1.5  1.5 1.5  1.5 1.5  1.5 1.5

qn1

4.1667 3.8333 3.5 3.1667 3.1667 2.8333 4.5 4.1667

qn2

4.1667 3.8333 5.5 5.1667 5.1667 4.8333 4.5 4.1667

pn1

13.75 14.25 13.75 14.250 13.250 13.750 14.25 14.75

disrupted event. Therefore, the optimal total supply chain profit keeps steady and the supply chain only need to adjust the retail prices to avoid the production deviation cost. It illustrates that the original production quantity is robust. From the second case, when the market demands are simultaneously decreasing and the P production cost is increasing and satisfies Dc ni¼ 1 Dai =n 4cs , based on Proposition 1 and Proposition 2 given in Section 4, the manufacturer must decrease the total production quantity P  P2 2 n 0 i ¼ 1 qi o i ¼ 1 qi in order to meet the decreasing market demand. From the seventh case, when the demand scale is increasing P and the production cost is decreasing and satisfies Dc ni¼ 1 Dai =n o cu , from Proposition 1 and Proposition 2, the manufacturer P  P2 2 n 0 must increase the production quantity i ¼ 1 qi 4 i ¼ 1 qi in order to meet the increasing market demand. From the third case to the sixth case, we know that when the demand and production cost change in different directions and exceed some range, according to Proposition 2, the manufacturer should adjust the production plan to ensure that the supply chain achieves the optimal profit. Based on Proposition 3, we can obtain the feasible revenue sharing contracts under different disruptions, which are also listed in Table 1. In addition, we further consider the cases where one market demand is disrupted while the other market have no disruption, which can be easily obtained by setting one retailer’s demand disruption Da to 0, from Proposition 2 and Proposition 3. Table 2 illustrates the effects of one demand and cost disruptions on the supply chain decisions and coordination. Obviously, from Table 2, we know the optimal quantities of all retailers changed when there is one retailer’s demand unchanged, because the optimal quantity is related with all the retailers’ demands and cost disruptions according to Proposition 2. This also illustrates the managerial complexity when multi-retailers competing. Both Tables 1 and 2 illustrate that the total supply chain profit under new policy is larger than that with original policy. It indicates the significance and validity of the improved revenue sharing contract to cope with demand and cost disruptions.

7. Conclusions In this paper, we have developed the revenue sharing coordination contract of a supply chain with one manufacturer and n Cournot competing retailers when demand and production cost are disrupted simultaneously. The influences of the demands and cost disruptions on optimal decisions and coordination are analyzed comprehensively. In particular, when the cost and demand disruptions occur simultaneously, if the degree of disruptions are within a certain range, the order and supply plans do not need to change, and only the revenue sharing fraction for the retailers should be changed; if the degree of disruptions exceeds

pn2

12.75 13.25 13.75 14.25 13.25 13.75 13.25 13.75

wn1

2.15 3.8674 2.569 4.25 2.45 4.2676 2.3241 3.95

wn2

2.55 4.337 2.4106 4.05 2.25 3.9638 2.6648 4.35

Total supply chain profit Original policy

New policy

58.5 35.5 69.8333 42.8333 59.5 36.5 68.8333 41.8333

60.4167 35.75 70.0833 44.75 61.4167 36.75 69.0833 43.75

the range, both the plans and the revenue sharing fraction should be changed for the possible coordination of the supply chain. We clearly show that if the manufacturer is able to adjust his revenue sharing contract according to the demands and cost disruptions in a timely way, the supply chain can achieve optimal profit and realize arbitrary allocation of optimal supply chain profit. Furthermore, the revenue sharing contract has demonstrated a certain degree of robustness, and it is optimal for the supply chain to keep the original coordination mechanism if the variation of disruptions is sufficiently small. There are several extensions of this work that could be considered for future research. It would be meaningful to introduce other contracts to coordinate a multi-retailers or multisuppliers supply chain under disruptions. Another interesting aspect for future research is to consider supply chain disruption management under asymmetric information.

Acknowledgements This research is supported by the National Science Foundation of China under Grant 71001035 and 70925006, the National Basic Research Program of China (973 Program) under Grants 2012CB315805 and the Major Special Program for Science and Technology of Hunan Province, China under Grant 2009FJ1003. It is partially supported by The Hong Kong Polytechnic University grant No. ZV3E. Finally, the authors would like to thank the anonymous reviewers for their comments and suggestions. References Cachon, G.P., Lariviere, M.A., 2005. Supply chain coordination with revenuesharing contracts: strength and limitations. Management Science 51 (1), 30–44. Cachon, G.P., 2003. Supply chain coordination with contracts. In: de Kok, A.G., Graves, S.C. (Eds.), Handbooks in Operations Research and Management Science. Supply Chain Management, New York: North Holland. Clausen, J., Hansen, J., Larsen, J., 2001. Disruption management. OR/MS Today 28 (5), 40–43. Dana, J.D., Spier, K.E., 2001. Revenue sharing and vertical control in the video rental industry. Journal of Industrial Economics 49 (3), 223–245. Gerchak, Y., Wang, Y., 2004. Revenue-sharing vs wholesale-price contracts in assembly systems with random demand. Production and Operation Management 13 (1), 23–33. Giannoccaro, I., Pontrandolfo, P., 2004. Supply chain coordination by revenue sharing contracts. International Journal of Production Economics 89 (2), 131–139. Hou, J., Zeng, A.Z., Zhao, L., 2010. Coordination with a backup supplier through buy-back contract under supply disruption. Transportation Research Part E: Logistics and Transportation Review 46 (6), 881–895. Huang, C.C., Yu, G., Wang, S., et al., 2006. Disruption management for supply chain coordination with exponential demand function. Acta Mathematica Scientia 26B (4), 655–669. Lei, D., Li, J., Liu, Z., 2012. Supply chain contracts under demand and cost disruptions with asymmetric information. International Journal of Production Economics 139 (1), 116–126.

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