Strategy Analysis for Retailer-Leading Supply Chain under Buyback Contract with Focus Theory of Choice

Abstract

This paper investigates a retailer-leading two-tier supply chain with a buyback contract under market demand uncertainty, where the retailer first announces a potential maximal order quantity and the supplier then provides a unit wholesale price to influence the retailer’s order quantity. In recent years, an increasing number of experimental studies have reported that even in repeated multi-turn games, the decisions of suppliers viewed as newsvendors deviate significantly from the expectation-maximizing options. In light of this observation, we employ the focus theory of choice to characterize suppliers’ behavioral tendencies and theoretically derive optimal unit wholesale prices based on suppliers’ focus preferences. With these results, we further explore how suppliers’ foci may influence the interactions between the retailer and the supplier. We find that when the supplier takes the positive evaluation system as the decision criterion, optimism degree and confidence level have a negative effect on the wholesale price and a positive effect on the final order quantity, and the final order quantity must be located between the most possible market demand and the upper limit of the market demand. This paper provides a behavioral perspective to analyze suppliers’ optimal responses and their influences on retailers’ decision-making. Theoretical and numerical analyses gain managerial insights for retailers to make decisions when faced with suppliers with different focus preferences.

1. Introduction

Buyback contracts, as an effective mechanism to reduce retailers’ inventory costs, thereby encouraging retailers to order sufficient products to satisfy uncertain market demand, have a wide range of applications in the field of supply chain, such as the publishing industry, computer software and hardware, pharmaceutical, clothing, cosmetics and other industries [1]. In the buyback contract, retailers order products from suppliers at a wholesale price before the selling season, and all unsold products at the end-of-selling period are returned to suppliers at an agreed buyback price [2]. Through buyback contracts, the risk of leftover inventory is transferred from retailers to suppliers. On the other hand, many studies on marketing and inventory management report that power and status in a supply chain are gradually shifting from suppliers to retailers [3,4,5,6,7]. Based on these observations, this paper investigates a retailer-leading two-tier supply chain with a buyback contract. The supply chain consists of a single retailer and a single supplier playing a Stackelberg game, where the retailer as the leader first announces a potential maximal order quantity under market demand uncertainty, and then the supplier as the follower provides a unit wholesale price to influence the retailer’s order quantity.

For the retailer-leading buyback contract supply chain under uncertainty, the classical models assume that followers (suppliers) are completely rational and make decisions according to the principle of expectation maximization. However, many experimental studies have shown that even in repeated multi-turn games, the choices of suppliers regarded as newsvendors deviate significantly from theoretical predictions based on expectation maximization [8,9,10,11,12]. In recent years, a growing body of behavioral literature has shown that salient (attention-grabbing) information rather than the expected value/utility plays a critical role in human decision-making [13,14,15,16,17,18,19,20]. With this understanding, Guo [21] argues that human decision-makers are boundedly rational and suffering from bounded attention, and hence proposes the focus theory of choice (FTC). This theory suggests that individuals do not choose the expected-value/utility-maximizing option under risk or uncertainty, but evaluate a decision based on some associated events (called the foci of a decision) which are most salient to the decision-makers. The FTC models and axiomatizes the procedural rationality articulated first by Simon [22] and resolves several well-known anomalies, such as the St. Petersburg, Allais, and Ellsberg paradoxes, preference reversals, and the violations of stochastic dominance and transitivity. The core of FTC is that the most salient event corresponds to the most preferred decision, where salience depends on the specific focus preference of the decision-maker, reflecting different behavioral patterns in human decision-making processes. This assertion is supported by the results of psychological experiments [23,24]. The FTC has been widely applied to solve several practical problems, such as the newsvendor problem [25,26], the sealed-bid auction problem [27], the production planning problem [28], and the supply chain problem with recycling strategies [29].

As previously mentioned, although it is well-known that behavioral factors are quite important in operational research, it is still difficult to incorporate personality characteristics of decision-makers into the mathematical models. FTC provides a theoretical basis to construct the behavioral models in operational research. Using FTC, the model for a retailer-leading supply chain buyback contract can be built while considering decision-makers’ behavioral features. In addition, decision bias found in experimental studies can be explained by personality characteristics in FTC. In this paper, we adopt the FTC to illustrate how the introduction of focus preferences at the supplier level may influence the retailer’s decision-making in a retailer-leading supply chain with a buyback contract. Within the framework of FTC, we assume that the supplier makes a decision under market demand uncertainty through the following two-step strategy. In the first step, for every possible wholesale price, the supplier focuses on his/her most salient demands by examining the payoff of each possible demand and its corresponding probability, referred to as the foci of the wholesale price. In the second step, the supplier chooses the optimal wholesale price by evaluating all possible wholesale prices and their foci. We theoretically derive optimal wholesale prices with different focus preferences, and further discuss the implied decision-making insights from the perspective of behavioral patterns. Finally, by constructing the Stackelberg game model, we analyze the influence of suppliers’ optimal responses based on the focus preferences on the decision-making of risk-neutral retailers in the retailer-leading supply chain with a buyback contract. Numerical comparisons between the classical buyback contract model and the proposed model are also carried out through illustrative examples. Theoretical and numerical analyses gain managerial insights for retailers to make decisions when faced with suppliers with different focus preferences.

The remainder of this paper is organized as follows. In Section 2, we review the classical retailer-leading buyback contract model. In Section 3, we establish a supplier’s decision model under the positive evaluation system. In Section 4, we theoretically derive optimal solutions based on the supplier’s positive foci, as Figure 1 illustrates. In Section 5, we apply the proposed supplier models to the retailer-leading buyback contract and analyze the influence of the supplier’s focus preferences on the retailer’s decision-making. Numerical comparisons between the classical buyback contract model and the proposed models are carried out in Section 6. Finally, Section 7 concludes the paper.

2. Classical Buyback Contract Model

This paper considers a retailer-leading supply chain with a buyback contract in a single retailer-supplier setting, where the retailer first announces a potential maximal order quantity $q$ to influence the unit wholesale price $w$ offered by the supplier, and then determines an order quantity $q-\beta w$, where $\beta$ is the sensitivity of the retailer’s order quantity to the supplier’s wholesale price. At the end-of-selling period, all unsold products will be repurchased by the supplier at a pre-agreed unit buyback price $b$. Based on the above description, we understand that the retailer and supplier play a Stackelberg game, where the retailer as the leader decides the potential maximal order quantity and the supplier as the follower determines the unit wholesale price.

We assume that the market demand is a random variable $X$ with probability density function $f\left(·\right)$ and cumulative distribution function $F\left(·\right)$. We further make the following assumptions on the uncertain market demand:

(i)

The market demand lies on an interval $\left. [l,h\right]$ where $0\le l<h$.

(ii)

The probability density function $f\left(x\right)>0$ is continuous and strictly quasi-concave in the interval $\left. [l,h\right]$, and $\exists m\in \left. [l,h\right]$ such that

$$f\left(m\right)={\max }_{x\in \left. [l,h\right]}f\left(x\right)$$

Since the supplier determines the unit wholesale price after observing the potential maximal order quantity given by the retailer, we analyze the supplier’s decision first. For a given $q$ by the retailer and a realization of the random market demand $x$, the supplier’s payoff denoted by $v\left(x,q,w\right)$ can be given as follows:

$$v\left(x,q,w\right)=\left\{\begin{array}{cc}\left(w-c\right)\left(q-\beta w\right)-b\left(q-\beta w-X\right),& if X<q-\beta w,\\ \left(w-c\right)\left(q-\beta w\right),& if X\ge q-\beta w,\end{array}\right.$$

(1)

where $c$ denotes the supplier’s unit production cost, $b$ denotes the supplier’s unit buyback price, which is determined in advance ($0<b<c$), $w$ is the unit wholesale price which is the supplier’s decision variable.

Let $w\left(q\right)$ be the supplier’s optimal response. Then, the retailer’s payoff denoted by $r\left(x,q,w\right)$ is given by

$$r\left(x,q,w\right)=\left\{\begin{array}{cc}pX-w\left(q\right)\left(q-\beta w\left(q\right)\right)+b\left(q-\beta w\left(q\right)-X\right),& if X<q-\beta w\left(q\right),\\ \left(p-w\left(q\right)\right)\left(q-\beta w\left(q\right)\right),& if X\ge q-\beta w\left(q\right),\end{array}\right.$$

(2)

where $p>0$ is the retailer’s unit selling price, $q$ is the potential maximal order quantity which is the retailer’s decision variable, and $q-\beta w\left(q\right)$ is the order quantity.

For the retailer, we assume $q\in \left. [l+\beta \left(p-b\right),h+\beta c\right]$ where $0<\beta <\frac{h-l}{p-b-c}$. For the supplier, we understand that $w\in \left. [c,p-b\right]$ where the upper bound can avoid the retailer from profiting directly from the buyback. Thus, we have

$$l\le l+\beta \left(p-b-w\right)\le q-\beta w\le h+\beta \left(c-w\right)\le h.$$

Since $X$ is a random variable, we know that the payoffs (1) and (2) are both random variables. The standard model assumes that the players are perfectly rational and self-interested, and the objective of each player is to maximize his/her expected payoff. Denote the expected values of $v\left(x,q,w\right)$ and $r\left(x,q,w\right)$ as $E_v$ and $E_r$, respectively. We have

$$E_v=\left(w-c\right)\left(q-\beta w\right)-b{{\displaystyle \int }}_l^{q-\beta w}\left(q-\beta w-x\right)f\left(x\right)\mathrm{d}x,$$

and

$$E_r=\left(p-w\left(q\right)\right)\left(q-\beta w\left(q\right)\right)-\left(p-b\right){{\displaystyle \int }}_l^{q-\beta w\left(q\right)}F\left(x\right)dx.$$

The first-order and second-order derivatives of $E_s$ with respect to $w$ are

$$\frac{\partial E_v}{\partial w}=q-2\beta w+\beta c+b\beta F\left(q-\beta w\right),$$

and

$$\frac{{\partial }^2{E}_v}{\partial w^2}=-2\beta -b{\beta }^{2}f\left(q-\beta w\right),$$

respectively. Since $\frac{{\partial }^2{E}_v}{\partial w^2}<0$ and $\frac{\partial E_v}{\partial w}>0$ at $w=c$, the optimal unit wholesale price based on the expected-value-maximization is determined by its first-order condition, i.e., $w\left(q\right)$ is the unique solution to the following implicit function equation on $w$:

$$q-2\beta w+\beta c+b\beta F\left(q-\beta w\right)=0,$$

or $w\left(q\right)$ takes the upper bound, i.e.,

$$w\left(q\right)=p-b \mathrm{when} q-2\beta \left(p-b\right)+\beta c+b\beta F\left(q-\beta \left(p-b\right)\right)>0.$$

Substitute $w\left(q\right)$ into the retailer’s expected payoff function $E_r$. The first derivative of $E_r$ with respect to $q$ can be given as follows:

$$\frac{\partial E_r}{\partial q}=p-c-\frac{2\left(q-\beta w\right)}{\beta }-b\left(q-\beta w\right)f\left(q-\beta w\right)-pF\left(q-\beta w\right)$$

$$=\left. [1-F\left(q-\beta w\right)\right]\left. [p-b\frac{\left(q-\beta w\right)f\left(q-\beta w\right)}{1-F\left(q-\beta w\right)}-\frac{2}{\beta }\frac{q-\beta w}{1-F\left(q-\beta w\right)}-\frac{c}{1-F\left(q-\beta w\right)}\right].$$

As in [30,31,32], we define $g\left(x\right)=xf\left(x\right)/\overline{F}\left(x\right)$ and say that a distribution has an increasing generalized failure rate (IGFR) if $g\left(x\right)$ is weakly increasing for all $x$ such that $F\left(x\right)<1$. Since $F\left(x\right)$ is an IGFR, $\frac{d\frac{\left(q-\beta w\right)f\left( q-\beta w \right)}{1-F\left(q-\beta w\right)} }{d\left(q-\beta w\right)}>0$, we can obtain $\frac{d^2{E}_r}{d{\left(q-\beta w\right)}^2}<0$, and the optimal solution determined is unique at this time. The retailer’s optimal potential maximal order quantity $q^{*}$ is determined by the equation

$$p-b\frac{\left(q^{*}-\beta w\right)f\left(q^{*}-\beta w\right)}{1-F\left(q^{*}-\beta w\right)}-\frac{2}{\beta }\frac{q^{*}-\beta w}{1-F\left(q^{*}-\beta w\right)}-\frac{c}{1-F\left(q^{*}-\beta w\right)}=0.$$

In recent years, a large number of experimental studies have shown that the real decision-making behavior systematically deviates from the results predicted by the expected-value-maximization theory, and the results of the normative model cannot reflect the real decision-making behavior [8,9,10,11,12]. This means that the decision-making of supply chain members is not completely rational and will be affected by many factors. So far, there is less literature that studies how wholesale price affects the optimal decision of supply chain members under behavioral preference in the retailer-leading buyback contract model. Hence, this paper will take the focus theory of choice as the supplier’s decision-making criteria to conduct research, so as to improve the explanatory power and guiding force of the theoretical results on the decision-making behavior of the supply chain parts.

3. Supplier’s Decision Model with the Focus Theory of Choice

It can be seen from (1) that for any given $q$, $v_s$ reaches its maximum value at $x=q-\beta w$. To reconstruct the buyback contract model under the positive evaluation system, the probability density function is converted into a relative likelihood function, and the supplier’s profit function is converted into a satisfaction function.

Definition 1.

([21]) Let $f:\left. [l,h\right]\to {ℝ}_{+}$ be the density function of stochastic market demand.$\pi :\left. [l, h\right]\to \left. [0,1\right]$ is called a relative likelihood function if

$$\pi \left(x_1\right)>\pi \left(x_2\right)⟺f\left(x_1\right)>f\left(x_2\right), \forall x_1,x_2\in \left. [l, h\right],$$

and $\exists x_c\in \left. [l, h\right]$ such that $\pi \left(x_c\right)={\max }_{x\in \left. [l, h\right]}\pi \left(x\right)=1$.

For any $x\in \left. [l,h\right]$, $\pi \left(x\right)$ is called the relative likelihood degree of $x$. The relative likelihood function is a normalized probability density function. In this research, the relative likelihood function can be defined as follows:

$$\pi \left(x\right)=\frac{f\left(x\right)}{{\max }_{x\in \left. [l, h\right]}f\left(x\right)}$$

(3)

Since $f\left(x\right)$ is strictly quasi-concave, we know that $\pi \left(x\right)$ strictly increases on $\left. [l,m\right]$ and strictly decreases on $\left. [m,h\right]$, $\pi \left(x\right)$ reaches the unique maximum at $x=m$.

Definition 2.

([21]) Let $q$ be given and $V\left(q\right)$ be the range of the supplier’s payoff function $v\left(x,q,w\right)$ on $\left(x,w\right)\in \left. [l,h\right]\times \left. [c, p-b\right]$. $u:V\to \left. [0,1\right]$ is called the satisfaction function if $\forall v_1 , v_2\in V$ satisfies $v_1>v_2⟺u\left(v_1\right)>u\left(v_2\right)$ and $\exists v_0\in V$ such that $u\left(v_0\right)=ma{x}_{v\in V}u\left(v\right)=1$.

For any given potential maximal order quantity $q$, $u\left(x,q,w\right)$ represents the supplier’s satisfaction level to the payoff when the market demand is 𝑥. In this research, the satisfaction function is defined as

$$u\left(x,q,w\right)=\frac{v\left(x,q,w\right)-\underset{x,w}{\min }v\left(x,q,w\right)}{\underset{x,w}{\max }v\left(x,q,w\right)-\underset{x,w}{\min }v\left(x,q,w\right)}$$

(4)

After the payoff and probability functions are converted to the satisfaction and relative likelihood functions, the decision problem in the focus theory of choice will take satisfaction function and relative likelihood function as basic inputs. The supplier chooses the most salient outcomes $\left(\pi \left(x\right),u\left(x,q,w\right)\right)$ from all given possible decision actions, namely the focus, which usually has a relative high satisfaction level and a relative high likelihood degree.

For any given potential maximal order quantity $q$ under the positive evaluation system, denote $X\left(q,w\right)$ as the set of the following optimal solutions:

$$\underset{x\in \left. [l,h\right]}{\max }\min \left\{\phi \ast \pi \left(x\right),u\left(x,q,w\right)\right\}$$

(5)

The parameter $\phi$ is a positive real number and denoted as a scaling factor that directly affects whether the focus has a higher likelihood or satisfaction. When $\phi$ increases, $\phi \ast \pi \left(x\right)$ will increase, making $u\left(x,q,w\right)$ more likely to appear in minimization, resulting in an optimal solution with a relative high satisfaction level and a relative low likelihood degree. Therefore, $\phi$ can be used to measure the supplier’s emphasis on profit, that is, the degree of optimism. Next, we define the positive focus of the wholesale price $w$.

Definition 3.

For any given$q$, if there is only one element in$X\left(q,w\right)$, then it is the positive focus of the wholesale price$w$, denoted by$x_{+}\left(q,w\right)$. If there exists more than one element in$X\left(q,w\right)$and$\nexists x_{+}\in X\left(q,w\right)$such that$\pi \left(x_{+}\right)>\pi \left(x_{+}\left(q,w\right)\right)$,$u\left(x_{+},q,w\right)\ge u\left(x_{+}\left(w\right),q,w\right)$or$\pi \left(x_{+}\right)\ge \pi \left(x_{+}\left(q,w\right)\right)$,$u\left(x_{+},q,w\right)>u\left(x_{+}\left(w\right),q,w\right)$for$x_{+}\left(q,w\right)\in X\left(q,w\right)$, then$x_{+}\left(q,w\right)$is called the positive focus of the wholesale price$w$.

From Definition 3, we know that $x_{+}\left(q,w\right)$ is the most favorable focus for wholesale price $w$. If multiple positive foci exist for a given wholesale price $w$, the set of positive foci $x_{+}\left(q,w\right)$ is denoted as $X_{+}\left(q,w\right)$ where the subscript $“+”$ indicates the positive evaluation system.

After the above procedure, the optimal wholesale price will be determined in the next step. Since $\pi \left(x\right)$ and $u\left(x,q,w\right)$ are both strictly quasi-concave continuous functions for $x\in X$, there is one unique element in $X_{+}\left(q,w\right)$ denoted by $x_{+}\left(q,w\right)$ and referred to the positive focus of wholesale price $w$. Next, among all the positive foci for different actions (wholesale price), we seek the optimal $w_{+}\left(q\right)$ by the following optimization problem:

$$\underset{w\in \left. [c, p-b\right]}{\max }\min \left\{\kappa \ast \pi \left(x_{+}\left(q,w\right)\right),u\left(x_{+}\left(q,w\right),q,w\right)\right\},$$

(6)

where parameter $\kappa$ is a positive real number. Similar to the interpretation of parameter $\phi$, as $\kappa$ increases, $\kappa \ast \pi \left(x_{+}\left(q,w\right)\right)$ increases relative to $u\left(x_{+}\left(q,w\right),q,w\right)$, resulting in a positive focus that has a relative high satisfaction level but a relative low likelihood degree. Hence, the parameter $\kappa$ can measure the confidence index of the supplier to his/her decision; increasing $\kappa$ represents that the supplier is willing to somewhat sacrifice probability to pursue a higher payoff, and decreasing $\kappa$ represents that the supplier is willing to somewhat sacrifice payoff to pursue a higher probability.

The set of optimal wholesale price $w_{+}\left(q\right)$ is denoted by $W_{+}\left(q\right)$. For any $w_1,w_2\in \left. [c,p-b\right]$, if $\pi \left(x_{+}\left(q,w_1\right)\right)\ge \pi \left(x_{+}\left(q,w_2\right)\right)$ and $u\left(x_{+}\left(q,w_1\right),q,w_1\right)\ge u\left(x_{+}\left(q,w_2\right),q,w_2\right)$, then $\min \left\{\kappa \ast \pi \left(x_{+}\left(q,w_1\right)\right),u\left(x_{+}\left(q,w_1\right),\mathrm{q},w_1\right)\right\}\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q,w_2\right)\right),u\left(x_{+}\left(q,w_2\right),q,w_2\right)\right\}$. This means that equation (6) generates the optimal wholesale price, whose focus has a relative high probability and a relative high satisfaction.

Definition 4.

For any given $q$, if there is only one element in$W_{+}\left(q\right)$, then it is the optimal wholesale price under the positive evaluation system, denoted by$w_{+}^{*}\left(q\right)$. If there exists more than one element in$W_{+}\left(q\right)$and$\nexists w_{+}\in W_{+}\left(q\right)$such that$\pi \left(x_{+}\left(q,w_{+}\right)\right)>\pi \left(x_{+}\left(q,w_{+}^{*}\left(q\right)\right)\right)$,$u\left(x_{+}\left(q,w_{+}\right),q,w_{+}\right)\ge u\left(x_{+}\left(q,w_{+}^{*}\left(q\right)\right), q,w_{+}^{*}\left(q\right)\right)$or$\pi \left(x_{+}\left(q,w_{+}\right)\right)\ge \pi \left(x_{+}\left(q,w_{+}^{*}\left(q\right)\right)\right)$,$u\left(x_{+}\left(q,w_{+}\right),q,w_{+}\right)>u\left(x_{+}\left(q,w_{+}^{*}\left(q\right)\right),q, w_{+}^{*}\left(q\right)\right)$for$w_{+}^{*}\left(q\right)\in W_{+}\left(q\right)$, then$w_{+}^{*}\left(q\right)$is the optimal wholesale price under the positive evaluation system.

For any given $q$, from Definition 4, we know that the optimal wholesale price $w_{+}^{*}\left(q\right)$ weakly dominates all other elements in $W_{+}^{*}\left(q\right)$ if it contains multiple foci under the positive evaluation system.

The retailer-leading buyback contract model based on the focus theory of choice in this paper consisting of (5) and (6) is a bilevel programming problem, where (5) is the lower-level program and (6) is the upper-level program. Since both the upper- and lower-level optimization problems are non-smooth, the proposed model cannot be solved by existing optimization methods. In the following section, the optimal solution will be theoretically derived and the properties of the optimal solution will be explained.

4. Supplier’s Optimal Wholesale Price with Focus Preference

In this section, some lemmas and theorems are proposed to derive the optimal solution firstly, and the properties of the optimal solution are described by the positive evaluation system. As described below, Lemma 1 will be proposed to characterize the focus demand for any given wholesale price under the positive evaluation system, which is also the solution to the lower-level optimization problem (5).

Lemma 1.

Let$q$be given, for any wholesale price$w\in \left. [c, p-b\right]$, the positive focus$x_{+}\left(q,w\right)$is characterized as follows:

(i) When $\phi >\frac{u\left(q-\beta w, q,w\right)}{\pi \left(q-\beta w\right)}$, then

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}q-\beta w,& if w\in \left. [c,\frac{q-m}{\beta }\right),\\ m,& if w\in \left. [\frac{q-m}{\beta },p-b\right],\end{array}\right.$$

(ii) When $\frac{u\left(m,q,w\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(q-\beta w,q,w\right)}{\pi \left(q-\beta w\right)}$, then

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}{x}_r\left(q,w\right) ,& if w\in \left. [c,\frac{q-m}{\beta }\right),\\ m,& if w\in \left. [\frac{q-m}{\beta },p-b\right], \end{array}\right.$$

where $x_r\left(q,w\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ on $\left. [m,q-\beta w\right]$.

(iii) When $0<\phi <\frac{u\left(m,q,w\right)}{\pi \left(m\right)}$, then $x_{+}\left(q,w\right)=m$.

Proof of Lemma 1.

For any potential maximal order quantity $q$, as $u\left(q-\beta w,q,w\right)\ge u\left(m,w\right)$ and $0\le \pi \left(q-\beta w\right)\le \pi \left(m\right)$, we have $\frac{u\left(q-\beta w,q,w\right)}{\pi \left(q-\beta w\right)}\ge \frac{u\left(m,q,w\right)}{\pi \left(m\right)}$.

(i) If any potential maximal order quantity $q$ satisfies $q-\beta w\in \left. [l,m\right]$, we know that $u\left(x,q,w\right)$ stays the same for $q-\beta w$ on $\left. [q-\beta w,m\right]$ and $\phi \ast \pi \left(x\right)$ strictly increase on $x\in \left. [q-\beta w,m\right]$ for the monotonicity of $u\left(x,q,w\right)$ and $\pi \left(x\right)$. For any $x\ne m$, as $\phi \ast \pi \left(q-\beta w\right)>u\left(q-\beta w,q,w\right)$, we have $\phi \ast \pi \left(m\right)>\phi \ast \pi \left(x\right)$, $\phi \ast \pi \left(m\right)>u\left(m,q,w\right)$ and $u\left(m,q,w\right)=u\left(x,q,w\right)$ for $x\in \left. [q-\beta w,m\right]$. Hence, we can obtain that $\min \left\{\phi \ast \pi \left(m\right),u\left(m,q,w\right)\right\}=u\left(m,q,w\right)\ge u\left(x,q,w\right)\ge \min \left\{\phi \ast \pi \left(x\right),u\left(x,q,w\right)\right\}$. Based on (6) and the definition of the positive focus, we have $x_{+}\left(q,w\right)=m$. Then, if any order quantity $q-\beta w\in \left(m,h\right]$, for any $x\ne q-\beta w$, as $\phi \ast \pi \left(q-\beta w\right)>u\left(q-\beta w,q,w\right)$, we have $\min \left\{\phi \ast \pi \left(q-\beta w\right),u\left(q-\beta w,q,w\right)\right\}=u\left(q-\beta w,q,w\right)>u\left(x,q,w\right)\ge \min \left\{\phi \ast \pi \left(x\right),u\left(x,q,w\right)\right\}$. Based on (6) and the definition of the positive focus, we have the positive focus $x_{+}\left(q,w\right)=q-\beta w$.

(ii) If any potential maximal order quantity $q$ satisfies $q-\beta w\in \left. [l,m\right]$, for any $x\ne m$, as $\phi \ast \pi \left(m\right)\ge u\left(m,q,w\right)$ and $\phi \ast \pi \left(q-\beta w\right)\le u\left(q-\beta w,q,w\right)$, we know that $u\left(x,q,w\right)$ stays the same for $q-\beta w$ on $\left. [q-\beta w,m\right]$ and $\phi \ast \pi \left(x\right)$ strictly increases for $x$ on $\left. [q-\beta w,m\right]$ for the monotonicity of $u\left(x,q,w\right)$ and $\pi \left(x\right)$. Thus, we have $\min \left\{\phi \ast \pi \left(m\right),u\left(m,q,w\right)\right\}=u\left(m,q,w\right)\ge u\left(x,q,w\right)\ge \min \left\{\phi \ast \pi \left(x\right),u\left(x,q,w\right)\right\}$. Based on (6) and the definition of the positive focus, we have $x_{+}\left(q,w\right)=m$. For any potential maximal order quantity $q$ satisfies $q-\beta w\in \left(m,h\right]$, as $u\left(q-\beta w,q,w\right)\ge \phi \ast \pi \left(q-\beta w\right)$ and $u\left(m,q,w\right)\le \phi \ast \pi \left(m\right)$, we know that $u\left(x,q,w\right)$ strictly increases for $q-\beta w$ on $\left(m,q-\beta w\right]$ and $\phi \ast \pi \left(x\right)$ strictly decreases for $q-\beta w$ on $\left(m,q-\beta w\right]$. Thus, there exists a unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ on $\left. [m,q-\beta w\right]$, denoted by $x_r\left(w,\phi \right)$. For any $x\ne x_r\left(w,\phi \right)$, we have $\min \left\{\phi \ast \pi \left(x_r\left(w,\phi \right)\right),u\left(x_r\left(w,\phi \right),q,w\right)\right\}>\min \left\{\phi \ast \pi \left(x\right),u\left(x,q,w\right)\right\}$, which means $x_{+}\left(q,w\right)=x_r\left(w,\phi \right)$.

(iii) For any demand $x\ne m$, as $\phi \ast \pi \left(m\right)<u\left(m,q,w\right)$, we can have $\min \left\{\phi \ast \pi \left(m\right),u\left(m,q,w\right)\right\}=\phi \ast \pi \left(m\right)>\phi \ast \pi \left(x\right)\ge \min \left\{\phi \ast \pi \left(x\right),u\left(x,q,w\right)\right\}$. It means that $x_{+}\left(q,w\right)=m$. □

Lemma 1 characterizes the importance of the parameter $\phi$ in determining the positive focus for any given wholesale price $w\in \left. [c,\frac{q-m}{\beta }\right)$. Based on Lemma 1, the following results can be confirmed.

Theorem 1.

(1) If $\frac{c}{2}+\frac{q}{2\beta }\ge p$, the positive focus $x_{+}\left(q,w\right)$ decreases with wholesale price $w$ in the interval $\left. [c,\frac{q-m}{\beta }\right]$ whenever $\phi >0$, and equals to $m$ for any $w\in \left. [\frac{q-m}{\beta },p-b\right]$.

(2) If $\beta c+2h\le q\le \beta \left(c+b\right)+2l$, the positive focus $x_{+}\left(q,w\right)$ decreases with wholesale price $w$ in the interval $\left. [c,\frac{q-m}{\beta }\right]$ whenever $\phi >0$, and equals to $m$ for any $w\in \left. [\frac{q-m}{\beta },p-b\right]$.

Proof of Theorem 1.

(1) According to Lemma 1, for any $q-\beta w\in \left. [l,m\right]$, we know that $x_{+}\left(q,w\right)=m$ whatever $\phi >0$. Thus, the positive focus $x_{+}\left(q,w\right)$ is independent of wholesale price $w$ in this case.

In the case of $q-\beta w\in \left. [m,h\right]$, we have $x_{+}\left(q,\frac{q-m}{\beta }\right)=m$ whenever $\phi >0$. let $q_1,q_2\in \left. [m,h\right]$ and $q_1<q_2$. From Lemma 1, we know $x_{+}\left(q,\frac{q-q_i}{\beta }\right)\in \left. [m,q_i\right]$ for $i=1, 2$. Then, contradictions can be used to show the proof in the following. Suppose $x_{+}\left(q,\frac{q-q_1}{\beta }\right)>x_{+}\left(q,\frac{q-q_2}{\beta }\right)$, then we have $m\le x_{+}\left(q,\frac{q-q_2}{\beta }\right)<x_{+}\left(q,\frac{q-q_1}{\beta }\right)\le q_1<q_2\le h$. By the definitions of $v\left(x,q,w\right)$ and $u\left(x,q,w\right)$, it is easy to verify that

$$u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_1}{\beta }\right)>u\left(x_{+}\left(q,\frac{q-q_2}{\beta }\right),q,\frac{q-q_1}{\beta }\right)$$

(7)

and

$$u\left(x_{+}\left(q,\frac{q-q_2}{\beta }\right),q,\frac{q-q_2}{\beta }\right)=u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_2}{\beta }\right)$$

(8)

Considering $x_{+}\left(q,\frac{q-q_1}{\beta }\right)\ge m$, it follows from Lemma 1 that this situation is

$$\mathrm{either} \phi >\frac{u\left(q_1,q,\frac{q-q_1}{\beta }\right)}{\pi \left(q_1\right)} \mathrm{or} \frac{u\left(m,q,\frac{q-q_1}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(q_1,q,\frac{q-q_1}{\beta }\right)}{\pi \left(q_1\right)}$$

and if $\phi >\frac{u\left(q_1,q,\frac{q-q_1}{\beta }\right)}{\pi \left(q_1\right)}$, we have $x_{+}\left(q,\frac{q-q_1}{\beta }\right)=q_1$, otherwise $x_{+}\left(q,\frac{q-q_1}{\beta }\right)$ satisfies the equation $u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_1}{\beta }\right)=\phi \ast \pi \left(x_{+}\left(q,\frac{q-q_1}{\beta }\right)\right)$. In either above situation, we have

$$u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_1}{\beta }\right)\le \phi \ast \pi \left(x_{+}\left(q,\frac{q-q_1}{\beta }\right)\right)$$

(9)

Combining (9) and (7), it results in $u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_2}{\beta }\right)<\phi \ast \pi \left(x_{+}\left(q,\frac{q-q_1}{\beta }\right)\right)$. Since $x_{+}\left(q,\frac{q-q_2}{\beta }\right)\in X_{+}\left(q,\frac{q-q_2}{\beta }\right)$ and $x_{+}\left(q,\frac{q-q_1}{\beta }\right)\ne x_{+}\left(q,\frac{q-q_2}{\beta }\right)$, we further have

$$\begin{array}{c}u\left(x_{+}\left(q,\frac{q-q_2}{\beta }\right),q,\frac{q-q_2}{\beta }\right)\ge \min \left\{\phi \ast \pi \left(x_{+}\left(q,\frac{q-q_2}{\beta }\right)\right),u\left(x_{+}\left(q,\frac{q-q_2}{\beta }\right),q,\frac{q-q_2}{\beta }\right)\right\}\\ >\min \left\{\phi \ast \pi \left(x_{+}\left(q,\frac{q-q_1}{\beta }\right)\right),u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_2}{\beta }\right)\right\}\\ =u\left(x_{+}\left(q,\frac{q-q_1}{\beta }\right),q,\frac{q-q_2}{\beta }\right)\hfill \end{array}$$

It is clear that this formula and (8) clearly contradict each other. Thus, for any $q_1,q_2\in \left. [m,h\right]$, if $q_1<q_2$, then

$$x_{+}\left(q,\frac{q-m}{\beta }\right)\le x_{+}\left(q,\frac{q-q_1}{\beta }\right)<x_{+}\left(q,\frac{q-q_2}{\beta }\right).$$

Over the above discussion, the Theorem 1 can be proved.

(2) The proof is similar to (1). □

Theorem 1 demonstrates the relationship between the potential maximal order quantity and the positive focus. It shows that for any given wholesale price $w_i$ satisfying $q-\beta w_i\in \left. [l,h\right]$, $i=1,2,$ if $w_1<w_2$, then $q-\beta w_1>q-\beta w_2$ and the positive focus of $w_1$ is bigger than or equal to the positive focus of $w_2$. Meanwhile, this implies that for any wholesale price $w\in \left. [c,\frac{q-m}{\beta }\right]$, the positive decision-maker tends to pay more attention to the larger market demand when the wholesale price is lower. Since $\pi \left(x\right)$ is strictly increasing over $\left. [l,m\right]$ and strictly decreasing over $\left. [m,h\right]$, the following result is obvious.

Theorem 2.

(1) If $\frac{c}{2}+\frac{q}{2\beta }\ge p$, $\pi \left(x_{+}\left(q,w\right)\right)$ increases with wholesale price $w$ on $\left. [c,\frac{q-m}{\beta }\right]$ whenever $\phi >0$, and equals to $\pi \left(m\right)$ for any $w\in \left. [\frac{q-m}{\beta },p-b\right]$.

(2) If $\beta c+2h\le q\le \beta \left(c+b\right)+2l$, $\pi \left(x_{+}\left(q,w\right)\right)$ increases with wholesale price $w$ on $\left. [c,\frac{q-m}{\beta }\right]$ whenever $\phi >0$, and equals to $\pi \left(m\right)$ for any $w\in \left. [\frac{q-m}{\beta },p-b\right]$.

Theorem 2 illustrates that the relative likelihood function of the positive focus of $w$ is a quasi-concave function. When $\phi$ is sufficiently small as in Lemma 1(iii), it also holds when $0<\phi <\frac{u\left(m,q,w\right)}{\pi \left(m\right)}$ for any $w\in \left. [c, p-b\right]$. Meanwhile, it still holds for any wholesale price $w\in \left. [\frac{q-m}{\beta },p-b\right]$ regardless of the value of $\phi$, in this case that we have $\pi \left(x_{+}\left(q,w\right)\right)=\pi \left(m\right)$. To study the monotonicity of the function $u\left(x_{+}\left(q,w\right),q,w\right)$, the following lemma is given.

Lemma 2.

(1) If $\frac{c}{2}+\frac{q}{2\beta }\ge p$, for wholesale price $w\in \left. [c,\frac{q-m}{\beta }\right]$, the positive focus $x_{+}\left(q,w\right)$ is determined as follows:

(i) When $\phi <\min \left\{\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)},\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}\right\}$, then

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}{x}_r\left(q\right),\hfill & w\in \left. [c,w_{\phi }\right], \hfill \\ m,\hfill & w\in \left(w_{\phi },\frac{q-m}{\beta }\right],\hfill \end{array}\right.$$

where $w_{\phi }$ is the unique solution to the equation $\phi \ast \pi \left(m\right)=u\left(m,q,w\right)$ for $w$ in the interval $\left. [c,\frac{q-m}{\beta }\right]$, $x_r\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ in the interval $\left. [m,q-\beta w\right]$.

(ii) When $\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}{x}_r\left(q\right),\hfill & w\in \left. [c,w_0\right], \hfill \\ q-\beta w,\hfill & w\in \left(w_0,\frac{q-m}{\beta }\right], \hfill \end{array}\right.$$

where $w_0$ is the unique solution to the equation $\phi \ast \pi \left(q-\beta w\right)=u\left(q-\beta w,q,w\right)$ for $w$ in the interval $\left. [c,\frac{q-m}{\beta }\right]$, $x_r\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ in the interval $\left. [m,q-\beta w\right]$.

(iii) When $\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\ge \phi \ge \frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}q-\beta w,& w\in \left. [c,w_0\right], \\ x_r\left(q\right),& w\in \left(w_0,w_{\phi }\right], \\ m,& w\in \left(w_{\phi },\frac{q-m}{\beta }\right], \end{array}\right.$$

where $w_{\phi }$ is the unique solution to the equation $\phi \ast \pi \left(m\right)=u\left(m,q,w\right)$ for $w$ in the interval $\left. [c,\frac{q-m}{\beta }\right]$, $w_0$ is the unique solution to the equation $\phi \ast \pi \left(q-\beta w\right)=u\left(q-\beta w,q,w\right)$ for $w$ in the interval $\left. [c,\frac{q-m}{\beta }\right]$, $x_r\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ in the interval $\left. [m,q-\beta w\right]$.

(iv) When $\phi \ge \max \left\{\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)},\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}\right\}$, then $x_{+}\left(q,w\right)=q-\beta w$.

(2) If $\beta c+2h\le q\le \beta \left(c+b\right)+2l$, for wholesale price $w\in \left. [c,\frac{q-m}{\beta }\right]$, the positive focus $x_{+}\left(q,w\right)$ is determined as follows:

(i) When $\phi >\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then $x_{+}\left(q,w\right)=q-\beta w$.

(ii) When $\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then there is a unique solution $x_{\phi }\left(q\right)$ for $x$ in the interval $\left. [m,h\right]$ to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,\frac{q-x}{\beta }\right)$, such that

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}{x}_r\left(q\right),\hfill & w\in \left. [c,\frac{q-x_{\phi }}{\beta }\right], \hfill \\ q-\beta w,\hfill & w\in \left(\frac{q-x_{\phi }}{\beta },\frac{q-m}{\beta }\right]\hfill \end{array}\right.$$

where $x_r\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ in the interval $\left. [m,q-\beta w\right]$.

(iii) When $\frac{u\left(m,q,\frac{q-h}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, then there is a unique solution $w_{\phi }$ for $x$ in the interval $\left. [m,h\right]$ to the equation $\phi \ast \pi \left(m\right)=u\left(m,q,w\right)$, such that

$$x_{+}\left(q,w\right)=\left\{\begin{array}{cc}{x}_r\left(q\right) ,& w\in \left. [c,w_{\phi }\right], \\ m,& w\in \left(w_{\phi },\frac{q-m}{\beta }\right],\end{array}\right.$$

where $x_r\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for $x$ in the interval $\left. [m,q-\beta w\right]$.

(iv) When $0<\phi <\frac{u\left(m,q,\frac{q-h}{\beta }\right)}{\pi \left(m\right)}$, then $x_{+}\left(q,w\right)=m$.

Proof of Lemma 2.

(1) According to the definitions of relative likelihood function and satisfaction functions, we know that $\pi \left(x\right)$ is strictly increasing and $u\left(x,q,\frac{q-x}{\beta }\right)$ is strictly increasing for $w$ on $\left. [c,\frac{q-m}{\beta }\right]$.

(i) If $\phi <\min \left\{\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)},\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}\right\}$, there is a unique solution $w_{\phi }$ for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$ to the equation $\phi \ast \pi \left(m\right)=u\left(m,q,w\right)$, then in the case of $w\in \left(w_{\phi },\frac{q-m}{\beta }\right]$, as $\phi \ast \pi \left(m\right)<u\left(m,q,w\right)$, $\phi <\frac{u\left(m,q,w\right)}{\pi \left(m\right)}$, we have $x_{+}\left(q,w\right)=m$ as the result of Lemma 1 (iii). In the case of $w\in \left. [c, w_{\phi }\right]$, there is a unique solution $x_r\left(q\right)$ for $x$ on $\left. [m,q-\beta w\right]$ to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for the monotonicity of $u\left(x,q,w\right)$ and $\pi \left(x\right)$, we have $\phi =\frac{u\left(x_r\left(q\right), q, \frac{q-x_r\left(q\right)}{\beta }\right)}{\pi \left(x_r\left(q\right)\right)}\le \frac{u\left(q-\beta w, q, w\right)}{\pi \left(q-\beta w\right)}$ and $\phi \ge \frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\ge \frac{u\left(m, q, w\right)}{\pi \left(m\right)}$. From Lemma 1 (ii), we know that $x_{+}\left(q,w\right)=x_r\left(q\right)$.

(ii) If $\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, since $u\left(x,q,\frac{q-x}{\beta }\right)$ is strictly decreasing for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$, then there exists a unique solution $w_0$ for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$ to the equation $\phi \ast \pi \left(q-\beta w\right)=u\left(q-\beta w, q, w\right)$. In the case of $w\in \left. [w_0, \frac{q-m}{\beta }\right]$, as $\phi \ast \pi \left(q-\beta w\right)>u\left(q-\beta w, q, w\right)$, $\phi >\frac{u\left(q-\beta w, q, w\right)}{\pi \left(q-\beta w\right)}$, it can be referred as Lemma 1 (i) that $x_{+}\left(q,w\right)=q-\beta w$. In the case of $w\in \left. [c, w_0\right]$, there is a unique solution $x_r\left(q\right)$ for $x$ on $\left. [m,q-\beta w\right]$ to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for the monotonicity of $u\left(x,q,w\right)$ and $\pi \left(x\right)$, we can have $\phi =\frac{u\left(x_r\left(q\right),q,\frac{q-x_r\left(q\right)}{\beta }\right)}{\pi \left(x_r\left(q\right)\right)}\le \frac{u\left(q-\beta w,q,w\right)}{\pi \left(q-\beta w\right)}$ and $\phi \ge \frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\ge \frac{u\left(m,q,w\right)}{\pi \left(m\right)}$. From Lemma 1 (ii), we have $x_{+}\left(q,w\right)=x_r\left(q\right)$.

(iii) If $\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\ge \phi \ge \frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, there is a unique solution $w_{\phi }$ for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$ to the equation $\phi \ast \pi \left(m\right)=u\left(m,q,w\right)$, then in the case of $w\in \left(w_{\phi },\frac{q-m}{\beta }\right]$, as $\phi \ast \pi \left(m\right)<u\left(m,q,w\right)$, we have $\phi <\frac{u\left(m,q,w\right)}{\pi \left(m\right)}$, it can be referred as Lemma 1 (iii) that $x_{+}\left(q, w\right)=m$. Since $u\left(x,q,\frac{q-x}{\beta }\right)$ is strictly decreasing, then there exists a unique solution $w_0$ for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$ to the equation $\phi \ast \pi \left(q-\beta w\right)=u\left(q-\beta w,q,w\right)$. In the case of $w\in \left. [w_0,w_{\phi }\right]$, there is a unique solution $x_r\left(q\right)$ for $x$ on $\left. [m,q-\beta w\right]$ to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,w\right)$ for the monotonicity of $u\left(x,q,w\right)$ and $\pi \left(x\right)$, we can have $\phi =\frac{u\left(x_r\left(q\right),q,\frac{q-x_r\left(q\right)}{\beta }\right)}{\pi \left(x_r\left(q\right)\right)}\le \frac{u\left(q-\beta w,q,w\right)}{\pi \left(q-\beta w\right)}$ and $\phi \ge \frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\ge \frac{u\left(m,q,w\right)}{\pi \left(m\right)}$, it can be referred as Lemma 1 (ii) that $x_{+}\left(q,w\right)=x_r\left(q\right)$. In the case of $w\in \left. [c, w_0\right]$, as $\phi \ast \pi \left(q-\beta w\right)>u\left(q-\beta w,q,w\right)$, we can have $\phi >\frac{u\left(q-\beta w,q,w\right)}{\pi \left(q-\beta w\right)}$. From Lemma 1 (i), we have $x_{+}\left(q,w\right)=q-\beta w$.

(iv) If $\phi \ge \max \left\{\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)},\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}\right\}$, then we have $\phi \ast \pi \left(q-\beta w\right)>u\left(q-\beta w,q,w\right)$, $\phi >\frac{u\left(q-\beta w,q,w\right)}{\pi \left(q-\beta w\right)}$. From Lemma 1 (i), we have $x_{+}\left(q,w\right)=q-\beta w$.

(2) According to the definitions of relative likelihood function and satisfaction functions, we know that $\pi \left(x\right)$ is strictly increasing and $u\left(x,q,\frac{q-x}{\beta }\right)$ is strictly decreasing for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$.

(i) If $\phi >\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$ and $w\in \left. [c, \frac{q-m}{\beta }\right]$, we have $\phi >\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}\ge \frac{u\left(q-\beta w,q,w\right)}{\pi \left(h\right)}$. According to Lemma 1 (i), we know that $x_{+}\left(q,w\right)=q-\beta w$. Thus, $u\left(x_{+}\left(q,w\right),q,w\right)=u\left(q-\beta w,q,w\right)$.

(ii) If $\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then there is a unique solution $x_{\phi }\left(q\right)$ for $x$ on $\left. [m,h\right]$ to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,\frac{q-x}{\beta }\right)$ for the monotonicity of $u\left(x,q,\frac{q-x}{\beta }\right)$ and $\pi \left(x\right)$. In the case of $w\in \left(\frac{q-x_{\phi }}{\beta }, \frac{q-m}{\beta }\right]$, as $\phi \ast \pi \left(q-\beta w\right)>u\left(q-\beta w,q,w\right)$, we have $x_{+}\left(q, w\right)=q-\beta w$ as the result of Lemma 1 (i). In the case of $w\in \left. [c, \frac{q-x_{\phi }}{\beta }\right]$, we have $\phi =\frac{u\left(x_{\phi }\left(q\right), q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)}{\pi \left(x_{\phi }\left(q\right)\right)}\le \frac{u\left(q-\beta w, q, w\right)}{\pi \left(q-\beta w\right)}$ and $\phi \ge \frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\ge \frac{u\left(m, q, w\right)}{\pi \left(m\right)}$ From Lemma 1 (ii), we know that there is a unique solution $x_{\gamma }\left(q\right)$ to the equation of $\phi \ast \pi \left(x\right)=u\left(x, q, w\right)$ for $x$ on $\left. [m,q-\beta w\right]$. Then, we have $x_{+}\left(q, w\right)=x_{\gamma }\left(q\right)$. In addition, $x_{+}\left(q,\frac{q-x_{\phi }\left(q\right)}{\beta }\right)=x_{\gamma }\left(q\right)$.

(iii) If $\frac{u\left(m, q, \frac{q-h}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, since $u\left(m, q, \frac{q-x}{\beta }\right)$ is strictly decreasing for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$, then there exists a unique solution $w_{\phi }$ for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$ to the equation $\phi \ast \pi \left(m\right)=u\left(m, q, w\right)$ In the case of $w\in \left(w_{\phi }, \frac{q-m}{\beta }\right]$, as $\phi \ast \pi \left(m\right)<u\left(m, q, w\right)$, it can be referred as Lemma 1 (iii) that $x_{+}\left(q, w\right)=m$. In the case of $w\in \left. [c, w_{\phi }\right]$, as $\phi \ast \pi \left(m\right)=u\left(m, q, w_{\phi }\right)$, we can have $\phi =\frac{u\left(m, q, w_{\phi }\right)}{\pi \left(m\right)}\ge \frac{u\left(m, q, w\right)}{\pi \left(m\right)}$ and $\phi =\frac{u\left(m, q, w_{\phi }\right)}{\pi \left(m\right)}\le \frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \frac{u\left(q-\beta w, q, w\right)}{\pi \left(q-\beta w\right)}$. From Lemma 1 (ii), we have $x_{+}\left(q, w\right)=x_{\gamma }\left(q\right)$. Additionally, we have $x_{+}\left(q,w_{\phi }\right)=x_{\gamma }\left(q\right)$.

(iv) If $0<\phi <\frac{u\left(m, q, \frac{q-h}{\beta }\right)}{\pi \left(m\right)}$, then we have $\phi \ast \pi \left(m\right)<u\left(m, q, \frac{q-h}{\beta }\right)\le u\left(m, q, w\right)$ for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$. From Lemma 1 (iii), we know that $x_{+}\left(q, w\right)=m$. □

Lemma 2 further derives the positive focus of the wholesale price $w$ in the interval $\left. [c,p-b\right]$. On the base of Lemma 1, it can be known that different positive foci appear in different intervals of the parameter $\phi$. In addition, the critical value of $\phi$ can be determined to be independent of wholesale price. Based on Lemma 1, Theorem 1, and Lemma 2, the following results can be further derived.

Theorem 3.

(1) If $\frac{c}{2}+\frac{q}{2\beta }\ge p$, the function $u\left(x_{+}\left(q,w\right),q,w\right)$ is continuous and increasing of $w$ in the interval $\left. [c, p-b\right]$.

(2) If $\beta c+2h\le q\le \beta \left(c+b\right)+2l$, the monotonicity of the function $u\left(x_{+}\left(q,w\right),q,w\right)$ with respect to $w$ in the interval $\left. [c, p-b\right]$ is as follows:

(i) When $\phi >\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then $u\left(x_{+}\left(q,w\right),q,w\right)$ is strictly decreasing in the interval $\left. [c, p-b\right]$.

(ii) When $\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then $u\left(x_{+}\left(q,w\right),q,w\right)$ is strictly increasing in the interval $\left. [c,\frac{q-x_{\phi }}{\beta }\right]$ and decreasing in the interval $\left. [\frac{q-x_{\phi }}{\beta },p-b\right]$, where $x_{\phi }\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,\frac{q-x}{\beta }\right)$ for $x$ in the interval $\left. [m,h\right]$.

(iii) When $\frac{u\left(m,q,\frac{q-h}{\beta }\right)}{\pi \left(m\right)}\le \phi <\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, then $u\left(x_{+}\left(q,w\right),q,w\right)$ is strictly increasing in the interval $\left. [c,\frac{q-m}{\beta }\right]$ and decreasing in the interval $\left. [\frac{q-m}{\beta },p-b\right]$.

(iv) When $0<\phi <\frac{u\left(m,q,\frac{q-h}{\beta }\right)}{\pi \left(m\right)}$, then $u\left(x_{+}\left(q,w\right),q,w\right)$ is strictly increasing in the interval $\left. [c,\frac{q-m}{\beta }\right]$ and decreasing in the interval $\left. [\frac{q-m}{\beta },p-b\right]$.

Proof of Theorem 3.

(1) According to (1), we know that $v\left(x_{+}\left(q, w\right), q, w\right)=\min \left\{\left(w-c-b\right)\ast \left(q-\beta w\right)+b\ast x_{+}\left(q, w\right), \left(w-c\right)\ast \left(q-\beta w\right)\right\}$. Since $x_{+}\left(q, w\right)$ is continuous on $\left. [l,h\right]$, then $v\left(x_{+}\left(q, w\right), q, w\right)$ is also continuous. Considering the definition of satisfaction function, we know that $u\left(x_{+}\left(q, w\right), q, w\right)$ is also a continuous function of $w$ on $\left. [c, p-b\right]$.

In order to show the monotonicity of $u\left(x_{+}\left(q, w\right), q, w\right)$, we separate $\left. [c, p-b\right]$ into two intervals: $\left. [c, \frac{q-m}{\beta }\right]$ and $\left. [\frac{q-m}{\beta }, p-b\right]$. In the following proof, we consider the two cases respectively.

Case 1. For $w\in \left. [\frac{q-m}{\beta }, p-b\right]$, we have $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(m, q, w\right)$ as Lemma 1 described. Thus, $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [\frac{q-m}{\beta }, p-b\right]$ whenever $\phi >0$ for the monotonicity of $v\left(x_{+}\left(q, w\right), q, w\right)$.

Case 2. It follows from Lemma 1 that $x_{+}\left(q, \frac{q-m}{\beta }\right)=m$ whenever $\phi >0$. In the following, as Lemma 2 described above, we are considering the monotonicity of $u\left(x_{+}\left(q, w\right), q, w\right)$ for cases (i), (ii), (iii) and (iv), respectively.

(i) Since $m\le w_{\phi }\le h$, we separate $\left. [c, \frac{q-m}{\beta }\right]$ into two intervals: $\left. [c, w_{\phi }\right]$ and $\left(w_{\phi }, \frac{q-m}{\beta }\right]$. In the following proof, we consider the two cases, respectively.

(i.a) Let $w_1, w_2\in \left. [c, w_{\phi }\right]$ and $w_1>w_2$. As per Lemma 1, we have $x_{+}\left(q, w_i\right)\le q-\beta w_i$ for $i=1, 2$. By the definition of (1), for $i=1, 2$, we have

$$v\left(x_{+}\left(q, w_i\right), q, w\right)=\left(w-c-b\right)\ast \left(q-\beta w\right)+b\ast x_{+}\left(q, w_i\right)$$

(10)

If $x_{+}\left(q, w_1\right)<x_{+}\left(q, w_2\right)$, as per Lemma 2 (1.iii), we have $u\left(x_{+}\left(q, w_i\right), q, w\right)=\phi \ast \pi \left(x_{+}\left(q, w_i\right)\right)$ for $i=1, 2$ and hence $u\left(x_{+}\left(q, w_1\right), q, w_1\right)=\phi \ast \pi \left(x_{+}\left(q, w_1\right)\right)>\phi \ast \pi \left(x_{+}\left(q, w_2\right)\right)=u\left(x_{+}\left(q, w_2\right), q, w_2\right)$. If $x_{+}\left(q, w_1\right)=x_{+}\left(q, w_2\right)$, (10) results in $u\left(x_{+}\left(q, w_1\right), q, w_1\right)>u\left(x_{+}\left(q, w_2\right), q, w_2\right)$ due to $w_1>w_2$ and hence, we have $u\left(x_{+}\left(q, w_1\right), q, w_1\right)>u\left(x_{+}\left(q, w_1\right), q, w_2\right)$.

(i.b) For any $w\in \left(w_{\phi }, \frac{q-m}{\beta }\right]$, it follows from Lemma 2 (1.i) that $x_{+}\left(q, w\right)=m$. Thus, we know that $u\left(m, q, w\right)$ is strictly increasing on $\left(w_{\phi }, \frac{q-m}{\beta }\right]$ according to the definitions of $v\left(x, q, w\right)$ and $u\left(x, q, w\right)$.

In summary, Case (i.a) and (i.b) show that the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$.

(ii) Since $m\le w_0\le h$, we separate $\left. [c, \frac{q-m}{\beta }\right]$ into the following two intervals: $\left. [c, w_0\right)$ and $\left. [w_0, \frac{q-m}{\beta }\right]$. In the following proof, we consider the two cases, respectively.

(ii.a) Let $w_3, w_4\in \left. [c, w_0\right)$ and $w_3>w_4$. As per Lemma 1, we have $x_{+}\left(q, w_i\right)\le q-\beta w_i$ for $i=3, 4$. By the definition of (1), for $i=3, 4$, we have

$$v\left(x_{+}\left(q, w_i\right), q, w\right)=\left(w-c-b\right)\ast \left(q-\beta w\right)+b\ast x_{+}\left(q, w_i\right)$$

(11)

If $x_{+}\left(q, w_3\right)<x_{+}\left(q, w_4\right)$, as per Lemma 2 (1.iii), we have $u\left(x_{+}\left(q, w_i\right), q, w\right)=\phi \ast \pi \left(x_{+}\left(q, w_i\right)\right)$ for $i=3, 4$ and hence $u\left(x_{+}\left(q, w_3\right), q, w_3\right)=\phi \ast \pi \left(x_{+}\left(q, w_3\right)\right)>\phi \ast \pi \left(x_{+}\left(q, w_4\right)\right)=u\left(x_{+}\left(q, w_4\right), q, w_4\right)$. If $x_{+}\left(q, w_3\right)=x_{+}\left(q, w_4\right)$, (11) results in $u\left(x_{+}\left(q, w_3\right), q, w_3\right)>u\left(x_{+}\left(q, w_4\right), q, w_4\right)$ due to $w_3>w_4$ and hence, we have $u\left(x_{+}\left(q, w_3\right), q, w_3\right)>u\left(x_{+}\left(q, w_3\right), q, w_4\right)$.

(ii.b) For any $w\in \left. [w_0, \frac{q-m}{\beta }\right]$, it follows from Lemma 2 (1.ii) that $x_{+}\left(q, w\right)=q-\beta w$. Thus, $u\left(q-\beta w, q, w\right)$ is strictly increasing on $\left. [w_0, \frac{q-m}{\beta }\right]$ for the definition of $v\left(x, q, w\right)$ and $u\left(x, q, w\right)$.

In summary, Case (ii.a) and (ii.b) show that the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$.

(iii) Since $m<w_0\le w_{\phi }\le h$, we divide $\left. [c, \frac{q-m}{\beta }\right]$ into the following three intervals: $\left. [c, w_0\right]$, $\left(w_0, w_{\phi }\right]$ and $\left(w_{\phi }, \frac{q-m}{\beta }\right]$. In the following proof, we consider the three cases, respectively.

(iii.a) For any $w\in \left. [c, w_0\right]$ it follows from Lemma 2 (1.iii) that $x_{+}\left(q, w\right)=q-\beta w$. Clearly, $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(q-\beta w, q, w\right)$ is strictly increasing on $\left. [c, w_0\right]$.

(iii.b) Let $w_5, w_6\in \left(w_0, w_{\phi }\right]$ and $w_5>w_6$. As per Lemma 1, we have $x_{+}\left(q, w_i\right)\le q-\beta w_i$ for $i=5, 6$. By the definition of (1), we know that (9) holds for $i=5, 6$. If $x_{+}\left(q, w_5\right)<x_{+}\left(q, w_6\right)$, as per Lemma 2 (1.iii), we can have $u\left(x_{+}\left(q, w_i\right), q, w_i\right)=\phi \ast \pi \left(x_{+}\left(q, w_i\right)\right)$ for $i=5, 6$ and hence $u\left(x_{+}\left(q, w_5\right), q, w_5\right)=\phi \ast \pi \left(x_{+}\left(q, w_5\right)\right)>\phi \ast \pi \left(x_{+}\left(q, w_6\right)\right)=u\left(x_{+}\left(q, w_6\right), q, w_6\right)$. If $x_{+}\left(q, w_5\right)=x_{+}\left(q, w_6\right)$, (9) results in $u\left(x_{+}\left(q, w_5\right), q, w_5\right)>u\left(x_{+}\left(q, w_6\right), q, w_6\right)$ due to $w_5>w_6$ and hence, we have $u\left(x_{+}\left(q, w_5\right), q, w_5\right)>u\left(x_{+}\left(q, w_5\right), q, w_6\right)$.

(iii.c) For any $w\in \left(w_{\phi }, \frac{q-m}{\beta }\right]$, it follows from Lemma 2 (1.iii) that $x_{+}\left(q, w\right)=m$. Clearly, $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(m, q, w\right)$ is strictly increasing on $\left(w_{\phi }, \frac{q-m}{\beta }\right]$.

In summary, Case (iii.a), (iii.b) and (iii.c) show that the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$.

(iv) If $\phi \ge \max \left\{\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}, \frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}\right\}$ and $w\in \left. [c, \frac{q-m}{\beta }\right]$, then $x_{+}\left(q, w\right)=q-\beta w$. As described above, $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(q-\beta w, q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$. Considering the continuity of $u\left(x_{+}\left(q, w\right), q, w\right)$ and case 1, we have $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, p-b\right]$.

(2) According to (1), we know that $v\left(x_{+}\left(q, w\right), q, w\right)=\min \left\{\left(w-c-b\right)\ast \left(q-\beta w\right)+b\ast x_{+}\left(q, w\right), \left(w-c\right)\ast \left(q-\beta w\right)\right\}$. Since $x_{+}\left(q, w\right)$ is continuous on $\left. [l, h\right]$, then $v\left(x_{+}\left(q, w\right), q, w\right)$ is also continuous. Considering the definition of satisfaction function, we know that $u\left(x_{+}\left(q, w\right), q, w\right)$ is also a continuous function of $w$ on $\left. [c, p-b\right]$.

In order to show the monotonicity of $u\left(x_{+}\left(q, w\right), q, w\right)$, we separate $\left. [c, p-b\right]$ into two intervals: $\left. [c, \frac{q-m}{\beta }\right]$ and $\left. [\frac{q-m}{\beta }, p-b\right]$. In the following proof, we consider the two cases, respectively.

Case 1. It follows from Lemma 1 that $x_{+}\left(q, \frac{q-m}{\beta }\right)=m$ whenever $\phi >0$. In the following, as Lemma 2 described above, we considering the monotonicity of $u\left(x_{+}\left(q, w\right), q, w\right)$ for cases (i), (ii), (iii) and (iv), respectively.

(i) If $\phi >\frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$ and $w\in \left. [c, \frac{q-m}{\beta }\right]$, we have the focus demand $x_{+}\left(q, w\right)=q-\beta w$. As noted earlier, $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(q-\beta w, q, w\right)$ is strictly decreasing on $\left. [c, \frac{q-m}{\beta }\right]$. Considering the continuity of $u\left(x_{+}\left(q, w\right), q, w\right)$ and Case 1, we know that $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly decreasing on $\left. [c, p-b\right]$.

(ii) Since $m<x_{\phi }\left(q\right)\le h$, we separate $\left. [c, \frac{q-m}{\beta }\right]$ into two intervals: $\left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ and $\left. [c, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)$. In the following proof, we consider the two cases, respectively.

(ii.a) Let $w_7, w_8\in \left. [c, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)$ and $w_7>w_8$. As per Lemma 1, we have $x_{+}\left(q, w_i\right)\le q-\beta w_i$ for $i=7, 8$. By the definition of (1), we know that (9) holds for $i=7, 8$. If $x_{+}\left(q, w_7\right)<x_{+}\left(q, w_8\right)$, as per Lemma 2 (ii), we can have $u\left(x_{+}\left(q, w_i\right), q, w_i\right)=\phi \ast \pi \left(x_{+}\left(q, w_i\right)\right)$ for $i=7, 8$ and hence $u\left(x_{+}\left(q, w_7\right), q, w_7\right)=\phi \ast \pi \left(x_{+}\left(q, w_7\right)\right)>\phi \ast \pi \left(x_{+}\left(q, w_8\right)\right)=u\left(x_{+}\left(q, w_8\right), q, w_8\right)$. If $x_{+}\left(q, w_7\right)=x_{+}\left(q, w_8\right)$, (9) results in $u\left(x_{+}\left(q, w_7\right), q, w_7\right)>u\left(x_{+}\left(q, w_8\right), q, w_8\right)$ due to $w_7>w_8$ and hence, we have $u\left(x_{+}\left(q, w_7\right), q, w_7\right)>u\left(x_{+}\left(q, w_7\right), q, w_8\right)$.

(ii.b) For any $w\in \left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$, it follows from Lemma 2 (ii) that $x_{+}\left(q, w\right)=q-\beta w$. Clearly, $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(q-\beta w, q, w\right)$ is strictly decreasing on $\left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$.

Case 2. For $w\in \left. [\frac{q-m}{\beta }, p-b\right]$, we have $u\left(x_{+}\left(q, w\right), q, w\right)=u\left(m, q, w\right)$ as Lemma 1 described. Thus, $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly decreasing on $\left. [\frac{q-m}{\beta }, p-b\right]$ whenever $\phi >0$ for the monotonicity of $v\left(x_{+}\left(q, w\right), q, w\right)$.

In summary, case (ii.a) shows that the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, \frac{q-x_{\phi }\left(q\right)}{\beta }\right]$, (i) (ii.b) in Case 1 and Case 2 show that $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly decreasing on $\left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, p-b\right]$.

(iii) Since $m\le q_{\phi }\left(q\right)\le h$, we separate $\left. [c, \frac{q-m}{\beta }\right]$ into two intervals: $\left. [c, \frac{q-q_{\phi }\left(q\right)}{\beta }\right)$ and $\left. [\frac{q-q_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$. In the following proof, we consider the two cases, respectively.

(iii.a) Let $w_9, w_{10}\in \left. [c, \frac{q-q_{\phi }\left(q\right)}{\beta }\right)$ and $w_9>w_{10}$. As per Lemma 1, we have $x_{+}\left(q, w_i\right)\le q-\beta w_i$ for $i=9, 10$. By the definition of (1), for $i=9, 10$, we have

$$v\left(x_{+}\left(q, w_i\right), q, w\right)=\left(w-c-b\right)\ast \left(q-\beta w\right)+b\ast x_{+}\left(q, w_i\right)$$

(12)

If $x_{+}\left(q, w_9\right)<x_{+}\left(q, w_{10}\right)$, as per Lemma 2 (iii), we have $u\left(x_{+}\left(q, w_i\right), q, w\right)=\phi \ast \pi \left(x_{+}\left(q, w_i\right)\right)$ for $i=9, 10$ and hence $u\left(x_{+}\left(q, w_9\right), q, w_9\right)=\phi \ast \pi \left(x_{+}\left(q, w_9\right)\right)>\phi \ast \pi \left(x_{+}\left(q, w_{10}\right)\right)=u\left(x_{+}\left(q, w_{10}\right), q, w_{10}\right)$. If $x_{+}\left(q, w_9\right)=x_{+}\left(q, w_{10}\right)$, (12) results in $u\left(x_{+}\left(q, w_9\right), q, w_9\right)>u\left(x_{+}\left(q, w_{10}\right), q, w_{10}\right)$ due to $w_9>w_{10}$ and hence, we have $u\left(x_{+}\left(q, w_9\right), q, w_9\right)>u\left(x_{+}\left(q, w_9\right), q, w_{10}\right)$.

(iii.b) For any $w\in \left. [\frac{q-q_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$, as Lemma 2 (iii) described, then $x_{+}\left(q, w\right)=m$. Thus, $u\left(m, q, w\right)$ is strictly increasing on $\left. [\frac{q-q_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ for the definition of $v\left(x, q, w\right)$ and $u\left(x, q, w\right)$.

In summary, (iii) shows that $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$, and Case 1 shows that the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly decreasing on $\left. [\frac{q-m}{\beta }, p-b\right]$.

(iv) For any $w\in \left. [c, \frac{q-m}{\beta }\right]$, it follows from Lemma 2 (iv) that $x_{+}\left(q, w\right)=m$. It is easy to verify from the definition of (1) that $v\left(x, q, w\right)=v\left(m, q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$. Considering the definition of satisfaction function, we know that $u\left(m, q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$. In summary, the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly increasing on $\left. [c, \frac{q-m}{\beta }\right]$ and strictly decreasing on $\left. [\frac{q-m}{\beta }, p-b\right]$. □

The optimal wholesale price of the supplier in the buyback contract model is derived below, which is the solution to the upper-level program (6).

Theorem 4.

(1) If $\frac{c}{2}+\frac{q}{2\beta }\ge p$, since the function $u\left(x_{+}\left(q,w\right),q,w\right)$ is continuous and increasing of $w$ in the interval $\left. [c, p-b\right]$, the positive focus is unique. Therefore, the optimal wholesale price $w_{+}\left(q\right)=\frac{q-m}{\beta }$ and the corresponding positive focus $x_{+}\left(q,w_{+}\left(q\right)\right)=m$ under the positive evaluation system regardless of the value of $\kappa$.

(2) If $\beta c+2h\le q\le \beta \left(c+b\right)+2l$, the optimal wholesale price $w_{+}\left(q\right)$ is defined as follows:

(i) When $\phi >\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then

$$w_{+}\left(q\right)= \left\{\begin{array}{cc}\frac{q-h}{\beta },\hfill & if \kappa >\frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)},\\ \frac{q-x_{ \kappa }\left(q\right)}{\beta } ,\hfill & if \frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \kappa \le \frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}, \\ \frac{q-m}{\beta },\hfill & if 0<\kappa <\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}. \end{array}\right.$$

(ii) When $\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h,q,\frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, then

$$w_{+}\left(q\right)=\left\{\begin{array}{cc}\frac{q-x_{\phi }\left(q\right)}{\beta },\hfill & \kappa >\phi , \hfill \\ \frac{q-x_{ \kappa }\left(q\right)}{\beta },\hfill & \frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \kappa \le \phi ,\hfill \\ \frac{q-m}{\beta },\hfill & 0<\kappa <\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}.\hfill \end{array}\right.$$

(iii) When $0<\phi <\frac{u\left(m,q,\frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, then

$$w_{+}\left(q\right)=\frac{q-m}{\beta }, \forall \kappa >0.$$

where $x_{\phi }\left(q\right)$ is the unique solution to the equation $\phi \ast \pi \left(x\right)=u\left(x,q,\frac{q-x}{\beta }\right)$ for $x$ in the interval $\left. [m,h\right]$, $x_{ \kappa }\left(q\right)$ is the unique solution to the equation $\kappa \ast \pi \left(x\right)=u\left(x,q,\frac{q-x}{\beta }\right)$ for $x$ in the interval $\left. [m,h\right]$.

Proof of Theorem 4.

(1) In view of the above lemmas and theorems, we know that $\pi \left(x_{+}\left(q, w\right)\right)=\pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right)$ for $w$ on $\left(\frac{q-m}{\beta }, p-b\right]$ and $u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)>u\left(x_{+}\left(q, w\right), q, w\right)$ whenever $\phi >0$. This means that $w\left(q\right)$ will not exist in the interval $\left(\frac{q-m}{\beta }, p-b\right]$. Thus, we only need to consider the interval $\left. [c, \frac{q-m}{\beta }\right]$ in the following proof.

It can be known from Theorem 3 that the function $u\left(x_{+}\left(q, w\right), q, w\right)$ is continuous and strictly increasing for $w$ on $\left. [c, \frac{q-m}{\beta }\right]$, and it can be known from Theorem 2 that $\pi \left(x_{+}\left(q, w\right)\right)$ strictly increases as $w$ increases on $\left. [c, \frac{q-m}{\beta }\right]$, so in this case $w\left(q\right)=\frac{q-m}{\beta }$ and the corresponding positive focus $x_{+}\left(q, w\left(q\right)\right)=m$.

(2) In view of above lemmas and theorems, we know that $\pi \left(x_{+}\left(q, w\right)\right)=\pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right)$ for $w$ on $\left(\frac{q-m}{\beta }, p-b\right]$, and $u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)>u\left(x_{+}\left(q, w\right), q, w\right)$ whenever $\phi >0$. This means that $w\left(q\right)$ will not exist in the interval $\left(\frac{q-m}{\beta }, p-b\right]$. Thus, we only need to consider the interval $\left. [c, \frac{q-m}{\beta }\right]$ in the following proof.

(i) When the parameter $\phi >\frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, we have $x_{+}\left(q, w\right)=q-\beta w$ for any $w\in \left. [c, \frac{q-m}{\beta }\right]$ referring to Lemma 2 (2.i). As described above, $\pi \left(x\right)$ is strictly increasing and $u\left(x, q, \frac{q-x}{\beta }\right)$ is strictly increasing in the interval $\left. [c, \frac{q-m}{\beta }\right]$.

(i.a) If $\kappa >\frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, we have $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-h}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-h}{\beta }\right), q, \frac{q-h}{\beta }\right)\right\}=u\left(h, q, \frac{q-h}{\beta }\right)>u\left(x_{+}\left(q, w\right), q, w\right)\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q, w\right)\right), u\left(x_{+}\left(q, w\right), q, w\right)\right\}$. Based on Lemma 2, this means $w\left(q\right)=x_{+}\left(q, w\left(q\right)\right)=h$.

(i.b) If $\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \kappa \le \frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, there exists a unique solution $x_{ \kappa }\left(q\right)$ to the equation $\kappa \ast \pi \left(x\right)=u\left(x, q, \frac{q-x}{\beta }\right)$ on $\left. [m,h\right]$. For any $w\in \left. [c, \frac{q-x_{ \kappa }\left(q\right)}{\beta }\right)\cup \left(\frac{q-x_{ \kappa }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$, we can also have $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-x_{ \kappa }\left(q\right)}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-x_{ \kappa }\left(q\right)}{\beta }\right), q, \frac{q-x_{ \kappa }\left(q\right)}{\beta }\right)\right\}=\min \left\{\kappa \ast \pi \left(x_{ \kappa }\left(q\right)\right), u\left(x_{ \kappa }\left(q\right), q, \frac{q-x_{ \kappa }\left(q\right)}{\beta }\right)\right\}>\min \left\{\kappa \ast \pi \left(q-\beta w\right), u\left(q-\beta w, q, w\right)\right\}=\min \left\{\kappa \ast \pi \left(x_{+}\left(q, w\right)\right), u\left(x_{+}\left(q, w\right), q, w\right)\right\}$. This means $w\left(q\right)=x_{+}\left(q, w\left(q\right)\right)=x_{ \kappa }\left(q\right)$ in this situation.

(i.c) If $0<\kappa <\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, for any $w\in \left. [c, \frac{q-m}{\beta }\right)$, we can have $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)\right\}=\kappa \ast \pi \left(m\right)>\kappa \ast \pi \left(q-\beta w\right)\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)\right\}$. This means $w\left(q\right)=x_{+}\left(q, w\left(q\right)\right)=m$ in this situation.

(ii) When $\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \phi \le \frac{u\left(h, q, \frac{q-h}{\beta }\right)}{\pi \left(h\right)}$, we know that $\pi \left(x_{+}\left(q, w\right)\right)$ is decreasing in the interval $\left. [c, \frac{q-m}{\beta }\right]$ as Theorem 2 described, and $u\left(x_{+}\left(q, w\right), q, w\right)$ is strictly decreasing on $\left. [c, \frac{q-x_{\phi }\left(q\right)}{\beta }\right]$ and strictly increasing on $\left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ as Theorem 3 described. Thus, this means that $w\left(q\right)$ will only lie in the interval $\left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$. We only consider this case in the following proof. Moreover, we have $x_{+}\left(q, w\right)=q-\beta w$ for any $w\in \left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ as Lemma 2 (2.ii) described.

(ii.a) If $\kappa >\phi =\frac{u\left(x_{\phi }\left(q\right), q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)}{\pi \left(x_{\phi }\left(q\right)\right)}$, we can have $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right), q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)\right\}=u\left(x_{+}\left(q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right), \frac{q-x_{\phi }\left(q\right)}{\beta }\right)>u\left(x_{+}\left(q, w\right), q, w\right)\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q, w\right)\right), u\left(x_{+}\left(q, w\right), q, w\right)\right\}$ for any $w\in \left(\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$. Thus, this means $w\left(q\right)=\frac{q-x_{+}\left(q, w\left(q\right)\right)}{\beta }=\frac{q-x_{\phi }\left(q\right)}{\beta }$ in this situation.

(ii.b) If $\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}\le \kappa \le \phi$, we have $x_{+}\left(q, w\right)=q-\beta w$ for any $w\in \left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ as Lemma 2 (2.ii) described. Since $\kappa \ast \pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right)=\kappa \ast \pi \left(m\right)\ge u\left(m, q, \frac{q-m}{\beta }\right)=u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)$ and $\kappa \ast \pi \left(x_{+}\left(q, x_{\phi }\left(q\right)\right)\right)=\kappa \ast \pi \left(x_{\phi }\left(q\right)\right)\le \phi \ast \pi \left(x_{\phi }\left(q\right)\right)=u\left(x_{\phi }\left(q\right), q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)=u\left(x_{+}\left(q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right), q, \frac{q-x_{\phi }\left(q\right)}{\beta }\right)$, there exists a unique solution $x_{ \kappa }\left(q\right)$ to the equation $\kappa \ast \pi \left(x\right)=u\left(x, q, \frac{q-x}{\beta }\right)$. Thus, we can have $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right), q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)\right\}=\kappa \ast \pi \left(x_{+}\left(q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)\right)=\kappa \ast \pi \left(x_{\kappa }\left(q\right)\right)>\kappa \ast \pi \left(q-\beta w\right)=\kappa \ast \pi \left(x_{+}\left(q, w\right)\right)\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q, w\right)\right), u\left(x_{+}\left(q, w\right), q, w\right)\right\}$ for any $w\in \left(\frac{q-x_{\kappa }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ and $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right), q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)\right\}=u\left(x_{+}\left(q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right), q, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)>u\left(x_{+}\left(q, w\right), q, w\right)\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q, w\right)\right), u\left(x_{+}\left(q, w\right), q, w\right)\right\}$ for any $w\in \left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-x_{\kappa }\left(q\right)}{\beta }\right)$. This means $w\left(q\right)=\frac{q-x_{+}\left(q, w\left(q\right)\right)}{\beta }=\frac{q-x_{\kappa }\left(q\right)}{\beta }$ in this situation.

(ii.c) If $0<\kappa <\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, we have $x_{+}\left(q, w\right)=q-\beta w$ for any $w\in \left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right]$ as Lemma 2 (2.ii) described. Since $\min \left\{\kappa \ast \pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right), u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)\right\}=\kappa \ast \pi \left(m\right)>\kappa \ast \pi \left(q-\beta w\right)=\kappa \ast \pi \left(x_{+}\left(q, w\right)\right)\ge \min \left\{\kappa \ast \pi \left(x_{+}\left(q, w\right)\right), u\left(x_{+}\left(q, w\right), q, w\right)\right\}$ for any $w\in \left. [\frac{q-x_{\phi }\left(q\right)}{\beta }, \frac{q-m}{\beta }\right)$, we have $w\left(q\right)=\frac{q-x_{+}\left(q, w\left(q\right)\right)}{\beta }=\frac{q-m}{\beta }$ in this situation.

(iii) When $0<\phi <\frac{u\left(m, q, \frac{q-m}{\beta }\right)}{\pi \left(m\right)}$, we have $\pi \left(x_{+}\left(q, \frac{q-m}{\beta }\right)\right)\ge \pi \left(x_{+}\left(q, w\right)\right)$ for any $w\in \left. [c, \frac{q-m}{\beta }\right)$ as Theorem 2 and $u\left(x_{+}\left(q, \frac{q-m}{\beta }\right), q, \frac{q-m}{\beta }\right)>u\left(x_{+}\left(q, w\right), q, w\right)$ as Theorem 3 (2.i)–(2.ii). Thus, we have $w\left(q\right)=\frac{q-x_{+}\left(q, w\left(q\right)\right)}{\beta }=\frac{q-m}{\beta }$ for any $\kappa >0$ in this case. □

Theorem 4 gives the decision results after suppliers determine the positive focus in the first stage under the positive evaluation system in the retailer-leading buyback contract model. It shows that the wholesale price $w$ must be equal to $\frac{q-m}{\beta }$ when $\frac{c}{2}+\frac{q}{2\beta }\ge p$. In fact, this conclusion is intuitive and easy to understand in the decision-making process, because when the wholesale price $w$ is in the interval $\left. [\frac{q-m}{\beta },p-b\right]$, the wholesale price $\frac{q-m}{\beta }$ brings both the highest relative likelihood degree and the highest satisfaction level, and when the wholesale price $w$ is in the interval $\left. [c,\frac{q-m}{\beta }\right]$, the wholesale price $\frac{q-m}{\beta }$ brings both the highest relative likelihood degree and the highest satisfaction level. Similarly, when $\beta c+2h\le q\le \beta \left(c+b\right)+2l$, the order quantity must be in the interval $\left. [m,h\right]$ under the positive evaluation system. These results of Theorem 4 explain the behavioral pattern of the supplier’s decision-making in the buyback contract model effectively.

According to the analysis of the supply chain model based on the focus theory of choice above, it is possible to understand how suppliers make decisions in the buyback contract model by considering optimism and confidence levels and behavioral patterns. For any given potential maximal order quantity $q$, the supplier’s positive focus $x_{+}\left(q,w_{+}\left(q\right)\right)$ and the optimal wholesale price $w_{+}\left(q\right)$ can be specifically obtained through functions (5) and (6) under the positive evaluation system. The numerical results in Section 6 can help us to clearly understand how suppliers make decisions by considering his/her optimism and confidence level and behavioral pattern.

5. Stackelberg Game Model and Retailer’s Optimal Order Quantity

After obtaining the supplier’s optimal wholesale price based on the focus theory of choice, the equilibrium solution of the retailer-leading buyback contract will be solved in the Stackelberg game framework in the following: Among them, the retailer is the leader and first announces the potential maximal order quantity $q$; the supplier determines the wholesale price based on the positive evaluation system; finally, the retailer determines the optimal order quantity $q^{*}$.

By conjecturing the supplier’s optimal wholesale price $w_{+}^{*}\left(q\right)$, the retailer’s profit is

$$r\left(x,q,w_{+}^{*}\left(q\right)\right)=\left\{\begin{array}{c}px+b\left(q-\beta w_{+}^{*}\left(q\right)-x\right)-w_{+}^{*}\left(q\right)\left(q-\beta w_{+}^{*}\left(q\right)\right), if x<q-\beta w_{+}^{*}\left(q\right),\hfill \\ \left(p-w_{+}^{*}\left(q\right)\right)\left(q-\beta w_{+}^{*}\left(q\right)\right), if x\ge q-\beta w_{+}^{*}\left(q\right).\hfill \end{array}\right.$$

Assume that the retailer is risk-neutral and maximizes the expected payoff function to determine his/her optimal order quantity:

$$\begin{array}{cc}\hfill H\left(q\right)=& {{\displaystyle \int }}_l^{q-\beta w_{+}^{*}\left(q\right)}\left. [px+b\left(q-\beta w_{+}^{*}\left(q\right)-x\right)-w_{+}^{*}\left(q\right)\left(q-\beta w_{+}^{*}\left(q\right)\right)\right]f\left(x\right)dx+\hfill \\ & {{\displaystyle \int }}_{q-\beta w_{+}^{*}\left(q\right)}^h\left. [\left(p-w_{+}^{*}\left(q\right)\right)\left(q-\beta w_{+}^{*}\left(q\right)\right)\right]f\left(x\right)dx\hfill \end{array}$$

The optimization problem is as follows:

$$\underset{q\in \left. [l+\beta c,h+\beta \left(p-b\right)\right]}{\max } H\left(q\right)$$

(13)

After the retailer announces the potential maximal order quantity $q$, the supplier as a follower determines his/her wholesale price $w_{+}^{*}\left(q\right)$, and finally the retailer determines the optimal order quantity according to the wholesale price. In the Stackelberg game of the buyback contract model, the retailer needs to determine the optimal potential maximal order quantity by speculating on the supplier’s decision. Therefore, the retailer’s optimal potential maximal order quantity $q^{*}$ in the game problem can be obtained by maximizing the expected profit as (13). Then, the existence of the optimal potential maximal order quantity is analyzed by solving the first and second derivatives of the objective function of problem (14). The first and second derivatives of $H\left(q\right)$ with respect to $q$ can be given as follows:

$$H\left(q\right)=\left. [\left(p-w_{+}^{*}\left(q\right)\right)\ast \left(q-\beta w_{+}^{*}\left(q\right)\right)\right]-\left(p-b\right){{\displaystyle \int }}_l^{q-\beta w_{+}^{*}\left(q\right)}F\left(x\right)dx,$$

(14)

$$\begin{array}{ll}{H}^{\prime }\left(q\right)=& \left. [-q{\left(w_{+}^{*}\left(q\right)\right)}^{\prime }+2\beta w_{+}^{*}\left(q\right){\left(w_{+}^{*}\left(q\right)\right)}^{\prime }-w_{+}^{*}\left(q\right)q^{\prime }+p{q}^{\prime }-p\beta {\left(w_{+}^{*}\left(q\right)\right)}^{\prime }\right]\\ & -\left(p-b\right)\left. [F\left(q-\beta w_{+}^{*}\left(q\right)\right)\left(q^{\prime }-\beta {\left(w_{+}^{*}\left(q\right)\right)}^{\prime }\right)\right],\end{array}$$

(15)

$$\begin{array}{cc}\hfill H^{″}\left(q\right)=& [-q{\left(w_{+}^{*}\left(q\right)\right)}^{″}+2\beta {\left({\left(w_{+}^{*}\left(q\right)\right)}^{\prime }\right)}^2+2\beta w_{+}^{*}\left(q\right){\left(w_{+}^{*}\left(q\right)\right)}^{″}-w_{+}^{*}\left(q\right)q^{″}]\hfill \\ & +p(q^{″}-\beta {\left(w_{+}^{*}\left(q\right)\right)}^{″}) \hfill \\ & -\left(p-b\right)[F\left(q-\beta w_{+}^{*}\left(q\right)\right)\left(q^{″}-\beta {\left(w_{+}^{*}\left(q\right)\right)}^{″}\right)\hfill \\ & +F^{\prime }\left(q-\beta w_{+}^{*}\left(q\right)\right){(q^{\prime }-\beta {\left(w_{+}^{*}\left(q\right)\right)}^{\prime })}^2]. \hfill \end{array}$$

(16)

Based on the equations of (15) and (16), the following three possible cases are mainly considered to analyze the existence of the optimal potential maximal order quantity $q$.

(i) If the objective function $H\left(q\right)$ is strictly increasing with respect to the potential maximal order quantity $q$, that is, the first derivative of $H\left(q\right)$ is strictly greater than 0 ($H^{\prime }\left(q\right)>0$), it shows that the retailer’s expected profit increases as the potential maximal order quantity increases. Thus, the retailer needs to provide a maximal potential maximal order quantity to maximize his/her expected profit in this case.

(ii) If the objective function $H\left(q\right)$ is strictly decreasing with respect to the potential maximal order quantity $q$, that is, the first derivative of $H\left(q\right)$ is strictly less than 0 ($H^{\prime }\left(q\right)>0$), it shows that the retailer’s expected profit decreases as the potential maximal order quantity increases. Thus, the retailer needs to provide a minimal potential maximal order quantity to maximize his/her expected profit in this case.

(iii) If the objective function $H\left(q\right)$ is strictly concave with respect to the potential maximal order quantity $q$, that is, the second derivative of $H\left(q\right)$ is less than 0 ($H^{″}\left(q\right)<0$), and there exists a potential maximal order quantity such that $H^{\prime }\left(q\right)=0$, this situation shows that there exists a unique potential maximal order quantity to maximize the retailer’s expected profit.

In the following, the triangular distribution is taken as an example for specific analysis, by setting specific parameters to solve the retailer’s optimal potential maximal order quantity and the supplier’s optimal wholesale price and then analyzing the possible relationship between the two.

6. Numerical Examples and Result Analyses

6.1. Basic Settings of the Example and Numerical Results

In the numerical examples, we assume that retailer’s unit selling price $p=40$, the supplier’s unit production cost $c=10$, the unit buyback price $b=10$, and the sensitivity of the retailer’s order quantity to the supplier’s wholesale price $\beta =100$. The random market demand is assumed to satisfy the triangular distribution, which is one of the quasi-concave functions in the numerical examples, and its support set is $\left. [0,200\right]$, the probability density function $f\left(x\right)$ is symmetric centered on $x=100$, then the most possible market demand is $m=100$. In the two-tier supply chain, the retailer takes expected-value-maximization theory as the decision criterion, and the supplier is positive; that is, the positive evaluation system is taken as the decision criterion. According to the above model parameter settings, the supplier’s satisfaction level function can be described as

$$u\left(x,q,w\right)=\left\{\begin{array}{cc}\frac{\left(w-20\right)\left(q-100w-200\right)+10x}{2000},& if x<q-100\ast w,\\ \frac{\left(w-10\right)\left(q-100w-200\right)+2000}{2000},& if x\ge q-100\ast w.\end{array}\right.$$

(17)

The probability density function is

$$f\left(x\right)=\left\{\begin{array}{cc}8\times {10}^{-5}x+0.001,& if 0\le x\le 100,\\ -8\times {10}^{-5}x+0.017,& if 100<x\le 200.\end{array}\right.$$

(18)

The distribution function is

$$F\left(x\right)=\left\{\begin{array}{cc}4\times {10}^{-5}{x}^2+{10}^{-3}x,& if 0\le x\le 100,\\ -4\times {10}^{-5}{x}^2+1.7\times {10}^{-2}x-0.8,& if 100<x\le 200.\end{array}\right.$$

(19)

The relative likelihood function of the supplier is

$$\pi \left(x\right)=\left\{\begin{array}{cc}\frac{2}{225}x+\frac{1}{9},& if 0\le x\le 100,\\ -\frac{2}{225}x+\frac{17}{9},& if 100<x\le 200.\end{array}\right.$$

(20)

According to the settings of the above relevant parameters, this section will specifically solve the game equilibrium solution $\left\{q^{*},w\left(q^{*}\right)\right\}$ according to the buyback contract model under the Stackelberg game; that is, the retailer’s optimal potential maximal order quantity and the supplier’s optimal wholesale price under the positive evaluation system.

Consistent with the above, the supplier’s response function $w\left(q\right)$ and the corresponding positive focus $x_{+}\left(q,w\left(q\right)\right)$ under the positive evaluation system can be first determined based on the backward induction method. According to Theorem 4, it can be known that the supplier’s optimal wholesale price and positive focus will change with the change of the positive level. Therefore, the critical points of $\phi$ and $\kappa$ need to be first determined in the specific solution process. Then, the optimal wholesale price and positive focus can be determined when suppliers take the positive evaluation system as the decision criterion based on the results of Theorem 4 in different ranges. Therefore, according to functions (17) and (20), the supplier’s optimal wholesale price $w\left(q\right)$ and the corresponding positive focus $x_{+}\left(q,w\left(q\right)\right)$ can be expressed as follows:

$$w\left(q\right)= \left\{\begin{array}{cc}\frac{q-200}{100},\hfill & \kappa >9, \hfill \\ \frac{q+800-\frac{16000}{9}\kappa +\sqrt{{\left(q-800+\frac{16000}{9}\kappa \right)}^2+4\left(4\times {10}^5-200q-\frac{34\times {10}^5}{9}\kappa \right)}}{200},\hfill & \frac{31}{20}-\frac{q}{2000}\le \kappa \le 9,\hfill \\ \frac{q-100}{100},\hfill & 0<\kappa <\frac{31}{20}-\frac{q}{2000},\hfill \end{array}\right.$$

and

$$x_{+}\left(q,w\left(q\right)\right)= \left\{\begin{array}{cc}200,\hfill & \kappa >9, \hfill \\ \frac{q-800+\frac{16000}{9}\kappa -\sqrt{{\left(q-800+\frac{16000}{9}\kappa \right)}^2+4\left(4\times {10}^5-200q-\frac{34\times {10}^5}{9}\kappa \right)}}{2},\hfill & \frac{31}{20}-\frac{q}{2000}\le \kappa \le 9,\hfill \\ 100,\hfill & 0<\kappa <\frac{31}{20}-\frac{q}{2000}.\hfill \end{array}\right.$$

(ii) When $\frac{31}{20}-\frac{q}{2000}\le \phi \le 9$,

$$w\left(q\right)= \left\{\begin{array}{cc}\frac{q+800-\frac{16000}{9}\phi +\sqrt{{\left(q-800+\frac{16000}{9}\phi \right)}^2+4\left(4\times {10}^5-200q-\frac{34\times {10}^5}{9}\phi \right)}}{200},\hfill & \kappa >\phi , \hfill \\ \frac{q+800-\frac{16000}{9}\kappa +\sqrt{{\left(q-800+\frac{16000}{9}\kappa \right)}^2+4\left(4\times {10}^5-200q-\frac{34\times {10}^5}{9}\kappa \right)}}{200},\hfill & \frac{31}{20}-\frac{q}{2000}\le \kappa \le \phi ,\hfill \\ \frac{q-100}{100},\hfill & 0<\kappa <\frac{31}{20}-\frac{q}{2000},\hfill \end{array}\right.$$

and

$$x_{+}\left(q,w\left(q\right)\right)= \left\{\begin{array}{cc}\frac{q-800+\frac{16000}{9}\phi -\sqrt{{\left(q-800+\frac{16000}{9}\phi \right)}^2+4\left(4\times {10}^5-200q-\frac{34\times {10}^5}{9}\phi \right)}}{2},\hfill & \kappa >\phi , \hfill \\ \frac{q-800+\frac{16000}{9}\kappa -\sqrt{{\left(q-800+\frac{16000}{9}\kappa \right)}^2+4\left(4\times {10}^5-200q-\frac{34\times {10}^5}{9}\kappa \right)}}{2},\hfill & \frac{31}{20}-\frac{q}{2000}\le \kappa \le \phi ,\hfill \\ 100,\hfill & 0<\kappa <\frac{31}{20}-\frac{q}{2000}.\hfill \end{array}\right.$$

(iii) When $0<\phi <\frac{31}{20}-\frac{q}{2000}$,

$$w\left(q\right)=\frac{q-100}{100}, \forall \kappa >0,$$

and

$$x_{+}\left(q,w\left(q\right)\right)=100, \forall \kappa >0.$$

According to the value range of $q$, we can derive $\frac{31}{20}-\frac{q}{2000}\in \left. [\frac{11}{20},\frac{17}{20}\right]$.

Referring to the supplier’s response function $w\left(q\right)$ and Equation (13), the retailer’s optimal potential maximal order quantity $q^{*}$ can be specifically obtained in the retailer-leading buyback contract model. Then, by substituting $q^{*}$ into the supplier’s reaction function, the supplier’s optimal wholesale price $w\left(q^{*}\right)$ can be obtained, and the equilibrium solution of the game can be expressed as $\left\{q^{*},w\left(q^{*}\right)\right\}$.

According to Theorem 4, both the supplier’s positive focus and the optimal wholesale price change with the change in the positive level. Therefore, the retailer’s optimal potential maximal order quantity $q^{*}$, the optimal wholesale price $w\left(q^{*}\right)$, the retailer’s profit $H\left(q^{*}\right)$, the supplier’s satisfaction level $u\left(x_{+}\left(q^{*},w\left(q^{*}\right)\right),w\left(q^{*}\right)\right)$, and relative likelihood degree $\pi \left(x_{+}\left(q^{*},w\left(q^{*}\right)\right)\right)$ will be obtained by setting the values of $\phi$ and $\kappa$. Next, setting $\phi =0.1, 0.5, 1, 2$ and $10,$ the following results can be obtained with $\kappa =0.1, 0.5, 1, 2$ and $10$, shown in

Table 1. Solutions of the proposed model with $\mathbf{\phi }=\mathbf{0.1}$.

$$\mathit{\kappa }$$	$${\mathit{q}}^{*}$$	$$\mathit{w}\left({\mathit{q}}^{*}\right)$$	$$\mathit{H}\left({\mathit{q}}^{*}\right)$$	$$\mathit{u}\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right),\mathit{w}\left({\mathit{q}}^{*}\right)\right)$$	$$\mathit{\pi }\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right)\right)$$
0.1	1400	13	2150	0.85	1
0.5	1400	13	2150	0.85	1
1	1400	13	2150	0.85	1
2	1400	13	2150	0.85	1
10	1400	13	2150	0.85	1

,

Table 2. Solutions of the proposed model with $\mathbf{\phi }=\mathbf{0.5}$.

$$\mathit{\kappa }$$	$${\mathit{q}}^{*}$$	$$\mathit{w}\left({\mathit{q}}^{*}\right)$$	$$\mathit{H}\left({\mathit{q}}^{*}\right)$$	$$\mathit{u}\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right),\mathit{w}\left({\mathit{q}}^{*}\right)\right)$$	$$\mathit{\pi }\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right)\right)$$
0.1	1400	13	2150	0.85	1
0.5	1400	13	2150	0.85	1
1	1400	13	2150	0.85	1
2	1400	13	2150	0.85	1
10	1400	13	2150	0.85	1

,

Table 3. Solutions of the proposed model with $\mathbf{\phi }=\mathbf{1}$.

$$\mathit{\kappa }$$	$${\mathit{q}}^{*}$$	$$\mathit{w}\left({\mathit{q}}^{*}\right)$$	$$\mathit{H}\left({\mathit{q}}^{*}\right)$$	$$\mathit{u}\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right),\mathit{w}\left({\mathit{q}}^{*}\right)\right)$$	$$\mathit{\pi }\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right)\right)$$
0.1	1400	13	2150	0.85	1
0.5	1400	13	2150	0.85	1
1	1400	12.86	2308.60	0.88	0.88
2	1400	12.86	2308.60	0.88	0.88
10	1400	12.86	2308.60	0.88	0.88

,

Table 4. Solutions of the proposed model with $\mathbf{\phi }=\mathbf{2}$.

$\mathit{\kappa }$	${\mathit{q}}^{*}$	$\mathit{w}\left({\mathit{q}}^{*}\right)$	$\mathit{H}\left({\mathit{q}}^{*}\right)$	$\mathit{u}\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right),\mathit{w}\left({\mathit{q}}^{*}\right)\right)$	$\mathit{\pi }\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right)\right)$
0.1	1400	13	2150	0.85	1
0.5	1400	13	2150	0.85	1
1	1400	12.86	2308.60	0.88	0.88
2	1400	12.41	2564.03	0.95	0.48
10	1400	12.41	2564.03	0.95	0.48

and

Table 5. Solutions of the proposed model with $\mathbf{\phi }=\mathbf{10}$.

$\mathit{\kappa }$	${\mathit{q}}^{*}$	$\mathit{w}\left({\mathit{q}}^{*}\right)$	$\mathit{H}\left({\mathit{q}}^{*}\right)$	$\mathit{u}\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right),\mathit{w}\left({\mathit{q}}^{*}\right)\right)$	$\mathit{\pi }\left({\mathit{x}}_{+}\left({\mathit{q}}^{*},\mathit{w}\left({\mathit{q}}^{*}\right)\right)\right)$
0.1	1400	13	2150	0.85	1
0.5	1400	13	2150	0.85	1
1	1400	12.86	2308.60	0.88	0.88
2	1400	12.41	2564.03	0.95	0.48
10	1400	12	2600	1	0.11

, respectively.

From the numerical results in Table 1, Table 2, Table 3, Table 4 and Table 5, it can be found that the optimal wholesale prices of suppliers with different positive levels will be different under the positive evaluation system. Meanwhile, the calculation results also reveal the retailer’s optimal potential maximal order quantity, the supplier’s optimal wholesale price, the retailer’s expected profit, the supplier’s satisfaction level and the relative likelihood degree under different φ and κ values. Therefore, further analysis of the results can be obtained by combining theoretical analysis and numerical results.

6.2. Results Analysis and Managerial Insights

According to Table 1, Table 2, Table 3, Table 4 and Table 5, the specific numerical results of the retailer-leading buyback contract model and the influence trend of $\kappa$ or $\phi$ on the three different results when $\phi$ or $\kappa$ is fixed can be obtained. The numerical results and the influence trend figure can provide managerial insights for the retailer’s strategy choice when facing suppliers with different personality traits in the retailer-leading buyback contract.

In the positive evaluation system, the supplier’s optimism level $\phi$ and confidence level $\kappa$ play an important role in determining the optimal wholesale price. For example, when a supplier has a low optimism level ($\phi =0.1, 0.5$), as shown in Table 1 and Table 2, then regardless of the confidence level $\kappa$ the supplier will choose wholesale price 13 as the optimal wholesale price, and the corresponding final order quantity is equal to the most possible realized market demand. In this case, retailers face more optimistic suppliers. Since the pricing strategy is negatively related to the potential maximal order quantity, the retailer will choose a lower potential maximal order quantity in order to maximize the expected profit. The optimal wholesale price, supplier satisfaction level, and relative likelihood degree do not change.

When the supplier has a moderate optimism level ($\phi =1, 2$), as shown in Table 3 and Table 4, if the supplier is not confident about the pricing strategy ($\kappa =0.1, 0.5$), then the supplier will still choose the most possible realizable market demand as the focus, and wholesale price 13 as the optimal wholesale price. If the supplier feels more confident about the pricing strategy ($\kappa =1, 2, 10$), then the supplier’s optimal wholesale price will decrease with the increase of the confidence level, and the final order quantity increases with the increase of confidence level. The optimal wholesale price, supplier’s satisfaction level, and relative likelihood degree vary moderately.

When the supplier has a high optimism level ($\phi =10$), as shown in Table 5, if the supplier is not confident in the pricing strategy ($\kappa =0.1, 0.5$), the supplier will still choose the most possible market demand as the focus, and wholesale price 13 as the optimal wholesale price. If the supplier feels more confident in the pricing strategy ($\kappa =1, 2, 10$), the supplier’s optimal wholesale price will decrease with the increase of the confidence level, and the corresponding final order quantity will increase with the increase of the confidence level until the highest market demand 200. The optimal wholesale price, the supplier’s satisfaction level, and the relative likelihood degree have the largest changes.

Similarly, the larger the value of $\kappa$, the greater the influence of $\phi$, and the more obvious the change trends of the optimal wholesale price, the supplier’s satisfaction level, and the relative likelihood degree. It can be seen that the supplier’s optimism and confidence levels generally have a negative impact on the pricing strategies. This is understandable and is consistent with reality, and can help to understand the behavioral decisions of suppliers in retailer-leading buyback contracts and also provide certain decision-making guidance and management insights for retailers facing suppliers with different personality characteristics.

In the positive evaluation system, the potential maximal order quantity offered by the retailer has a positive impact on the supplier’s pricing strategy. At the same time, the supplier’s personality characteristics will also affect the retailer’s ordering decision. According to the numerical results in Table 1, Table 2, Table 3, Table 4 and Table 5, since the retailer’s expected profit function is a monotonically decreasing function with respect to $q$, the retailer offers the lowest potential maximal order quantity 1400. When the supplier is more optimistic and confident, he/she will choose the wholesale price that is higher than the most possible market demand as the focus; that is, the wholesale price is lower than 13. When the supplier is not too optimistic and confident, he/she will choose the wholesale price corresponding to the most likely realized market demand as the focus—that is, the wholesale price is equal to 13—and the retailer will offer a lower potential maximal order quantity to pursue maximizing of expected profits.

7. Conclusions

7.1. Main Findings

This paper reconstructs a new retailer-leading buyback contract model by introducing the focus theory of choice, in which the supplier takes the positive evaluation system as the decision criterion. The supplier’s decision process is divided into two steps: In the first step, for every possible wholesale price, the supplier selects the positive focus by considering both the relative likelihood degree and the satisfaction level. In the second step, the supplier decides the optimal wholesale price based on all positive foci. Then, the Stackelberg game model is constructed in a two-tier supply chain system consisting of one supplier and one retailer, and the equilibrium solution is solved. Finally, the differences in the decision-making of suppliers with different personality characteristics and the effect of optimism degree and confidence level on the results are discussed in numerical examples and results analyses.

Based on the theoretical analysis and numerical examples, it can be found that when the supplier takes the positive evaluation system as the decision criterion, optimism degree and confidence level have a negative effect on the wholesale price and a positive effect on the final order quantity, and the final order quantity must be located between the most possible market demand and the upper limit of the market demand. When the optimism degree or confidence level is higher, the wholesale price will be lower, and the final order quantity will be closer to the upper limit of the market demand. On the other hand, when the optimism degree or confidence level is lower, the wholesale price will be higher, and the final order quantity will be closer to the most possible market demand. Therefore, the decision results in this interval can be explained by the optimism degree and confidence level based on the positive evaluation system of the focus theory of choice. The range of final order quantity taking the positive evaluation system as the decision criterion can cover part of the market demand. This theoretical analysis provides a new idea for the supplier’s behavior analysis and provides managerial insights for the retailer to make decisions when faced with a supplier with different personality characteristics.

7.2. Limitations and Future Directions

This research enriches the literature of buyback contract models and can be extended in several other directions. Firstly, this paper only considers the positive evaluation system of the focus theory of choice, and the negative evaluation system can also be considered. Secondly, this paper only studies the retailer-leading buyback contract, which is only one form of abundant supply chain contracts. Therefore, the application of the focus theory of choice to supply chain contracts can be extended to other contracts or combinations of contracts led by different decision-makers. Thirdly, this paper constructs a two-level supply chain contract model consisting of one supplier and one retailer that is a specific and simple form. Therefore, the research can be extended to complex situations where the supply chain system has more participants or products in the future. Finally, this paper only considers the case that the supplier takes the focus theory of choice as the decision criterion and the retailer takes expected profit maximization as the decision criterion. Therefore, the research can be extended to the case where the retailer is also bounded rational. There are abundant theories in the field of behavioral decision-making, such as fairness concern and prospect theory; the focus theory of choice is only one of them, so the results of the focus theory of choice can be compared and analyzed with the results of other theories. In future research, behavioral decision analysis will play an important role.