1 Introduction

With the development of the e-commerce, many manufacturers who traditionally sell their products through retailers have started to use the Internet as a direct channel to sell products. This type of dual-channel distribution system is widely used in various industries. Many brand name manufacturers, like Sony Electronics, Nike, Apple, Dell, etc., have added direct channel alongside the traditional channel to sell their products (Dumrongsiri et al. 2008; Hua et al. 2010; Ding et al. 2016). However, there are both opportunity and threat when the manufacturer introduces the direct channel (Kurata et al. 2007). On the one hand, the sales volume could be increased as the Internet is accessible for more and more consumers. On the other hand, adding a direct channel may place the manufacturer in competition with the existing retailers (Tsay and Agrawal 2004; Lu and Liu 2015). Thus, traditional optimization models for single-channel supply chain are not sufficient for developing the insights into the performance of the manufacturer and retailer.

Motivated by the above observation, in this paper we study optimization of the dual-channel problem. There are some fundamental research questions remaining to be answered, especially when the demands are uncertain. Addressing such questions will help both the manufacturer and retailer to adopt the most appropriate polices in a dual-channel supply chain.

First, pricing competition between the traditional channel and direct channel is inevitable in a dual-channel supply chain. So the first fundamental research question is what are the optimal pricing policies for both the manufacturer and retailer. Some previous researches on dual-channel problem have focused on the pricing policy with deterministic demand. For instance, Chiang et al. (2003), Cattani et al. (2006) and Ding et al. (2016) show that under some conditions, adding the direct channel may benefit both the manufacturer and retailer, with appropriate pricing policies. However, the optimal pricing policy under the stochastic demand still needs to be addressed.

Second, inventory policy is another fundamental research question that needs to be addressed for the dual-channel problem with stochastic demand. Adding a direct channel alongside the existing traditional channel may cause the havoc on the demand structure (Chiang and Monahan 2005). It requires the retailer to redesign the ordering policy. In addition, the manufacturer also needs to determine the amount of inventory allocated to the direct channel, to meet its stochastic demand. From the perspective of the inventory policy, in this paper, we seek to find the balance for the stock levels in two channels.

Although the pricing and inventory policies are two fundamental research questions for the dual-channel problem, it cannot be treated as separate optimization problems when the demands are uncertain. Therefore, in this paper we study the joint decision of the pricing and inventory policies for the dual-channel problem with stochastic demand. Specifically, we seek to answer the following questions: First, what are the optimal retail prices in the traditional and direct channels? Second, what is the optimal wholesale price should be charged to the retailer? Third, what are the optimal order quantity and total inventory capacity of the retailer and manufacturer, respectively? Fourth, what are the effects of the demand uncertainties and price sensitivities on the optimal solutions and performance?

To address these questions, we consider a dual-channel problem with stochastic demand, where the manufacturer and retailer play a Stackelberg game. The manufacturer, acting as the leader, decides wholesale price to the retailer, selling price in the direct channel, and the total inventory capacity. The retailer, acting as the follower, decides the order quantity and retail price in the traditional channel. We first model the manufacturer’s problem, which is followed by the retailer’s problem. To the best of our knowledge, this is the first study considering the dual-channel problem with the stochastic demand in a Stackelberg game framework.

Our analysis is composed of several steps. First, given the manufacturer’s decisions, the retailer’s problem can be treated as a modified newsvendor problem. Then we derive the retailer’s optimal responses of the price and inventory decisions, which have the same nice structure as those in Petruzzi and Dada (1999). For the manufacturer’s problem, we first obtain the optimal stocking factor for the direct channel, which is similar to the newsvendor problem with the fixed selling price. We find that there is a one-to-one relationship between the optimal stocking factor of the retailer and the wholesale price. Then we represent the optimal solution of retailer’s problem in terms of the retail price in the traditional channel and the wholesale price. It is worth noting that the optimal responses of the retail price in the traditional channel, wholesale price and total inventory capacity can be presented in closed-forms. After that, we reformulate the manufacturer’s problem and change to maximize the manufacturer’s expected profit over the selling price in the direct channel and stocking factor for the traditional channel. For the reformulated problem, we derive the optimal selling price in the direct channel and use an algorithm to find the optimal stocking factor for the traditional channel.

Our work contributes to the dual-channel literature as follows: First, we introduce a tractable method to solve the optimization problem, associated with the joint decision of pricing and inventory policies for a dual-channel supply chain with stochastic demand. It has not been solved in the literature under a general setting like ours. Second, regarding the optimal solutions, we find that the optimal inventory decisions for two channels have the similar structure with that for the newsvendor problem. In addition, although we consider a general setting for dual-channel problem with five decision variables, the optimal solutions of three decision variables, i.e., the retail price in the traditional channel, wholesale price and total inventory capacity, can be presented in closed-forms. Finally, by numerically studying the effects of the demand uncertainties and price sensitivities on the decisions of both the manufacturer and retailer, we discuss some managerial insights in the paper.

The remainder of this paper is organized as follows. In Sect. 2, we provide a brief review of the related literature. In Sect. 3, we introduce the modelling. In Sect. 4, we derive the optimal solutions for both the manufacturer and retailer. In Sect. 5, we conduct some numerical studies to investigate the effects of the demand uncertainties and price sensitivities. We conclude the paper and suggest some future research topics in Sect. 6. All proofs are relegated to the Appendix.

2 Literature review

Our work is related to two streams of research in the literature of dual-channel supply chain. The first stream is the literature on pricing strategies, and the second stream is the literature on production and inventory strategies.

We first review the literature on the dual-channel problem with the consideration of pricing strategies. Chiang et al. (2003) study a price-competition game in a dual-channel supply chain. Their results show that adding a direct channel may benefit both the retailer and manufacturer. Park and Keh (2003) compare the equilibriums under the traditional channel structure and centralized supply chain with the equilibrium under the dual-channel structure with respect to the marketing decision variables, such as prices. By focusing on the channel conflict and coordination of the dual-channel supply chain, Tsay and Agrawal (2004) also show that adding a direct channel alongside a traditional retail channel may benefit to the retailer. Viswanathan (2005) studies the impact of differences in channel flexibility, network externalities, and switching costs on the competition across online, traditional and hybrid channels, by using a variant of circular city model. Yao and Liu (2005) consider price competition between two channels in a dual-channel supply chain under two power structures. In the first power structure, the manufacturer and retailer simultaneously determine the retail prices in the direct and traditional channels, respectively; and in the second power structure, the manufacturer acting as a leader decides the price in the direct channel, and the retailer acting as a follower decides the retail price in the traditional channel. Cattani et al. (2006) develop a model to study the effect of adding a direct channel on profits and market shares of the manufacturer and retailer, by using consumer utility theory. Three pricing strategies are compared in their paper, i.e., wholesale prices are kept as before, retail prices are kept as before, and manufacturer chooses wholesale price to maximize profit. Kurata et al. (2007) investigate channel pricing in multiple distribution channels with the consideration of competition between a national brand (NB) and a store brand (SB). In their setting, the NB operates both the direct channel and traditional channel, and the SB only operates the traditional channel. Huang and Swaminathan (2009) study the optimal pricing strategies for a dual-channel problem with deterministic demand. Four pricing strategies that differ in the degree of autonomy for the direct channel are investigated in their paper. Recently, Ding et al. (2016) examine the pricing strategies for a dual-channel problem with a hierarchical pricing decision process. All the above papers consider the deterministic demand and focus on the pricing strategies. However, we consider that the demands in two channels are stochastic, thus we not only study the pricing strategies but also the stocking policies for both the manufacturer and retailer.

We next review the literature on the dual-channel problem with the consideration of production and inventory strategies. Boyaci (2005) studies the optimal stocking policy for both the manufacturer and retailer with stochastic demand. He considers that all prices are exogenously given in the paper. Chiang and Monahan (2005) consider an inventory management problem in a two-echelon dual-channel supply chain. Their numerical results show that dual-channel strategy outperforms the single traditional channel strategy and single direct channel strategy in most cases. Alptekinoglu and Tang (2005) instigate a multi-channel distribution system with multiple depots and multiple sales locations with the consideration of stochastic demand. They propose a decomposition scheme that can be used to develop a near-optimal distribution policy with minimum total expected distribution cost. Geng and Mallik (2007) study inventory competition between a manufacturer and a retailer in the presence of the capacity constraint, for a dual-channel supply chain. Yao et al. (2009) investigate strategic inventory deployment for retail and e-tail stores in a dual channel supply chain, by comparing three different inventory strategies, i.e., centralized inventory strategy, a Stackelberg inventory strategy, and a strategy where the operations of e-tail store is outsourced to a third party logistics provider. Chiang (2010) examines the impact of stock-out based substitution on the product availability and the channel efficiency of a dual-channel supply chain, where the supplier and retailer simultaneously choose their own base-stock levels to satisfy the stochastic demand. Takahashi et al. (2011) study the inventory control policy for a dual-channel supply chain with setup of production and delivery. Li et al. (2017) focus on the ordering policy for a dual-channel supply chain with the consideration of a risk-averse retailer. In all papers considering the production and inventory strategies mentioned above, the market prices are fixed. On the contrary, we consider that the prices in both channels are decision variables.

Recently, there has been growing literature on the joint decision of the pricing and inventory policies. Liu et al. (2010) investigate the joint decision of pricing and production policies for a dual-channel supply chain under information asymmetry. Niu et al. (2012) examine the joint pricing and inventory policies for a dual-channel supply chain, by incorporating intra-product line price interaction in the EOQ model. Huang et al. (2012) analyze the pricing and production decisions for a dual-channel supply chain with the consideration of demand disruptions. Hsieh et al. (2014) consider a multiple-channel supply chain involving multiple manufacturers and a common retailer. They obtain supply chain members’ equilibrium decisions of prices and quantities, and explore the impact of parameters and channel structure on supply chain members’ decisions and performance. Dumrongsiri et al. (2008) and Ryan et al. (2013) are two papers closely related to our study. Dumrongsiri et al. (2008) study the pricing and stocking policies for a dual-channel supply chain, where the manufacturer decides the selling price in the direct channel and the retailer decides the selling price in the traditional channel and order quantity. Ryan et al. (2013) investigate the coordination of a dual-channel supply chain with stochastic demand. Similar to our model, they use linear demand substitution functions to model the stochastic demand. Our paper differs from these two papers in the following ways: First, Dumrongsiri et al. (2008) consider that the manufacturer only needs to decide the selling price in the direct channel, while our paper considers that the manufacturer not only needs to make the pricing decision but also needs to decide the total inventory capacity. Second, both Dumrongsiri et al. (2008) and Ryan et al. (2013) consider that the manufacturer and retailer play a simultaneous move game, for a given wholesale price. However, we consider a Stackelberg game, where the manufacturer acting as the leader decides the total inventory capacity, wholesale price and selling price in the direct channel, and the retailer acting as the follower decides the order quantity and retail price in the traditional channel.

Fig. 1
figure 1

Dual channels with stochastic demands

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3 Modelling

Consider a dual-channel problem where a manufacturer sells a single type of products to customers through the traditional and direct channels simultaneously. We define channels 1 and 2 as the traditional and direct channels, respectively. For each item sold through the traditional channel, customers pay a retail price \(p_1\) to the retailer, who in turn pays a wholesale price w to the manufacturer. For each item sold through the direct channel, customers pay a selling price \(p_2\) to the manufacturer. We consider that the demands in both channels are stochastic. Let \(D_i\) denote the demand of channel i, \(i=1,2\). Let \(q_1\) denote the order quantity of the retailer, and N denote the total inventory capacity of the products in two channels. The goal of the manufacturer is to maximize his profit by optimally setting the prices w, \(p_2\) and total inventory capacity N, and the goal of the retailer is to maximize her profit by optimally setting the retail price in the traditional channel \(p_1\) and order quantity \(q_1\). Figure 1 illustrates the dual channels with stochastic demands.

For the demands in the two channels, we consider the following general linear demand functions:

$$\begin{aligned} D_1 (p_1, p_2, \epsilon _1)= & {} y_1(p_1, p_2)+\epsilon _1 = a_1-b_{11}p_1+b_{12}p_2+\epsilon _1, \end{aligned}$$
(1)
$$\begin{aligned} D_2 (p_1, p_2, \epsilon _2)= & {} y_2(p_1, p_2)+\epsilon _2 = a_2-b_{22}p_2+b_{21}p_1+\epsilon _2. \end{aligned}$$
(2)

Here, \(y_i(p_1, p_2)\), \(i=1, 2\), is the deterministic part of the demand, which indicates the dependency of the demand and prices. In this part, \(a_i>0\) represents the market potential of channel i, \(b_{ii}\ge 0, i=1,2\), is the price-sensitivity coefficient, and \(b_{ij}\ge 0, i,j=1,2\) and \(i\ne j\), is the cross-price-sensitivity coefficients. For the tractability of the model, we assume that the price sensitivity coefficients satisfy the condition: \(b_{ii}\ge b_{jk}, j\ne k\). This assumption is called the dominance of price-sensitivity coefficients, which is wildly used and discussed in the literature, e.g., Maglaras and Meissner (2006), Horn and Johnson (2012) and Ding et al. (2016). It indicates that the demand of channel i is more sensitive to \(p_i\) than \(p_j\), meanwhile, \(p_i\) has a larger impact on the demand of channel i than the demand of channel j, for \(j\ne i\). \(\epsilon _i\) is the random part of the demand. It is defined on the rang \(\left[ A_i,B_i\right] \), with probability density function \(f_i(\cdot )\), cumulative distribution function \(F_i(\cdot )\) , mean \(\mu _i\) and standard deviation \(\sigma _i\). In order to assure that positive demand is possible for some range of \(p_i\), we require that \(A_i>-a_i\). Define \(r_i(\cdot )\equiv {f_i(\cdot )}/ \left[ {1-F_i(\cdot )}\right] \) as the hazard rate. We assume that \(F_i(\cdot )\) is a distribution function satisfying the condition \(2r_i(\cdot )^2+r_i'(\cdot )>0\). This assumption is not unreasonable, and the condition will be satisfied by various distributions, e.g., all nondecreasing hazard rate distributions (Petruzzi and Dada 1999). Besides, we assume that \(\epsilon _i\) and \(\epsilon _j\) (\(i\ne j\)) are independent. In the literature, the linear demand functions of dual-channel problem are commonly used, e.g., Ryan et al. (2013), Ding et al. (2016) and Li et al. (2016). For modeling the demand randomness, the additive forms can also be found in Mills (1959), Petruzzi and Dada (1999), Dumrongsiri et al. (2008) and Ryan et al. (2013).

It is worth noting that in the literature some papers consider that two channels are sharing a total market potential, i.e., \(a_1+a_2\) is equal to a constant (see, e.g., Yue and Liu 2006; Hua et al. 2010; Huang et al. 2012; Li et al. 2016). Our demand functions can also be adjusted to consider that setting, by defining a and \(\theta \) as the total market potential and market share of the traditional channel, respectively. All the analytical results in our paper keep unchanged when we let \(a_1=\theta a\) and \(a_2=(1-\theta ) a\). Note that the cross-price-sensitivity coefficients \(b_{12}\) and \(b_{21}\) reflect the degree to which the products sold through the two channels are substitute (Hua et al. 2010). Similarly, all the analytical results keep unchanged if we let \(b_{12}=b_{21}\), which indicates that the substitutability of the products in two channels is the same. However, to keep the generality, in our model we do not assume that \(a_1+a_2\) is equal to a constant and \(b_{12}=b_{21}\). Alternatively, in the numerical study, we first set \(b_{12}=b_{21}\) and investigate the effects of the demand variabilities and price-sensitivities, and then examine the effects of the cross-price-sensitivities by changing the value of \(b_{12}\) and \(b_{21}\).

Consider a Stackelberg game, where the manufacturer acts as the leader and the retailer acts as the follower. We now outline the sequence of events for our problem as follows: (1) Prior to the start of the selling season, the manufacturer sets the wholesale price w charged to the retailer, selling price \(p_2\) for customers in the direct channel, and total inventory capacity for two channels N; (2) The retailer orders \(q_1\le N\) from the manufacturer and sets the retail price in the traditional channel \(p_1\); (3) The manufacturer allocates the other \(N-q_1\) inventory capacity to the direct channel; and (4) Customers simultaneously come to the traditional and direct channels.

Then, the optimization problem can be divided into two sub-problems: the manufacturer’s problem and the retailer’s problem. We first present the modelling for the manufacturer’s problem, which is followed by the retailer’s problem.

3.1 The manufacturer’s problem

The manufacturer’s profit consists of two parts: the profits obtained from selling the products through the retailer, and the profits obtained through selling the products directly to the customers. To facilitate the exposition, we assume that there are no salvage value for the unsold products and no extra penalty for the shortage. Then for the manufacturer, the leftover cost for per unit product is equal to the unit production cost c, and the shortage cost for per unit product is equal to the unit profit margin, i.e., \(p_2-c\). Since the demands are stochastic, the profits depend on whether the demand in the direct channel \(D_2\) exceeds the number of the product allocated to this channel \(N-q_1\). Specifically, we have

$$\begin{aligned}&{\varPi }_m(N,p_2,w)\\&\quad = \left\{ \begin{array}{l l} p_2D_2(p_1,p_2,\epsilon _2)-c(N-q_1)+(w-c)q_1, &{}\quad D_2(p_1,p_2,\epsilon _2)\le N-q_1,\\ (p_2-c)(N-q_1)+(w-c)q_1, &{}\quad D_2(p_1,p_2,\epsilon _2)> N-q_1. \end{array}\right. \end{aligned}$$

The manufacturer maximizes his profit by solving the following optimization problem:

$$\begin{aligned} \max _{(N,p_2,w)\in R_m}{{\mathbb {E}}[{\varPi }_m(N,p_2,w)]}, \end{aligned}$$

where \(R_m=\{(N,p_2,w):N\ge 0,p_2\ge c,w\ge c\}\).

3.2 The retailer’s problem

Similarly, we assume that there is no salvage value for the unsold products and no extra penalty for the shortage. Then for the retailer, the leftover cost for per unit product is equal to the wholesale price w, and the shortage cost for per unit product is equal to the unit profit margin, i.e., \(p_1-w\). The profit of the retailer depends on whether the demand \(D_1\) is larger than the order quantity \(q_1\). Specifically, we have

$$\begin{aligned} {\varPi }_r(q_1,p_1) = \left\{ \begin{array}{l l} p_1D_1(p_1, p_2, \epsilon _1)-wq_1, &{}\quad D_1(p_1, p_2, \epsilon _1)\le q_1,\\ p_1q_1-wq_1, &{}\quad D_1(p_1, p_2, \epsilon _1) > q_1. \end{array}\right. \end{aligned}$$
(3)

The retailer maximizes her profit by solving the following optimization problem:

$$\begin{aligned} \max _{(q_1,p_1)\in R_r}{{\mathbb {E}}[{\varPi }_r(q_1,p_1)]}, \end{aligned}$$

where \(R_r=\{(q_1,p_1):0\le q_1 \le N, p_1\ge 0\}\).

Table 1 summarizes some major notations used in this paper, where \(i=1\) and 2 represents the traditional channel and direct channel, respectively. Notice that \(a_i\), \(b_{ij}\) and c are all non-negative.

Table 1 Notation
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4 Analysis and solutions

In this section, we show the analysis and solutions. We use the backward induction approach to solve the optimization problems. That is, we first solve the retailer’s problem and then the manufacturer’s problem. We use a method, such as that in Petruzzi and Dada (1999), to determine the joint pricing and inventory decisions, for the retailer’s problem. For the manufacturer’s problem, we first obtain the optimal total inventory capacity of the manufacturer, by substituting the optimal responses of retailer’s decisions into the manufacturer’s objective function. However, due to the stochastic demands, it is intractable to obtain the optimal selling price in the direct channel and the wholesale price, by substituting the optimal total inventory capacity back into the manufacturer’s objective function. Then we define an equivalent reformulation as in the Sect. 4.2.2 to solve the manufacturer’s problem.

4.1 Retailer’s decision

Note that the retailer’s order quantity cannot exceed manufacturer’s total inventory capacity, i.e., \(q_1\le N\). However, in the following approach, we first solve the problem without considering the total inventory capacity constraint. It means that the total inventory capacity N is large enough, i.e., \(q_1(p_2, w)< N\), where \(q_1(p_2, w)\) is the optimal response of order quantity for given \(p_2\) and w. After that, we consider the case that if the optimal response of the order quantity is larger than the total inventory capacity, i.e., \(q_1(p_2, w)\ge N\).

Following Ernst (1970), Thowsen (1975) and Petruzzi and Dada (1999), we define \(z_1=q_1-y_1(p_1, p_2)\) as the stocking factor of the retailer, which represents the amount of product for satisfying the stochastic portion of the demand in the traditional channel. Then, the profit function in Eq. (3) can be presented as follows:

$$\begin{aligned} {\varPi }_r(z_1,p_1) = \left\{ \begin{array}{l l} p_1[y_1(p_1, p_2)+\epsilon _1]-w[y_1(p_1, p_2)+z_1], &{}\quad \epsilon _1 \le z_1,\\ p_1[y_1(p_1, p_2)+z_1]-w[y_1(p_1, p_2)+z_1], &{}\quad \epsilon _1 > z_1. \end{array}\right. \end{aligned}$$

It shows that after this transformation, the profit of the retailer depends on the value of \(z_1\) and the realization of \(\epsilon _1\). Meanwhile, as explained in Petruzzi and Dada (1999), after this transformation the ordering decisions can be interpreted in an alternative way: If \(z_1\) is larger than the realization of \(\epsilon _1\), then the leftovers occur; otherwise, shortages occur. Then the joint decision of the pricing and ordering policy for the retailer is to order \(q_1^* = y_1(p^*_1, p_2) + z_1\) units and sell them at unit retail price \(p_1^*\). The expected profit of the retailer is given as follows:

$$\begin{aligned} {\mathbb {E}}[{\varPi }_r(z_1,p_1)]= & {} \int ^{z_1}_{A_1}(p_1[y_1(p_1, p_2)+u])f_1(u)du \nonumber \\&+\,\int ^{B_1}_{z_1}(p_1[y_1(p_1, p_2)+z_1])f_1(u)du-w[y_1(p_1, p_2)+z_1]. \end{aligned}$$

Defining \({\varLambda }_1(z_1)=\int ^{z_1}_{A_1}(z_1-u)f_1(u)du\) and \({\varTheta }_1(z_1)=\int ^{B_1}_{z_1}(u-z_1)f_1(u)du\), then the expected profit can be written as:

$$\begin{aligned} {\mathbb {E}}[{\varPi }_r(z_1,p_1)] = {\varPsi }_1(p_1)-L_1(z_1,p_1), \end{aligned}$$
(4)

where \({\varPsi }_1(p_1)=(p_1-w)[y_1(p_1, p_2)+\mu _1]\) and \(L_1(z_1,p_1)=w{\varLambda }_1(z_1)+(p_1-w){\varTheta }_1(z_1)\).

Then we change to maximize the expected profit of the retailer over \(z_1\) and \(p_1\):

$$\begin{aligned} \max _{z_1,p_1}{{\mathbb {E}}\left[ {\varPi }_r(z_1,p_1)\right] }. \end{aligned}$$

Taking the first and second partial derivatives of \({\mathbb {E}}[{\varPi }_r(z_1,p_1)]\) with respect to \(z_1\) and \(p_1\), we have

$$\begin{aligned} \frac{\partial {\mathbb {E}}[{\varPi }_r(z_1,p_1)]}{\partial z_1}= & {} -w+p_1[1-F_1(z_1)], \end{aligned}$$
(5)
$$\begin{aligned} \frac{\partial ^2 {\mathbb {E}}[{\varPi }_r(z_1,p_1)]}{\partial z_1^2}= & {} -p_1f_1(z_1),\nonumber \\ \frac{\partial {\mathbb {E}}[{\varPi }_r(z_1,p_1)]}{\partial p_1}= & {} 2b_{11}(p_1^0-p_1)-{\varTheta }_1(z_1), \end{aligned}$$
(6)
$$\begin{aligned} \frac{\partial ^2 {\mathbb {E}}[{\varPi }_r(z_1,p_1)]}{\partial p^2_1}= & {} -2b_{11}, \end{aligned}$$
(7)

where \(p_1^0=\frac{a_1+b_{11}w+\mu _1+b_{12}p_2}{2b_{11}}\) is the optimal risk-less price that maximizes \({\varPsi }_1(p_1)\).

Notice from (7) that \({\mathbb {E}}[{\varPi }_r(z_1,p_1)]\) is concave in \(p_1\) for a given \(z_1\), then we can have the following result directly from (6) and (7):

Lemma 1

For a given \(z_1\), the optimal retail price in the traditional channel is determined uniquely as a function of \(z_1\):

$$\begin{aligned} p_1(z_1)=p_1^0-\frac{{\varTheta }_1(z_1)}{2b_{11}}. \end{aligned}$$

Lemma 1 indicates that for a given \(z_1\), the optimal response of the retail price in the traditional channel consists of two parts. The first part is the optimal risk-less price \(p_1^0\), and the second part is the term related to the demand uncertainty. Note that \({\varTheta }_1(z_1)\) is non-negative. It means that, compared with the deterministic demand, the retailer should set a lower retail price when the demand is uncertain.

Lemma 2

Let \(q_1(z_1)=y_1(p_1(z_1),p_2)+z_1\), then \(d{{q}_{1}}\left( {{z}_{1}} \right) / d {z}_{1}>0\).

Lemma 2 indicates that after obtaining the optimal response of the retailer price \(p_1\), there is a positive correlation between the order quantity \(q_1\) and \(z_1\).

For solving optimal \(z_1\), we substitute \(p^*_1=p_1(z_1)\) back into the expected profit function. The retailer’s optimization problem becomes as follows with the single decision variable \(z_1\):

$$\begin{aligned} \max _{z_1}{{\mathbb {E}}\left[ {\varPi }_r(z_1,p_1(z_1))\right] }. \end{aligned}$$

Similar to Petruzzi and Dada (1999), we show in Proposition 1 that \({\mathbb {E}}[{\varPi }_r(z_1,p_1(z_1))]\) might have multiple points that satisfy the first-order optimality condition, depending on the demand distribution.

Proposition 1

Given the manufacturer’s selling price in the direct channel \(p_2\) and wholesale price w, the optimal ordering and pricing policy for the retailer is to stock \(q_1^*=y_1(p_1(z_1^*), p_2)+z_1^*\) units and sell at the unit price \(p_1^*\), where \(p_1^*\) is given in Lemma 1 and \(z_1^*\) is determined as follows: If \({{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}>0\), then \(z_1^*\) is the unique \(z_1\) in the region \([A_1, B_1]\) that satisfies \({d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]}/{dz_1}=0\); otherwise, \(z_1^*\) is the largest \(z_1\) in the region \([A_1, B_1]\) that satisfies \({d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]}/{dz_1}=0\).

The condition in Proposition 1 ensures that \({\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]\) is unimodal in \(z_1\) and thus has a unique solution. It can be satisfied if the base demand is large, e.g., \(a_1>{{b}_{11}}w-{{b}_{12}}{{p}_{2}}-{{A}_{1}}\), or if the wholesale price is small, e.g., \(w<\left( {{a}_{1}}+{{b}_{12}}{{p}_{2}}+{{A}_{1}} \right) /b_{11}\).

In the above analysis, we do not consider the condition \(q_1\le N\). Next, we consider the case that if the optimal solutions in Proposition 1 lead to \(q_1(p_2,w)> N\). We define \({\bar{z}}_1\) satisfying \(N=y_1(p_1({\bar{z}}_1), p_2)+{\bar{z}}_1\). Note from Lemma 2 that \(y_1(p_1(z_1), p_2)+z_1\) increases in \(z_1\). So there is only one \(z_1\) satisfying this equation. We define \(\hat{z_1}\) satisfying \({d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]}/{dz_1}=0\) if \({{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}>0\), and define \(\hat{z_1}\) as the largest \(z_1\) in the region \([A_1, B_1]\) that satisfies \({d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]}/{dz_1}=0\) if \({{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}\le 0\). Then we have,

Corollary 1

Given that when the optimal solutions in Proposition 1 lead to \(q_1(p_2,w)> N\), then optimal \(z_1\) is given as follows: if \({{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}>0\), then \(z_1^*={\bar{z}}_1\); otherwise, \(z_1^*={\bar{z}}_1\) if \({\mathbb {E}}[{\varPi }_r({\bar{z}}_1, p_1({\bar{z}}_1))]>{\mathbb {E}}[{\varPi }_r(A_1, p_1(A_1))]\), and \(z_1^*=A_1\) if \({\mathbb {E}}[{\varPi }_r({\bar{z}}_1, p_1(\bar{z}_1))]\le {\mathbb {E}}[{\varPi }_r(A_1, p_1(A_1))]\).

4.2 Manufacturer’s decision

In this section, we solve the manufacturer’s problem. Note that in Sect. 4.1 we have obtained optimal responses of the order quantity and retail price in the traditional channel, for given the total inventory capacity, selling price in the direct channel and wholesale price. Then we substitute the optimal responses into manufacturer’s objective function and solve the manufacturer’s problem.

The manufacturer’s profit is now given as follows:

$$\begin{aligned} {\varPi }_m(N,p_2,w) = \left\{ \begin{array}{l l} p_2D_2(p_1(p_2, w),p_2,\epsilon _2)-c(N-q_1(p_2, w))+(w-c)q_1(p_2, w), \\ \qquad \qquad \qquad \qquad \qquad \mathrm {if} D_2(p_1(p_2, w),p_2,\epsilon _2)\le N-q_1(p_2, w), \\ (p_2-c)(N-q_1(p_2, w))+(w-c)q_1(p_2, w), \\ \qquad \qquad \qquad \qquad \qquad \mathrm {if} D_2(p_1(p_2, w),p_2,\epsilon _2)> N-q_1(p_2, w). \end{array}\right. \end{aligned}$$

Here, \(q_1(p_2, w)\) and \(p_1(p_2, w)\) are the optimal responses of the order quantity and retail price in the traditional channel, which are given in Proposition 1 and Corollary 1.

4.2.1 Optimal total inventory capacity \(N^*\)

Note from Proposition 1 and Corollary 1 that given the total inventory capacity N, the optimal response of the order quantity by the retailer \(q_1(p_2, w)\) may exceed the total inventory capacity. Next we first show that the manufacturer will only set the total inventory capacity larger than the optimal response of the order quantity.

Lemma 3

Given the selling price in the direct channel \(p_2\) and wholesale price w, the manufacturer will only choose the total inventory capacity N in the region \(R_N=\{N: N\ge q_1(p_2,w)\}\).

Lemma 3 indicates that the manufacturer should always prepare enough total inventory capacity to satisfy the retailer’s ordering demand. In other words, the total inventory capacity should not be a constraint to the retailer’s optimal response of the order quantity. This is because given the wholesale price, the manufacturer can obtain more profit from the traditional channel when more inventory ordered by the retailer. Besides, Lemma 3 implies that for the optimal response of the order quantity, we only need to consider the case where the optimal value is obtained by the first-order condition of the objective function.

Note that the total inventory capacity of the manufacturer equals to the total inventories allocated to the traditional and direct channels, i.e., \(N=q_1+q_2\). In the retailer’s problem, we have obtained the optimal response of order quantity \(q_1(p_2, w)\). Then for the optimal N, we only need to obtain the optimal inventory allocated to the direct channel \(q_2\).

Similar to the retailer’s problem, we define \({{z}_{2}}={{q}_{2}}-{{y}_{2}}\). For notation convenience, we use \(q_1\), \(p_1\) and \(z_1\) to represent their optimal responses, i.e., \(q_1=q_1(p_2, w)\), \(p_1=p_1(p_2, w)\) and \(z_1=z_1(p_2, w)\). Then we have

$$\begin{aligned} {{{\varPi }}_{m}}({{z}_{2}},{{p}_{2}},w)=\left\{ \begin{array}{*{35}{l}} {{p}_{2}}\left( {{y}_{2}}+{{\epsilon }_{2}} \right) -c\left( {{y}_{2}}+{{z}_{2}} \right) +(w-c){{q}_{1}}, &{}\quad {{\epsilon }_{2}}\le {{z}_{2}}, \\ {{p}_{2}}\left( {{y}_{2}}+{{z}_{2}} \right) -c\left( {{y}_{2}}+{{z}_{2}} \right) +(w-c){{q}_{1}}, &{}\quad {{\epsilon }_{2}}>{{z}_{2}}. \\ \end{array} \right. \end{aligned}$$

The manufacturer’s expected profit is given as follows:

$$\begin{aligned} {\mathbb {E}}\left[ {{{\varPi }}_{m}}({{z}_{2}},{{p}_{2}},w) \right]= & {} ({{p}_{2}}-c){{y}_{2}}-c\int _{{{A}_{2}}}^{{{z}_{2}}}{({{z}_{2}}-u)}{{f}_{2}}(u)du\\&-\,({{p}_{2}}-c)\int _{{{z}_{2}}}^{{{B}_{2}}}{(u-{{z}_{2}})}{{f}_{2}}(u)du+(w-c)({{y}_{1}}+{{z}_{1}}). \end{aligned}$$

Defining \({\varLambda }_2(z_2)=\int ^{z_2}_{A_2}(z_2-u)f_2(u)du\) and \({\varTheta }_2(z_2)=\int ^{B_2}_{z_2}(u-z_2)f_2(u)du\), we can write:

$$\begin{aligned} {\mathbb {E}}[{\varPi }_m(z_2,p_2,w)]= & {} {\varPsi }_2(p_2)-L_2(z_2,p_2)+\left( w-c\right) \left( y_1+z_1\right) , \end{aligned}$$
(8)

where \({\varPsi }_2(p_2)=(p_2-c)[y_2(p_2)+\mu _2]\) and \(L_2(z_2,p_2)=c{\varLambda }_2(z_2)+(p_2-c){\varTheta }_2(z_2)\).

The manufacturer’s objective is to maximize his expected profit over \(z_2\), \(p_2\) and w:

$$\begin{aligned} \max _{z_2,p_2,w}{{\mathbb {E}}\left[ {\varPi }_m(z_2,p_2,w)\right] }. \end{aligned}$$

We use a sequential decision approach introduced by Whitin (1995) to determine the optimal solutions of \(z_2\) and \(p_2\). That is, we first solve the optimal value of \(z_2\) as a function of \(p_2\) and w, and then substitute the optimal response of \(z_2(p_2, w)\) back into the objective function. Taking the first and second partial derivatives of \({\mathbb {E}}[{\varPi }_m(z_2,p_2,w)]\) with respect to \(z_2\), we have

$$\begin{aligned} \frac{\partial {\mathbb {E}}[{\varPi }_m(z_2,p_2,w)]}{\partial z_2}= & {} -c+p_2[1-F_2(z_2)], \nonumber \\ \frac{\partial ^2{\mathbb {E}}[{\varPi }_m(z_2,p_2,w))]}{\partial z_2^2}= & {} -p_2f_2(z_2). \end{aligned}$$
(9)

Notice from (9) that \({\mathbb {E}}[{\varPi }_m(z_2,p_2,w)]\) is concave in \(z_2\) for given \(p_2\) and w. Then we can directly obtain the following results:

Proposition 2

Given \(p_2\) and w, the manufacturer’s optimal stocking factor for the direct channel is

$$\begin{aligned} z_2(p_2, w) = F_2^{-1}\left( \frac{p_2-c}{p_2}\right) . \end{aligned}$$
(10)

Proposition 2 shows manufacturer’s optimal stocking factor for the direct channel. We observe that the optimal stocking factor is the same as that for the newsvendor problem. Specifically, given the selling price and production cost, the optimal solution only depends on the demand distribution. Combining (10) with \(z_2=q_2-y_2\) and \(N=q_1+q_2\), we obtain that \(N^*=F_2^{-1}\left( (p_2-c)/{p_2}\right) +q_1+y_2\).

For now, we have obtained the optimal total inventory capacity for the manufacturer’s problem. Then substituting (10) into (8), the manufacturer’s problem is simplified to a maximization over the two variables \(p_2\) and w:

$$\begin{aligned} {{{\text {P}}}_{m}}:\underset{{{p}_{2}},w}{\mathop {\max }}\,{\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( {{p}_{2}},w \right) ,{{p}_{2}},w \right) \right] , \end{aligned}$$
(11)

subject to

$$\begin{aligned}&p_1=\frac{a_1+b_{11}w+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)}{2b_{11}}, \end{aligned}$$
(12)
$$\begin{aligned}&\qquad -w+p_1[1-F_1(z_1)]=0,\\&a_1+b_{11}w+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)-\frac{1-F_1(z_1)}{r_1(z_1)}\ge 0,\nonumber \\&\qquad z_2(p_2, w) = F_2^{-1}\left( \frac{p_2-c}{p_2}\right) .\nonumber \end{aligned}$$
(13)

In the above optimization problem, the first and second constraints are the retailer’s optimal responses associated with the stocking and pricing policy. Specifically, (12) follows from Lemma 1, and (13) follows from (5). The third constraint follows from Proposition 1. It guarantees that \(z_1\) is the largest one in the region \([A_1, B_1]\) that satisfies \({d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]}/{dz_1}=0\). The last constraint is the optimal stocking factor for the direct channel, which follows from Proposition 2.

4.2.2 A reformulated manufacturer’s problem

Recalling that in the previous sections, we have obtained the retailer’s pricing and stocking policy for given the manufacturer’s decisions, as presented in Proposition 1, and the manufacturer’s optimal stocking policy for given the selling price in the direct channel \(p_2\) and wholesale price w, as presented in Proposition 2. Then, there are two remaining decision variables to be determined for the manufacturer’s problem, i.e. \(p_2\) and w. However, it is intractable to directly obtain the optimal \(p_2\) and w. In order to solve our problem and obtain some nice properties of the optimal solutions, we use an alternative method to reformulate the problem in the following.

Before presenting the reformulated problem, we first show that given the retailer’s optimal stocking and pricing policy, optimal solutions of the retailer’s problem can be presented in terms of \(p_1\) and w.

Corollary 2

Given \(p_1\) and \(z_1\) as the optimal solutions to the retailer’s problem, we can obtain

$$\begin{aligned} w= & {} \frac{[a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)][1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]}, \end{aligned}$$
(14)
$$\begin{aligned} p_1= & {} \frac{a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)}{b_{11}[1+F_1(z_1)]}. \end{aligned}$$
(15)

According to Corollary 2, we find that \(z_1\) and w has a one-to-one corresponding relationship. It is because that the retailer orders products according to the wholesale price offered by the manufacturer. Thus, we can use w to represent \(z_1\). Then, plugging w and \(p_1\) obtained in Corollary 2 and \(z_2\) obtained in Proposition 2 into the manufacturer’s objective function, there are two decision variables, i.e., \(p_2\) and \(z_1\), remaining unsolved in our model. Accordingly, we can define

$$\begin{aligned} {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]={\mathbb {E}}[{\varPi }_m(z_2(p_2,w(p_2,z_1)),p_2,w(p_2,z_1))], \end{aligned}$$

and the manufacturer’s problem can be reformulated as follows:

$$\begin{aligned} {{{\text {P}}}_{new}}:\underset{{{p}_{2}},{{z}_{1}}}{\mathop {\max }}\,{\mathbb {E}}\left[ {{{\varPi }}_{new}}\left( p_2,z_1 \right) \right] , \end{aligned}$$
(16)

subject to

$$\begin{aligned} w= & {} \frac{[a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)] [1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]}\ge c, \end{aligned}$$
(17)
$$\begin{aligned} p_1= & {} \frac{a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)}{b_{11}[1+F_1(z_1)]},\nonumber \\&a_1+b_{11}w+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)-\frac{1-F_1(z_1)}{r_1(z_1)}\ge 0, \\&z_2= F_2^{-1}\left( \frac{p_2-c}{p_2}\right) .\nonumber \end{aligned}$$
(18)

Similar to (11), in the optimization problem (16), the first three constraints follow from the optimal solutions of the retailer’s problem, and the last constraint is the optimal stocking factor for the direct channel. Then we have the following results:

Proposition 3

The optimization problems (11) and (16) are equivalent.

4.2.3 Optimal selling price in the direct channel \(p_2^*\)

Given \(z_1\), in this subsection we obtain the optimal selling price in the direct channel \(p_2\) for the reformulated optimization problem (16). Before presenting the optimal solution, we first show an upper bound for the optimal selling price in the direct channel in the following lemma.

Lemma 4

Given \(z_1\), the optimal selling price in the direct channel \(p_2^*\) is always less than or equal to \(p_2^0={\mathbb {M}}/{\mathbb {N}}\), where

$$\begin{aligned} {\mathbb {M}}=&-2\left\{ {{b}_{12}}{{b}_{21}}-{{b}_{11}}{{b}_{22}}[1+{{F}_{1}}({{z}_{1}})]+\frac{b_{12}^{2}{{F}_{1}}({{z}_{1}})[1-{{F}_{1}}({{z}_{1}})]}{1+{{F}_{1}}({{z}_{1}})} \right\} ,\\ {\mathbb {N}}=\,&{{a}_{2}}{{b}_{11}}\left[ 1+{{F}_{1}}({{z}_{1}}) \right] +{{b}_{21}}\left[ {{a}_{1}}-{{{\varTheta }}_{1}}\left( {{z}_{1}} \right) \right] -c\left\{ {{b}_{12}}{{b}_{21}}-{{b}_{11}}{{b}_{22}\left[ 1+F_1\left( z_1\right) \right] +b_{11}b_{22}F_1\left( z_1\right) } \right\} \\&+\,{{b}_{12}}\left[ 1-{{F}_{1}}({{z}_{1}}) \right] \left( {{a}_{1}}+{{z}_{1}} \right) -\frac{{{b}_{12}}{{\left[ 1-{{F}_{1}}({{z}_{1}}) \right] }^{2}}\left[ {{a}_{1}}-{{{\varTheta }}_{1}}\left( {{z}_{1}} \right) \right] }{1+{{F}_{1}}({{z}_{1}})}, \end{aligned}$$

and \(p_2^*\) equals to \(p_2^0\) if and only if the demand in the direct channel is deterministic (the risk-less case).

Lemma 4 shows that the optimal price with demand uncertainty is small than or equal to the risk-less price. This is because as the direct channel demand becomes uncertain, the manufacturer takes additional overstocking and under-stocking risks. Consequently, the manufacturer would like to reduce the impact of variability, by making the uncertain demand be a smaller part of the total demand in the direct channel. This can be achieved by reducing the price, and thus increasing the deterministic part of the total demand (Dumrongsiri et al. 2008). Similar relationship was first demonstrated by Mills (1959) for the single-channel supply chain. Combining with Lemma 1, we show that the similar results hold for the dual-channel supply chain.

Similar to the retailer’s problem, here we also show that given \(z_1\), \({\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]\) might have multiple points that satisfy the first-order optimality condition.

Proposition 4

Given \(z_1\), the optimal selling price in the direct channel \(p_2^*\) can be determined as follows: If \(\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]/\partial p_2|_{p_2=c}>0\), then \(p_2^*\) is the unique \(p_2\) in the region \([c,p_2^0]\) that satisfies \(\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]/\partial p_2=0\); otherwise, \(p_2^*\) is the largest \(p_2\) in the region \([c,p_2^0]\) that satisfies \(\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]/\partial p_2=0\).

4.2.4 Optimal stocking factor for the traditional channel \(z_1^*\)

In the above analysis, by solving the retailer’s problem, we obtain the optimal pricing and stocking policy \(p_1^*\) and \(z_1^*\). For the manufacturer’s problem, we first obtain the optimal total inventory capacity \(N^*\). We find that there is a one-to-one relationship between \(z_1\) and w. Then we represent the optimal solutions of the retailer’s problem in terms of \(p_1^*\) and \(w^*\). Consequently, we reformulate the manufacturer’s problem and change to maximize the manufacturer’s profit over \(p_2\) and \(z_1\) instead of \(p_2\) and w. For the reformulated problem, we fist solve the optimal selling price in the direct channel \(p_2^*\) for a given \(z_1\). After substituting it back into the manufacturer’s objective function, the reformulated optimization problem becomes a maximization over the single variable \(z_1\), i.e., \(\max \limits _{{{z}_{1}}}\,{\mathbb {E}}\left[ {{{\varPi }}_{new}}\left( p_2(z_1),z_1\right) \right] \). Unfortunately, it is intractable to obtain the analytical optimal solution for this optimization problem. Therefore, we use an exhaustive search algorithm to find the optimal \(z_1\).

Proposition 5

The optimal \(z_1^*\) can be obtained by exhaustively searching the \(\left( p_2(z_1),{{z}_{1}}\right) \in {{{\varOmega }}_{{\text {new}}}}\) that maximizes \({\mathbb {E}}\left[ {\varPi }_{new}\left( p_2(z_1),z_1\right) \right] \) in the region \([A_1,B_1]\).

Now we have completed the analysis of the optimal pricing and inventory control policies for our dual-channel problem. In order to clearly show how to solve our problem, we summarize the procedure of the analysis in the form of the pseudo-code as follows:

Procedure

Input: \({{a}_{1}}\), \({{a}_{2}}\), \({{b}_{11}}\), \({{b}_{12}}\), \({{b}_{21}}\), \({{b}_{22}}\), c, \({{\mu }_{1}}\), \({{\mu }_{2}}\), \({{\sigma }_{1}}\), \({{\sigma }_{2}}\), \({{A}_{1}}\), \({{B}_{1}}\), \({{A}_{2}}\), \({{B}_{2}}\), and stocking factor increment \({\varDelta }=\frac{{{B}_{1}}-{{A}_{1}}}{n}\).

Outputs: manufacturer’s optimal total inventory capacity \({{N}^{*}}\), optimal selling price in the direct channel \(p_{2}^{*}\), optimal wholesale price \({{w}^{*}}\), retailer’s optimal order quantity \(q_{1}^{*}\), optimal retail price in the traditional channel \(p_{1}^{*}\), manufacturer’s optimal profit \({\varPi }_{m}^{*}\), and retailer’s optimal profit \({\varPi }_{r}^{*}\).

  1. 1.

    Set \({\varPi }_{m}^{*}=0\).

  2. 2.

    for \({{z}_{1}}={{A}_{1}},{{A}_{1}}+{\varDelta },{{A}_{1}}+2{\varDelta },...,{{A}_{1}}+n{\varDelta }\) do

    1. (a)

      compute \({{p}_{2}}\) by Proposition 4.

    2. (b)

      compute \({{z}_{2}}\), w and \({{p}_{1}}\) by (10), (14) and (15), respectively.

    3. (c)

      if \(a_1+b_{11}w+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)-\frac{1-F_1(z_1)}{r_1(z_1)}\ge 0\) then

      1. i.

        compute \({\mathbb {E}}\left[ {{{\varPi }}_{m}}({{z}_{2}},{{p}_{2}},w) \right] \) by (8).

      2. ii.

        if \({\mathbb {E}}\left[ {{{\varPi }}_{m}}({{z}_{2}},{{p}_{2}},w) \right] \ge {\varPi }_{m}^{*}\) then set \({\varPi }_{m}^{*}={\mathbb {E}}\left[ {{{\varPi }}_{m}}({{z}_{2}},{{p}_{2}},w) \right] \), \(z_2^*=z_2\), \(p_2^*=p_2\), \(w^*=w\), \(z_1^*=z_1\) and \(p_1^*=p_1\); compute \(y_1\), \(y_2\) and \({\mathbb {E}}\left[ {{{\varPi }}_{r}}({{z}_{1}},{{p}_{1}}) \right] \) by (1), (2) and (4), respectively, and set \(q_1^*=y_1+z_1\), \(N^*=y_1+z_1+y_2+z_2\) and \({\varPi }_{r}^{*}={\mathbb {E}}\left[ {{{\varPi }}_{r}}({{z}_{1}},{{p}_{1}}) \right] \).

  3. 3.

    end for

  4. 4.

    Return \(\left( {{N}^{*}},p_{2}^{*},{{w}^{*}},q_{1}^{*},p_{1}^{*},{\varPi }_{m}^{*},{\varPi }_{r}^{*} \right) \).

    End

5 The effects of demand uncertainties and price sensitivities

In this section, we numerically study the effects of demand uncertainties and price sensitivities. We assume that the demands follow the normal distributions, that is \({{\epsilon }_{i}}\sim N\left( 0,\sigma _{i}^{2} \right) \), where \(i\in \{1, 2\}\). Note that the random variable \(\epsilon _i\) is defined in the range \([A_i, B_i]\). Besides, for the normal distribution, \(99.73\%\) of the values lie within three standard deviation from the mean. Thus, we use truncated normal distribution as approximations, and set \(A_i=-3\sigma _i\) and \(B_i=3\sigma _i\). Following Ding et al. (2016), we first set \(b_{11}=65\), \(b_{22}=65\), \(b_{12}=25\), \(b_{21}=25\) and \(c=1\). For the values of the base demands, we set \(a_1=600\) and \(a_2=1200\), which are larger than the basic settings for these two parameters in their paper. The reason is that the demands in their paper are deterministic, whereas we consider the stochastic demand; and with stochastic demand, there may exist infeasible solutions if the base demand is small. Besides, note that we let the base demand in the direct channel be larger than that in the traditional channel, because the sales volume from the direct channel can be significant as the Internet is accessible for more and more consumers (Ding et al. 2016).

5.1 Effects of demand uncertainties

5.1.1 Effects of the demand variability in the traditional channel \(\sigma _1\)

In this subsection, we study the effects of the demand variability in the traditional channel \(\sigma _1\). We set \(\sigma _2=100\), and change \(\sigma _1\) from 40 to 200. The results are shown in Fig. 2.

Fig. 2
figure 2

Effects of the change of \(\sigma _1\) a Performance of price b Performance of stocking factor c Performance of inventory capacity d Performance of expected profit e Performance of inventory percentage f Performance of profit percentage

Full size image

From Fig. 2a, we can see that all the three prices decrease in the demand variability in the traditional channel. This is because when \(\sigma _1\) increases, the retailer takes more overstocking and under-stocking risks. Facing these risks, the retailer will reduce the price and thus increase the deterministic part of the demand. In this way, the proportion of uncertainty over the demand will be smaller. Anticipating the retailer’s action, the manufacturer will also reduce its wholesale price and selling price in the direct channel.

We can see from Fig. 2b that the stocking factor of the retailer \(z_1\) may decrease in the demand variability in the traditional channel. This can be explained as follows: On the one hand, for a given \(z_1\), the expected shortage increases as the demand variability increases. Then the stocking factor is expected to be increased. On the other hand, as we can see from Fig. 2a that the price is decreased when the demand variability increases. Then the shortage cost for not having enough stock is lower, which leads to a decrease in the stocking factor. Thus, combining with these increasing and decreasing effects, the stocking factor of the retailer may be decreased when the demand variability increases (Dumrongsiri et al. 2008). From Fig. 2c, we can see that the order quantity of the retailer will be increased. This is because that the expected demand will be increased due to the decrease of the retail price in the traditional channel. However, both the stocking factor for the direct channel and the inventory allocated to the direct channel will be decreased when \(\sigma _1\) increases. It may be due to the price competition between the traditional channel and direct channel. As we can see from Fig. 2a, the decrease rate of the retail price in the traditional channel is larger than that of the selling price in the direct channel. So the demand in the direct channel may be decreased when \(\sigma _1\) increases. It may lead to the decreases in the stocking factor for the direct channel and inventory allocated to the direct channel. And the total inventory capacity will be increased, as the increasing effect from the order quantity of the retailer dominates the decreasing effect from the optimal inventory allocated to the direct channel. Figure 2e shows the effects on the percentages of the quantities in the traditional and direct channels over the total inventory capacity. Similarly, due to the increase of the order quantity in the traditional channel and decrease of the quantity in the direct channel, the percentage of the quantity in the traditional (direct) channel increases (decreases) in \(\sigma _1\).

It is interesting to show in Fig. 2d that the retailer’s expected profit may be unexpectedly increased in the demand variability. One reason is that when \(\sigma _1\) increases, the expected demand will be increased due to the decease of the price. In addition, as we discussed in the above, the shortage cost for not having the enough stock will be lower when \(\sigma _1\) increases. Thus, due to these two effects, the expected profit may be increased, although the price is decreased. For the expected profit of the manufacturer, we can observe that it is decreasing in \(\sigma _1\). Then, the percentage of the manufacturer’s (retailer’s) profit over the total profit of the supply chain is decreased (increased), as shown in Fig. 2f.

5.1.2 Effects of the demand variability in the direct channel \(\sigma _2\)

In this subsection, we study the effects of the demand variability in the direct channel \(\sigma _2\). We set \(\sigma _1=100\), and change \(\sigma _2\) from 40 to 200. The results are shown in Fig. 3.

Fig. 3
figure 3

Effects of the change of \(\sigma _2\) a Performance of price b Performance of stocking factor c Performance of inventory capacity d Performance of expected profit e Performance of inventory percentage f Performance of profit percentage

Full size image

It is worth noting that the retail price in the traditional channel, and the stocking factor, the order quantity and the expected profit of the retailer are relatively stable when \(\sigma _2\) changes. This implies that without losing too many profits, the retailer can keep the pricing and stocking policy unchanged when the demand variability in the direct channel increases.

Regarding the effects on the manufacturer, we can see from Fig.  3b, c that the stocking factor for the direct channel, inventory allocated to the direct channel and total inventory capacity increase in \(\sigma _2\). The main reason is that the manufacturer should increase the stocking factor to mitigate the under-stocking risk when \(\sigma _2\) increases. Consequently, the inventory allocated to the direct channel will be increased, which has positive effect on the total inventory capacity. Figure 3e shows the effects on the percentages of the quantities in the traditional channel and direct channel over the total inventory capacity. It is straightforward to see that the percentage of the quantity in the direct (traditional) channel increases (decreases) in \(\sigma _2\). For the effects on the expected profits of the manufacturer, we can see from Fig. 3d that it decreases in \(\sigma _2\). Besides, note from Fig. 3f that the manufacturer takes up a lion share of the total profit. One reason is that the retailer makes profit only from the traditional channel, whereas the manufacturer makes profit from both channels. Another reason is due to our numerical setting that the base demand in the direct channel is larger than that in the traditional channel.

5.2 Effects of the price sensitivities

To explore the effects of the price sensitivities, we set \(\sigma _1=\sigma _2=100\) and change the values of \(b_{11}\), \(b_{22}\), \(b_{12}\) and \(b_{21}\) respectively.

5.2.1 Effects of the price-sensitivity coefficient \(b_{11}\)

In this subsection, we study the effects of the price-sensitivity coefficient \(b_{11}\) by changing it from 40 to 80. The results are shown in Fig. 4.

Fig. 4
figure 4

Effects of the change of \(b_{11}\) a Performance of price b Performance of stocking factor c Performance of inventory capacity d Performance of expected profit e Performance of inventory percentage f Performance of profit percentage

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From Fig. 4a we can see that all the three prices decrease in \(b_{11}\). This is because the increase in \(b_{11}\) indicates that the customers in the traditional channel are more sensitive to the price, and the traditional channel are less attractive or convenient to customers. Then, the retailer will decrease selling price to attract more customers to avoid the profit loss. Anticipating the retailer’s behavior, the manufacturer will also reduce its wholesale price and selling price in the direct channel. The decreasing effect on the selling price in the traditional channel is significant than that in the direct channel, as \(b_{11}\) directly affects the demand in the traditional channel. Besides, from Proposition 2 we can see that \(z_2\) increases in \(p_2\), so it is easy to understand that the stocking factor for the direct channel decreases in \(b_{11}\) in Fig. 4b. In addition, we can see from Fig. 4b that the stocking factor for the traditional channel is relatively stable. It implies that when customers become more sensitive to the selling price in the traditional channel, the retailer can keep the stocking policy relatively stable by adjusting the selling price.

From Fig. 4c we can see that the order quantity of the retailer, the inventory allocated to the direct channel and the total inventory capacity decrease in \(b_{11}\). This is because the less attractive traditional channel directly leads to the less order quantity of the retailer, which reduces the inventory allocated to the direct channel due to the competition between the traditional and direct channels. Consequently, the total inventory capacity is decreasing in \(b_{11}\). Then, it is straightforward to see from Fig. 4d that both the expected profit of the retailer and the manufacturer will be decreased when customers are more sensitive to the price in the traditional channel.

From Fig. 4e we can see that the inventory percentages of the quantities in the traditional and direct channels over the total inventory capacity are relatively stable, because the decrease rate of \(q_1\) almost equals to that of \(q_2\), as shown in Fig. 4c. From Fig. 4f, we can see that the percentage of the manufacturer’s (retailer’s) profit over the total profit of the supply chain increases (decreases) in \(b_{11}\). This is because the increase in \(b_{11}\) eventually leads to that the manufacturer has more market power over the retailer, as customers are more sensitive to the retailer’s selling price.

5.2.2 Effects of the price-sensitivity coefficient \(b_{22}\)

In this subsection, we study the effects of the price-sensitivity coefficient \(b_{22}\) by changing it from 40 to 80. The results are shown in Fig. 5.

Fig. 5
figure 5

Effects of the change of \(b_{22}\) a Performance of price b Performance of stocking factor c Performance of inventory capacity d Performance of expected profit e Performance of inventory percentage f Performance of profit percentage

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Similar to the analysis for the price-coefficient \(b_{11}\), we can see that all the three prices, the stocking factor for the direct channel, the product quantities, the total inventory capacity, and both the expected profits of the retailer and manufacturer decrease in \(b_{22}\). The inventory percentages of the quantities in the traditional and direct channels over the total inventory capacity are relatively stable, and the percentage of the manufacturer’s (retailer’s) profit over the total profit of the supply chain is decreasing (increasing) in \(b_{22}\). However, it is interesting to show in Fig. 5b that the stocking factor for the traditional channel increases in \(b_{22}\). The reason is that when \(b_{22}\) increases, to a certain extent, the traditional channel is more attractive than the direct channel. It leads to the increase of the stocking factor for the traditional channel to avoid the inventory shortage.

5.2.3 Effects of the cross-price-sensitivity coefficient \(b_{12}\)

In this subsection, we study the effects of the cross-price-sensitivity coefficient \(b_{12}\) by changing it from 5 to 45. The results are shown in Fig. 6.

Fig. 6
figure 6

Effects of the change of \(b_{12}\) a Performance of price b Performance of stocking factor c Performance of inventory capacity d Performance of expected profit e Performance of inventory percentage f Performance of profit percentage

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From Fig. 6a we can see that all the three prices increase in \(b_{12}\). This is because \(b_{12}\) reflects the degree to which the products sold through the two channels are substitute (Yue and Liu 2006; Hua et al. 2010; Huang et al. 2012). The increase in \(b_{12}\) indicates that the customers in the traditional channel become more sensitive to the change of the selling price in the direct channel. Then, due to the price competition in the two channels, the retailer can increase the selling price in the traditional channel without worrying about the loss of the demands when \(b_{12}\) increases. Anticipating the retailer’s behavior, the manufacturer can also increase its wholesale price and selling price in the direct channel. The increasing effects on the selling price in the traditional channel is significant than that in the direct channel, as \(b_{12}\) directly affects the demand in the traditional channel. Besides, from Proposition 2 we find that \(z_2\) increases in \(p_2\), thus it is easy to understand that the stocking factor for the direct channel increases in \(b_{12}\) in Fig. 6b. In addition, we can see from Fig. 6b that the stocking factor for the traditional channel is relatively stable. It implies that when the cross-price-sensitivity \(b_{12}\) increases, the retailer can still keep the stocking policy relatively stable by adjusting the selling price.

From Fig. 6c we can see that the order quantity of the retailer and the total inventory capacity increase in \(b_{12}\), as \(b_{12}\) has a directly increasing effect on the demand in the traditional channel. However, the inventory allocated to the direct channel will be decreased, because the increase of the price in the direct channel has a dominate decreasing effect on the demand in the direct channel. Thus, the percentage of the quantities in the traditional (direct) channel increases (decreases) in \(b_{12}\), as shown in Fig. 6e. Besides, as shown in Fig. 6d, the increase in \(b_{12}\) indicates that both the manufacturer and retailer can obtain more profits from the traditional channel, which leads to the increases in both the retailer’s and manufacturer’s profits. However, the percentage of the manufacturer’s (retailer’s) profit over the total profit of the supply chain decreases (increases) in \(b_{12}\).

5.2.4 Effects of the cross-price-sensitivity coefficient \(b_{21}\)

In this subsection, we study the effects of the cross-price-sensitivity coefficient \(b_{21}\) by changing it from 5 to 45. The results are shown in Fig. 7.

Fig. 7
figure 7

Effects of the change of \(b_{21}\) a Performance of price b Performance of stocking factor c Performance of inventory capacity d Performance of expected profit e Performance of inventory percentage f Performance of profit percentage

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Similar to the analysis for the cross-price-coefficient \(b_{12}\), we can see that all the three prices and the stocking factor for the direct channel increase in \(b_{21}\). However, we can see from Fig. 7b that the stocking factor for the traditional channel is decreased. The reason is that when \(b_{21}\) increases, to a certain extent, the direct channel is more attractive than the traditional channel. Customers may prefer to buy from the direct channel instead of the traditional channel, leading to the decrease of the stocking factor for the traditional channel to avoid the inventory overstocking.

From Fig. 7c we can see that the inventory allocated to the direct channel and the total inventory capacity increase in \(b_{21}\), as \(b_{21}\) has a directly increasing effect on the demand in the direct channel. However, the order quantity of the retailer will be decreased, because the increase of the price in the traditional channel has a dominate decreasing effect on the demand in the traditional channel. Thus, the percentage of the quantities in the traditional (direct) channel decreases (increases) in \(b_{12}\), as shown in Fig. 7e. Besides, the increase in \(b_{21}\) also indicates the manufacturer could extract the retailer’s profit from the traditional channel by selling the products in the direct channel, leading to the increase (decrease) in the manufacturer’s (retailer’s) profit, as shown in Fig. 7d. Then, it is straightforward to see that the percentage of the manufacturer’s (retailer’s) profit over the total profit of the supply chain increases (decreases) in \(b_{21}\), as shown in Fig. 7f.

6 Conclusions

In this paper, we study the optimization problem associated with the joint decision of the pricing and inventory policies, for a dual-channel supply chain consisting of one manufacturer and one retailer. The manufacturer and retailer play a Stackelberg game with the manufacturer as the leader and the retailer as the follower. We consider a general model setting with five decision variables, i.e., the total inventory capacity of the manufacturer, the selling price in the direct channel, the wholesale price to the retailer, the order quantity of the retailer and the retail price in the traditional channel. We develop a mechanism based on the chain rule to solve the optimization problem.

We derive the optimal solutions for both the manufacturer’s and retailer’s problems. We find that given the selling price in direct channel and the wholesale price, the retailer’s decisions on order quantity is similar to that for the newsvendor problem (Petruzzi and Dada 1999). Meanwhile, the manufacturer’s decision on the inventory allocated to the direct channel also has the similar structure. Regarding the format of the optimal solutions, we find that in our analysis framework, the retail price in the traditional channel, the wholesale price and the total inventory capacity can be presented in closed-forms.

Besides, we conduct numerical studies to show the effects of the demand variabilities and price sensitivities on the optimal solutions and performance of the manufacturer and retailer. One might expect that the demand variability will reduce firm’s profit. However, we show that the retailer’s expected profit may be unexpectedly increased when the demand variability in the traditional channel increases. One reason is that when the demand variability in the traditional channel increases, the expected demand will be increased due to the decease of the price. In addition, the shortage cost for not having the enough stock will be lower when the demand variability increases, because the price is decreased. Thus, due to these two effects, the expect profit may be increased, although the price is decreased. Moreover, we observe that without losing too many profits, the retailer can keep the pricing and stocking policy unchanged, when the demand variability in the direct channel increases. For the effects of the price sensitivities, we find that all the prices, the inventories in two channels, the total inventory capacity, and the profits are decreasing in the price-sensitivity coefficients, whereas most of them are increasing in the cross-price-sensitivity coefficients.

For future research, one extension is to consider that there are multiple retailers ordering products from the manufacturer. Another possible future research direction is to investigate the competition between two manufacturers. Moreover, it would be interesting to consider the risk attitude of the retailer or manufacturer, as a future research direction.