Pricing Analysis of Risk-Averse Supply Chains with Supply Disruption Considering Reference Price Effect

Abstract

This paper examines the impact of the reference price effect on pricing decisions in a risk-averse supply chain with a dual-sourcing procurement strategy, particularly during single-sourcing supply disruption. To analyze supply chain pricing decisions under non-disrupted and disrupted scenarios, we innovatively use semivariance as a risk measure to effectively avoid the limitations of the traditional variance approach and integrate it into Stackelberg game models. Based on these models, we analyze the impact of the reference price effect, risk aversion, and single-sourcing supply disruption on supply chain members’ pricing decisions. The main findings include the following: the single-sourcing supply disruption degree may increase the price of non-disrupted products and then increase the non-disrupted supplier’s utility; the strength of the reference price effect positively influences retailer utility but negatively impacts product pricing for supply chain members; the pricing decisions and utility of supply chain members are influenced by their risk aversion, and supply chain members with higher risk aversion adopt more conservative pricing strategies and consequently obtain lower utility; and equilibrium decisions generally demonstrate a degree of robustness. These insights may help supply chain managers respond rationally to supply disruptions and properly develop pricing strategies by taking into account the reference price effect.

1. Introduction

With the rapid development of internet and information technology, the process of economic globalization is accelerating. Due to significant differences in political, economic, and environmental factors, as well as unforeseeable global events, supply chain disruptions have become a very common phenomenon [1]. For instance, during the third quarter of 2021, Nike faced significant challenges due to the COVID-19 pandemic, which shut down almost all shoe factories in Vietnam and prevented the normal supply of numerous Nike shoes. In addition, the production and supply of Adidas shoes, Apple smartphones, and Samsung smartphones were also severely impacted by the COVID-19 pandemic (https://www.bbc.com/news/business-57902093, accessed on 16 May 2024). The Russo-Ukrainian war began in 2022 and the Israeli–Palestinian conflict erupted in 2023, both of which had a significant impact on global supply chain transport due to their locations as key intersections of maritime and land transport (http://world.people.com.cn/n1/2023/1220/c1002-40142968.html, accessed on 16 May 2024). Hence, there is an urgent need for the international community to establish and enhance a global supply chain governance system to facilitate the steady progress of globalization. Countries are actively promoting supply chain diversification to mitigate compounded and overlapping risks to supply chains and to deal with global uncertainties. Indeed, strengthening the resilience of supply chains is a high priority in mitigating risks.

Dual sourcing is a widely used risk management strategy in supply chain management. It aims to increase supply chain stability and mitigate losses due to supply disruption [1,2]. Briefly speaking, dual sourcing is a procurement strategy in which a retailer purchases the same or alternative products from two suppliers. It ensures that if one supplier experiences a disruption in the supply chain, the other supplier can continue to supply products normally to meet market demand, effectively reducing the risk of supply shortages and providing the retailer with a stable source of products. A growing body of research has considered dual-sourcing strategies when exploring supply disruptions [2,3,4,5]. By embracing dual sourcing, a retailer can diversify its sourcing options, protect itself against risks, and reduce losses caused by supply disruption. This strategy is especially valuable when the products being dual-sourced are bidirectionally substitutable. Therefore, in supply chain risk management, the dual-sourcing strategy is usually combined with a product substitution strategy to mitigate losses from supply disruption. The work by Gupta et al. on supply chain disruptions also highlights the synergy between dual sourcing and product substitution strategies. In particular, the authors analyze how these two strategies work together to dynamically cope with supply chain uncertainty [6]. A practical example is that Lenovo buys processors from both Intel and Advanced Micro Devices, and the dual-sourcing strategy can help reduce Lenovo’s reliance on a single supplier, reducing the risk of production downtime due to supply disruptions. In contrast to existing studies that primarily focus on dual sourcing and product substitution in the context of supply disruptions, this research also incorporates the risk attitudes of supply chain members. By considering these risk preferences, we aim to provide a more comprehensive understanding of how dual sourcing and product substitution can be effectively leveraged in supply chain risk management.

In the context of bidirectional substitutability among dual-sourcing products, consumers invariably engage in price comparisons when arriving at a purchasing decision. This behavior is predominantly influenced by the reference price effect [7,8]. The reference price effect refers to one phenomenon where consumers are more price-sensitive when a product’s price is higher than others they know well. Conversely, consumers are less price-sensitive when the price of a product is lower compared to others known to them. Presently, the rapid development of internet technology has provided consumers with more diversified ways to obtain product prices. This improves the accessibility and promptness of information for consumers, and the impact of the reference price effect on supply chain members’ pricing decisions becomes increasingly influential. In practice, more and more companies use substitutable products as a crucial factor in pricing decisions. For example, in 2024, BYD Company Limited announced a reduction in the prices of several vehicles, which had a significant impact on the vehicle market, forcing other vehicle manufacturers to adjust their prices in order to remain competitive in the market (https://www.whichev.net/2024/03/27/byd-prepares-to-move-aggressively-on-price-reductions/, accessed on 10 January 2025). Moreover, to enhance consumer purchasing intentions, e-commerce platforms like Taobao and Amazon capitalize on reference price effects by actively providing a variety of comparative pricing information on their homepages. Related studies also support this theoretical framework. For example, Ding and Liu demonstrated that reference price has a significant impact on supply chain pricing through regression analyses based on real transaction data [9]. Zhao et al. explored the impact of random demand disruptions and the reference price effect on supply chain pricing, further revealing the importance of reference prices in pricing decisions [10]. These studies reinforce the importance of the reference price effect in today’s market environment and emphasize that enterprises should fully consider the impact of this factor when formulating pricing strategies. Distinguished from previous studies, in this paper, we conduct a thorough analysis of the interplay among the reference price effect, risk aversion, and single-sourcing supply disruptions in shaping pricing decisions. This multidimensional analytical framework presents a theoretical perspective for comprehending pricing strategies in intricate market environments.

Motivated by the above discussion, this study examines supply chain members’ pricing decisions under single-sourcing supply disruption by taking the reference price effect into account. We mainly focus on the following research questions: (1) In a dual-sourcing supply chain, what pricing decisions should supply chain members adopt to maximize their own utility when disruption occurs in one single sourcing? (2) How do supply chain members develop optimal pricing decisions for a risk-averse supply chain? (3) How do the strength of the reference price effect, the degree of risk aversion, and the degree of single-sourcing supply disruption affect the pricing decisions and utility of supply chain members? To solve the above problems, we construct some game models in non-disrupted and disrupted situations, and the main contributions are as follows:

• This paper focuses on the impact of single-sourcing supply disruption on suppliers’ and retailers’ pricing decisions in dual sourcing. Differently from the existing works only considering the influence of single or partial factors on pricing decisions, we comprehensively analyze the influence of the reference price effect, risk aversion, and single-sourcing supply disruption on pricing decisions.
• In terms of model construction, unlike the existing studies on supply chain risk management, this paper is based on the mean semivariance method to measure the risk-averse behavior of supply chain members. Compared to the traditional mean variance method, this method does not limit excess profits above expected returns so as to control losses below expected returns more reasonably and hence can measure the risk-averse behavior of supply chain members more effectively.

The remainder of this paper is organized as follows. Section 2 reviews some of the relevant literature. A description of the problem is laid out in Section 3. Section 4 constructs game models and derives equilibrium solutions. Section 5 performs a sensitivity analysis through numerical experiments. In Section 6, we conclude the paper with future research directions. All proofs are given in an Appendix A.

2. Literature Review

This paper is related to three different streams of the literature, namely, supply chain disruption, the reference price effect, and risk measurement methodology. To highlight our contributions, we show the main differences between this study and previous highly relevant studies in

Table 1. Main differences between this study and relevant studies.

Reference	Pricing Decision	Supply Chain Disruption	Risk Aversion	Reference Price Effect	Mean– Semivarance
Shu et al. (2015) [11]		√	√
Wang (2016) [12]	√			√
Li et al. (2017) [13]	√		√
Choi et al. (2019) [14]	√		√
Basu et al. (2019) [15]			√		√
Zhang and Chiang (2020) [16]	√			√
Gupta and Ivanov (2020) [17]	√	√	√
Gupta et al. (2021) [6]	√	√
Ding and Liu (2021) [9]	√			√
Zhao et al. (2022) [10]	√	√		√
Han et al. (2023) [18]		√	√
Rajabzadeh et al. (2023) [3]	√	√
Li et al. (2024) [19]	√	√
This paper	√	√	√	√	√

.

2.1. Pricing Decision

Pricing decisions, as a core component of a supply chain, directly influence the profitability and efficiency of the entire chain. How to formulate suitable pricing strategies to ensure the stability and efficient operation of a supply chain is always a shared concern for both researchers and practitioners. Alptekinoğlu et al. explored the role of product diversity and pricing strategies in addressing the risk of supply disruption. They showed that when a certain product experiences a supply disruption, the optimal pricing of another product will decrease and interrelationships between two products should be considered when developing pricing strategies [20]. Chen and Liu focused on a supply chain involving two competing suppliers with different reliability levels and one retailer by analyzing the manufacturers’ pricing strategy. Their results showed that responsive pricing strategies were consistently beneficial for manufacturers and could enhance their willingness to adopt a dual-sourcing procurement strategy. This finding emphasized the importance of flexible pricing strategies for manufacturers in the context of supply disruptions [21]. Shan et al. provided an important perspective for understanding the impact of supply disruptions on retailers’ pricing decisions, as they studied the responsive pricing strategies of retailers when suppliers faced supply interruptions. They found that when the correlation of disruptions between two suppliers was high, the lower-cost supplier benefited from the disruption [22]. Li et al. constructed a supply chain coordination mechanism to analyze the impact of supply disruptions on retailers’ procurement strategies and suppliers’ pricing strategies. Their research aimed to assist supply chain members in maximizing profits, highlighting that reasonable pricing and procurement strategies could effectively mitigate risks during supply disruptions [19]. Mohsenzadeh Ledari et al. proposed pricing models by considering the risk of supply chain disruption for substitute and complementary products. The feasibility of the models was verified through numerical experiments, demonstrating that reasonable pricing was crucial for maintaining supply chain stability in the context of supply disruptions [23]. Gheibi and Fay explored the effect of supply disruption risk on retailers’ pricing and purchasing strategies in the presence of substitutes. They observed that when suppliers were unable to deliver regularly, it was beneficial for retailers to reduce the price of the substitute. It also provided empirical support for pricing strategies that retailers could adopt in the face of supply risks [24]. Wang and Yu noted that, under certain circumstances, contingent sourcing and responsive pricing could complement each other in addressing supply chain risks. Their findings indicated that effective pricing could mitigate risks associated with supply disruptions [2].

2.2. Supply Chain Disruption

Many scholars have studied supply chain disruption management from different perspectives. In particular, Parast and Subramanian categorized supply chain disruption into four types: demand disruption, supply disruption, process disruption, and environmental disruption. They demonstrated that different disruptions may have different effects on supply chain performance. Additionally, they found that disruption occurring upstream in a supply chain has greater impacts than that occurring downstream [25]. Wu et al. explored how different supply chain strategies impact consumer panic buying behavior during supply disruptions. They emphasize the need for adaptive strategies to cope with disruptions [26]. Chakraborty et al. examined the impact of supply disruption on the operational strategies of two suppliers and one retailer. Their findings indicated that retailers always prefer to utilize a backup supplier under supply disruption [27]. Gupta et al. investigated the impact of supply capacity disruption timing on the pricing decisions of substitute products in a supply chain with two suppliers and one retailer [6]. Liu et al. analyzed the interactions between product pricing and self-protection decisions from both aggregate and individual supply chain perspectives. They provide supply chain practitioners with managerial insights to better cope with the risk of supply disruption [28]. Xue et al. discussed the supply shortage caused by suppliers’ refusal to supply under dual sourcing. They proposed a subsidy mechanism to reduce the risk of supply shortages and verified its feasibility and effectiveness through validation experiments [29]. Moreover, Rajabzadeh et al. employed a game-theoretic approach to investigate the impact of pricing strategies in closed-loop supply chain under supply disruption. They discussed dual-sourcing and product recycling strategies in addressing disruption problems. Their study showed that dual-sourcing can assist retailers in maintaining market share during supply disruption [3].

Inspired by the above studies, this study considers a dual-sourcing supply chain system with two suppliers and one single retailer. Additionally, we uniquely explore the joint influence of reference price effects, single-sourcing supply disruption, and risk aversion on pricing decisions under single-sourcing supply disruption, which may help to provide more effective theoretical guidance for practical managerial pricing decisions.

2.3. Risk Aversion

As supply chain instability increases and market competition intensifies, business managers are becoming more sensitive to supply chain risk. Managers’ attitudes towards risk not only affect their decision-making process but also have an impact on the overall performance of the supply chain. As a result, more and more scholars have begun to consider the risk aversion of supply chain members in supply chain research. Chernonog and Tatyana discussed the effects of risk aversion on retailer strategies and overall supply chains. Their findings indicated that the optimal order quantity for retailers did not always decrease with an increase in their level of risk aversion [30]. Xie et al. examined the impact of the risk-aversion behaviors of supply chain members on supply chain quality investments and pricing strategies. Their research demonstrated that in a supplier-led supply chain strategy, the product quality of risk-averse supply chains was superior to that of risk-neutral supply chains. This study highlights the importance of risk aversion in influencing decision-making in supply chains [31]. Zhu et al. explored the dual-channel supply chain coordination issues faced by risk-averse retailers under production and demand uncertainty. They proposed a joint supply chain contract and proved its effectiveness [32]. Liu et al. investigated the coordination issues in supplier-led and retailer-led supply chains under option contracts. They found that in retailer-led supply chains, risk-averse retailers tended to set option prices as low as possible to transfer more risk to suppliers [33].

This paper examines the risk aversion of supply chain members as one of the key factors in this study.

2.4. Reference Price Effect

The reference price effect is a crucial factor in influencing consumer purchasing behavior and supply chain members’ pricing strategies. In this respect, Zhang et al. developed a Stackelberg differential game model by considering the reference price effect in a bilateral monopoly environment and showed that managers can capture more benefits if there is a higher initial reference price and consumers exhibit greater price sensitivity and stronger product brand loyalty [34]. Cao and Duan introduced the reference price effect into the pricing problem of joint dynamic production and discovered that as the reference price effect factor or customers’ memory parameter increases, the production level decreases but the product price increases [35]. Zhao et al. investigated the impact of the reference price effect and price-matching on retailer’s pricing decisions. They found that when the strength of the reference price effect is large, ignoring the reference price effect and only considering price matching may harm retailer’s revenue [36]. In a two-stage dynamic pricing model, Chen et al. considered the reference price effect in order to address the pricing strategy of a seller when dealing with strategic consumers. The study demonstrated that when strategic consumer behavior was low, the seller’s profit increased with the reference price effect [37]. Zhang and Chiang analyzed the pricing strategy of durable goods by considering the reference price effect and demonstrated that the optimal pricing strategy for forward-looking sellers depends on the potential market and reference price effect [16]. Colombo and Labrecciosa analyzed the pricing problem in an oligopoly model with the reference price effect. They argued that for loss-averse consumers, there exists an initial reference price interval such that price and quantity competition lead to the same outcome [38]. Duan and Feng conducted a study to examine the impact of social network effects and reference price effects on monopoly pricing strategies. They demonstrated that both factors play a significant role in the formulation of pricing strategies for social networks [39]. Ding and Liu investigated the optimal pricing strategies employed by retailers when selling two substitutable products, which highlighted the significance of reference and alternative prices in influencing consumer demand [9]. Zhao et al. investigated the impact of random demand disruption and the reference price effect on pricing, information-gathering, and sharing decisions in supply chain operations. Their findings indicated that inaccurate disruptive information results in significant losses in the supply chain and the reference price effect may mitigate supplier’s losses but exacerbate the losses for the platform [10]. Liu and Popkowski Leszczyc explored the reference price effect of historical price lists on final auction prices in auction processes. They showed that the maximum price in historical price lists has a positive effect on the auction’s final price, while the price range has a negative effect on auction’s final price [40].

Actually, there is a paucity of research examining the impact of the reference price effect under supply disruption. This study aims to analyze the impact of the reference price effect and risk aversion under single-sourcing supply disruption to develop more rational pricing decisions to mitigate supply chain risks.

2.5. Risk Measurement Methodology

2.5.1. Mean Variance/Mean Standard Variance

In the context of escalating supply chain instability and the intensification of market competition, business managers are becoming increasingly cognizant of the risks inherent in supply chains. The attitudes of these managers towards risk have been shown to exert a significant influence on their decision-making processes and the overall performance of supply chains [33,41,42]. In the context of supply chain risk management, the mean variance (MV) and mean standard variance (MSD) methods are commonly used to measure risk. The MV method was proposed by Markowitz in 1959 and is one of the most widely used risk measures [43,44]. Hung et al. discussed a two-stage risk-averse supply chain problem and proposed a new operating mechanism to meet demand more efficiently [45]. Shu et al. employed the MV method to examine the inventory and procurement strategy of risk-averse retailers under unreliable supply and stochastic demand. Their findings indicated that retailers with high levels of risk aversion exhibit great sensitivity to parameters affecting order quantity [11]. Li et al. analyzed the dual-channel supply chain pricing problem under information asymmetry and explored the impact of retailers’ risk-averse behavior on supply chain members’ profits and utility using the MV method [13]. Wang et al. studied the influence of agents’ risk preference on pricing strategies in dual-channel supply chains using the MV method to measure the risk aversion of the agents, and their findings revealed that the pricing decisions obtained differ according to agents’ risk attitudes [46]. Choi et al. investigated the pricing problem in a mass customization supply chain using the MSD method and revealed the relationship between the level of manufacturer’s risk aversion and retailers in the supply chain and optimal pricing strategies [14]. Gupta and Ivanov studied the pricing problem of substitute products in dual sourcing under supply disruption using the MSD method [17].

2.5.2. Mean Semivariance

The innovation of using semivariance as a risk measurement tool lies in its focus on downside risk, which has been proven to be superior in the supply chain context according to existing studies. For example, Basu et al. selected the semivariance method to measure supply chain risk. They proposed a buyback contract with linear prices to coordinate supply chain performance and their results showed that semivariance has attractive properties as a risk measure, especially when demand uncertainty in a supply chain is high, providing theoretical support for the application of semivariance in the supply chain domain [15]. In addition to the mean semivariance approach, Value at Risk (VaR) and Conditional Value at Risk (CVaR) are common measures of downside risk. However, the limitation of VaR is that it can only measure the potential loss at a certain confidence level and the solution process is relatively complex. To address this shortcoming, Roekafeller and Uryasev proposed CVaR to provide a more comprehensive assessment of risk. Based on the CVaR theory, Zhao et al. applied the CVaR theory to study the effects of service level and risk aversion on each member of the supply chain and found that the optimal conditional value at risk of a retailer increases with increasing risk aversion, while the optimal expected profit of a supplier decreases with decreasing risk aversion [47]. In contrast to these methods, the mean semivariance approach is flexible and can be combined with other optimization models to better deal with complex decision scenarios in supply chains. By adjusting the risk aversion coefficients, the semivariance approach can effectively reflect the individual risk preferences of different decision-makers, which greatly enhances its potential application in supply chain risk management.

Although extant studies have widely recognized the advantages of the semivariance approach in supply chain management, especially in the context of significant demand volatility, the extant literature still focuses on the variance as the risk measure in the research of risk-averse supply chain pricing in the context of supply disruptions. In view of this, this paper introduces semivariance as a risk measurement to more accurately assess and address the downside risks associated with supply disruptions, thereby facilitating a deeper exploration of pricing strategies for risk-averse supply chains.

3. Problem Formulation

Consider a supply chain system consisting of two suppliers and a retailer, where both suppliers and the retailer are risk-averse. Assume that the suppliers are competitive with each other by providing bidirectional, completely substitutable products. We focus on the impact of the reference price effect, risk aversion, and single-sourcing supply disruption on pricing decisions and the utility of all members.

Table 2. Notations.

Notation	Explanation
${\omega }_i$	Wholesale price of product $i$$; i=1,2$
$p_i$	Retail price of product $i$$; i=1,2$
$\alpha$	$\mathrm{Strength} \mathrm{of} \mathrm{reference} \mathrm{price} \mathrm{effects}; 0\le \alpha \le 1$
$\eta$	$\mathrm{Random} \mathrm{demand} \mathrm{disturbance} \mathrm{with} \mathrm{mean} \overline{\eta }>0$
$\lambda$	Degree of single-sourcing $\mathrm{supply} \mathrm{disruption}; \lambda \in (0,1]$
${\phi }_{{\mathrm{S}}_i}$	Degree of supplier $i$$’\mathrm{s} \mathrm{risk} \mathrm{aversion}; {\phi }_{{\mathrm{S}}_i}>0, i=1,2$
${\phi }_R$	$\mathrm{Degree} \mathrm{of} \mathrm{retailer}’\mathrm{s} \mathrm{risk} \mathrm{aversion}; {\phi }_R>0$
$Q_i$	Market demand for product $i$$; i=1,2$
${\mathrm{\Pi }}_{{\mathrm{S}}_i}$	Profit of supplier $i$; $i=1,2$
${\mathrm{\Pi }}_{\mathrm{R}}$	Profit of retailer
$U_{S_i}$	Utility of supplier $i$$; i=1,2$
$U_R$	Utility of retailer

displays the relevant notation discussed in this paper. The equilibrium results in non-disrupted and disrupted situations are indicated by the superscripts $N$ and $D$, respectively. The equilibrium outcomes for the suppliers and retailer are distinguished by the subscripts ${\mathrm{S}}_i$ and $\mathrm{R}$, respectively.

In the non-disrupted situation, the suppliers’ decision variables are their wholesale prices for their respective products. The retailer’s decision variables are its retail prices for two products. The retailer decides the purchase quantities $Q_1$ and $Q_2$ from the two suppliers and subsequently sells them to consumers in the market. The structure of the non-disrupted supply chain system is shown in Figure 1.

In the disrupted situation, the decisions for suppliers and the retailer remain the wholesale prices and the retail prices, respectively. Without loss of generality, we suppose that supply disruption occurs at supplier 1. We denote by $\lambda \in (0,1]$ the degree of single-sourcing supply disruption. That is, supplier 1 can only supply $\left(1-\lambda \right)Q_1$ to the retailer.

To meet market demand consistently, the retailer needs to purchase product 2 from the non-disrupted supplier 2, mitigating the risk of stockouts caused by single-sourcing supply disruption. The structure of the disrupted supply chain system is shown in Figure 2.

Assume that the market demands for products 1 and 2 are random. Referring to the existing literature [10,12], we adopt linear demand functions:

$$Q_i=1-p_i+\alpha \left(p_j-p_i\right)+\eta , \left(i,j=1,2, i\ne j\right),$$

where $0\le \alpha \le 1$ can be viewed as the strength of consumers’ reference price effects in two products and $\eta$ is a random disturbance in the market with the probability density $f\left(\eta \right)$ and distribution function $F\left(\eta \right)$. Assume that $f\left(\eta \right)$ is a nonnegative integrable function and continuous in $\eta \in (-\infty ,a]$ and ${\displaystyle {\int }_{-\infty }^{a}f\left(\eta \right)}d\eta$ converges. Similarly to [15,48], we use the mean semivariance method to give the utility functions of the suppliers and retailer as

$$\mathrm{Supplier} i: U_{{\mathrm{S}}_i}=E\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)-{\phi }_{{\mathrm{S}}_i}S\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right), i=1,2,$$

$$\mathrm{Retailer}: U_R=E\left({\mathrm{\Pi }}_{\mathrm{R}}\right)-{\phi }_{\mathrm{R}}S\left({\mathrm{\Pi }}_{\mathrm{R}}\right),$$

where ${\phi }_{{\mathrm{S}}_i}$ and ${\phi }_{\mathrm{R}}$ represent the degrees of risk aversion of supplier $i$ and the retailer, respectively; ${\mathrm{\Pi }}_{{\mathrm{S}}_i}$ and ${\mathrm{\Pi }}_{\mathrm{R}}$ represent the profit functions for supplier $i$ and the retailer, respectively; $S\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)=E\left[{\left({\left(E({\mathrm{\Pi }}_{{\mathrm{S}}_i})-{\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)}^{+}\right)}^2\right]$ represents the semivariance of the suppliers’ profit; and $S\left({\mathrm{\Pi }}_{\mathrm{R}}\right)=E\left[{\left({\left(E({\mathrm{\Pi }}_{\mathrm{R}})-{\mathrm{\Pi }}_{\mathrm{R}}\right)}^{+}\right)}^2\right]$ means the semivariance of the retailer’s profit.

Model Assumption

Similarly to the existing literature, we make the following basic hypotheses:

(1)

Products 1 and 2 are bidirectional, completely substitutable products.

(2)

Supply chain members are all risk-averse [49].

(3)

The suppliers’ marginal cost is zero [50].

(4)

The random disturbance $\eta$ follows a Gamma distribution [51].

In many cases, consumers choose similar alternatives based on price or other characteristics, thus reflecting the actual competitive environment of the market. For analyzing pricing strategies and consumer behavior, implementing the assumption of two-way substitutability helps to simplify the model, and similar assumptions are used in the literature [6]. The results of a questionnaire survey of company managers showed that managers always maintain a cautious attitude when facing risks, i.e., their risk attitude as a whole is more risk-averse, so this paper assumes that supply chain members are all risk-averse [49]. To keep notational parsimony, without loss of generality, we assume that production costs for suppliers are normalized to be zero, as assumed by Shen et al. [50]. The use of the Gamma distribution allows for the consideration of heterogeneity in demand fluctuations, which is essential for analyzing pricing strategies and their resulting utility implications.

4. Game-Theoretic Models

In this section, we construct game models for both non-disrupted and disrupted situations. These models are then solved to determine optimal decisions for suppliers and the retailer.

4.1. Non-Disrupted Situation

In the non-disrupted situation, we construct a Stackelberg game model with the suppliers as leaders and the retailer as a follower. At first, suppliers 1 and 2 set the wholesale prices ${\omega }_1$ and ${\omega }_2$ for products 1 and 2. Based on the wholesale prices and the market demands, the retailer sets the retail prices $p_1$ and $p_2$, which are both market-clearing. The suppliers’ and retailer’s objectives are to maximize their own utility.

We solve the Stackelberg game model by the backward induction method. The lower-level model is the retailer’s decision-making problem, that is,

$$\begin{array}{l}\underset{p_1,p_2}{\max }\hspace{1em}{U}_R=E\left({\mathrm{\Pi }}_{\mathrm{R}}\right)-{\phi }_{\mathrm{R}}S\left({\mathrm{\Pi }}_{\mathrm{R}}\right)\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{p}_1\ge {\omega }_1, p_2\ge {\omega }_2,\end{array}$$

where the retailer’s profit function ${\mathrm{\Pi }}_{\mathrm{R}}=\left(p_1-{\omega }_1\right)Q_1+\left(p_2-{\omega }_2\right)Q_2$, which is the sum of the profits from selling products 1 and 2.

We denote by $g\left({\mathrm{\Pi }}_{\mathrm{R}}\right)$ the probability density function of ${\mathrm{\Pi }}_{\mathrm{R}}$. Since $E\left({\mathrm{\Pi }}_R\right)-{\mathrm{\Pi }}_R=\left(p_1-{\omega }_1+p_2-{\omega }_2\right)\left(\overline{\eta }-\eta \right)$, we can easily obtain

$$S\left({\mathrm{\Pi }}_{\mathrm{R}}\right)={\displaystyle {\int }_{-\infty }^{E({\mathrm{\Pi }}_{\mathrm{R}})}{\left(E({\mathrm{\Pi }}_{\mathrm{R}})-{\mathrm{\Pi }}_{\mathrm{R}}\right)}^2}g({\mathrm{\Pi }}_{\mathrm{R}})d{\mathrm{\Pi }}_{\mathrm{R}}={\left(p_1-{\omega }_1+p_2-{\omega }_2\right)}^2{\displaystyle {\int }_{-\infty }^{E({\mathrm{\Pi }}_{\mathrm{R}})}{\left(\overline{\eta }-\eta \right)}^2}g({\mathrm{\Pi }}_{\mathrm{R}})d{\mathrm{\Pi }}_{\mathrm{R}}.$$

Since ${\mathrm{\Pi }}_R=\left(p_1-{\omega }_1\right)\left(1-p_1+\alpha \left(p_2-p_1\right)+\eta \right)+\left(p_2-{\omega }_2\right)\left(1-p_2+\alpha \left(p_1-p_2\right)+\eta \right)$ and $f\left(\eta \right)$ is the probability density function of $\eta$, we have

$$\begin{array}{cc}g({\mathrm{\Pi }}_{\mathrm{R}})& =f\left({\displaystyle \frac{{\mathrm{\Pi }}_{\mathrm{R}}-\left(p_1-{\omega }_1\right)\left(1-p_1+\alpha \left(p_2-p_1\right)\right)-\left(p_2-{\omega }_2\right)\left(1-p_2+\alpha \left(p_1-p_2\right)\right)}{p_1-{\omega }_1+p_2-{\omega }_2}}\right)\cdot \left({\displaystyle \frac{1}{p_1-{\omega }_1+p_2-{\omega }_2}}\right)\hfill \\ & =f\left(\eta \right)\cdot \left({\displaystyle \frac{1}{p_1-{\omega }_1+p_2-{\omega }_2}}\right),\hfill \end{array}$$

and $d{\mathrm{\Pi }}_{\mathrm{R}}=(p_1-{\omega }_1+p_2-{\omega }_2)d\eta$. Thus, we obtain

$$S\left({\mathrm{\Pi }}_{\mathrm{R}}\right)={\left(p_1-{\omega }_1+p_2-{\omega }_2\right)}^2{\displaystyle {\int }_{-\infty }^{\overline{\eta }}{\left(\overline{\eta }-\eta \right)}^2}f(\eta )d\eta .$$

Solving the retailer’s decision-making problem yields the following proposition, whose proof is given in Appendix A.

Proposition 1.

In the non-disrupted situation, $U_R\left(p_1,p_2\right)$ is jointly concave in $\left(p_1,p_2\right)$ for any given wholesale prices $\left({\omega }_1,{\omega }_2\right)$ . When $0\le {\omega }_i\le \overline{\eta }+1 \left(i=1,2\right)$, the optimal retail prices for products 1 and 2 are

$$p_1^N\left({\omega }_1,{\omega }_2\right)={\displaystyle \frac{{\omega }_1+3M{\phi }_{\mathrm{R}}{\omega }_1+M{\phi }_{\mathrm{R}}{\omega }_2+\overline{\eta }+1}{2\left(2M{\phi }_{\mathrm{R}}+1\right)}} \mathit{and} p_2^N\left({\omega }_1,{\omega }_2\right)={\displaystyle \frac{{\omega }_2+3M{\phi }_{\mathrm{R}}{\omega }_2+M{\phi }_{\mathrm{R}}{\omega }_1+\overline{\eta }+1}{2\left(2M{\phi }_{\mathrm{R}}+1\right)}}$$

where $M={\displaystyle {\int }_{-\infty }^{\overline{\eta }}{\left(\overline{\eta }-\eta \right)}^{2}f\left(\eta \right)d\eta }$.

Next, we analyze the upper-level models (i.e., the suppliers’ decision-making problems):

$$\begin{array}{l}\underset{{\omega }_1}{\max }{U}_{{\mathrm{S}}_1}=E\left({\mathrm{\Pi }}_{{\mathrm{S}}_1}\right)-{\phi }_{{\mathrm{S}}_1}S\left({\mathrm{\Pi }}_{{\mathrm{S}}_1}\right)\\ \mathrm{s}.\mathrm{t}.{\omega }_1\ge 0,\end{array}\hspace{1em}\begin{array}{l}\underset{{\omega }_2}{\max }{U}_{{\mathrm{S}}_2}=E\left({\mathrm{\Pi }}_{{\mathrm{S}}_2}\right)-{\phi }_{{\mathrm{S}}_2}S\left({\mathrm{\Pi }}_{{\mathrm{S}}_2}\right)\\ \mathrm{s}.t.{\omega }_2\ge 0,\end{array}$$

where ${\mathrm{\Pi }}_{S_i}={\omega }_i{Q}_i$ is supplier $i$’s profit function $(i=1,2)$.

In order to facilitate the analysis, we let $h\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)$ be the probability density function of ${\mathrm{\Pi }}_{{\mathrm{S}}_i}$. Note that $E\left({\mathrm{\Pi }}_{S_i}\right)-{\mathrm{\Pi }}_{S_i}={\omega }_i\left(\overline{\eta }-\eta \right)$ and $S\left({\mathrm{\Pi }}_{S_i}\right)=E\left[{\left({\left(E({\mathrm{\Pi }}_{S_i})-{\mathrm{\Pi }}_{S_i}\right)}^{+}\right)}^2\right]$; we can easily obtain

$$S\left({\mathrm{\Pi }}_{S_i}\right)={\displaystyle {\int }_{-\infty }^{E\left({\mathrm{\Pi }}_{S_i}\right)}{\left(E({\mathrm{\Pi }}_{S_i})-{\mathrm{\Pi }}_{S_i}\right)}^2}h\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)d{\mathrm{\Pi }}_{{\mathrm{S}}_i}={{\omega }_i}^2{\displaystyle {\int }_{-\infty }^{E\left({\mathrm{\Pi }}_{S_i}\right)}{\left(\overline{\eta }-\eta \right)}^2}h({\mathrm{\Pi }}_{{\mathrm{S}}_i})d{\mathrm{\Pi }}_{{\mathrm{S}}_i}.$$

Since ${\mathrm{\Pi }}_{S_i}={\omega }_i\left(1-p_i+\alpha \left(p_j-p_i\right)+\eta \right)$ and $f\left(\eta \right)$ is the probability density function of $\eta$, we have

$$h({\mathrm{\Pi }}_{{\mathrm{S}}_i})=f\left({\displaystyle \frac{{\mathrm{\Pi }}_{{\mathrm{S}}_i}-{\omega }_i\left(1-p_i+\alpha \left(p_j-p_i\right)\right)}{{\omega }_i}}\right)\cdot {\displaystyle \frac{1}{{\omega }_i}}=f\left(\eta \right)\cdot {\displaystyle \frac{1}{{\omega }_i}},$$

and $d{\mathrm{\Pi }}_{{\mathrm{S}}_i}={\omega }_{i}d\eta$, where $i,j=1,2, i\ne j$. Then, we have $S\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)={{\omega }_i}^2{\displaystyle {\int }_{-\infty }^{\overline{\eta }}{\left(\overline{\eta }-\eta \right)}^2}f(\eta )d\eta$.

By substituting $p_1^N\left({\omega }_1,{\omega }_2\right)$ and $p_2^N\left({\omega }_1,{\omega }_2\right)$ into $U_{S_1}$ and $U_{{\mathrm{S}}_2}$ in Proposition 1, we obtain the following Nash game models:

$$\begin{array}{l}\underset{{\omega }_1}{\max } U_{S_1}={\omega }_1\overline{\eta }+{\omega }_1\left(1-{\displaystyle \frac{{\omega }_1+3M{\phi }_{\mathrm{R}}{\omega }_1+M{\phi }_{\mathrm{R}}{\omega }_2+\overline{\eta }+1}{2\left(2M{\phi }_{\mathrm{R}}+1\right)}}+{\displaystyle \frac{{\omega }_2-{\omega }_1}{2}}\alpha \right)-M{\phi }_{{\mathrm{S}}_1}{\omega }_1^2\\ \mathrm{s}.\mathrm{t}.0\le {\omega }_1\le \overline{\eta }+1,\end{array}$$

$$\begin{array}{l}\underset{{\omega }_2}{\max } U_{{\mathrm{S}}_2}={\omega }_2\overline{\eta }+{\omega }_2\left(1-{\displaystyle \frac{{\omega }_2+3M{\phi }_{\mathrm{R}}{\omega }_2+M{\phi }_{\mathrm{R}}{\omega }_1+\overline{\eta }+1}{2\left(2M{\phi }_{\mathrm{R}}+1\right)}}+{\displaystyle \frac{{\omega }_1-{\omega }_2}{2}}\alpha \right)-M{\phi }_{{\mathrm{S}}_2}{{\omega }_2}^2\\ \mathrm{s}.\mathrm{t}.0\le {\omega }_2\le \overline{\eta }+1.\end{array}$$

Using the backward induction method to solve the above problems, we obtain the following proposition, whose proof is given in Appendix A.

Proposition 2.

In the non-disrupted situation, when $0<{\phi }_{{\mathrm{S}}_1}\le m_1$ and $-2A_1{\phi }_{{\mathrm{S}}_2}<\left(4M{\phi }_{\mathrm{R}}+1-2A_1\right){\phi }_{{\mathrm{S}}_2}\le m_2$ , each supplier’s utility function is concave in its wholesale price. The optimal decisions of the suppliers and the retailer are

$${\omega }_1^N={\displaystyle \frac{t_1}{8M{A}_1\left(2M{\phi }_{\mathrm{R}}+1\right){\phi }_{{\mathrm{S}}_2}+A_2}}, {\omega }_2^N={\displaystyle \frac{t_2}{8M{A}_1\left(2M{\phi }_{\mathrm{R}}+1\right){\phi }_{{\mathrm{S}}_2}+A_2}},$$

$$p_1^N={\displaystyle \frac{t_3}{16M{A}_1{\left(2M{\phi }_{\mathrm{R}}+1\right)}^2{\phi }_{{\mathrm{S}}_2}+2A_2\left(2M{\phi }_{\mathrm{R}}+1\right)}}, p_2^N={\displaystyle \frac{t_4}{16M{A}_1{\left(2M{\phi }_{\mathrm{R}}+1\right)}^2{\phi }_{{\mathrm{S}}_2}+2A_2\left(2M{\phi }_{\mathrm{R}}+1\right)}},$$

respectively, where $m_1$, $m_2$, $t_1~t_4$, $A_1$, and $A_2$ are defined in

Table A1. Notation and explanation.

Notation	Explanation
$a_1$	$12M^2{\phi }_R^2+\left(4\alpha +13\right)M{\phi }_R+2\alpha +3$
$a_2$	$4M\left(2M{\phi }_R+1\right)\left(4M^2{\phi }_R^2+4M\alpha {\phi }_R+7M{\phi }_R+2\alpha +2\right)$
$a_3$	$16\left(6\alpha +5\right)M^3{\phi }_R^3+\left(12{\alpha }^2+148\alpha +107\right)M^2{\phi }_R^2+3\left(4{\alpha }^2+24\alpha +15\right)M{\phi }_R+3{\alpha }^2+11\alpha +16$
$a_4$	$-2M\left(2\alpha +1\right)\left({\left(\lambda -2\right)}^2-1\right){\phi }_{\mathrm{R}}+\left(\lambda -1\right)\left(\left(4-\lambda \right)\alpha +2\right)$
$a_5$	$2M\left(2\alpha +1\right)\left({\lambda }^2-1\right){\phi }_{\mathrm{R}}-\lambda \left(\left(1-\lambda \right)\alpha +1\right)$
$a_6$	$\left(\lambda -1\right)\left(-2M\left(2\alpha +1\right)\left(\lambda -1\right){\phi }_{\mathrm{R}}-\lambda \left(\alpha +1\right)\right)$
$a_7$	$2M\left(2\alpha +1\right)\left({\left(\lambda +1\right)}^2-4\right){\phi }_{\mathrm{R}}+\left(\alpha +1\right){\lambda }^2+\left(3\alpha +2\right)\lambda -2\left(2\alpha +1\right)$
$m$	${\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)$
$m_1$	${\displaystyle \frac{8M{A}_1\left(2M{\phi }_{\mathrm{R}}+1\right){\phi }_{{\mathrm{S}}_2}+A_2-\left(\left(6\alpha +5\right)M{\phi }_{\mathrm{R}}+3\alpha +2\right)\left(4M{\phi }_{\mathrm{R}}+1\right)}{4M\left(2M{\phi }_{\mathrm{R}}+1\right)\left(4M{\phi }_{\mathrm{R}}+1\right)}}$
$m_2$	${\displaystyle \frac{A_2-\left(\left(6\alpha +5\right)M{\phi }_{\mathrm{R}}+3\alpha +2\right)\left(4M{\phi }_{\mathrm{R}}+1\right)}{4M\left(2M{\phi }_{\mathrm{R}}+1\right)}}$
$m_3$	${\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)-a_4$
$m_4$	$\left(\left(4\alpha +\lambda \right)\left(\lambda -1\right)+4\lambda -2\right)\left(\overline{\eta }+1\right)$
$m_5$	${\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)-a_7$
$m_6$	$\left(\lambda -1\right)\left(4\alpha +\lambda +2\right)\left(\overline{\eta }+1\right)$
$b_1$	$2M{\left(\lambda +1\right)}^2\left(-64M^2\left(2\alpha +1\right){\left(\lambda -1\right)}^2{\phi }_R^2-8M\left(\lambda -1\right)\left({\lambda }^2+\left(2\alpha +1\right)\left(5\lambda -6\right)\right){\phi }_R-\left(\lambda -2\right)m\right)$
$b_2$	$-16M^2\left(\lambda -1\right)\left({\lambda }^2+2\left(1-\alpha \right)\lambda +6\alpha +5\right)$
$b_3$	$2M\left({\lambda }^3+2\left(1-5\alpha \right){\lambda }^2+\left(46\alpha +9\right)\lambda -36\alpha -26\right)$
$b_4$	$\left(2\alpha -1\right){\lambda }^2-2\left(7\alpha +3\right)\lambda +12\alpha +8$
$b_5$	$2M\left(\lambda -1\right)\left(64M^2\left(2\alpha +1\right)\left(\lambda -1\right)\left(\lambda +1\right){\phi }_R^2+8M\left({\lambda }^3+2\left(5\alpha +3\right){\lambda }^2+\left(2\alpha +1\right)\left(\lambda -6\right)\right){\phi }_R+\left(\lambda +2\right)m\right)$
$b_6$	$16M^2\left(\lambda -1\right)\left(-{\lambda }^2+2\left(\alpha +1\right)\lambda +6\alpha +5\right)$
$b_7$	$2M\left(-{\lambda }^3+2\left(5\alpha +6\right){\lambda }^2+\left(26\alpha +17\right)\lambda -36\alpha -26\right)$
$b_8$	$\left(2\alpha +3\right){\lambda }^2+2\left(5\alpha +3\right)\lambda -12\alpha -8$
$A_1$	$M\left(4M{\phi }_{{\mathrm{S}}_1}+2\alpha +3\right){\phi }_R+2M{\phi }_{{\mathrm{S}}_1}+\alpha +1$
$A_2$	$\begin{array}{l}{M}^2\left(16M\left(2\alpha +3\right){\phi }_{{\mathrm{S}}_1}+\left(2\alpha +7\right)\left(6\alpha +5\right)\right){\phi }_R^2+2M\left(4M\left(4\alpha +5\right){\phi }_{{\mathrm{S}}_1}+6{\alpha }^2+21\alpha +12\right){\phi }_R\\ +8M\left(\alpha +1\right){\phi }_{{\mathrm{S}}_1}+\left(\alpha +2\right)\left(3\alpha +2\right)\end{array}$
$A_3$	$\begin{array}{l}2M\left(2\alpha +1\right)\left(8mM\left(\lambda -1\right){\phi }_{{\mathrm{S}}_1}+{\lambda }^3+\left(6\alpha +1\right){\lambda }^2+4\left(2\alpha +1\right)\left(-2\left(2\alpha +3\right)\left(\lambda -1\right)-1\right)\right){\phi }_R\\ +16M^2{\left(2\alpha +1\right)}^2\left(\lambda -1\right)\left(4M\left(\lambda -1\right){\phi }_{{\mathrm{S}}_1}-\left(2\alpha -\lambda +3\right)\right){\phi }_R^2+M{m}^2{\phi }_{{\mathrm{S}}_1}-2m\left(2\alpha +1\right)\left(\alpha +1\right)\end{array}$
$A_4$	$\left(2\alpha +1\right)\left(\begin{array}{l}{A}_5+A_6-8mM\left(\lambda -1\right)\left(\alpha \left(\lambda -1\right)-1\right){\phi }_{{\mathrm{S}}_1}\\ +\left(2\alpha +1\right)\left(\left(12{\alpha }^2+12\alpha -1\right){\lambda }^2-4\left(2\alpha +1\right)\left(3\alpha +4\right)\lambda +4\left(3\alpha +2\right)\left(\alpha +2\right)\right)\end{array}\right)$
$A_5$	$4M^2\left(2\alpha +1\right)\left(\lambda -1\right)\left(\begin{array}{l}16M\left(\lambda -1\right)\left({\lambda }^2+2\left(1-\alpha \right)\lambda +2\alpha +3\right){\phi }_{{\mathrm{S}}_1}+3{\lambda }^3-3\left(4\alpha +1\right){\lambda }^2\\ +3\left(2\alpha -1\right)\left(2\alpha +3\right)\lambda -\left(2\alpha +7\right)\left(6\alpha +5\right)\end{array}\right){\phi }_R^2$
$A_6$	$4M\left(\begin{array}{l}2M\left(\lambda -1\right)\left({\lambda }^4+6\left(\alpha +1\right){\lambda }^3-\left(32{\alpha }^2+6\alpha -7\right){\lambda }^2+4\left(2\alpha +1\right)(\left(8\alpha +3\right)\lambda -4\alpha -5)\right){\phi }_{{\mathrm{S}}_1}\\ -\left(2\alpha +1\right)\left(3\left(2\alpha +1\right){\lambda }^3-\left(12{\alpha }^2+18\alpha -1\right){\lambda }^2+3\left(2\alpha +1\right)\left(4\alpha +7\right)\lambda -6\left(2{\alpha }^2+7\alpha +4\right)\right)\end{array}\right){\phi }_R$
$c_1$	$16M^2\left(\lambda -1\right){\left(2\alpha +1\right)}^2\left(\begin{array}{l}8M\left(\lambda -1\right)\left({\lambda }^2+\left(4\alpha +3\right)\left(\lambda -1\right)+1\right){\phi }_{{\mathrm{S}}_1}+3{\lambda }^3\\ +2\left(9\alpha +1\right)-\left(16{\alpha }^2+106\alpha +51\right)\lambda +2\left(2\alpha +1\right)\left(4\alpha +19\right)\end{array}\right)$
$c_2$	$2M\left(2\alpha +1\right)\left(\begin{array}{l}16M\left(\lambda -1\right)\left({\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\right)\left({\lambda }^2+\left(4\alpha +3\right)\left(\lambda -1\right)+1\right){\phi }_{{\mathrm{S}}_1}\\ +2{\lambda }^5+\left(26\alpha +9\right){\lambda }^4+\left(40{\alpha }^2-86\alpha -55\right){\lambda }^3-2\left(2\alpha +1\right)\left(64{\alpha }^2+226\alpha +79\right){\lambda }^2\\ +4\left(2\alpha +1\right)\left(64{\alpha }^2+200\alpha +87\right)\lambda -8{\left(2\alpha +1\right)}^2\left(8\alpha +19\right)\end{array}\right)$
$c_3$	$\begin{array}{l}2M\left({\lambda }^2+\left(4\alpha +3\right)\left(\lambda -1\right)+1\right){\left({\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\right)}^2{\phi }_{{\mathrm{S}}_1}\\ -\left(2\alpha +1\right)\left(\begin{array}{l}\left(3\alpha +4\right){\lambda }^4+2\left(20{\alpha }^2+39\alpha +15\right){\lambda }^3+4\left(2\alpha +1\right)\left(16{\alpha }^2+29\alpha +7\right){\lambda }^2\\ -8\left(2\alpha +1\right)\left(16{\alpha }^2+33\alpha +13\right)\lambda +16\left(2\alpha +3\right){\left(2\alpha +1\right)}^2\end{array}\right)\end{array}$
$c_4$	$64M^3{\left(2\alpha +1\right)}^2{\left(\lambda -1\right)}^2\left(4M\left(\lambda -1\right){\left(\lambda +1\right)}^2{\phi }_{{\mathrm{S}}_1}+{\lambda }^3-\left(2\alpha +1\right){\lambda }^2-2\left(2-\alpha \right)\lambda -12\alpha -10\right)$
$c_5$	$4M^2\left(2\alpha +1\right)\left(\lambda -1\right)\left(\begin{array}{l}8M\left(\lambda -1\right)\left(\begin{array}{l}3{\lambda }^4+\left(18\alpha +17\right){\lambda }^3-\left(16{\alpha }^2-12\alpha -17\right){\lambda }^2\\ +\left(32{\alpha }^2+14\alpha +1\right)\lambda -2\left(2\alpha +1\right)\left(4\alpha +9\right)\end{array}\right){\phi }_{{\mathrm{S}}_1}\\ +35{\lambda }^5+4\left(3\alpha +2\right){\lambda }^4-4\left(23{\alpha }^2+29\alpha +12\right){\lambda }^3+8\left(6{\alpha }^3+31{\alpha }^2-9\right){\lambda }^2\\ -\left(96{\alpha }^3+772{\alpha }^2+684\alpha +165\right)\lambda +48{\alpha }^3+616{\alpha }^2+724\alpha +214\end{array}\right)$
$c_6$	$2M\left(\begin{array}{l}2M\left(\lambda -1\right)\left(\begin{array}{l}8\left(5{\lambda }^2-12\lambda +76\right){\left(\lambda -1\right)}^2{\alpha }^2+2\alpha \left(\lambda -1\right)\left(13{\lambda }^4+65{\lambda }^3+72{\lambda }^2+172\lambda -208\right)\\ +2{\lambda }^6+19{\lambda }^5+48{\lambda }^4+21{\lambda }^3-14{\lambda }^2-148\lambda +88-256{\left(\lambda -1\right)}^3{\alpha }^3\end{array}\right){\phi }_{{\mathrm{S}}_1}\\ +\left(2\alpha +1\right)\left(\begin{array}{l}-\left(52{\alpha }^2+48\alpha +25\right){\lambda }^4+2\left(48{\alpha }^3+160{\alpha }^2+41\alpha -15\right){\lambda }^3-12\left(2\alpha +1\right)\left(4{\alpha }^2+24\alpha +15\right)\\ -\left(288{\alpha }^3+1108{\alpha }^2+724\alpha +109\right){\lambda }^2+4\left(73{\alpha }^3+366{\alpha }^2+337\alpha +87\right)\lambda -2\left(5\alpha +3\right){\lambda }^5\end{array}\right)\end{array}\right)$
$c_7$	$\begin{array}{l}2M\left(\lambda -1\right)\left(\begin{array}{l}-3\alpha {\lambda }^5-\left(5\alpha +3\right)\left(8\alpha -1\right){\lambda }^4-2\left(64{\alpha }^3+32{\alpha }^2-21\alpha -11\right){\lambda }^3\\ +4\left(2\alpha +1\right)\left(48{\alpha }^2+39\alpha +5\right){\lambda }^2-8\left(2\alpha +1\right)\left(24{\alpha }^2+29\alpha +9\right)\lambda +32{\left(2\alpha +1\right)}^2\left(\alpha +1\right)\end{array}\right){\phi }_{{\mathrm{S}}_1}\\ +\left(2\alpha +1\right)\left(\begin{array}{l}\left(8{\alpha }^2+10\alpha -1\right){\lambda }^4+2\left(24{\alpha }^3+38{\alpha }^2-4\alpha -9\right){\lambda }^3-2\left(72{\alpha }^3+188{\alpha }^2+113\alpha +17\right){\lambda }^2\\ +4\left(36{\alpha }^3+123{\alpha }^2+102\alpha +25\right)\lambda -8\left(2\alpha +1\right)\left(3\alpha +2\right)\left(\alpha +3\right)\end{array}\right)\end{array}$
$d_1$	$16M^2\left(\lambda -1\right){\left(2\alpha +1\right)}^2\left(\begin{array}{l}8M\left(\lambda -1\right){\left(\lambda +1\right)}^2\left(\lambda +4\alpha +2\right){\phi }_{{\mathrm{S}}_2}+3{\lambda }^4+\left(18\alpha +19\right){\lambda }^3\\ -\left(16{\alpha }^2-56\alpha -37\right){\lambda }^2+\left(32{\alpha }^2+10\alpha -5\right)\lambda -2\left(2\alpha +1\right)\left(4\alpha +19\right)\end{array}\right)$
$d_2$	$2M\left(2\alpha +1\right)\left(\begin{array}{l}16M\left(\lambda -1\right){\left(\lambda +1\right)}^2\left(\lambda +4\alpha +2\right)\left({\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\right){\phi }_{{\mathrm{S}}_2}+2{\lambda }^6+\left(26\alpha +19\right){\lambda }^5\\ +2\left(20{\alpha }^2+70\alpha +31\right){\lambda }^4+\left(-256{\alpha }^3+144{\alpha }^2+282\alpha +71\right){\lambda }^3+6\left(2\alpha +1\right)\left(64{\alpha }^2+6\alpha -21\right){\lambda }^2\\ -4\left(2\alpha +1\right)\left(96{\alpha }^2+140\alpha +43\right)\lambda +8{\left(2\alpha +1\right)}^2\left(8\alpha +19\right)\end{array}\right)$
$d_3$	$\left({\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\right)\left(\begin{array}{l}2M{\left(\lambda +1\right)}^2\left(\lambda +4\alpha +2\right)\left({\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\right){\phi }_{{\mathrm{S}}_2}\\ -\left(2\alpha +1\right)\left(\left(3\alpha -1\right){\lambda }^3+\left(16{\alpha }^2-5\alpha -8\right){\lambda }^2-2\left(16{\alpha }^2+15\alpha +3\right)\lambda +4\left(2\alpha +1\right)\left(2\alpha +3\right)\right)\end{array}\right)$
$d_4$	$64M^3{\left(2\alpha +1\right)}^2{\left(\lambda -1\right)}^2\left(4M\left(\lambda -1\right){\left(\lambda +1\right)}^2{\phi }_{{\mathrm{S}}_2}+{\lambda }^3-\left(2\alpha -1\right){\lambda }^2-2\left(2+\alpha \right)\lambda -12\alpha -10\right)$
$d_5$	$4M^2\left(2\alpha +1\right)\left(\lambda -1\right)\left(\begin{array}{l}-8M\left(\lambda -1\right){\left(\lambda +1\right)}^2\left(-3{\lambda }^2-\left(18\alpha +7\right)\lambda +2\left(2\alpha +1\right)\left(4\alpha +9\right)\right){\phi }_{{\mathrm{S}}_2}\\ +3{\lambda }^5+2\left(6\alpha +5\right){\lambda }^4-2\left(46{\alpha }^2+34\alpha +11\right){\lambda }^3+2\left(24{\alpha }^3+40{\alpha }^2-84\alpha -53\right){\lambda }^2\\ -\left(96{\alpha }^3+604{\alpha }^2+548\alpha +131\right)\lambda +2\left(2\alpha +1\right)\left(12{\alpha }^2+148\alpha +107\right)\end{array}\right)$
$d_6$	$2M\left(\begin{array}{l}2M\left(\lambda -1\right){\left(\lambda +1\right)}^2\left(\begin{array}{l}2{\lambda }^4+\left(26\alpha +11\right){\lambda }^3+2\left(2\alpha +1\right)\left(10\alpha -11\right){\lambda }^2\\ -4\left(2\alpha +1\right)\left(32{\alpha }^2+68\alpha +23\right)\lambda +8{\left(2\alpha +1\right)}^2\left(8\alpha +11\right)\end{array}\right){\phi }_{{\mathrm{S}}_2}\\ -\left(2\alpha +1\right)\left(\begin{array}{l}\left(52{\alpha }^2+36\alpha +19\right){\lambda }^4-2\left(48{\alpha }^3-88{\alpha }^2-37\alpha -33\right){\lambda }^3+\left(288{\alpha }^3+820{\alpha }^2+428\alpha +33\right){\lambda }^2\\ +2\left(5\alpha +2\right){\lambda }^5-4\left(72{\alpha }^3+330{\alpha }^2+299\alpha +75\right)\lambda +12\left(2\alpha +1\right)\left(4{\alpha }^2+24\alpha +15\right)\end{array}\right)\end{array}\right)$
$d_7$	$\begin{array}{l}2M{\left(\lambda +1\right)}^2\left(\alpha +1\right)\left(3{\lambda }^4+2\left(20\alpha +9\right){\lambda }^3+4\left(2\alpha +1\right)\left(16\alpha +1\right){\lambda }^2-8\left(2\alpha +1\right)\left(16\alpha +7\right)\lambda +32{\left(2\alpha +1\right)}^2\right){\phi }_{{\mathrm{S}}_1}\\ +\left(2\alpha +1\right)\left(\begin{array}{l}-\left(8{\alpha }^2+6\alpha -3\right){\lambda }^4-4\left(12{\alpha }^3+9{\alpha }^2-11\alpha -6\right){\lambda }^3+2\left(9\alpha +5\right)\left(8{\alpha }^2+12\alpha +1\right){\lambda }^2\\ -4\left(36{\alpha }^3+113{\alpha }^2+90\alpha +21\right)\lambda +8\left(2\alpha +1\right)\left(3\alpha +2\right)\left(\alpha +3\right)\end{array}\right)\end{array}$
$N_1$	$128M^3{\left(2\alpha +1\right)}^3{\left(\lambda -1\right)}^2\left(\lambda -3\right){\phi }_R^3+c_1{\phi }_R^2+c_2{\phi }_R+c_3$
$N_2$	$c_4{\phi }_R^3+c_5{\phi }_R^2+c_6{\phi }_R+c_7$
$N_3$	$128M^3{\left(2\alpha +1\right)}^3{\left(\lambda -1\right)}^2\left(\lambda +3\right)\left(\lambda +1\right){\phi }_R^3+d_1{\phi }_R^2+d_2{\phi }_R+d_3$
$N_4$	$d_4{\phi }_R^3+d_5{\phi }_R^2+d_6{\phi }_R-d_7$
$n_1$	$b_1{m}_3\left(2\alpha +1\right)\left(\overline{\eta }+1\right)$
$n_2$	$m_3{\left(2\alpha +1\right)}^2\left(\lambda -1\right)\left(\overline{\eta }+1\right)\left(b_2{\phi }_{\mathrm{R}}^2-b_3{\phi }_{\mathrm{R}}+b_4\right)$
$n_3$	$4m_{4}M{\left(\lambda +1\right)}^2{A}_3$
$n_4$	$m_4{A}_4+a_5\left(2\alpha +1\right)\left(\overline{\eta }+1\right)\left(b_5{\phi }_{{\mathrm{S}}_1}-\left(2\alpha +1\right)\left(b_6{\phi }_{\mathrm{R}}^2+b_7{\phi }_{\mathrm{R}}+b_8\right)\right)$
$n_5$	$m_5\left(2\alpha +1\right)\left(\overline{\eta }+1\right)\left(b_5{\phi }_{{\mathrm{S}}_1}-\left(2\alpha +1\right)\left(b_6{\phi }_{\mathrm{R}}^2+b_7{\phi }_{\mathrm{R}}+b_8\right)\right)$
$n_6$	$4m_6{A}_{3}M\left(\lambda -1\right){\left(\lambda +1\right)}^2+a_6{b}_1\left(2\alpha +1\right)\left(\overline{\eta }+1\right)$
$n_7$	$m_6{A}_4\left(\lambda -1\right)+a_6{\left(2\alpha +1\right)}^2\left(\lambda -1\right)\left(\overline{\eta }+1\right)\left(b_2{\phi }_{\mathrm{R}}^2-b_3{\phi }_{\mathrm{R}}+b_4\right)$
$k_1$	$4M{A}_3{\left(\lambda +1\right)}^2\left(\lambda -1\right)$
$k_2$	$\left(\lambda -1\right)A_4$
$k_3$	$4M{\left(\lambda +1\right)}^2{A}_3{\phi }_{{\mathrm{S}}_2}$
$t_1$	$\left(\overline{\eta }+1\right)\left(4M{\phi }_{\mathrm{R}}+1\right)\left(4M\left(2M{\phi }_{\mathrm{R}}+1\right){\phi }_{{\mathrm{S}}_2}+\left(6\alpha +5\right)M{\phi }_{\mathrm{R}}+3\alpha +2\right)$
$t_2$	$\left(\overline{\eta }+1\right)\left(4M{\phi }_{\mathrm{R}}+1\right)\left(4M\left(2M{\phi }_{\mathrm{R}}+1\right){\phi }_{{\mathrm{S}}_1}+\left(6\alpha +5\right)M{\phi }_{\mathrm{R}}+3\alpha +2\right)$
$t_3$	$\left(\overline{\eta }+1\right)\left(4M\left(2M{\phi }_{\mathrm{R}}+1\right)\left(4M\left(2{\phi }_{\mathrm{R}}^2+1\right){\phi }_{{\mathrm{S}}_1}+a_1\right){\phi }_{{\mathrm{S}}_2}+a_2{\phi }_{{\mathrm{S}}_1}+a_3\right)$
$t_4$	$\left(\overline{\eta }+1\right)\left(4M\left(2M{\phi }_{\mathrm{R}}+1\right)\left(4M\left(2{\phi }_{\mathrm{R}}^2+1\right){\phi }_{{\mathrm{S}}_2}+a_1\right){\phi }_{{\mathrm{S}}_1}+a_2{\phi }_{{\mathrm{S}}_2}+a_3\right)$
$T_1$	$\left(2\alpha +1\right)\left(\overline{\eta }+1\right)\left(b_1{\phi }_{{\mathrm{S}}_2}+\left(2\alpha +1\right)\left(\lambda -1\right)\left(b_2{\phi }_{\mathrm{R}}^2-b_3{\phi }_{\mathrm{R}}+b_4\right)\right)$
$T_2$	$\left(2\alpha +1\right)\left(\overline{\eta }+1\right)\left(b_5{\phi }_{{\mathrm{S}}_1}-\left(2\alpha +1\right)\left(b_6{\phi }_{\mathrm{R}}^2+b_7{\phi }_{\mathrm{R}}+b_8\right)\right)$
$T_3$	$\left(\overline{\eta }+1\right)\left(2M{\left(\lambda +1\right)}^2{N}_1{\phi }_{{\mathrm{S}}_2}+\left(2\alpha +1\right)N_2\right)$
$T_4$	$\left(\overline{\eta }+1\right)\left(2M\left(\lambda -1\right)N_3{\phi }_{{\mathrm{S}}_1}+\left(2\alpha +1\right)N_4\right)$
$T_5$	${\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)$

 of Appendix B.

4.2. Disrupted Situation

In the disrupted situation, we still suppose that the suppliers are leaders and the retailer is a follower. Firstly, suppliers 1 and 2 simultaneously set their wholesale prices ${\omega }_1$ and ${\omega }_2$. Then, the retailer receives information about the suppliers’ decisions and, based on market demand, determines its retail prices $p_1$ and $p_2$. We assume that the market demands for products 1 and 2 are $Q_1$ and $Q_2$, respectively. Since supplier 1 can only deliver $\left(1-\lambda \right)Q_1$ to the retailer, the retailer needs to purchase $\lambda Q_1+Q_2$ of product 2 from supplier 2 to supplement the market demand that is not covered by product 1.

All supply chain members’ objectives are to maximize their own utility. We use backward induction to solve the Stackelberg game models. We first consider the lower-level model, which is the retailer’s decision-making problem:

$$\begin{array}{l}\underset{p_1,p_2}{\max }\hspace{1em}{U}_R=E\left({\mathrm{\Pi }}_{\mathrm{R}}\right)-{\phi }_{\mathrm{R}}S\left({\mathrm{\Pi }}_{\mathrm{R}}\right)\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{p}_1\ge {\omega }_1, p_2\ge {\omega }_2,\end{array}$$

where ${\mathrm{\Pi }}_{\mathrm{R}}=(p_1-{\omega }_1)\left(1-\lambda \right)Q_1+(p_2-{\omega }_2)\left(Q_2+\lambda Q_1\right)$. Similarly to in Section 4.1, we can obtain $S\left({\mathrm{\Pi }}_{\mathrm{R}}\right)={\left(\left(1-\lambda \right)\left(p_1-{\omega }_1\right)+\left(1+\lambda \right)\left(p_2-{\omega }_2\right)\right)}^2{\displaystyle {\int }_{-\infty }^{\overline{\eta }}{\left(\overline{\eta }-\eta \right)}^2}f(\eta )d\eta$.

Then, we have the following proposition, whose proof is presented in Appendix A.

Proposition 3.

In the disrupted situation, when $\lambda \in \left(0, 2\sqrt{2}-2\right]$ , $U_R\left(p_1,p_2\right)$ is jointly concave in $\left(p_1,p_2\right)$ for the given wholesale price $\left({\omega }_1,{\omega }_2\right)$ . The optimal retail prices for products 1 and 2 are

$$p_1^D\left({\omega }_1,{\omega }_2\right)={\displaystyle \frac{a_4{\omega }_1+a_5{\omega }_2+\left(\left(4\alpha +\lambda \right)\left(\lambda -1\right)+4\lambda -2\right)\left(\overline{\eta }+1\right)}{{\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)}},$$

$$p_2^D\left({\omega }_1,{\omega }_2\right)={\displaystyle \frac{a_6{\omega }_1+a_7{\omega }_2+\left(\lambda -1\right)\left(4\alpha +\lambda +2\right)\left(\overline{\eta }+1\right)}{{\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)}},$$

where $a_4~a_7$ are defined in Table A1 of Appendix B.

Next, we consider the upper-level models, namely, the suppliers’ decision-making problems:

$$\begin{array}{l}\underset{{\omega }_1}{\max }\hspace{1em}{U}_{{\mathrm{S}}_1}=E\left({\mathrm{\Pi }}_{{\mathrm{S}}_1}\right)-{\phi }_{{\mathrm{S}}_1}S\left({\mathrm{\Pi }}_{{\mathrm{S}}_1}\right)\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{\omega }_1\ge 0,\end{array}\hspace{1em}\begin{array}{l}\underset{{\omega }_2}{\max }\hspace{1em}{U}_{{\mathrm{S}}_2}=E\left({\mathrm{\Pi }}_{{\mathrm{S}}_2}\right)-{\phi }_{{\mathrm{S}}_2}S\left({\mathrm{\Pi }}_{{\mathrm{S}}_2}\right)\\ \mathrm{s}.\mathrm{t}.\hspace{1em}{\omega }_2\ge 0,\end{array}$$

where ${\mathrm{\Pi }}_{{\mathrm{S}}_1}={\omega }_1\left(1-\lambda \right)Q_1$ and ${\mathrm{\Pi }}_{{\mathrm{S}}_2}={\omega }_2\left(Q_2+\lambda Q_1\right)$. Similarly to in Section 4.1, we have $S\left({\mathrm{\Pi }}_{{\mathrm{S}}_i}\right)={{\omega }_i}^2{\left(1-\lambda \right)}^2{\displaystyle {\int }_{-\infty }^{\overline{\eta }}{\left(\overline{\eta }-\eta \right)}^2}f(\eta )d\eta$, $i,j=1,2, i\ne j$. Substituting $p_1^D\left({\omega }_1,{\omega }_2\right)$ and $p_2^D\left({\omega }_1,{\omega }_2\right)$ into $U_{S_1}$ and $U_{{\mathrm{S}}_2}$ in Proposition 3 yields the following Nash game model:

$$\begin{array}{ll}\hfill \underset{{\omega }_1}{\max }\hspace{1em}{U}_{S_1}=& {\omega }_1\left(1-\lambda \right)\left({\displaystyle \frac{\left(\left(a_6-a_4\right)\alpha -a_4\right){\omega }_1+\left(\left(a_7-a_5\right)\alpha -a_5\right){\omega }_2-\left({\lambda }^2+\left(2\alpha +1\right)\left(3\lambda -2\right)\right)\left(\overline{\eta }+1\right)}{{\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)}}+\overline{\eta }+1\right)\hfill \\ & -{\phi }_{{\mathrm{S}}_1}{{\omega }_1}^2{\left(1-\lambda \right)}^{2}M\hfill \\ \hfill \mathrm{s}.\mathrm{t}.\hspace{1em}{m}_3{\omega }_1\ge a_5{\omega }_2+m_4,\\ \hfill \underset{{\omega }_2}{\max }\hspace{1em}{U}_{{\mathrm{S}}_2}=& {\omega }_2\left({\displaystyle \frac{\left(a_4\alpha -\left(\alpha +1\right)a_6\right){\omega }_1+\left(a_5\alpha -\left(\alpha +1\right)a_7\right){\omega }_2-\left({\lambda }^2+\left(2\alpha +1\right)\left(\lambda -2\right)\right)\left(\overline{\eta }+1\right)}{{\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)}}+\overline{\eta }+1\right)\hfill \\ & +\lambda {\omega }_2\left({\displaystyle \frac{\left(\left(a_6-a_4\right)\alpha -a_4\right){\omega }_1+\left(\left(a_7-a_5\right)\alpha -a_5\right){\omega }_2-\left({\lambda }^2+\left(2\alpha +1\right)\left(3\lambda -2\right)\right)\left(\overline{\eta }+1\right)}{{\lambda }^2+4\left(2\alpha +1\right)\left(\lambda -1\right)\left(2M{\phi }_{\mathrm{R}}+1\right)}}+\overline{\eta }+1\right)\hfill \\ & -{\phi }_{{\mathrm{S}}_2}{{\omega }_2}^2{\left(1+\lambda \right)}^{2}M\hfill \\ \hfill \mathrm{s}.\mathrm{t}.\hspace{1em}{m}_5{\omega }_2\ge a_6{\omega }_1+m_6,\end{array}$$

where $m_3~m_6$ are defined in Table A1 in Appendix B.

Through the backward induction method, we can obtain the following proposition, whose proof is given in Appendix A.

Proposition 4.

In the disrupted situation, when ${\displaystyle \frac{n_1{\phi }_{{\mathrm{S}}_2}+n_2}{k_1{\phi }_{{\mathrm{S}}_2}+k_2}}\ge {\displaystyle \frac{n_3{\phi }_{{\mathrm{S}}_2}+n_4}{k_3{\phi }_{{\mathrm{S}}_2}+A_4}}$ , ${\displaystyle \frac{n_5}{k_3{\phi }_{{\mathrm{S}}_2}+A_4}}\ge {\displaystyle \frac{n_6{\phi }_{{\mathrm{S}}_2}+n_7}{k_1{\phi }_{{\mathrm{S}}_2}+k_2}}$ and $\lambda \in \left(0, 2\sqrt{2}-2\right]$ , each supplier’s utility function is concave in its wholesale price. The optimal decisions of the suppliers and the retailer are

$${\omega }_1^D={\displaystyle \frac{T_1}{\left(\lambda -1\right)\left(4M{\left(\lambda +1\right)}^2{A}_3{\phi }_{{\mathrm{S}}_2}+A_4\right)}},{\omega }_2^D={\displaystyle \frac{T_2}{4M{\left(\lambda +1\right)}^2{A}_3{\phi }_{{\mathrm{S}}_2}+A_4}},$$

$$p_1^D={\displaystyle \frac{T_3}{T_5\left(4M{\left(\lambda +1\right)}^2{A}_3{\phi }_{{\mathrm{S}}_2}+A_4\right)}},p_2^D={\displaystyle \frac{T_4}{T_5\left(4M{\left(\lambda +1\right)}^2{A}_3{\phi }_{{\mathrm{S}}_2}+A_4\right)}},$$

where $n_1~n_7$, $k_1~k_3$, $A_3$, $A_4$, and $T_1~T_5$ are defined in Table A1 of Appendix B.

5. Model Analysis

5.1. Case Study

Based on the above theoretical analysis, we find that the reference price effect has important impacts on supply chain members’ pricing decisions. In fact, the reference price effect is also widely applied in real-life scenarios. For example, specialized online shopping price comparison apps are now available in the market. These apps provide consumers with various forms of price comparison functions to help them select satisfactory products. This study uses Alibaba’s Yitao App and Shangyi Networks’ Manmanbuy App as examples to show how the reference price effect influences suppliers’ pricing strategies and consumers’ buying behavior. These apps are seen as typical examples in the field of applications that leverage the reference price effect to offer real-time product price comparisons.

Price comparison platforms underscore the importance and practicality of the reference price effect. Specifically, Yitao App offers real-time product price comparisons. With a large user base and a well-designed personalized recommendation feature, it caters to consumers’ diverse needs. Yitao App tracks price changes in a specific product over time and compares price differences among different suppliers for the same product. It also displays price information for related alternative products. As shown in Figure 3, in both non-disrupted and disrupted scenarios, Yitao App effectively helps consumers identify and compare price variations among similar products. Its user-friendly interface makes it easy for consumers to access detailed price information for each product, allowing them to visualize potential price savings across various options. This highlights that the reference price effect significantly influences consumers’ purchasing decisions in all scenarios. This case study suggests that Yitao App’s price comparison approach not only emphasizes the role of reference price in decision-making but also aids consumers in making more rational shopping choices.

The Manmanbuy App is another platform that integrates the reference price effect into its core features. As depicted in Figure 4, even during supply disruption, consumers are still able to access information regarding price trends and the lowest price in the history of the product over time through the Manmanbuy App. This functionality is crucial for understanding price fluctuations and providing consumers with a solid basis for making informed purchasing decisions. Moreover, should there be a disruption in the current supply channel, the Manmanbuy app offers consumers alternative recommendations for the same or similar products from different channels. This combination of handling supply disruptions and providing product price information empowers consumers, enabling them to make more informed decisions and enhancing their overall shopping experience. For suppliers, observing the pricing of the same/similar products through apps can inform decisions on implementing promotional campaigns or discount strategies aimed at attracting more consumers. Further, the alternative product recommendation feature of price comparison apps can increase the visibility of suppliers’ products, exposing them to a wider audience of potential consumers and thereby increasing sales opportunities.

5.2. Numerical Analysis

In the subsequent subsections, we will use numerical analyses to further illustrate the impact of different factors on the pricing and utility of supply chain members.

5.2.1. Impact of Reference Price Effects and Risk Aversion

This subsection focuses on analyzing the impact of the strength of reference price effects and the degree of risk aversion on wholesale prices, retail prices, and utility. Based on the results in Propositions 1–4, we assume that $\lambda =0.3$, $\alpha \in \left[0,1\right]$, $\eta ~Gamma\left(3,1\right)$, and ${\phi }_R=10$. We consider two control cases of risk aversion for supplier $i$ $(i=1,2)$: low risk aversion with ${\phi }_{{\mathrm{S}}_i}=1$ and high risk aversion with ${\phi }_{{\mathrm{S}}_i}=5$.

We first examine the impact of reference price effects and risk aversion on the wholesale prices of products 1 and 2. The numerical results, illustrated in Figure 5, reveal that, in both disrupted and non-disrupted situations, the wholesale prices for both products tend to decrease as the strength of the reference price effect increases. This decrease occurs because consumers become more sensitive to the pricing of alternative products when the reference price effect is more pronounced. As a result, suppliers are compelled to lower their wholesale prices to remain competitive and protect their market share. This strategy encourages retailers to purchase and promote their products, ensuring that consumers are more likely to choose them over alternatives. Regardless of supplier 1’s degree of risk aversion, its wholesale price in the disrupted situation consistently exceeds that in the non-disrupted situation. As for supplier 2, in the case of low risk aversion, its wholesale price in the disrupted situation is consistently higher than that in the non-disrupted situation. This phenomenon primarily arises from an imbalance between product supply and market demand in the disrupted situation, which results in an inadequate supply of the product in the market. Consequently, the supplier implements a strategy of increasing the wholesale price to deal with this situation. However, in the case of high risk aversion, in the scenario with a weak reference price effect under single-sourcing supply disruption, supplier 2 sets a lower wholesale price than that in the non-disrupted situation. This indicates that, upon recognizing potential risk inherent in the market, suppliers with high risk aversion tend to mitigate their risk by implementing a strategy of reducing wholesale prices so as to maintain a robust market position in a risky market environment.

Figure 6 illustrates the impact of the reference price effect and risk aversion on retail prices. In both disrupted and non-disrupted situations, it can be observed that as the strength of the reference price effect increases, the retail prices of products 1 and 2 decrease. This phenomenon is mainly attributed to the growing influence of the reference price effect, which leads to increased price sensitivity among consumers and a great propensity to buy products with low prices. Thus, the retailer is compelled to adopt the strategy of lowering the prices to attract more consumers and ultimately enhance its revenue. In the non-disrupted situation, when the suppliers exhibit a homogeneous degree of risk aversion, the retailer establishes identical retail prices for both products 1 and 2. However, in the disrupted situation, the retailer establishes a high retail price for product 2 compared to product 1. The rationality behind this can be understood by examining the impact of single-sourcing supply disruption on product 1 and the broader market context. Specifically, when supply disruption occurs with supplier 1, the availability of product 1 decreases, but overall market demand remains relatively stable. This situation presents a potential advantage for supplier 2. In response, the retailer increases the price of product 2 to preserve its revenue streams. By performing so, the retailer aims to capitalize on the reduced availability of product 1 while ensuring that they can maintain normal profit levels despite the disruptions in supply. Furthermore, it can be observed that when ${\phi }_{{\mathrm{S}}_1}=5$ and ${\phi }_{{\mathrm{S}}_2}=1$, product 2’s retail price is higher than product 1’s retail price in both disrupted and non-disrupted scenarios. When ${\phi }_{{\mathrm{S}}_1}=1$ and ${\phi }_{{\mathrm{S}}_2}=5$, product 1’s retail price is higher than product 2’s retail price in both disrupted and non-disrupted situations. This indicates that when the retailer maintains a constant degree of risk aversion, the retail prices of products 1 and 2 are inversely proportional to the risk aversion degree of suppliers 1 and 2. This may be because suppliers with high risk aversion tend to set low wholesale prices to maintain market shares, which in turn leads to low retail prices.

Figure 7 reflects the impact of the reference price effect and risk aversion on utility. For both suppliers 1 and 2, regardless of variations in their degrees of risk aversion, supplier 1 consistently exhibits higher utility in the non-disrupted scenario than the disrupted scenario, whereas supplier 2 demonstrates the opposite trend. This implies that single-sourcing supply disruption in product 1 may prompt some consumers to switch to product 2 so as to enable supplier 2 to benefit from the disruption. For the retailer, its utility is positively related to the strength of the reference price effect in both non-disrupted and disrupted situations. This suggests that when the market is more competitive, the retailer can achieve high returns through flexible pricing decisions. Particularly, in Figure 7d, the retailer’s utility in the disrupted situation is higher than that in the non-disrupted situation when the strength of the reference price effect is weak. This indicates that the occurrence of single-sourcing supply disruption does not necessarily always harm the retailer and, on the contrary, it may even lead to an enhancement in the retailer’s utility in some circumstances. This phenomenon can be explained by the fact that when single-sourcing supply disruption occurs for a product, the demand for the substitute product increases. This is to say that the retailer may increase the retail price of the substitute product to obtain high utility, particularly when there is large consumer demand for the product.

5.2.2. Impact of Single-Sourcing Supply Disruption and Risk Aversion

We devote this subsection to analyzing the impact of single-sourcing supply disruption and risk aversion on pricing decisions and utility. Based on the results in Propositions 1–4, we set $\alpha =0.1$, $\lambda \in \left(0,0.4\right]$, $\eta ~Gamma\left(3,1\right)$, and ${\phi }_R=10$. We also consider two control cases of risk aversion for supplier $i$$(i=1,2)$: low risk aversion with ${\phi }_{{\mathrm{S}}_i}=1$ and high risk aversion with ${\phi }_{{\mathrm{S}}_i}=5$.

Firstly, we analyze the sensitivity of wholesale prices to the degrees of single-sourcing supply disruption and risk aversion. Figure 8 reveals that, regardless of the degree of the risk aversion of supply chain members, in the disrupted situation, as the single-sourcing supply disruption increases, the wholesale price set by the disruptor supplier rises and is higher than that in the non-disrupted situation. One reason is that after single-sourcing supply disruption occurs, the market anticipates elevated prices for the products and the perception is that consumers may engage in hoarding behavior and be willing to pay high prices. For supplier 2, when supplier 2 has low risk aversion, the wholesale price of product 2 increases as the degree of single-sourcing supply disruption rises. In this case, it is consistently higher than the wholesale price in the non-disrupted situation. When supplier 2 has high risk aversion, the wholesale price of product 2 in the disrupted situation is negatively correlated with the degree of single-sourcing supply disruption and is lower than the wholesale price in the non-disrupted situation. It can also be observed that in the presence of market risks, a non-disrupted supplier with high risk aversion tends to decrease the wholesale price to incentivize the retailer to more actively procure and sell their products. This strategy enables suppliers, to some extent, to consolidate the market position and increase the market penetration of their products.

We analyze the impact of the degrees of single-sourcing supply disruption and risk aversion on products’ retail prices. As shown in Figure 9, the retail price of product 2 in the disrupted situation is positively correlated with the degree of single-sourcing supply disruption and is consistently higher than product 2’s retail price in the non-disrupted situation. This phenomenon is mainly due to the bidirectional substitutability between products 1 and 2. In particular, when there is disruption in the supply of product 1, consumers faced with a shortage of product 1 tend to switch to product 2. This leads to a significant increase in the market demand for product 2. Therefore, as the demand for product 2 increases, its selling price also increases. This mechanism can be explained by the theory of market fit redistribution in economics. When the supply of a product is disrupted, consumer choice leads to a reallocation of market resources. In this process, the disruption of product 1 leads directly to a shift in its consumers to product 2, an increase in the competing demand for that product, and, in turn, a price increase. The retail price of product 2 reflects not only real-time changes in consumer demand but also how the market adjusts and reallocates through the price mechanism in the presence of supply chain instability. For product 1, when the degree of single-sourcing supply disruption is great, product 1’s retail price in the disruption situation shows a decreasing trend with the increasing severity of disruption. This is attributed to the retailer engaging in secondary procurement from supplier 2 after single-sourcing supply disruption to increase the supply of substitute product 1 and consequently cause a decline in its retail price in the disrupted situation. When the degree of single-sourcing supply disruption is low and supplier 1 has low risk aversion, the retail price of product 1 in the disruption situation is higher overall than its retail price in the non-disruption situation. This indicates that, despite the existence of bidirectional, complete substitutes, disruption in the supply of any one product can still potentially exert a negative impact on consumer purchasing power.

Figure 10 illustrates the correlation of single-sourcing supply disruption, risk aversion, and supply chain members’ utility. Under different levels of risk aversion, supplier 1’s utility in the disrupted situation decreases as the degree of single-sourcing supply disruption increases. It can be identified that in the presence of substitute products, suppliers facing supply disruptions may experience customer loss due to an inability to maintain normal product supply, resulting in a decrease in sales and utility. In a competitive market, customers’ resources are limited and the supply disruption to supplier 1 causes its customers to switch to supplier 2, resulting in a loss of market share. This reallocation of market share is based on the optimal allocation of resources under the market mechanism, i.e., in the event of supply disruption, the market automatically shifts demand from the supplier who is unable to supply to other suppliers who are able to provide alternative products. Therefore, suppliers should contemplate establishing a supply chain that is more adaptable and resilient to effectively address uncertainties in supply chain operations. In contrast, supplier 2’s utility in the disrupted situation is positively related to the degree of single-sourcing supply disruption under different combinations of risk aversion levels. This suggests that supplier 2 is able to capture a large market share and thus increase its utility by satisfying the market demand for substitute products. This phenomenon reflects the market’s self-adjusting mechanism in the face of supply disruptions, i.e., when a supplier’s supply capacity is constrained, the market will rebalance supply and demand in the market by reallocating resources and redirecting demand to other suppliers. Furthermore, amid single-sourcing supply disruption, suppliers unaffected by disruption can escalate their investments in marketing and promotional activities. This strategic move enables suppliers unaffected by disruption to cultivate a positive brand perception among consumers, ultimately bolstering their long-term competitiveness and profitability. For the retailer, although its utility in the disrupted situation exhibits a general decrease compared to that in the non-disrupted scenario, Figure 10d presents a scenario where the retailer’s utility in the disruption scenario surpasses that in the non-disruption scenario. This observation implies that the retailer has the potential to leverage single-sourcing supply disruption to their advantage through agile adjustments to its pricing decisions.

5.2.3. Impact of Multiple Factors on Supply Chain Members’ Utility

This subsection examines the combined impact of the reference price effect, risk aversion, and single-sourcing supply disruption on supply chain members’ utility. We set ${\phi }_{{\mathrm{S}}_1}\in \left[0,10\right]$; ${\phi }_{{\mathrm{S}}_2}=5$; ${\phi }_{\mathrm{R}}=10$; $\lambda =0.3$; $\eta ~Gamma\left(3,1\right)$; and $\alpha \in \left[0,1\right]$. The numerical results related to the impact of the reference price effect and supplier 1’s degree of risk aversion on the suppliers and the retailer’s utility are shown in Figure 11.

Figure 11a reflects the impact on the suppliers’ and retailer’s utility in the non-disrupted situation. When the risk-averse degrees of suppliers 1 and 2 are equal, their utilities are also equal in the non-disrupted situation. Notably, the utility of supplier 1 diminishes as its risk aversion degree increases. According to the theory of risk aversion in economics, the more risk-averse a supplier is, the more likely it is to adopt a conservative pricing strategy. This conservatism is reflected in the fact that the supplier will lower the expected return on the product in exchange for a more stable profit, leading to a reduction in the utility of the product. Moreover, the suppliers encourage the retailer to procure their products through reducing wholesale prices to achieve great utility. As the strength of the reference price effect increases, the retailer’s utility also tends to rise, primarily due to consumers’ heightened sensitivity to product prices in the presence of a strong reference price effect. As a result, price competition between the suppliers becomes more intense, which, in turn, benefits the retailer. As shown in Figure 11b, the conclusion that the strength of the reference price effect positively affects the retailer’s utility and supplier 1’s utility is negatively correlated with its own risk aversion in the disrupted situation still holds. In addition, supplier 2 is able to maximize its benefits from single-sourcing supply disruption among supply chain members, which is consistent with practical experience. Thus, it is important for the retailer to cultivate strong cooperative partnerships with the suppliers, maintain effective communication, and engage in cooperation to safeguard the stability of the whole supply chain.

The analysis in the previous two subsections is based on the assumption that ${\phi }_{\mathrm{R}}\ge {\phi }_{{\mathrm{S}}_i}(i=1,2)$. Next, we discuss the impact of various parameters on utility under the assumption that ${\phi }_{\mathrm{R}}<{\phi }_{{\mathrm{S}}_i}(i=1,2)$. The numerical results are presented in

Table 3. Impact of multiple factors on supply chain members’ utility.

$\alpha =0.1$
$\left({\phi }_{\mathrm{R}}, {\phi }_1, {\phi }_2\right)$	$\lambda =0.1$			$\lambda =0.3$			$\lambda =0.5$
	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$
(1, 5, 5)	3.02	1.53	1.59	2.96	1.53	1.65	3.00	1.68	1.76
(1, 10, 10)	3.80	1.23	1.22	3.78	1.31	1.22	3.83	1.55	1.26
(1, 10, 5)	3.36	1.23	1.59	3.23	1.28	1.67	3.19	1.46	1.80
(1, 5, 10)	3.46	1.53	1.22	3.50	1.57	1.20	3.62	1.77	1.23
$\alpha =0.5$
$\left({\phi }_{\mathrm{R}}, {\phi }_1, {\phi }_2\right)$	$\lambda =0.1$			$\lambda =0.3$			$\lambda =0.5$
	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$
(1, 5, 5)	3.40	1.49	1.59	3.28	1.44	1.68	3.24	1.46	1.79
(1, 10, 10)	4.07	1.20	1.23	4.01	1.21	1.25	4.02	1.31	1.29
(1, 10, 5)	3.69	1.25	1.54	3.50	1.24	1.66	3.39	1.31	1.79
(1, 5, 10)	3.79	1.43	1.27	3.80	1.40	1.27	3.88	1.46	1.28
$\alpha =0.9$
$\left({\phi }_{\mathrm{R}}, {\phi }_1, {\phi }_2\right)$	$\lambda =0.1$			$\lambda =0.3$			$\lambda =0.5$
	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$	$U_{\mathrm{R}}^D$	$U_{{\mathrm{S}}_1}^D$	$U_{{\mathrm{S}}_2}^D$
(1, 5, 5)	3.72	1.44	1.55	3.56	1.38	1.68	3.46	1.35	1.79
(1, 10, 10)	4.30	1.16	1.21	4.21	1.15	1.26	4.19	1.19	1.30
(1, 10, 5)	3.97	1.24	1.48	3.74	1.22	1.64	3.58	1.23	1.78
(1, 5, 10)	4.07	1.36	1.27	4.04	1.30	1.29	4.07	1.30	1.31

. It can be observed that, under a consistent retailer’s risk aversion level, ${\phi }_{\mathrm{R}}=1$, and equivalent suppliers’ risk aversion levels, ${\phi }_{{\mathrm{S}}_1}={\phi }_{{\mathrm{S}}_2}$, the retailer’s utility exhibits a positive correlation with the suppliers’ risk aversion levels, whereas the suppliers’ utility demonstrates a negative correlation with the retailer’s risk aversion level. This is mainly because risk-averse suppliers usually adopt a low pricing strategy to protect themselves against risk, which may cause suppliers to miss some profit opportunities by being overly risk-averse. Moreover, regardless of the combination of the degrees of risk aversion and single-sourcing supply disruption, the retailer’s utility is always positively related to the strength of the reference price effect. Furthermore, supplier 2’s utility increases with the degree of single-sourcing supply disruption for product 1. These findings imply that the existence of the reference price effect is advantageous for the retailer, while supplier 2 can benefit from disruption in the competitor by adopting adaptive pricing decisions. In addition, supplier 1’s utility is inversely proportional to the strength of the reference price effect. If supplier 2 exhibits lower risk aversion relative to supplier 1 $({\phi }_{{\mathrm{S}}_1}=10, {\phi }_{{\mathrm{S}}_2}=5)$, supplier 2’s utility diminishes as the strength of the reference price effect increases. Conversely, when supplier 2 demonstrates higher risk aversion compared to supplier 1 $({\phi }_{{\mathrm{S}}_1}=5,{\phi }_{{\mathrm{S}}_2}=10)$, supplier 2’s utility rises as the strength of the reference price effect increases. The intricate interplay between the reference price effect and the suppliers’ utility underscore the complexity of this relationship. To mitigate the adverse consequences of the reference price effect, the suppliers should adjust their product pricing strategies in a timely manner based on market conditions and competitors’ prices. Such proactive adjustments may facilitate enhanced adaptability to market dynamics, elevate responsiveness to market shifts, bolster market competitiveness, and culminate in sustained profitability and business advancement.

5.3. Robustness Analysis

In this section, we examine the robustness of models when faced with random fluctuations in demand. The goal is to enhance the practical applicability of these models.

Drawing from the findings in Propositions 1 through 4, we maintain the baseline parameters as previously established; Figure 12 illustrates how the robustness of the pricing models varies with different levels of risk aversion.

We consider ${\eta }_r$ a random variable with the mean $\widehat{\eta }$; then, the change rate of random demand disturbance is ${\displaystyle \frac{\widehat{\eta }-\overline{\eta }}{\overline{\eta }}}$ and the change rates of equilibrium prices for the suppliers and the retailer are ${\displaystyle \frac{{\omega }_i^j\left(\widehat{\eta }\right)-{\omega }_i^j\left(\overline{\eta }\right)}{{\omega }_i^j\left(\overline{\eta }\right)}}$ and ${\displaystyle \frac{p_i^j\left(\widehat{\eta }\right)-p_i^j\left(\overline{\eta }\right)}{p_i^j\left(\overline{\eta }\right)}}$, respectively, where $i=1,2$ and $j\in \left\{N,D\right\}$. Based on the results in Propositions 1–4, we set the base parameters as before, that is, $\alpha =0.1$; $\lambda =0.3$; and $\overline{\eta }=3$. The robustness of the pricing models with different degrees of risk aversion is shown in Figure 12.

As demonstrated in Figure 12, regardless of the variations in risk aversion among supply chain members, the change rate of equilibrium prices of both suppliers and the retailer is lower than the change rate of random demand disturbance when the change rate of random demand disturbance is positive. This indicates that the models presented in Section 4 are robust in terms of equilibrium decisions under varying degrees of risk aversion, particularly in scenarios where the random demand disturbance is positive. Therefore, supply chain managers can mitigate the risk by adjusting their pricing strategy in response to a surge in market demand. However, when the change in random demand disturbances is negative and relatively small, the change rate of equilibrium prices of the suppliers and the retailer is significant and exceeds the change rate of random demand disturbances. This finding suggests that supply chain members implement substantial price adjustments in response to a modest decline in demand. This phenomenon can be attributed to the fact that a smaller drop in demand results in a smaller expected loss for the companies. Consequently, managers are more inclined to make aggressive price adjustments to maintain market share and revenue. Conversely, when market demand experiences a substantial decline, companies are observed to adopt more conservative price adjustments. Thus, in the instance where the change rate of random demand disturbances is both negative and substantial, the change rate of equilibrium decisions for the suppliers and the retailer is notably low.

6. Conclusions

This paper focuses on a risk-averse supply chain system consisting of two suppliers and one retailer. Some Stackelberg game models, with the suppliers as the leaders and the retailer as the follower, are constructed based on the mean semivariance method under two distinct situations, namely, the non-disrupted situation and the disrupted situation. We analyze the influence of various factors such as reference price effects, risk aversion, and single-sourcing supply disruption on the pricing decisions and utility of supply chain members. The main research conclusions and management insights are as follows:

• In the context of dual sourcing, when one supplier is subject to supply disruption, the retailer employs a pricing escalation strategy for regularly supplied products. The retail prices incrementally increase in accordance with the degree of single-sourcing supply disruption. Simultaneously, the non-disrupted supplier may benefit from the disruption of the competitor, with its utility increasing with the degree of single-sourcing supply disruption. During single-sourcing supply disruption, the non-disrupted supplier should seize the opportunity to strategize and implement targeted marketing campaigns to enhance its brand reputation and foster customer loyalty. This proactive approach may yield enduring advantages for the organization in the long term.
• Under various combinations of risk aversion degree and single-sourcing supply disruption degree, the strength of the reference price effect inversely affects wholesale prices and retail prices. Specifically, product pricing decreases as the strength of the reference price effect increases. It can be observed that the reference price effect exerts a significant influence on product pricing and so enterprises should regularly monitor the pricing strategy of competing products and flexibly adjust product prices to adapt to market changes and maintain product competitiveness.
• All members’ pricing decisions and utility are influenced by their risk-aversion degrees. Typically, supply chain members with great risk-aversion degrees tend to adopt more conservative pricing decisions, which may result in low utility. Hence, it is suggested to vigilantly monitor market dynamics and promptly address potential issues to mitigate the adverse impact of supply chain uncertainty. Simultaneously, maintaining an objective and rational stance toward supply disruption is essential for averting undue risk aversion that may result in avoidable losses.
• The models generally exhibit robustness regarding equilibrium decisions among supply chain members across different degrees of risk aversion. However, there is minor negative fluctuation observed under random demand conditions.

The dual-sourcing supply chain system considered in this paper is based on the assumption that products are bidirectional completely substitutable. Nevertheless, the actual market demand is generally influenced by a number of factors such as consumer brand loyalty, and it is more prevalent for products to exhibit a partial substitution relationship. Consequently, in the future, we may consider more factors such as consumer preferences to further investigate the pricing problem of risk-averse supply chains under supply disruption. In fact, actual market demand may be nonlinear and factors such as consumer preferences and information asymmetry may result in asymmetric substitution effects. Therefore, in future work, we plan to incorporate more complex demand structures to test the robustness of our models. Moreover, we standardize the production cost to zero, and future research can introduce more complex cost structures and explore pricing decisions under different cost scenarios. This may provide theoretical support for supply chain cost management. It is worth mentioning that this study currently only analyzes the dual-sourcing strategy of a single retailer and does not cover the competitive dynamics among retailers. Therefore, future research may explore the pricing issues of multiple suppliers and multiple competing retailers in dual-sourcing situations to fully understand the impact of market competition on supply chain decisions and thus provide more targeted management recommendations. In addition, we also plan to extend the models to include risk-sharing mechanisms and analyze their interactions with pricing strategies, supply chain members’ utilities, and their performance under scenarios involving the reference price effect, risk aversion, and supply disruptions.