Pricing and production decisions in dual-channel supply chains with demand disruptions

Computers & Industrial Engineering 62 (2012) 70–83

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Computers & Industrial Engineering

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Pricing and production decisions in dual-channel supply chains withdemand disruptions q

Song Huang a,⇑, Chao Yang a, Xi Zhang b

a School of Management, Huazhong University of Science & Technology, Wuhan 430074, PR Chinab School of Management, Wuhan Institute of Technology, Wuhan 430205, PR China

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

Article history:Received 1 December 2010Received in revised form 10 May 2011Accepted 22 August 2011Available online 31 August 2011

Keywords:Supply chain managementDual-channelDisruption managementGame theory

0360-8352/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.08.017

q This manuscript was processed by Area Editor Jos⇑ Corresponding author. Tel.: +86 27 87556487.

E-mail address: [email protected] (S. Huang).

This paper develops a two-period pricing and production decision model in a one- manufacturer-one-retailer dual-channel supply chain that experiences a disruption in demand during the planning horizon.While disruption management has long been a key research issue in supply chain management, littleattention has been given to disruption management in a dual-channel supply chain once the original pro-duction plan has been made. Generally, changes to the original production plan induced by a disruptionmay impose considerable deviation costs throughout the supply chain system. In this paper, we examinehow to adjust the prices and the production plan so that the potential maximal profit is obtained under adisruption scenario. We first study the scenario where the manufacturer and the retailer are verticallyintegrated with demand disruptions. Then we further assume that the manufacturer bears the deviationcosts and obtain the manufacturer’s and the retailer’s individual optimal pricing decision, as well as themanufacturer’s optimal production quantity in a decentralized decision-making setting. We derive con-ditions under which the maximum profit can be achieved. The results indicate that the optimal produc-tion quantity has some robustness under a demand disruption, in both centralized and decentralizeddual-channel supply chains. We also find that the optimal pricing decisions are affected by customers’preference for the direct channel and the market scale change, in both centralized and decentralizeddual-channel supply chains.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Channel design is one of the most important aspects of market-ing decisions. There are several different kinds of channel structuresin current marketing systems, such as the traditional retail channel,the direct channel through the Internet, and the dual-channel,which is a combination of the first two channels. In the traditionalretail channel, manufacturers sell the products to retailers whothen sell the products to end customers. This had been a commonchannel design until the commencement of the Internet. In recentyears, with the rapid development of the TPL firms such as FedExand UPS, customers are increasingly accustomed to purchasingproducts online. A large number of prominent examples of compa-nies that are selling directly to end customers through the directchannel include IBM, Dell, the former Compaq, Hewlett–Packard,Nike, Barnes & Nobel, Fnac, Pioneer Electronics, Levi Strauss, EsteeLauder and Wal-mart (Beck, 2000; Chiang, Chhajed, & Hess, 2003;Kaufman, 1999; Tedeschi, 2000; Tsay & Agrawal, 2004).

ll rights reserved.

eph Geunes.

The Internet has significantly influenced customers’ purchasepatterns. Some customers may prefer purchasing online, while oth-ers may prefer shopping in stores. Consequently, different customerpurchase patterns have encouraged more and more manufacturersto redesign their traditional channel structures by engaging in di-rect sales to reach different customer segments that cannot bereached by the traditional retail channel, giving birth to the dual-channel. However, as manufactures compete with the retailers inmarket, retailers may claim that the orders placed through themanufacturers’ direct channels were orders that should have beenplaced through them (Chiang et al., 2003). This leads to ‘‘channelconflict’’. Although it appears that the direct channel is likely to can-nibalize the retailer’s market share, the retailer may not be worseoff. Because the manufacturer’s participation in the direct channelis usually accompanied by wholesale price reduction, this may ben-efit the retailers in the end. Furthermore, as some customers prefershopping online and the dual-channel is a strategic necessity toserve customers in the aspect of marketing, it is not wise for the re-tailer to boycott the direct channel and drive customers to buy else-where (Hanover, 1999). In this paper, we assume that the dual-channel already exists even in the presence of the channel conflict,so we will not analyze the necessity of introducing a direct channel(Hua, Wang, & Cheng, 2010).

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S. Huang et al. / Computers & Industrial Engineering 62 (2012) 70–83 71

In recent a couple of years, the research on dual-channel supplychain management had gained much attention among the market-ing and supply chain management researchers. Chiang et al. (2003)studied a price-setting game between a manufacturer and a retai-ler in a dual-channel supply chain based on consumer choice mod-el. They found that the direct channel could not always bedetrimental to the retailer because it would be accompanied by awholesale price reduction. In a paper closely related to Chianget al. (2003), Yan and Pei (2009) investigated the impact of retailservices on a dual-channel supply chain and found that the directchannel could not always be detrimental to the retailer because ofimproved retail services. Chen, Kaya, and Özer (2008) investigatedservice competition in a dual-channel supply chain based on con-sumer channel choice behavior. Tsay and Agrawal (2004) usedgame theory to study the channel conflict and coordination be-tween the manufacturer and the retailer in a dual-channel supplychain and proposed policies that could coordinate the actions ofchannel members. Seifert, Thonemann, and Sieke (2006) developedmodels for both a dedicated and integrated supply chain, and ana-lyzed how to coordinate the supply chain and allocate the supplychain’s profit between the manufacturer and the retailer underdecentralized decision-making environment. Cai (2010) investi-gated the impact of channel structures on the supplier, the retailer,and the entire supply chain in the context of two single-channeland two dual-channel supply chains. Mukhopadhyay, Zhu, andYue (2008) investigated the optimal contract design problem in amixed channels supply chain when the manufacturer had incom-plete information about the retailer’s cost of adding value. Huaet al. (2010) examined the optimal decisions of delivery lead timeand prices in centralized and decentralized dual-channel supplychains. However, litter attention has been given to the issue ofdual-channel management under a disruption scenario. In practice,perfect information about the market demand may be difficult orimpossible to obtain in the planning period for short lifecycle prod-ucts, and disruptions in demands are very common. Thus, this pa-per investigates the two-period pricing and production quantitydecisions model and focuses primarily on the second period.

Our work is also related to previous research on disruptionmanagement. Generally speaking, disruptions will result in devia-tion penalties for the original production quantity change. Qi, Bard,and Yu (2004) first introduced the disruption management intosupply chain management. They examined how the original pro-duction plan should be adjusted after demand disruptions oc-curred and how to coordinate the supply chain by means ofwholesale quantity discount policies. Xiao, Yu, Sheng, and Xia(2005) further studied the coordination of the supply chain withdemand disruptions and considered a price-subsidy rate contractto coordinate the investments of the competing retailers with salespromotion opportunities and demand disruptions. Huang, Yu,Wang, and Wang (2006) investigated the coordination problemof a supply chain under demand disruptions. However, they usedan exponential demand function instead of a linear function. Xiaoand Yu (2006) developed an indirect evolutionary game modelwith two vertically integrated channels to study evolutionarily sta-ble strategies (ESS) of retailers in the quantity-setting duopoly sit-uation with homogeneous goods and analyzed the effects of thedemand and raw material supply disruptions on the retailers’strategies. Xiao, Qi, and Yu (2007) and Xiao and Qi (2008) studiedhow to coordinate a supply chain with one manufacturer and twocompeting retailers by means of quantity discount schedule whendemands are disrupted. Chen and Xiao (2009) further studied thecoordination scheme, the linear quantity discount schedule, andthe Groves wholesale price schedule in a supply chain with a dom-inant retailer. Yang, Qi, and Yu (2005) studied the problem ofrecovering a production plan after a disruption. They proposed adynamic programming method for the problem for the general-

cost case, and considered the cost disruption first and then the de-mand disruption for the convex-cost case. This paper is distinctfrom the literatures above in the following two aspects. Firstly,we investigate the pricing and production quantity decisions in atwo-period dual-channel supply chain when it experiences a de-mand disruption, while the literatures above focus mainly on thedisruption management in a single channel. Secondly, we considerboth the centralized and decentralized scenarios.

The framework of the paper is mostly related to Qi et al. (2004).They studied a two-period pricing and production model for a prod-uct with a short lifecycle. In the first period, a production plan isdeveloped before demand is known. In the second period, demanduncertainty is resolved and products are sold with the possibilitythat adjustments of prices and production quantity had to be madeto the original plan. As is pointed out by Qi et al. (2004), in the tra-ditional models, the retailer’s ordering decision is usually made be-fore the demand is known. In the context of the two-period model,this means that the retailer has to submit the order before the firstperiod. In today’s business environment, a powerful retailer may re-fuse to make such a risky commitment, and not place an order untilthe demand is known. Therefore, the manufacturer has to deter-mine the production quantities of the products with long produc-tion lead times and short lifecycles before the selling seasonstarts. The production plan for the product is conducted based onhistorical data, if they can be obtained, and demand forecast. Longproduction lead time, as well as short lifecycle, material acquisition,capacity constraint, makes the determination of the productionplan in the first period more complex.

The main purpose of this paper is to illustrate the impact on atwo-period dual-channel supply chain consisting of a manufac-turer and a retailer when the original plan cannot be implementedas originally formulated. In the first period, a production plan isdeveloped based on demand forecast, and we assume that no saleoccurs in the first period and it is merely the plan and productionperiod. In the second period, demand uncertainty is resolved andproducts are sold with the possibility that adjustments have tobe made to the original production quantity and prices to compen-sate for the deviation in market scale. This is very common whenlittle information about disruptions in demand can be acquiredin advance before the beginning of the second period. Examples in-clude ‘‘hot’’ electronic gadgets such as Apple’s iPhone and Ninten-do’s Wii video game console, perishable goods, new magazines, etc.Because the disruption is usually unpredictable, and the distribu-tion probability of the occurrence of the demand disruption usuallycannot be obtained. For example, in the 2010 Yushu Earthquake inChina, the demands for tents and some medicines increased con-siderably, which could not be forecasted in advance. In such cir-cumstance, production plans and pricing decisions have to beadjusted correspondingly to compensate for these disruptions. Thisanalytical context has been widely used by Qi et al. (2004), Xiaoet al. (2005, 2007), and Huang et al. (2006).

We study the disruption model in the context of centralized anddecentralized two-period dual-channel supply chains, respectively,with emphasis on the second period. In the centralized dual-chan-nel supply chain, the central decision-maker determines the retailprice, the direct sale price and the production quantity aiming atsupply chain profit maximization under a disruption scenario. Inthe decentralized dual-channel supply chain where the manufac-ture and the retailer are independent decision-makers who seekto maximize their individual profit, we assume that the manufac-ture acts as the Stackelberg leader who determines the wholesaleprice and direct sale price first, and the retailer acts as the Stackel-berg follower who determines the retail price in response to themanufacturer’s pricing decision. We analyze their individual pric-ing decision and the production quantity given a demand disrup-tion occurs in the second period.

72 S. Huang et al. / Computers & Industrial Engineering 62 (2012) 70–83

We have some interesting observations. First, the manufac-turer’s original production plan has some robustness under ademand disruption, in both centralized and decentralized dual-channel supply chains. When the market scale is mildly perturbed,the original production plan should be kept unchanged, and adjust-ments in prices alone can be made to compensate for any deviationcosts. Only when the change of the market scale exceeds somethresholds, should both the original production quantity and theprices be adjusted. Second, in the centralized dual-channel, it is al-ways beneficial for the central decision-maker to utilize the ad-justed pricing decisions under a disruption in market scale. Atlast, in the decentralized dual-channel supply chain, the optimalpricing policies depend on both customers’ preference for the di-rect channel and the market scale change.

The rest of the paper is organized as follows. Section 2 intro-duces the notations and formulates the decision models. In section3 we analyze the pricing and production decisions for the central-ized dual-channel supply chain with and without demand disrup-tions. Furthermore, we examine the value of knowing of demanddisruptions on the manufacturer’s and the retailer’s individualoptimal profit. In Section 4 we analyze the scenario of a decentral-ized dual-channel supply chain with and without demand disrup-tions, respectively. We conclude the results and limitations inSection 5.

2. Model description

We examine a dual-channel supply chain composed of a manu-facturer and a retailer. The manufacturer produces a product witha long production lead time and a short lifecycle. The productionplan has to be made before the selling season starts. The manufac-turer may sell the products to the retailer, as well as to end cus-tomers directly (see Fig. 1, which is presented at the end of themanuscript). Assume that the price-demand relationship is deter-ministic and known. Let c be the manufacturer’s unit productioncost. The manufacturer sells the products to the retailer at whole-sale price w. The retailer sells the products to end customers at re-tail price pr. The manufacturer sells the products to end customersdirectly at direct sale price pd.

We consider a two-period dual-channel supply chain. In thefirst period, the production plan is developed before demand isrealized. In the second period, market demand is resolved and nec-essary adjustments in production quantity and prices have to bemade based on the original plan. We refer to the differential be-tween the actual demand resolved in the second period and theoriginal production plan in the first period as the demand disrup-tion. The problem that needs consideration now is how the pricesand production quantity should be adjusted after the demand is re-solved. We consider the problem in a centralized and decentralizedenvironment, respectively.

Direct Sale Price pd

Retail Price pr

Wholesale Price w

Manufacturer

Market Demand

Retailer

Fig. 1. Dual-channel supply chain structure.

Following Choi (1996), Raju and Abhik (2000), Yue and Liu(2006), Huang and Swaminathan (2009), and Hua et al. (2010),we assume that, for model simplicity, the channel demand func-tions in the two channels are linear in self-price and cross-price ef-fects, but with different parameters for each channel. The demandfunctions in two channels are formulated as follows:

Dr ¼ ð1� qÞa� a1pr þ b1pd; ð1ÞDd ¼ qa� a2pd þ b2pr: ð2Þ

In the above formulas, the subscripts r and d denote the retailchannel and the direct channel, respectively. The expressions in(1) and (2) indicate that the demand in the retail channel Dr andthe demand in the direct channel Dd both depend on the direct salepricepr and the retail price pd. a represents the forecasted potentialdemand if the products are free of charge. The share of the demandgoes to the direct channel is q, and the rest 1 � q goes to the retailchannel, when pr and pd are zeros. q captures customers’ prefer-ence for the direct channel when the products are free of charge.The larger q is, the more intense the channel conflict is. This is be-cause if the direct channel captures more demand, the retailer willthink that the manufacturer corrodes its own market share. a1 anda2 are the coefficients of self-price elasticity of Dr and Dd, respec-tively. The cross-price sensitivities b1 and b2 reflect the degree towhich the products sold via the two channels are substitutes. As-sume that ai > bi for i = 1, 2, so that the self-price effects are greaterthan the cross-price effects. To maintain analytical tractability, fol-lowing Raju and Abhik (2000), Yue and Liu (2006), and Hua et al.(2010), we assume that the cross-price effects are symmetric, thatis, b1 = b2 = b. The total demand in two channels is given byDsc = a � (a1 � b) pr � (a2 � b)pd.

Following Chiang et al. (2003) and Hua et al. (2010), we assumethat both the manufacturer and the retailer have no merchandisingcosts associated with selling the products to end customers foranalytical simplicity. With the above notations and assumptions,the retailer’s profit function is determined by

pr ¼ ðpr �wÞDr ; ð3Þ

the manufacturer’s profit function is determined by

pm ¼ ðw� cÞDr þ ðpd � cÞDd; ð4Þ

and the dual-channel supply chain’s total profit function is deter-mined by

psc ¼ pr þ pm ¼ ðpr � cÞDr þ ðpd � cÞDd: ð5Þ

As mentioned above, the disruption model being consideredcontains two periods. In the first period, a production plan is builton given the assumption that the potential demand is a. In the sec-ond period, the realized demand is found to be a + Da, where thedisruption is captured by Da. Thus, the demand functions in twochannels are formulated as follows:

Dr ¼ ð1� qÞðaþ DaÞ � a1�pr þ b�pd; ð6ÞDd ¼ qðaþ DaÞ � a2�pd þ b�pr ; ð7Þ

and the total demand in two channels is given by Dsc ¼ Dr þ Dd ¼ðaþ DaÞ � ða1 � bÞ�pr � ða2 � bÞ�pd. Throughout the paper, we usethe bar to denote the case of demand disruptions.

3. Centralized dual-channel supply chain

In this section, we consider the pricing and production quantitydecisions in a centralized dual-channel supply chain where themanufacture and the retailer are vertically integrated. Assume thatthere exists a central decision-maker who seeks to maximize thesupply chain’s total profit. The central decision-maker determinesthe retail price pr and the direct sale price pdsimultaneously. In or-


der to study the impact of a demand disruption on the pricing andproduction decisions, we first study the scenario where no demanddisruption occurs as the baseline case in Section 3.1. In Section 3.2we study the scenario where a demand disruption occurs.

3.1. Baseline case: centralized decisions without disruptions

If the actual demand resolved in the second period is equal tothe original production plan, the problem can be simplified tothe basic dual-channel supply chain model. In order to maximizepsc, we first examine the property regarding psc.

Lemma 1. The dual-channel supply chain’s total profit psc is jointlyconcave in pr and pd. The optimal retail price p�r and optimal direct saleprice p�d are given by

p�r ¼a2ð1� qÞ þ bq

2ða1a2 � b2Þaþ c

2; ð8Þ

p�d ¼a1qþ bð1� qÞ


2; ð9Þ

respectively, and the dual-channel supply chain’s optimal profit is givenby

p�sc ¼a1q2 þ 2bqð1� qÞ þ a2ð1� qÞ2

4ða1a2 � b2Þa2 � 1

2ac

þ a1 þ a2 � 2b4

c2: ð10Þ

The optimal production quantity is given by

Q �sc ¼ D�sc ¼12½a� ða1 þ a2 � 2bÞc�: ð11Þ

The proof of Lemma 1, as well as the proofs of other lemmas andpropositions, is relegated in Appendix. The expressions given in (8)and (9) are similar in structure and symmetric to some extent. Thisis because we assume that the cross-price effects in two channeldemand functions are symmetric.

3.2. Centralized decisions with demand disruptions

In this subsection, we analyze the pricing and production deci-sions in a dual-channel supply chain with demand disruptions. Toavoid triviality, we assume that a + Da > 0 and both channels havepositive demands. Intuitively, more profit may be made if the cen-tral decision-maker raises the production quantity and priceswhen Da > 0, and when Da < 0, the opposite is true (Qi et al.,2004). When the demand uncertainty is resolved in the secondperiod, the question at this point is how the central decision-makershould adjust the pricing and production decisions accordingly.

Similar to Qi et al. (2004), we consider this problem in two sce-narios, i.e., Da > 0 and Da < 0. If Da > 0, the actual demand resolvedis larger than the original plan. Given prices and production quan-tity unchanged, lost sales will occur and there will be underagecost. While Da < 0, the actual demand resolved is less than the ori-ginal plan. Given prices and production quantity unchanged, therewill be disposal costs for the leftover inventory. In subsequent, wediscuss the two scenarios separately to examine how to adjust theprices and the production quantity to maximize the supply chain’stotal profit. Let l1 and l2 denote the marginal underage cost andmarginal disposal cost, respectively. In practice, the underage costand disposal cost are usually much less than the marginal produc-tion cost. In order to avoid triviality, we make the followingassumption that max{l1, l2} 6 c, which is similar to Qi et al.(2004).

From the central decision-maker’s point of view, given the ori-ginal production quantity Q �sc , the supply chain’s total profit undera demand disruption can be formulated as

�psc ¼ ð�pr � cÞDr þ ð�pd � cÞDd � l1 Dsc � Q �sc

� �þ � l2 Q �sc � Dsc� �þ

;

where the first term denotes the profit obtained in the retail chan-nel, the second term denotes the profit obtained in the direct chan-nel, the third term denotes the possible underage cost and thefourth term denotes the possible disposal cost. Intuitively, whenmarket scale increases, the production level should also increase,and vice versa. This is stated in the following lemma.

Lemma 2. Suppose Q�sc is defined in (11). Then Dsc P Q�sc if Da > 0,and Dsc 6 Q �sc if Da < 0.

3.2.1. Da > 0When Da > 0, from Lemma 2 the central decision-maker’s profit

�psc reduces to

�psc ¼ ð�pr � cÞDr þ ð�pd � cÞDd � l1 Dsc � Q �sc

� �þ; ð12Þ

simplifying (12), the central decision-maker’s problem can be for-mulated as

max �psc¼ð�pr�cÞ½ð1�qÞðaþDaÞ�a1�prþb�pd�þð�pd�cÞ½qðaþDaÞ�a2�pdþb�pr ��l1

� Da�ða1�bÞ�pr�ða2�bÞ�pdþ12

aþ12ða1þa2�2bÞc

� �; ð13Þ

s:t: Da� ða1 � bÞ�pr � ða2 � bÞ�pd þ12

aþ 12ða1 þ a2 � 2bÞc P 0:ð14Þ

The constraint given in (14) indicates that only when the actualdemand resolved is larger than the original production quantity,will the underage cost takes place. The following proposition char-acterizes the central decision-maker’s optimal prices and produc-tion quantity.

Proposition 1. .Given a disruption in market demand Da > 0, thecentralized dual-channel supply chain’s optimal retail price and directsale price are given by

�p�r ¼

a2ð1�qÞþbq2ða1a2�b2Þ ðaþ DaÞ þ c

2þ Da2ða1þa2�2bÞ

if 0 < Da < ða1 þ a2 � 2bÞl1;a2ð1�qÞþbq2ða1a2�b2Þ ðaþ DaÞ þ cþl1

2

if Da P ða1 þ a2 � 2bÞl1;

8>>>>><>>>>>:

ð15Þ

�p�d ¼

a1qþbð1�qÞ2ða1a2�b2Þ ðaþ DaÞ þ c


if 0 < Da < ða1 þ a2 � 2bÞl1;a1qþbð1�qÞ2ða1a2�b2Þ ðaþ DaÞ þ cþl1

2

if Da P ða1 þ a2 � 2bÞl1;

8>>>>><>>>>>:

ð16Þ

respectively, and the optimal production quantity is given by

Q �sc ¼

12 a� ða1 þ a2 � 2bÞc½ �

if 0 < Da < ða1 þ a2 � 2bÞl1;12 ðaþ DaÞ � ða1 þ a2 � 2bÞðc þ l1Þ� �

if Da P ða1 þ a2 � 2bÞl1:

8>>><>>>:

ð17Þ

Proposition 1 implies that the original optimal production planin a centralized dual-channel supply chain is robust to some extentwith a demand disruption Da > 0. No change in production quan-


tity is required when Da is smaller than (a1 + a2 � 2b)l1. Onlywhen Da is larger than (a1 + a2 � 2b)l1, should the decision-makerraise the production quantity. Intuitively, this is because it is easierfor the central decision-maker to adjust prices than to adjust pro-duction quantity. When the market scale is mildly perturbed, priceadjustments alone can be utilized to compensate for the marketscale change. The expression defined in (17) also leads to the fol-lowing observation. If Da is larger than (a1 + a2 � 2b)l1, comparedwith the expression defined in (11), Q �sc can be interpreted as theoptimal production quantity in a market with market scalea + Da and marginal production cost c + l1 without demanddisruptions.

3.2.2. Da < 0When Da <0, from Lemma 2 the central decision-maker’s profit

�psc reduces to

�psc ¼ ð�pr � cÞDr þ ð�pd � cÞDd � l2 Q �sc � Dsc� �þ

; ð18Þ

simplifying (18), the central decision-maker’s decision problem canbe formulated as

max �psc¼ð�pr�cÞ½ð1�qÞðaþDaÞ�a1�prþb�pd�þð�pd�cÞ½qðaþDaÞ�a2�pdþb�pr �þl2

� Da�ða1�bÞ�pr�ða2�bÞ�pdþ12

aþ12ða1þa2�2bÞc

� �; ð19Þ

s:t: Da�ða1� bÞ�pr �ða2� bÞ�pdþ12

aþ12ða1þa2�2bÞc6 0: ð20Þ

The constraint given in (20) indicates that only when the actualdemand resolved is smaller than the original production quantity,will the disposal cost takes place. The following proposition char-acterizes the central decision-maker’s optimal prices and produc-tion quantity.

Proposition 2. Given a disruption in market demand Da < 0, thecentralized dual-channel supply chain’s optimal retail price and directsale price are given by

�p�r ¼



if � ða1 þ a2 � 2bÞl2 < Da < 0;a2ð1�qÞþbq2ða1a2�b2Þ ðaþ DaÞ þ c�l2

2

if Da 6 �ða1 þ a2 � 2bÞl2;

8>>>>><>>>>>:

ð21Þ

�p�d ¼



if � ða1 þ a2 � 2bÞl2 < Da < 0;a1qþbð1�qÞ2ða1a2�b2Þ ðaþ DaÞ þ c�l2

2


8>>>>><>>>>>:

ð22Þ

respectively, and the optimal production quantity is given by

Q �sc ¼

12 ½a� ða1 þ a2 � 2bÞc�

if � ða1 þ a2 � 2bÞl2 < Da < 0;12 ½ðaþ DaÞ � ða1 þ a2 � 2bÞðc � l2Þ�

if Da 6 �ða1 þ a2 � 2bÞl2:

8>>><>>>:

ð23Þ

Proposition 2 implies that the original optimal production planin the centralized dual-channel supply chain is robust to some ex-tent with a demand disruption Da < 0. No change in productionquantity is required when Da is larger than �(a1 + a2 � 2b)l2. Onlywhen the market scale decease Da is smaller than

�(a1 + a2 � 2b)l2, should the central decision-maker reduce theproduction quantity. Proposition 2 also leads to the followingobservation. If Da is smaller than �(a1 + a2 � 2b)l2, the optimalproduction quantity Q �sc can be interpreted as the optimal produc-tion quantity in a market with market scale a + Da and marginalproduction cost c � l2 without demand disruptions.

In summary, we have the following proposition.

Proposition 3. Given a disruption in market demand D a, thecentralized dual-channel supply chain’s total profit is maximized atthe following values of the retail price and direct sale price:

�p�r ¼

p�r þa2ð1�qÞþbq2ða1a2�b2Þ Da� l2

2

h iif Da 6 �ða1 þ a2 � 2bÞl2;

p�r þa2ð1�qÞþbq2ða1a2�b2Þ Daþ Da

2ða1þa2�2bÞ

h iif � ða1 þ a2 � 2bÞl2 < Da < ða1 þ a2 � 2bÞl1;

p�r þa2ð1�qÞþbq2ða1a2�b2Þ Daþ l1

2

h iif Da P ða1 þ a2 � 2bÞl1;

8>>>>>>>>>><>>>>>>>>>>:

ð24Þ

�p�d ¼

p�d þa1qþbð1�qÞ2ða1a2�b2Þ Da� l2

2

h iif Da 6 �ða1 þ a2 � 2bÞl2;

p�d þa1qþbð1�qÞ2ða1a2�b2Þ Daþ Da

2ða1þa2�2bÞ

h iif � ða1 þ a2 � 2bÞl2 < Da < ða1 þ a2 � 2bÞl1;

p�d þa1qþbð1�qÞ2ða1a2�b2Þ Daþ l1

2

h iif Da P ða1 þ a2 � 2bÞl1;

8>>>>>>>>>>>><>>>>>>>>>>>>:

�

ð25Þcorrespondingly, the centralized dual-channel supply chain’s optimalproduction quantity is given by

Q �sc¼Q�sc�1

2 ½�Da�ða1þa2�2bÞl2� if Da6�ða1þa2�2bÞl2;

Q�sc if �ða1þa2�2bÞl2 <Da< ða1þa2�2bÞl1;

Q�scþ12 ½Da�ða1þa2�2bÞl1� if DaP ða1þa2�2bÞl1:

8><>:

ð26ÞThe expression given in (26) leads to another observation. No

change in production quantity is required when Da is larger than�(a1 + a2 � 2b)l2 and smaller than (a1 + a2 � 2b)l1. When the mar-ket scale decrease is smaller than�(a1 + a2 � 2b)l2, the optimal pro-duction quantity Q �sc equals to the original optimal productionquantity Q �sc minus an adjustment term [�Da � (a1 + a2 � 2b)l2]/2, which can be interpreted as the optimal production quantity in anew market with market scale �Da and marginal production costl2 without demand disruptions. When market scale increase is lar-ger than (a1 + a2 � 2 b)l1, the optimal production quantity Q �sc

equals to the original optimal production quantity Q �sc plus an adjust-ment term [D a � (a1 + a2 � 2b)l1]/2, which can be interpreted asthe optimal production quantity in a new market with market scaleDa and marginal production cost l1 without demand disruptions.

The expression in (24) and (25) imply that the optimal retailprice �p�r and direct sale price �p�d can be obtained by adding anadjustment to the original retail price p�r and direct sale price p�d,respectively. For example, when Da P (a1 + a2 � 2b)l1, the adjust-ment term can be interpreted as the optimal retail price in a newmarket with market scale Da and marginal production cost l1

without demand disruptions. It can also be found that the optimalretail price �p�r and direct sale price �p�d given in (24) and (25) alwayslinearly increase with Da.

The total profit of the centralized dual-channel supply chainafter the disruption in market scale is resolved is given by


�p�sc ¼

a1q2þ2bqð1�qÞþa2ð1�qÞ2

4ða1a2�b2Þ ðaþ DaÞ2 � 12 ðaþ DaÞc

þ a1þa2�2b4 ðl2

2 þ c2Þ þ 12 Dal2




þ a1þa2�2b4 c2 � 1

4ða1þa2�2bÞ ðDaÞ2

if � ða1 þ a2 � 2bÞl2 < Da < ða1 þ a2 � 2bÞl1;



þ a1þa2�2b4 ðl2

1 þ c2Þ � 12 Dal1

if Da P ða1 þ a2 � 2bÞl1:

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

ð27Þ

3.3. Value of knowing of demand disruptions

Next, we would like to examine the value of knowing of the ac-tual demand disruptions in the centralized dual-channel supplychain. If a demand disruption occurs but the central decision-ma-ker does not know the value of it, he can only use the original opti-mal pricing and production decisions, which is likely to besuboptimal. To illustrate the differential, we compare the policyproposed in Proposition 3 with the policy that maintains the sameprices given in Lemma 1. Define p̂sc as the centralized dual-channelsupply chain’s total profit obtained by using the original retail priceand direct sale price defined in Lemma 1 when a demand disrup-tion occurs. We obtain

p̂sc ¼ p�r � c� �

ð1� qÞðaþ DaÞ � a1p�r þ bp�d� �

þ ðp�d� cÞ qðaþ DaÞ � a2p�d þ bp�r

� �� l1ðDaÞþ � l2ðDaÞ�;

after some algebra manipulation, we have

p̂sc¼a1q2þ2bqð1�qÞþa2 ð1�qÞ2

4ða1a2�b2Þ aðaþ2DaÞ�12ðaþDaÞcþ a1þa2�2b

4 c2þDal2 if Da<0;

a1q2þ2bqð1�qÞþa2 ð1�qÞ2

4ða1a2�b2Þ aðaþ2DaÞ�12ðaþDaÞcþ a1þa2�2b

4 c2�Dal1 if Da>0:

8<:

ð28Þ

By comparing the profit p̂sc defined in (28) with the profit �p�sc

given in Proposition 3, we have the following results presentedin Table 1.

Table 1 lists the profit differential in four cases, respectively.The calculations show that knowing of demand disruptions andusing the prices defined in Proposition 3 is always beneficial tothe central decision-maker. Therefore, when a demand disruptionoccurs, it is always beneficial for the central decision-maker to col-lect more market information to get to know the actual demanddisruption. Moreover, for a given demand disruption Da, whenDa is larger than (a1 + a2 � 2b)l1 or smaller than �(a1 +a2 � 2b)l2, the profit differential quadratically increases with l1

and l2, respectively. When the demand disruption Da satisfies0 < Da < (a1 + a2 � 2b)l1, the profit differential linearly increaseswith l1. When the demand disruption Da satisfies �(a1 + a2 � 2b)l2 < Da < 0, the profit differential linearly increases with l2.

Table 1Value of knowing of demand disruptions.

Demand disruptionDa Profit differenti

Da 6 � (a1 + a2 � 2b)l2 a1q2þ2bqð1�qÞþa2ð4ða1a2�b2Þ

�(a1 + a2 � 2b)l2 < Da < 0 a1q2þ2bqð1�qÞþa2

4ða1a2�b2Þ

h0 < Da < (a1 + a2 � 2b)l1 a1q2þ2bqð1�qÞþa2

4ða1a2�b2Þ

hDa P (a1 + a2 � 2b)l1 a1q2þ2bqð1�qÞþa2ð

4ða1a2�b2Þ

4. Decentralized dual-channel supply chain

We now analyze a decentralized dual-channel supply chainwhere the manufacturer and the retailer make decisions indepen-dently to maximize their individual profit. The manufacturer as-sumes the role of Stackelberg game leader and the retailerassumes the role of Stackelberg game follower. The decision pro-cess is assumed to follow the following sequence: the manufac-turer plays first by declaring the wholesale price w and the directsale price pd, and then the retailer reacts to that policy by deter-mining what retail price pr to set. In order to avoid triviality, the di-rect sale price pd must not be smaller than the wholesale price w,which is the no arbitrary condition. Otherwise, if pd < w, the retailercan obtain the products from the direct channel at a lower price.

Usually, the deviation costs are borne by the manufacture in or-der to simplify the models and analysis. In some cases, the devia-tion costs are borne by the retailer rather than the manufacturer(Xiao et al., 2007). In this section, we assume that the manufacturerassumes all related deviation costs. We first analyze the optimalprices and production quantity for the manufacturer and the retai-ler, respectively, under no demand disruptions. We then analyzetheir individual optimal pricing decision after a disruption in mar-ket scale is resolved. At last, we analyze the impact of demand dis-ruptions on the manufacture’s profit and the retailer’s profit.

4.1. Baseline case: decentralized decisions without demand disruptions

When the demand resolved is equal to the original productionplan, no demand disruption occurs. The manufacturer and the re-tailer determine their individual optimal price aiming at maximiz-ing their profit, respectively. The retailer’s profit and themanufacturer’s profit are determined by (3) and (4). We begin withconsidering the retailer’s optimal pricing strategy by use of back-ward induction. The retailer has control over only one variable tomaximize her profit, the retail price pr. As the retailer plays asthe follower in the Stackelberg game, given the manufacturerdetermining the wholesale price w and direct price pd, the retailerdetermines the optimal retail price pr in response to the manufac-turer’s pricing decisions. The following Lemma 3 characterizes theretailer’s optimal responsive pricing strategy.

Lemma 3. Given the manufacturer determining the wholesale price wand direct sale price pd, the retailer’s optimal responsive pricingstrategy is given by

pr ¼1

2a1½ð1� qÞaþ bpd þ a1w�: ð29Þ

Next, we consider the manufacturer’s optimal pricing strategy.As the manufacturer plays as the Stackelberg leader, he knows thatthe retailer’s optimal responsive pricing strategy will be given by(29). Substituting pr defined in (29) into the manufacturer’s profitfunction defined in (4), the manufacturer’s decision problem canbe formulated as

al ¼ �p�sc � p̂sc

1�qÞ2 ðDaÞ2 þ l24 ½ða1 þ a2 � 2bÞl2 � 2Da� > 0

ð1�qÞ2 � 14ða1þa2�2bÞ

iðDaÞ2 � Dal2 > 0

ð1�qÞ2 � 14ða1þa2�2bÞ

iðDaÞ2 þ Dal1 > 0

1�qÞ2 ðDaÞ2 þ l14 ½ða1 þ a2 � 2bÞl1 þ 2Da� > 0

ustrial Engineering 62 (2012) 70–83

maxpdPw

pm ¼12ðw� cÞ½ð1� qÞaþ bpd � a1w� þ ðpd

� cÞ qa� a2pd þb

2a1½ð1� qÞaþ bpd þ a1w�

� : ð30Þ

76 S. Huang et al. / Computers & Ind

Lemma 4. The manufacturer’s profit function pm is jointly concave inpd and w.

Lemma 4 implies that pm is jointly concave in pd and w. And theconstraint pd P w is linear, too. The following proposition character-izes the manufacturer’s optimal pricing strategy.

Proposition 4. In the decentralized dual-channel supply chain with-out demand disruptions, for the retailer’s optimal responsive pricingstrategy given by Lemma 3, the manufacturer’s optimal pricing strat-egy is given by

w ¼ða1þbÞþða1�bÞq

2 a21�b2þ2a1a2�2a1bð Þ aþ

c2 if 0 < q < a2�b

a1þa2�2b ;

a2ð1�qÞþbq2ða1a2�b2Þ aþ c

2 if a2�ba1þa2�2b 6 q < 1;

8<: ð31Þ

pd ¼ða1þbÞþða1�bÞq

2 a21�b2þ2a1a2�2a1bð Þ aþ

c2 if 0 < q < a2�b

a1þa2�2b ;

a1qþbð1�qÞ2ða1a2�b2Þ aþ c

2 if a2�ba1þa2�2b 6 q < 1;

8<: ð32Þ

correspondingly, the manufacturer’s optimal profit is given by

pm ¼

½ða1þbÞþða1�bÞq�a�ða21�b2þ2a1a2�2a1bÞcf g2

8a1 a21�b2þ2a1a2�2a1bð Þ

if 0 < q < a2�ba1þa2�2b ;

2a21q

2þ4a1bqð1�qÞþða1a2þb2Þð1�qÞ2

8a1ða1a2�b2Þ a2 � ða1þbÞþða1�bÞq4a1

ac

þ a21�b2þ2a1a2�2a1b

8a1c2

if a2�ba1þa2�2b 6 q < 1:

8>>>>>>>>>>><>>>>>>>>>>>:

ð33Þ

Proposition 4 implies that when q is larger than the thresholda2�b

a1þa2�2b, the manufacturer’s optimal direct sale price is identicalto the optimal direct sale price in the centralized dual-channel sup-ply chain, and the manufacturer’s optimal wholesale price is iden-tical to the optimal retail price in the centralized dual-channelsupply chain, when no demand disruption occurs. In other words,when customers’ preference for the direct channel is larger thanthe threshold a2�b

a1þa2�2b, the manufacturer will treat the retailer asan end customer and set the wholesale price w equal to the opti-mal retail price in the centralized supply chain, and keep the directsale price unchanged. However, when customers’ preference forthe direct channel is smaller than the threshold a2�b

a1þa2�2b, the manu-facturer will set the wholesale price equal to the direct sale price.This is because when customers prefer the retail channel, the man-ufacturer has to lower the direct sale price to attract more custom-ers to purchase. And note that w is the lower bound for pd.

Integrating Lemma 3 and Proposition 4, the retailer’s optimalretail price is given by

pr ¼

ð3a21�b2þ4a1a2�2a1bÞ�ða2

1�b2þ4a1a2�4a1bÞq4a1 a2

1�b2þ2a1a2�2a1bð Þ aþ a1þb4a1

c

if 0 < q < a2�ba1þa2�2b ;

ð3a1a2�b2Þ�ð3a1a2�2a1b�b2Þq4a1ða1a2�b2Þ aþ a1þb

4a1c

if a2�ba1þa2�2b 6 q < 1;

8>>>>>>><>>>>>>>:

ð34Þ

the retailer’s optimal profit is given by

pr¼

116a1

a21�b2þ4a1a2�4a1bð Þ� 3a2

1�b2þ4a1a2�6a1bð Þqa2

1�b2þ2a1a2�2a1ba�ða1�bÞc

� �2

if 0<q< a2�ba1þa2�2b ;

116a1½ð1�qÞa�ða1�bÞc�2

if a2�ba1þa2�2b6q<1;

8>>>>>>><>>>>>>>:

ð35Þ

and the optimal production quantity is given by

Q0sc ¼

14a1

½ða1 þ bÞ þ ða1 � bÞq�a� a21 � b2 þ 2a1a2 � 2a1b

� �c

�: ð36Þ

It is interesting to note that the manufacturer’s optimal pricingdecision and the retailer’s optimal retail price all depend on thescope of the parameter q. However, the dual-channel supplychain’s optimal production quantity depends only on the value ofparameter q, and has nothing to do with the scope of it. In otherwords, the manufacturer’s optimal wholesale price and direct saleprice, as well as the retailer’s retail price, are piecewise continuousfunction with respect to the parameter q, while the supply chain’soptimal production quantity is continuous with respect to theparameter q.

4.2. Decentralized decisions with demand disruptions

In this subsection, we analyze the retailer’s and the manufac-turer’s optimal pricing decisions in a decentralized dual-channelsupply chain with demand disruptions. Firstly, assume that boththe retailer and the manufacturer are aware of the demand disrup-tion if it occurs. This is reasonable because the manufacturer ownsa direct channel and is able to know the actual demand. Secondly,assume that the manufacturer and the retailer know the same de-mand disruption. This may not be as strict as it seems, because theactual demand can be easily recorded as long as the manufacturerand the retailer can contact the potential customers, directly orindirectly. Thirdly, in order to simplify the analysis, we further as-sume that only the manufacturer assumes the possible underagecost and disposal cost. This is common in practice, for example,some products laid in the supermarket are owned by the manufac-turer. And the supermarket pays the manufacturer only after theproducts are sold and won’t assume any disposal cost or underagecost. After a demand disruption is detected, both the retailer andthe manufacturer have to adjust their pricing strategies, in orderto achieve their individual profit maximization.

Subsequently, we consider the problem in two cases, i.e., Da > 0and Da < 0, similar to the case of the centralized dual-channel sup-ply chain scenario. With the assumptions and notations above, gi-ven the demand disruption Da, the retailer’s profit is given by

�pr ¼ ð�pr �wÞDr ¼ ð�pr �wÞ½ð1� qÞðaþ DaÞ � a1�pr þ b�pd�; ð37Þ

by utilizing the first-order condition, the retailer’s optimal pricingstrategy is given by

�pr ¼1

2a1½ð1� qÞðaþ DaÞ þ b�pd þ a1w�: ð38Þ

The manufacturer’s profit is given by

�pm ¼ ðw� cÞDr þ ð�pd � cÞDd � l1 Dsc � Q0sc

� þ� l2 Q 0

sc � Dsc

� þ;

where the third term denotes the possible underage cost, and thefourth term denotes the possible disposal cost. Intuitively, whenmarket scale increases, the production level should also increase,and vice versa. This is stated in the following lemma.

Lemma 5. Suppose Q0sc is defined in (36). Then Dsc P Q0

sc if Da > 0,and Dsc 6 Q0

sc if Da < 0.

4.2.1. Da > 0From Lemma 5, the manufacturer’s profit function can be writ-

ten as

�pm ¼ ðw� cÞDr þ ð�pd � cÞDd � l1 Dsc � Q0sc

� þ; ð39Þ


plugging (36) and (38) into (39), the manufacturer’s decision prob-lem can be formulated as follows:

max �pm¼12ðw�cÞ½ð1�qÞðaþDaÞþb�pd�a1w�þð�pd�cÞ

� qðaþDaÞ�a2�pdþb

2a1½ð1�qÞðaþDaÞþb�pdþa1w�

�

� l1

2a1ða1þbÞþða1�bÞq½ �ðaþDaÞ�ð2a1a2�a1b�b2Þ�pd

�a1ða1�bÞw�2a1Q 0

sc

o; ð40Þ

s:t: �pd P w; ð41Þ½ða1 þ bÞ þ ða1 � bÞq�ðaþ DaÞ � ð2a1a2 � a1b� b2Þ�pd � a1

ða1 � bÞw� 2a1Q 0sc P 0: ð42Þ

The constraint given in (41) is the non-arbitrary condition, andthe constraint given in (42) indicates that only when the actual de-mand is larger than the original plan, will the manufacturer sufferthe underage cost. It is easy to verify that �pm is jointly concave withrespect to pd and w. The following proposition characterizes themanufacturer’s optimal pricing decisions in a manufacturer Stac-kelberg game when Da > 0.

Proposition 5. In a manufacturer Stackelberg game when Da > 0, themanufacturer maximizes its own profit with the direct sale price andwholesale price:

Case 1: Da P a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l1

�pcase1d ¼

ða1þbÞþða1�bÞq2ða2

1�b2þ2a1a2�2a1bÞ ðaþ DaÞ þ cþl12

if 0 < q < a2�ba1þa2�2b ;

a1qþbð1�qÞ2ða1a2�b2Þ ðaþ DaÞ þ cþl1

2

if a2�ba1þa2�2b 6 q < 1;

8>>>>>><>>>>>>:

ð43Þ

wcase1 ¼


1�b2þ2a1a2�2a1bÞ ðaþ DaÞ þ cþl12

if 0 < q < a2�ba1þa2�2b ;

a2ð1�qÞþbq2ða1a2�b2Þ ðaþ DaÞ þ cþl1

2

if a2�ba1þa2�2b 6 q < 1;

8>>>>>><>>>>>>:

ð44Þ

Case 2: 0 < Da <a2

1�b2þ2a1a2�2a1b


�pcase2d ¼

ða1þbÞþða1�bÞq2 a2

1�b2þ2a1a2�2a1bð Þðaþ2DaÞþ c2


a1qþbð1�qÞ2ða1a2�b2Þ ðaþDaÞþ c

2þða1þbÞþða1�bÞq

2 a21�b2þ2a1a2�2a1bð ÞDa

if a2�ba1þa2�2b6q<1;

8>>>>>>><>>>>>>>:

ð45Þ

wcase2¼


1�b2þ2a1a2�2a1bÞðaþ2DaÞþ c

2


a2ð1�qÞþbq2ða1a2�b2Þ ðaþDaÞþ c



if a2�ba1þa2�2b6q<1:

8>>>>>><>>>>>>:

ð46Þ

Proposition 5 indicates that if Da P a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l1, the con-straint given in (42) is not binding, which means that the optimalproduction quantity should be raised to compensate for the devia-tion from the market scale. On the contrary, if0 < Da < a2


ða1þbÞþða1�bÞq l1, the constraint given in (42) is binding,

which means that the original optimal plan should be kept un-changed. Furthermore, for any given disruption Da in the marketscale, the manufacturer’s wholesale price and direct sale price bothdepend on the parameter q. To be specific, if a2�b

a1þa2�2b 6 q < 1, themanufacturer can set different wholesale prices and direct saleprices, while if 0 < q < a2�b

a1þa2�2b, the manufacturer should set thewholesale price equal to the direct sale price.

4.2.2. Da < 0From Lemma 5, the manufacturer’s profit can be written as

�pm ¼ ðw� cÞDr þ ð�pd � cÞDd � l2 Q 0sc � Dsc

� þ; ð47Þ

plugging (36) and (38) into (47), we obtain the manufacturer’s deci-sion problem as follows:

max �pm¼12ðw�cÞ½ð1�qÞðaþDaÞþb�pd�a1w�þð�pd

�cÞ qðaþDaÞ�a2�pdþb

2a1½ð1�qÞðaþDaÞþb�pdþa1w�

�

þ l2

2a1½ða1þbÞþða1�bÞq�ðaþDaÞf �ð2a1a2�a1b

�b2Þ�pd�a1ða1�bÞw�2a1Q 0scg;

ð48Þ

s:t: �pd P w; ð49Þ

½ða1 þ bÞ þ ða1 � bÞq�ðaþ DaÞ � ð2a1a2 � a1b� b2Þ�pd

� a1ða1 � bÞw� 2a1Q 0sc

6 0: ð50Þ

The constraint defined in (50) indicates that the disposal costoccurs only when the actual demand is less than the original plan.It is easy to examine the joint concavity of �pm with respect to �pd

and w. The following proposition characterizes the manufacturer’soptimal pricing decisions when the demand disruption Da < 0.

Proposition 6. In a manufacturer Stackelberg game when Da < 0, themanufacturer maximizes its own profit with the direct sale price andwholesale price:

Case 3: � a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l2 < Da < 0

�pcase3d ¼

ða1þbÞþða1�bÞq2 a2

1�b2þ2a1a2�2a1bð Þ ðaþ 2DaÞ þ c2

if 0 < q < a2�ba1þa2�2b ;




if a2�ba1þa2�2b 6 q < 1;

8>>>>>><>>>>>>:

ð51Þ

wcase3 ¼


1�b2þ2a1a2�2a1bÞ ðaþ 2DaÞ þ c2

if 0 < q < a2�ba1þa2�2b ;




if a2�ba1þa2�2b 6 q < 1:

8>>>>>><>>>>>>:

ð52Þ

Case 4: Da 6 � a21�b2þ2a1a2�2a1b


�pcase4d ¼


1�b2þ2a1a2�2a1bÞ ðaþ DaÞ þ c�l22

if 0 < q < a2�ba1þa2�2b ;

a2qþbð1�qÞ2ða1a2�b2Þ ðaþ DaÞ þ c�l2

2

if a2�ba1þa2�2b 6 q < 1;

8>>>>>><>>>>>>:

ð53Þ

�pcase1r

�pcase2r

�pcase4r


wcase4 ¼


1�b2þ2a1a2�2a1bÞ ðaþ DaÞ þ c�l22

if 0 < q < a2�ba1þa2�2b ;

a1ð1�qÞþbq2ða1a2�b2Þ ðaþ DaÞ þ c�l2

2

if a2�ba1þa2�2b 6 q < 1:

8>>>>>><>>>>>>:

ð54Þ

Combing (38), Propositions 5 and 6, we can obtain the retailer’soptimal retail price and the decentralized dual-channel supplychain’s optimal production quantity when disruption Da in marketdemand is resolved. They are summarized by the followingproposition.

Proposition 7. In a manufacturer Stackelberg game for givendisruption D a in market scale, the retailer’s optimal retail price isgiven by

�pcase1r ¼

ð3a21�b2þ4a1a2�2a1bÞ�ða2

1�b2þ4a1a2�4a1bÞq4a1ða2

1�b2þ2a1a2�2a1bÞ ðaþ DaÞ þ a1þb4a1ðc þ l1Þ

if 0 < q < a2�ba1þa2�2b ;

ð3a1a2�b2Þ�ð3a1a2�2a1b�b2Þq4a1ða1a2�b2Þ ðaþ DaÞ þ a1þb

4a1ðc þ l1Þ

if a2�ba1þa2�2b 6 q < 1;

8>>>>>>><>>>>>>>:

ð55Þ

�pcase2r ¼ �pcase3

r ¼

3a21�b2þ4a1a2�2a1bð Þ� a2

1�b2þ4a1a2�4a1bð Þq4a1 a2

1�b2þ2a1a2�2a1bð Þ aþ ða1þa2Þ�ða2�bÞqa2

1�b2þ2a1a2�2a1bDaþa1þb

4a1c


ð3a1a2�b2Þ�ð3a1a2�2a1b�b2Þq4a1ða1a2�b2Þ ðaþDaÞþ ða1þbÞ2þ a2

1�b2ð Þq4a1 a2

1�b2þ2a1a2�2a1bð ÞDaþa1þb4a1

c

if a2�ba1þa2�2b6q<1:

8>>>>>>>><>>>>>>>>:

ð56Þ

¼1

16a1




ðaþ DaÞ � ða1 � bÞðc þ l1Þ� �2

if 0 < q < a2�ba1þa2�2b ;

116a1½ð1� qÞðaþ DaÞ � ða1 � bÞðc þ l1Þ�

2 if a2�ba1þa2�2b 6 q < 1;

8><>: ð61Þ

¼ �pcase3r ¼

116a1

ða21�b2þ4a1a2�4a1bÞ� 3a2


1�b2þ2a1a2�2a1baþ 4a1 ½ða2�bÞ�ða1þa2�2bÞq�

a21�b2þ2a1a2�2a1b

Da� ða1 � bÞc� �2

if 0 < q < a2�ba1þa2�2b ;

116a1

ð1� qÞðaþ DaÞ � a21�b2ð Þþða1�bÞ2q


Da� ða1 � bÞc� �2

if a2�ba1þa2�2b 6 q < 1;

8>>><>>>:

ð62Þ

¼1

16a1



1�b2þ2a1a2�2a1bðaþ DaÞ � ða1 � bÞðc � l2Þ

� �2

if 0 < q < a2�ba1þa2�2b ;

116a1½ð1� qÞðaþ DaÞ � ða1 � bÞðc � l2Þ�

2 if a2�ba1þa2�2b 6 q < 1:

8><>: ð63Þ

�pcase4r ¼

ð3a21�b2þ4a1a2�2a1bÞ� a2

1�b2þ4a1a2�4a1bð Þq4a1ða2

1�b2þ2a1a2�2a1bÞ ðaþ DaÞ þ a1þb4a1ðc � l2Þ

if 0 < q < a2�ba1þa2�2b ;

ð3a1a2�b2Þ�ð3a1a2�2a1b�b2Þq4a1ða1a2�b2Þ ðaþ DaÞ þ a1þb

4a1ðc � l2Þ

if a2�ba1þa2�2b 6 q < 1;

8>>>>><>>>>>:

ð57Þand the decentralized dual-channel supply chain’s optimal productionquantity in the four cases above are given by

Qcase1 ¼ 14a1

½ða1 þ bÞ þ ða1 � bÞq�ðaþ DaÞf

� a21 � b2 þ 2a1a2 � 2a1b

� �ðc þ l1Þ

�; ð58Þ

Qcase2 ¼ Q case3

¼ 14a1

½ða1 þ bÞ þ ða1 � bÞq�a� a21 � b2 þ 2a1a2 � 2a1b

� �c

�;

ð59Þ

Qcase4 ¼ 14a1

½ða1 þ bÞ þ ða1 � bÞq�ðaþ DaÞf

� a21 � b2 þ 2a1a2 � 2a1b

� �ðc � l2Þ

�: ð60Þ

It can be found from (58)–(60) that in the decentralized dual-channel supply chain, the original optimal production quantityhas some robustness under a disrupted market condition. Whenthe market scale is mildly perturbed, no change in productionquantity is required, and adjustments in prices can be utilized tocompensate for deviation in the market scale. Only when the dis-ruption in market scale exceeds the thresholds defined in Proposi-tions 5 and 6, will it be necessary for the manufacturer to changethe production quantity. Furthermore, in case 1, the optimal pro-duction quantity defined in (58) can be interpreted as the optimalproduction quantity in a market with market scale a + Da and mar-ginal production cost c + l1 under no disrupted market scale. Sim-ilarly, in case 4, the optimal production quantity defined in (60)can be interpreted as the optimal production quantity in a marketwith market scale a + Da and marginal production cost c � l2 un-der no disrupted market scale.

From Proposition 7, the retailer’s optimal profit in the decen-tralized dual-channel supply chain is given by

Compared (61)–(63) with (35), we obtain the following obser-vations. In case 1, the retailer’s optimal profit can be interpreted

as the optimal profit in a market with market scale a + Da and mar-ginal production cost c + l1 under no disrupted market scale.When the market scale is mildly perturbed, the retailer’s optimalprofit is not affected by parameter l1and l2. This is because themanufacturer won’t change his optimal production quantity. Incase 4, the retailer’s optimal profit can be interpreted as the opti-mal profit in a market with market scale a + Da and marginal pro-duction cost c � l2.

The manufacturer’s optimal profit is given by

�pcase1m ¼

½ða1þbÞþða1�bÞq�2

8a1 a21�b2þ2a1a2�2a1bð Þ ðaþ DaÞ2 � ða1þbÞþða1�bÞq

4a1½ðaþ DaÞc þ Dal1� þ


8a1c2 þ l2

1

� �if 0 < q < a2�b

a1þa2�2b ;

2a21q


8a1ða1a2�b2Þ ðaþ DaÞ2 � ða1þbÞþða1�bÞq4a1

½ðaþ DaÞc þ Dal1� þa2


8a1c2 þ l2

1

� �if a2�b

a1þa2�2b 6 q < 1;

8><>: ð64Þ

�pcase2m ¼ �pcase3

m ¼½ða1þbÞþða1�bÞq�2

8a1 a21�b2þ2a1a2�2a1bð Þaðaþ2DaÞ�ða1þbÞþða1�bÞq

4a1ðaþDaÞcþa2


8a1c2 if 0<q< a2�b

a1þa2�2b ;

2a21q


8a1ða1a2�b2Þ ðaþDaÞ2�ða1þbÞþða1�bÞq4a1

ðaþDaÞcþa21�b2þ2a1a2�2a1b

8a1c2� ½ða1þbÞþða1�bÞq�2

8a1 a21�b2þ2a1a2�2a1bð ÞðDaÞ2 if a2�b

a1þa2�2b6q<1;

8><>:

ð65Þ

�pcase4m ¼

½ða1þbÞþða1�bÞq�2

8a1 a21�b2þ2a1a2�2a1bð Þ ðaþ DaÞ2 � ða1þbÞþða1�bÞq

4a1½ðaþ DaÞc � Dal2� þ


8a1c2 þ l2

2

� �if 0 < q < a2�b

a1þa2�2b ;

2a21q


8a1ða1a2�b2Þ ðaþ DaÞ2 � ða1þbÞþða1�bÞq4a1

½ðaþ DaÞc � Dal2� þa2


8a1c2 þ l2

2

� �if a2�b

a1þa2�2b 6 q < 1:

8><>: ð66Þ


We next analyze the impact of demand disruptions on the opti-mal profit for the manufacturer and the retailer, respectively. Table2 calculates the impact of demand disruptions on the optimal prof-it for the manufacturer. Table 3 calculates the impacts of demanddisruptions on the optimal profit for the retailer.

At last, we would like to study which kind of decision mode ismore robust for the optimal production plan under demand dis-ruptions. From the analysis above, we know that when the marketscale is mildly perturbed, in both the centralized and decentralized

Table 2Profit differential for the manufacturer with demand disruptions.

Cases q Profit

Case 1 0 < q < a2�ba1þa2�2b ða1þbÞ

4

a2�ba1þa2�2b 6 q < 1 2a2

1q2þ


4

a2�ba1þa2�2b 6 q < 1 2a2

1q2þ


4

a2�ba1þa2�2b 6 q < 1 2a2

1q2þ


4

a2�ba1þa2�2b 6 q < 1 2a2

1q2þ

Table 3Profit differential for the retailer with demand disruptions.

Cases q Profit differe

Case1 0 < q < a2�ba1þa2�2b 1

16a1

ða21�b2þ4

�

� ða21�b2þ4a1a2

a

�a2�b

a1þa2�2b 6 q < 1 116a1½ð1� qÞð

Case2 0 < q < a2�ba1þa2�2b ½ða2�bÞ�ða1þa2�

2 a21�b2þ2a1a2ð

a2�ba1þa2�2b 6 q < 1 ½ða2�bÞ�ða1þa2�

4ða21�b2þ2a1a2

Case3 0 < q < a2�ba1þa2�2b ½ða2�bÞ�ða1þa2�

2ða21�b2þ2a1a2

a2�ba1þa2�2b 6 q < 1 ½ða2�bÞ�ða1þa2�

4 a21�b2þ2a1a2ð

Case4 0 < q < a2�ba1þa2�2b 1

16a1

ða21�b2þ4

�

� ða21�b2þ4a1a2

a

�a2�b

a1þa2�2b 6 q < 1 116a1½ð1� qÞð

dual-channel supply chain, the optimal production decision is ro-bust with demand disruptions, i.e., the original production planshould be kept unchanged. The following proposition gives theresult.

Proposition 8. The optimal production plan in the centralized dual-channel supply chain is more robust than that in the decentralizeddual-channel supply chain under demand disruptions when

a2�ba1þa2�2b 6 q < 1; the opposite is true when 0 < q < a2�b

a1þa2�2b.

differential ¼ �pcaseim � pm

þða1�bÞqa1

ðDaÞ ða1þbÞþða1�bÞqa2

1�b2þ2a1a2�2a1ba� c

� �þ ½ða1þbÞþða1�bÞq�Da�ða2

1�b2þ2a1a2�2a1bÞl1f g2

8a1ða21�b2þ2a1a2�2a1bÞ

4a1bqð1�qÞþða1a2þb2Þð1�qÞ2

8a1ða1a2�b2Þ ð2aþ DaÞDa� ða1þbÞþða1�bÞq4a1

ðc þ l1ÞDaþ a21�b2þ2a1a2�2a1b

8a1l2

1

þða1�bÞqa1



� �4a1bqð1�qÞþða1a2þb2Þð1�qÞ2


cDa� ½ða1þbÞþða1�bÞq�2

8a1 a21�b2þ2a1a2�2a1bð Þ ðDaÞ2

þða1�bÞqa1



� �4a1bqð1�qÞþða1a2þb2Þð1�qÞ2


cDa� ½ða1þbÞþða1�bÞq�2

8a1 a21�b2þ2a1a2�2a1bð Þ ðDaÞ2

þða1�bÞqa1



� �þ ½ða1þbÞþða1�bÞq�Daþða2

1�b2þ2a1a2�2a1bÞl2f g2

8a1 a21�b2þ2a1a2�2a1bð Þ

4a1bqð1�qÞþða1a2þb2Þð1�qÞ2


ðc � l2ÞDaþ a21�b2þ2a1a2�2a1b

8a1l2

2

ntial ¼ �pcaseir � pr

a1a2�4a1bÞ�ð3a21�b2þ4a1a2�6a1bÞq


ð2aþ DaÞ � ða1 � bÞð2c þ l1Þ�

�4a1bÞ�ð3a21�b2þ4a1a2�6a1bÞq

21�b2þ2a1a2�2a1b

Da� ða1 � bÞl1

�2aþ DaÞ � ða1 � bÞð2c þ l1Þ�½ð1� qÞDa� ða1 � bÞl1�

2bÞq�Da�2a1bÞ

ða21�b2þ4a1a2�4a1bÞ�ð3a2

1�b2þ4a1a2�6a1bÞqa2



Da� ða1 � bÞc� �

2bÞq�Da�2a1bÞ ð1� qÞaþ 2a1 ½ða2�bÞ�ða1þa2�2bÞq�



2bÞq�Da�2a1bÞ

ða21�b2þ4a1a2�4a1bÞ�ð3a2

1�b2þ4a1a2�6a1bÞqa2




2bÞq�Da�2a1bÞ ½ð1� qÞaþ 2a1 ½ða2�bÞ�ða1þa2�2bÞq�


Da� ða1 � bÞc�

a1a2�4a1bÞ�ð3a21�b2þ4a1a2�6a1bÞq


ð2aþ DaÞ � ða1 � bÞð2c � l2Þ�

�4a1bÞ�ð3a21�b2þ4a1a2�6a1bÞq

21�b2þ2a1a2�2a1b

Daþ ða1 � bÞl2

�2aþ DaÞ � ða1 � bÞð2c � l2Þ�½ð1� qÞDaþ ða1 � bÞl2�


Proposition 8 above indicates that for the two decision modes,the centralized decision and the decentralized decision, neither isalways more robust than the other. It depends on customers’ pref-erence for the direct channel. When customers’ preference for thedirect channel exceeds the threshold a2�b

a1þa2�2b, the centralized dual-channel supply chain is more robust under demand disruptions.When customers’ preference for the direct channel is less thanthe threshold a2�b

a1þa2�2b, the decentralized dual-channel supply chainis more robust. This also indicates that customers’ preference forthe direct channel has great impact on the robustness of the twodecision modes.

5. Concluding remarks

In this paper, we studied the pricing and production problem ina two-period dual-channel supply chain consisting of one manu-facturer and one retailer with demand disruptions. To be specific,we modeled the costs of deviating from the original plan and ana-lyzed how the pricing and production decisions should be adjustedto achieve profit maximization. We first considered the problem ina centralized dual-channel supply chain setting, and then in adecentralized dual-channel supply chain setting in which the man-ufacturer assumed the role of Stackelberg leader and the retailerassumed the role of Stackelberg follower. In both settings we usedthe scenario where no demand disruption takes place as the base-line. We calculated the value of knowing of the demand disrup-tions in the centralized dual-channel supply chain. We alsocalculated the profit differentials for the manufacturer and the re-tailer, respectively, in a decentralized dual-channel supply chain.We made the following observations.

Firstly, in the centralized dual-channel supply chain, the opti-mal production plan is robust with demand disruptions when themarket scale is mildly perturbed. Only when the demand disrup-tions exceed some thresholds, should the central decision-makerchange the original production plan. Once a demand disruption isresolved, prices should be adjusted to compensate for the deviationin market scale. What’s more, it is always beneficial for the centraldecision-maker to adopt the adjusted optimal pricing decisionswhen a demand disruption occurs. Secondly, in the decentralizeddual-channel supply chain, similar to the centralized dual-channelsupply chain scenario, the optimal production plan is also robustwith a disruption when the market scale is mildly perturbed. Theoptimal retail price determined by the retailer and the optimalwholesale price and direct sale price determined by the manufac-turer should be adjusted accordingly to compensate for the devia-tion in the market scale. Thirdly, in the decentralized supply chain,the retailer’s pricing strategy and the manufacturer’s pricing strat-egy both depend on customers’ preference for the direct channeland the scale of demand disruption.

Our analysis might also have some limitations. We assume aparticular demand function in this paper and that the cross-priceeffect is symmetric for analytical simplicity. While this particulardemand model has been used extensively in economics and oper-ation management literatures, we still believe that further examin-ations are needed to investigate whether the qualitativeimplications can be generalized to other demand functions, evenif this may potentially require more complex technical skills. Thisresearch can be extended in several directions in future work.Firstly, we only consider one-manufacture-one-retailer dual-chan-nel supply chain. More research can be extended to multiple com-peting retailers. Secondly, we assume the demand disruptiondetected by the manufacturer and the retailer is identical. How-ever, the pricing and production decisions problem in a dual-chan-nel supply chain when the retailer and the manufacturer inspectdifferent demand disruptions is also a subject worth furtherinvestigation.

Acknowledgements

We would like to thank the anonymous referees for theirinsightful comments and suggestions. This research was supportedby the National Natural Science Foundation of China under Grantnumber 70871044.

Appendix A

Proof of Lemma 1. Taking the second-order partial derivatives ofpsc with respect to pr and pd, respectively, we have the followingHessian matrix

H ¼@2psc=@p2

r @2psc=@pr@pd

@2psc=@pd@pr @2psc=@p2d

!¼�2a1 2b

2b �2a2

� �

As a1 > b > 0 and a2 > b > 0, we can obtain that �2a1 < 0,(�2a1)(�2a2) � (2b)(2b) = 4(a1a2 � b2) > 0. Therefore, psc is jointlyconcave in pr and pd. Taking the first-order partial derivatives of psc

with respect to pr and pd, and letting the derivatives be zeros, weobtain@psc=@pr ¼ ð1� qÞa� 2a1pr þ 2bpd þ ða1 � bÞc ¼ 0;@psc=@pd ¼ qa� 2a2pd þ 2bpr þ ða2 � bÞc ¼ 0:

�

Solving the equations above, we obtain (8) and (9). Substituting(8) and (9) into psc, we obtain (10). h

Proof of Lemma 2. When Da > 0, we first suppose that Dsc 6 Q �sc.Then the central decision-maker’s problem is given in (19), (20).From Proposition 2, we know that Dsc ¼ Q �sc if Da > 0. In otherwords, Dsc ¼ Q �sc always dominates Dsc < Q �sc if Da > 0. Therefore,we know that Dsc P Q �sc if Da > 0. When Da < 0, the result can beproved in a similar way. h

Proof of Proposition 1. It is easy to verify that �psc is jointly con-cave in �pr and �pd. In order to solve the constrained optimizationproblem defined in (13) and (14), we introduce the Lagrange mul-tiplier k P 0 and relax the constraint given in (14), the KKT condi-tions of which are

� ð1� qÞðaþ DaÞ þ 2a1�pr � 2b�pd � ða1 � bÞðc þ l1Þþ kða1 � bÞ ¼ 0; ðA1Þ

�qðaþDaÞþ2a2�pd�2b�pr�ða2�bÞðcþl1Þþkða2�bÞ¼0; ðA2Þ

k Da� ða1 � bÞ�pr � ða2 � bÞ�pd þ12

aþ 12ða1 þ a2 � 2bÞc

� �¼ 0; ðA3Þ

k P 0: ðA4Þ

(1) Whenk = 0, we obtain

�pr ¼a2ð1� qÞ þ bq

2ða1a2 � b2Þðaþ DaÞ þ c þ l1

2;

�pd ¼a1qþ bð1� qÞ


2:

The constraint given in (14) reduces to Da P (a1 + a2 � 2b)l1. Thus,�pr and �pd defined above are the optimal solutions when D a P(a1 + a2 � 2b)l1.

(2) When k > 0, from with (A1), (A2) and (A3), we obtain



2ða1a2 � b2Þðaþ DaÞ þ c

2þ Da

2ða1 þ a2 � 2bÞ ;



2þ Da

2ða1 þ a2 � 2bÞ ;

k ¼ l1 �Da

a1 þ a2 � 2b:

Note that k > 0 is equivalent to 0 < D a < (a1 + a2 � 2b)l1, whichindicates that that �pr and �pd defined above are the optimal solu-tions when 0 < Da < (a1 + a2 � 2b)l1. Combining with the twocases, we obtain (15) and (16). Substituting (15) and (16) into Dsc

and simplifying, we obtain (17). h

Proof of Proposition 2. It can be verified that �psc is jointly concavein �pr and �pd. In order to solve the constrained optimization problemgiven in (19) and (20), we introduce the Lagrange multiplier k P 0and relax the constraint given in (20), the KKT conditions of whichare

�ð1�qÞðaþDaÞþ2a1�pr�2b�pd�ða1�bÞðc�l2Þ�kða1�bÞ¼0; ðA5Þ

�qðaþDaÞþ2a2�pd�2b�pr�ða2�bÞðc�l2Þ�kða2�bÞ¼0; ðA6Þ

k Da� ða1 � bÞ�pr � ða2 � bÞ�pd þ12ða1 þ a2 � 2bÞc þ 1

2a

� �¼ 0; ðA7Þ

k P 0: ðA8Þ

(1) Whenk = 0, we obtain


2ða1a2 � b2Þðaþ DaÞ þ c � l2

2;



2:

The constraint given in (20) reduces to Da 6 � (a1 + a2 � 2b)l2.Thus, �pr and �pd defined above are the optimal solutions whenDa 6 � (a1 + a2 � 2b)l2.(2) Whenk > 0, we have Da� ða1 � bÞ�pr � ða2 � bÞ�pd þ 1

2 ða1 þa2 � 2bÞc þ 1

2 a ¼ 0 from (A7). Combining with (A5) and (A6), we ob-tain�pr ¼

a2ð1� qÞ þ bq2ða1a2 � b2Þ

ðaþ DaÞ þ c2þ Da

2ða1 þ a2 � 2bÞ ;



2þ Da

2ða1 þ a2 � 2bÞ ;

k ¼ l2 þDa

a1 þ a2 � 2b:

Note that k > 0 is equivalent to �(a1 + a2 � 2b)l2 < Da < 0. Com-bining with the two cases above, we obtain (21) and (22). Substi-tuting (21) and (22) into Dsc and simplifying, we obtain (23). h

Proof of Proposition 3. The results can be directly obtained byintegrating Propositions 1 and 2 after some algebra manipula-tions. h

Proof of Lemma 3. It is easy to verify that the retailer’s profitfunction given by (3) is concave inpr. By using the first-order con-dition, we can obtain (29). h

Proof of Lemma 4. Taking the second-order partial derivatives ofpm with respect to pd and w, we have the following Hessian matrix

H ¼@2pm=@p2

d @2pm=@pd@w

@2pm=@w@pd @2pm=@w2

!¼ ð�2a1a2 þ b2Þ=a1 b

b �a1

!

As @2pm=@p2d ¼ ð�2a1a2 þ b2Þ=a1 < 0; @2pm=@w2 ¼ �a1 < 0, and

jHj = 2(a1a2 � b2) > 0, therefore, pm is jointly concave with respectto pd and w. h

Proof of Proposition 4. Relaxing the constraint pd P w, and byusing the Lagrange multiplier k P 0, the KKT condition is given by

ð1� qÞa� 2a1wþ 2bpd þ ða1 � bÞc � 2k ¼ 0; ðA9Þ

½2a1qþ bð1� qÞ�aþ 2ðb2 � 2a1a2Þpd þ 2a1bw

þ ð2a1a2 � a1b� b2Þc þ 2a1k ¼ 0; ðA10Þ

kðpd �wÞ ¼ 0; ðA11Þ

k P 0: ðA12Þ

(1) If k = 0, substituting it to (A9) and (A10), we have

w ¼ a2ð1� qÞ þ bq2ða1a2 � b2Þ

aþ c2;

pd ¼a1qþ bð1� qÞ


2:

As the constraint pd P w must be satisfied, after some algebramanipulation, we know that pd P w is equivalent to q P a2�b

a1þa2�2b.That is, when q P a2�b

a1þa2�2b is satisfied, the constraint pd P w is notbinding.(2) If k > 0, from (A11), we have pd = w, substituting it to (A9) and

(A10), we havew ¼ pd ¼

ða1 þ bÞ þ ða1 � bÞq2 a2

1 � b2 þ 2a1a2 � 2a1b� � aþ c

2;

k ¼ a1½ða2 � bÞ � ða1 þ a2 � 2bÞq�a2

1 � b2 þ 2a1a2 � 2a1ba:

From (A12), we obtain 0 < q < a2�ba1þa2�2b, that is, when

0 < q < a2�ba1þa2�2b is satisfied, the constraint pd P w is binding. Inte-

grating the two cases above, the results hold. h

Proof of Lemma 5. When Da > 0, we first suppose that Dsc 6 Q 0sc.

Then the central decision-maker’s problem is given in (48)–(50).From Propositions 6 and 7, we know that Dsc ¼ Q0

sc if Da > 0. Inother words, Dsc ¼ Q 0

sc always dominates Dsc < Q 0sc if Da > 0. There-

fore, we know that Dsc P Q 0sc if Da > 0. When Da < 0, the result can

be proved in a similar way. h

Proof of Proposition 5. Relaxing the constraint given in (41) and(42), and by using the Lagrange multiplier k1 P 0 and k2 P 0,respectively, the KKT condition is given by

½2a1qþ bð1� qÞ�ðaþ DaÞ þ 2ðb2 � 2a1a2Þ�pd þ 2a1bw

þ ð2a1a2 � a1b� b2Þðc þ l1 � 2a1k2Þ þ 2a1k1 ¼ 0; ðA13Þ

ð1� qÞðaþ DaÞ þ 2b�pd � 2a1wþ ða1 � bÞðc þ l1Þ� 2k1 � 2a1ða1 � bÞk2 ¼ 0; ðA14Þ


k1ð�pd �wÞ ¼ 0; ðA15Þ

k2f½ða1 þ bÞ þ ða1 � bÞq�ðaþ DaÞ � ð2a1a2 � a1b� b2Þ�pd

� a1ða1 � bÞw� 2a1Q 0scg ¼ 0; ðA16Þ

k1 P 0; k2 P 0: ðA17Þ

(1) When k1 = 0 and k2 = 0, solving the Eqs. (A13)–(A16), weobtain


ðaþ DaÞ þ c þ l1

2;



2:

Note that the constraint defined in (42) is equivalent toDa P a2


ða1þbÞþða1�bÞq l1, and the constraint given in (41) is equiva-lent to a2�b

a1þa2�2b < q < 1.(2) When k1 > 0 and k2 = 0, solving the Eqs. (A13)–(A16), we

obtain

�pd ¼ w ¼ ða1 þ bÞ þ ða1 � bÞq2 a2

1 � b2 þ 2a1a2 � 2a1b� � ðaþ DaÞ þ c þ l1

2;

k1 ¼a1½ða2 � bÞ � ða2 þ a2 � 2bÞq�

a21 � b2 þ 2a1a2 � 2a1b

ðaþ DaÞ:

Note that the constraint defined in (42) is equivalent toDa P a2


ða1þbÞþða1�bÞq l1, and the constraint k1 > 0 is equivalent to0 < q < a2�b

a1þa2�2b. Combining the two scenarios above, we obtain(43) and (44).

(3) When k1 = 0 and k2 > 0, solving the Eqs. (A13)–(A16), weobtain

w¼a2ð1�qÞþbq2ða1a2�b2Þ

ðaþDaÞþ c2þ ða1þbÞþða1�bÞq

2ða21�b2þ2a1a2�2a1bÞ

Da;

�pd¼a1qþbð1�qÞ2ða1a2�b2Þ


2ða21�b2þ2a1a2�2a1bÞ

Da;

k2¼l1

2a1� ða1þbÞþða1�bÞq

2a1 a21�b2þ2a1a2�2a1b

� �Da:

Note that the constraint k2 > 0 is equivalent to

Da <a2


ða1þbÞþða1�bÞq l1, and the constraint defined in (41) is equiva-

lent to a2�ba1þa2�2b < q < 1.

(4) When k1 > 0 and k2 > 0, solving the Eqs. (A13)–(A16), weobtain


1 � b2 þ 2a1a2 � 2a1b� � ðaþ 2DaÞ þ c

2;

k1 ¼a1½ða2 � bÞ � ða2 þ a2 � 2bÞq�

a21 � b2 þ 2a1a2 � 2a1b

ðaþ DaÞ;

k2 ¼l1

2a1� ða1 þ bÞ þ ða1 � bÞq

2a1 a21 � b2 þ 2a1a2 � 2a1b

� �Da:

Note that the constraint k1 > 0 is equivalent to 0 < q < a2�ba1þa2�2b,

and the constraint k2 > 0 is equivalent to Da <a2


ða1þbÞþða1�bÞq l1.

Combining the two scenarios above, we obtain (45) and (46). h

Proof of Proposition 6. Relaxing the constraint given in (49) and(50), and by using the Lagrange multiplier k1 P 0 and k2 P 0,respectively, the KKT condition is given by

½2a1qþ bð1� qÞ�ðaþ DaÞ þ 2ðb2 � 2a1a2Þ�pd

þ 2a1bwþ ð2a1a2 � a1b� b2Þðc � l2 þ 2a1k2Þ þ 2a1k1 ¼ 0; ðA18Þ

ð1� qÞðaþ DaÞ þ 2b�pd � 2a1wþ ða1 � bÞðc � l2Þ � 2k1

þ 2a1ða1 � bÞk2 ¼ 0; ðA19Þ

k1ð�pd �wÞ ¼ 0; ðA20Þ

k2f½ða1 þ bÞ þ ða1 � bÞq�ðaþ DaÞ � ð2a1a2 � a1b� b2Þ�pd

� a1ða1 � bÞw� 2a1Q 0scg ¼ 0; ðA21Þ

k1 P 0; k2 P 0: ðA22Þ

(1) When k1 = 0 and k2 = 0, solving the Eqs. (A18)–(A21), weobtain


ðaþ DaÞ þ c � l2

2;



2:

Note that the constraint defined in (50) is equivalent to

Da 6 � a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l2. The constraint given in (49) is equivalent

to a2�ba1þa2�2b < q < 1.

(2) When k1 > 0 and k2 = 0, solving the Eqs. (A18)–(A21), weobtain


1 � b2 þ 2a1a2 � 2a1b� � ðaþ DaÞ þ c � l2

2;

k1 ¼a1½ða2 � bÞ � ða2 þ a2 � 2bÞq�

a21 � b2 þ 2a1a2 � 2a1b

ðaþ DaÞ:

Note that the constraint defined in (50) is equivalent

toDa 6 � a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l2, and k1 > 0 is equivalent to

0 < q < a2�ba1þa2�2b. Combining the two scenarios above, we obtain

(51) and (52).(3) When k1 = 0 and k2 > 0, solving the Eqs. (A18)–(A21), we

obtain

w¼a2ð1�qÞþbq2ða1a2�b2Þ


2 a21�b2þ2a1a2�2a1b

� �Da;

�pd¼a1qþbð1�qÞ2ða1a2�b2Þ


2 a21�b2þ2a1a2�2a1b

� �Da;

k2¼l2

2a1þ ða1þbÞþða1�bÞq

2a1ða21�b2þ2a1a2�2a1bÞ

Da:

Note that the constraint k2 > 0 is equivalent to

Da > � a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l2, and the constraint defined in (49) is equiv-

alent to a2�ba1þa2�2b < q < 1.

(4) When k1 > 0 and k2 > 0, solving the Eqs. (A18)–(A21), weobtain



1 � b2 þ 2a1a2 � 2a1b� � ðaþ 2DaÞ þ c

2;

k1 ¼a1½ða2 � bÞ � ða2 þ a2 � 2bÞq�

a21 � b2 þ 2a1a2 � 2a1b

ðaþ DaÞ;

k2 ¼l2

2a1þ ða1 þ bÞ þ ða1 � bÞq

2a1 a21 � b2 þ 2a1a2 � 2a1b

� �Da:

Note that the constraint k1 > 0 is equivalent to 0 < q < a2�ba1þa2�2b,

and the constraint k2 > 0 is equivalent to Da > � a21�b2þ2a1a2�2a1b

ða1þbÞþða1�bÞq l2.

Combining the two scenarios above, we obtain (53) and (54). h

Proof of Proposition 7. The results can be easily obtained bysubstituting the results given in Propositions 5 and 6 into (38)and simplifying, so we omit it here. h

Proof of Proposition 8. Compare the threshold value given inPropositions 3, 5 and 6, we obtain ða1 þ a2 � 2bÞ�a2


ða1þbÞþða1�bÞq ¼ða1�bÞ½ða1þa2�2bÞq�ða2�bÞ�

ða1þbÞþða1�bÞq . It is easy to verify that theresults hold. h

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Pricing and production decisions in dual-channel supply chains with demand disruptions