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

Accepted Manuscript

Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand Nikunja Mohan Modak , Peter Kelle PII: DOI: Reference:

S0377-2217(18)30486-7 10.1016/j.ejor.2018.05.067 EOR 15180

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

13 May 2017 25 May 2018 29 May 2018

Please cite this article as: Nikunja Mohan Modak , Peter Kelle , Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research (2018), doi: 10.1016/j.ejor.2018.05.067

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Highlights

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Manufacturers combine retail channel with online channel to address a wider range of customers. Consider dual-channel supply chain under price and delivery-time dependent stochastic customer demand. Demand uncertainty requires inventories that affect the optimal price and lead time. Effect of customers’ channel preference on the optimal operation of dual-channel supply chains. Centralized and decentralized decisions for known distribution and distribution-free approach.

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

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EOR_15180 Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand Nikunja Mohan Modak

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Palpara Vidyamandir Chakdaha-741222, West Bengal, India Email: [email protected]

Peter Kelle

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SEIS Dept., E J Ourso College of Business

Louisiana State University, Baton Rouge, LA 70803, USA Email: [email protected] Corresponding author

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Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand Abstract: Several leading manufacturers recently combined the traditional retail channel with a

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direct online channel to reach a wider range of customers. We examine such a dual-channel supply chain under price and delivery-time dependent stochastic customer demand. We consider five decision variables, the price and order quantity for both the retail and the online channels and the delivery time for

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the online channel. Uncertainty frequently arises in both retail and online channels and so additional inventory management is required to control shortage or overstock and that has an effect on the optimal

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order quantity, price, and lead time. We developed mathematical models with the profit maximization motive. We analyze both centralized and decentralized systems for unknown distribution function of the random variables through a distribution-free approach and also for known distribution function. We

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examine the effect of delivery lead time and customers’ channel preference on the optimal operation. For supply chain coordination a hybrid all-unit quantity discount along a franchise fee contract is used. Moreover, we use the generalized asymmetric Nash bargaining for surplus profit distribution. A numerical example illustrates the findings of the model and the managerial insights are summarized for centralized, decentralized, and coordinated scenarios.

Key words: Supply chain management; Supply chain coordination; Dual-channel supply chain;

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Price dependent demand; Lead time dependent demand; Stochastic demand; Distribution-free approach 1. Introduction To reap the benefits of online sales, leading companies including Hewlett-Packard, IBM, Eastman Kodak, Nike, and Apple have added online direct channels for customers to their traditional retail channels (Tsay and Agrawal 2004). The main reason behind the establishment of the dual-channel

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supply chain is to address a wider range of consumers. There are different types of customers. Some dislike shopping in retail stores due to their busy schedules or some inconvenience in retail shops like long queue, mismanagement, behavior of the retailer, weather conditions, etc. Other costumers dislike online shopping because they prefer the ability to personally inspect the merchandise, ask for advice and assistance and be able to take their purchases home immediately rather than having to pay shipping costs and wait for delivery. This type of customer behavior is called as customers’ channel preference or

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compatibility index. To address both types of consumers many manufacturers and distributors sell their products through dual-channel, combining the traditional brick and mortar channel with online sales (Nair and Pleasance, 2005). A survey report reveals that about 42% of the top industrial suppliers are selling also directly to consumers through online channel (Dan et al., 2012).

Price and delivery time are two key factors in the customers’ choice. The speed of shipping is

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very important to develop online customers’ loyalty. A survey conducted by Dotcom Distribution found that 87% of online shoppers identified shipping speed as a key factor in the decision to shop online (mhlnews.com). This survey describes that delivery time is more important than price as 67% of online

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shoppers would pay more to get same-day delivery. Thus, the delivery time of online channel needs to manage carefully to achieve competitiveness.

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This paper examines a two-echelon (manufacturer and retailer) dual-channel (retail and direct online) supply chain with stochastic demand which depends on selling prices (retail and online) and online delivery lead time. To address uncertainties, this article assumes stochastic demand for both the

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retail and online channels and shows the effect of demand uncertainty on the optimal decision extending existing literature. Total supply chain profit maximization is the objective. We developed mathematical

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models combining issues handled separately in literature including dual-channel supply chain, price and delivery-time dependent demand and uncertainty in demand. We examined the effect of the customers’ channel preference of the product (product compatibility) on the optimal operation of dual-channel supply chains. We analyze both centralized and decentralized systems when the random variables follow a known distribution function and also with unknown distribution function of the random variables through a distribution-free approach. The decentralized channel interactions between the manufacturer and the retailer are studied under the Stackelberg game setting. We also address the situation when the channel members are independent and they make their decision through a Nash game. A hybrid all-unit quantity

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discount along a franchise fee contract is used for supply chain coordination. Moreover, we use the generalized asymmetric Nash bargaining for surplus profit distribution. A numerical example illustrates the theoretical findings of the model for centralized, decentralized, and coordinated scenarios. Finally, we determined a closed form solution of the decentralized system under deterministic demand assumption. We also investigated the following managerial questions: What is the effect of the delivery lead time on the optimal selling prices? How does the transfer rate of consumers from online channel to retail channel

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affect optimal decision? How to coordinate decentralized channel to eliminate double marginalization? How to distribute surplus profit between the channel members? What is the feasible range of product compatibility for successful operation of a dual-channel supply chain?

The rest of the paper is organized as follows. Section 2 discusses relevant literature, Section 3 provides the notations and assumptions, and Section 4 develops mathematical formulations and analysis

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of the proposed dual-channel supply chain. Subsection 4.1 analyzes centralized decisions while subsection 4.2 deals with decentralized decisions. Section 5 discusses channel coordination and surplus profit distribution and Section 6 numerically illustrates the models. Section 7 summarizes the managerial findings of the paper. Finally, Section 8 provides the conclusions and future research suggestions.

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2. Literature review

We summarize the relevant literature handling the different supply chain characteristics of our models including dual-channel supply chain with product compatibility, price and delivery-time

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dependent demand, uncertainty in demand in both retail and online channels, and distribution-free approach.

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Several researchers and practitioners have focused on dual-channel supply chains during the last decade. Huang and Swaminathan (2009) considered the introduction of online channel as a second channel to sell a product and analyzed its effect on pricing and profit. Takahashi et al. (2011) and Tsao

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and Su (2012) determined the optimal price and warranty length to provide for end customers in a dualchannel supply chain. Chen and Cao (2012) found that the customer channel migration behavior strongly

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influences the manufacturer’s and the retailer’s pricing strategies and profits. Huang et al. (2012) analyzed centralized and decentralized production and pricing decisions in a two-echelon dual channel supply chain model under demand disruption. Panda et al. (2015) examined pricing and shipment policies in a dual-channel supply chain for products with decreasing unit cost. Their analysis revealed the effects of product compatibility on the successful operation of dual-channel supply chain. Xiao and Shi (2016) investigated the pricing and channel priority strategy of a dual-channel supply chain facing potential supply shortage caused by yield uncertainty and they explored the effects of decentralization of the supply chain on the channel priority strategy. Yan et al. (2017) developed a two-period game-theoretic model to

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investigate the effect of product durability on dual channel operation. The latest publications in dualchannel supply chain extend the research in different areas including product distribution strategy (Matsui, 2016), quality improvement (Chen et al., 2017a), recycling and environmental effects (Feng et al., 2017, Batarfi et al., 2017, Ji et al., 2017), timing effect of the price announcements (Matsui, 2017), and how to obtain equilibrium prices for a retailer in a Stackelberg dual-channel supply chain (Chen et al., 2017b).The papers listed in this paragraph all assume deterministic demand that is extended to stochastic

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demand in our paper.

Although there are number of articles in the literature on dual-channel supply chain but few of them have considered the effect of delivery time decisions on the online channel. Hua et al. (2010) determined optimal decisions of delivery time and prices in a centralized and a decentralized dual-channel supply chain and analyzed the impacts of delivery time and product compatibility on the manufacturer’s

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and retailer’s pricing decision for deterministic demand. We extend this research for stochastic demand case expressing its effect on the optimal decisions including the order quantities as additional decision variables. Xu et al. (2012) followed a threshold policy in both forms of ownership to investigate how price and delivery lead time decisions affect channel configuration strategy and how the choice of channel structure depends on customer acceptance of the online channel, also assuming known demand. Considering price and delivery lead-time sensitive demand Pekgün et al. (2016) explored price and lead-

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time competition of two firms in a common market. We extend these studies to random demand. Stochastic demand has been considered also in dual-channel supply chain research. Chiang and

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Monahan (2005) dealt with the inventories in a dual-channel supply chain that receives stochastic demand from retail and online channels but no price dependent demand is considered. Yao et al. (2005) discussed the benefits of sharing demand forecast information in a two-level dual-channel supply chain. They

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analyzed the model for both the make-to-order and the make-to-stock scenarios and showed that the direct channel has a negative impact on the retailer’s performance. Yue and Liu (2006) explored the hybrid

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make-to-order and make-to-stock (MTO-MTS) scenario in a manufacturer-retailer dual channel supply chain based on the assumption of information sharing between the channel members. Recently, under the

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hybrid MTO-MTS production systems Beemsterboer et al. (2016) analyzed the job shop control problem while Beemsterboer et al. (2017) examined the effect of lot size flexibility but they did not consider price and time dependent demand. The present study adopts a hybrid MTO-MTS production system extending previous research to price and lead time dependent random demand. Dumrongsiri et al. (2008) showed that the equilibrium prices and the manufacturer’s motivation for opening a direct channel have been highly influenced by the demand variation. Under stochastic demand environment Yu and Liu (2012) analyzed how a manufacturer can motivate the retailer to expand its business adopting dual sales channel. They showed that the promotional effort of the retailer heavily depends on the cost sharing decision of the

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manufacturer. Roy et al. (2016) developed a two-echelon dual-channel supply chain to determine the optimal stock level, prices, promotional effort and service level under stochastic demand environment. All the papers in this paragraph deal with either pricing or inventory aspects considering random demand but disregard the dependence of demand on the delivery time that is explicitly handled in our paper besides the inventory requirement due to random demand. Sometimes randomness of demand may follow a well-known distribution but in practice usually

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there is limited information about the randomness of the demand, only the mean and variance of the random variables are estimated. There is a tendency of using the normal distribution in this case disregarding other distributions having the same mean and variance (Moon et al. 2016). Scarf (1958) solved a newsboy problem where only the mean and variance of the demand are known. Without knowing any further information about the form of distribution function of the demand they considered

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the maximum of the lower bounds of expected profit for all possible distribution function. Based on Scarf’s model, researchers extended the results in different areas including continuous review inventory models (Gallego and Moon, 1993, Moon and Gallego, 1994), continuous review inventory model with a service level constraint (Moon and Choi, 1994), customers balk (Moon and Choi, 1995), multi-item newsboy problem (Vairaktarakis, 2000; Moon and Silver, 2000), resalable returns (Mostard et al., 2005) and multiple discounts and upgrades (Moon et al. 2016).

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Motivated by the practice and the relevant research in supply chain management, this article considers dual-channel operations under delivery lead time and price dependent stochastic demand and

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deals also with safety inventories. Since uncertainty frequently appear in both sales channels, shortage and overstock may occur that makes necessary to include the ordering/inventory decisions for retail and online channel and delivery lead time for online channel. The decision variables considered are the

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delivery lead time of the online channel, and the pricing and inventory decisions on both cannels under demand uncertainties. The models we handle are quite complex but the joint consideration of the

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different effects can provide major improvements as we show in the paper. Proper management against uncertainty provides more channel profit than the decision made under the assumption of deterministic

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demand, disregarding uncertainty. The demand randomness also influences the optimal price and delivery time decisions through the effect of the expected shortage that results in decreasing expected demand. Thus, considering the expected demand only will lead to inflated prices and longer delivery time decision. This paper contributes to the literature in several aspects. To the best of our knowledge, no article

in literature considered price and delivery-time dependent stochastic demand in dual-channel supply chain. There are publications on dual-channel supply chain considering decisions of delivery lead time and prices (for example Hua et al., 2010; Xu, et al., 2012; Saha et al. 2018) but under deterministic demand assumption without inventory consideration. Modak (2017) developed a two-level omni-channel

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supply chain under price and delivery time sensitive additive stochastic demand for known distribution function of the random variables. Yang et al. (2016) proposed a newsvendor model in a dual-channel supply chain with consideration of the delivery lead time of the online channel. In this model they examined the switching behavior of the consumer based on stockouts and lead-time but neglected the effects of the retail and online channel price on demand functions. Differing from the Yang et al. (2016) model the present study deals with the retail channel price, online channel price and delivery lead time

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dependent stochastic demand to determine optimal pricing and delivery lead time policies for both known and unknown distribution of random variables. Moreover, this work applies also distribution-free approach.

Under uncertainty, centralized and decentralized models are developed in our paper for both known distribution of the random demand as well as applying distribution-free approach which is also

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new in literature. In decentralized scenario, interaction between the manufacturer and the retailer are determined through Nash game. This research provides new mathematical models and managerial findings advancing the present literature.

3. Assumptions and Notations

a)

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3.1 Assumptions

Our models consider a two-level dual-channel supply chain for a single period and single product. The manufacturer supplies the product to the retailer and also sells the product directly

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to the consumers through online channel. The consumers may choose the retail channel or online channel to obtain the product. Channel members know all the related information. That is, we assume information symmetry

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b)

between the manufacturer and the retailer.

d) e)

Cost of operating retail and online channels are normalized to zero.

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c)

Demand is dependent on retail price, online price and delivery lead time of the online channel. Price change in one channel switches a portion of demand from that channel to the other

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channel.

f)

Online channel will lose some demand when its delivery lead time increases, of which some amount may transfer to the retail channel and some 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 delivery lead time.

3.2 Notations

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Following notations are used to develop the model. Cost parameters c

unit cost of the manufacturer

h

per unit holding cost at the end of the period

ς

per unit shortage cost at the end of the period delivery time dependent cost parameters of the online channel Demand parameters

Do

deterministic demand of online channel

Dos

stochastic demand of online channel

Dr

deterministic demand of retail channel

Drs

stochastic demand of retail channel

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r0 & r1

market potential of the product

k

is the customers’ channel preference index (0
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a

known as product compatibility index a1

number of customers prefer the online channel [a1=(1-k)a]

a2

number of customers prefer the retailer channel [a2 = ka]

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Sensitivity parameters

price sensitivity in online channel

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price sensitivity in retail channel delivery time sensitivity parameter of the demand in online channel delivery time sensitivity parameter of the demand in retail channel δ

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number of customers switching from the retail channel to the online channel per unit increase in the price difference between

and

. It is also called as transferred

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demand rate

zr

ordered quantity to satisfy the stochastic portion of the demand in the retail channel

zo

ordered quantity to satisfy the stochastic portion of the demand in the online channel

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Decision variables and objectives

selling prices of the product in the online channel selling prices of the product in the retail channel

L

delivery lead time of the product in the online channel

w

wholesale price for the retailer set by the manufacturer profit functions of integrated channel, manufacturer and retailer expected profit function

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4. Model formulation The channel structure of the proposed model is presented in Figure 1.

Retailer

Selling Price

Retail Channel

Uncertainty

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Online Channel

Manufacturer

Delivery lead time

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Demand

Consumer

Figure1: Channel structure

To obtain the demand functions of the online channel (Do) and the retail channel (Dr), this work

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extends the framework established by Huang and Swaminathan (2009) and Hua et al. (2010). We also employ demand functions with linear self and cross-price effects but under the influence of delivery time.

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The form of the demand functions for the online and retail channels are: and

.The demand is positive in both channels if 0 . If the delivery time L increases then

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will be lost from the online channel of which

(

of demand

) units will transfer to the retail channel and

units will be lost from the two channels. For the delivery cost of the manufacturer we assume

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a quadratic form

with r0/r1>L which is common in literature (see Desiraju and Moorthy,

1997, Savaskan and Wassenhove, 2006).

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In addition to the deterministic demand functions, we consider randomness in demand that can be accomplished in two ways, multiplicative and additive forms (Mills, 1959, Petruzzi and Dada, 1999). As the additive case offers better tractability, we add a different exogenous random variable for both online and retail demand. The form of stochastic demand functions are as follows:

where

and

are random variables defined on the ranges [

,

] and [

,

] with the mean

and

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, standard deviations σ0 and σr and cumulative distribution functions

and

respectively. Due

to uncertainty in demand, the supply chain will face some additional overstocking and understocking costs. First, we discuss the integrated supply chain under centralized decision making context and then the decentralized decisions where supply chain members make their decisions through Stackelberg or Nash games. Sensitivity analysis is presented in Section 6 and the managerial insights are discussed in

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Section 7.

4.1 Centralized decision making through distribution-free approach

Under centralized decision making, all actions are controlled by one decision maker or alternatively, all the channel members are willing to cooperate and want to implement a joint decision. In

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this case the product is manufactured in a single batch and is sold to the customers through the retail and online channels. The expected profit function of the integrated supply chain (using subscript I) ∫

∫

where

and

∫

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∫

respectively represent the quantity ordered and produced to satisfy the stochastic portion

with the notation

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Equivalently, we have

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of the demand for the retail and online channels and the total order quantities are Dr+ zr and Do+z0.

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∫

∫

expressing the expected value of the random demand satisfied in retail and online channel, and ∫

and

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∫

expressing the expected value of the random demand lost in the two channels. The two random variables

and

may follow some well-known distribution but in most

practical cases the information of the distribution function is limited to the expected value and variance estimates. Following Gallego and Moon (1993), in this case we maximize the expected profit for the worst possible distribution (that gives the minimum profit) of the random variables (

,

) and the variances (

,

and

with means

).We use the inequality (Lemma-1 of Gallego and Moon, 1993)

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, so for any distribution function of the random variables

and

, the

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minimum expected profit of the integrated channel can be derived from equation (3):

The optimization problem under the distribution-free approach can be presented as: Problem (P1): ,

, ,

and

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s.t.:

The necessary conditions given in Appendix-1 (available online)will provide the optimal solutions of the expected profit function of the integrated channel if the Hessian matrix, definite at the stationery point (

,

,

,

and

is negative

) (also presented in Appendix-1). The model

considers five decision variables jointly. Although the above model can provide optimal equilibrium

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decision under concavity conditions, but it provides limited insights for managers because of the factors such as the random, price-sensitive demand, delivery lead time, and dual channels interact with each

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other. To separate these effects, in the following sub-sections we analyze the model under different conditions.

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4.1.1. Delivery lead time sensitive demand To analyze the effects of the delivery lead time more precisely first consider that delivery lead

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time of the online channel as the only decision variable of the channel. Following the approach of Porteus (1990), the maximization of

with respect to delivery lead time. Thus,

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of optimization of

is evaluated from equation (4) by using the necessary conditions

Moreover notice that,

Also note that,

and

, so . Further,

yields

is a concave function of . , and

.

From these results we have the following proposition. Proposition-1: (i) For given selling prices (

and

) and order quantities to satisfy random

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demand ( ,

),

is a concave function of the online delivery lead time and the optimal value

is (ii) The optimal lead time for delivery will increase (decrease) if price of the product increases in retail (online) channel. (iii) The optimal lead time for delivery will increase (decrease) if delivery time sensitivity

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parameter, γ increases in retail (online) channel.

Demand of the product in retail and online channel are realized as follows

Comparing DoL and DrL we get

if √

and

if

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√

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(7)

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where

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and

Figure 2 (a)

√

√

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Figure 2 (b)

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Figure 2. Effect of the delivery sensitivity parameters of online marketing on the demand of retail and online channels

Figure 2(a) and 2(b) respectively present the effect of consumers’ delivery sensitivity parameters,

and

, on the retail and online channel demand. Online channel demand will outperform

retail channel demand if

lies below the threshold √

or above

. Notice that,

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the threshold

√

hence retail channel √

demand will outperform online channel demand if

Online

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(ii)

(i)

Retail

channel

demand

outperforms

online

√

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Proposition-2:

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. From the above discussion we have the following proposition.

channel √

demand

channel

demand

if

√ outperforms {

retail

channel √

demand }

if

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Figure 3. Dependency of the retail (blue, upper surface) and online (red, lower surface) channels demand on parameters

and .

Figure 3 is a graphical representation of demand of retail and online channels for different (see the Figure 2(a)), i.e.,

and it attains its minimum at

channel demand increases.

is a

and above this threshold the online

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convex function of

Note that,

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under the condition that

and

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4.1.2. Price sensitive demand

This sub-section assumes that only and

are decision variables. For given order quantities

) and lead time (L), solving

and

for

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(

and

and

we

have the following proposition. Proposition-3: (i) For given order quantities to satisfy random demand ( ,

) and lead time (L)

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the optimum selling prices in retail and online channel are given by √

√

(9) √

√

(10)

Note that Proposition-4 will maximize

is a concave function of when

,

and

and

. Hence, the selling prices described in

are given. Also note that,

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and clear that

since

. It is

. On the other hand,

if

i.e., if

. That means, if the rate of lost sale with respect to the sale transfer to retail channel (due to increment of the delivery lead time) is less than that

then

will increase if

increases. Note

,

,

and

.

That

is,

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although prices in both the retail and online channels are decreasing (increasing) with the increment of ( ), but the rate of decrement (increment) is much higher in the online (retail) channel than in the retail (online) channel.

So far the discussions are following the procedure of Porteus (1990) where maximization of is evaluated based on the assumption that the value of some decision variables are given. The

variables

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above results can easily be applied for known distribution functions, Fr(.) and Fo(.), of the random . The key results are presented in the following propositions.

Proposition-4: Optimal value of the decision variables in the integrated decision (

,

,

(

)

(

)

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) satisfy the following five equations simultaneously:

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and

,

)

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(

Proposition-5: (i) The optimal stocking decision for demand uncertainty in retail (online)

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channel is increasing with retail (online) price. (ii) Selling price of the product in retail channel is greater than that in online channel if

compatibility of the product ( ) lies in ( , 1) and selling price of the product in online channel is greater than that in retail channel if

. Where kI =

.

Detailed discussion and proofs are presented in Supplementary document available online.

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The centralized scenario is the benchmark for decision making in supply chain. Centralized channel always outperform decentralized channel as there is no ‘double marginalization’ having a single decision maker. Although managers or decisions makers want to apply optimal supply chain decisions but in reality they face several difficulties due to decentralized channel structure analyzed in the next section.

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4.2. Decentralized channels The manufacturer and the retailer are independent in decentralized channels and they try to maximize their own expected profit without considering the total supply chain's expected profit. The retailer has to determine

and

to maximize its expected profit. On the other hand,

,

and

are

the decision variables of the manufacturer. Expected profit functions of the retailer and the manufacturer

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under the non-cooperative scenario are respectively as follows

Using the same approach as described in centralized scenario, the minimum expected profit of the retailer is given by

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for any distribution of the random variable

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The minimum expected profit of the manufacturer for any distribution of the random variable

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by

is given

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Following the line of the classic literature in dual-channel, interactions between the manufacturer and the retailer are studied under the Stackelberg game setting. The manufacturer acts as the Stackelberg

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leader of the channel. The downstream channel member (retailer) acts as the follower. The manufacturer first announces the wholesale price, online price, delivery lead time and stocking decision of the online channel. Based on the manufacturer’s decision, the retailer determines the retail price and stocking decision of retail channel. Here, optimal decisions are determined through the method of backward induction. Solving the first order necessary conditions to maximize

based on the assumption

that the values of some decision variables are given, the following proposition is valid. Proposition-6: In decentralized decision making context under the Stackelberg game setting (i)

For given

, the minimum expected profit function of the retailer is a concave function of

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and optimal value of

is given by √

(ii)

For given

, the minimum expected profit function of the retailer is a concave function of

and optimal value of

is given by

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√

(iii) The minimum expected profit function of the retailer is a concave function of if

(

(

)

)

√

. Optimal value ( ,

equations √

where,

.

Proof: See Appendix-2 (available online).

and the rate of increment of

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Note that from equation (20) to

) will satisfy the following

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√

and

is linearly proportional

. Considering the available information of the retailer’s decision, the manufacturer leader of the

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channel will now optimize its profit function. Solving the first order necessary conditions after substituting retailer decision to maximize ,

and

are given, the following proposition is valid.

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decision variables

, based on the assumption that the values of the

Proposition-7: In decentralized decision making context under the Stackelberg game setting For given

the minimum expected profit function of the manufacturer is a concave function

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and L if

and

. Optimal value of

and L are given by

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of

,

√

(

Proof. First, substituting

in (19) and then solving

√

)

and

we

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have the optimal value of wholesale price and delivery lead time presented in Proposition-7. Moreover

note

that,

,

, Thus,

function of

and L if

and

,

is a concave

. ;

. These results prove that the

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Note that,

behavior of the delivery lead time in decentralized channel with respect to

and

remain the same as it

is obtained in the centralized channel. In decentralized channel, interaction between the manufacturer and the retailer are determined through wholesale price, which is the most important decentralized decision variable. Notice that,

, so increasing consumers sensitivity to delivery lead

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time will decrease the optimal wholesale price and the rate of decrease mainly depends on the value of

.

For higher sensitivity to delivery lead time more consumers will leave marketing through the online channel and attract those consumers towards retail channel the manufacturer will reduce the wholesale price. This move effectively forces the retailer to reduce its retail price as (

)

. On the other hand,

, so the wholesale price increases with the increasing online price if ). Considering

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2 ( the following proposition.

, and L as decision variables we have

given

,

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Proposition-8: In decentralized decision making context under the Stackelberg game setting, for the minimum expected profit function of the manufacturer is a concave function of

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Optimal value of online).

.

L, and the proof of the proposition are given in Appendix-3 (available

From Proposition-8 we note that the optimal value of

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Substituting optimal values of

and L are functions of

( ,

Figure 4 and Figure 5.

) and

.

and L in (18) and (19), we have the minimum expected profit

function of the manufacturer and the retailer as function of the

and

and

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L if

,

( ,

) with respect to

only. Graphical representation of and

are respectively presented in

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Figure 4. Nature of the minimum expected profit function of the manufacturer

Figure 5. Nature of the minimum expected profit function of the retailer

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Figure 4 clearly shows the concavity of with the increasing value of

) with respect to

. On the other hand, Figure 5 depicts that

, while it increases ( ,

) is a concave

. Moreover, Figure 6(a) shows that if the retailer accepts the cooperative order decision of

AC

function of

( ,

integrated channel (

), then the manufacturer will receive its highest expected profit

accepting the cooperative order decision of integrated channel i.e., If

. On the other

hand, under the decentralized decision making, the retailer will receive maximum profit ordering when the manufacturer orders the integrated order quantity (see Figure 6(b)).

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(a) Manufacturer

(b) Retailer

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Figure 6. Graphical representation of the minimum expected profit function with respect to zo and Extended analysis of the above decentralized model for known distribution functions of the random variables as well as for the distribution-free approach under the Nash game provides the following main results presented in Supplementary documents S1, S2 and S3 with detailed discussions. and

if

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Proposition-9: (i) The expected profit function of the retailer is a concave function of

The retailer's optimal decision following the Nash game will satisfy the following equations

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)

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(

Proposition-10: (i) The expected profit function of the manufacturer is a concave function of if

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and

,

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(ii) The manufacturer's optimal decision under Nash game will satisfy the following equations

(

)

Propositions 9 and 10 outline the response of each member given the other member’s decisions.At an equilibrium point, each party must respond optimally given the other party's decisions. Therefore, such a point must satisfy both Propositions 9 and 10. Under the distribution-free approach the following propositions are valid.

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Proposition-11: In decentralized decision making context if the retailer and manufacturer follow Nash game then the minimum expected profit function of the retailer is a concave function of ( (

)

√

)

. The optimal values

,

and

if

will satisfy the following

√ √

where,

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equations

.

,

) following Nash game

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Proposition-12: The manufacturer's optimal decision ( , under distribution-free approach will satisfy the following equations

√

)

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√

√

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(

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√

If the manufacturer and the retailer are playing the game simultaneously that is, one member

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making its decisions without responding to the other one's decision, the equilibrium solution can be obtained from Proposition-11 and Proposition-12. In order to determine closed form of the equilibrium solution and to analyze effect of various factors on the operation of the dual-channel supply chain, next a

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special case is considered where demand is deterministic.

Special case: deterministic demand in decentralized case Consider the special case where the demand is deterministic in both channels. That is, and we normalize both

and

to be 0, we derived the following propositions. Detailed discussions

are presented in the Supplementary document S4, available online. Proposition-13: Under deterministic demand environment if the manufacturer and the retailer

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make their decentralized decisions through Nash game then (i) the equilibrium online selling price of the manufacturer decreases (increases) if delivery lead time increases (decreases). (ii) the equilibrium retail price increases (decreases) with the increment of delivery lead time if (

)

make their decentralized decisions through Nash game then (i)

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Proposition-14: Under deterministic demand environment if the manufacturer and the retailer

it is not economic to operate business through dual-channel mode if product compatibility k lies in

the equilibrium retail price is less than the equilibrium online selling price if

(iii)

the equilibrium online channel price is less than the equilibrium retail price if

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(ii)

where the threshold values of kr, ko, and k1 are expressed as functions of the system parameters in Supplementary document S4, available online.

In decentralized setting, there is no reason for the retailer to order the centralized optimum,

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unless it receives some surplus profit. To induce retailer for the adaptation of the centralized decisions various well established coordination contract mechanism can be applied. As the retailer has two decision and

), a hybrid coordination contract has the ability to coordinate the channel and provide

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variables (

win-win outcome. Next section analyzes the issue of supply chain coordination.

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5. Channel coordination through distribution-free approach using all unit quantity discount along with franchise fee contract

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For channel coordination, assume that the manufacturer provides an all unit quantity discount to the retailer and charges a franchise fee. We apply these two coordination contract mechanisms jointly for channel coordination. Suppose that the manufacturer supplies the product to the retailer at a wholesale

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price

and charges a franchise fee,

retailer and manufacturer are

. Then the minimum expected profit function of

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The retailers necessary condition

to maximize its profit yields

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√ The proposed coordination contract resolves channel conflict only when optimal value of the quantity discount fraction,

, which means that the

√

(

)

The proposed hybrid contract can be implemented successfully only if the minimum expected

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profit of the retailer and the manufacturer in the proposed mechanism is greater than or equal to their respective decentralized minimum expected profit, i.e.,

, which after simplification provides the upper and lower bounds,

and

on the

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franchise fee, fd

and

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From the above discussion we have the following proposition.

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Proposition-15: An all unit quantity discounts along with franchise fee contract coordinates the supply chain for ( ,

) and the optimal value of the quantity discount fraction

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lies in (

) and the channel members get win-win opportunity when the franchise fee ( )

Channel coordination generates surplus profit equal to (

is expressed in (39).

).

Surplus profit will be shared between the manufacturer and the retailer through settlement of the franchise fee ( ) within the range (

,

). This settlement can be realized using various factors like their

decentralized profit ratio, respective bargaining powers etc. To divide surplus profit between the channel members, in the next subsection, we incorporate the generalized asymmetric Nash bargaining solution (Nash, 1950) technique.

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5.1 Surplus profit division through bargaining In this sub-section, we use the generalized asymmetric Nash bargaining solution to distribute surplus profit generated through cooperation. Suppose the bargaining power of the manufacturer and the retailer are θ and (1-θ), respectively. Here, the objective is to maximize the following Nash bargaining

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product (42)

The optimal bargaining solution of the franchise fee is found from (42) as follows:

Where,

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and

. Now, substituting

in (36) and

(37), the profits of the channel members after bargaining can be found as [

denotes channel surplus generated through coordination. In the next section, we

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Note that,

]

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present numerical illustration of the model.

6. Numerical illustration

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We apply normal distributed demand in the first subsection and then the distribution-free approach in the next subsection using the same numerical values of the parameters: , ,

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,

,

,

,

,

,

,

,

,

,

, ,

,

. In our numerical setting the Hessian matrix of the minimum expected profit

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function of the integrated channel is given by

(

The eigenvalues of the matrix

) are all negative (-98.0077, -26.8744, -13.755, -2.34108, -1.54245)

which confirms the concavity of the minimum expected profit function of the integrated channel. The optimal values of the decision variables under distribution-free approach under the centralized,

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decentralized and coordinated scenarios are presented in Table 1. From Table 1 note that the decentralized channel receives a profit of 18322.2 which is 13.3% less than the centralized profit. Optimal value of the quantity discount fraction (

) is 0.447592. Thus, effective

discounted wholesale price under the coordination contract is equal to 73.754. Table 1. Optimal decision in the centralized, decentralized and coordinated scenarios for γ2=0.5 Decentralized Retailer

Manufacturer

165.434

209.081

_

-

_

164.779

168.769

_

168.588

9.7406

_

9.506

32.6321

(

27.1571

_

φ

-

-

,

-

-

20760.4

1603.9

Retailer

Manufacturer

165.434

_

_

164.779

_

168.769

_

9.7406

_

32.6321

27.157

-

27.1571

-

0.447592

-

(4000.0, 6438.2)

16718.3

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) MinE(πx)

Coordinated

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w

Centralized

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Optimal

Channel members get win-win profit when

(1603.9, 4042.1)

(16718.3, 19156.5)

. The range of profit of the retailer and

the manufacturer under the proposed all unit quantity discounts along with franchise fee contract are

PT

(1603.9, 4042.1) and (16718.3, 19156.5) respectively. Surplus profit share between the channel members depends on their respective bargaining power, . In particular, if

then minimum expected profit

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of the retailer and the manufacturer will be 2416.6 and 18343.8 respectively.

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Table 2. Optimal decision in the centralized, decentralized and coordinated scenarios for γ2=0.2 Optimal

w

Centralize

Decentralized

Coordinated

d

Retailer

Manufacturer

Retailer

Manufacturer

163.895

206.87

_

163.895

_

-

_

163.291

_

163.291

168.576

_

168.352

_

168.576

9.385

_

_

9.385

32.602 φ (

,

)

_

_

32.602

27.154

_

27.154

_

27.154

-

-

-

0.451435

-

-

-

(3864.2, 6228.7)

20431.5

1542.2

16524.8

(1542.2, 3906.7)

(16524.8, 18889.3)

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MinE(πx)

In Table 1 we assume γ2=0.5 i.e., more than 70% out going unsatisfied customers of online channel return to the retail channel. Table 2 depicts the situation when γ2=0.2, i.e., less than 30% outgoing unsatisfied customers of online channel return to the retail channel. Table 2 shows that the manufacturer decreases the delivery lead time due to low return rate of outgoing unsatisfied customers. Also note that, total profit

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decreases while quantity discount fraction increases in Table 2 compared to the results of Table 1.

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(a) quantity discount fraction(b) franchise fee

Figure 7. Effect of mean of the random variable on the coordination contract parameters

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Figure 7(a) and 7(b) respectively depict the effect of

on the quantity discount fraction and franchise

fee range. Bargaining space increases but quantity discount fraction decreases if

increases. Table 3

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discusses effect of the mean of the random variables on optimal decisions, profits and also on the coordination contract parameters.

d

e

iz

Optimal al

tr

n

Model

Table3. Optimal decision in the centralized, decentralized and coordinated

e

C

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5

5

40 5

150.1

153.98

168.5

1.5 40 172.3

10

.6 166.06

1.3 165.84

164.73

164.44

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174.30

178.1

169.28

167.90

169.05

167.61

9.7610

9.6427

9.7088

9.7402

9.7488

9.7350

9.7438

7.32

7.40

42.69

42.76

30.79

30.63

35.24

35.22

6.94 15828.2

42.24

7.0012

25.86

30.36

25.70

19840.7

19654

42.30 23801.1

21281.4

20562.3

20561.9

30.35 19790.5

191.5

218.4

211.5

211.39

205.78

205.51

153.4

171.6

165.8

165.65

163.38

163.10

174.1

177.9

169.3

167.92

168.59

167.16

9.5799

9.4615

9.5044

9.5129

9.5026

9.5115

1.21

36.14

36.08

28.86

29.09

22.15

22.16

6.94

42.24

7.0012

42.30

25.86

30.36

25.70

30.35

MinE(πr)

1108.6

1102.6

1828.84

1822.8

1973.4

2005.47

1084.9

1085.67

MinE(πm)

12874.8

16877

15140.6

19276.8

17081

16348.3

16737

15967.4

0.4881

0.4277

0.4407

0.4453

0.4442

0.4518

0.4508

2965.6

2940.3

4481.7

4447.2

4192.96

4210.14

3716.34

3721.21

4810.4

4801.4

7166.3

7148.67

6419.9

6418.7

6456.32

6458.58

φ*

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9.4405

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Coordinated

Decentralized

w

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MinE(πI)

154.8

Table 3 depicts that the minimum expected profit of the integrated channel attains its lowest value

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when the means are small. The reason behind it is straightforward, the demand decreases due to small value of the means and as a result the expected channel profit decreases. A lower mean of the random

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variables results in lower selling price in both channels but longer online delivery lead time. Thus, irrespective of the nature of random variables, if the mean of uncertainties decrease then selling prices decrease and the expected channel profit decreases. Table3 highlights that expected channel profit

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decreases with the increase of the variances. The reason is that for larger

and

the channels have to

stock more product to manage uncertainty, raising the total cost.

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To illustrate the improvement provided by our model we compare the above results to the outcome of making the best decision without considering the uncertainty of demand. Disregarding randomness in demand, the optimal value of the decision variables are

=147.95,

𝑜=152.55,

and

=9.76. Now putting these decision variables into the random demand case and optimizing the total profit

provides the optimal values

= 32.27 and

𝑜=

26.89. The total minimum expected integrated channel

profit is equal to 20,319.7. That is, the monetary gain of considering the uncertainty in demand has the numerical value 20,760.4 – 20,319.7 = 440.7. So, we can achieve 2.17% higher expected profit in the above numerical example considering demand uncertainty in our decision. Assuming the same parameter

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values as Table 2 provides a 2.2% monetary gain. The optimal selling prices in both retail and online channel are lower than for the stochastic demand but delivery lead time, L is higher if we disregard randomness in demand. 6.1 Sensitivity analysis To understand the impact of parameters on the optimal decisions and on the corresponding total

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channel profit, sensitivity analysis is conducted here changing the price elasticity, channel potential and marginal manufacturing cost parameters. The results of the sensitivity analysis are illustrated in Figures 8 to 10.The five graphs show how the percent changes in the elasticity parameters influence the optimal value of the five decision variables and the minimum expected total profit. All changes are in percentages

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relative to the base case solution.

Figure 8(b). Effect of

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Figure 8(a). Effect of

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Figure 8. Effect of online price elasticity ( and expected profit. Figures 8(a) and 8(b) illustrate that the increase of

) on the optimal decisions

has a significant decreasing effect in the

online-channel selling price and less impact on retail-channel selling price while

has a similar impact

AC

on retail-channel selling price but less impact on online-channel selling price. Both

and

(the

quantities ordered to satisfy the stochastic portion of the demand in retail and online) decrease slowly with the increment of (

and

. The optimal online lead time increases (decreases) with increasing

). The channel profit heavily depends on the price elasticity parameters, it is a decreasing function of

both elasticity parameters.

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Figure 9(a). Effect of a1Figure 9(b). Effect of a2

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Figure 9. The effect of the market potential parameters on optimal decisions and profits

Figure 10. Effect of the marginal manufacturing cost (c) on optimal decisions and profits

( ,

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Figures 9(a) and 9(b) show the effect of the retail and online channel market potential parameters ) while Figure 10 demonstrates the effect of the marginal manufacturing cost (c) on optimal value

of the decision variables and expected channel profit. Both (

with the increase of

have large effect on expected

has higher impact on po(pr)than pr (po). Optimal value of L decreases (increases)

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channel profit.

and

(

Selling price in both channels increases but channel profit decreases with

AC

increasing . Optimal stocking decisions (zr and zo) to manage demand uncertainty decrease if marginal manufacturing cost increases. The effect of c on optimal delivery lead time is negligible. 7. Managerial insights 7.1 Centralized decision making can serve as a benchmark. Centralized channel optimum always outperforms decentralized channel optimum as there is no ‘double marginalization’ having a single decision maker. It needs a central control or complete cooperation between producer and retailer. What is the effect of online channel lead time, lead time sensitivity and customers’ channel preference on

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the optimal channel structure and price setting? In centralized system, as the online channel lead time of the product increases the optimal online selling price will decrease. Hence the lead time has an adverse effect on online selling price as well as on the profit of the online channel. For increasing online lead time sensitivity, more customers leave the online channel and a part of them turn to the retail channel depending on the sensitivity parameters of the retail channel. In this

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situation, the optimal delivery time will increase and as a result, demand in retail channel will increase. This happens since the centralized channel can increase profit by controlling both the retail and online channels. Increment in the online lead time sensitivity parameter indicates higher expectation of the consumers for better service that lessen the optimal delivery time, which in turn will help to increase the online channel demand. We could specify two threshold values that define the ranges of the lead time

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sensitivity parameters where the retail or the online channel has a higher demand.

We specified the ranges for the customers’ channel preference index toward retail channel as a function of the elasticity parameters to determine the optimal channel and price structure. We found the range where it is not economic to operate dual-channel mode, the range where the equilibrium retail price

7.2 Decentralized decisions

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is less than the equilibrium online selling price and where the online price is less.

Under a decentralized setting (by Stackelberg as well as Nash game) we checked how

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decentralization modifies the system behavior compared to the centralized system. The manufacturer as the Stackelberg game leader, first announces the wholesale price, online price, delivery lead time, and stocking decision of the online channel. The retailer acts as the follower and

PT

based on the manufacturer’s decision, determines the retail price and stocking decision of the retail channel. The retailer will increase its selling price if online lead time increases since for longer waiting

CE

time of online purchase the customer often switch over to retail channel. In decentralized setting, interaction between the manufacturer and the retailer are determined through the wholesale price, so it is the most important decision variable in a decentralized system.

AC

Increasing consumers’ sensitivity to delivery lead time will decrease the optimal wholesale price and the rate of decrease mainly depends on the retail channel delivery time sensitivity parameter. Managerial intuition behind this result is: for higher sensitivity to delivery lead time more consumers will leave the online channel and attract those consumers towards retail channel, so the manufacturer will reduce the wholesale price. In the Nash game setting the wholesaler cooperates with the retailer and provides initiatives to motivate the retailer to order the integrated optimal quantity. We determined the optimal response of each

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member given the other member’s decision. The online selling price will decrease but delivery lead time will increase if the retail channel lead time sensitivity parameter increases. This happens because the lost sales of the online channel turning to the retail channel and in order to attract consumers to the online channel the manufacturer needs to decrease its online selling price. Thus, the manufacturer will take more time to deliver the product and will save when the lost sale returns to retail store due to longer delivery

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time.

7.3 Coordinated decision making

We examined how a decentralized channel can be coordinated to eliminate double marginalization. In decentralized setting, the retailer orders its optimal quantity not accepting the integrated optimal quantity unless it receives additional benefits. To induce retailer to adapt the

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centralized optimum, various well-established coordination contract mechanisms can be applied. In our channel coordination, the manufacturer provides an all-unit quantity discount to motivate the retailer to order the optimal order quantity and charges a franchise fee to eliminate channel conflict. The proposed hybrid contract can be implemented successfully only if all the channel members get win-win outcome. That is, the minimum expected profit of the retailer and the manufacturer in the proposed mechanism must be at least as much as their respective decentralized minimum expected profit.

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We provide the discount rate and the lower and upper bounds on the franchise fee that ensures win-win situation. Surplus profit will be shared between the manufacturer and the retailer through the settlement of

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the franchise fee within the specified range. To divide surplus profit between the channel members, the generalized asymmetric Nash bargaining solution (Nash, 1950) technique is used to find the appropriate franchise fee. We expressed the franchise fee as a function of the bargaining power. The surplus profit for

PT

the retailer and the manufacturer also depend on their respective bargaining powers. We determined a closed form of the Nash game equilibrium for the deterministic demand case

CE

and analyzed the effect of parameter changes on the equilibrium decision. The online delivery lead time and transferred demand from online channel to retail channel plays an important role in the equilibrium

AC

retail price. If the transfer from the lost demand in online channel is higher than a threshold (a function of price and demand sensitivity parameters), then the retailer’s optimal price will increase with the increment of the delivery lead time. On the other hand, if the transfer rate from the lost demand of online channel is less than the threshold, the retailer needs to decrease its selling price with the increment of the delivery lead time to attract consumers towards its channel. A numerical example illustrates that the decentralized channel system can receive considerably lower profit than the centralized one. The optimal online delivery lead time in decentralized channel is less than in centralized channel. With decreasing retail lead time sensitivity the manufacturer decreases

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the delivery lead time due to the low return rate of outgoing unsatisfied customers.

7.4 Effect of random demand The expected channel profit decreases with the increase of the variances because stocking more product to manage uncertainty is raising the total cost. This study reveals that the optimal selling prices of the product in both channels increase due to randomness in demand because both the retail and online

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channel require to carry some additional stock. The optimal delivery lead time for the online channel is shorter in uncertain demand environment than in the deterministic demand environment.

Proper management against uncertainty provides more channel profit than the decision made under the assumption of deterministic demand, disregarding uncertainty. If variances of random variables increase then the inventory to manage uncertainty increases and results a decrease in the expected channel

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profit. For both channels the optimal size of its production/order to satisfy the stochastic demand increases if the selling price of the product increases and thereby the optimal service level increases. The demand randomness also influences the optimal price and delivery time decisions through the effect of the expected shortage that results in the decrease of expected demand. Thus, considering the expected demand only, neglecting uncertainty will lead to inflated prices and longer delivery lead time decision.

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8. Conclusions and future research

Combining retail and online sales channels is getting more popular in business. Our research not

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only integrated different research streams in this area but also resulted in new managerial insight mostly by investigating the effect of uncertain demand on the optimal pricing and ordering policy for a dualchannel business with price and lead-time dependent demand. Uncertainty is common in both retail and

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online channel and so additional inventory management is required to control shortage or overstock and

CE

that affects the optimal price and lead time. Our dual-channel supply chain models were developed for known distribution as well as for unknown distribution where demand is stochastic additive as well as price and delivery time sensitive. We

AC

also examined the decentralized setting where the channel members are independent and make their decision through a Stackelberg or Nash game. We specified the optimal response of each member given the other member’s decisions. The intersections of the simultaneous move in the game between the retailer and the manufacturer provide the equilibrium point of the Nash game. At an equilibrium point, each party must respond optimally given the other party's decisions. We provided the simultaneous equations that specify the equilibrium point for both known distribution and distribution-free case.

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There are a number of extensions possible to this study. The single-period setting could be extended to a multi-period inventory model. The interaction between the retailer and the manufacturer can be studied under Stackelberg game perspective considering different leader of the channel. Another extension to this research can be the consideration of the channel structure with competitive manufacturers, duopoly retailers and multiple retailers. Channel coordination can also be considered as a research topic to improve decentralized channel performance. Equilibrium solution of the decentralized

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channel can be studied using the concept of strategic bargaining and sub-game perfect equilibrium.

Acknowledgment: The author is grateful to the editor and the anonymous referees for their extremely useful suggestions to improve the quality of the paper. The first author is also thankful forever

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to Shri Bibhas Cha’pndra Das for giving endless encouragement.

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