Pricing and production decisions in a dual-channel supply chain when production costs are disrupted

Economic Modelling 30 (2013) 521–538

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Pricing and production decisions in a dual-channel supply chain when production costs are disrupted Song Huang a, b,⁎, Chao Yang b, Hui Liu b a b

College of Economics & Management, South China Agricultural University, Guangzhou 510642, China School of Management, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Accepted 14 October 2012 Keywords: Supply chain management Dual-channel Production cost disruptions Game theory

a b s t r a c t This paper studies a pricing and production problem in a dual-channel supply chain when production costs are disrupted. When a production cost disruption occurs, the original production plan which is designed based on the initially estimated production cost needs to be revised. It is necessary to explicitly consider possible related deviation costs caused by changes of the original production plan. We consider this problem in the centralized and decentralized dual-channel supply chain, respectively. The optimal prices and production quantity under production cost disruptions are derived. We find that the original production plan has some robustness with production cost disruptions. Only when the production cost disruption exceeds some thresholds will the decision-maker change the production quantity. The production cost disruption robustness region in the centralized dualchannel supply chain is the same as that in the decentralized dual-channel supply chain. In the centralized dual-channel supply chain, it is always beneficial for the central decision-maker to take timely response to the disruptions and utilize the revised strategy under production cost disruptions. In the decentralized dual-channel supply chain we characterize a threshold. If customers' preference for the direct channel is below the threshold, the optimal direct sale price equals the wholesale price; otherwise, the optimal direct sale price and wholesale price equals the optimal direct sale price and retail price in the centralized dual-channel supply chain. The manufacturer always gets better off if he takes timely response to the production cost disruptions, while the retailer gets better off only if the production cost disruption is negative and gets worse off otherwise. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Customers are becoming more and more accustomed to purchasing online in the passing decades. The rapid development of the Internet and emerging technology play a significant role in the development of this new business model. By means of an online channel, firms can deal with customers' orders, have control over the distribution and pricing of goods, have a greater understanding of customers' preference and be closer to customers (Mukhopadhyay et al., 2008). Many firms have engaged in online channels to reach customers who cannot be reached by the traditional retail channel. It is reported that about 42% of the top suppliers such as IBM, Nike, Pioneer Electronics, Estee Lauder, and Dell are selling to customers through an online channel (Chiang et al., 2003; Tsay and Agrawal, 2004). However, the introduction of the online direct channels has made the pricing and production decisions more complicated when facing unexpected risks or disruptions. On the one hand, the manufacturer's and the retailer's pricing strategies are

⁎ Corresponding author at: College of Economics & Management, South China Agricultural University, Guangzhou 510642, China. E-mail address: [email protected] (S. Huang). 0264-9993/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2012.10.009

competitive to some extent; on the other hand, they have to respond to unexpected production risks such as machine breakdown and raw material shortage to maximize their individual profits. Generally, disruptive events in a supply chain will significantly affect the performance of the supply chain. It is reported that publicly trade firms experiencing disruptive events received negative stock market reactions with the magnitude of decline in market capitalization being as large as 10% (Hendricks and Singhal, 2003, 2005). Disruptive events have made companies realize the importance of disruption management. The disruption management has gained much attention in the communities of academics and practitioners. Most literature is concerned with how to coordinate members' decisions to maximize the supply chain's profit in the traditional channel. Little attention has been paid to the disruption management in a dual-channel supply chain. In a dual-channel supply chain the interaction of member's pricing decisions and its impact on channel demands are even more complicated when there are possible disruptions. Once a disruption occurs, how the members should revise their decisions to maximize their individual profits is a problem worth of investigation. This paper considers the pricing and production decisions in a dual-channel supply chain consisting of one manufacturer and one retailer when the production cost experiences disruptions. The objective of this paper is not only to analyze how to revise the pricing and

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S. Huang et al. / Economic Modelling 30 (2013) 521–538

production quantity in response to production cost disruptions, but also to investigate whether the manufacturer who has access to the accurate production cost information is willing to take timely response to the disruptions and what the value of this reaction is. This paper is closely related to disruption management. Qi et al. (2004) first introduced the idea of disruption management to supply chain management. In their paper, they considered two kinds of deviation costs. If the demand exceeds the original production quantity, underage cost may happen. Otherwise disposal cost may happen. Based on Qi et al. (2004), many researchers had extended disruption management in supply chain management in various scenarios. Yang et al. (2005) proposed a dynamic programming method for recovering a production plan after a disruption. Xu et al. (2006) discussed the supply coordination problem with production cost disruptions. Qi et al. (2006) investigated the disruption management in machine scheduling. Huang et al. (2006) investigated the coordination problem of a supply chain with exponential function under demand disruptions. Xiao and Yu (2006) developed an indirect evolutionary game model with two vertically integrated channels to study evolutionarily stable strategies of retailers in the quantity-setting duopoly situation. Xiao et al. (2005, 2007) and Xiao and Qi (2008) studied how to coordinate a supply chain with one manufacturer and two competing retailers by means of quantity discount schedule when demands are disrupted. Chen and Xiao (2009) studied how to coordinate the supply chain with a dominant retailer after demand is disrupted. Chen and Zhuang (2011) further considered the coordination model for a dominant retailer with constant demand-stimulating cost under demand disruptions. Lei et al. (2012) studied how to devise linear contract menus to coordinate the supply chain with asymmetric information under both demand and production cost disruptions. This paper is distinct from the disruption management literature above in two aspects. Firstly, the literature above considered a single channel and did not considered the disruption management in a dual-channel setting, with the only exception of Huang et al. (2012), which addressed the disruption management in a dual-channel supply chain under demand disruptions. However, in this paper we investigate the pricing and production quantity decisions in a dual-channel supply chain under production cost disruptions. Secondly, we examine the value of timely response to production cost disruptions in both centralized and decentralized dual-channel supply chain settings. The research on dual-channel supply chain management had obtained much attention among the marketing and supply chain management researchers in recent years. For dual-channels, Chiang et al. (2003) studied a price-setting game between a manufacturer and a retailer in a dual-channel supply chain based on consumer choice model. Tsay and Agrawal (2004) used game theory to study the channel conflict and coordination between the manufacturer and the retailer in a dual-channel supply chain. Mukhopadhyay et al. (2006) proposed a way of alleviating the possibility of channel conflict when the retailer is allowed to add value to the product at a cost. Based on Chiang' work, Yan (2008) used a game theory approach to investigate the strategic role of profit sharing and found that both the manufacturer and the retailer will benefit from a dual-channel profit-sharing strategy. Cai et al. (2009) and Cai (2010) studied the influence of channel structures and pricing schemes on the dual-channel supply chain. Chen et al. (2008) investigated service competition in a dual-channel supply chain based on consumer channel choice behavior. Mukhopadhyay et al. (2008) investigated the optimal contract design problem in a mixed channels supply chain where the manufacturer had incomplete information about the retailer's cost of adding value. Zhang et al. (2012) studied the effect of product substitutability and relative channel status on pricing decisions under different power structures. However, the literature above all assumed that the production cost was given and known. Little attention has been paid to the issue of production

costs disruption management in a dual-channel supply chain. Usually, accurate production costs information cannot be obtained in advance if no mass production has been conducted before. As a result of long production lead time, manufacturers usually make the production plans according to the anticipated production costs. Many factors, such as increased labor cost, changed auxiliary material price, changed transportation cost, and interest rate fluctuation, will lead production cost to vary in practice. Once the real production cost is not equal to the original anticipated one, necessary adjustments have to be executed to mitigate the impact of production cost disruptions and achieve profit maximization. The framework of this paper is most related to Qi et al. (2004), Xu et al. (2006) and Huang et al. (2012). We consider this problem in a two-period dual-channel supply chain. In the first period, a production plan is made based on the anticipated production cost. Raw material purchasing and preprocessing are conducted in this period. In the second period, the actual production cost is resolved, and a new production plan has to be made accordingly. Production cost variety will lead to selling price change, which will eventually lead to market demand change. Once the market demand changes, some related deviation costs will possibly occur. Especially, if the demand exceeds the original production quantity more raw materials will have to be purchased in the physical market at higher prices, while if the demand is less than the original production quantity the excessive materials will have to be disposed. Either of them will add deviation costs to the supply chain. We would like to know how the decision-maker should revise the original production plan in a dual-channel supply chain under production cost disruptions. This paper contributes to the literature in two aspects. Firstly, this paper complements the disruption management in a dual-channel supply chain. As far as we know, the studies on supply chain disruption management usually consider the traditional retail channel, with the exception of Huang et al. (2012). This paper is distinct from Huang et al. (2012) in that we focus on production cost disruption management in a dual-channel supply chain, while they considered the demand disruption management in a dual-channel supply chain. As the impacts of demand disruption and production cost disruption on the optimal decisions are different, the main results of this paper are also different from theirs. As more and more manufacturers engage in dual channels and the environment in which the supply chain operates changes quickly, it is necessary to investigate how to respond to disruptions in a dual-channel supply chain. This paper studies how a dual-channel supply chain should revise its pricing and production quantity decisions to deal with production cost disruptions and derives the optimal pricing and production decisions. Secondly, this paper complements the dual-channel supply chain management by incorporating the disruption management. Most studies on dual-channel supply chain management focused on channel structure choice, channel conflict and coordination and joint decisions with other factors, such as retail service, cooperative advertising and lead time (Hua et al., 2010; Xie and Wei, 2009; Yan, 2008). Little attention has been paid to the issue that how a dual-channel supply chain should respond to disruptions caused by changing outside environment or inside operation. This paper contributes to this stream of literature by considering production cost disruptions in a dual-channel supply chain and derives optimal pricing decisions for the supply chain. The remainder of this paper is organized as follows. The notations and the models are introduced in Section 2. In Section 3, we analyze the optimal pricing and production quantity decisions in the centralized and decentralized dual-channel supply chain without production cost disruptions as the baseline cases. Section 4 and Section 5 examine the optimal pricing and production quantity decisions in a centralized and decentralized dual-channel supply chain under production cost disruptions, respectively. Section 6 presents some numerical examples to illustrate the results. Section 7 summarizes the results and limitations and points out directions for future research.

S. Huang et al. / Economic Modelling 30 (2013) 521–538

Following Yao and Liu (2005), Yue and Liu (2006), Kurata et al. (2007), Hua et al. (2010) and Huang et al. (2012), we assume that, for model's simplicity, the channel demand functions in the two channels are linear in self-price and cross-price effects. The demand functions in two channels are formulated as follows

Manufacturer Wholesale Price w Retailer

523

Direct Sale Price pd

Retail Price pr Market Demand Fig 1. Dual-channel supply chain structure.

2. Model description We examine a dual-channel supply chain composed of one manufacturer and one retailer. The manufacturer may sell the products to the retailer, as well as to end customers directly (see Fig. 1). Suppose that the manufacturer produces a single type of product with a long raw material purchase lead time and the market demand decreases with price and the price-demand relationship is deterministic and known. The manufacturer sells the products to the retailer at wholesale price w. The retailer sells the products to end customers at retail price pr. The manufacturer sells the products to end customers directly at direct sale price pd. Let c be the manufacturer's anticipated production cost. We further assume that the manufacturer and the retailer make decisions independently to maximize their individual profits. Following conventional assumptions, we examine this problem in a Stackelberg game with the manufacturer leader and the retailer follower. Consider a two-period dual-channel supply chain. Similar to Xu et al. (2006), we assume that the production requires two sequential periods with the first period followed by the second period. In the first period, the manufacturer purchases some raw materials with some preprocessing, and the second period converts the materials into final products. If the raw material has a long purchase lead time or the preprocessing takes a long time, the first period is necessary for reducing the production time and improving efficiency. The manufacturer makes the production plan based on the information of production costs obtained from the two periods. As the production in the first period is followed by the second period, the cost information in the first period is accurate. However, when the second period begins, the actual cost information is only an estimation. Various factors will lead to production cost disruptions, which will make the original production plan made in the first period suboptimal. To solve this issue, some necessary adjustments have to be made accordingly. The problem that needs consideration now is how to revise the prices and production quantity when the production cost disruptions are resolved.

Dr ¼ ð1−ρÞa−a1 pr þ βpd

ð1Þ

Dd ¼ ρa  a2 pd þ βpr

ð2Þ

In the above formulas (1) and (2), the subscripts r and d denote the retail channel and the direct channel, respectively. The expressions in Eqs. (1) and (2) indicate that the demand in the two channels both depend on the retail price pr and the direct sale price pd. a represents the potential demand if the products are free. The parameter ρ captures customers' preference for the direct channel when the products are free. α1 and α2 are the coefficients of self-price elasticity of the demand in the retail channel and of the demand in the direct channel, respectively. To maintain analytical tractability, following Yue and Liu (2006), Hua et al. (2010), and Huang et al. (2012), we assume that the cross-price effects are symmetric. The cross-price sensitivity β reflects the degree to which the products sold via the two channels are substitutes. Assume that αi > β for i = 1, 2, which indicates that the self-price effects are greater than the cross-price effect, which is very common in economics and operations management literatures. Obviously, all the parameters should be positive. The total demand in two channels is given by Dsc = a − (α1 − β)pr − (α2 − β)pd. The disruption model considered in this paper contains two periods. The original production plan is built on the assumption that the anticipated production cost is c in the first period, and in the second period, the real production cost is found to be c + Δc, where the production cost disruption is captured by Δc. Under the production cost disruption scenario, the selling price may change. Thus, let p
r and p
d be the revised retail price and direct sale price. Throughout the paper, we use the bar to denote the case of production cost disruptions. Following Chiang et al. (2003), Hua et al. (2010) and Huang et al. (2012), we assume that both the manufacturer and the retailer have no merchandising costs associated with selling the products to end customers for analytical simplicity. With the above notations and assumptions, the retailer's profit function is determined by πr ¼ ðpr  wÞDr ;

ð3Þ

and the manufacturer's profit function is determined by πm ¼ ðw−cÞDr þ ðpd −cÞDd

ð4Þ

The dual-channel supply chain's total profit function is determined by πsc ¼ πr þ πm ¼ ðpr −cÞDr þ ðpd −cÞDd :

ð5Þ

3. Baseline cases In order to study the impact of production cost disruptions on the original pricing and production decisions, we study the scenario when no production cost disruption occurs as the baseline case first. The centralized and decentralized dual-channel supply chains are studied in Section 3.1 and Section 3.2, respectively, without production cost disruptions.

3.1. Baseline case: centralized decisions without production cost disruptions In a centralized dual-channel supply chain where the manufacturer and the retailer are vertically integrated, there exists a central decision-maker who determines the retail price pr and the direct sale price pd simultaneously to maximize the supply chain's total profit. If the actual production cost resolved in the second period is equal to what is anticipated, the problem can be simplified to the basic dual-channel supply chain model. In order to maximize πsc, we examine the property regarding πsc first. The following two lemmas are presented in Huang et al. (2012). We list them here to make the paper self-contained, but we also add more insight and explanations to cast on the results.

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Lemma 1. In a centralized dual-channel supply chain, the total profit function πsc is jointly concave in pr and pd, and the optimal retail price pr* and optimal direct sale price pd* are given by 

α 2 ð1−ρÞ þ βρ
 2 α 1 α 2 −β 2

aþ ;

c 2

ð6Þ



α 1 ρ þ βð1−ρÞ
 2 α 1 α 2 −β 2

aþ ;

c 2

ð7Þ

pr ¼

pd ¼

respectively, and the centralized dual-channel supply chain's optimal profit is given by 

πsc ¼

α 1 ρ2 þ 2βρð1−ρÞ þ α 2 ð1−ρÞ2
 4 α 1 α 2 −β 2

1 2

2

a − ac þ

α 1 þ α 2 −2β 2 c : 4

ð8Þ

The optimal production quantity is given by 



1 2

Q sc ¼ Dsc ¼ ½a−ðα 1 þ α 2 −2βÞc:

ð9Þ

Lemma 1 implies that the centralized dual-channel supply chain's total profit function πsc is jointly concave in pr and pd. Therefore, the optimal retail price and direct price can be obtained by means of first-order conditions. From Eqs. (6)–(7), we can find that the optimal retail price and direct price are symmetric in structure to some extent, because the demand functions in the two channels are similar in structure and the cross-price effects are symmetric. Note that the retail price may be higher/lower than the direct sale price, depending on customers' preference for the direct channel. 3.2. Baseline case: decentralized decisions without production cost disruptions In what follows, we consider a decentralized dual-channel supply chain where the manufacturer and the retailer make decisions independently. The manufacturer acts as the leader who declares the wholesale price w and the direct sale price pd first. Then the retailer acts as the follower responds to it and determines the retail price pr. In order to avoid triviality, we assume that the direct sale price pd must not be smaller than the wholesale price w, which is the no arbitrary condition. Otherwise, if pd b w, the retailer can obtain the products from the direct channel at a lower price (Hua et al., 2010), which is unreasonable in practice. We consider the problem by means of backward induction. As the retailer plays the follower, given that the manufacturer determines the wholesale price w and direct price pd, the retailer's optimal responsive pricing strategy is pr = [(1 − ρ)a + βpd + α1w] / 2α1. Substituting the retailer's optimal responsive pricing strategy into the manufacturer's profit function defined in Eq. (4), the manufacturer's decision problem can be formulated as  1 max πm ðpd ; wÞ ¼ ðw−cÞ½ð1−ρÞa þ βpd −α 1 w þ ðpd −cÞ ρa−α 2 pd þ 2

pd ≥w

β ½ð1−ρÞa þ βpd 2α 1

 þ α 1 w :

ð10Þ

^ ¼ ðα 2 −βÞ=ðα 1 þ α 2 −2βÞ. In the decentralized dual-channel supply chain without production cost disruptions, the manufacturer's Lemma 2. Define ρ profit function πm is jointly concave in pd and w, and the manufacturer's optimal pricing strategy and the retailer's optimal pricing strategy are given by 0

w ¼

8 < 2α1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c2

^; if 0 < ρ < ρ

: α 2 ð1−ρÞ þ βρ a þ c2

^ ≤ ρ b 1; if ρ

8 þ ðα 1 þ β Þð1−ρÞ < 2α21αρ −β a þ c2 þ2α α −2α β

^; if 0 < ρ < ρ

Þ : α 1 ρ þ βð1−ρ a þ c2 2ðα 1 α 2 −β 2 Þ

^ ≤ ρ b 1; if ρ

2ðα 21 −β 2 þ2α1 α2 −2α1 βÞ 2ðα 1 α 2 −β 2 Þ

0 pd

0 pr

ð

¼

¼

2 1

2

1

2

1

Þ

ð11Þ

ð12Þ

8   2 2 > 2α ðα þ β Þρ þ 3α 1 −β þ 4α 1 α 2 −2α 1 β ð1−ρÞ > < 1 1 ^; a þ α14αþ βc if 0 < ρ < ρ 4α ðα −β þ2α α −2α β Þ 

1

2 1

2



1

2

4α 1 ðα 1 α 2 −β 2 Þ

ð13Þ

1

1

2 > 2α βρ þ 3α 1 α 2 −β ð1−ρÞ > : 1 a þ α 1 þ βc

^ ≤ ρ b 1; if ρ

4α 1

and the manufacturer's optimal profit and the retailer's optimal profit are given by

0

πm ¼

8n   o2 2 2 > ½2α 1 ρ þ ðα 1 þ β Þð1−ρÞa− α 1 −β þ 2α 1 α 2 −2α 1 β c > < 8α 1 ðα 21 −β 2 þ2α 1 α 2 −2α 1 β Þ   2 2 α 1 α 2 þ β ð1−ρÞ

2 2 > > : 2α 1 ρ þ 4α 1 βρð1−ρÞ þ

8α 1 ðα1 α2 −β

0

πr ¼

8 > > < > > :

" 1

16α1

2

Þ

2 16α ½ð1−ρÞa−ðα 1 −β Þc

1

1

1 þ β Þð1−ρÞ ac þ a2 −2α1 ρ þ ðα4α

  2 2 2α 1 ðβ−α 1 Þρ þ α 1 −β þ 4α 1 α 2 −4α 1 β ð1−ρÞ α 21 −β2 þ2α 1 α 2 −2α 1 β

^; if 0 < ρ < ρ 2 2 α 1 −β

þ 2α 1 α 2 −2α 1 β 2

1

8α1

c

ð14Þ

^ ≤ ρ b 1; if ρ

#2 a−ðα 1 −βÞc

^; if 0 < ρ < ρ ^ ≤ ρ b 1; if ρ

ð15Þ

S. Huang et al. / Economic Modelling 30 (2013) 521–538

525

respectively, and correspondingly, the optimal production quantity is given by 0

Q sc ¼

1 4α 1

n   o 2 2 ½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa− α 1 −β þ 2α 1 α 2 −2α 1 β c :

ð16Þ

^ for the parameter ρ. We can interpret them respectively as follows. The parameter ρ can be interpreted as Lemma 2 defines a critical threshold ρ the static channel demand allocation rate, because the parameter ρ allocates the market demand in the two channels when the products are free of ^ can be interpreted as the dynamic channel demand allocation rate, because the threshold ρ ^ captures how the changed decharge. The threshold ρ mand caused by unit prices (direct sale price and retail price) change is allocated in the two channels. To be specific, if the retail price and direct sale price both decrease one unit, the total demand will increase α1 + α2 − 2β with α2 − β coming from the direct channel and α1 − β coming from the ^ show how much the demand goes to the direct channel from two different perspectives. retail channel. The parameter ρ and the threshold ρ ^ ≤ρb1, The results presented in Lemma 2 indicate that the value of ρ has significant impact on the central decision-maker's decisions. When ρ the manufacturer's optimal direct sale price and wholesale price in the decentralized supply chain are identical to the optimal direct sale price and retail price in the centralized supply chain. In other words, the manufacturer will treat the retailer as an end customer and set the wholesale ^ , the manufacturer price equal the optimal retail price in the centralized supply chain, and keep the direct sale price unchanged. When 0bρ≤ρ will set the wholesale price equal to the direct sale price, because when customers prefer the retail channel, the manufacturer has to cut the direct sale price to attract more customers to purchase from the direct channel. Additionally, the optimal production quantity of the supply ^ has no direct effect on it. In other words, the manufacturer's optimal wholesale chain depends only on the value of ρ. And the threshold ρ price and direct sale price, as well as the retailer's retail price, are piecewise continuous in ρ, while the supply chain's optimal production quantity is continuous in ρ.

4. Centralized decisions with production cost disruptions In this section, we analyze the pricing and production decisions in the centralized dual-channel supply chain with production cost disruptions. We assume that Δc is not too large to simplify the analysis. When the actual production cost uncertainty is resolved in the second period, the question at this point is how the central decision-maker should revise the pricing and production decisions accordingly. Similar to Qi et al. (2004), Xu et al. (2006) and Huang et al. (2012), we analyze this problem in two scenarios, i.e., Δc > 0 and Δc b 0. If Δc > 0, the actual production cost is larger than anticipated. Given that the production quantity is unchanged, the central decision-maker will raise prices, then fewer customers will purchase the products and there will be disposal cost for the leftover inventory. If Δc b 0, the actual production cost is less than that anticipated. Given that the production quantity is unchanged, the central decision-maker will lower prices to attract more customers to purchase and there will be underage costs for unmet demand. In this section and the following section, we discuss these two scenarios separately to examine how to revise the prices and production quantity. Let μ1 and μ2 denote the marginal underage cost and marginal disposal cost, respectively. In practice, the underage cost and disposal cost are usually much lower than the marginal production cost. In order to avoid triviality, we make the following assumption that max{μ1,μ2}≤ c, which is similar to Qi et al. (2004) and Huang et al. (2012). From the central decision-maker's point of view, given the original production quantity Qsc*, the supply chain's total profit under production cost disruptions can be formulated as


−Q  þ −μ Q  −D
þ ðp
−c−ΔcÞD
−μ D
þ; π
sc ¼ ðp
r −c−ΔcÞD r d d sc sc sc sc 1 2

ð17Þ

where the first two terms represent the profit obtained from the retail channel and the direct channel without considering production cost disruptions; the third term is the possible underage cost and the fourth term denotes the possible disposal cost. Intuitively, the demand will increase if the production cost decreases, and vice versa. This is stated in the following lemma.
sc ≥Q  if Δc b 0, and D
sc ≤Q  if Δc > 0. Lemma 3. Suppose Qsc* is defined in Eq. (9). Then D sc sc All proofs are relegated in Appendix A. Lemma 3 indicates that if the actual production cost is lower than anticipated, the realized demand will exceed the original production quantity. And if the actual production cost is higher than anticipated, the realized demand will be less than the original production quantity. The results in Lemma 3 are intuitively reasonable. This is because if the production cost decreases, the central decision-maker may lower the selling prices, which will induce more customers to purchase and the demand resolved will exceed the original production quantity. If the production cost increases, the central decision-maker will have to raise the selling prices, which will induce fewer customers to purchase and the resolved demand will be less than the original production quantity. 4.1. Δc b 0 When Δc b 0, from Lemma 3 we know that the central decision-maker will bear possible related underage cost. After some simple algebra manipulating, the central decision-maker's problem can be formulated as maxπ
sc ¼ ðp
−c−ΔcÞ½ð1−ρÞa−α p
þ βp
 þ ðp
−c−ΔcÞ½ρa−α p
þ βp
−μ r 1 r d d 2 d r 1 p
r ;p
d

h

1 2

i 1 a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ ðα 1 þ α 2 −2βÞc ; 2

ð18Þ

s.t. 1 2

1 2

a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ ðα 1 þ α 2 −2βÞc≥0:

ð19Þ

The constraint given in Eq. (19) indicates that only when the resolved actual demand is larger than the original production quantity will the underage cost occur. The following proposition characterizes the central decision-maker's optimal prices and production quantity.

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S. Huang et al. / Economic Modelling 30 (2013) 521–538

Proposition 1. Given production cost disruption Δc b 0, the centralized dual-channel supply chain's optimal retail price and direct sale price are given by  p
r ¼

8 < α 2 ð1−ρÞ þ βρ a þ c þ Δc2 þ μ 1

if Δc ≤ −μ 1 ;

: α 2 ð1−ρÞ þ βρ a þ c2

if −μ 1 b Δc b 0;

8 < α 1 ρ þ βð1−ρÞ a þ c þ Δc2 þ μ 1

if Δc ≤ −μ 1 ;

: α 1 ρ þ βð1−ρÞ a þ c2

if −μ 1 b Δc b 0;

2ðα1 α2 −β2 Þ 2ðα1 α2 −β2 Þ

 p
d

¼

2ðα 1 α2 −β2 Þ 2ðα 1 α2 −β2 Þ

ð20Þ

ð21Þ

respectively, and the optimal production quantity is given by
 ¼ Q sc



1½a−ðα 1 2 1½a−ðα 2

1

þ α 2 −2βÞðc þ Δc þ μ 1 Þ þ α 2 −2βÞc

if Δc ≤ −μ 1 ; if −μ 1 b Δc b 0:

ð22Þ

From Proposition 1, we find that the original optimal production plan in the centralized dual-channel supply chain has some robustness when Δc b 0. No change in prices and production quantity are required when − μ1 b Δc b 0. Only when the production cost disruption Δc ≤ − μ1 should the central decision-maker raise the production quantity and lower the prices. It can be intuitively explained as follows. When the production cost decreases, Δc b 0, the manufacturer may have the motivation to increase the production quantity for a higher profit, which will cause underage cost. If − μ1 b Δc b 0, the deviation cost exceeds the saving caused by the production cost decrease. Thus, it is optimal to keep the original plan rather than produce more. If Δc ≤ −μ1, the saving caused by the production cost decrease exceeds the deviation cost. Thus, it is optimal to produce more rather than keep the original plan unchanged. The expression defined in Eq. (22) also leads to the following observation. If Δc ≤ −μ1, compared with the expression defined in Eq. (9), Q
sc can be interpreted as the optimal production quantity in a market with market scale a and marginal production cost c + Δc + μ1 without production cost disruptions. 4.2. Δc > 0 When Δc > 0, from Lemma 3 we know that the central decision-maker will bear possible related overage cost. After some simple algebra manipulating, the central decision-maker's problem can be formulated as maxπ
sc ¼ ðp
−c−ΔcÞ½ð1−ρÞa−α p
þ βp
 þ ðp
−c−ΔcÞ½ρa−α p
þ βp
 þ μ r 1 r d d 2 d r 2 p
r ;p
d

h

1 2

i 1 a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ ðα 1 þ α 2 −2βÞc ; 2

ð23Þ

s.t. 1 2

1 2

a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ ðα 1 þ α 2 −2βÞc≤0:

ð24Þ

The constraint given in Eq. (24) indicates that only when the resolved actual demand is smaller than the original production quantity, will the disposal cost occur. The following proposition characterizes the central decision-maker's optimal prices and production quantity. Proposition 2. Given production cost disruption Δc > 0, the centralized dual-channel supply chain's optimal retail price and direct sale price are given by  p
r

 p
d

¼

¼

8 < α 2 ð1−ρÞ þ βρ a þ c2 2ðα1 α2 −β2 Þ : α 2 ð1−ρÞ þ2 βρ 2ðα1 α2 −β Þ

aþ

if 0 b Δc ≤ μ 2 ;

c þ Δc−μ 2 2

if Δc > μ 2 ;

8 < α 1 ρ þ βð1−ρÞ a þ c2

if 0 b Δc ≤ μ 2 ;

2 : α 1 ρ þ βð1−ρÞ a þ c þ Δc−μ 2

if Δc > μ 2 ;

2ðα 1 α2 −β2 Þ 2ðα 1 α2 −β

2

Þ

ð25Þ

ð26Þ

respectively, and the optimal production quantity is given by
 ¼ Q sc



1½a−ðα 1 2 1½a−ðα 2

1

þ α 2 −2βÞc if 0 b Δc ≤ μ 2 ; þ α 2 −2βÞðc þ Δc−μ 2 Þ if Δc > μ 2 :

ð27Þ

Proposition 2 shows that the original production plan in the centralized dual-channel supply chain also has some robustness when Δc > 0. No change in prices and production quantity are required when 0 b Δc ≤ μ2. Only when the production cost disruption Δc ≥ μ2 should the central decision-maker reduce the production quantity and raise the prices. It can be intuitively explained as follows. When the production cost increases, Δc > 0, the manufacturer may have the motivation to reduce the production quantity, which will cause disposal cost. If 0 b Δc b μ2, the deviation cost exceeds the production cost increase. Thus, it is optimal to keep the original plan unchanged. If Δc ≥ μ2, the production cost increase exceeds the deviation cost. Thus, it is optimal to revise the plan with a deviation cost being incurred. Proposition 2 also leads to the fol
 can be interpreted as the optimal production quantity in a market with lowing observation. If Δc ≥ μ2, the optimal production quantity Q sc market scale a and marginal production cost c + Δc − μ2 without production cost disruptions. In summary, the following proposition characterizes the optimal pricing strategy and production quantity for the centralized dual-channel supply chain and compares the revised optimal decisions with the original optimal decisions explicitly.

S. Huang et al. / Economic Modelling 30 (2013) 521–538

527

Proposition 3. Given production cost disruption Δc, the centralized dual-channel supply chain's total profit is maximized at the following values of the retail price and direct sale price:   8  Δc þ μ 1 > > < pr − − 2  p
r ¼ pr   > > : p þ Δc−μ 2 r

if Δc b −μ 1 ; if −μ 1 ≤ Δc ≤ μ 2 ;

  8  Δc þ μ 1 > > < pd − − 2  p
d ¼ pd   > > : p þ Δc−μ 2 d

ð28Þ

if Δc > μ 2 ;

2

if Δc b −μ 1 ; if −μ 1 ≤ Δc ≤ μ 2 ;

ð29Þ

if Δc > μ 2 ;

2

and the centralized dual-channel supply chain's optimal production quantity is given by
 Q sc

8  < Q sc þ 12 ½−ðα 1 þ α 2 −2βÞðΔc þ μ 1 Þ if Δc b −μ 1 ; ¼ Q sc if −μ 1 ≤ Δc ≤ μ 2 ; :  1 Q sc − 2 ½ðα 1 þ α 2 −2βÞðΔc−μ 2 Þ if Δc > μ 2 :

ð30Þ

The expressions in Eqs. (28) and (29) imply that if the production cost disruption Δc b −μ1 or Δc >μ2, the optimal retail price p
r and direct sale price p
d can be obtained by adding an adjustment term to the original retail price pr* and direct sale price pd*, respectively. Specifically, when Δc b −μ1, the adjustment term is given by (Δc+μ1)/2; when Δc >μ2, the adjustment term is given by (Δc −μ2)/2. Note that the optimal retail price p
r and direct sale price p
d given in Eqs. (28) and (29) are always linearly increasing with Δc. The expression given in Eq. (30) leads to another observation. No change in production quantity is required when −μ1 ≤Δc ≤μ2. When the production cost disruption Δc ≤−μ1, the optimal production quantity Q
sc equals the original optimal production quantity Qsc* plus an adjustment term [−(α1 +α2 −2β)(Δc +μ1)] /2. When the production cost disruption Δc ≥μ2, the
 equals the original optimal production quantity Qsc* minus an adjustment term [(α1 +α2 −2β)(Δc −μ2)]/2. optimal production quantity Q sc The results in Proposition 3 are similar to those in Xu et al. (2006), where they found that the optimal retail prices in the traditional retail channel under production cost disruptions can be obtained by adding some adjustment terms, which are the same as those given in Proposition 3, to the original retail price. The optimal production quantity in the traditional retail channel under production cost disruptions can also be obtained by adding some adjustment terms to the original production quantity. However, the adjustment terms in theirs are different from those given in Proposition 3 due to channel structure difference. The range [− μ1,μ2] is referred to as the production cost disruption robust region in Xu et al. (2006). It is interesting to find that the production cost disruption robust region in the centralized dual-channel supply chains is the same as that in the centralized retail channel presented in Xu et al. (2006). The results in Proposition 3 are also similar to those in Huang et al. (2012), where the optimal prices and production quantity in dual-channel supply chains under demand disruptions can be obtained by adding some adjustment terms to the original production plan; however, the production cost robust region in this paper is different from the demand disruption robustness region in Huang et al. (2012). The results presented in Proposition 3 complement the literature by indicating that the original production plan in the dual-channel supply chain is robust with production cost disruptions to some extent. The total profit of the centralized dual-channel supply chain after the disruption in production cost is resolved is given by

 π
sc

8 h i 2 2 α 1 ρ þ 2βρð1−ρÞ þ α 2 ð1−ρÞ 2 1 > > if Δc < −μ 1 ; a − 2 aðc þ ΔcÞ þ α 1 þ α42 −2β ðc þ Δc þ μ 1 Þ2 −2cμ 1 > 4ðα α −β Þ > < 2 2 2 α ρ þ 2βρ ð 1−ρ Þ þ α ð 1−ρ Þ α þ α −2β 1 2 ¼ a −12 aðc þ ΔcÞ þ 1 42 cðc þ 2ΔcÞ if −μ 1 ≤ Δc ≤ μ 2 ; 4ðα α −β Þ > > h i > 2 2 > 2 2 α ρ þ 2βρ ð 1−ρ Þ þ α ð 1−ρ Þ α þ α −2β : 1 2 if Δc > μ 2 : a −12 aðc þ ΔcÞ þ 1 42 ðc þ Δc−μ 2 Þ þ 2cμ 2 4ðα α −β Þ 1

2

1

2

1

2

2

2

ð31Þ

2

4.3. Value of reaction to production cost disruptions Subsequently, it is necessary to examine the value of taking timely response to the production cost disruptions in the centralized dual-channel supply chain. If a production cost disruption occurs while the central decision-maker cannot react to it immediately, he can only maintain the original optimal pricing and production decisions, which is likely to be suboptimal. We compare the policy proposed in Proposition 3 with the policy that maintains the same prices given in Lemma 1. To illustrate the difference, define π^ sc as the centralized dual-channel supply chain's total profit, which is obtained by maintaining the original retail price and direct sale price defined in Lemma 1 when a production cost disruption occurs. We obtain
 
 
      π^ sc ¼ pr −c−Δc ð1−ρÞa−α 1 pr þ βpd þ pd −c−Δc ρa−α 2 pd þ βpr ¼ πsc −ΔcQ sc :

ð32Þ

The first term denotes the profit from the retail channel, and the second term denotes the profit from the direct channel. Note that the deviation costs are absent in Eq. (32). This is because if the central decision-maker uses the pricing decisions defined in Lemma 1, the demand will be the same, and no deviation cost will occur. By comparing the profit π^ sc defined in Eq. (32) with the profit π
sc defined in Proposition 3, we have the following results presented in Table 1. Table 1 lists the profit difference in different cases, respectively. The calculations show that taking timely response to production cost disruptions and using the prices defined in Proposition 3 is always beneficial (in non-strict sense) to the supply chain. Thus, when a production cost disruption occurs, it is always beneficial for the central decision-maker to react to it immediately. Moreover, given production cost disruption Δc, the profit difference quadratically decreases in μ1 and Δc when Δc b − μ1, and the profit difference quadratically increases in Δc and decreases in μ2 when Δc > μ2, highlighting the importance of incorporating disruption management into the dual-channel supply chain management. The analysis on the value of reaction to production cost disruptions in the traditional retail channel is absent in Xu et al. (2006). The analysis above is similar to that in Huang et al. (2012), where they found that it was always beneficial for the central decision-maker to utilize the revised

528

S. Huang et al. / Economic Modelling 30 (2013) 521–538

Table 1 Value of reaction to production cost disruptions in the centralized dual-channel supply chain. Production cost disruption Δc

Profit difference = π
sc −π^ sc

(Δc)′ impact

Δc b − μ1 − μ1 ≤Δc ≤ μ2 Δc > μ2

α 1 þ α 2 −2β ðΔc þ μ 1 Þ2 4

α 1 þ α 2 −2β ðΔc þ μ 1 Þ 2

b0

α 1 þ α 2 −2β ðΔc−μ 2 Þ 2

>0

>0

0

α 1 þ α 2 −2β ðΔc−μ 2 Þ2 4

(μ1 orμ2)′ impact

0

>0

α 1 þ α 2 −2β ðΔc þ μ 1 Þ 2

b0

0

α 1 þ α 2 −2β ðμ 2 −ΔcÞ 2

b0

decisions under demand disruptions in a dual-channel supply chain. The analysis above also implies that the central decision-maker will always prefer using the revised decisions under production cost disruptions in a dual-channel supply chain.

5. Decentralized decisions with production cost disruptions The above section analyzes the optimal pricing and production decisions in the centralized dual-channel supply chain with production cost disruptions. Next, we move on to the scenario of the decentralized dual-channel supply chain. Usually, the deviation costs are borne by the manufacturer to simplify the models and analysis (Chen and Zhuang, 2011). In some cases, the deviation costs are borne by the retailer rather than the manufacturer (Xiao et al., 2007). In order to simplify the analysis, we assume that the manufacturer assumes the deviation costs in this paper. This is common in practice, for example, some products in the supermarket are owned by the manufacturer. The supermarket pays the manufacturer only after the products are sold and won't assume any disposal cost or underage cost (Huang et al., 2012). After a production cost disruption is detected, both the retailer and the manufacturer will have to revise their pricing strategies in order to maximize their individual profits. We first analyze their individual optimal pricing decisions after the production cost disruption is resolved. Then we analyze the impact of production cost disruptions on the manufacturer's profit and the retailer's profit, respectively. Analogously, we consider the problem in two cases, i.e., Δc b 0 and Δc > 0. With the assumptions and notations above, given the production cost disruption Δc, the retailer's profit is denoted by
¼ ðp
−w
ÞD
Þ½ð1−ρÞa−α 1 p
r þ βp
d ; π
r ¼ ðp
r −w r r

ð33Þ

by utilizing the first-order condition, we derive the retailer's optimal pricing strategy as follows p
r ¼

1 ½ð1−ρÞa þ βp
d 2α 1


: þ α1 w

ð34Þ

The manufacturer's profit is denoted by    
−Q 0 þ −μ Q 0 −D
þ ðp
−c−ΔcÞD
−μ D
þ;
ÞD π
m ¼ ðw−c−Δc sc sc r d d sc sc 1 2

ð35Þ

where the first term denotes the profit from the retail channel, the second term denotes the profit from the direct channel, the third term denotes the possible underage cost, and the fourth term denotes the possible disposal cost. Intuitively, when production cost decreases the market demand will increase, and vice versa. This is stated in the following lemma. 0
sc ≥Q 0 if Δc b 0, and D
sc ≤Q 0 if Δc > 0. Lemma 4. Suppose Qsc is defined in Eq. (16). Then D sc sc

Lemma 4 is similar to Lemma 3. The result in Lemma 4 is also intuitively reasonable. When the production cost decreases, the manufacturer can lower the wholesale price and direct sale price correspondingly. Then the retailer can lower the retail price, which will induce more customers to purchase and the demand will exceeds the original production quantity. When the production cost increases, the manufacturer will have to raise the wholesale price and direct sale price correspondingly. Then the retailer can raise the retail price, which will induce fewer customers to purchase and the resolved demand will be less than the original production quantity. 5.1. Δc b 0 When Δc b 0, from Lemma 4 we know that the manufacturer may bear possible underage cost. Putting Eqs. (16) and (34) into Eq. (35), the manufacturer's decision problem can be formulated as follows n


d þ 2αβ ½ð1−ρÞa þ βp
d þ α 1 w

 þ ðp
d −c−ΔcÞ ρa−α 2 p
½ð1−ρÞa þ βp
d −α 1 w maxπ
m ¼ 12 ðw−c−ΔcÞ 1
p
d w;

n   o 0 μ1
−2α ½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa− 2α 1 α 2 −α 1 β−β2 p
d −α 1 ðα 1 −βÞw−2α 1 Q sc ;

o ð36Þ

1

s.t.
p
d ≥w;

ð37Þ

  2 0
½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa− 2α 1 α 2 −α 1 β−β p
d −α 1 ðα 1 −βÞw−2α 1 Q sc ≥ 0:

ð38Þ

S. Huang et al. / Economic Modelling 30 (2013) 521–538

529

The constraint given in Eq. (37) is the non-arbitrary condition, and the constraint given in Eq. (38) indicates that only when the actual demand is larger than the original plan, will the underage cost occur. The following proposition characterizes the manufacturer's optimal pricing decisions when Δc b 0. Proposition 4. In a manufacturer Stackelberg game when Δc b 0, the manufacturer maximizes its profit with the following direct sale price and wholesale price: Case 1 Δc b − μ1 8 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c þ Δc2 þ μ 1 2 α −β þ2α α −2α βÞ case1 p
d ¼ α ρð þ βð1−ρÞ : 1 a þ c þ Δc2 þ μ 1 2ðα α −β Þ

^; if 0 b ρ b ρ

8 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c þ Δc2 þ μ 1

^; if 0 b ρ b ρ

2 1

1


w

case1

¼

2

2

1

2

1

^ ≤ ρ b 1; if ρ

2

2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ : α 2 ð1−ρÞ þ2 βρ a þ c þ Δc2 þ μ 1 2ðα 1 α 2 −β Þ

^ ≤ ρ b 1; if ρ

ð39Þ

ð40Þ

Case 2 − μ1 ≤ Δc b 0 case2 p
d

¼

8 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c2

^; if 0 b ρ b ρ

: α 1 ρ þ βð1−ρÞ a þ c2

^ ≤ ρ b 1; if ρ

8 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c2

^; if 0 b ρ b ρ

: α 2 ð1−ρÞ þ βρ a þ c2

^ ≤ ρ b 1: if ρ

2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ 2ðα 1 α 2 −β 2 Þ


w

case2

¼

2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ 2ðα 1 α 2 −β 2 Þ

ð41Þ

ð42Þ

Proposition 4 indicates that if Δc b − μ1, the constraint given in Eq. (38) is not binding, which means that the optimal production quantity should be raised. While if − μ1 ≤ Δc b 0, the constraint given in Eq. (38) is binding, which implies that the original optimal plan should be kept unchanged. Furthermore, for any given production cost disruption Δc b 0, if customers' preference for the direct channel ρ is smaller than a threshold, the manufacturer should set the wholesale price equal the direct sale price and consider the retailer as an end customer. If customers' preference for the direct channel ρ exceeds a threshold, the manufacturer should set different wholesale price and direct sale price (Hua et al., 2010; Huang et al., 2012). To be specific, the manufacturer's optimal direct sale price is identical to the optimal direct sale price in the centralized dual-channel supply chain, and the optimal wholesale price is identical to the optimal retail price in the centralized dual-channel supply chain when no production cost disruption occurs. When Δc b − μ1, the optimal direct sale price and the optimal wholesale price and can be regarded as the optimal direct sale price and optimal wholesale price in a new market with market scale a and marginal production cost c + Δc + μ1 in the decentralized dual-channel supply chain under no production cost disruptions. 5.2. Δc > 0 When Δc > 0, from Lemma 4 we know that the manufacturer may bear possible overage cost. Putting Eqs. (16) and (34) into Eq. (35), the manufacturer's decision problem can be formulated as follows  
m ¼ 1ðw−c−ΔcÞ

 þ ðp
d −c−ΔcÞ ρa−α 2 p
d þ β ½ð1−ρÞa þ βp
d þ α 1 w
 maxw; ½ð1−ρÞa þ βp
d −α 1 w
p
d π 2 2α 1 n   o 2 0 μ
þ 2 ½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa− 2α 1 α 2 −α 1 β−β p
d −α 1 ðα 1 −βÞw−2α 1 Q sc ;

ð43Þ


p
d ≥w;

ð44Þ

  2 0
½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa− 2α 1 α 2 −α 1 β−β p
d −α 1 ðα 1 −βÞw−2α 1 Q sc ≤ 0:

ð45Þ

2α

s.t.

The constraint defined in Eq. (45) indicates that the disposal cost occurs only when the actual demand is less than the original production quantity. The following proposition characterizes the manufacturer's optimal pricing decisions when the production cost disruption Δc > 0. Proposition 5. In a manufacturer Stackelberg game when Δc > 0, the manufacturer maximizes its profit with the following direct sale price and wholesale price: Case 3 0 b Δc ≤ μ2 case3 ¼ p
d


w

case3

¼

8 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c2 2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ Þ : α 1 ρ þ βð1−ρ a þ c2 2ðα 1 α 2 −β 2 Þ

8 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c2 2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ : α 2 ð1−ρÞ þ2 βρ a þ c2 2ðα 1 α 2 −β Þ

^; if 0 b ρ b ρ ^ ≤ ρ b 1; if ρ ^; if 0 b ρ b ρ ^ ≤ ρ b 1; if ρ

ð46Þ

ð47Þ

530

S. Huang et al. / Economic Modelling 30 (2013) 521–538

Case 4 Δc > μ2 case4 p
d ¼

8 2 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c þ Δc−μ 2

^; if 0 b ρ b ρ

2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ Þ 2 : α 1 ρ þ βð1−ρ a þ c þ Δc−μ 2 2ðα 1 α 2 −β 2 Þ


case4 ¼ w

ð48Þ

^ ≤ ρ b 1; if ρ

8 2 < 2α 1 ρ þ ðα 1 þ βÞð1−ρÞ a þ c þ Δc−μ 2

^; if 0 b ρ b ρ

2 : α 2 ð1−ρÞ þ βρ a þ c þ Δc−μ 2

^ ≤ ρ b 1: if ρ

2ðα 21 −β2 þ2α 1 α2 −2α 1 βÞ 2ðα 1 α 2 −β 2 Þ

ð49Þ

Similarly, Proposition 5 indicates that if 0 b Δc b μ2, the constraint given in Eq. (45) is binding, which implies that the original optimal plan should be kept unchanged. While if Δc > μ2 the constraint given in Eq. (45) is not binding, which indicates that the optimal production quantity should be reduced, and the optimal prices should be changed accordingly. Besides, when Δc > μ2, the optimal direct sale price and the optimal wholesale price can be regarded as the optimal direct sale price and optimal wholesale price in a new market with market scale a and marginal production cost c + Δc − μ2 in the decentralized dual-channel supply chain under no production cost disruptions. Note that when Δc > μ2, the optimal direct sale price and wholesale price under production cost disruptions are larger than those without production cost disruptions. From Propositions 4 and 5 we can obtain the retailer's optimal retail price and the decentralized dual-channel supply chain's optimal production quantity when production cost disruption Δc is resolved. They are summarized in the following proposition. Proposition 6. In a manufacturer Stackelberg game for given production cost disruption Δc, the retailer's optimal retail price is given by

case1 p
r ¼

8   2 2 > 2α ðα þ β Þρ þ 3α 1 −β þ 4α 1 α 2 −2α 1 β ð1−ρÞ > < 1 1 a þ α 14αþ βðc þ Δc þ μ 1 Þ

^; if 0 b ρ b ρ

> 2α βρ þ > : 1

^ ≤ ρ b 1; if ρ

case2 case3 p
r ¼ p
r ¼

case4 p
r ¼



4α 1 ðα21 −β2 þ2α 1 α 2 −2α 1 βÞ  2

3α 1 α 2 −β

4α 1 ðα 1 α 2 −β

2

ð1−ρÞ

Þ

1

a þ α 14αþ βðc þ Δc þ μ 1 Þ 1

8   2 2 > 2α ðα þ β Þρ þ 3α 1 −β þ 4α 1 α 2 −2α 1 β ð1−ρÞ > < 1 1 a þ α 14αþ βc 4α 1 ðα 21 −β 2 þ2α 1 α 2 −2α 1 β Þ   2 > 2α 1 βρ þ 3α 1 α 2 −β ð1−ρÞ > : a þ α 14αþ1 βc 4α 1 ðα1 α2 −β2 Þ

^; if 0 b ρ b ρ

1



4α 1 ðα21 −β2 þ2α 1 α 2 −2α 1 βÞ  2

3α 1 α 2 −β

4α 1 ðα 1 α 2 −β

2

Þ

ð1−ρÞ

ð51Þ

^ ≤ ρ b1: if ρ

8   2 2 > 2α ðα þ β Þρ þ 3α 1 −β þ 4α 1 α 2 −2α 1 β ð1−ρÞ > < 1 1 a þ α 14αþ βðc þ Δc−μ 2 Þ > 2α βρ þ > : 1

ð50Þ

^; if 0 b ρ b ρ

1

^ ≤ρb1; a þ α 14αþ βðc þ Δc−μ 2 Þif ρ

ð52Þ

1

and the decentralized dual-channel supply chain's optimal production quantity are given by   o 8 n 2 2 1 > if Δc b −μ 1 ; > > 4α n½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa−α 1 −β þ 2α 1 α 2 −2α 1 βðcoþ Δc þ μ 1 Þ < 2 2 1 ¼ 4α ½2α 1 ρ þ ðα 1 þ βÞð1−ρÞa− α 1 −β þ 2α 1 α 2 −2α 1 β c if −μ 1 ≤ Δc ≤ μ 2 ; > n   o > > : 1 ½2α ρ þ ðα þ βÞð1−ρÞa− α 2 −β2 þ 2α α −2α β ðc þ Δc−μ Þ if Δc > μ 2 : 1 1 1 2 1 1 4α 2 1


Q sc

1

ð53Þ

1

Note from Eq. (53) that the original plan also has some robustness under a disrupted production cost scenario in the decentralized dual-channel supply chain. When the production cost is mildly perturbed (− μ1 ≤ Δc ≤ μ2), no change in production quantity is required, and the optimal direct sale price and retail price should be kept unchanged. And the manufacturer and the retailer should treat it as normal and do not need to take actions to deal with it. Only when the production cost disruption Δc b − μ1 or Δc > μ2, will it be necessary for the manufacturer to change the production quantity. Furthermore, if Δc b − μ1, the optimal production quantity can be regarded as the optimal production quantity in a market with market scale a and marginal production cost c + Δc + μ1 under no disrupted production costs. Analogously, if Δc > μ2, the optimal production quantity can be regarded as the optimal production quantity in a market with market scale a and marginal production cost c + Δc − μ2 under no disrupted production costs. The robustness of the original production plan in the decentralized retail channel under production cost disruptions is not explicitly addressed in Xu et al. (2006). Proposition 6 indicates that the original production plan also has some robustness under production cost disruptions in the decentralized dual-channel supply chain. It is interesting to find that the production cost disruption robust region [− μ1,μ2] in the decentralized dual-channel supply chain is the same as that in the centralized dual-channel supply chain. This also indicates that the production cost disruption robust region is a general concept both in the centralized and decentralized dual-channel supply chain. The retailer's optimal profit in the decentralized dual-channel supply chain is given by

case1 π
r ¼

8 > > < > > :

" 1

16α 1

1

16α 1

#2

  2 2 2α 1 ðβ−α 1 Þρ þ α 1 −β þ 4α 1 α 2 −4α 1 β ð1−ρÞ α 21 −β2 þ 2α 1 α 2 −2α 1 β

a−ðα 1 −βÞðc þ Δc þ μ 1 Þ

case2 case3 π
r ¼ π
r ¼

> > :

" 1

  2 2 2α 1 ðβ−α 1 Þρ þ α 1 −β þ 4α 1 α 2 −4α 1 β ð1−ρÞ

1

2

16α 1

16α 1

α21 −β2 þ2α 1 α 2 −2α 1 β

½ð1−ρÞa−ðα 1 −βÞc

ð54Þ

^ ≤ ρ b 1; if ρ

½ð1−ρÞa−ðα 1 −βÞðc þ Δc þ μ 1 Þ2 8 > > <

^; if 0 < ρ < ρ

#2 a−ðα 1 −βÞc

^; if 0 < ρ < ρ ^ ≤ ρ b 1; if ρ

ð55Þ

S. Huang et al. / Economic Modelling 30 (2013) 521–538

case4 ¼ π
r

8 > > < > > :

" 1

  2 2 2α 1 ðβ−α 1 Þρ þ α 1 −β þ 4α 1 α 2 −4α 1 β ð1−ρÞ α 21 −β 2 þ2α 1 α 2 −2α 1 β

16α1

1

16α1

½ð1−ρÞa−ðα 1 −βÞðc þ Δc−μ 2 Þ

531

#2 a−ðα 1 −βÞðc þ Δc−μ 2 Þ

^; if 0 < ρ < ρ

ð56Þ

^ ≤ ρ b 1: if ρ

2

Comparing Eqs. (54)–(56) with Eq. (15), we obtain the following observations. In case 1, the retailer's optimal profit can be interpreted as the optimal profit in a market with market scale a and marginal production cost c +Δc +μ1 under no disrupted production cost. Analogously, In case 4, the retailer's optimal profit can be interpreted as the optimal profit in a market with market scale a and marginal production cost c+Δc−μ2 under no disrupted production cost. When the production cost is mildly perturbed, the retailer's optimal profit is independent of the parameters μ1 and μ2. This is because the manufacturer will not change his optimal production quantity, and both the manufacturer and the retailer will keep their original pricing decisions unchanged. Therefore, the demand will be the same. The manufacturer's optimal profit is given by 2

case1 0 0 α −β π
m ¼ πm −ΔcQ sc þ 1

2

2 þ 2α 1 α 2 −2α 1 β ðΔc þ μ 1 Þ ; 8α 1

ð57Þ

case2 case3 0 0 π
m ¼ π
m ¼ πm −ΔcQ sc ; 2

case4 0 0 α −β π
m ¼ πm −ΔcQ sc þ 1

2

ð58Þ

2 þ 2α 1 α 2 −2α 1 β ðΔc−μ 2 Þ : 8α 1

ð59Þ

From Eqs. (57)–(59), we find that in case 2 (case 3), if the manufacturer keeps the original production quantity unchanged and both the 0 manufacturer and the retailer maintain their original pricing policies, the manufacturer obtains (suffers) a profit (loss) of ΔcQsc . In other words, the production cost disruption Δc leads to a profit (loss) of Δc for each item the manufacturer produced. When Δc b − μ1, the manufac0 turer also suffers a loss of ΔcQsc , though the manufacturer changes the production quantity and both the manufacturer and the retailer change   their pricing policies. However, there is an additional term denoted by α 21 −β2 þ 2α 1 α 2 −2α 1 β ðΔc þ μ 1 Þ2 =8α 1 , which can be regarded as the payoff from timely response to production cost disruptions by the manufacturer. When Δc > μ2, the additional term is given by   α 21 −β2 þ 2α 1 α 2 −2α 1 β ðΔc−μ 2 Þ2 =8α 1 . 5.3. Value of reaction to production cost disruptions Next, we examine the value of reaction to the actual production cost disruptions in the decentralized dual-channel supply chain. We use π^ m to denote the manufacturer's profit obtained by maintaining the original production plan given in Lemma 2 when a production cost disruption occurs. i    h  0 0 0 0 0 0 0 0 π^ m ¼ w −c−Δc ð1−ρÞa−α 1 pr þ βpd þ pr −c−Δc ρa−α 2 pd þ βpr ¼ πm −ΔcQ sc :

ð60Þ

The profit difference between π
casei and π^ m for the manufacturer when a production cost disruption occurs is illustrated in the following m Table 2. In the decentralized supply chain it is the manufacturer who is able to detect the production cost disruptions. The production cost disruptions have direct impact on the manufacturer's profit, because the manufacturer's profit comes from both channels and the margin in each channel decreases with production cost. Therefore, intuitively, the manufacturer has the motivation to reveal the production cost disruption information to the retailer and take timely response to it. The results in Table 2 indicate that the manufacturer will always take timely response to the production cost disruptions and utilize the pricing strategy defined in Propositions 4 and 5. When − μ1 ≤ Δc ≤ μ2, the profit difference is zero because the manufacturer and the retailer will not change their pricing decisions. The profit difference increases with Δc and decreases with μ2 when Δc > μ2, and decreases with Δc and μ1 when Δc b −μ1. Similarly, we use π^ r to denote the retailer's profit obtained by maintaining the original production plan defined in Lemma 2 when a production cost disruption occurs.  h i 0 0 0 0 case2 case3 ð1−ρÞa−α 1 pr þ βpr ¼ π
r ¼ π
r : π^ r ¼ pr −w

ð61Þ

When a production cost disruption occurs, the profit difference between π
casei and π^ r for the retailer is illustrated in Table 3. From the analysis r above we know that the manufacturer will always reveal the production cost disruption information to the retailer and take timely response to it once a production cost disruption occurs. Therefore, the retailer will utilize the retail price defined in Proposition 6. From Table 3 we know that the retailer benefits when Δc b − μ1 and suffers when Δc > μ2. This is because when Δc b − μ1, if the production cost decreases and the Table 2 Value of reaction to Δc for the manufacturer in the decentralized dual-channel supply chain. Cases

^ Profit difference = π
casei m −π m

(Δc)′ impact

(μ1 orμ2)′ impact

Case 1

α 21 −β 2 þ 2α 1 α 2 −2α 1 β ðΔc þ μ 1 Þ2 8α 1

α 21 −β 2 þ 2α 1 α 2 −2α 1 β ðΔc þ μ 1 Þb0 4α 1

α 21 −β 2 þ 2α 1 α 2 −2α 1 β ðΔc þ μ 1 Þb0 4α 1

α 1 −β 2 þ 2α 1 α 2 −2α 1 β ðΔc−μ 2 Þ > 0 α1

α 21 −β 2 þ 2α 1 α 2 −2α 1 β ðμ 2 −ΔcÞb0 α1

Case 2 Case 3 Case 4

0 02

>0

α 1 −β 2 þ 2α 1 α 2 −2α 1 β ðΔc−μ 2 Þ2 > 0 8α 1

0 02

0 0

532

S. Huang et al. / Economic Modelling 30 (2013) 521–538

Table 3 Value of reaction to Δc for the retailer in the decentralized dual-channel supply chain. Cases

ρ

Case 1

^ 0bρbρ

Case 2 Case 3 Case 4

^ ≤ρb1 ρ ^ 0bρbρ ^ ≤ρb1 ρ ^ 0bρbρ ^ ≤ρb1 ρ ^ 0bρbρ

^ ≤ρb1 ρ

Profit difference = π
casei −π^ r r "  −ðα 1 −β ÞðΔc þ μ 1 Þ 16α 1

2

   α 21 −β2 þ 4α 1 α 2 −4α 1 β − 3α 21 −β 2 þ 4α 1 α 2 −6α 1 β ρ α 21 −β 2

þ 2α 1 α 2 −2α 1 β

−ðα 1 −β ÞðΔc þ μ 1 Þ ½2ð1−ρÞa−ðα 1 −βÞð2c þ Δc þ μ 1 Þ 16α 1

0 0 0 0

"

−ðα 1 −β ÞðΔc−μ 2 Þ 16α 1

2

>0

    α 21 −β 2 þ 4α 1 α 2 −4α 1 β − 3α 21 −β 2 þ 4α 1 α 2 −6α 1 β ρ α 21 −β 2

#

a−ðα 1 −β Þð2c þ Δc þ μ 1 Þ > 0

þ 2α 1 α 2 −2α 1 β

# a−ðα 1 −β Þð2c þ Δc−μ 2 Þ b0

−ðα 1 −β ÞðΔc−μ 2 Þ ½2ð1−ρÞa−ðα 1 −βÞð2c þ Δc−μ 2 Þb0 16α 1

manufacturer and the retailer both take timely response to it, both members can lower the prices to attract more customers to purchase and will benefit from market demand increase. However, when Δc > μ2, the manufacturer will raise the wholesale price, and the retailer will have to raise the retail price to make a positive profit. But the market demand from the retail channel decreases and leads to profit loss for the retailer. In summary, the above results can be stated in the following proposition. Proposition 7. The manufacturer will utilize the revised pricing strategy defined in Propositions 4 and 5 whenever a production cost disruption occurs. The manufacturer will benefit from it and the retailer gets better off only when Δc b − μ1 and worse off when Δc > μ2. At last, we would like to study which decision mode, centralized or decentralized, is more robust for the optimal production plan under production cost disruptions. From the analysis above, we know that the production cost disruption robust region is [− μ1,μ2] in both the centralized and decentralized dual-channel supply chain. The following proposition gives the result. Proposition 8. The robustness of the original production plan in the centralized dual-channel supply chain is the same as that in the decentralized dual-channel supply chain under production cost disruptions. Proposition 8 above indicates that the two decision modes, the centralized decision and the decentralized decision, have equal robustness. The production cost disruption robust region depends on the parameters of μ1 and μ2, and has nothing to do with customers' preference for the direct channel. In other words, customers' preference for the direct channel has no impact on the robustness of the two decision modes. Next, we would like to compare the result given in Proposition 8 with what presented in Huang et al. (2012). In Huang et al. (2012), they pointed out that regarding the two decision modes, the centralized decision mode or the decentralized decision mode, neither is always ^ ≤ρb1, the centralized more robust than the other one under demand disruptions. It depends on customers' preference for the direct channel. If ρ ^ , the decentralized dual-channel supply chain is more robust with dual-channel supply chain is more robust with demand disruptions. If 0bρbρ demand disruptions. This also indicates that customers' preference for the direct channel has great impact on the robustness of the two decision modes under demand disruptions. The different results regarding the robustness can be explained as follows. In the scenario of production cost disruptions, as marginal deviation cost μ1, μ2 and production cost disruptions Δc are both cost related terms, they affect the pricing strategy directly. In the scenario of demand disruptions, since the demand disruption Δa is not a cost related term, the transformation of demand disruption Δa from a demand related term to a cost related term is only possible by means of ρ. 6. Numerical analysis According to the theoretical analysis above, we find that the production cost disruption has an important impact on the optimal decisions of the supply chain members and the profitability of the dual-channel supply chain. Hereby we present some numerical examples to illustrate the effect of production cost disruption on the optimal pricing and production decisions for the dual-channel supply chain. In the following example, we assume the market scale a = 150, the originally estimated unit production cost c = 10, the price elasticity coefficient α1 = 4, α2 = 3, β = 1, the unit underage cost μ1 = 2, the unit disposal cost μ2 = 2, customers' preference for the direct channel ρ = 0.2, and the production cost disruption Δc ∈ [−6,6]. We first analyze the impact of production cost disruption on the optimal pricing and production quantity decisions, which is depicted in Figs. 2 and 3. Fig. 2 depicts that when the production cost disruption is small (Δc ∈ [−2,2]), the optimal prices in the centralized and decentralized dual-channel supply chain should be kept unchanged. However, when the production cost disruption Δc > 2 or Δc b −2, in the centralized dual-channel supply chain, the central decision-maker

should change the retail price and direct sale price. And both the optimal retail price and the optimal direct sale price are increasing with the production cost disruption even if there are deviation costs. In the decentralized dual-channel supply chain, the optimal direct sale price and the optimal wholesale price, together with the optimal retail price are all increasing with Δc. Note that the optimal direct sale price and wholesale price in the decentralized supply chain are identical when ρ= 0.2. We also observe that the direct sale price and the retail price in the decentralized dual-channel supply chain are larger than those in the centralized dual-channel supply chain, respectively, due to double marginalization effect. Fig. 3 illustrates the centralized and decentralized dual-channel supply chain's optimal production quantities, respectively. When the production cost disruption is small (Δc ∈ [−2,2]), the optimal production quantity should be kept unchanged both in the centralized and decentralized supply chain. However, when the production cost disruption Δc > 2 or Δc b −2, the optimal production quantities in the centralized and decentralized dual-channel supply chain are both decreasing with production cost disruption Δc. This is intuitive because if the actual production cost increases, the selling prices will increase, which will

S. Huang et al. / Economic Modelling 30 (2013) 521–538

34

533

900 Retail price pr*

32

* Supply chain's profit πsc

Direct sale price pd*

Retailer's profit πr

800

Manufacturer's profit πm

30

Retail price pr

28

Direct sale price pd

700 600

Optimal profit

Optimal prices

Wholesale price w 26 24 22

500 400

20 300 18 200

16 14 -6

-4

-2

0

2

4

100 -6

6

Production cost disruption Δc

-4

-2

0

2

4

6

Production cost disruption Δc

Fig. 2. Optimal prices versus production cost disruption Δc when ρ = 0.2.

Fig. 4. Optimal profits versus production cost disruption Δc when ρ = 0.2.

cause fewer customers to buy the products. From Figs. 2 and 3 we can find that the original production plan has some robustness with production cost disruptions. Both selling prices and production quantity will be kept unchanged when the production costs are mildly perturbed (Δc ∈ [−2,2]). Fig. 2 shows that the direct sale price is smaller than the retail price in both the centralized and decentralized dual-channel supply chain if customers' preference for the direct channel ρ is small. From Fig. 3 we also find that the curve of the optimal production quantity in the centralized dual-channel supply chain is steeper than that in the decentralized dual-channel supply chain, which implies that the influence of production cost disruptions on the optimal production quantity is more significant in the centralized dual-channel supply chain. Fig. 4 illustrates the centralized dual-channel supply chain's profit π
sc , the manufacturer's profit π
m and the retailer's profit π
r in decentralized dual-channel supply chain when ρ= 0.2. We observe that when Δc b −2 or Δc >2, π
sc , π
m and π
r are all quadratically decreasing with production cost disruption Δc. While when Δc ∈ (−2,2), π
sc

and π
m are both linearly decreasing with production cost disruption Δc, and π
r is a constant and equals the retailer's profit in decentralized dual-channel supply chain without production cost disruptions. From Figs. 2–4 we also find that the optimal retail price and direct sale price in the decentralized dual-channel supply chain are higher than those in the centralized dual-channel supply chain. The optimal production quantity in the decentralized dual-channel supply chain is larger than that in the centralized dual-channel supply chain, and the centralized dual-channel supply chain's total profit π
sc is larger than the sum of π
m and π
r in the decentralized dual-channel supply chain, which means that the decentralization will lead to system inefficiency resulting from double marginalization effect. Next, keeping the other parameters unchanged, we assume that ρ= 0.5. The optimal pricing decisions, the optimal production quantities and the optimal profits for the centralized and decentralized dualchannel supply chain are illustrated in Figs. 5–7, respectively. From

60

30

*

Retail price pr*

Production quantity Qsc Production quantity Qsc

Direct sale price pd*

28

Retail price pr 50

Direct sale price pd

26

Wholesale price w

Optimal prices

Optimal production quantity

55

45

40

24

22

35

20

30

18

25

-6

-4

-2

0

2

4

6

Production cost disruption Δc Fig. 3. Optimal production quantity versus production cost disruption Δc when ρ = 0.2.

16 -6

-4

-2

0

2

4

Production cost disruption Δc Fig. 5. Optimal prices versus production cost disruption Δc when ρ = 0.5.

6

534

S. Huang et al. / Economic Modelling 30 (2013) 521–538

60

decentralized dual-channel supply chain are different and are affected by the value of ρ. On the other hand, for any given ρ, the optimal production quantity in the decentralized supply chain is smaller than that in the centralized supply chain.

*

Production quantity Qsc Production quantity Qsc

Optimal production quantity

55

7. Concluding remarks

50

45

40

35

30

-6

-4

-2

0

2

4

6

Production cost disruption Δc Fig. 6. Optimal production quantity versus production cost disruption Δc when ρ = 0.5.

Fig. 5 we find that when ρ= 0.5, the direct sale price p
d in the decentralized dual-channel supply chain is equal to the direct sale price p
d in the centralized dual-channel supply chain under disrupted
in the decentralized production cost scenario, and the wholesale price w dual-channel supply chain is equal to the retail price p
r in the centralized dual-channel supply chain under disrupted production cost scenario. Comparing Fig. 5 with Fig. 2 we find that in the decentralized dual-channel supply chain, the direct sale price can be higher than the retail price when ρ is large, and the direct sale price will be always higher than or equal to the wholesale price. Comparing Fig. 6 with Fig. 3, we can obtain the following observations. The optimal production quantities under production cost disruptions in the centralized dual-channel supply chain are identical and have nothing to do with customers' preference for the direct channel. However, the optimal production quantities under production cost disruptions in the

900 * Supply chain's profit πsc

800

Retailer's profit πr Manufacturer's profit πm

700

Optimal profit

600 500 400 300

Disruption management is a very important issue in dual-channel supply chain management. This paper examines the pricing and production quantity decision in a dual-channel supply chain with one manufacturer and one retailer under production cost disruptions. We examine this problem in two periods. In the first period, a production plan is made based on anticipated production cost, and in the second period the actual production cost is resolved and the supply chain has to make some adjustments to the original production plan. We mainly focus on how to revise the pricing and production quantity decisions in response to the production cost disruptions to achieve profit maximization. We first consider the problem in a centralized dual-channel supply chain where there is a central decision-maker who determines the retail price and direct sale price simultaneously. We then consider the problem in a decentralized dual-channel supply chain where the two members make pricing decisions independently supposing that the manufacturer assumes the possible deviation costs. We use the scenarios when no production cost disruptions as the baseline cases and calculate the value of timely reaction to the production cost disruptions in the two settings. Firstly, we find that in the centralized and decentralized dual-channel supply chain the original production plans have some robustness with production cost disruptions. Only when the production cost disruption exceeds some thresholds will the supply chain change the production quantity and pricing strategy. Secondly, it is always beneficial to know the actual value of the production cost disruptions and take timely response to it in the centralized dual-channel supply chain. In the decentralized dual-channel supply chain where only the manufacturer knows the production cost disruptions, it is also always beneficial for the manufacturer to reveal this information to the retailer and both members should take timely response to it. Thirdly, in the decentralized dual-channel supply chain, we characterize a threshold. The wholesale price should be equal to the direct sale price if customers' acceptance for the direct channel is below this threshold. The numerical analysis also shows that the direct sale price may be higher (lower) than the retail price in decentralized dual-channel supply chain, depending on customers' acceptance for the direct channel. At last, we find that the centralized and decentralized dual-channel supply chains have the same robustness in face of production cost disruptions. However, there might also be some limitations. Firstly, we assume the deviation costs are borne by the manufacturer for analytical simplicity, however, the retailer and the manufacturer may cooperate to share the deviation costs in practice. We believe that further analysis is needed to examine whether the results still hold in such situations. Secondly, we do not address the coordination problem in this paper partly because both the coordination of a dual-channel supply chain (Chen et al., 2012) and the disruption management in a dualchannel supply chain are somewhat complicated. The analysis may potentially require more complex technical skills. We believe that the coordination of a dual-channel supply chain with production cost disruptions is a subject worth further investigation.

200

Acknowledgments 100

0

-6

-4

-2

0

2

4

Production cost disruption Δc Fig. 7. Optimal profits versus production cost disruption Δc when ρ = 0.5.

6

The authors are grateful to Editors and the two anonymous referees for their valuable comments and suggestions, which have significantly helped improving the quality of this paper. This research was supported by the National Natural Science Foundation of China under grant numbers 70871044 and 71172093.

S. Huang et al. / Economic Modelling 30 (2013) 521–538

535

Appendix A
sc ≤Q  when Δc b 0, then the central decision-maker's decision problem is given by the problem Proof of Lemma 3. We first suppose that D sc
sc ¼ Q  when Δc b 0 if we relax Δc to be non-positive. In other words, defined in Eq. (23)–(24). From Proposition 2 we can easily verify that D sc
sc bQ  if Δc b 0. Therefore, we know that D
sc ≥Q  if Δc b 0. When Δc > 0, the result can be proved in a similar
sc ¼ Q  always dominates D D sc sc sc way, so we omit it here. Proof of Proposition 1. We can verify that the supply chain's total profit function π
sc is jointly concave in p
r and p
d , and the constraint given in Eq. (19) is linear, so there is only one pair of optimal solutions to this problem. In order to solve the constrained optimization problem, we introduce the Lagrange multiplier λ≥0 and relax the constraint given in Eq. (19). The KKT conditions of the problem defined in Eqs. (18)–(19) are 8 ð1−ρÞa−2α 1 p
r þ 2βp
d þ ðα 1 −βÞðc þ ΔcÞ−ðα 1 −βÞðλ−μ 1 Þ ¼ 0; > > > ρa þ 2βp
−2α >
<  r 2 p d þ þðα 2 −β Þðc þ ΔcÞ−ðα 2 −β Þð λ−μ 1 Þ ¼ 0; λ 12 a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ 12 ðα 1 þ α 2 −2βÞc ¼ 0; > > > 12 a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ 12 ðα 1 þ α 2 −2βÞc≥0; > : λ≥0:

ðA1Þ ðA2Þ ðA3Þ ðA4Þ ðA5Þ

(1) When λ>0, we know that Eq. (A4) is binding, from Eqs. (A1), (A2) and (A4), we obtain  α ð1−ρÞ þ βρ c  aþ ; p
r ¼ 2
2

2 α 1 α 2 −β

 α ρ þ βð1−ρÞ c  aþ ; p
d ¼ 1
2

2 α 1 α 2 −β

2

λ ¼ Δc þ μ 1 :

2

Note that λ > 0 is equivalent to Δc > − μ1, which indicates that p
r and p
d defined above are the optimal solutions when Δc > − μ1. (2) When λ = 0, we know that Eq. (A4) is not binding, plugging λ = 0 into Eqs. (A1) and (A2), we obtain  α ð1−ρÞ þ βρ c þ Δc þ μ 1  aþ ; p
r ¼ 2
2

2 α 1 α 2 −β

 α ρ þ βð1−ρÞ c þ Δc þ μ 1  aþ p
d ¼ 1
: 2

2 α 1 α 2 −β

2

2

After some simplification, the constraint given in Eq. (19) reduces to Δc ≤ −μ1. That is, p
r and p
d defined above are the optimal solutions when Δc ≤ −μ1. Combining the two scenarios above, we obtain Eqs. (20) and (21). From Eqs. (20) and (21) we obtain Eq. (22). Proof of Proposition 2. In order to solve the constrained optimization problem, we introduce the Lagrange multiplier λ ≥ 0 and relax the constraint given in Eq. (24). The KKT conditions of the problem defined in Eqs, (23)–(24) are 8 ð1−ρÞa−2α 1 p
r þ 2βp
d þ ðα 1 −βÞðc þ ΔcÞ þ ðα 1 −βÞðλ þ μ 2 Þ ¼ 0; > > > ρa þ 2βp
−2α >
<  r 2 p d þ þðα 2 −β Þðc þ ΔcÞ þ ðα 2 −β Þ ðλ þ μ 2 Þ ¼ 0; λ 12 a−ðα 1 −βÞp
r −ðα 2 −βÞp
d þ 12 ðα 1 þ α 2 −2βÞc ¼ 0; > > 1 a−ðα −β Þp
r −ðα 2 −βÞp
d þ 12 ðα 1 þ α 2 −2βÞc≤0; > > 1 :2 λ≥0:

ðA6Þ ðA7Þ ðA8Þ ðA9Þ ðA10Þ

(1) When λ > 0, we know that Eq. (A9) is binding, from Eqs. (A6), (A7) and (A9), we obtain  α ð1−ρÞ þ βρ c  aþ ; p
r ¼ 2
2

2 α 1 α 2 −β

 α ρ þ βð1−ρÞ c  aþ ; p
d ¼ 1
2

2 α 1 α 2 −β

2

λ ¼ −Δc þ μ 2 :

2

  Note that λ > 0 is equivalent to Δc b μ2, which indicates that p
r and p
d defined above are the optimal solutions when Δc b μ2. (2) When λ = 0, we know that Eq. (A9) is not binding, plugging λ = 0 into Eqs. (A6) and (A7), we obtain  α ð1−ρÞ þ βρ c þ Δc−μ 2  aþ ; p
r ¼ 2
2

2 α 1 α 2 −β

2

 α ρ þ βð1−ρÞ c þ Δc−μ 2  aþ p
d ¼ 1
: 2

2 α 1 α 2 −β

2

After some simplification, the constraint given in Eq. (24) reduces to Δc≥μ2. That is, p
r and p
d defined above are the optimal solutions when Δc≥μ2. Combining the two scenarios above, we obtain Eqs. (25) and (26). From Eqs. (25) and (26) we obtain Eq. (27). Proof of Proposition 3. The results can be easily obtained by integrating the results presented in Propositions 1 and 2 after some algebra manipulation. Proof of Lemma 4. The proof is similar to the proof of Lemma 3, so we omit it here.
Relaxing the constraints Proof of Proposition 4. It can be easily verified that π
m in a decentralized supply chain is jointly concave in p
d and w. defined in Eqs. (37) and (38) and introducing Lagrange multipliers λ1 ≥ 0 and λ2 ≥ 0, respectively, the KKT conditions are given by 8     >
þ 2α 1 α 2 −α 1 β−β2 ðc þ Δc þ μ 1 −2α 1 λ2 Þ þ 2α 1 λ1 ¼ 0; ½2α 1 ρ þ βð1−ρÞa−2 2α 1 α 2 −β2 p
d þ 2α 1 βw > > > > >
þ ðα 1 −βÞðc þ Δc þ μ 1 Þ−2λ1 −2α 1 ðα 1 −βÞλ2 ¼ 0; < ð1−ρÞa þ 2βp
d −2α 1 w
Þ ¼ 0; λ1 n ðp
d −w   o > > 0 >
> λ2 ½ðα 1 þ βÞ þ ðα 1 −βÞρa− 2α 1 α 2 −α 1 β−β2 p
d −α 1 ðα 1 −βÞw−2α 1 Q sc ¼ 0; > > : λ1 ≥0; λ2 ≥0:

ðA11Þ ðA12Þ ðA13Þ ðA14Þ ðA15Þ

536

S. Huang et al. / Economic Modelling 30 (2013) 521–538

(1) When λ1 = 0 and λ2 = 0, we know that the constraints given in Eqs. (37) and (38) are not binding. Solving Eqs. (A11) and (A12), we obtain α ρ þ β ð1−ρÞ c þ Δc þ μ 1  aþ p
d ¼ 1
; 2 2 α 1 α 2 −β

c þ Δc þ μ 1
¼ α 2
ð1−ρÞ þ βρ  aþ w : 2 2 α 1 α 2 −β

2

2

^ ≤ρb1, and the constraint given in Eq. (38) is equivalent to Δc ≤ − μ1. Note that the constraint given in Eq. (37) is equivalent to ρ (2) When λ1 > 0 and λ2 = 0, we know that the constraint given in Eq. (37) is binding and the constraint given in Eq. (38) is not binding. Solving Eqs. (A11)–(A14), we obtain
¼ p
d ¼ w

2α ρ þ ðα 1 þ β Þð1−ρÞ
1  2 α 21 −β 2 þ 2α 1 α 2 −2α 1 β

aþ

c þ Δc þ μ 1 ; 2

λ1 ¼

α 1 ½ðα 2 −βÞ−ðα 1 þ α 2 −2β Þρ a: α 21 −β 2 þ 2α 1 α 2 −2α 1 β

^ . Combining the two scenarios Note that the constraint given in Eq. (38) reduces to Δc ≤− μ1, and the constraint λ1 > 0 is equivalent to 0bρbρ above, we obtain Eqs. (39) and (40). (3) When λ1 =0 and λ2 > 0, we know that the constraint given in Eq. (37) is not binding and the constraint given in Eq. (38) is binding. Solving Eqs. (A11)–(A14), we obtain α ρ þ βð1−ρÞ c  aþ ; p
d ¼ 1
2 2 α 1 α 2 −β

c
¼ α 2
ð1−ρÞ þ βρ  aþ ; w 2 2 α 1 α 2 −β

2

λ2 ¼

2

Δc þ μ 1 : 2α 1

^ ≤ρb1, and the constraint λ2 > 0 is equivalent to − μ1 b Δc b 0. Note that the constraint given in Eq. (37) is equivalent to ρ (4) When λ1 > 0 and λ2 > 0, we know that the constraints given in Eqs. (37) and (38) are both binding. Solving Eqs. (A11)–(A14), we obtain
¼ p
d ¼ w

2α ρ þ ðα 1 þ β Þð1−ρÞ
1  2 α 21 −β 2 þ 2α 1 α 2 −2α 1 β

c 2

aþ ;

λ1 ¼

α 1 ½ðα 2 −β Þ−ðα 1 þ α 2 −2β Þρ a; α 21 −β 2 þ 2α 1 α 2 −2α 1 β

λ2 ¼

Δc þ μ 1 : 2α 1

^ , and λ2 > 0 reduces to − μ1 b Δc b 0. Combining the two scenarios above, we obtain Eqs. (41) and (42). Note that λ1 > 0 reduces to 0bρbρ Proof of Proposition 5. To solve the constrained optimization problem defined in Eqs. (43)–(45), we relax the constraints given in Eqs. (44) and (45) and introduce the Lagrange multipliers λ1 ≥ 0 and λ2 ≥ 0, respectively, the KKT conditions are given by 8     >
þ 2α 1 α 2 −α 1 β−β2 ðc þ Δc−μ 2 þ 2α 1 λ2 Þ þ 2α 1 λ1 ¼ 0; ½2α 1 ρ þ βð1−ρÞa−2 2α 1 α 2 −β2 p
d þ 2α 1 βw > > > > >
þ ðα 1 −βÞðc þ Δc−μ 2 Þ−2λ1 þ 2α 1 ðα 1 −βÞλ2 ¼ 0; < ð1−ρÞa þ 2βp
d −2α 1 w
Þ ¼ 0; λ1 n ðp
d −w   o > > 0 >
> λ2 ½ðα 1 þ βÞ þ ðα 1 −βÞρa− 2α 1 α 2 −α 1 β−β2 p
d −α 1 ðα 1 −βÞw−2α 1 Q sc ¼ 0; > > : λ1 ≥0; λ2 ≥0:

ðA16Þ ðA17Þ ðA18Þ ðA19Þ ðA20Þ

(1) When λ1 = 0 and λ2 = 0, we know that the constraints given in Eqs. (44) and (45) are not binding. Solving Eqs. (A16) and (A17), we obtain α ρ þ β ð1−ρÞ c þ Δc−μ 2  aþ ; p
d ¼ 1
2 2 α 1 α 2 −β

c þ Δc−μ 2
¼ α 2
ð1−ρÞ þ βρ  aþ w : 2 2 α 1 α 2 −β

2

2

^ ≤ρb1, and the constraint given in Eq. (45) is equivalent to Δc ≥ μ2. Note that the constraint given in Eq. (44) is equivalent to ρ (2) When λ1 > 0 and λ2 = 0, we know that the constraint given in Eq. (44) is binding and the constraint given in Eq. (45) is not binding. Solving Eqs. (A16)–(A19), we obtain
¼ p
d ¼ w

2α ρ þ ðα 1 þ β Þð1−ρÞ
1  2 α 21 −β 2 þ 2α 1 α 2 −2α 1 β

aþ

c þ Δc−μ 2 ; 2

λ1 ¼

α 1 ½ðα 2 −β Þ−ðα 1 þ α 2 −2βÞρ a: α 21 −β2 þ 2α 1 α 2 −2α 1 β

^ . Combining the two scenarios Note that the constraint given in Eq. (44) reduces to Δc ≥ μ2, and the constraint λ1 > 0 is equivalent to 0bρbρ above, we obtain Eqs. (46) and (47). (3) When λ1 = 0 and λ2 > 0, we know that the constraint given in Eq. (44) is not binding and the constraint given in Eq. (45) is binding. Solving Eqs. (A16)–(A19), we obtain α ρ þ βð1−ρÞ c  aþ ; p
d ¼ 1
2 2 α 1 α 2 −β

c
¼ α 2
ð1−ρÞ þ βρ  aþ ; w 2 2 α 1 α 2 −β

2

2

λ2 ¼

−Δc þ μ 2 : 2α 1

^ ≤ρb1, and the constraint λ2 > 0 is equivalent to 0 b Δc b μ2. Note that the constraint given in Eq. (44) is equivalent to ρ (4) When λ1 > 0 and λ2 > 0, we know that the constraints given in Eqs. (44) and (45) are both binding. Solving Eqs. (A16)–(A19), we obtain
¼ p
d ¼ w

2α ρ þ ðα 1 þ β Þð1−ρÞ
1  2 α 21 −β 2 þ 2α 1 α 2 −2α 1 β

c 2

aþ ;

λ1 ¼

α 1 ½ðα 2 −β Þ−ðα 1 þ α 2 −2β Þρ a; α 21 −β2 þ 2α 1 α 2 −2α 1 β

λ2 ¼

−Δc þ μ 2 : 2α 1

^ , and λ2 > 0 reduces to 0 b Δc b μ2. Combining the two scenarios above, we obtain Eqs. (48) and (49). Note that λ1 > 0 reduces to 0bρbρ Proof of Proposition 6. The results can be easily obtained by plugging the results of Propositions 4 and 5 into (34) and simplifying, so we omit the details here.

S. Huang et al. / Economic Modelling 30 (2013) 521–538

537

In this Appendix, we derive the retailer's optimal pricing and production quantity decisions in the traditional one-manufacturer-one-retailer supply chain. We analyze this problem in the centralized and decentralized supply chain, respectively, and compare them with the results obtained in the dual-channel supply chain. Similar to Qi et al. (2004) and Xu et al. (2006), with the notations above, we assume that the demand for the products in the traditional retail channel is given by

Comparing Theorem 1 with Proposition 3, we observe that the production cost disruption robust region in the traditional retail channel is identical with that in the dual-channel supply chain. The optimal retail price can be obtained by adding some adjustment terms to the original retail price if the production cost disruption exceeds some thresholds. The optimal production quantity should be kept unchanged if − μ1 ≤ Δc ≤ μ2. However, due to the difference in channel structure, the optimal production quantity in the traditional channel and that in the dual channels are quite different. What's more, the supply chain's profits in these two channel structures are also different.

Dr ¼ a−α 1 pr :

B2. Decentralized decisions in the traditional retail channel

Appendix B. Comparisons with the traditional retail channel

ðB1Þ

The retailer's, the manufacturer's and the supply chain's profits are given by πr ¼ ðpr −wÞDr ;

ðB2Þ

πm ¼ ðw−cÞDr ;

ðB3Þ

πsc ¼ π r þ πm ¼ ðpr −cÞDr :

ðB4Þ

B1. Centralized decisions in the traditional retail channel First, we analyze the scenario when the manufacturer and the retailer are vertically integrated. When there is no production cost disruption, it is easy to verify that the supply chain's profit πsc is concave in pr. By FOC, the supply chain's optimal retail price, optimal production quantity and optimal profit are given by (Qi et al., 2004; Xu et al., 2006) 

pr ¼

a þ α1 c 2α 1



;Q ¼

a−α 1 c 2



; πsc ¼

2

ða−α 1 cÞ 4α 1

:

In what follows, we analyze the scenario when the manufacturer and the retailer are independent decision-makers who seek to maximize their individual profits. When there is no production cost disruption, after the manufacturer determining the wholesale price w, the retailer's optimal responsive pricing strategy is given by pr = (a + α1w) / 2α1. Then the manufacturer's profit function reduces to 1 2

πm ¼ ðw−cÞða−α 1 wÞ:

By FOC, we can obtain the closed-form solutions for the problem defined in Eq. (B6). The supply chain's optimal wholesale price, optimal retail price, and the optimal production quantity are given by 0

w ¼

0




−Q  þ −μ Q  −D
−μ D
þ: π
sc ¼ ðp
r −c−ΔcÞD r r r 1 2

a þ α1 c 2α 1

0

; pr ¼

3a þ α 1 c 4α 1

0

;Q ¼

a−α 1 c : 4

The manufacturer's and the retailer's profits are given by πm ¼

When the actual production cost does not equal the anticipated one, some possible deviation costs may occur. Assume that the unit underage cost and unit disposal cost are μ1 and μ2, respectively, and the production cost disruption is captured by Δc. Then the demand function
r ¼ a−α 1 p
. is D r The supply chain's profit under production cost disruption is given by

ðB6Þ

ða−α 1 cÞ2 8α 1

0

; πr ¼

ða−α 1 cÞ2 : 16α 1

When the actual production cost does not equal the anticipated one, some possible deviation costs may occur. The manufacturer's profit under production cost disruption is given by    
−Q 0 þ −μ Q 0 −D
−μ D
þ:
π
m ¼ ðw−c−Δc ÞD r r r 1 2

ðB7Þ

ðB5Þ

The following Theorem 1 in Xu et al. (2006) gives the solutions to the problem defined in Eq. (B5).

We can obtain the closed-form solutions for the problem defined in Eq. (B7). The results are summarized in the following theorem.

Theorem 1. (Xu et al., 2006)

Theorem 2. When the manufacturer and the retailer make decisions independently, and there is a production cost disruption Δc, the manufacturer's profit is maximized at the optimal production quantity

When there is a production cost disruption Δc, the total supply chain profit is maximized at the optimal production quantity 8  > < a−α 1 ðc þ2 Δc þ μ 1 Þ ¼ Q −α 1 ðΔc2 þ μ 1 Þ
 ¼ a−α 1 c ¼ Q  Q 2 > : a−α 1 ðc þ Δc−μ 2 Þ 2Þ ¼ Q  −α 1 ðΔc−μ 2 2

if Δcb−μ 1 ; if −μ 1 ≤Δc≤μ 2 ; if Δc > μ 2 ;

with the corresponding retail price being: 8 a þ α ðc þ Δc þ μ Þ μ1 1 1 > ¼ pr þ Δc þ 2α 2 <   a þ α c 1 p
r ¼ ¼ pr 2α > : a þ α 1 ðc þ Δc−μ 2 Þ 2 ¼ pr þ Δc−μ 2α 2 1

if Δc > μ 2 ;

1

and a corresponding supply chain's profit of:

 π
sc ¼

8 2 2 ða−α 1 cÞ > 1 c þ α 1 ðΔc þ μ 1 Þ > −Δca−α > 4 2 < 4α 1

2

ða−α 1 cÞ 4α 1

1c −Δca−α 2

4α 1

1 c þ α 1 ðΔc−μ 2 Þ −Δca−α 4 2

> > > : ða−α 1 cÞ2

if Δcb−μ 1 ;

8 a þ α 1 ðc þ Δc þ μ 1 Þ μ1 > ¼ w0 þ Δc þ > 2α 2 < 0 0 a þ α c 1
¼ w ¼w 2α > > : a þ α 1 ðc þ Δc−μ 2 Þ ¼ w0 þ Δc−μ 2 2α 2 1

1

1

8 3a þ α 1 ðc þ Δc þ μ 1 Þ μ1 > ¼ p0r þ Δc þ > 4α 4 < 0 0 3a þ α c 1
pr ¼ ¼ pr 4α > > : 3a þ α 1 ðc þ Δc−μ 2 Þ ¼ p0 þ Δc−μ 2 r 4α 4 1

if −μ 1 ≤Δc≤μ 2 ; 2

if Δcb−μ 1 ; if −μ 1 ≤Δc≤μ 2 ; if Δc > μ 2 ;

with the wholesale price and retail price being:

if Δcb−μ 1 ; if −μ 1 ≤Δc≤μ 2 ;

1

8 a−α 1 ðc þ Δc þ μ 1 Þ > ¼ Q 0 −α 1 ðΔc4 þ μ 1 Þ < 4 0 0 a−α c
1 ¼ Q Q ¼ 4 > : a−α 1 ðc þ Δc−μ 2 Þ 0 2Þ ¼ Q −α 1 ðΔc−μ 4 4

if Δc > μ 2 :

1

1

if Δcb−μ 1 ; if −μ 1 ≤Δc≤μ 2 ; if Δc > μ 2 ;

if Δcb−μ 1 ; if −μ 1 ≤Δc≤μ 2 ; if Δc > μ 2 ;

538

S. Huang et al. / Economic Modelling 30 (2013) 521–538

and the manufacturer's and the retailer's optimal profits are given by

0 π
r ¼

8 2 ½a−α 1 ðc þ Δc þ μ 1 Þ > > > 16α < ða−α 1 cÞ

2

1

16α > > > : ½a−α 1 ðc þ Δc−μ 2 Þ2 1

16α 1

if −μ 1 ≤Δc≤μ 2 ; if Δc > μ 2 ;

8 2 2 ða−α 1 cÞ > 1 c þ α 1 ðΔc þ μ 1 Þ > −Δca−α > 8 4 < 8α 1

0 π
m ¼

if Δcb−μ 1 ;

2

ða−α 1 cÞ

8α > > > : ða−α 1 cÞ2 1

8α 1

1c −Δca−α 4

−Δc

a−α 1 c 4

if Δcb−μ 1 ; if −μ 1 ≤Δc≤μ 2 ;

þ

2

α 1 ðΔc−μ 2 Þ 8

if Δc > μ 2 :

Comparing Theorem 2 with Propositions 4, 5 and 6, we find that the production cost disruption robust region in the traditional retail channel is identical with that in the dual-channel supply chain in decentralized supply chains. When the production cost disruption Δc b −μ1 or Δc >μ2, the optimal production quantity should be changed and can be obtained by adding an adjustment term to the original optimal production quantity. In the dual-channel supply chain, the manufacturer's optimal retail price is affected by the direct sale price, and the wholesale price is the lower bound for the direct sale price. However, the constraint pd ≥ w is absent in the traditional retail channel. Compared with the traditional retail channel, the optimal prices in the dual-channel supply chain are more complicated. In the dual-channel supply chain, the optimal prices, the manufacturer's profit and the retailer's profit are both closely related to ρ. Besides, due to the difference in channel structure, the profits for the manufacturer and the retailer are also quite different. In sum, the optimal prices and production quantity in the dual-channel supply chain are different from those in the traditional retail channel due to channel structure difference, both in centralized and decentralized supply chains; though the production cost disruption robust regions are identical. From the analysis above, we conclude that the existence of the direct sales channel does change the decisions for the supply chain. Therefore, it is necessary to incorporate the production cost disruption into the dual-channel supply chain management. References Cai, G., 2010. Channel selection and coordination in dual-channel supply chains. Journal of Retailing 86 (1), 22–36. Cai, G., Zhang, Z.G., Zhang, M., 2009. Game theoretical perspectives on dual-channel supply chain competition with price discounts and pricing schemes. International Journal of Production Economics 117 (1), 80–95. Chen, K., Xiao, T., 2009. Demand disruption and coordination of the supply chain with a dominant retailer. European Journal of Operational Research 197 (1), 225–234. Chen, K., Zhuang, P., 2011. Disruption management for a dominant retailer with constant demand-stimulating service cost. Computers and Industrial Engineering 61 (4), 936–946. Chen, K.Y., Kaya, M., Özer, Ö., 2008. Dual sales channel management with service competition. Manufacturing & Service Operations Management 10 (4), 654–675.

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