Two-Step Pricing Decision Models for Manufacturer-Led Dual-Channel Supply Chains

Abstract

In this paper, we study the optimal profit change in a manufacturer-led dual-channel supply chain under centralized decision-making and decentralized decision-making scenarios. The supply chain is composed of only a single supplier, a manufacturer, and a retailer, and the manufacturer plays a leading role in the supply chain. Meanwhile, the following conditions exist simultaneously: the promotion levels of retailers in offline physical stores, the promotion compensation given by manufacturers to retailers, and channel competition. In order to coordinate the profits of channel members, a two-step pricing decision model is established. This research shows that using a two-step pricing decision model can make the retailer’s optimal promotion fees and the manufacturer’s optimal sales compensation fees the same as the centralized decision-making’s optimal values; however, the results are not good, because after the coordination, the retailer’s profits increase and the manufacturer’s profits decrease, and, as a consequence, the retailer needs to pay a fixed fee to the manufacturer within the validity of the contract, which is signed by both sides, to achieve a win–win situation for the channel members.

1. Introduction

With the development of online shopping platforms (Taobao, JD, etc.), e-commerce has developed an important role in national life and economic development, and product transactions in online and offline channels are very active [1]. Due to the existence of the “bilateral effect”, the problem of coordinated decision making in dual-channel supply chains has become a hot issue in the current supply chain management community. The research conducted by domestic and foreign scholars on the coordination of dual-channel supply chains is mainly carried out through the establishment of models that coordinate the interests of channel members. Esmaeili et al. [2] explored the optimal price and the optimal order quantity decision for different channels of different leaders. Shin [3] and Wu et al. [4] studied the impact of dual-channel supply chains on retailer pricing strategies. In the course of cooperation between manufacturers and retailers, the use of quantity discount contracts can achieve the coordination of dual-channel supply chains [5]. Wholesale price contracts have less coordination effects on the income of supply chain members than two-step pricing contracts and quantity discount contracts [6]. Building two optimal linear contracts can encourage members to invest together in order to achieve channel coordination [7]. In a single market, manufacturers and retailers cannot coordinate wholesale price contracts, revenue sharing mechanisms, or buyback contracts [8]. If the members of a dual-channel supply chain have fair preferences, then a single wholesale price contract can achieve interest coordination [9]. Chiang [10] proposed a combination mechanism between supply chain members to share the dual-channel cost and electronic direct sales channel revenue, and the mechanism achieved dual-channel supply chain coordination. When there are only two manufacturers and one retailer in a supply chain channel, quantity discount contracts, two-step pricing contracts, and wholesale price contracts cannot coordinate the three profits. The two manufacturers lose both sides through competition, but the retailer can achieve high profits [11]. Cai et al. [12] researched a variety of dual-channel supply chain structures and designed a price contract to enable the coordination of multiple dual channels. Wang et al. [13] designed a fixed pricing and transfer payment plan, and a quantity discount pricing plan to coordinate the decisions of each member of a supply chain. Some scholars start with two-step pricing contracts in dual-channel supply chain research. In a previous study, under the premise of two-step pricing contracts, a wholesale price contract was studied, and the optimal profit of a dual-channel supply chain under the wholesale price was obtained through a numerical analysis [14]. Through the study of two-step pricing, it can be concluded that, in the same market competition environment, two-step pricing contracts have a better coordination effect than other forms of contracts, such as quantity discount contracts, revenue sharing contracts, and repurchase contracts; they can promote the coordinated operation of all enterprises and improve the efficiency of the overall supply chain [15]. Shugan [16] investigated how to achieve the best interests of manufacturers in vertically integrated supply chains through two-step pricing contracts. Oliveira et al. [17] studied two-step pricing contracts and found that they can greatly reduce the “double marginalization effect” in the power industry and improve the overall supply chain operation performance. Moderate channel conflicts contribute to the competitiveness of retailers and manufacturers, while excessive channel conflicts lead to consumers losing their interest in shopping [18]. Zhang et al. [19] discussed the influence of consumers’ green preferences on supply chain pricing decisions. Based on a two-step dual-channel supply chain, Lei et al. [20] studied the influence of three factors, namely, consumer patience, direct sales channel acceptance, and delivery waiting time, on manufacturer-led Starkberg model pricing; Pu et al. [21] studied the influence of consumers’ sensitivity to product quality differences on product distribution strategies in dual-channel supply chains.

The above literature mainly discussed the various coordination contracts of supply chains, but most scholars focused their research on comparative coordination contracts and coordination problems in the case of uncertain demand. The promotion level of manufacturer-oriented supply chains and retailers needs to be further improved. Specifically, this study aims to answer the following research questions:

(1) What are the applicable conditions for the two-step pricing decision of a manufacturer-led dual-channel supply chain?

(2) What are the characteristics of the changes in the optimal profit values of supply chain members in centralized and decentralized decision-making scenarios?

(3) How does the two-step pricing coordination decision-making model achieve a win–win situation among supply chain members?

This paper focuses on the retail channel of a manufacturer and a retailer in a dual-channel supply chain. The retailer conducts product transactions through promotion activities in the retail channel, and the manufacturer sells the products online through the direct sales channel and gives the retailer corresponding sales compensation. Due to the existence of the “bilateral effect”, supply chain profit changes greatly in cases of centralized decision making and decentralized decision making. In order to effectively coordinate the profits of dual-channel supply chains, two-step pricing decision models are established to achieve the coordination of dual-channel supply chains and to achieve the Pareto optimality of the channel members’ interests. The contributions are as follows:

(1) When many factors, such as the promotion level of offline physical stores and the promotion compensation of online channel manufacturers, exist at the same time, we establish a two-step pricing decision-making model of the supply chain to study the pricing decision-making problem of the “bilateral effect”.

(2) Two-step pricing and retailers paying fixed fees can achieve the optimal profit of supply chain members. When the market share and fixed payment fees change, the overall optimal profit of the supply chain under centralized decision making is always better than that under decentralized decision making.

(3) The two-step pricing coordination decision-making methods can balance the retailer’s promotion cost and the manufacturer’s promotion compensation.

The remainder of this study is organized as follows: Section 2 presents the modeling methodology. A basic decision model of dual-channel supply chains is described in Section 3. Section 4 presents two-step pricing decision models under the coordinated decision of dual-channel supply chains. Then, the changes in the supply chain profit are illustrated in Section 5. The numerical results and analysis are presented in Section 6, and concluding remarks are presented in Section 7.

2. Modeling Methodology

2.1. Model Description

This paper combines actual cases of enterprises and assumes that the market consists of a competitive market composed of one supplier and one retailer. A structure diagram of the production and sales of homogeneous products, where the manufacturer is the market leader, is shown in Figure 1 (the dotted lines indicate the demand information flow).

In Figure 1, since customers have basic needs, it is assumed that there are loyal customers in both channels. After the supplier supplies the manufacturer, the manufacturer sells to the customer through both online and offline channels. In the online channel, the manufacturer sells the product to the customer $c_d$ at price $p_d$. In the offline channel, the manufacturer sells the product to the retailer at a wholesale price of $w$, and the retailer sells at physical stores at a price of $p_r$. The rapid development of e-commerce logistics has greatly reduced the sales of traditional store products. In order to increase the sales volume of offline physical stores, retailers decide to actively promote products with a promotion level of $u$. In order to improve the retailer’s promotion enthusiasm, the manufacturer decides to give a compensation of $uv$ to the retailer for promotion activities.

2.2. Symbol Description

To facilitate model calculations, all the symbols and variables are shown in

Table 1. List of symbols and variables.

Symbols and Variables	Definitions
s	The ratio of demand for products by offline retail customers
1-s	The ratio of demand for products by online direct customers
a	The total basic demand of the market
Pr	The product transaction price for offline retail channel retailers
Pd	The product transaction price for online direct channel manufacturers with customers
α	The sensitivity of product interaction between online and offline dual channels
β	The sensitivity to market demand in online and offline dual channels
Dr	Demand for the product by offline retail channel customers
Dd	Demand for online direct channel customers
πr	The profit of the retailer
πd	The profit of the manufacturer
πz	The overall profit of the supply chain
w	The wholesale price of the wholesale product of the retailer from the manufacturer
c	The cost of the manufacturer to produce the product (per unit)
u	Promotion level of the retailer’s offline retail product promotion
βu	The change in demand when the promotion level is $u$
uv	The compensation given by the manufacturer to the retailer after the retailer’s promotion
u2	The cost of the retailer’s online promotion
v2	The cost of the manufacturer’s compensation for retailer promotions
Superscript L	Two-step pricing decisions
Superscript j	Centralized decision model
Superscript f	Decentralized decision model

.

2.3. Establishment of Decision-Making Model

This paper assumes that the demand function is a linear function; channel demand is inversely proportional to the price of the same channel and is proportional to the price of another channel, which is $\alpha \succ 1$. The two channels are mutually competitive. The sales volume of offline channel retailers is proportional to the promotion level, which is set as $\beta \succ 1$. The relationship between online channel sales volume and promotion level is negligible. According to the model interpretation, the following demand and profit function models are established:

Retailer demand function:

$$D_r=sa-p_r+\alpha p_d+\beta u$$

(1)

Manufacturer demand function:

$$D_d=\left(1-s\right)a-p_d+\alpha p_r$$

(2)

In Equations (1) and (2), $\{\begin{array}{l}sa-p_r>0\\ (1-s)a-p_d>0\end{array}$ indicates that there are always potential buyers using offline and online channels.

Retailer profit function:

$${\pi }_r=\left(p_r-w\right)\times D_r-u^2+uv,\left(p_r\succ w\succ c\right)$$

(3)

Manufacturer profit function:

$${\pi }_d=\left(w-c\right)\times D_r+\left(p_d-c\right)\times D_d-v^2,\left(p_d\succ c\right)$$

(4)

3. Basic Decision Model of Dual-Channel Supply Chain

3.1. Centralized Decision Model of Dual-Channel Supply Chain

In the centralized decision-making scenario, online channels and offline channels cooperate to treat online and offline as a decision-making whole, and manufacturers and retailers pursue profit maximization together [17]. Therefore, the overall profit of the dual-channel supply chain under the centralized decision-making scenario is as follows:

$$\begin{array}{ll}{\pi }^j& ={\pi }_z={\pi }_r+{\pi }_d\\ & =\left(p_r-w\right)\times D_r-u^2+uv+\left(w-c\right)\times D_r+\left(p_d-c\right)\times D_d-v^2\\ & =\left(p_r-c\right)\times \left(sa-p_r+\alpha p_d+\beta u\right)+\left(p_d-c\right)\times \left[\left(1-s\right)a-p_d+\alpha p_r\right]-u^2-v^2+uv\end{array}$$

(5)

Equation (5) concerns the negative form of the Hessian matrix, and the Hessian matrix is

$$\left[\begin{array}{cccc}-2,& 2\alpha ,& \beta ,& 0\\ 2\alpha ,& -2,& 0,& 0\\ \beta ,& 0,& -2,& 1\\ 0,& 0,& 1,& -2\end{array}\right]$$

(6)

Since the supply chain as a whole seeks to maximize profits, for Equation (5), firstly use ${\pi }^j$ for the first-order partial derivative of $p_r$ and make it zero, and obtain the optimal sales price of the retailer under the concentrated situation; substituting this optimal sales price into Equation (1) can determine the optimal demand of the retailer. Then, use ${\pi }^j$ for the first-order partial guide of $u$ so that it is zero to obtain the retailer’s optimal promotion level. By substituting the retailer’s optimal selling price and the optimal demand into Equation (5), and by using ${\pi }^j$ for the first-order partial derivative of $p_d$ and $v$ and making it zero, the optimal sales of the manufacturers and retailers under the centralized decision-making scenario can be determined. The price and promotion levels are as follows:

$$p_r^{j\ast }=c+\frac{3\left[c-as+a\alpha \left(s-1\right)-c{\alpha }^2\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(7)

$$p_d^{j\ast }=\frac{c\left(1+a\right)\left(-3+3\alpha +{\beta }^2\right)+a\left[-3+{\beta }^2-s\left(-3+3\alpha +{\beta }^2\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(8)

$$u^{j\ast }=\frac{\beta \left[a\left(s+\alpha -s\alpha \right)+c\left({\alpha }^2-1\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(9)

$$v^{j\ast }=\frac{\beta \left[a\left(s+\alpha -s\alpha \right)+2c\left({\alpha }^2-1\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(10)

The retailer and the manufacturer’s optimal selling price, the optimal demand, the optimal promotional price, and the best promotional compensation price from Equations (7)–(10), respectively, can be used in Equation (5) to obtain the overall supply chain under the concentrated situation. The optimal profit is

$${\pi }^{j\ast }=\frac{\left\{a^2\left[-3+6s\left(s-1\right)\left(\alpha -1\right)\right]-6c^2{\left(\alpha -1\right)}^2\left(1+\alpha \right)-6ac\left({\alpha }^2-1\right)+{\beta }^2{\left[c+a\left(s-1\right)-ca\right]}^2\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}+\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{4\beta }$$

(11)

3.2. Decentralized Decision Model of Dual-Channel Supply Chain

Under the decentralized decision-making scenario, the two channels form a fierce competition pattern. Manufacturers and retailers are in charge of each other and pursue their own profits. This paper analyzes the decentralized decision based on the Stackelberg game model; the manufacturer is the market leader, and the retailer is the market follower. First, maximize the profit of the retailer, and then maximize the profit of the manufacturer. Their profit functions are as follows:

Retailer profit function:

$${\pi }_r^f=\left(p_r-w\right)\times D_r-u^2+uv,\left(p_r>w>c\right)$$

(12)

Manufacturer profit function:

$${\pi }_d^f=(w-c)\times D_r+\left(p_d-c\right)\times D_d-v^2,\left(p_d>c\right)$$

(13)

In Equation (13), retailers pursue profit maximization. The expression of ${\pi }_r^f$ is the negative form of the Hessian matrix, and the Hessian matrix is $\left[\begin{array}{cc}-2,& \beta \\ \beta ,& -2\end{array}\right]$. Then, use ${\pi }_r^f$ for the first order of $p_r$ and make it zero. The best response strategy of the retailer to the manufacturer’s strategy at this time is as follows:

$$p_r=\frac{as+w+u\beta +\alpha p_d}{2}$$

(14)

Substituting Equation (14) into Equations (1) and (2) can be used to determine the demand. The functions of the retailers and manufacturers are as follows:

$$D_r^{f\ast }=\frac{s\alpha -\omega +\upsilon \beta +\alpha p_d}{2}$$

(15)

$$D_d^{f\ast }=\frac{2\left(\alpha -p_d\right)+\alpha s\left(\alpha -2\right)+\alpha \left(w+\upsilon \beta +\alpha p_d\right)}{2}$$

(16)

Substitute Equations (14)–(16) into Equation (13), because the expression is ${\pi }_d^f$’s Hessian matrix:

$$\left[\begin{array}{ccc}\frac{4}{{\beta }^2-4},& \alpha ,& \frac{\beta }{4-{\beta }^2}\\ \alpha ,& \frac{8-4{\alpha }^2-2{\beta }^2}{{\beta }^2-4},& \frac{\alpha \beta }{4-{\beta }^2}\\ \frac{\beta }{4-{\beta }^2},& \frac{\alpha \beta }{4-{\beta }^2},& -2\end{array}\right]$$

(17)

The Hessian matrix is a negative definite form. Find the first-order partial derivatives of $w$, $p_d$, $v$, and $u$ and make them zero. The manufacturer’s optimal sales price, the wholesale price, the retailer’s optimal promotion level, and the manufacturer’s retailer-based optimal promotion compensation fee can be obtained in the case of a decentralized decision. They are

$$p_d^{f\ast }=\frac{16a\left\{\left[1+s\left(\alpha -1\right)\right]+a{\beta }^2\left[9+s\left(4\alpha -9\right)\right]+c\left[-32+9{\beta }^2+2{\alpha }^2\left(5{\beta }^2+{\beta }^4\right)\right]\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(18)

$$w^{f\ast }=\frac{c\left[a^2+\left(\alpha -2\right)\left({\beta }^2-3\right)+a^2\left({\beta }^2+6\right)\right]+\beta \left(2\alpha -1\right)}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(19)

$$u^{f\ast }=\frac{\beta \left[a\left(s+\alpha -s\alpha \right)+c\left({\alpha }^2-1\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}+{\alpha }^2\left(\beta -1\right)$$

(20)

$$v^{f\ast }=\frac{\beta \left[a\left(s+\alpha -s\alpha \right)+c\left({\alpha }^2-1\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}+\frac{{\alpha }^2\left(\beta -1\right)}{2\beta }$$

(21)

By comparing Equations (7)–(10) and (18)–(21), the retailer’s promotional expenses in the dispersed scenario increase by $u^{f\ast }-u^{j\ast }={\alpha }^2\left(\beta -1\right)$ compared to the concentrated scenario. The optimal promotion compensation fee increases by $v_f^{\ast }-v_j^{\ast }=\frac{{\alpha }^2\left(\beta -1\right)}{2\beta }$. Substituting Equations (18)–(21) into Equation (14) gives the retailer the best-selling price:

$$p_r^{f\ast }=\frac{{\beta }^2\left\{8c\left(\alpha -1\right){\left(\alpha +1\right)}^2+8a\left[-2\alpha +s\left(\alpha -1\right)\left(\alpha +3\right)\right]+\left[5a+as\left(4-5\alpha \right)\right]\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(22)

Substituting Equations (18)–(21) and (22) into Equations (15) and (16) shows that the optimal demand for the retailers and manufacturers is

$$D_r^{f\ast }=\frac{c\left(\alpha -1\right)\left[16+4{\alpha }^2\left({\beta }^2-4\right)\right]+\alpha \left\{-5\alpha {\beta }^2+\sigma \left[-16+4{\alpha }^2\left({\beta }^2-4\right)\right]\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(23)

$$D_d^{f\ast }=\frac{c\left\{\left[32+6\left(3{\beta }^2-8\right)\right]+\alpha \left(-16+11{\beta }^2\right)+2{\alpha }^2\left(8-5{\beta }^2\right)\right\}+\alpha \left[9{\beta }^2-4\left({\beta }^2-8\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(24)

Substituting Equations (18)–(21) and (22)–(24) into Equations (12) and (13) shows that the optimal profit of the retailers and manufacturers under the decentralized decision scenarios is

$${\pi }_r^{f\ast }=\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }$$

(25)

$${\pi }_d^{f\ast }=\frac{\left\{a^2\left[-3+6s\left(s-1\right)\left(\alpha -1\right)\right]-6c^2\left({\alpha }^2-1\right)+{\beta }^2{\left[c+a\left(s-1\right)-ca\right]}^2\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}+\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{8\beta }$$

(26)

By adding ${\pi }_r^{f\ast }$ and ${\pi }_d^{f\ast }$ in Equations (25) and (26), the overall optimal profit of the supply chain under the decentralized decision is

$${\pi }^{f\ast }=\frac{\left\{a^2\left[-3+6s\left(s-1\right)\left(\alpha -1\right)\right]-6c^2{\left(\alpha -1\right)}^2\left(1+\alpha \right)-6ac\left({\alpha }^2-1\right)+{\beta }^2{\left[c+a\left(s-1\right)-ca\right]}^2\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}+\frac{3{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }$$

(27)

According to the calculations of Equations (11) and (27), there is a difference in the overall optimal profit of the supply chain under the concentrated scenario and the dispersed scenario. Moreover, the overall optimization profit of the supply chain in the centralized decision scenario is greater than the overall optimization profit under the decentralized decision scenario, which is

$${\pi }^{j\ast }-{\pi }^{f\ast }=\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{4\beta }-\frac{3{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }=\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }>0$$

(28)

Equation (16) shows that the optimal profit of the supply chain is the overall optimal profit of the supply chain under the centralized decision scenario.

4. Two-Step Pricing Decision Models under the Coordinated Decisions of a Dual-Channel Supply Chain

Because there are phenomena in the dual-channel supply chain that aim to maximize the member’s respective interests, there is a conflict between the members of the supply chain and the overall optimal profit. Equations (7)–(10) and (18)–(21) show that the manufacturers and retailers differ in the optimal selling price, which is concentrated in the decentralized decision scenario. Equations (7)–(10) and (18)–(21) show that the retailer’s promotion level under the centralized decision-making scenario is less than that under the decentralized decision-making scenario. Therefore, it can be concluded that, in the case of cooperation between the two parties, unnecessary expenditures can be reduced, and the income of both parties can be increased. At the same time, the overall optimal profit of the supply chain under decentralized decision making is equal to the total profit of the supply chain under centralized decision making. For this purpose, we design two-step pricing contracts where the manufacturer sets the wholesale price of the market according to the market environment and then sets the direct selling price of the online channel according to the actual situation. It can be seen from Equation (14) that the retailer’s optimal response strategy is

$$p_r^{L\ast }=\frac{as+w+u\beta +\alpha p_d^{L\ast }}{2}$$

(29)

The main goal of dual-channel supply chain coordination is to make the overall optimal profit of the supply chain under the decentralized decision scenario equal to the optimal total profit of the supply chain under the centralized decision scenario. Therefore, make the optimal selling price of the retailer under the two-step pricing scenarios equal to the retail price of the retailer under the concentrated situation, and, at the same time, make the optimal selling price of the manufacturer under the two-step pricing equal to the optimal selling price of the manufacturer under the concentrated situation. That is,

$$p_r^{L\ast }=p_r^{j\ast }$$

(30)

$$p_d^{L\ast }=p_d^{j\ast }$$

(31)

The optimal wholesale price and promotion level for the two-step pricing cases are

$$w^{L\ast }=\frac{c\left[a^2+\left(\alpha -2\right)\left({\beta }^2-3\right)+{\alpha }^2\left({\beta }^2+6\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(32)

$$u^{L\ast }=\frac{\beta \left[a\left(s+\alpha -s\alpha \right)+c\left({\alpha }^2-1\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(33)

$$v^{L\ast }=\frac{\beta \left[a\left(s+\alpha -s\alpha \right)+2c\left({\alpha }^2-1\right)\right]}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(34)

By comparing Equations (7)–(10) and (32)–(34), it could be found that the retailer’s promotion level in the two-step pricing scenarios is equal to the retailer’s optimal promotion level under the centralized scenario, while the manufacturer’s promotion compensation fee for the retailer is the same under the two-step pricing and concentration scenarios. It is considered that two-step pricing can coordinate the profit of the dual-channel members considering the promotion level of the retailer and the manufacturer’s promotion compensation. Therefore, according to the above method of solving the optimal equilibrium solution using the decentralized decision, Equations (12) and (13) are substituted into the corresponding variables of Equations (32)–(34) to obtain the optimal profits of the retailers and manufacturers under the two-step pricing scenario as follows:

$${\pi }_r^{L^{\ast }}=\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{4\beta }$$

(35)

$${\pi }_d^{L^{\ast }}=\frac{\left\{a^2\left[-3+6s\left(s-1\right)\left(\alpha -1\right)\right]-6c^2\left(\alpha -1\right)2\left(1+\alpha \right)-6ac\left({\alpha }^2-1\right)+{\beta }^2{\left[c+a\left(s-1\right)-ca\right]}^2\right\}}{2\left(-3+3{\alpha }^2+{\beta }^2\right)}$$

(36)

5. Changes in Supply Chain Profit

After using two-step pricing contracts, Equation (13) shows that there is a significant change in profits between the manufacturers and retailers in the profit and decentralized decision scenarios, and the retailer’s profit changes are as follows:

$$\Delta {\pi }_r^L={\pi }_r^{L\ast }-{\pi }_r^{f\ast }=\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{4\beta }-\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }=\frac{3{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }>0$$

(37)

It can be seen from Equation (37) that the retailer’s profit can be changed by two-step pricing, and the added value of the retailer’s profit is

$$A=\frac{3{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }$$

(38)

The profit change obtained by the manufacturer is

$$\Delta {\pi }_d^L={\pi }_d^{L\ast }-{\pi }_d^{f\ast }=-\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{8\beta }<0$$

(39)

It can be seen from Equation (38) that the manufacturer’s profit reduction is

$$B=\frac{{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{8\beta }$$

(40)

Coordinating the decision-making mechanism achieves a win–win situation for the supply chain members, because the manufacturer’s profit is reduced in this contract decision scenario. In order to ensure that the two-step pricing contracts in the supply chain work better, we design a manufacturer-led transfer payment mechanism. The mechanism is that, when signing the contract, the retailer needs to pay a fixed fee to the manufacturer to make up for the manufacturer’s loss. Now, if the fixed fee is $G$ ($G\in \left(B,A\right)$), the profit of both parties can better achieve Pareto optimality.

Because, when the fixed fee is G,

$\Delta {\pi }_r^L=\frac{3{\left[sa+c\left(3\alpha -\beta \right)\right]}^2}{16\beta }>0$, which is, after two-step pricing contracts, the retailer’s profits increase.

When $-\frac{\left[sa+c\left(3\alpha -\beta \right)\right]2}{8\beta }<0$, after two-step pricing contracts, the profit of the manufacturer is reduced.

Thus, we obtain the optimal value of the retailer’s profit after the coordination of the two-step pricing contracts is ${\pi }_r^{L^{\ast }}-G$, and we obtain the optimal value of the manufacturing profit after the coordination of the two-step pricing contracts is ${\pi }_d^{L^{\ast }}+G$.

Under this decision mechanism, the sum of the optimal profit of the manufacturer and the retailer after the coordination of the two-step pricing contracts is equal to the overall optimal profit of the supply chain under the centralized scenario. The optimal profit of the supply chain under the two-step pricing contracts is equal to the overall optimal profit of the supply chain. When the fixed cost is $G$, the profit of both parties better achieves Pareto optimality.

Based on the above, in the dual-channel supply chain sales process, considering the promotion level and promotion compensation, the “bilateral effect” between channel members always exists. After using two-step pricing contracts, the manufacturer’s profit is reduced. Retailers need to pay a fixed fee to make up for the manufacturer’s losses so that the overall profit of the supply chain can be coordinated.

6. Numerical Results and Analysis

In order to verify the applicability and effectiveness of the model, the following numerical calculations and results analysis were performed on the manufacturer-led dual-channel supply chain coordination decision model:

The related coefficient parameter settings in the text are as follows: $a=10000$ (unit: piece), $c=10$ (unit: Yuan, RMB), $\alpha =\beta =10$. These parameter settings were analyzed in two situations, and the result of the calculation retains two decimal places.

Situation 1: The fixed cost is $G$ = 4 (unit: 10,000 Yuan, RMB), and the market share $s$ is taken from [0.1–0.8] at intervals of 0.1, taking 8 points. The optimal profit values are shown Table 1 and Figure 2 when the market share changes.

It can be seen in

Table 2. The optimal profit values when the market share changes.

$\mathit{s}$	$\mathit{G}$	${\mathit{\pi }}^{{\mathit{j}}^{\ast }}$	${\mathit{\pi }}_{\mathit{r}}^{{\mathit{f}}^{\ast }}$	${\mathit{\pi }}_{\mathit{d}}^{{\mathit{f}}^{\ast }}$	${\mathit{\pi }}^{{\mathit{f}}^{\ast }}$	${\mathit{\pi }}_{\mathit{r}}^{{\mathit{L}}^{\ast }}$	${\mathit{\pi }}_{\mathit{d}}^{{\mathit{L}}^{\ast }}$	${\mathit{\pi }}^{{\mathit{L}}^{\ast }}$
0.1	4	9.38	0.01	9.36	9.35	0.01	9.37	9.38
0.2	4	6.82	0.03	6.76	6.73	0.08	6.74	6.82
0.3	4	4.71	0.06	4.58	4.51	0.22	4.49	4.71
0.4	4	3.03	0.11	2.81	2.70	0.40	2.63	3.03
0.5	4	1.79	0.17	1.45	1.28	0.64	1.15	1.79
0.6	4	0.98	0.24	0.50	0.26	0.92	0.06	0.98
0.7	4	0.62	0.32	−0.03	−0.35	1.26	−0.64	0.62
0.8	4	0.69	0.42	−0.15	−0.57	1.64	−0.95	0.69

and Figure 2 that, as a whole, when the fixed cost of $G$ is unchanged, the change in the market share $s$ affects the overall profit of the manufacturers, retailers, and supply chains. Figure 2a shows the change in the optimal profit value with a market share of $s$ in the case of decentralized decision making and two-step pricing. When $s$ increases, that is, the market share of the retailer in the offline channel increases, the market share of the manufacturer decreases. Therefore, the optimal profit of the manufacturers in both scenarios shows a decreasing trend, and in the decentralized scenario, the profit of the manufacturer is always greater than the optimal profit of the manufacturer under two-step pricing. This means that the manufacturer’s profit is reduced after coordination, indicating the validity of Equation (39). Figure 2b shows that, when the market share $s$ increases, the retailer’s market share becomes larger. The retailer’s optimal profit value shows an increasing trend in the case of decentralized decision making and two-step pricing, and the retail optimal profit is always greater than the optimal profit in the dispersed scenario under the two-step pricing scenarios. That is, after the coordination of two-step pricing, the retailer’s profit increases, which demonstrates the effectiveness of Equation (37). Figure 2c shows that the market share of $s$ becomes larger, and the overall profit of the supply chain under the concentrated and dispersed scenario shows a decreasing trend, which indicates that the active channel in the e-commerce era increases the overall profit of the supply chain. After the two-step pricing coordination, the optimal profit is equal to the centralized decision-making optimal profit, and the overall profit of the supply chain under the centralized decision-making scenario is always greater than the distributed decision-making supply chain profit.

Situation 2: The market share $s$ = 0.5, and the fixed fee $G$ is given by the retailer from [4–4.7] (unit: 10,000 Yuan, RMB) at intervals of 0.1, taking 8 points. The optimal profit values are shown

Table 3. The optimal profit values when the fixed costs change (unit: million yuan).

$\mathit{s}$	$\mathit{G}$	${\mathit{\pi }}^{{\mathit{j}}^{\ast }}$	${\mathit{\pi }}_{\mathit{r}}^{{\mathit{f}}^{\ast }}$	${\mathit{\pi }}_{\mathit{d}}^{{\mathit{f}}^{\ast }}$	${\mathit{\pi }}^{{\mathit{f}}^{\ast }}$	${\mathit{\pi }}_{\mathit{r}}^{{\mathit{L}}^{\ast }}$	${\mathit{\pi }}_{\mathit{d}}^{{\mathit{L}}^{\ast }}$	${\mathit{\pi }}^{{\mathit{L}}^{\ast }}$
0.5	4	1.79	0.17	1.45	1.28	0.66	0.64	1.79
0.5	4.1	1.79	0.17	1.45	1.28	0.65	1.14	1.79
0.5	4.2	1.79	0.17	1.45	1.28	0.65	1.14	1.79
0.5	4.3	1.79	0.17	1.45	1.28	0.64	1.15	1.79
0.5	4.4	1.79	0.17	1.45	1.28	0.64	1.15	1.79
0.5	4.5	1.79	0.17	1.45	1.28	0.64	1.15	1.79
0.5	4.6	1.79	0.17	1.45	1.28	0.63	1.16	1.79
0.5	4.7	1.79	0.17	1.45	1.28	0.63	1.16	1.79

and Figure 3 when the fixed costs change.

It can be seen in Table 3 and Figure 3 that, when the fixed payment cost $G$ changes and the market share $s$ remains unchanged, the overall profit of the manufacturer, the retailer, and the supply chain changes. In Figure 3a, when the fixed cost G increases, the optimal profit of the manufacturer in the decentralized case remains unchanged, and the manufacturer’s profit increases after the two-step pricing coordination. Figure 3b shows that, when the fixed cost G increases, the retailer’s optimal profit in the decentralized case remains unchanged, and the two-step pricing coordination relatively reduces the retailer’s profit. Figure 3a,b show that the retailer’s and manufacturer’s profit changes are in line with the coordination goal. Figure 3c shows that, when the fixed payment cost changes and the market share is constant, the optimal profit of the supply chain under the decentralized decision is still less than the optimal profit of the supply chain under the centralized decision. After the coordination of the two-step pricing, the optimal profit after the coordination is equal to the optimal profit of the supply chain and achieves the coordination goal.

7. Conclusions

This paper takes a manufacturer-led dual-channel supply chain as the research object and considers the promotion level of offline physical stores and the promotion compensation of online channel manufacturers and other factors. By establishing a two-step pricing decision model, this paper focuses on the “bilateral effect” pricing decision problem of the channel and proves the validity of the model through a numerical analysis. The research results and contributions of this paper are as follows:

(1) After the combination of two-step pricing and fixed costs are paid by retailers, the profit of the supply chain members can be achieved by Pareto optimization.

(2) When the market share and fixed payment costs change, the overall optimal profit of the supply chain under centralized decision making is always better than the optimal profit under decentralized decision making.

(3) Considering the promotion level of offline physical stores and the promotion compensation of online channel manufacturers, the overall optimization profit of the supply chain under decentralized decision-making scenario is still less than that under the centralized decision-making scenario. At this time, the single two-step pricing decisions cannot coordinate the channel members’ profits. After the two-step pricing coordination, the retailer pays a fixed fee to the manufacturer to achieve a win–win situation for the supply chain members.

(4) In the case of decentralized decision making, the retailer’s promotion fee and the manufacturer’s promotion compensation are greater than the cost under centralized decision making. The coordination of the two-step pricing can make the retailer’s promotional expenses and the manufacturer’s promotional compensation the same as those under the centralized decision.

We only study the two-tier pricing decision model of a manufacturer-led dual-channel supply chain based on the previous literature considering the retailer’s promotion level and the manufacturer’s promotion compensation factors for retailers. The pricing coordination decision model for retailer-led, multi-manufacturer, and multi-retailer supply chains will be a topic of further study.