Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand

European Journal of Operational Research 272 (2019) 147–161
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European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and LogisticsManaging a dual-channel supply chain under price and delivery-timedependent stochastic demandNikunja Mohan Modak a, Peter Kelle b,∗a Palpara Vidyamandir, Chakdaha, 741222, West Bengal, Indiab SEIS Dept., E J Ourso College of Business, Louisiana State University, Baton Rouge, LA 70803, USAa r t i c l e i n f oArticle history:Received 13 May 2017Accepted 29 May 2018Available online 25 June 2018Keywords:Supply chain managementSupply chain coordinationDual-channel supply chainPrice dependent demandLead time dependent demandStochastic demandDistribution-free approacha b s t r a c tSeveral leading manufacturers recently combined the traditional retail channel with a direct online chan-nel to reach a wider range of customers. We examine such a dual-channel supply chain under price anddelivery-time dependent stochastic customer demand. We consider five decision variables, the price andorder quantity for both the retail and the online channels and the delivery time for the online channel.Uncertainty frequently arises in both retail and online channels and so additional inventory managementis required to control shortage or overstock and that has an effect on the optimal order quantity, price,and lead time. We developed mathematical models with the profit maximization motive. We analyzeboth centralized and decentralized systems for unknown distribution function of the random variablesthrough a distribution-free approach and also for known distribution function. We examine the effect ofdelivery lead time and customers’ channel preference on the optimal operation. For supply chain coor-dination a hybrid all-unit quantity discount along a franchise fee contract is used. Moreover, we use thegeneralized asymmetric Nash bargaining for surplus profit distribution. A numerical example illustratesthe findings of the model and the managerial insights are summarized for centralized, decentralized, andcoordinated scenarios.Published by Elsevier B.V.1. IntroductionTo reap the benefits of online sales, leading companies includ-ing Hewlett-Packard, IBM, Eastman Kodak, Nike, and Apple haveadded online direct channels for customers to their traditional re-tail channels (Tsay & Agrawal, 2004). The main reason behind theestablishment of the dual-channel supply chain is to address awider range of consumers. There are different types of customers.Some dislike shopping in retail stores due to their busy schedulesor some inconvenience in retail shops like long queue, mismanage-ment, behavior of the retailer, weather conditions, etc. Other cos-tumers dislike online shopping because they prefer the ability topersonally inspect the merchandise, ask for advice and assistanceand be able to take their purchases home immediately rather thanhaving to pay shipping costs and wait for delivery. This type of cus-tomer behavior is called as customers’ channel preference or com-patibility index. To address both types of consumers many manu-facturers and distributors sell their products through dual-channel,combining the traditional brick and mortar channel with online
∗ Corresponding author.
E-mail address: [e-mail] (P. Kelle).sales (Nair & Pleasance, 2005). A survey report reveals that about42% of the top industrial suppliers are selling also directly to con-sumers through online channel (Dan, Xu, & Liu, 2012).Price and delivery time are two key factors in the customers’choice. The speed of shipping is very important to develop on-line customers’ loyalty. A survey conducted by Dotcom Distribu-tion found that 87% of online shoppers identified shipping speedas a key factor in the decision to shop online (mhlnews.com). Thissurvey describes that delivery time is more important than priceas 67% of online shoppers would pay more to get same-day de-livery. Thus, the delivery time of online channel needs to managecarefully to achieve competitiveness.This paper examines a two-echelon (manufacturer and retailer)dual-channel (retail and direct online) supply chain with stochasticdemand which depends on selling prices (retail and online) andonline delivery lead time. To address uncertainties, this articleassumes stochastic demand for both the retail and online channelsand shows the effect of demand uncertainty on the optimaldecision extending existing literature. Total supply chain profitmaximization is the objective. We developed mathematical mod-els combining issues handled separately in literature includingdual-channel supply chain, price and delivery-time dependent de-mand and uncertainty in demand. We examined the effect of the
https://doi.org/10.1016/j.ejor.2018.05.067
0377-2217/Published by Elsevier B.V.

148 N.M. Modak, P. Kelle / European Journal of Operational Research 272 (2019) 147–161customers’ channel preference of the product (product compatibil-ity) on the optimal operation of dual-channel supply chains. Weanalyze both centralized and decentralized systems when the ran-dom variables follow a known distribution function and also withunknown distribution function of the random variables through adistribution-free approach. The decentralized channel interactionsbetween the manufacturer and the retailer are studied under theStackelberg game setting. We also address the situation when thechannel members are independent and they make their decisionthrough a Nash game. A hybrid all-unit quantity discount alonga franchise fee contract is used for supply chain coordination.Moreover, we use the generalized asymmetric Nash bargainingfor surplus profit distribution. A numerical example illustrates thetheoretical findings of the model for centralized, decentralized,and coordinated scenarios. Finally, we determined a closed formsolution of the decentralized system under deterministic demandassumption. We also investigated the following managerial ques-tions: What is the effect of the delivery lead time on the optimalselling prices? How does the transfer rate of consumers fromonline channel to retail channel affect optimal decision? How tocoordinate decentralized channel to eliminate double marginal-ization? How to distribute surplus profit between the channelmembers? What is the feasible range of product compatibility forsuccessful operation of a dual-channel supply chain?The rest of the paper is organized as follows. Section 2 dis-cusses relevant literature, Section 3 provides the notations and as-sumptions, and Section 4 develops mathematical formulations andanalysis of the proposed dual-channel supply chain. Section 4.1 an-alyzes centralized decisions while Section 4.2 deals with decentral-ized decisions. Section 5 discusses channel coordination and sur-plus profit distribution and Section 6 numerically illustrates themodels. Section 7 summarizes the managerial findings of the pa-per. Finally, Section 8 provides the conclusions and future researchsuggestions.2. Literature reviewWe summarize the relevant literature handling the differentsupply chain characteristics of our models including dual-channelsupply chain with product compatibility, price and delivery-timedependent demand, uncertainty in demand in both retail and on-line channels, and distribution-free approach.Several researchers and practitioners have focused on dual-channel supply chains during the last decade. Huang and Swami-nathan (2009) considered the introduction of online channel as asecond channel to sell a product and analyzed its effect on pricingand profit. Takahashi, Aoi, Hirotani, and Morikawa (2011) and Tsaoand Su (2012) determined the optimal price and warranty lengthto provide for end customers in a dual-channel supply chain. Chenand Cao (2012) found that the customer channel migration behav-ior strongly influences the manufacturer’s and the retailer’s pricingstrategies and profits. Huang, Yang, and Zhang (2012) analyzedcentralized and decentralized production and pricing decisions ina two-echelon dual channel supply chain model under demanddisruption. Panda, Modak, Sana, and Basu (2015) examined pricingand shipment policies in a dual-channel supply chain for productswith decreasing unit cost. Their analysis revealed the effects ofproduct compatibility on the successful operation of dual-channelsupply chain. Xiao and Shi (2016) investigated the pricing andchannel priority strategy of a dual-channel supply chain facingpotential supply shortage caused by yield uncertainty and theyexplored the effects of decentralization of the supply chain onthe channel priority strategy. Yan et al. (2018) developed a two-period game-theoretic model to investigate the effect of productdurability on dual channel operation. The latest publications indual-channel supply chain extend the research in different areasincluding product distribution strategy (Matsui, 2016), qualityimprovement (Chen, Liang, Yao, & Sun, 2017a), recycling and envi-ronmental effects (Batarfi, Jaber, & Aljazzar, 2017; Feng, Govindan,& Li, 2017; Ji et al., 2017), timing effect of the price announcements(Matsui, 2017), and how to obtain equilibrium prices for a retailerin a Stackelberg dual-channel supply chain (Chen, Zhang, Zhang,& Chen, 2017b).The papers listed in this paragraph all assumedeterministic demand that is extended to stochastic demand inour paper.Although there are number of articles in the literature on dual-channel supply chain but few of them have considered the effectof delivery time decisions on the online channel. Hua, Wang, andCheng (2010) determined optimal decisions of delivery time andprices in a centralized and a decentralized dual-channel supplychain and analyzed the impacts of delivery time and product com-patibility on the manufacturer’s and retailer’s pricing decision fordeterministic demand. We extend this research for stochastic de-mand case expressing its effect on the optimal decisions includ-ing the order quantities as additional decision variables. Xu, Liu,and Zhang (2012) followed a threshold policy in both forms ofownership to investigate how price and delivery lead time deci-sions affect channel configuration strategy and how the choice ofchannel structure depends on customer acceptance of the onlinechannel, also assuming known demand. Considering price and de-livery lead-time sensitive demand Pekgün, Griffin, and Keskinocak(2016) explored price and lead-time competition of two firms in acommon market. We extend these studies to random demand.Stochastic demand has been considered also in dual-channelsupply chain research. Chiang and Monahan (2005) dealt with theinventories in a dual-channel supply chain that receives stochasticdemand from retail and online channels but no price dependentdemand is considered. Yao, Yue, Wang, and Liu (2005) discussedthe benefits of sharing demand forecast information in a two-level dual-channel supply chain. They analyzed the model for boththe make-to-order and the make-to-stock scenarios and showedthat the direct channel has a negative impact on the retailer’sperformance. Yue and Liu (2006) explored the hybrid make-to-order and make-to-stock (MTO-MTS) scenario in a manufacturer-retailer dual channel supply chain based on the assumption of in-formation sharing between the channel members. Recently, un-der the hybrid MTO-MTS production systems Beemsterboer et al.(2017b) analyzed the job shop control problem while Beemsterboeret al. (2017a) examined the effect of lot size flexibility but theydid not consider price and time dependent demand. The presentstudy adopts a hybrid MTO-MTS production system extending pre-vious research to price and lead time dependent random demand.Dumrongsiri, Fan, Jain, and Moinzadeh (2008) showed that theequilibrium prices and the manufacturer’s motivation for openinga direct channel have been highly influenced by the demand varia-tion. Under stochastic demand environment Yu and Liu (2012) an-alyzed how a manufacturer can motivate the retailer to expand itsbusiness adopting dual sales channel. They showed that the pro-motional effort of the retailer heavily depends on the cost sharingdecision of the manufacturer. Roy, Sana, and Chaudhuri (2016) de-veloped a two-echelon dual-channel supply chain to determine theoptimal stock level, prices, promotional effort and service level un-der stochastic demand environment. All the papers in this para-graph deal with either pricing or inventory aspects consideringrandom demand but disregard the dependence of demand on thedelivery time that is explicitly handled in our paper besides theinventory requirement due to random demand.Sometimes randomness of demand may follow a well-knowndistribution but in practice usually there is limited informationabout the randomness of the demand, only the mean and variance

N.M. Modak, P. Kelle / European Journal of Operational Research 272 (2019) 147–161 149of the random variables are estimated. There is a tendency ofusing the normal distribution in this case disregarding other dis-tributions having the same mean and variance (Moon, Yoo, & Saha,2016). Scarf (1958) solved a newsboy problem where only themean and variance of the demand are known. Without knowingany further information about the form of distribution function ofthe demand they considered the maximum of the lower boundsof expected profit for all possible distribution function. Based onScarf’s model, researchers extended the results in different areasincluding continuous review inventory models (Gallego & Moon,1993; Moon & Gallego, 1994), continuous review inventory modelwith a service level constraint (Moon & Choi, 1994), customersbalk (Moon & Choi, 1995), multi-item newsboy problem (Moon& Silver, 20 0 0; Vairaktarakis, 20 0 0), resalable returns (Mostard,De Koster, & Teunter, 2005) and multiple discounts and upgrades(Moon et al., 2016).Motivated by the practice and the relevant research in supplychain management, this article considers dual-channel operationsunder delivery lead time and price dependent stochastic demandand deals also with safety inventories. Since uncertainty frequentlyappear in both sales channels, shortage and overstock may occurthat makes necessary to include the ordering/inventory decisionsfor retail and online channel and delivery lead time for onlinechannel. The decision variables considered are the delivery leadtime of the online channel, and the pricing and inventory deci-sions on both cannels under demand uncertainties. The modelswe handle are quite complex but the joint consideration of thedifferent effects can provide major improvements as we show inthe paper. Proper management against uncertainty provides morechannel profit than the decision made under the assumption ofdeterministic demand, disregarding uncertainty. The demand ran-domness also influences the optimal price and delivery time de-cisions through the effect of the expected shortage that results indecreasing expected demand. Thus, considering the expected de-mand only will lead to inflated prices and longer delivery timedecision.This paper contributes to the literature in several aspects. Tothe best of our knowledge, no article in literature considered priceand delivery-time dependent stochastic demand in dual-channelsupply chain. There are publications on dual-channel supply chainconsidering decisions of delivery lead time and prices (for ex-ample Hua et al., 2010; Saha, Modak, Panda, & Sana, 2018; Xuet al., 2012) but under deterministic demand assumption with-out inventory consideration. Modak (2017) developed a two-levelomni-channel supply chain under price and delivery time sensi-tive additive stochastic demand for known distribution function ofthe random variables. Yang et al. (2017) proposed a newsvendormodel in a dual-channel supply chain with consideration of thedelivery lead time of the online channel. In this model they ex-amined the switching behavior of the consumer based on stock-outs and lead-time but neglected the effects of the retail and on-line channel price on demand functions. Differing from the Yanget al. (2017) model the present study deals with the retail chan-nel price, online channel price and delivery lead time dependentstochastic demand to determine optimal pricing and delivery leadtime policies for both known and unknown distribution of ran-dom variables. Moreover, this work applies also distribution-freeapproach.Under uncertainty, centralized and decentralized models aredeveloped in our paper for both known distribution of therandom demand as well as applying distribution-free approachwhich is also new in literature. In decentralized scenario, in-teraction between the manufacturer and the retailer are deter-mined through Nash game. This research provides new mathe-matical models and managerial findings advancing the presentliterature.3. Assumptions and notations3.1. Assumptionsa) Our models consider a two-level dual-channel supply chain for a singleperiod and single product. The manufacturer supplies the product to theretailer and also sells the product directly to the consumers through onlinechannel. The consumers may choose the retail channel or online channelto obtain the product.b) Channel members know all the related information. That is, we assumeinformation symmetry between the manufacturer and the retailer.c) Cost of operating retail and online channels are normalized to zero.d) Demand is dependent on retail price, online price and delivery lead timeof the online channel.e) Price change in one channel switches a portion of demand from thatchannel to the other channel.f) Online channel will lose some demand when its delivery lead timeincreases, of which some amount may transfer to the retail channel andsome units may be lost from the system.g) If the offered delivery time is long, the manufacturer has to invest less,hence the delivery cost is considered as a decreasing function of deliverylead time.3.2. NotationsFollowing notations are used to develop the model.Cost parametersc unit cost of the manufacturerh per unit holding cost at the end of the periodς per unit shortage cost at the end of the periodr0 & r1 delivery time dependent cost parameters of the online channelDemand parametersDo deterministic demand of online channelDos stochastic demand of online channelDr deterministic demand of retail channelDrs stochastic demand of retail channela market potential of the productk is the customers’ channel preference index (0 < k < 1) toward retailchannel also known as product compatibility indexa1 number of customers prefer the online channel [a1 = (1-k)a]a2 number of customers prefer the retailer channel [a2 = ka]Sensitivity parametersβ1 price sensitivity in online channelβ2 price sensitivity in retail channelγ1 delivery time sensitivity parameter of the demand in online channelγ2 delivery time sensitivity parameter of the demand in retail channelδ number of customers switching from the retail channel to the onlinechannel per unit increase in the price difference between po and pr .It is also called as transferred demand rateDecision variables and objectiveszr ordered quantity to satisfy the stochastic portion of the demand inthe retail channelzo ordered quantity to satisfy the stochastic portion of the demand inthe online channelpo selling prices of the product in the online channelpr selling prices of the product in the retail channelL delivery lead time of the product in the online channelw wholesale price for the retailer set by the manufacturerπI , m r profit functions of integrated channel, manufacturer and retailerE[.] expected profit function4. Model formulationThe channel structure of the proposed model is presented inFig. 1.To obtain the demand functions of the online channel (Do )and the retail channel (Dr ), this work extends the frameworkestablished by Huang and Swaminathan (2009) and Hua et al.(2010). We also employ demand functions with linear self andcross-price effects but under the influence of delivery time. Theform of the demand functions for the online and retail chan-nels are: Do = a1 − β1 po − γ1 L − δ ( po − pr ) and Dr = a2 − β2 pr +γ2 L + δ ( po − pr ). The demand is positive in both channels if

150 N.M. Modak, P. Kelle / European Journal of Operational Research 272 (2019) 147–161 Fig. 1. Channel structure.0< 1 − (β1 +δ ) po −δ pr +γ1 La < k < (β2 +δ ) pr −δ po −γ2 La < 1. If the deliverytime L increases then γ1 L partof demand will be lost from the on-line channel of which γ2 L (γ2 ≤ γ1 ) units will transfer to the retailchannel and (γ1 − γ2 )L units will be lost from the two channels.For the delivery cost of the manufacturer we assume a quadraticform (r0 − r1 L )2 with r0 /r1 > L which is common in literature (seeDesiraju & Moorthy, 1997, Savaskan & Van Wassenhove, 2006).In addition to the deterministic demand functions, we considerrandomness in demand that can be accomplished in two ways,multiplicative and additive forms (Mills, 1959; Petruzzi & Dada,1999). As the additive case offers better tractability, we add a dif-ferent exogenous random variable for both online and retail de-mand. The form of stochastic demand functions are as follows:Dos = Do + εo = a1 − β1 po − γ1 L − δ ( po − pr ) + εo (1)Drs = Dr + εr = a2 − β2 pr + γ2 L + δ ( po − pr ) + εr (2)where εo and εr are random variables defined on the ranges [Ao,Bo] and [Ar , Br ] with the mean μo and μr , standard deviations σ 0and σ r and cumulative distribution functions Fo ( . ) and Fr ( . ), re-spectively. Due to uncertainty in demand, the supply chain willface some additional overstocking and understocking costs.First, we discuss the integrated supply chain under central-ized decision making context and then the decentralized decisionswhere supply chain members make their decisions through Stack-elberg or Nash games. Sensitivity analysis is presented in Section6 and the managerial insights are discussed in Section 7.4.1. Centralized decision making through distribution-free approachUnder centralized decision making, all actions are controlled byone decision maker or alternatively, all the channel members arewilling to cooperate and want to implement a joint decision. Inthis case the product is manufactured in a single batch and is soldto the customers through the retail and online channels. The ex-pected profit function of the integrated supply chain (using sub-script I)E [πI ] = ∫ zrAr [ pr (Dr + u ) − h(zr − u )] fr (u )du+ ∫ Brzr [ pr (Dr + zr ) − ς (u − zr )] fr (u )du − c (Dr + zr )+ ∫ zoAo [ po (Do + v ) − h(zo − v )] fo (v )dv+ ∫ Bozo [ po (Do + zo ) − ς (v − zo )] fo (v )dv−c (Do + zo ) − (r0 − r1 L )2where zr and zo respectively represent the quantity ordered andproduced to satisfy the stochastic portion of the demand for theretail and online channels and the total order quantities are Dr + zrand Do + z0 . Equivalently, we haveE [πI ] = ( pr − c )(Dr + μr ) + ( po − c )(D0 + μo )−(r0 − r1 L )2 − (c + h )( r + o ) − ( pr + ς − c ) r−( po + ς − c ) o(3)with the notationr =∫ zrAr(zr − u ) fr (u )du and o =∫ zoAo(zo − v ) fo (v )dvexpressing the expected value of the random demand satisfied inretail and online channel, andr =∫ Brzr(u − zr ) fr (u )du and o =∫ Bozo(v − zo ) fo (v ) dvexpressing the expected value of the random demand lost in thetwo channels.The two random variables εo and εr may follow some well-known distribution but in most practical cases the information ofthe distribution function is limited to the expected value and vari-ance estimates. Following Gallego and Moon (1993), in this casewe maximize the expected profit for the worst possible distribu-tion (that gives the minimum profit) of the random variables εoand εr with means (μo , μr ) and the variances (σ 2o , σ 2r ). We usethe inequality (Lemma-1 of Gallego & Moon, 1993) E (D − Q )+ ≤[σ 2 +(Q−μ )2 ]1/2 −(Q−μ )2 , so for any distribution function of the ran-dom variables εo and εr , the minimum expected profit of the inte-grated channel can be derived from Eq. (3):MinE [πI ] = pr (Dr + μr ) + po (D0 + μo ) − (r0 − r1 L )2− c (Dr + Do + zr + zo ) − h(zr − μr )− h(zo − μo ) − ( pr + ς + h )×[σ 2r + (zr − μr )2 ] 12− (zr − μr )2− ( po + ς + h )[σ 2o + (zo − μo )2 ] 12− (zo − μo )2 (4)The optimization problem under the distribution-free approachcan be presented as:Problem (P1 ):maxpr , po ,L,zr ,zoMin E [πI ]s.t. : pr , po , L, zr zo > 0(5)The necessary conditions given in Appendix-1 (available online)will provide the optimal solutions of the expected profit functionof the integrated channel if the Hessian matrix, HImin is negativedefinite at the stationery point ( p∗rI , p∗oI , L∗I ,z∗rI and z∗oI ) (also pre-sented in Appendix-1). The model considers five decision variablesjointly. Although the above model can provide optimal equilib-rium decision under concavity conditions, but it provides limitedinsights for managers because of the factors such as the random,price-sensitive demand, delivery lead time, and dual channels in-teract with each other. To separate these effects, in the followingsub-sections we analyze the model under different conditions.