# Strategic Customer Behavior and Pricing Strategy Based on the Horizontal Differentiation of Products

Authors: Yang Zhao; Guojun Ji; Yue Jiang; Xiaopei Dai
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Manufacturers produce products with horizontal differences to meet different needs of customers. This paper compares the influence of three different sales channels on strategic customers’ choice and the pricing strategy of products with horizontal differentiation. The results show that the strategic customers whose willingness to pay (WTP) is close to 1 will buy high-performance products and whose WTP is close to 0 will not purchase any kind of products in the two dual-channel models. If the manufacturers adopt dual channel to sell products with horizontal differences, the retailers agree that the manufacturers sell high-performance products in the traditional channel and sell low-performance products in the electronic channel. In dual-channel supply chain model I, the higher the satisfaction of high-performance products and the lower the satisfaction of low-performance products, the more conducive to the retailers.


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## Body

## 1. Introduction

In recent years, the rapid development of the logistics industry and electronic channels has brought new opportunities to the traditional manufacturers. The electronic channel acquaints the customer with the characteristics, price, and third-party evaluations of the products. Customers can obtain product information conveniently through the electronic channel. The reduced cost of searching for information and discrepant channel price lead to free-rider behavior between channels. The customers acquire information from the retailer and then purchase products at low prices in the electronic channel, which provides limited service. This is called free-rider behavior. Free-rider behavior between channels can operate in both directions. Customers can also search product information in the electronic channel, such as performance parameters, and choose to purchase in the retail channel [1]. Strategic customers select the buying channel according to the surplus utility maximization, known as the channel behavior of strategic customers. The consequences of widespread free-rider behavior are serious because free-rider behavior does not achieve the group goal or the optimal goals. And the free-rider behavior does not fully utilize limited resources. More seriously, this behavior can spread among customers and damage retailer profits [2].How to alleviate the conflicts between channels has become a hot topic among scholars. A manufacturer can sell the same products in electronic and retail channels and also sell two kinds of products, which have different performances. In the multichannel, product differentiation strategy is not only the key factor in merchants’ game but also the role of alleviating channel conflicts has been concerned by scholars at home and abroad. In the existing product differentiation model, the high differentiation will weaken the market competition and break the original stable Nash equilibrium [3–5]. The manufacturer producing two differentiated products can sell in these models as follows: first, the single retail channel sells these differentiated products. Second, the manufacturer sells the lower performance product in its own electronic channel and sells the higher performance product in the retail channel; we call this dual-channel model I. Third, the manufacturer sells the higher performance product in its own electronic channel and sells the lower performance product in the retail channel; we call this dual-channel model II.In this paper, we seek to get optimal prices in the above three models and analyze how the satisfaction of strategic customers with differentiated products affects the optimal decisions. To answer these questions, we review the related literature and introduce our research method as follows.


## 2. Literature Review

For our research, the literature review can be divided into three parts: the product differentiation, the dual-channel, and the strategic customer behavior. We elaborate on them below:(a)
Products differentiation: initially, the most researches focused on the two manufacturers selling differentiated products through traditional channels. Gabszewicz and Thisse established a duopoly model and pointed out that when the income gap among customers is large enough, the two enterprises will expand the gap in product quality to avoid fierce price competition and improve their respective profits [6]. Shake and Sutton further pointed out that when there were only two manufacturers in the market, manufacturers producing high-quality products would make higher profits [7]. Moorthy and Png found that in order to avoid the market share of high-quality products being plundered by low-quality products, manufacturers would introduce high-quality products first and then low-quality products [8]. At the same time, some scholars found that when the market share of low-quality products is larger or the high-quality products have substitutes, they will first introduce low-quality products and then high-quality products [9, 10]. Zhao et al. discussed the pricing strategy and product differentiation strategy, and the pricing strategy is easier to change, belonging to the short-term market behavior, and differentiation strategy belongs to the long-term market behavior [11]. Liu and Zhang studied a dynamic pricing competition model which retailers offer vertically differentiated products to strategic customers, focusing on the role of product performance and the value of price commitment [12]. When manufacturers produce similar or differentiated products and sell them at different prices in diversified markets, Kim and Bell concluded that revenue managers and supply-chain coordinators set prices and product quantities according to the costs of each market segment [13].(b)
Differentiated products in dual channel: With the change of sales patterns, Yan found when customers are insensitive to prices, the greater the product differentiation, the more conducive to increase corporate profits [14]. Chen and Liu pointed out that greater degree of product differentiation can bring higher profits to the manufacturers who dominate the game in the dual-channel supply chain [15]. Andaluz and Jarne defined a demand model about differentiated products and found that if the complementary products and substituted products exist, the stability of Nash equilibrium will improve with the increase of the product independence [16]. Freyberger develops asymptotic theory in differentiated product demand systems according to a small number of products and a large number of markets [17]. However, few of the above literatures deal with the strategic customers. Chen et al. concerned price and quality decisions in dual-channel supply chains and discussed the effects of the quality sensitivity parameters of different channels on price and product quality, as well as profits and consumer surplus [18]. Huang et al. considered the parallel importers (PIs) and developed a game-theoretic model for a dual-channel supply chain, and the results suggested that the manufacturer should choose to design the lower quality products to weaken the PI’s competitiveness [19].(c)
Strategic customer behavior: in the dual-channel supply chain, the free-rider behavior of customers is actually a kind of strategic behavior. When the sales services and the sales offered by retailers themselves can be separated, and the different retailers provide different services and prices, then this will result in the free-rider behavior. It can damage the profits of the retailer that provide high-quality services [20]. Cubitt et al. analyzed how social preferences affect free-rider motivation and behavior directly or indirectly. Some scholars design the experiment about free-rider behavior to test the effectiveness of the control mechanism [21]. For example, Ertan et al. studied the punishment mechanism [22], Coats et al. researched the compensation mechanism [23], and Kiyonari and Barclay discussed the reward mechanism [24]. The literature above finds that punishment and reward mechanisms can significantly control the free-rider behavior, while the compensation mechanism reduces free riding only in simultaneous institution. Scholars study the problems of information free-rider behavior and of coordinating contract design between a single manufacturer and retailer. For example, Ding and Liu discussed the optimal prices when free-rider behavior exists within centralized and decentralized structures and found the revenue-sharing contract can achieve dual-channel supply-chain coordination [25].Huang and Yang pointed out that retailers set different prices during normal and discounted sales periods and influence strategic customer behavior through capacity selection. Strategic customers choose the best purchase opportunity according to price and capacity [26]. Bi et al. held that strategic customer behavior would reduce the additional revenue obtained through dynamic pricing [27]. Li and Wei studied the strategic customer behavior and the price guarantee contract. The results show that the strategic customer behavior will reduce the supply-chain profits and the price guarantee contract can alleviate but cannot eliminate the strategic customer behavior completely [28]. Correa et al. found that retailers can publish product prices ahead of time to alleviate strategic customer behavior [29]. Ji et al. studied the effect of strategic customer behavior on the performance of manufacturers and retailers. The results show that strategic customers are more inclined to the retailer-dominated market with lower prices. If enterprises face up to strategic customer behavior, they can achieve a win-win situation [30]. Chen constructed a supply-chain coordination model for short-life-cycle products based on strategic customer behavior from the perspective of rational expected utility theory and prospect theory [31]. Zhang et al. studied the retailer’s return policy with strategic customers and pointed out that only when the return channel is more efficient, the retailer accepting the returned products and selling them in the normal sales period can bring higher profits [32]. Some scholars begin to integrate strategic customer behavior and product differentiation. Parlaktuk studied the value of manufacturers adopting product differentiation strategy based on strategic customer behavior [33]. Yang and Ji discussed the intertemporal choice of strategic customers between the original products of the advanced market entrants and the imitated products of the late market entrants [34].The above literatures either focus on the combination of dual channel and product differentiation, or consider the combination of the strategic customer behavior and product differentiation and rarely studied product differentiation strategy in dual channel based on strategic customer behavior. And most of the literatures analyze vertical differentiation strategy, which is products having quality differences. However, this paper discusses the product horizontal differences, which is put forward by Hotelling, i.e., some customers prefer high-performance products, some customers feel that low-performance products can meet their needs, and the products themselves do not exhibit quality differences. Under the background of horizontal differences, which is expressed by high-performance productH and low-performance product L, this paper studies the choice behavior of strategic customers and the corresponding optimal pricing strategy in the single traditional channel and dual channel, respectively. The purpose is to make manufacturers and retailers understand strategic customer behavior deeply and better design sales channels and pricing strategy.


## 3. Models

### 3.1. Sell ProductsH and L in the Single Retail Channel

This model includes one manufacturer, one retailer, and strategic customers. We useH to express the higher performance products and L to express the lower performance products. The manufacturer sells products H and L in a single retail channel. The sales process is shown in Figure 1. Assume PH represents the price of higher performance products and PL represents the price of lower performance products, and PH>PL.Figure 1
Single retail channel model.The willingness to pay of strategic customers is represented byx∈0,1, subjected to uniform distribution [35]. The satisfaction of strategic customers about products H is denoted by θH and about products L is denoted by θL. The parameters should meet the criteria 0<θL<θH<1. The utility function is UH=θHx−PH for buying the higher performance products and UL=θLx−PL for buying the lower performance products. The selection function of strategic customers is V=maxUH,UL,0=maxθHx−PH,θLx−PL,0, and then we can get the following expression by comparative analysis:(1)V=θHx−PH,x≥PH−PLθH−θL,x≥PHθH,θLx−PL,x≤PH−PLθH−θL,x≥PLθL,0,x≤PHθH,x≤PLθL.When the parameters meet the conditionPL/θL≤PH/θH, i.e., PH−PL/θH−θL≥PH/θH≥PL/θL, the expression (1) can be derived as follows:(2)V=θHx−PH,x≥PH−PLθH−θL,θLx−PL,PLθL≤x≤PH−PLθH−θL,0,x≤PLθL.When the parameters meet the conditionPL/θL≥PH/θH, i.e., PH−PL/θH−θL≤PH/θH≤PL/θL, the expression (1) is changed into (3). The strategic customers will not purchase the lower performance products L:(3)V=θHx−PH,x≥PHθH,0,x≤PHθH.Lemma 1.
The strategic customers can select to buy the lower performance productsL, when 0≤PL/θL≤PH/θH≤PH−PL/θH−θL≤1.The selections of strategic customers under expression (2) are discussed, and the details are illustrated in Figure 2.The number of strategic customers that choose to buy the higher performance products isQH=∫PH−PL/θH−θL1dx=1−PH−PL/θH−θL, while the number that chooses to purchase the lower performance products is QL=∫PL/θLPH−PL/θH−θLdx=PH−PL/θH−θL−PL/θL, and the number that does not buy any products is Q0=PL/θL.Figure 2
The decision of strategic customers.Lemma 2.
When WTP is close to 1, the best choice for strategic customers is to buy the higher performance products. When WTP nears zero, strategic customers will leave the market.


#### 3.1.1. Decentralized Decision

The production costs of the higher performance products areCH¯ and the lower performance products are CL¯. The wholesale prices of these two products are represented by WH and WL, respectively. Suppose the manufacturer is the leader and the retailer is the follower, and the two sides follow the Stackelberg game. According to the choices of strategic customers, the profit function of the retailer is(4)ΠR=QHPH−WH+QLPL−WL=1−PH−PLθH−θLPH−WH+PH−PLθH−θL−PLθLPL−WL.The profit function of the manufacturer is(5)ΠM=QHWH−CH¯+QLWL−CL¯=1−PH−PLθH−θLWH−CH¯+PH−PLθH−θL−PLθLWL−CL¯.Proposition 1.
The manufacturer sells higher performance productsH and lower performance products L via the single retail channel, and the optimal prices are WH∗=CH¯+θH/2, WL∗=CL¯+θL/2, PH∗=CH¯+3θH/4, and PL∗=CL¯+3θL/4. The retailer profit is ΠR⟶DD∗=θH−θL−CH¯+CL¯θH−CH¯/16θH−θL+θLCH¯−θHCL¯θL−CL¯/16θLθH−θL, and the manufacturer profit is ΠM⟶DD∗=θH−θL−CH¯+CL¯θH−CH¯/8θH−θL+θLCH¯−θHCL¯θL−CL¯/8θLθH−θL=2ΠR⟶DD∗.Substitute the optimal prices into the expression of the strategic customer number, and we can getQH=1/4−CH¯−CL¯/4θH−θL, QL=θLCH¯−θHCL¯/4θLθH−θL, and the number of strategic customers who do not buy any products is Q0=3/4+CL¯/4θL.


#### 3.1.2. Centralized Decision

The profit function of the supply chain isΠZ=QHPH−CH¯+QLPL−CL¯=1−PH−PL/θH−θLPH−CH¯+PH−PL/θH−θL−PL/θLPL−CL¯.Proposition 2.
If the single retail channel supply chain is centralized, the optimal price of higher performance products isPZH∗=CH¯+θH/2, and the optimal price of lower performance products is PZL∗=CL¯+θL/2. The maximum profit of the supply chain is ΠZ⟶CD∗=θLCH¯2+θHCL¯2−2θLCH¯CL¯−2θLθH−θLCH¯+θHθLθH−θL/4θLθH−θL.Put the optimal prices into the expression of the strategic customer number, and we can getQH=1/2−CH¯−CL¯/2θH−θL, QL=CH¯−CL¯/2θH−θL−CL/2θL, and the number of strategic customers who do not buy any products is Q0=1/2+CL/2θL.


### 3.2. Sell ProductH in Retail Channel and Product L in Electronic Channel

In dual-channel supply-chain model I, the manufacturer sells productsH via the retail channel and sells products L in the electronic channel. The sales process is shown in Figure 3.Figure 3
The dual-channel supply-chain model I.The price of the retail channel is represented byPS and of the electronic channel is represented by PN. The prices meet the criteria PS>PN. The preference degree of strategic customers about the electronic channel is denoted by μ, and 0<μ<1. The utility function is US=θHx−PS for buying the higher performance products via the retail channel and UN=μθLx−PN for buying the lower performance products via the electronic channel. The selection function of strategic customers is V=maxUS,UN,0=maxθHx−PS,μθLx−PN,0, and then we can get the following expression by comparative analysis:(6)V=θHx−PS,x≥PS−PNθH−μθL,x≥PSθH,μθLx−PN,x≤PS−PNθH−μθL,x≥PNμθL,0,x≤PSθH,x≤PNμθL.When the parameters meet the conditionPN/μθL≤PS/θH, i.e., PS−PN/θH−μθL≥PS/θH≥PN/μθL, the expression (6) can be derived as follows:(7)V=θHx−PS,x≥PS−PNθH−μθL,μθLx−PN,PNμθL≤x≤PS−PNθH−μθL,0,x≤PNμθL.When the parameters meet the conditionPN/μθL≥PS/θH, i.e., PS−PN/θH−μθL≤PS/θH≤PN/μθL, the expression (6) is changed into (8). The strategic customers do not buy products via the electronic channel:(8)V=θHx−PS,x≥PSθH,0,x≤PSθH.Lemma 3.
If the strategic customers can buy the lower performance products via the electronic channel, it needs to meet the prerequisite:0≤PN/μθL≤PS/θH≤PS−PN/θH−μθL≤1.The selections of strategic customers under expression (7) are discussed, and the details are illustrated in Figure 4.The number of strategic customers that choose to buy the higher performance products via the retail channel isQSH=∫PS−PN/θH−μθL1dx=1−PS−PN/θH−μθL, while the number that choose to purchase the lower performance products via the electronic channel is QNL=∫PN/μθLPS−PN/θH−μθLdx=PS−PN/θH−μθL−PN/μθL, and the number that does not buy any products is Q0=PN/μθL.Figure 4
The decision of strategic customers.Lemma 4.
When WTP is close to 1, the best choice for strategic customers is to buy the higher performance products in the retail channel. When WTP nears zero, strategic customers do not buy any products.


#### 3.2.1. Decentralized Decision

According to the choices of strategic customers, the profit function of the retailer is(9)ΠRHL=QSHPS−WH=1−PS−PNθH−μθLPS−WH.The profit function of the manufacturer is(10)ΠMHL=QSHWH−CH¯+QNLPN−CL¯=1−PS−PNθH−μθLWH−CH¯+PS−PNθH−μθL−PNμθLPN−CL¯.Proposition 3.
The manufacturer sells higher performance productH in the retail channel and lower performance product L via the electronic channel, and the optimal prices is WH∗=CH¯+θH/2, PNHL∗=CL¯+μθL/2, and PSHL∗=CH¯+CL¯+3θH−μθL/4. The retailer profit is ΠR⟶DDHL∗=θH−μθL−CH¯+CL¯2/16θH−μθL, and the manufacturer profit is ΠM⟶DDHL∗=CH¯2/8θH−μθL+2θH−μθLCL¯2/2μθLθH−μθL−CH¯CL¯/4θH−μθL−CH¯+CL¯/4+θH+μθL/8.Then substitute the optimal prices into the expression of the strategic customer number, and we can getQSH=1/4−CH¯−CL¯/4θH−μθL, QNL=1/4+μθLCH¯−2θH−μθLCL¯/4μθLθH−μθL, and the number of strategic customers who do not buy any products Q0=1/2+CL¯/2μθL.


#### 3.2.2. Centralized Decision

The profit function of the supply chain isΠZHL=QSHPS−CH¯+QNLPN−CL¯=1−PS−PN/θH−μθLPS−CH¯+PS−PN/θH−μθL−PN/μθLPN−CL¯.Proposition 4.
If the dual-channel supply chain is centralized, the optimal price of higher performance products in the retail channel isPZSHL∗=CH¯+θH/2, and the optimal price of lower performance products in electronic channel is PZNHL∗=CL¯+μθL/2. The maximum profit of the supply chain is ΠZ⟶CDHL∗=CH¯2/4θH−μθL+θHCL¯2/4μθLθH−μθL−CH¯CL¯/2θH−μθL−CH¯/2+θH/4.Put the optimal prices into the expression of strategic customer number, and we can getQSH=1/2−CH¯−CL¯/2θH−μθL, QNL=CH¯−CL¯/2θH−μθL−CL¯/2μθL, and the number of strategic customers who do not buy any products is Q0=1/2+CL¯/2μθL.


### 3.3. Sell ProductL in Retail Channel and Product H in Electronic Channel

In the dual-channel supply-chain model II, the manufacturer sells productsL via the retail channel and sells products H in the electronic channel. The sales process is shown in Figure 5.Figure 5
The dual-channel supply-chain model II.The utility function for buying the higher performance products via the retail channel isUS=θLx−PS and for buying the lower performance products via the electronic channel is UN=μθHx−PN. Suppose sale prices meet the requirement PN>PS. The selection function of strategic customers is V=maxUS,UN,0=maxθLx−PS,μθHx−PN,0.When the parameters meet the conditionPS/θL≤PN/μθH, i.e., PN−PS/μθH−θL≥PN/μθH≥PS/θL, the selection function can be derived as follows:(11)V=θLx−PS,PSθL≤x≤PN−PSμθH−θL,μθHx−PN,x≥PN−PSμθH−θL,0,x≤PSθL.When the parameters meet the conditionPS/θL≥PN/μθH, i.e., PN−PS/μθH−θL≤PN/μθH≤PS/θL, the selection function is changed into (12). The strategic customers do not buy lower performance products in the retail channel:(12)V=μθHx−PN,x≥PNμθH,0,x≤PNμθH.Lemma 5.
If the strategic customers can purchase products from the retail channel and electronic channel, it needs to meet the prerequisite:0≤PS/θL≤PN/μθH≤PN−PS/μθH−θL≤1.The selections of strategic customers under expression (11) are discussed, and the details are illustrated in Figure 6.The number of strategic customers that chooses to buy the higher performance products via the electronic channel isQNH=∫PN−PS/μθH−θL1dx=1−PN−PS/μθH−θL, while the number that chooses to purchase the lower performance products via the retail channel is QSL=∫PS/θLPN−PS/μθH−θLdx=PN−PS/μθH−θL−PS/θL, and the number that does not buy any products is Q0=PS/θL.Figure 6
The decision of strategic customers.Lemma 6.
When WTP is close to 1, the best choice for strategic customers is to buy the higher performance products in the electronic channel. When WTP nears zero, the strategic customers do not buy any products.


#### 3.3.1. Decentralized Decision

According to the choices of strategic customers, the profit function of the retailer is(13)ΠRLH=QSLPS−WL=PN−PSμθH−θL−PSθLPS−WL.The profit function of the manufacturer is(14)ΠMLH=QSLWL−CL¯+QNHPN−CH¯=PN−PSμθH−θL−PSθLWL−CL¯+1−PN−PSμθH−θLPN−CH¯.Proposition 5.
The manufacturer sells lower performance productL in the retail channel and higher performance product H via electronic channel, and the optimal prices are WL∗=CL¯+θL/2, PNLH∗=CH¯+μθH/2, and PSLH∗=θLCH¯+μθHCL¯+2μθHθL/4μθH. The retailer profit is ΠR⟶DDLH=θLCH¯−μθHCL¯2/16μθHθLμθH−θL, and the maximum profit of the manufacturer is ΠM⟶DDLH=2μθH−θLCH¯2/8μθHμθH−θL+μθHCL¯2/8θLμθH−θL−CH¯CL¯/4μθH−θL−CH¯/2+μθH/4.Then, substitute the optimal prices into the expression of the strategic customer number, and we can getQNH=1/2−2μθH−θLCH¯−μθHCL¯/4μθHμθH−θL, QSL=θLCH¯−μθHCL¯/4θLμθH−θL, and the number of strategic customers who do not buy any products is Q0=1/2+θLCH¯+μθHCL¯/4μθHθL.


#### 3.3.2. Centralized Decision

The profit function of the supply chain isΠZLH=QSLPS−CL¯+QNHPN−CH¯=PN−PS/μθH−θL−PS/θLPS−CL¯+1−PN−PS/μθH−θLPN−CH¯.Proposition 6.
If the dual-channel supply chain is centralized, the optimal price of lower performance products in the retail channel isPZSLH∗=CL¯+θL/2, and the optimal price of higher performance products in the electronic channel is PZNLH∗=CH¯+μθH/2. The maximum profit of the supply chain is ΠZ⟶CDHL∗=CH¯2/4μθH−θL+μθHCL¯2/4θLμθH−θL−CH¯CL¯/2μθH−θL−CH¯/2+μθH/4.Put the optimal prices into the expression of the strategic customer number, and we can getQNH=1/2−CH¯−CL¯/2μθH−θL, QSL=CH¯−CL¯/2μθH−θL−CL¯/2θL, and the number of strategic customers who do not buy any products is Q0=1/2+CL¯/2θL.Based on the above analysis, this paper compares the influence of parametersθH and θL on optimal pricing strategies and customer numbers in three different models. The change trend is shown in Table 1.Table 1
The influence of parameters on optimal price strategy and number of customers.
Decentralized decision
Centralized decision
Optimal pricing strategies
Customer numbers
Optimal pricing strategies
Customer numbers
θ
H
↗
Single retail channel model
W
H
∗
↗，PH∗↗， WL∗¯，PL∗¯
Q
H
↗，QL↘， Q0¯
P
Z
H
∗
↗， PZL∗¯
Q
H
↗，QL↘， Q0¯
Dual-channel model I
W
H
∗
↗，PSHL∗↗， PNHL∗¯
Q
S
H
↗，QNL↘， Q0¯
P
Z
S
H
L
∗
↗， PZNHL∗¯
Q
S
H
↗，QNL↘， Q0¯
Dual-channel model II
W
L
∗
¯，PSLH∗↘， PNLH∗↗
Q
S
L
↘，QNH↗， Q0↘
P
Z
S
L
H
∗
¯， PZNLH∗↗
Q
S
L
↘，QNH↗ Q0¯
θ
L
↗
Single retail channel model
W
H
∗
¯，PH∗¯， WL∗↗，PL∗↗
Q
H
↘，QL↗， Q0↘
P
Z
H
∗
¯， PZL∗↗
Q
H
↘，QL↗ Q0↘
Dual-channel model I
P
N
HL
∗
↗，PSHL∗↘， WH∗¯
Q
S
H
↘，QNL↗， Q0↘
P
ZS
HL
∗
¯， PZNHL∗↗
Q
S
H
↘，QNL↗， Q0↘
Dual-channel model II
W
L
∗
↗，PSLH∗↗， PNLH∗¯
Q
S
L
↗，QNH↘， Q0↘
P
Z
S
L
H
∗
↗， PZNLH∗¯
Q
S
L
↗，QNH↘， Q0↘


## 3.1. Sell ProductsH and L in the Single Retail Channel

This model includes one manufacturer, one retailer, and strategic customers. We useH to express the higher performance products and L to express the lower performance products. The manufacturer sells products H and L in a single retail channel. The sales process is shown in Figure 1. Assume PH represents the price of higher performance products and PL represents the price of lower performance products, and PH>PL.Figure 1
Single retail channel model.The willingness to pay of strategic customers is represented byx∈0,1, subjected to uniform distribution [35]. The satisfaction of strategic customers about products H is denoted by θH and about products L is denoted by θL. The parameters should meet the criteria 0<θL<θH<1. The utility function is UH=θHx−PH for buying the higher performance products and UL=θLx−PL for buying the lower performance products. The selection function of strategic customers is V=maxUH,UL,0=maxθHx−PH,θLx−PL,0, and then we can get the following expression by comparative analysis:(1)V=θHx−PH,x≥PH−PLθH−θL,x≥PHθH,θLx−PL,x≤PH−PLθH−θL,x≥PLθL,0,x≤PHθH,x≤PLθL.When the parameters meet the conditionPL/θL≤PH/θH, i.e., PH−PL/θH−θL≥PH/θH≥PL/θL, the expression (1) can be derived as follows:(2)V=θHx−PH,x≥PH−PLθH−θL,θLx−PL,PLθL≤x≤PH−PLθH−θL,0,x≤PLθL.When the parameters meet the conditionPL/θL≥PH/θH, i.e., PH−PL/θH−θL≤PH/θH≤PL/θL, the expression (1) is changed into (3). The strategic customers will not purchase the lower performance products L:(3)V=θHx−PH,x≥PHθH,0,x≤PHθH.Lemma 1.
The strategic customers can select to buy the lower performance productsL, when 0≤PL/θL≤PH/θH≤PH−PL/θH−θL≤1.The selections of strategic customers under expression (2) are discussed, and the details are illustrated in Figure 2.The number of strategic customers that choose to buy the higher performance products isQH=∫PH−PL/θH−θL1dx=1−PH−PL/θH−θL, while the number that chooses to purchase the lower performance products is QL=∫PL/θLPH−PL/θH−θLdx=PH−PL/θH−θL−PL/θL, and the number that does not buy any products is Q0=PL/θL.Figure 2
The decision of strategic customers.Lemma 2.
When WTP is close to 1, the best choice for strategic customers is to buy the higher performance products. When WTP nears zero, strategic customers will leave the market.


### 3.1.1. Decentralized Decision

The production costs of the higher performance products areCH¯ and the lower performance products are CL¯. The wholesale prices of these two products are represented by WH and WL, respectively. Suppose the manufacturer is the leader and the retailer is the follower, and the two sides follow the Stackelberg game. According to the choices of strategic customers, the profit function of the retailer is(4)ΠR=QHPH−WH+QLPL−WL=1−PH−PLθH−θLPH−WH+PH−PLθH−θL−PLθLPL−WL.The profit function of the manufacturer is(5)ΠM=QHWH−CH¯+QLWL−CL¯=1−PH−PLθH−θLWH−CH¯+PH−PLθH−θL−PLθLWL−CL¯.Proposition 1.
The manufacturer sells higher performance productsH and lower performance products L via the single retail channel, and the optimal prices are WH∗=CH¯+θH/2, WL∗=CL¯+θL/2, PH∗=CH¯+3θH/4, and PL∗=CL¯+3θL/4. The retailer profit is ΠR⟶DD∗=θH−θL−CH¯+CL¯θH−CH¯/16θH−θL+θLCH¯−θHCL¯θL−CL¯/16θLθH−θL, and the manufacturer profit is ΠM⟶DD∗=θH−θL−CH¯+CL¯θH−CH¯/8θH−θL+θLCH¯−θHCL¯θL−CL¯/8θLθH−θL=2ΠR⟶DD∗.Substitute the optimal prices into the expression of the strategic customer number, and we can getQH=1/4−CH¯−CL¯/4θH−θL, QL=θLCH¯−θHCL¯/4θLθH−θL, and the number of strategic customers who do not buy any products is Q0=3/4+CL¯/4θL.


### 3.1.2. Centralized Decision

The profit function of the supply chain isΠZ=QHPH−CH¯+QLPL−CL¯=1−PH−PL/θH−θLPH−CH¯+PH−PL/θH−θL−PL/θLPL−CL¯.Proposition 2.
If the single retail channel supply chain is centralized, the optimal price of higher performance products isPZH∗=CH¯+θH/2, and the optimal price of lower performance products is PZL∗=CL¯+θL/2. The maximum profit of the supply chain is ΠZ⟶CD∗=θLCH¯2+θHCL¯2−2θLCH¯CL¯−2θLθH−θLCH¯+θHθLθH−θL/4θLθH−θL.Put the optimal prices into the expression of the strategic customer number, and we can getQH=1/2−CH¯−CL¯/2θH−θL, QL=CH¯−CL¯/2θH−θL−CL/2θL, and the number of strategic customers who do not buy any products is Q0=1/2+CL/2θL.


## 3.1.1. Decentralized Decision

The production costs of the higher performance products areCH¯ and the lower performance products are CL¯. The wholesale prices of these two products are represented by WH and WL, respectively. Suppose the manufacturer is the leader and the retailer is the follower, and the two sides follow the Stackelberg game. According to the choices of strategic customers, the profit function of the retailer is(4)ΠR=QHPH−WH+QLPL−WL=1−PH−PLθH−θLPH−WH+PH−PLθH−θL−PLθLPL−WL.The profit function of the manufacturer is(5)ΠM=QHWH−CH¯+QLWL−CL¯=1−PH−PLθH−θLWH−CH¯+PH−PLθH−θL−PLθLWL−CL¯.Proposition 1.
The manufacturer sells higher performance productsH and lower performance products L via the single retail channel, and the optimal prices are WH∗=CH¯+θH/2, WL∗=CL¯+θL/2, PH∗=CH¯+3θH/4, and PL∗=CL¯+3θL/4. The retailer profit is ΠR⟶DD∗=θH−θL−CH¯+CL¯θH−CH¯/16θH−θL+θLCH¯−θHCL¯θL−CL¯/16θLθH−θL, and the manufacturer profit is ΠM⟶DD∗=θH−θL−CH¯+CL¯θH−CH¯/8θH−θL+θLCH¯−θHCL¯θL−CL¯/8θLθH−θL=2ΠR⟶DD∗.Substitute the optimal prices into the expression of the strategic customer number, and we can getQH=1/4−CH¯−CL¯/4θH−θL, QL=θLCH¯−θHCL¯/4θLθH−θL, and the number of strategic customers who do not buy any products is Q0=3/4+CL¯/4θL.


## 3.1.2. Centralized Decision

The profit function of the supply chain isΠZ=QHPH−CH¯+QLPL−CL¯=1−PH−PL/θH−θLPH−CH¯+PH−PL/θH−θL−PL/θLPL−CL¯.Proposition 2.
If the single retail channel supply chain is centralized, the optimal price of higher performance products isPZH∗=CH¯+θH/2, and the optimal price of lower performance products is PZL∗=CL¯+θL/2. The maximum profit of the supply chain is ΠZ⟶CD∗=θLCH¯2+θHCL¯2−2θLCH¯CL¯−2θLθH−θLCH¯+θHθLθH−θL/4θLθH−θL.Put the optimal prices into the expression of the strategic customer number, and we can getQH=1/2−CH¯−CL¯/2θH−θL, QL=CH¯−CL¯/2θH−θL−CL/2θL, and the number of strategic customers who do not buy any products is Q0=1/2+CL/2θL.


## 3.2. Sell ProductH in Retail Channel and Product L in Electronic Channel

In dual-channel supply-chain model I, the manufacturer sells productsH via the retail channel and sells products L in the electronic channel. The sales process is shown in Figure 3.Figure 3
The dual-channel supply-chain model I.The price of the retail channel is represented byPS and of the electronic channel is represented by PN. The prices meet the criteria PS>PN. The preference degree of strategic customers about the electronic channel is denoted by μ, and 0<μ<1. The utility function is US=θHx−PS for buying the higher performance products via the retail channel and UN=μθLx−PN for buying the lower performance products via the electronic channel. The selection function of strategic customers is V=maxUS,UN,0=maxθHx−PS,μθLx−PN,0, and then we can get the following expression by comparative analysis:(6)V=θHx−PS,x≥PS−PNθH−μθL,x≥PSθH,μθLx−PN,x≤PS−PNθH−μθL,x≥PNμθL,0,x≤PSθH,x≤PNμθL.When the parameters meet the conditionPN/μθL≤PS/θH, i.e., PS−PN/θH−μθL≥PS/θH≥PN/μθL, the expression (6) can be derived as follows:(7)V=θHx−PS,x≥PS−PNθH−μθL,μθLx−PN,PNμθL≤x≤PS−PNθH−μθL,0,x≤PNμθL.When the parameters meet the conditionPN/μθL≥PS/θH, i.e., PS−PN/θH−μθL≤PS/θH≤PN/μθL, the expression (6) is changed into (8). The strategic customers do not buy products via the electronic channel:(8)V=θHx−PS,x≥PSθH,0,x≤PSθH.Lemma 3.
If the strategic customers can buy the lower performance products via the electronic channel, it needs to meet the prerequisite:0≤PN/μθL≤PS/θH≤PS−PN/θH−μθL≤1.The selections of strategic customers under expression (7) are discussed, and the details are illustrated in Figure 4.The number of strategic customers that choose to buy the higher performance products via the retail channel isQSH=∫PS−PN/θH−μθL1dx=1−PS−PN/θH−μθL, while the number that choose to purchase the lower performance products via the electronic channel is QNL=∫PN/μθLPS−PN/θH−μθLdx=PS−PN/θH−μθL−PN/μθL, and the number that does not buy any products is Q0=PN/μθL.Figure 4
The decision of strategic customers.Lemma 4.
When WTP is close to 1, the best choice for strategic customers is to buy the higher performance products in the retail channel. When WTP nears zero, strategic customers do not buy any products.


### 3.2.1. Decentralized Decision

According to the choices of strategic customers, the profit function of the retailer is(9)ΠRHL=QSHPS−WH=1−PS−PNθH−μθLPS−WH.The profit function of the manufacturer is(10)ΠMHL=QSHWH−CH¯+QNLPN−CL¯=1−PS−PNθH−μθLWH−CH¯+PS−PNθH−μθL−PNμθLPN−CL¯.Proposition 3.
The manufacturer sells higher performance productH in the retail channel and lower performance product L via the electronic channel, and the optimal prices is WH∗=CH¯+θH/2, PNHL∗=CL¯+μθL/2, and PSHL∗=CH¯+CL¯+3θH−μθL/4. The retailer profit is ΠR⟶DDHL∗=θH−μθL−CH¯+CL¯2/16θH−μθL, and the manufacturer profit is ΠM⟶DDHL∗=CH¯2/8θH−μθL+2θH−μθLCL¯2/2μθLθH−μθL−CH¯CL¯/4θH−μθL−CH¯+CL¯/4+θH+μθL/8.Then substitute the optimal prices into the expression of the strategic customer number, and we can getQSH=1/4−CH¯−CL¯/4θH−μθL, QNL=1/4+μθLCH¯−2θH−μθLCL¯/4μθLθH−μθL, and the number of strategic customers who do not buy any products Q0=1/2+CL¯/2μθL.


### 3.2.2. Centralized Decision

The profit function of the supply chain isΠZHL=QSHPS−CH¯+QNLPN−CL¯=1−PS−PN/θH−μθLPS−CH¯+PS−PN/θH−μθL−PN/μθLPN−CL¯.Proposition 4.
If the dual-channel supply chain is centralized, the optimal price of higher performance products in the retail channel isPZSHL∗=CH¯+θH/2, and the optimal price of lower performance products in electronic channel is PZNHL∗=CL¯+μθL/2. The maximum profit of the supply chain is ΠZ⟶CDHL∗=CH¯2/4θH−μθL+θHCL¯2/4μθLθH−μθL−CH¯CL¯/2θH−μθL−CH¯/2+θH/4.Put the optimal prices into the expression of strategic customer number, and we can getQSH=1/2−CH¯−CL¯/2θH−μθL, QNL=CH¯−CL¯/2θH−μθL−CL¯/2μθL, and the number of strategic customers who do not buy any products is Q0=1/2+CL¯/2μθL.


## 3.2.1. Decentralized Decision

According to the choices of strategic customers, the profit function of the retailer is(9)ΠRHL=QSHPS−WH=1−PS−PNθH−μθLPS−WH.The profit function of the manufacturer is(10)ΠMHL=QSHWH−CH¯+QNLPN−CL¯=1−PS−PNθH−μθLWH−CH¯+PS−PNθH−μθL−PNμθLPN−CL¯.Proposition 3.
The manufacturer sells higher performance productH in the retail channel and lower performance product L via the electronic channel, and the optimal prices is WH∗=CH¯+θH/2, PNHL∗=CL¯+μθL/2, and PSHL∗=CH¯+CL¯+3θH−μθL/4. The retailer profit is ΠR⟶DDHL∗=θH−μθL−CH¯+CL¯2/16θH−μθL, and the manufacturer profit is ΠM⟶DDHL∗=CH¯2/8θH−μθL+2θH−μθLCL¯2/2μθLθH−μθL−CH¯CL¯/4θH−μθL−CH¯+CL¯/4+θH+μθL/8.Then substitute the optimal prices into the expression of the strategic customer number, and we can getQSH=1/4−CH¯−CL¯/4θH−μθL, QNL=1/4+μθLCH¯−2θH−μθLCL¯/4μθLθH−μθL, and the number of strategic customers who do not buy any products Q0=1/2+CL¯/2μθL.


## 3.2.2. Centralized Decision

The profit function of the supply chain isΠZHL=QSHPS−CH¯+QNLPN−CL¯=1−PS−PN/θH−μθLPS−CH¯+PS−PN/θH−μθL−PN/μθLPN−CL¯.Proposition 4.
If the dual-channel supply chain is centralized, the optimal price of higher performance products in the retail channel isPZSHL∗=CH¯+θH/2, and the optimal price of lower performance products in electronic channel is PZNHL∗=CL¯+μθL/2. The maximum profit of the supply chain is ΠZ⟶CDHL∗=CH¯2/4θH−μθL+θHCL¯2/4μθLθH−μθL−CH¯CL¯/2θH−μθL−CH¯/2+θH/4.Put the optimal prices into the expression of strategic customer number, and we can getQSH=1/2−CH¯−CL¯/2θH−μθL, QNL=CH¯−CL¯/2θH−μθL−CL¯/2μθL, and the number of strategic customers who do not buy any products is Q0=1/2+CL¯/2μθL.


## 3.3. Sell ProductL in Retail Channel and Product H in Electronic Channel

In the dual-channel supply-chain model II, the manufacturer sells productsL via the retail channel and sells products H in the electronic channel. The sales process is shown in Figure 5.Figure 5
The dual-channel supply-chain model II.The utility function for buying the higher performance products via the retail channel isUS=θLx−PS and for buying the lower performance products via the electronic channel is UN=μθHx−PN. Suppose sale prices meet the requirement PN>PS. The selection function of strategic customers is V=maxUS,UN,0=maxθLx−PS,μθHx−PN,0.When the parameters meet the conditionPS/θL≤PN/μθH, i.e., PN−PS/μθH−θL≥PN/μθH≥PS/θL, the selection function can be derived as follows:(11)V=θLx−PS,PSθL≤x≤PN−PSμθH−θL,μθHx−PN,x≥PN−PSμθH−θL,0,x≤PSθL.When the parameters meet the conditionPS/θL≥PN/μθH, i.e., PN−PS/μθH−θL≤PN/μθH≤PS/θL, the selection function is changed into (12). The strategic customers do not buy lower performance products in the retail channel:(12)V=μθHx−PN,x≥PNμθH,0,x≤PNμθH.Lemma 5.
If the strategic customers can purchase products from the retail channel and electronic channel, it needs to meet the prerequisite:0≤PS/θL≤PN/μθH≤PN−PS/μθH−θL≤1.The selections of strategic customers under expression (11) are discussed, and the details are illustrated in Figure 6.The number of strategic customers that chooses to buy the higher performance products via the electronic channel isQNH=∫PN−PS/μθH−θL1dx=1−PN−PS/μθH−θL, while the number that chooses to purchase the lower performance products via the retail channel is QSL=∫PS/θLPN−PS/μθH−θLdx=PN−PS/μθH−θL−PS/θL, and the number that does not buy any products is Q0=PS/θL.Figure 6
The decision of strategic customers.Lemma 6.
When WTP is close to 1, the best choice for strategic customers is to buy the higher performance products in the electronic channel. When WTP nears zero, the strategic customers do not buy any products.


### 3.3.1. Decentralized Decision

According to the choices of strategic customers, the profit function of the retailer is(13)ΠRLH=QSLPS−WL=PN−PSμθH−θL−PSθLPS−WL.The profit function of the manufacturer is(14)ΠMLH=QSLWL−CL¯+QNHPN−CH¯=PN−PSμθH−θL−PSθLWL−CL¯+1−PN−PSμθH−θLPN−CH¯.Proposition 5.
The manufacturer sells lower performance productL in the retail channel and higher performance product H via electronic channel, and the optimal prices are WL∗=CL¯+θL/2, PNLH∗=CH¯+μθH/2, and PSLH∗=θLCH¯+μθHCL¯+2μθHθL/4μθH. The retailer profit is ΠR⟶DDLH=θLCH¯−μθHCL¯2/16μθHθLμθH−θL, and the maximum profit of the manufacturer is ΠM⟶DDLH=2μθH−θLCH¯2/8μθHμθH−θL+μθHCL¯2/8θLμθH−θL−CH¯CL¯/4μθH−θL−CH¯/2+μθH/4.Then, substitute the optimal prices into the expression of the strategic customer number, and we can getQNH=1/2−2μθH−θLCH¯−μθHCL¯/4μθHμθH−θL, QSL=θLCH¯−μθHCL¯/4θLμθH−θL, and the number of strategic customers who do not buy any products is Q0=1/2+θLCH¯+μθHCL¯/4μθHθL.


### 3.3.2. Centralized Decision

The profit function of the supply chain isΠZLH=QSLPS−CL¯+QNHPN−CH¯=PN−PS/μθH−θL−PS/θLPS−CL¯+1−PN−PS/μθH−θLPN−CH¯.Proposition 6.
If the dual-channel supply chain is centralized, the optimal price of lower performance products in the retail channel isPZSLH∗=CL¯+θL/2, and the optimal price of higher performance products in the electronic channel is PZNLH∗=CH¯+μθH/2. The maximum profit of the supply chain is ΠZ⟶CDHL∗=CH¯2/4μθH−θL+μθHCL¯2/4θLμθH−θL−CH¯CL¯/2μθH−θL−CH¯/2+μθH/4.Put the optimal prices into the expression of the strategic customer number, and we can getQNH=1/2−CH¯−CL¯/2μθH−θL, QSL=CH¯−CL¯/2μθH−θL−CL¯/2θL, and the number of strategic customers who do not buy any products is Q0=1/2+CL¯/2θL.Based on the above analysis, this paper compares the influence of parametersθH and θL on optimal pricing strategies and customer numbers in three different models. The change trend is shown in Table 1.Table 1
The influence of parameters on optimal price strategy and number of customers.
Decentralized decision
Centralized decision
Optimal pricing strategies
Customer numbers
Optimal pricing strategies
Customer numbers
θ
H
↗
Single retail channel model
W
H
∗
↗，PH∗↗， WL∗¯，PL∗¯
Q
H
↗，QL↘， Q0¯
P
Z
H
∗
↗， PZL∗¯
Q
H
↗，QL↘， Q0¯
Dual-channel model I
W
H
∗
↗，PSHL∗↗， PNHL∗¯
Q
S
H
↗，QNL↘， Q0¯
P
Z
S
H
L
∗
↗， PZNHL∗¯
Q
S
H
↗，QNL↘， Q0¯
Dual-channel model II
W
L
∗
¯，PSLH∗↘， PNLH∗↗
Q
S
L
↘，QNH↗， Q0↘
P
Z
S
L
H
∗
¯， PZNLH∗↗
Q
S
L
↘，QNH↗ Q0¯
θ
L
↗
Single retail channel model
W
H
∗
¯，PH∗¯， WL∗↗，PL∗↗
Q
H
↘，QL↗， Q0↘
P
Z
H
∗
¯， PZL∗↗
Q
H
↘，QL↗ Q0↘
Dual-channel model I
P
N
HL
∗
↗，PSHL∗↘， WH∗¯
Q
S
H
↘，QNL↗， Q0↘
P
ZS
HL
∗
¯， PZNHL∗↗
Q
S
H
↘，QNL↗， Q0↘
Dual-channel model II
W
L
∗
↗，PSLH∗↗， PNLH∗¯
Q
S
L
↗，QNH↘， Q0↘
P
Z
S
L
H
∗
↗， PZNLH∗¯
Q
S
L
↗，QNH↘， Q0↘


## 3.3.1. Decentralized Decision

According to the choices of strategic customers, the profit function of the retailer is(13)ΠRLH=QSLPS−WL=PN−PSμθH−θL−PSθLPS−WL.The profit function of the manufacturer is(14)ΠMLH=QSLWL−CL¯+QNHPN−CH¯=PN−PSμθH−θL−PSθLWL−CL¯+1−PN−PSμθH−θLPN−CH¯.Proposition 5.
The manufacturer sells lower performance productL in the retail channel and higher performance product H via electronic channel, and the optimal prices are WL∗=CL¯+θL/2, PNLH∗=CH¯+μθH/2, and PSLH∗=θLCH¯+μθHCL¯+2μθHθL/4μθH. The retailer profit is ΠR⟶DDLH=θLCH¯−μθHCL¯2/16μθHθLμθH−θL, and the maximum profit of the manufacturer is ΠM⟶DDLH=2μθH−θLCH¯2/8μθHμθH−θL+μθHCL¯2/8θLμθH−θL−CH¯CL¯/4μθH−θL−CH¯/2+μθH/4.Then, substitute the optimal prices into the expression of the strategic customer number, and we can getQNH=1/2−2μθH−θLCH¯−μθHCL¯/4μθHμθH−θL, QSL=θLCH¯−μθHCL¯/4θLμθH−θL, and the number of strategic customers who do not buy any products is Q0=1/2+θLCH¯+μθHCL¯/4μθHθL.


## 3.3.2. Centralized Decision

The profit function of the supply chain isΠZLH=QSLPS−CL¯+QNHPN−CH¯=PN−PS/μθH−θL−PS/θLPS−CL¯+1−PN−PS/μθH−θLPN−CH¯.Proposition 6.
If the dual-channel supply chain is centralized, the optimal price of lower performance products in the retail channel isPZSLH∗=CL¯+θL/2, and the optimal price of higher performance products in the electronic channel is PZNLH∗=CH¯+μθH/2. The maximum profit of the supply chain is ΠZ⟶CDHL∗=CH¯2/4μθH−θL+μθHCL¯2/4θLμθH−θL−CH¯CL¯/2μθH−θL−CH¯/2+μθH/4.Put the optimal prices into the expression of the strategic customer number, and we can getQNH=1/2−CH¯−CL¯/2μθH−θL, QSL=CH¯−CL¯/2μθH−θL−CL¯/2θL, and the number of strategic customers who do not buy any products is Q0=1/2+CL¯/2θL.Based on the above analysis, this paper compares the influence of parametersθH and θL on optimal pricing strategies and customer numbers in three different models. The change trend is shown in Table 1.Table 1
The influence of parameters on optimal price strategy and number of customers.
Decentralized decision
Centralized decision
Optimal pricing strategies
Customer numbers
Optimal pricing strategies
Customer numbers
θ
H
↗
Single retail channel model
W
H
∗
↗，PH∗↗， WL∗¯，PL∗¯
Q
H
↗，QL↘， Q0¯
P
Z
H
∗
↗， PZL∗¯
Q
H
↗，QL↘， Q0¯
Dual-channel model I
W
H
∗
↗，PSHL∗↗， PNHL∗¯
Q
S
H
↗，QNL↘， Q0¯
P
Z
S
H
L
∗
↗， PZNHL∗¯
Q
S
H
↗，QNL↘， Q0¯
Dual-channel model II
W
L
∗
¯，PSLH∗↘， PNLH∗↗
Q
S
L
↘，QNH↗， Q0↘
P
Z
S
L
H
∗
¯， PZNLH∗↗
Q
S
L
↘，QNH↗ Q0¯
θ
L
↗
Single retail channel model
W
H
∗
¯，PH∗¯， WL∗↗，PL∗↗
Q
H
↘，QL↗， Q0↘
P
Z
H
∗
¯， PZL∗↗
Q
H
↘，QL↗ Q0↘
Dual-channel model I
P
N
HL
∗
↗，PSHL∗↘， WH∗¯
Q
S
H
↘，QNL↗， Q0↘
P
ZS
HL
∗
¯， PZNHL∗↗
Q
S
H
↘，QNL↗， Q0↘
Dual-channel model II
W
L
∗
↗，PSLH∗↗， PNLH∗¯
Q
S
L
↗，QNH↘， Q0↘
P
Z
S
L
H
∗
↗， PZNLH∗¯
Q
S
L
↗，QNH↘， Q0↘


## 4. Numerical Analysis

### 4.1. The Influence of Parameters on the Single Retail Channel Model

#### 4.1.1. The Influence of ParameterθH on Optimal Decisions

We setCH¯=0.4, CL¯=0.3, and θL=0.7, and then the value range of the parameter θH is θL,1. In the decentralized dual-channel supply chain, WH∗=0.4+θH/2, WL∗=0.5, PH∗=0.4+3θH/4, and PL∗=0.6, the profit function of the retailer is ΠR⟶DD∗=θH2−1.35θH+0.46/14θH−0.7, and the profit function of the manufacturer is ΠM⟶DD∗=θH2−1.35θH+0.46/7θH−0.7. In centralized dual-channel supply chains, PZH∗=0.4+θH/2 and PZL∗=0.5, the supply-chain profit is ΠZ⟶CD∗=0.7θH2−0.96θH+0.336/2.8θH−0.7. The influence of parameter θH on optimal prices and the optimal profits is shown by Figure 7.Figure 7
The influence of parameterθH on optimal decisions.
(a)
(b)In this type, both the higher performance products and the lower performances products are selected by strategic customers. According to Lemma1, we can deduce that PL/θL≤PH/θH and PH−PL/θH−θL≤1 and then get the range 0.8≤θH≤0.93. In this interval, the optimal prices meet the following relationship: PH∗>WH∗>PL∗>WL∗. The wholesale price of higher performance products is more than the sale price of lower performance products. With the parameter θH increasing, these profit functions will increase, the gap will widen between manufacturer profit and retailer profit, and the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL has no obvious change.


#### 4.1.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, and θH=0.9, and we can get the value range of θL is 0,θH. In the decentralized dual-channel supply chain, WH∗=0.65, WL∗=0.3+θL/2, PH∗=0.775, and PL∗=0.3+3θL/4, the maximum profit function of the retailer is ΠR⟶DD∗=−0.1θL2+0.01θL+0.081/16θL0.9−θL and the profit function of the manufacturer is ΠM⟶DD∗=−0.1θL2+0.01θL+0.081/8θL0.9−θL. In centralized dual-channel supply chains, PZH∗=0.65 and PZL∗=0.3+θL/2, the supply-chain profit is ΠZ⟶DD∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of parameter θL on optimal prices and the optimal profits is shown in Figure 8.Figure 8
The influence of parameterθL on optimal decisions.
(a)
(b)According to Lemma1, we can deduce that 0.675≤θL≤0.8. In range 0.675,0.77, the optimal prices have following relationship: PH∗>WH∗>PL∗>WL∗; in range 0.675,0.8, the optimal prices have the following relationship: PH∗>PL∗>WH∗>WL∗, the sale price of lower performance products exceeds the wholesale price of higher performance products. As parameter θL increases, these profit functions are all essentially unchanged, and ΠZ⟶DD∗=2ΠR⟶DD∗=4ΠR⟶DD∗.


### 4.2. The Influence of Parameters on the Dual-Channel Model I

#### 4.2.1. The Influence of ParameterθH on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θL=0.6, and μ=1, the range of the parameter θH is θL,1. When the dual-channel supply chain is decentralized, the optimal prices are WH∗=0.4+θH/2, PNHL∗=0.45, and PSHL∗=0.1+3θH/4. The profit function of the retailer is ΠR⟶DDHL∗=θH−0.72/16θH−0.6, the profit of the manufacturer is ΠM⟶DDHL∗=θH2−1.1θH+0.31/8θH−0.6, and the supply-chain profit is ΠM⟶DDHL∗=3θH2−3.6θH+1.11/16θH−0.6. In the centralized supply chain, the optimal prices PZSHL∗=0.4+θH/2 and PZNHL∗=0.45, and the maximum profit of supply chain is ΠZ⟶CDHL∗=θH2−1.25θH+0.4/4θH−0.6. The influence of the parameter θH on optimal prices and the optimal profits is shown in Figure 9.Figure 9
The influence of parameterθH on optimal decisions.
(a)
(b)On the basis of Lemma3, when the strategic customers select to buy products via the electronic channel, it needs to meet the following criteria: PN/θL≤PS/θH and PS−PN/θH−θL≤1, and we can get the range θH≥0.7. In the interval, the optimal prices have the following relationship: PSHL∗>WH∗>PNHL∗. The wholesale price of higher performance products in the retail channel is higher than the sale price of lower performance products in the electronic channel, and the manufacturer profit is more than the retailer profit, and the profit of centralized decision is superior to the profit of decentralized decision. With the parameter θH increasing, prices WH∗ and PSHL∗ will increase, and PNHL∗ will be no change; the profit functions will increase, and the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL will be bigger and bigger.


#### 4.2.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θH=0.9, and μ=1, and the range of the parameter θL is 0,θH. In the decentralized supply chain, the prices WH∗=0.65, PNHL∗=0.3+θL/2, and PSHL∗=3.4−θL/4. The retailer profit is ΠR⟶DDHL∗=0.8−θL2/160.9−θL, the manufacturer profit is ΠM⟶DDHL∗=−θL3+1.4θL2−0.62θL+0.162/8θL0.9−θL, and the profit of the supply chain is ΠZ⟶DDHL∗=−θL3+1.2θL2−0.6θL+0.324/16θL0.9−θL. In the centralized supply chain, the optimal prices are PZSHL∗=0.65 and PZNHL∗=0.3+θL/2, and the profit of the supply chain is ΠZ⟶CDHL∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of the parameter θL on optimal prices and the optimal profits is shown in Figure 10.Figure 10
The influence of parameterθL on optimal decisions.
(a)
(b)Similarly, we can deduce that the range of the parameterθL is 0.48<θL<0.8, according to Lemma 3. In the interval 0.48,0.8, the optimal prices have the following relationship: PSHL∗>WH∗>PNHL∗; the profit of the centralized supply chain is superior to the profit of the decentralized supply chain, and the manufacturer profit is significantly higher than the retailer profit. As the parameter θL grows, the sale price of higher performance products in retail channel PSHL∗ decreases, and the sale price of lower performance products in electronic channel PNHL∗ increases, but the wholesale price WH∗ remains the same; the manufacturer profit and the profit of the decentralized supply chain increase, while the retailer profit and the profit of centralized supply chain decrease. And the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL will be smaller and smaller.


### 4.3. The Influence of Parameters on the Dual-Channel Model II

#### 4.3.1. The Influence of ParameterθH on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θL=0.6, and μ=1, and the range of parameter θH is θL,1. In the decentralized supply chain, the optimal prices are WL∗=0.45, PNLH∗=0.4+θH/2, and PSLH∗=0.24+1.5θH/4θH, and then the maximum profit of the retailer is ΠR⟶DDLH=0.24−0.3θH2/9.6θHθH−0.6; meanwhile, the maximum profit of the manufacturer is ΠM⟶DDLH=1.2θH3−1.59θH2+0.624θH−0.0576/4.8θHθH−0.6, and the supply-chain profit is ΠZ⟶DDHL∗=θH3−1.2875θH2+0.46θH−0.024/4θHθH−0.6. In the centralized supply chain, the optimal prices are PZSLH∗=0.45 and PZNLH∗=0.4+θH/2, and the supply-chain profit is ΠZ⟶CDHL∗=θH2−1.25θH+0.4/4θH−0.6. The influence of the parameter θH on optimal prices and the optimal profits is shown in Figure 11.Figure 11
The influence of parameterθH on optimal decisions.
(a)
(b)According to Lemma5, it needs to meet the following prerequisite: PS/θL≤PN/θH and PN−PS/θH−θL≤1, and then we can deduce that the range of the parameter θH is 0.67<θH<0.8. In this interval, the optimal prices have the following relationship: PNLH∗>PSLH∗>WL∗, the price of higher performance products in the electronic channel is more than the price of lower performance products in the retailer channel, also more than the wholesale price of lower performance products. The manufacturer profit is obviously higher than retailer profit, and the profit of the centralized supply chain is slightly above the profit of the decentralized supply chain. As the parameter θH increases, PNLH∗ increases, but PSLH∗ decreases, and WL∗ remains the same; the retailer profit reduces gradually, while the other three profits firstly decrease and then increase and, finally, come together. And the difference between the total profit of centralized decision and of decentralized decision ΔΠZLH will be smaller and smaller.


#### 4.3.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θH=0.9, and μ=1, and the range of parameter θL is 0,θH. When the supply chain is decentralized, the optimal prices are WL∗=0.3+θL/2, PNLH∗=0.65, and PSLH∗=2.2θL+0.27/3.6, and the retailer profit is ΠR⟶DDLH=0.4θL−0.272/14.4θL0.9−θL, the manufacturer profit is ΠM⟶DDLH=−0.34θL2+0.234θL+0.0729/7.2θL0.9−θL, and the supply-chain profit is ΠZ⟶DDHL∗=−0.52θL2+0.252θL+0.2187/14.4θL0.9−θL. When the supply chain is centralized, the optimal prices are PZSLH∗=0.3+θL/2 and PZNLH∗=0.65, and the supply-chain profit is ΠZ⟶CDHL∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of the parameter θL on optimal prices and the optimal profits is shown in Figure 12.Figure 12
The influence of parameterθL on optimal decisions.
(a)
(b)Similarly, according to Lemma5, we can get the range of parameter θL is 0.68<θL<0.84. In this interval, the optimal prices have following relationship: PNLH∗>PSLH∗>WL∗. The sale price of higher performance products in the electronic channel is significantly more than the sale price of lower performance products in the retail channel, while the price of the retail channel is slightly higher than wholesale price WL∗; and the manufacturer profit is higher than the retailer profit. As the parameter θL increases, PSLH∗ and WL∗ increase, but PNLH∗ remains the same; all of the profit functions increase, and the speed of increasing is faster and faster. When the value of the parameter θL exceeds 0.8, the retailer profit increases from zero, and the total profit of centralized decision is slightly higher than the total profit of decentralized decision.Based on the analysis shown above, we find that the retailer profits of model I is more than that of model II, and the retailer profits is almost zero in model II. Although the manufacturer is the leader, it needs to take into account the retailer profits. Therefore, the manufacturer selects model II, i.e., sell higher performance products in the retail channel, and sell lower performance products in the electronic channel.


## 4.1. The Influence of Parameters on the Single Retail Channel Model

### 4.1.1. The Influence of ParameterθH on Optimal Decisions

We setCH¯=0.4, CL¯=0.3, and θL=0.7, and then the value range of the parameter θH is θL,1. In the decentralized dual-channel supply chain, WH∗=0.4+θH/2, WL∗=0.5, PH∗=0.4+3θH/4, and PL∗=0.6, the profit function of the retailer is ΠR⟶DD∗=θH2−1.35θH+0.46/14θH−0.7, and the profit function of the manufacturer is ΠM⟶DD∗=θH2−1.35θH+0.46/7θH−0.7. In centralized dual-channel supply chains, PZH∗=0.4+θH/2 and PZL∗=0.5, the supply-chain profit is ΠZ⟶CD∗=0.7θH2−0.96θH+0.336/2.8θH−0.7. The influence of parameter θH on optimal prices and the optimal profits is shown by Figure 7.Figure 7
The influence of parameterθH on optimal decisions.
(a)
(b)In this type, both the higher performance products and the lower performances products are selected by strategic customers. According to Lemma1, we can deduce that PL/θL≤PH/θH and PH−PL/θH−θL≤1 and then get the range 0.8≤θH≤0.93. In this interval, the optimal prices meet the following relationship: PH∗>WH∗>PL∗>WL∗. The wholesale price of higher performance products is more than the sale price of lower performance products. With the parameter θH increasing, these profit functions will increase, the gap will widen between manufacturer profit and retailer profit, and the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL has no obvious change.


### 4.1.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, and θH=0.9, and we can get the value range of θL is 0,θH. In the decentralized dual-channel supply chain, WH∗=0.65, WL∗=0.3+θL/2, PH∗=0.775, and PL∗=0.3+3θL/4, the maximum profit function of the retailer is ΠR⟶DD∗=−0.1θL2+0.01θL+0.081/16θL0.9−θL and the profit function of the manufacturer is ΠM⟶DD∗=−0.1θL2+0.01θL+0.081/8θL0.9−θL. In centralized dual-channel supply chains, PZH∗=0.65 and PZL∗=0.3+θL/2, the supply-chain profit is ΠZ⟶DD∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of parameter θL on optimal prices and the optimal profits is shown in Figure 8.Figure 8
The influence of parameterθL on optimal decisions.
(a)
(b)According to Lemma1, we can deduce that 0.675≤θL≤0.8. In range 0.675,0.77, the optimal prices have following relationship: PH∗>WH∗>PL∗>WL∗; in range 0.675,0.8, the optimal prices have the following relationship: PH∗>PL∗>WH∗>WL∗, the sale price of lower performance products exceeds the wholesale price of higher performance products. As parameter θL increases, these profit functions are all essentially unchanged, and ΠZ⟶DD∗=2ΠR⟶DD∗=4ΠR⟶DD∗.


## 4.1.1. The Influence of ParameterθH on Optimal Decisions

We setCH¯=0.4, CL¯=0.3, and θL=0.7, and then the value range of the parameter θH is θL,1. In the decentralized dual-channel supply chain, WH∗=0.4+θH/2, WL∗=0.5, PH∗=0.4+3θH/4, and PL∗=0.6, the profit function of the retailer is ΠR⟶DD∗=θH2−1.35θH+0.46/14θH−0.7, and the profit function of the manufacturer is ΠM⟶DD∗=θH2−1.35θH+0.46/7θH−0.7. In centralized dual-channel supply chains, PZH∗=0.4+θH/2 and PZL∗=0.5, the supply-chain profit is ΠZ⟶CD∗=0.7θH2−0.96θH+0.336/2.8θH−0.7. The influence of parameter θH on optimal prices and the optimal profits is shown by Figure 7.Figure 7
The influence of parameterθH on optimal decisions.
(a)
(b)In this type, both the higher performance products and the lower performances products are selected by strategic customers. According to Lemma1, we can deduce that PL/θL≤PH/θH and PH−PL/θH−θL≤1 and then get the range 0.8≤θH≤0.93. In this interval, the optimal prices meet the following relationship: PH∗>WH∗>PL∗>WL∗. The wholesale price of higher performance products is more than the sale price of lower performance products. With the parameter θH increasing, these profit functions will increase, the gap will widen between manufacturer profit and retailer profit, and the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL has no obvious change.


## 4.1.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, and θH=0.9, and we can get the value range of θL is 0,θH. In the decentralized dual-channel supply chain, WH∗=0.65, WL∗=0.3+θL/2, PH∗=0.775, and PL∗=0.3+3θL/4, the maximum profit function of the retailer is ΠR⟶DD∗=−0.1θL2+0.01θL+0.081/16θL0.9−θL and the profit function of the manufacturer is ΠM⟶DD∗=−0.1θL2+0.01θL+0.081/8θL0.9−θL. In centralized dual-channel supply chains, PZH∗=0.65 and PZL∗=0.3+θL/2, the supply-chain profit is ΠZ⟶DD∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of parameter θL on optimal prices and the optimal profits is shown in Figure 8.Figure 8
The influence of parameterθL on optimal decisions.
(a)
(b)According to Lemma1, we can deduce that 0.675≤θL≤0.8. In range 0.675,0.77, the optimal prices have following relationship: PH∗>WH∗>PL∗>WL∗; in range 0.675,0.8, the optimal prices have the following relationship: PH∗>PL∗>WH∗>WL∗, the sale price of lower performance products exceeds the wholesale price of higher performance products. As parameter θL increases, these profit functions are all essentially unchanged, and ΠZ⟶DD∗=2ΠR⟶DD∗=4ΠR⟶DD∗.


## 4.2. The Influence of Parameters on the Dual-Channel Model I

### 4.2.1. The Influence of ParameterθH on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θL=0.6, and μ=1, the range of the parameter θH is θL,1. When the dual-channel supply chain is decentralized, the optimal prices are WH∗=0.4+θH/2, PNHL∗=0.45, and PSHL∗=0.1+3θH/4. The profit function of the retailer is ΠR⟶DDHL∗=θH−0.72/16θH−0.6, the profit of the manufacturer is ΠM⟶DDHL∗=θH2−1.1θH+0.31/8θH−0.6, and the supply-chain profit is ΠM⟶DDHL∗=3θH2−3.6θH+1.11/16θH−0.6. In the centralized supply chain, the optimal prices PZSHL∗=0.4+θH/2 and PZNHL∗=0.45, and the maximum profit of supply chain is ΠZ⟶CDHL∗=θH2−1.25θH+0.4/4θH−0.6. The influence of the parameter θH on optimal prices and the optimal profits is shown in Figure 9.Figure 9
The influence of parameterθH on optimal decisions.
(a)
(b)On the basis of Lemma3, when the strategic customers select to buy products via the electronic channel, it needs to meet the following criteria: PN/θL≤PS/θH and PS−PN/θH−θL≤1, and we can get the range θH≥0.7. In the interval, the optimal prices have the following relationship: PSHL∗>WH∗>PNHL∗. The wholesale price of higher performance products in the retail channel is higher than the sale price of lower performance products in the electronic channel, and the manufacturer profit is more than the retailer profit, and the profit of centralized decision is superior to the profit of decentralized decision. With the parameter θH increasing, prices WH∗ and PSHL∗ will increase, and PNHL∗ will be no change; the profit functions will increase, and the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL will be bigger and bigger.


### 4.2.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θH=0.9, and μ=1, and the range of the parameter θL is 0,θH. In the decentralized supply chain, the prices WH∗=0.65, PNHL∗=0.3+θL/2, and PSHL∗=3.4−θL/4. The retailer profit is ΠR⟶DDHL∗=0.8−θL2/160.9−θL, the manufacturer profit is ΠM⟶DDHL∗=−θL3+1.4θL2−0.62θL+0.162/8θL0.9−θL, and the profit of the supply chain is ΠZ⟶DDHL∗=−θL3+1.2θL2−0.6θL+0.324/16θL0.9−θL. In the centralized supply chain, the optimal prices are PZSHL∗=0.65 and PZNHL∗=0.3+θL/2, and the profit of the supply chain is ΠZ⟶CDHL∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of the parameter θL on optimal prices and the optimal profits is shown in Figure 10.Figure 10
The influence of parameterθL on optimal decisions.
(a)
(b)Similarly, we can deduce that the range of the parameterθL is 0.48<θL<0.8, according to Lemma 3. In the interval 0.48,0.8, the optimal prices have the following relationship: PSHL∗>WH∗>PNHL∗; the profit of the centralized supply chain is superior to the profit of the decentralized supply chain, and the manufacturer profit is significantly higher than the retailer profit. As the parameter θL grows, the sale price of higher performance products in retail channel PSHL∗ decreases, and the sale price of lower performance products in electronic channel PNHL∗ increases, but the wholesale price WH∗ remains the same; the manufacturer profit and the profit of the decentralized supply chain increase, while the retailer profit and the profit of centralized supply chain decrease. And the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL will be smaller and smaller.


## 4.2.1. The Influence of ParameterθH on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θL=0.6, and μ=1, the range of the parameter θH is θL,1. When the dual-channel supply chain is decentralized, the optimal prices are WH∗=0.4+θH/2, PNHL∗=0.45, and PSHL∗=0.1+3θH/4. The profit function of the retailer is ΠR⟶DDHL∗=θH−0.72/16θH−0.6, the profit of the manufacturer is ΠM⟶DDHL∗=θH2−1.1θH+0.31/8θH−0.6, and the supply-chain profit is ΠM⟶DDHL∗=3θH2−3.6θH+1.11/16θH−0.6. In the centralized supply chain, the optimal prices PZSHL∗=0.4+θH/2 and PZNHL∗=0.45, and the maximum profit of supply chain is ΠZ⟶CDHL∗=θH2−1.25θH+0.4/4θH−0.6. The influence of the parameter θH on optimal prices and the optimal profits is shown in Figure 9.Figure 9
The influence of parameterθH on optimal decisions.
(a)
(b)On the basis of Lemma3, when the strategic customers select to buy products via the electronic channel, it needs to meet the following criteria: PN/θL≤PS/θH and PS−PN/θH−θL≤1, and we can get the range θH≥0.7. In the interval, the optimal prices have the following relationship: PSHL∗>WH∗>PNHL∗. The wholesale price of higher performance products in the retail channel is higher than the sale price of lower performance products in the electronic channel, and the manufacturer profit is more than the retailer profit, and the profit of centralized decision is superior to the profit of decentralized decision. With the parameter θH increasing, prices WH∗ and PSHL∗ will increase, and PNHL∗ will be no change; the profit functions will increase, and the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL will be bigger and bigger.


## 4.2.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θH=0.9, and μ=1, and the range of the parameter θL is 0,θH. In the decentralized supply chain, the prices WH∗=0.65, PNHL∗=0.3+θL/2, and PSHL∗=3.4−θL/4. The retailer profit is ΠR⟶DDHL∗=0.8−θL2/160.9−θL, the manufacturer profit is ΠM⟶DDHL∗=−θL3+1.4θL2−0.62θL+0.162/8θL0.9−θL, and the profit of the supply chain is ΠZ⟶DDHL∗=−θL3+1.2θL2−0.6θL+0.324/16θL0.9−θL. In the centralized supply chain, the optimal prices are PZSHL∗=0.65 and PZNHL∗=0.3+θL/2, and the profit of the supply chain is ΠZ⟶CDHL∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of the parameter θL on optimal prices and the optimal profits is shown in Figure 10.Figure 10
The influence of parameterθL on optimal decisions.
(a)
(b)Similarly, we can deduce that the range of the parameterθL is 0.48<θL<0.8, according to Lemma 3. In the interval 0.48,0.8, the optimal prices have the following relationship: PSHL∗>WH∗>PNHL∗; the profit of the centralized supply chain is superior to the profit of the decentralized supply chain, and the manufacturer profit is significantly higher than the retailer profit. As the parameter θL grows, the sale price of higher performance products in retail channel PSHL∗ decreases, and the sale price of lower performance products in electronic channel PNHL∗ increases, but the wholesale price WH∗ remains the same; the manufacturer profit and the profit of the decentralized supply chain increase, while the retailer profit and the profit of centralized supply chain decrease. And the difference between the total profit of centralized decision and of decentralized decision ΔΠZHL will be smaller and smaller.


## 4.3. The Influence of Parameters on the Dual-Channel Model II

### 4.3.1. The Influence of ParameterθH on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θL=0.6, and μ=1, and the range of parameter θH is θL,1. In the decentralized supply chain, the optimal prices are WL∗=0.45, PNLH∗=0.4+θH/2, and PSLH∗=0.24+1.5θH/4θH, and then the maximum profit of the retailer is ΠR⟶DDLH=0.24−0.3θH2/9.6θHθH−0.6; meanwhile, the maximum profit of the manufacturer is ΠM⟶DDLH=1.2θH3−1.59θH2+0.624θH−0.0576/4.8θHθH−0.6, and the supply-chain profit is ΠZ⟶DDHL∗=θH3−1.2875θH2+0.46θH−0.024/4θHθH−0.6. In the centralized supply chain, the optimal prices are PZSLH∗=0.45 and PZNLH∗=0.4+θH/2, and the supply-chain profit is ΠZ⟶CDHL∗=θH2−1.25θH+0.4/4θH−0.6. The influence of the parameter θH on optimal prices and the optimal profits is shown in Figure 11.Figure 11
The influence of parameterθH on optimal decisions.
(a)
(b)According to Lemma5, it needs to meet the following prerequisite: PS/θL≤PN/θH and PN−PS/θH−θL≤1, and then we can deduce that the range of the parameter θH is 0.67<θH<0.8. In this interval, the optimal prices have the following relationship: PNLH∗>PSLH∗>WL∗, the price of higher performance products in the electronic channel is more than the price of lower performance products in the retailer channel, also more than the wholesale price of lower performance products. The manufacturer profit is obviously higher than retailer profit, and the profit of the centralized supply chain is slightly above the profit of the decentralized supply chain. As the parameter θH increases, PNLH∗ increases, but PSLH∗ decreases, and WL∗ remains the same; the retailer profit reduces gradually, while the other three profits firstly decrease and then increase and, finally, come together. And the difference between the total profit of centralized decision and of decentralized decision ΔΠZLH will be smaller and smaller.


### 4.3.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θH=0.9, and μ=1, and the range of parameter θL is 0,θH. When the supply chain is decentralized, the optimal prices are WL∗=0.3+θL/2, PNLH∗=0.65, and PSLH∗=2.2θL+0.27/3.6, and the retailer profit is ΠR⟶DDLH=0.4θL−0.272/14.4θL0.9−θL, the manufacturer profit is ΠM⟶DDLH=−0.34θL2+0.234θL+0.0729/7.2θL0.9−θL, and the supply-chain profit is ΠZ⟶DDHL∗=−0.52θL2+0.252θL+0.2187/14.4θL0.9−θL. When the supply chain is centralized, the optimal prices are PZSLH∗=0.3+θL/2 and PZNLH∗=0.65, and the supply-chain profit is ΠZ⟶CDHL∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of the parameter θL on optimal prices and the optimal profits is shown in Figure 12.Figure 12
The influence of parameterθL on optimal decisions.
(a)
(b)Similarly, according to Lemma5, we can get the range of parameter θL is 0.68<θL<0.84. In this interval, the optimal prices have following relationship: PNLH∗>PSLH∗>WL∗. The sale price of higher performance products in the electronic channel is significantly more than the sale price of lower performance products in the retail channel, while the price of the retail channel is slightly higher than wholesale price WL∗; and the manufacturer profit is higher than the retailer profit. As the parameter θL increases, PSLH∗ and WL∗ increase, but PNLH∗ remains the same; all of the profit functions increase, and the speed of increasing is faster and faster. When the value of the parameter θL exceeds 0.8, the retailer profit increases from zero, and the total profit of centralized decision is slightly higher than the total profit of decentralized decision.Based on the analysis shown above, we find that the retailer profits of model I is more than that of model II, and the retailer profits is almost zero in model II. Although the manufacturer is the leader, it needs to take into account the retailer profits. Therefore, the manufacturer selects model II, i.e., sell higher performance products in the retail channel, and sell lower performance products in the electronic channel.


## 4.3.1. The Influence of ParameterθH on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θL=0.6, and μ=1, and the range of parameter θH is θL,1. In the decentralized supply chain, the optimal prices are WL∗=0.45, PNLH∗=0.4+θH/2, and PSLH∗=0.24+1.5θH/4θH, and then the maximum profit of the retailer is ΠR⟶DDLH=0.24−0.3θH2/9.6θHθH−0.6; meanwhile, the maximum profit of the manufacturer is ΠM⟶DDLH=1.2θH3−1.59θH2+0.624θH−0.0576/4.8θHθH−0.6, and the supply-chain profit is ΠZ⟶DDHL∗=θH3−1.2875θH2+0.46θH−0.024/4θHθH−0.6. In the centralized supply chain, the optimal prices are PZSLH∗=0.45 and PZNLH∗=0.4+θH/2, and the supply-chain profit is ΠZ⟶CDHL∗=θH2−1.25θH+0.4/4θH−0.6. The influence of the parameter θH on optimal prices and the optimal profits is shown in Figure 11.Figure 11
The influence of parameterθH on optimal decisions.
(a)
(b)According to Lemma5, it needs to meet the following prerequisite: PS/θL≤PN/θH and PN−PS/θH−θL≤1, and then we can deduce that the range of the parameter θH is 0.67<θH<0.8. In this interval, the optimal prices have the following relationship: PNLH∗>PSLH∗>WL∗, the price of higher performance products in the electronic channel is more than the price of lower performance products in the retailer channel, also more than the wholesale price of lower performance products. The manufacturer profit is obviously higher than retailer profit, and the profit of the centralized supply chain is slightly above the profit of the decentralized supply chain. As the parameter θH increases, PNLH∗ increases, but PSLH∗ decreases, and WL∗ remains the same; the retailer profit reduces gradually, while the other three profits firstly decrease and then increase and, finally, come together. And the difference between the total profit of centralized decision and of decentralized decision ΔΠZLH will be smaller and smaller.


## 4.3.2. The Influence of ParameterθL on Optimal Decisions

SetCH¯=0.4, CL¯=0.3, θH=0.9, and μ=1, and the range of parameter θL is 0,θH. When the supply chain is decentralized, the optimal prices are WL∗=0.3+θL/2, PNLH∗=0.65, and PSLH∗=2.2θL+0.27/3.6, and the retailer profit is ΠR⟶DDLH=0.4θL−0.272/14.4θL0.9−θL, the manufacturer profit is ΠM⟶DDLH=−0.34θL2+0.234θL+0.0729/7.2θL0.9−θL, and the supply-chain profit is ΠZ⟶DDHL∗=−0.52θL2+0.252θL+0.2187/14.4θL0.9−θL. When the supply chain is centralized, the optimal prices are PZSLH∗=0.3+θL/2 and PZNLH∗=0.65, and the supply-chain profit is ΠZ⟶CDHL∗=−0.1θL2+0.01θL+0.081/4θL0.9−θL. The influence of the parameter θL on optimal prices and the optimal profits is shown in Figure 12.Figure 12
The influence of parameterθL on optimal decisions.
(a)
(b)Similarly, according to Lemma5, we can get the range of parameter θL is 0.68<θL<0.84. In this interval, the optimal prices have following relationship: PNLH∗>PSLH∗>WL∗. The sale price of higher performance products in the electronic channel is significantly more than the sale price of lower performance products in the retail channel, while the price of the retail channel is slightly higher than wholesale price WL∗; and the manufacturer profit is higher than the retailer profit. As the parameter θL increases, PSLH∗ and WL∗ increase, but PNLH∗ remains the same; all of the profit functions increase, and the speed of increasing is faster and faster. When the value of the parameter θL exceeds 0.8, the retailer profit increases from zero, and the total profit of centralized decision is slightly higher than the total profit of decentralized decision.Based on the analysis shown above, we find that the retailer profits of model I is more than that of model II, and the retailer profits is almost zero in model II. Although the manufacturer is the leader, it needs to take into account the retailer profits. Therefore, the manufacturer selects model II, i.e., sell higher performance products in the retail channel, and sell lower performance products in the electronic channel.


## 5. Conclusions and Managerial Implications

Based on the supply chain selling two differentiated products, this paper discusses optimal prices for the manufacturer and the retailer, in the single retail channel and dual-channel supply chains, respectively. Then, we can get the influence of parameters on optimal prices and optimal profit by numerical analysis. We identify five important managerial implications as follows:First, in model I and model II, when WTP is close to 1, the best choice for strategic customers is to buy the higher performance products. When WTP nears zero, strategic customers will leave the market.Second, in the single retail channel and dual-channel supply-chain model I, if the manufacturer wants to expand the product demand, it needs to improve the satisfaction of strategic customers about lower performance productsθL. While the satisfaction of strategic customers about higher performance products θH is increasing, the product demand remains the same, i.e., some strategic customers who buy the lower performance products change to buy the higher performance products.Third, in the dual-channel supply-chain model II, when the supply chain is decentralized, the manufacturer needs to improve the satisfaction of strategic customers with higher performance productsθH or with lower performance products θL to expand the product demand. But when the supply chain is centralized, the manufacturer needs to increase satisfaction with lower performance products θL to expand the product demand.Fourth, when the manufacturer is dominant in the Stackelberg model, the retailer does not want the manufacturer to open the electronic channel. Through numerical analysis, we can find that the difference between retailer profit and manufacturer profit in the single retail channel is less than that in the dual channel. Once the manufacturer decides to open an electronic channel, i.e., adopt a dual-channel strategy, the retailer wants to sell the higher performance products rather than the lower performance products because the retailer profit is almost zero in dual-channel model II.Fifth, in dual-channel model I, the satisfaction of strategic customers with higher performance productsθH is higher and the parameter θL is lower, and then the retailer can get more benefits.Based on the above conclusion, we can get that if the manufacturer wants to expand the product demand in the single traditional channel and dual-channel model I, the key is to improve the satisfaction of lower performance productsθL. While there are two cases in the dual-channel model I, if the decision is decentralized, the manufacturer should improve the satisfaction of higher performance products θH or the satisfaction of lower performance products θL to expand the product demand, and if the decision is centralized, the manufacturer can only expand the market demand by improving the satisfaction of lower performance products θL. From the perspective of enterprise profit, the retailer does not want the dominated manufacturer to open electronic channels. If the manufacturer must open electronic channels, the retailer also wants to use dual-channel model I. If the manufacturer adopts dual-channel model II, the parameter θH is higher and θL is lower, and the profits of the retailer are higher.In this paper, our study considers all customers are strategic; actually, the customers own individual differences, so the customers are not all strategic in a real market. Further research is needed in this area. In addition, we assume that the retail channel sells one kind of product, while the electronic channel sells another kind of product. The effects of selling two kinds of differentiated products in every channel simultaneously should be researched.



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*Source: 9475150-2020-01-18.xml*