Achieving Resilience: Resilient Price and Quality Strategies of Fresh Food Dual-Channel Supply Chain Considering the Disruption

Abstract

In the face of demand disruptions, dual-channel supply chains (SCs) that lack resilience may be more vulnerable. Reaching moderate SC resilience through coordination is essential for dealing with disruptions. This paper investigates the operation management of a dual-channel fresh-food SC (FSC) under disruption. The centralized and decentralized decision models propose joint quality efforts based on the consideration of quality preference and loss. From the perspective of SC resilience, we analyze how SC members can optimally make price, quality, and quantity decisions resiliently and robustly under the disruption of quality preference. The results show that (1) no matter the kind of decision model, considering quality preference disruptions can significantly increase the SC profit; (2) there is a resilience range in decisions with the influence of the disruption cost. The original optimal decisions in the resilience range are robust and sustain SC performance without change; and (3) the disruption significantly impacts offline channel retailers, who are at a disadvantage when competing with online channels. A centralized decision model can achieve higher profits and quality levels in response to demand disruptions. This paper extends the concept of resilience to the FSC and provides suggestions for fresh-food enterprises to conduct quality efforts and cope with demand interruption.

1. Introduction

Recently, e-commerce has gradually changed the consumption patterns of consumers and the sales methods of fresh-food enterprises. Especially since the beginning of the COVID-19 pandemic, buying fresh food through online channels has become a mainstream method [1]. More and more fresh-food enterprises have adopted the combination of online and offline dual-channel supply chains (SCs) as their essential strategy. However, fresh food is significantly different from other packaged foods, as it is perishable, and a large volume will be lost in the operation of the SC [2]. Statistical analysis revealed that the loss of fresh food in China has reached 25–35%, whereas it is 15% in developed countries [3]. Furthermore, spoilage in fresh food can easily be perceived by consumers and affect their purchasing decisions. According to a report by iResearch (https://report.iresearch.cn/report/202105/3776.shtml, accessed on 15 May 2021), consumers’ preferences for price and quality are important factors affecting the operation of the fresh-food SC (FSC) with the improvement in consumption level [4]. It is worth noting that the consumers in first- and second-tier cities with a solid demand for fresh-food e-commerce are more sensitive to quality. Sustainable SC management is becoming an emerging consensus in various industries and enterprises [5].

In today’s dynamically changing SC environment, the various challenges of the FSC were analyzed by Yadav et al. [6], and these authors divided these challenges into four categories: sustainability, food waste, food safety and security, and miscellaneous challenges. With limited resources, meeting this exponentially growing demand for food and maintaining sustainable practices in the FSC is a huge challenge [7]. Sustainability challenges are mainly reflected in efforts to maintain freshness, weaker environmental regulation and enforcement, etc. [8]. Food waste and loss are often seen as a problem, accounting for one-third of total food production, and are related to poor storage facilities, cold chain deficiencies, etc. [9]. The challenges of food safety and security mainly come from consumer awareness, etc. [10]. Miscellaneous challenges manifest in many ways due to the complexity of the SC. Recent incidents of cybersecurity breaches highlight that cybersecurity is now a SC issue, creating new challenges for SC management [11]. Furthermore, challenges such as uncertainty, supply and demand mismatch, poor irrigation, and drought management exist in each stage of the SC, hindering the development of FSC [12].

Industry 4.0 is a continuous trend of using information technology to promote industrial transformation, including advanced technologies such as big data management, blockchain technology (BCT), the Internet of Things (IoT), etc. [13,14]. In recent years, these technologies have started to gain popularity in SC innovation and have led to the formation of SC 4.0 [15]. Gayialis et al. [16] proposed a BCT-based approach to ensure the origin, quality, and authenticity of wines and spirits. Kechagias et al. [17] applied systems thinking to the pharmaceutical industry, using Ventana’s simulation system dynamics method to simulate and optimize the quality control process. To drive digital transformation, many information systems have been established in urban distribution systems [18], maritime network security systems [19], etc. Technological advances have facilitated the sustainable development of the FSC. Track and trace facilities with new technology have been applied to efficiently check product quality and inventory levels, resulting in improved performance and less damage [20].

The FSC also calls for flexibility in terms of response capacity and improvement in SC resilience against disruptions due to changing customer preferences [21]. Risk management is vitally important to the FSC compared to a typical SC. Fresh food has three unique properties: perishability, a long production cycle, and seasonal production. Perishability means that the FSC is time-critical and requires quality effort from all SC members [22]. Due to the perishability of fresh food, their quality issues are closely watched by consumers. The current FSC faces challenges, including growing food demand and consumer preferences for food safety and quality [6]. The disruption of consumer quality preference has become a significant problem in the operation management of the FSC [23,24]. For example, the 2019 African Swine Fever epidemic severely affected pork quality, causing pork consumption to plummet. There have been frequent occurrences of unsellable fruits in recent years, mainly fruits that do not meet consumer quality preference [25].

Resilience is essential for effective SC risk management and aims to produce or foster moderate resilience in the SC to deal with the risk of disruption [26]. Because fresh food consumption is year-round, whereas production is seasonal and long-term, there is an imbalance between product supply and demand cycles. Long supply lead times limit the possibility of quick replacements in the case of a shortage [27]. A resilient FSC can quickly return to its original state or move to a new, more desirable state after a disruption [28]. Therefore, it is necessary to apply the concept of resilience during the decision-making stage to planning for the harvesting, storage, and transport of fresh produce. SC coordination is an effective means to build moderate resilience in the SC [29]. This involves designing an appropriate coordination incentive mechanism based on factors such as logistics, capital flow, and information flow among SC members. Regarding the research on the contract selection of coordinated SC coordination, a typical example is a revenue sharing contract. The revenue sharing provided by the contract demonstrates a good incentive and pre-control effect on the quality input and level of the SC [30]. Compared with the no-contract situation, the contract can achieve a “win-win” situation for revenue and quality and resilience to disruption [31].

This paper investigates the price and quality policy in a dual-channel FSC under the disruption of consumer quality preference and attempts to answer the following research questions:

• How can we formulate resilient price and quality strategies in a fresh-food dual-channel SC under the disruption of quality preference?
• What would be the implications for supply chain performance considering quality preference disruptions?
• How will a resilient supply chain strategy be affected, and how will it differ from a non-resilient decision?

To solve the above problems, this paper investigates an online and offline dual-channel FSC composed of one manufacturer and one retailer. Rahmani et al. [32] considered this type of SC structure as the basic dual-channel SC model, which could be regarded as the baseline for studying SCs with multiple manufacturers and multiple retailers. First, we introduce consumers’ quality preference into the market demand function. Regarding the quality of fresh food, we consider the joint efforts of manufacturers and retailers and the loss of quality. For quality preference disruption, we design the corresponding disruption cost to obtain each profit function. Second, based on the Stackelberg game, we solve the centralized and decentralized decision-making modes’ optimal price, quality, and quantity strategies. In various disruption scenarios, we analyze the impact of quality preference disruption on profits. In view of the inability to provide specific profit distribution among members under centralized decision-making, a revenue sharing contract is proposed to realize SC coordination. Finally, we explore the scope for resilient SC decision-making and corresponding management recommendations.

The contributions of this paper are as follows: first, we consider the impact of consumer quality preference disruption on market demand. Most FSC decision studies focus only on the effects of sales price and quality, ignoring the disruption of quality preference. Second, this paper investigates a dual-channel FSC with a comparative analysis of optimal price, quality, and quantity decisions in baseline and disruption scenarios. For the disruption situation under centralized decision-making, a revenue-sharing contract is proposed for SC coordination. Finally, differing from previous studies on SC resilience that focus on qualitative research, we quantitatively analyze how to make resilient production decisions under the disruption of quality preferences.

The structure of this paper is as follows: Section 2 provides a literature review. Section 3 establishes the dual-channel food SC model, setting the demands, disruption costs, and profits. Section 4 and Section 5 are divided into two cases, baseline and disruption, to solve the optimal price and quality strategies of the centralized and decentralized decision model. Section 6 proposes and proves propositions related to the operation management of the FSC based on the model. Section 7 proposes a revenue-sharing contract model. Section 8 describes and discusses the resilience range of the dual-channel FSC through numerical simulation. Finally, Section 9 summarizes the whole work and reaches conclusions. All proofs are in Appendix A.

2. Literature Review

2.1. Dual-Channel FSC

The dual-channel FSC could be mainly divided into three models: the manufacturer dual-channel, the retailer dual-channel, and the mixed dual-channel model [24]. In the manufacturer dual-channel model, the manufacturer plans to establish an additional direct online channel in addition to the retailer’s traditional offline sales channel [33]. The manufacturer first receives the order placed by the consumer on the online sales channel and then directly distributes the fresh food to the consumer from the production base [34]. For example, Pagoda and Dolly Farm have adopted the manufacturer dual-channel model. The retailer dual-channel model means that the manufacturer sells fresh food wholesale to the retailers who open online channels [35]. Typical companies using this model are Hema Fresh, Super Species, etc. As for the mixed dual-channel model, it can be seen as a combination of the above two models, where both manufacturers and retailers are developing online channels [36]. This paper will mainly study the manufacturer dual-channel model.

2.2. Consumer Preference

The price preference has always been the main foothold of scholars’ research. Still, quality preference cannot be ignored in the operation of FSC to improve consumers’ quality awareness. Yang et al. [36] developed FSC under three sales modes, considering the quality preference. Cai et al. [37] only considered the impact of quality preference on the SC through third-party logistics of fresh food. Lambertini [38] considered both quality and price preference. Ma et al. [39] proposed a three-echelon FSC considering freshness-keeping preference and effort with asymmetric information. Zhang et al. [40] found that the high sensitivity of quality preference can improve the preservation effect of the SC to a certain extent, thereby reducing product loss. Zhang et al. [41] argued that a price discount contract with an effort adjustment strategy or cost-sharing strategy can coordinate the SC. In addition, some scholars have considered low-carbon preference based on quality preference [24]. Therefore, due to the quality loss of fresh food, the models that only consider price preference are inaccurate, and quality preference should be added to improve SC performance.

2.3. Quality Effort

Among different supply chain entities, there are two main ways to improve quality efforts [25]. For manufacturers, these are mainly reflected in quality improvement efforts in production. Some fresh-food enterprises build their production bases to produce and process high-quality products or purchase high-quality upstream products with the help of a professional buyer team by establishing high-quality standards, such as Dingdong, Yiguo, etc.

Retailers mainly focus on freshness-keeping efforts at the sales stage. Hema and Super Species improve the freshness of their products by investing in complete cold chain logistics and fresh-food warehouses. In addition, well-equipped freshness-keeping equipment in stores and delivery services within 30 min improve product quality.

However, many scholars have focused on manufacturers’ efforts, ignoring the retailer’s efforts. Wu et al. [42] considered an outsourcing logistics channel whose quality effort and price affect the fresh food’s sellable quantity and quality. Zheng et al. [43] designed a freshness-keeping cost-sharing contract and revenue-sharing contract. Chuang and Wu [44] proposed an integrated supplier–retailer SC model considering quality effort with an asymmetric tolerance design and allowable shortage. As fresh food is perishable, it is vital to improve fresh-food quality from the production and distribution sides. Furthermore, considering the joint quality efforts of manufacturers and retailers in the SC decision model has important implications for the quality of fresh food instead of available manufactured products.

2.4. SC Disruption

This study believes that SC disruption refers to the state in which SC operation deviates from expectations due to external or internal disruption factors, thereby bringing various risks to enterprises. This could lead to other SC disruptions, mainly in terms of demand, consumer preference, price, cost, etc.

Huang et al. [45] were the first to study disruption management in a dual-channel SC. They focused on the impact of production cost disruptions on the SC but only considered the effect of price factors on demand. Subsequently, Huang et al. [46] explored in great depth the situation where demand and cost are disturbed at the same time. Giri et al. [47] focused on coordination under demand and supply disruption in a three-layer SC. Huang et al. [48] studied the price and inventory policy of the food SC with production disruption and controllable deterioration. Wan et al. [23] analyzed the coordination of a fresh agricultural product SC when the production cost and the loss rate are disrupted simultaneously. Mondal et al. [49] adopted interval a type-2 Pythagorean fuzzy set to describe uncertainty in SC risk management. Ali et al. [50] proposed a multi-objective mixed-integer nonlinear SC coordination model by studying the complexities of spoilage, transportation, demand uncertainty, price volatility, and transportation inefficiency. Many scholars have also studied SC disruptions under mixed uncertainty during the COVID-19 pandemic [51].

Based on the above research, the current research mainly focuses on the disruption of total market demand or cost. There are still few studies on the disruption of consumer preference. Rahmani et al. [32] explored a dual-channel SC affected by price and green level under demand disruption. However, the disruption of consumers’ green preference was ignored. Wang et al. [52] studied a single-channel SC situation in response to consumer preference disruptions. Si et al. [53] found an optimization SC strategy and analyzed the channel competition effect under the disruption of coupon promotion input and consumer preference. Therefore, it is necessary to conduct further research on the disruption of quality preference.

2.5. SC Resilience

SC resilience refers to the ability of the SC to return to its original state or an ideal state after a disruption, which mainly includes four dimensions: reliability, robustness, recovery, and reconfigurability [54]. Leat et al. [28] argued that supply chain resilience improved through horizontal collaboration among producers and cooperation with processors and retailers. Ivanov et al. [55] examined the range of disruption-based changes in supply structures and parameters needed to remain resilient for the dairy supply chain in Australia. Aviral et al. [56] defined the robustness of the SC based on four aspects, optimized the structure of the SC from the perspective of the emergency inventory, and provided research ideas for SC optimization under moderate flexibility. Wicaksono et al. [57] improved the resilience of food supply chains by applying the development of quality function deployment methods. Liu et al. [58] concluded that SC resilience is the key for cross-border e-commerce companies to gain a competitive advantage continuously. Few studies have quantitatively obtained how a dual-channel fresh-food supply chain makes resilient decisions in response to disruptions.

2.6. Research Gap

For the FSC, many studies have been conducted on inventory management, pricing, and ordering strategies. The literature related to this field is compared in

Table 1. Literature comparison.

	Dual-Channel	SC Type	Consumer Preference	SC Disruption	Quality Effort	Resilience
Chen et al. [34]	✓	Fashion	Quality	-	Manufacturers Only	-
Zhang et al. [40]	✓	Fresh food	Fresh-keeping	-	Retailer Only	-
Xie et al. [24]	✓	Fresh food	Low-Carbon	-	-	-
Gu et al. [25]	✓	Fresh Food	Quality	-	Both	-
Huang et al. [45]	✓	Common	-	Demand	-	-
Rahmani et al. [32]	✓	Common	Green Level	Demand	-	-
Wang et al. [52]	-	Fashion	Quality	Quality Preference	Manufacturers Only	-
Behzadi et al. [59]	-	Fresh Food	-	Demand	-	✓
This study	✓	Fresh Food	Quality	Quality Preference	Both	✓

.

According to the existing literature, this paper combines the channel characteristics and pricing strategies of the dual-channel FSC under quality preference disruption and quality effort based on SC resilience. The previous literature tends to focus on price preference in the research on consumer preference. However, for fresh food, quality preference is also significant to the operation of the supply chain and should be taken into account. In the research in this field, the research on disruption is mainly on demand and production cost, and few studies have considered the disruption of quality preference. In terms of quality effort, more effort is considered for a single member, and less is considered for joint efforts. Most studies on SC resilience focus on qualitative evaluation, factor analysis, etc. Few studies have considered how to achieve SC resilience from the production decision (price, quality, quantity, etc.).

Therefore, the innovation of this paper is mainly in the research into a dual-channel FSC, focusing on the perspective of quality management while considering quality preference, quality preference disruption, and quality joint effort. In addition to deciding on the optimal price, quality, and quantity, this paper focuses on decision-making with resilience and studies its influencing factors, application scope, and corresponding management decisions.

3. Model Description

As shown in Figure 1, the FSC model is divided into three stages: decision, production, and sales. The time proportion of the three stages is uneven, the production of fresh food takes the longest, and each type of fresh food generally has a corresponding suitable planting season. In the decision stage, centralized and decentralized decision-making are the two most typical decision-making methods in dual-channel SCs [33]. It is assumed that the SC will be divided into two decision-making methods to determine the price, quality, and quantity of fresh food. The production decisions are based on the original market demand and consumer quality preference. During the production stage, the changes in consumer quality preference may cause demand disruptions, so the original decisions may not adapt to the new quality preferences in the sales stage. Thus, due to the disruption of quality preference, it is possible that the original decision will be suboptimal. Inevitably, the disruption will generate additional disruption costs, which will reduce the profit of the SC.

To facilitate unified expression, the notations are defined in

Table 2. Symbol description.

Symbol	Description
Parameter
$i=0,1,2$	0 denotes the baseline case, 1 denotes the offline channel, and 2 denotes the online channel.
$j=M,R,C$	M refers to the manufacturer, R refers to the retailer, and C refers to the centralized decision.
$\rho$	Consumer preferences of offline channel, correspondingly, $\left(1-\rho \right)$ represents the consumer preferences of online channel, where $0<\rho <1$.
$D_i$	Demand for fresh food in channel i.
$\widetilde{D_i}$	Demand for fresh food in channel i under disruption.
${\epsilon }_i$	Price sensitivity of the demand in channel i.
${\lambda }_i$	Parameter of quality effort cost in channel i.
$\theta$	Quality sensitivity of the demand.
$\Delta \theta$	Disruption of quality sensitivity of the demand.
$k_i$	Loss ratio of fresh food in channel i.
${\mu }_i^n$	Unit penalty costs associated with the increase or decrease in demand $\left(n=+,-\right)$
${{\rm M}}_i$	Additional cost of the demand disruption in channel i.
${\Lambda }_i$	Effort cost for quality in channel i.
$\gamma$	Parameter of revenue sharing contract
Decision Variables
$p_i$	Price in channel i.
${\eta }_i$	Quality effort level in channel i
$w$	Wholesale price for the retailer.
$Q_j$	Production quantity for j.
$q_i$	Quality level of the fresh food in channel i.
${\pi }_j^i$	Profit of the j in channel i.

.

The model assumptions are as follows:

• The market information of the manufacturer and retailer is symmetrical, and their risk preference is assumed to be neutral and entirely rational. In this study, the FSC targets one kind of fresh food and operates for a cycle. It is assumed that the manufacturer has the ability to meet market demand and the retailer does not have inventory.
• In the literature on operations management and marketing, mainstream scholars have proposed linear demand assumptions for price and non-price variables [60,61]. To simplify the model and simple operation, we assume that the main factors influencing channels’ demand are price and quality when consumers purchase fresh food.
• Investment in quality efforts is a one-time expense. For the convenience of calculation, it is assumed that the manufacturer’s production cost is 0. This assumption is similar to Li et al. [62].

In the decision stage, the consumer’s quality preference is assumed to be $\theta$, whereas the quality preference may change and become $\theta +\Delta \theta$ in the sales stage. When $\Delta \theta >0$, it means that the current quality of fresh food conforms to the consumer preference trend. In contrast, when $\Delta \theta <0$, it means that consumer preference is decreasing. Based on [63,64], adopting a consistent pricing strategy in dual-channel can alleviate the conflict of competition between the channels. Similar practices are seen in companies such as Hema Fresh and Super Species [36].

The demands of online and offline channels are defined as follows:

$$\widetilde{D_1}=\rho D_0-{\epsilon }_1\tilde{p}+\left(\theta +\Delta \theta \right)\widetilde{q_1}$$

(1)

$$\widetilde{D_2}=\left(1-\rho \right)D_0-{\epsilon }_2\tilde{p}+\left(\theta +\Delta \theta \right)\widetilde{q_2}$$

(2)

The total demand of all channels in the centralized decision is defined as follows:

$$\widetilde{D_C}=D_0-\left({\epsilon }_1+{\epsilon }_2\right)\tilde{p}+\left(\theta +\Delta \theta \right)\left(\widetilde{q_1}+\widetilde{q_2}\right)$$

(3)

It is assumed that the manufacturer produces fresh food of the same $q_0$ and launches it on the market. Both channels will make quality efforts to increase sales, so that the quality levels when they reach consumers are $q_1$ and $q_2$. Following Krishnan et al. [65], the setting of the fresh food quality effort level and wastage rate under quality effort is expressed as: $q_i=q_0{\eta }_i{k}_i$, where the constraints are $0<{\eta }_i<1$ and $k_1,k_2>0$. The value range of this function is $\left(0,q_0\right)$. When $q_i\approx 0$, it means that the fresh food has decayed and cannot be sold, whereas there is almost no change in the quality level of fresh food when $q_i\approx 1$. According to the marginal beneficial characteristics of investment, the quality effort level function is a concave function. It is positively correlated with the cost but stabilizes after reaching a certain degree, expressed as: ${\Lambda }_i=\frac{1}{2}{\lambda }_i{{\eta }_i}^2$ [66,67].

In the sales stage, the actual demand for fresh food $\widetilde{D_i}$ may be different from the original production quantity ${Q_i}^{\ast }$. Due to demand disruptions, the SC will incur disruption costs: when $\widetilde{D_i}>{Q_i}^{\ast }$, the SC has to increase order quantity or other methods to meet additional demand, resulting in costs such as being out of stock. When $\widetilde{D_i}<{Q_i}^{\ast }$, the SC will incur inventory costs, processing costs caused by deterioration, etc. Similar to Wang et al. [52] and Qi et al. [68], the model will set ${\mu }_i^{+}$ and ${\mu }_i^{-}$ to represent the disruption cost when demand rises or falls. Suppose $\mu$ satisfies $0<{\mu }_i^{+}<p$, $0<{\mu }_i^{-}<\frac{1}{2}{\lambda }_i{{\eta }_i}^2$, thus the whole additional cost of the demand disruption can be expressed as:

$$\widetilde{{{\rm M}}_i}={\mu }_i^{+}{(\widetilde{D_i}-{Q_i}^{\ast })}^{+}+{\mu }_i^{-}{({Q_i}^{\ast }-\widetilde{D_i})}^{+}$$

(4)

where ${(x)}^{+}$ means $\max \left(x,0\right)$.

Due to the perishability of fresh food and the high quality requirements, the main costs of manufacturers and retailers in various channels are caused by quality efforts. Moreover, the sales cycle is short, and the inventory cost expenditure is mainly in the performance improvement of the inventory facility. Therefore, only the cost of quality improvement will be considered in this model, whereas the cost of production and operation will be ignored in each channel.

When dual-channel sales are at $\tilde{p}$, the demand is $\widetilde{D_1}$ and $\widetilde{D_2}$, respectively, and the related quality improvement costs and demand disruption costs are assumed. The profit function is:

$$\widetilde{{\pi }^C}=\tilde{p}\widetilde{D_C}-\frac{1}{2}{\lambda }_1{\widetilde{{\eta }_1}}^2-\frac{1}{2}{\lambda }_2{\widetilde{{\eta }_2}}^2-{\mu }_C^{+}{(\widetilde{D_C}-{Q_C}^{\ast })}^{+}-{\mu }_C^{-}{({Q_C}^{\ast }-\widetilde{D_C})}^{+}$$

(5)

$$\widetilde{{\pi }_M}=\tilde{w}\widetilde{D_1}+\tilde{p}\widetilde{D_2}-\frac{1}{2}{\lambda }_2{\widetilde{{\eta }_2}}^2-{\mu }_M^{+}{(\widetilde{D_2}-{Q_2}^{\ast })}^{+}-{\mu }_M^{-}{({Q_2}^{\ast }-\widetilde{D_2})}^{+}$$

(6)

$$\widetilde{{\pi }_R}=\left(\tilde{p}-\tilde{w}\right)\widetilde{D_1}-\frac{1}{2}{\lambda }_1{\widetilde{{\eta }_1}}^2-{\mu }_R^{+}{(\widetilde{D_1}-{Q_1}^{\ast })}^{+}-{\mu }_R^{-}{({Q_1}^{\ast }-\widetilde{D_1})}^{+}$$

(7)

For the model to be established, it supposes that all the parameters are positive. In order to signify the demand disruption case, we use the current notations with wavy lines.

4. Centralized Decision Model

In the centralized decision model, the manufacturer and the retailer are treated as a whole system. This means that an overall decision-maker decides the price and quality of fresh food to obtain the total optimal profit.

4.1. Baseline Case

In the baseline case, the original quantity at the time of decision is equal to the actual demand.

Lemma 1.

In a centralized dual-channel FSC without demand disruptions, if${\pi }^C$satisfies the condition of$2\left({\epsilon }_1+{\epsilon }_2\right){\lambda }_1-{(\theta q_0{k}_1)}^2>0$and$-2\left({\epsilon }_1+{\epsilon }_2\right){\lambda }_1{\lambda }_2+{\lambda }_1{(\theta q_0{k}_2)}^2+{\lambda }_2{(\theta q_0{k}_1)}^2<0$, it is a concavefunction with respect to$\left(p,{\eta }_1,{\eta }_2\right)$. The optimal sale price andquality effort levelin each channel, respectively, are given by:

$$p^{\ast }=\frac{D_0{\lambda }_1{\lambda }_2}{{\rm X}}$$

(8)

$${{\eta }_1}^{\ast }=\frac{D_0{\lambda }_2\theta q_0{k}_1}{{\rm X}}$$

(9)

$${{\eta }_2}^{\ast }=\frac{D_0{\lambda }_1\theta q_0{k}_2}{{\rm X}}$$

(10)

where ${\rm X}=2\left({\epsilon }_1+{\epsilon }_2\right){\lambda }_1{\lambda }_2-{\lambda }_2{(\theta q_0{k}_1)}^2-{\lambda }_1{(\theta q_0{k}_2)}^2$. Substituting the obtained Equations (8) and (10) into Equations (3) and (5), the optimal production quantity and total profit of the centralized dual-channel FSC are given by:

$${Q_C}^{\ast }={D_C}^{\ast }=\frac{\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2}{{\rm X}}$$

(11)

$${{\pi }^C}^{\ast }=\frac{1}{2}\frac{{D_0}^2{\lambda }_1{\lambda }_2}{{\rm X}}$$

(12)

The proof is presented in Appendix A.

4.2. Disruption Case

After predicting the actual demand for fresh food, the SC decision-maker should adjust the expected quantity under the original decision to maximize the total profit of the SC system.

Scenario 1: When $\Delta \theta >0$, then $\widetilde{D_C}\ge {Q_C}^{\ast }$, where ${Q_C}^{\ast }$ is defined in Equation (11). The total profit function of the centralized FSC in this time can be expressed as:

$$\widetilde{{\pi }^C}=\tilde{p}\widetilde{D_C}-\frac{1}{2}{\lambda }_1{\widetilde{{\eta }_1}}^2-\frac{1}{2}{\lambda }_2{\widetilde{{\eta }_2}}^2-{\mu }_C^{+}{(\widetilde{D_C}-{Q_C}^{\ast })}^{+}$$

(13)

$$s.t.D_0-\left({\epsilon }_1+{\epsilon }_2\right)\tilde{p}+\left(\theta +\Delta \theta \right)\left(\widetilde{q_1}+\widetilde{q_2}\right)-\frac{\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2}{{\rm X}}\ge 0$$

(14)

Due to $\widetilde{D_C}\ge {Q_C}^{\ast }$ and $\theta +\Delta \theta >0$, $\Delta \theta$ needs to conform to the value range of $\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$ in scenario 1, where ${\rm T}=\frac{{\rm X}\left({\epsilon }_1+{\epsilon }_2\right)}{[{(q_0{k}_1)}^2{\lambda }_2+(q_0{k}_2{)}^2{\lambda }_1]D_0}$.

Scenario 2: When $\Delta \theta <0$, then $\widetilde{D_C}\le {Q_C}^{\ast }$. The total profit function of the centralized FSC in this time can be expressed as:

$$\widetilde{{\pi }^C}=\tilde{p}\widetilde{D_C}-\frac{1}{2}{\lambda }_1{\widetilde{{\eta }_1}}^2-\frac{1}{2}{\lambda }_2{\widetilde{{\eta }_2}}^2-{\mu }_C^{+}{(\widetilde{D_C}-{Q_C}^{\ast })}^{+}$$

(15)

$$s.t.D_0-\left({\epsilon }_1+{\epsilon }_2\right)\tilde{p}+\left(\theta +\Delta \theta \right)\left(\widetilde{q_1}+\widetilde{q_2}\right)-\frac{\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2}{{\rm X}}\ge 0$$

(16)

Similarly, $\Delta \theta$ needs to conform to the value range of $-\theta \le \Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}-\theta$ in scenario 2.

Scenario 3: From the range of values derived from scenario 1 and 2, it can be found that there is a value range of $\sqrt{1-{\rm T}{\mu }_C^{-}}-\theta <\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$ between scenario 1 and 2. This shows that when demand disruption occurs and $\Delta \theta$ is in a certain range, the difference between $\widetilde{D_C}$ and ${Q_C}^{\ast }$ is not large, which may make production decisions resilient. This will be demonstrated later.

Lemma 2.

In the case of demand disruptions, the optimal selling price and the optimal online and offline channel quality level in the centralized dual-channel FSC are determined as follows:

$${\tilde{p}}^{\ast }=\{\begin{array}{cc}\frac{{\lambda }_1{\lambda }_2{D}_0}{\Delta {\rm X}}-{\mu }_C^{-}+\frac{\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{-}{\lambda }_1{\lambda }_2}{\Delta {\rm X}}\hfill & \hfill 0\le \theta +\Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}\\ \frac{{\lambda }_1{\lambda }_2{D}_0}{\Delta {\rm X}}\hfill & \hfill \sqrt{1-{\rm T}{\mu }_C^{-}}<\theta +\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}\\ \frac{{\lambda }_1{\lambda }_2{D}_0}{\Delta {\rm X}}+{\mu }_C^{+}-\frac{\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{+}{\lambda }_1{\lambda }_2}{\Delta {\rm X}}\hfill & \hfill \theta +\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}\end{array}$$

(17)

$${\widetilde{{\eta }_1}}^{\ast }=\{\begin{array}{cc}\frac{{\lambda }_2\left(\theta +\Delta \theta \right)q_0{k}_1\left[D_0+\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{-}\right]}{\Delta {\rm X}}\hfill & \hfill 0\le \theta +\Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}\\ \frac{{\lambda }_2\left(\theta +\Delta \theta \right)q_0{k}_1{D}_0}{\Delta {\rm X}}\hfill & \hfill \sqrt{1-{\rm T}{\mu }_C^{-}}<\theta +\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}\\ \frac{{\lambda }_2\left(\theta +\Delta \theta \right)q_0{k}_1\left[D_0-\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{+}\right]}{\Delta {\rm X}}\hfill & \hfill \theta +\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}\end{array}$$

(18)

$${\widetilde{{\eta }_2}}^{\ast }=\{\begin{array}{cc}\frac{{\lambda }_1\left(\theta +\Delta \theta \right)q_0{k}_2\left[D_0+\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{-}\right]}{\Delta {\rm X}}\hfill & \hfill 0\le \theta +\Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}\\ \frac{{\lambda }_1\left(\theta +\Delta \theta \right)q_0{k}_2{D}_0}{\Delta {\rm X}}\hfill & \hfill \sqrt{1-{\rm T}{\mu }_C^{-}}<\theta +\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}\\ \frac{{\lambda }_1\left(\theta +\Delta \theta \right)q_0{k}_2\left[D_0-\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{+}\right]}{\Delta {\rm X}}\hfill & \hfill \theta +\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}\end{array}$$

(19)

Comparing the ${\widetilde{Q_C}}^{\ast }$ obtained under the demand disruption with the ${Q_C}^{\ast }$ under the baseline case, the optimal production quantity of the three scenarios is obtained:

$${\widetilde{Q_C}}^{\ast }=\{\begin{array}{cc}{Q_C}^{\ast }+\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2\frac{{\rm X}-\Delta {\rm X}}{{\rm X}\Delta {\rm X}}+{\mu }_C^{-}\frac{{({\epsilon }_1+{\epsilon }_2)}^2{\lambda }_1{\lambda }_2}{\Delta {\rm X}}\hfill & \hfill 0\le \theta +\Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}\\ {Q_C}^{\ast }+\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2\frac{{\rm X}-\Delta {\rm X}}{{\rm X}\Delta {\rm X}}\hfill & \hfill \sqrt{1-{\rm T}{\mu }_C^{-}}<\theta +\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}\\ {Q_C}^{\ast }+\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2\frac{{\rm X}-\Delta {\rm X}}{{\rm X}\Delta {\rm X}}-{\mu }_C^{+}\frac{{({\epsilon }_1+{\epsilon }_2)}^2{\lambda }_1{\lambda }_2}{\Delta {\rm X}}\hfill & \hfill \theta +\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}\end{array}$$

(20)

where $\Delta {\rm X}=2\left({\epsilon }_1+{\epsilon }_2\right){\lambda }_1{\lambda }_2-{(\theta +\Delta \theta )}^2{(q_0{k}_1)}^2{\lambda }_2-{(\theta +\Delta \theta )}^2{(q_0{k}_2)}^2{\lambda }_1$.

The optimal profits of the SC system under the three disruption scenarios are given by:

$${\widetilde{{\pi }_C}}^{\ast }=\{\begin{array}{cc}\frac{{[D_0+\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{-}]}^2{\lambda }_1{\lambda }_2}{2\Delta {\rm X}}-\frac{{\mu }_C^{-}\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2}{{\rm X}}\hfill & \hfill 0\le \theta +\Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}\\ \frac{{D_0}^2{\lambda }_1{\lambda }_2}{2\Delta {\rm X}}\hfill & \hfill \sqrt{1-{\rm T}{\mu }_C^{-}}<\theta +\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}\\ \frac{{[D_0-\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{+}]}^2{\lambda }_1{\lambda }_2}{2\Delta {\rm X}}+\frac{{\mu }_C^{+}\left({\epsilon }_1+{\epsilon }_2\right)D_0{\lambda }_1{\lambda }_2}{{\rm X}}\hfill & \hfill \theta +\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}\end{array}$$

(21)

5. Decentralized Decision Model

In the decentralized decision-making model, manufacturers and retailers can make independent decisions and seek to maximize their own profits. Due to the unequal competitive position in line with the characteristics of the Stackelberg game, the manufacturer, as the leader of the Stackelberg game, determines the wholesale price and quality effort level first, and then the retailer determines its price as a follower.

In order to ensure that the channel demand and the profits obtained by the manufacturers and retailers are not negative, the following conditions must be met:

$$0\le w\le p\le \frac{\rho D_0+\left(\theta +\Delta \theta \right)q_0{k}_1{\eta }_1}{{\epsilon }_1}$$

(22)

$$0\le w\le p\le \frac{\left(1-\rho \right)D_0+\left(\theta +\Delta \theta \right)q_0{k}_2{\eta }_2}{{\epsilon }_2}$$

(23)

5.1. Baseline Case

In the baseline case, there is no demand disruption, and the reverse derivation method will be used to solve the problem of decentralized decision-making. In the solution process, the retailer first determines the offline channel quality effort level, and then the manufacturer decides the price and online channel quality effort level.

Lemma 3.

In a decentralized dual-channel FSC with demand disruptions, if${{\pi }_M}^{\prime }$satisfies the condition of$\frac{2{(\theta q_0{k}_1)}^2}{{\lambda }_1}>0$, $4{\epsilon }_2{\lambda }_1{(\theta q_0{k}_1)}^2-{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2>0$ and $2{\lambda }_1{(\theta q_0{k}_1)}^2{[{(\theta q_0{k}_2)}^2-2{\epsilon }_2{\lambda }_2]+[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2<0$ , it is a concave function with respect to $\left(w,p,{\eta }_2\right)$ . The optimal wholesale price for the retailer, sale price, and quality effort level of manufacturer are given by:

$$w^{\ast }=\frac{\left[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]\rho D_0{{\lambda }_1}^2-\right[\left(\theta q_0{k}_1{)}^2-{\epsilon }_1{\lambda }_1\right]\left(1-\rho \right)D_0{\lambda }_1{\lambda }_2}{2{\lambda }_1[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{(\theta q_0{k}_1)}^2+{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2}$$

(24)

$$p^{\ast }=-\frac{\{2{(\theta q_0{k}_1)}^2\left(1-\rho \right)+\rho [\left(\theta q_0{k}_1{)}^2-{\epsilon }_1{\lambda }_1\right]\}{D}_0{\lambda }_1{\lambda }_2}{2{\lambda }_1[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{(\theta q_0{k}_1)}^2+{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2}$$

(25)

$${{\eta }_2}^{\ast }=-\frac{\theta q_0{k}_2}{{\lambda }_2}\frac{\{2{(\theta q_0{k}_1)}^2\left(1-\rho \right)+\rho [\left(\theta q_0{k}_1{)}^2-{\epsilon }_1{\lambda }_1\right]\}{D}_0{\lambda }_1{\lambda }_2}{2{\lambda }_1[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{(\theta q_0{k}_1)}^2+{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2}$$

(26)

Replacing Equations (24)–(26) into the first-order condition of Equation (7) ${{\eta }_1}^{\ast }=\frac{\left(p-w\right)\theta q_0{k}_1}{{\lambda }_1}$, we can obtain the optimal quality effort level of a retailer:

$${{\eta }_1}^{\ast }=-\frac{\{[{(\theta q_0{k}_2)}^2-2{\epsilon }_2{\lambda }_2]\rho {\lambda }_1+[(\theta q_0{k}_1{)}^2+{\epsilon }_1{\lambda }_1]{\lambda }_2-2\rho {\epsilon }_1{\lambda }_1{\lambda }_2\}{D}_0\theta q_0{k}_1}{2{\lambda }_1[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{(\theta q_0{k}_1)}^2+{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2}$$

(27)

The optimal quantity of decentralized decision-making in the baseline case is given by:

$${Q_1}^{\ast }=\frac{\rho D_0[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{\lambda }_1{(\theta q_0{k}_1)}^2-\left(1-\rho \right)D_0[\left(\theta q_0{k}_1{)}^2-{\epsilon }_1{\lambda }_1\right]{\lambda }_2{(\theta q_0{k}_1)}^2}{2{\lambda }_1[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{(\theta q_0{k}_1)}^2+{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2}$$

(28)

$${Q_2}^{\ast }=\frac{(1-\rho )D_0{\lambda }_2\{{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2-2{\lambda }_1{\epsilon }_2{(\theta q_0{k}_1)}^2\}-{\lambda }_1\rho D_0\left[\left(\theta q_0{k}_1{)}^2-{\epsilon }_1{\lambda }_1\right]\right[\left(\theta q_0{k}_2{)}^2-{\epsilon }_2{\lambda }_2\right]}{2{\lambda }_1[\left(\theta q_0{k}_2{)}^2-2{\epsilon }_2{\lambda }_2\right]{(\theta q_0{k}_1)}^2+{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2}$$

(29)

Lemma 4.

In the decentralized decision model, in order to satisfy$w^{\ast }$,$p^{\ast }$,${{\eta }_1}^{\ast }$, and${{\eta }_2}^{\ast }$are solvable. In other words, the consistent pricing strategy would be feasible, and there is an optimal production decision for this problem. The range of consumer preferences is given by:$\mathit{max}\{{\rho }_1,0\}<\rho <{\rho }_2$, where${\rho }_1=\frac{[(\theta q_0{k}_1{)}^2+{\epsilon }_1{\lambda }_1]{\lambda }_2}{[2{\lambda }_2({\epsilon }_1+{\epsilon }_2)-(\theta q_0{k}_2{)}^2]{\lambda }_1}$,${\rho }_2=\frac{{[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1]}^2{\lambda }_2-2{(\theta q_0{k}_1)}^2{\epsilon }_2{\lambda }_1{\lambda }_2}{{\lambda }_1[{\epsilon }_1{\lambda }_1-(\theta q_0{k}_1{)}^2][{\epsilon }_2{\lambda }_2-{(\theta q_0{k}_2)}^2]+{\lambda }_2[{(\theta q_0{k}_1)}^2-{\epsilon }_1{\lambda }_1{]}^2-2{\epsilon }_2{\lambda }_2{\lambda }_1{(\theta q_0{k}_1)}^2}$.

From Lemma 4, it can be concluded that in decentralized decision-making, changes in consumer channel preferences will affect the implementation of channel strategies and supply chain members. Therefore, effective market research or big data analysis can help us to understand consumers’ channel preferences promptly and actively adjust market supply and channel strategies to enhance competitiveness.

5.2. Disruption Case

In the case of demand disruptions, according to the actual situation of the fresh food industry, the manufacturer and the retailer will bear the cost of disruption caused by demand disruptions. After demand disruptions occur, the manufacturer and the retailer should adjust their decisions to maximize their profits.

Scenario 1: When $\Delta \theta >0$, then $\widetilde{D_1}>{Q_1}^{\ast }$ and $\widetilde{D_2}>{Q_2}^{\ast }$, where ${Q_1}^{\ast }$ and ${Q_2}^{\ast }$ are defined in Equations (28) and (29). According to Equation (7), we obtain the first-order condition of ${\eta }_1$ as follows:

$${{\eta }_1}^{\ast }=\frac{\left(\tilde{p}-\tilde{w}-{\mu }_R^{+}\right)\left(\theta +\Delta \theta \right)q_0{k}_1}{{\lambda }_1}$$

(30)

when the demand disruption occurs and the demand of the product increases, we obtain ${\widetilde{{\pi }_M}}^{\prime }$ from Equation (6) and then obtain the first-order partial derivative of $\tilde{w},\tilde{p},\mathrm{and}$ $\widetilde{{\eta }_2}$ with respect to ${\widetilde{{\pi }_M}}^{\prime }$. Therefore, $\tilde{w}$, $\tilde{p}$, and $\widetilde{{\eta }_2}$ are given by:

$${\tilde{w}}^{\ast }=\frac{{\lambda }_1\left\{\left[\left(1-\rho \right)D_0{\lambda }_2-{\mu }_M^{+}{A}_2\right]A_1+\left(\rho D_0{\lambda }_1-{\mu }_R^{+}{{\rm E}}_1\right){{\rm B}}_2\right\}}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(31)

$${\tilde{p}}^{\ast }=\frac{\left(\rho D_0{\lambda }_1-{{\rm E}}_1{\mu }_R^{+}\right){\lambda }_2{A}_1{+2E}_1{\lambda }_1\left[\left(1-\rho \right)D_0{\lambda }_2-{\mu }_M^{+}{A}_2\right]}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(32)

$${\widetilde{{\eta }_2}}^{\ast }=\sqrt{E_2}\frac{\left\{2E_1{\lambda }_1\right[(1-\rho )D_0-{\epsilon }_2{\mu }_M^{+}]+A_1(\rho D_0{\lambda }_1-E_1{\mu }_R^{+})+{A_1}^2{\mu }_M^{+}\}}{2{\lambda }_1{E}_1{B}_2-{\lambda }_2{A_1}^2}$$

(33)

where $A_1={(\theta +\Delta \theta )}^2{\left(q_0{k}_1\right)}^2-{\epsilon }_1{\lambda }_1$, $A_2={(\theta +\Delta \theta )}^2{\left(q_0{k}_2\right)}^2-{\epsilon }_2{\lambda }_2$, $E_1={(\theta +\Delta \theta )}^2{(q_0{k}_1)}^2$, $E_2={(\theta +\Delta \theta )}^2{(q_0{k}_2)}^2$, and $B_2=2{\epsilon }_2{\lambda }_2-{(\theta +\Delta \theta )}^2{(q_0{k}_2)}^2$. The optimal quality level of manufacturer in this scenario is as follows:

$$\widetilde{{\eta }_1}=\frac{\sqrt{{{\rm E}}_1}}{{\lambda }_1}\left[\frac{{\lambda }_1\left({{\rm E}}_1+{\epsilon }_1{\lambda }_1\right)\left[\left(1-\rho \right)D_0{\lambda }_2-{\mu }_M^{+}{A}_2\right]+\left(\rho D_0{\lambda }_1-{\mu }_R^{+}{{\rm E}}_1\right)\left({\lambda }_2-{\lambda }_1{{\rm B}}_2\right)}{2\lambda {{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}-{\mu }_R^{+}\right]$$

(34)

Scenario 2: When $\Delta \theta <0$, then $\widetilde{D_1}<{Q_1}^{\ast }$ and $\widetilde{D_2}<{Q_2}^{\ast }$. The first-order condition of ${\eta }_1$ is given by:

$${{\eta }_1}^{\ast }=\frac{\left(\tilde{p}-\tilde{w}+{\mu }_R^{-}\right)\left(\theta +\Delta \theta \right)q_0{k}_1}{{\lambda }_1}$$

(35)

Similarly, $\tilde{w}$, $\tilde{p}$, $\widetilde{{\eta }_1}$, and $\widetilde{{\eta }_2}$ are as follows:

$${\tilde{w}}^{\ast }=\frac{{\lambda }_1\left\{\left[\left(1-\rho \right)D_0{\lambda }_2+{\mu }_M^{-}{A}_2\right]A_1+\left(\rho D_0{\lambda }_1+{\mu }_R^{-}{{\rm E}}_1\right){{\rm B}}_2\right\}}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(36)

$${\tilde{p}}^{\ast }=\frac{\left\{\left(\rho D_0{\lambda }_1+{{\rm E}}_1{\mu }_R^{-}\right){\lambda }_2{A}_1{+2E}_1{\lambda }_1\left[\left(1-\rho \right)D_0+{\mu }_M^{-}{A}_2\right]\right\}}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(37)

$${\widetilde{{\eta }_2}}^{\ast }=\sqrt{{{\rm E}}_2}\frac{\left\{{2E}_1{\lambda }_1\left[\left(1-\rho \right)D_0+{\epsilon }_2{\mu }_M^{-}\right]+A_1\left(\rho D_0{\lambda }_1+{{\rm E}}_1{\mu }_R^{-}\right)+{A_1}^2{\mu }_M^{-}\right\}}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(38)

$${\widetilde{{\eta }_1}}^{\ast }=\frac{\sqrt{{{\rm E}}_1}}{{\lambda }_1}\left[\frac{{\lambda }_1\left({{\rm E}}_1+{\epsilon }_1{\lambda }_1\right)\left[\left(1-\rho \right)D_0{\lambda }_2+{\mu }_M^{+}\left({\epsilon }_2{\lambda }_2+{{\rm E}}_2\right)\right]+\left(\rho D_0{\lambda }_1+{\mu }_R^{-}{{\rm E}}_1\right)\left({\lambda }_2-{\lambda }_1{{\rm B}}_2\right)}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}+{\mu }_R^{-}\right]$$

(39)

6. Model Analyses

Proposition 1.

In the centralized decision model, when$\sqrt{1-{\rm T}{\mu }_C^{-}}-\theta <\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$, the optimal decisions regarding price, quality, and quantity are resilient.

Proposition 1 indicates that when $\Delta \theta$ is in this interval, the consumer’s quality preference fluctuates little, and there is no need to make changes to the original production decision. When making decisions, SC members should consider future changes in quality preferences so that the quality of fresh food produced can be in the “resilience range”. This could avoid the impact of quality preferences and achieve moderate resilience of the SC. When making decisions, adequate market research should be conducted, not just production but also quality preferences.

Proposition 2.

When$\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$,${\tilde{p}}^{\ast }$will decrease as$\Delta \theta$increases, and its maximum value is$p^{\ast }+{\mu }_C^{+}$. When$-\theta \le \Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}-\theta$,${\tilde{p}}^{\ast }$will increase as$\Delta \theta$decreases, and its minimum value is$p^{\ast }-{\mu }_C^{-}$.

Proposition 2 reveals that when $\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$, the quality of fresh food is very consistent with the current consumer quality preference trend, and the price can be increased appropriately to $p^{\ast }+{\mu }_C^{+}$. When $-\theta \le \Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}-\theta$, the quality has seriously deviated from the current consumer quality preference trend, and the price can be appropriately lowered. The minimum is $p^{\ast }-{\mu }_C^{-}$. Therefore, adjusting the price according to the disruption cost can obtain the maximum profit in the SC system as a whole.

Proposition 3.

When$\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$,${\widetilde{{\eta }_1}}^{\ast }$increases monotonically with$\Delta \theta$, and the optimal value is${{\eta }_1}^{\ast }+\frac{D_0{\lambda }_2{q}_0{k}_1\Delta \theta }{{\rm X}}$. When$-\theta \le \Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}-\theta$,${\widetilde{{\eta }_1}}^{\ast }$decreases monotonically with$\Delta \theta$, and the optimal value is${{\eta }_1}^{\ast }+\frac{D_0{\lambda }_2{q}_0{k}_1\Delta \theta }{{\rm X}}$. ${\widetilde{{\eta }_2}}^{\ast }$is the same.

Proposition 3 shows that in addition to considering $\Delta \theta$, when determining the level of quality effort, it is mainly affected by the loss ratio of fresh food of its own channel $(k_1,k_2)$, as well as by the influence of the coefficient of quality effort cost in another channel $({\lambda }_1,{\lambda }_2)$. For centralized decision makers, when considering the quality improvement in the channel, the quality improvement cost of the two channels must be considered at the same time, to avoid channel conflicts and maximize the overall benefits of the SC system.

Proposition 4.

When centralized decision makers make production decisions, considering future consumer quality preference disruptions will provide more benefits than not considering them.

Proof of Proposition 4.

If demand disruptions occur but the centralized decision makers are not aware of them, then we use the production decisions in the baseline case. However, in the actual disruption situation, these decisions may not be optimal. With the change in $\Delta \theta$, the market demand in the sales stage can be expressed as:

$$\widetilde{D_C}=D_0-\left({\epsilon }_1+{\epsilon }_2\right)p^{\ast }+\left(\theta +\Delta \theta \right)\left({q_1}^{\ast }+{q_2}^{\ast }\right)$$

(40)

Based on the original production decision, the SC system takes into account the additional costs caused by demand disruptions and obtains:

$$\widetilde{{\pi }^C}=p^{\ast }{Q_C}^{\ast }-\frac{1}{2}{\lambda }_1{{\eta }_1^{\ast }}^2-\frac{1}{2}{\lambda }_2{{\eta }_2^{\ast }}^2-{\mu }_C^{+}{(\widetilde{D_C}-{Q_C}^{\ast })}^{+}-{\mu }_C^{-}{({Q_C}^{\ast }-\widetilde{D_C})}^{+}$$

(41)

$$\widetilde{{\pi }^C}=\{\begin{array}{c}\frac{{D_0}^2{\lambda }_1{\lambda }_2-2{\mu }_C^{+}{D}_0\theta \Delta \theta [{\lambda }_2{(q_0{k}_1)}^2+{\lambda }_1(q_0{k}_2{)}^2]}{2{\rm X}}if\Delta \theta >0\\ \frac{{D_0}^2{\lambda }_1{\lambda }_2+2{\mu }_C^{-}{D}_0\theta \Delta \theta [{\lambda }_2{(q_0{k}_1)}^2+{\lambda }_1(q_0{k}_2{)}^2]}{2{\rm X}}if\Delta \theta <0\end{array}$$

(42)

In the actual situation, the profit after considering the demand disruption is compared with the profit without considering the demand disruption, as shown in

Table 3. Comparison of profits after considering demand disruptions.

Demand Disruption $\Delta \mathit{\theta }$	Profits Comparison $\tilde{{{\mathit{\pi }}^{\mathit{C}}}^{\ast }}-\widetilde{{\mathit{\pi }}^{\mathit{C}}}$
$\Delta \theta \ge \sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$	$\frac{{\left[\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{+}\right]}^2}{2\Delta {\rm X}}+\frac{2{\mu }_C^{+}{D}_0\theta \Delta \theta [{\lambda }_2{(q_0{k}_1)}^2+{\lambda }_1(q_0{k}_2{)}^2]}{2{\rm X}}>0$
$0<\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$	$\frac{2{\mu }_C^{+}{D}_0\theta \Delta \theta [{\lambda }_2{(q_0{k}_1)}^2+{\lambda }_1(q_0{k}_2{)}^2]}{2{\rm X}}>0$
$\sqrt{1-{\rm T}{\mu }_C^{-}}-\theta <\Delta \theta <0$	$-\frac{2{\mu }_C^{-}{D}_0\theta \Delta \theta [{\lambda }_2{(q_0{k}_1)}^2+{\lambda }_1(q_0{k}_2{)}^2]}{2{\rm X}}>0$
$-\theta \le \Delta \theta \le \sqrt{1-{\rm T}{\mu }_C^{-}}-\theta$	$\frac{{\left[\left({\epsilon }_1+{\epsilon }_2\right){\mu }_C^{-}\right]}^2}{2\Delta {\rm X}}-\frac{2{\mu }_C^{+}{D}_0\theta \Delta \theta [{\lambda }_2{(q_0{k}_1)}^2+{\lambda }_1(q_0{k}_2{)}^2]}{2{\rm X}}>0$

. □

Proposition 4 indicates that the quality preference of consumers for fresh food makes manufacturers and retailers more profitable. As a decision-maker in the FSC, it is vitally important to consider the disruption of consumer quality preference. Quality efforts should be aggressively improved to meet consumer quality preference. Regardless of whether it is a positive or negative quality preference disruption, considering the disruption management of quality preferences will increase the profit of the SC.

7. Revenue Sharing Contract

It is well known that the SC profit under the centralized decision model is more than that under the decentralized decision model, but it cannot provide a specific profit distribution between SC members. This paper proposes a revenue sharing contract. The contract parameters are $\left(w,\gamma \right)$. Manufacturers supply retailers at lower $w$, and $w$ may be lower than the cost of production. After the end of the sales period, the retailer needs to share a $\gamma$ proportion of the profit with the manufacturer to make up for the loss caused by $w$. Therefore, the profits of manufacturers and retailers are as follows:

$$\overline{{\pi }_M}=w\widetilde{D_1}+\left(1+\gamma \right)\overline{p}\widetilde{D_2}-\frac{1}{2}{\lambda }_2{\overline{{\eta }_2}}^2$$

(43)

$$\overline{{\pi }_R}=\left[\left(1-\gamma \right)\overline{p}-w\right]\widetilde{D_1}-\frac{1}{2}{\lambda }_1{\widetilde{{\eta }_1}}^2$$

(44)

The optimal decision under this contract is

$${\overline{p}}^{\ast }=\frac{{\lambda }_2\left\{\left[-{\lambda }_1{\epsilon }_1+\left(1-\gamma \right){{\rm E}}_1\right]w+{{\lambda }_1}^2\left(1-\rho \right)D_0\left(1+\gamma \right)\right\}}{{\lambda }_1\left(1+\gamma \right)\left\{2{\epsilon }_2{\lambda }_2-\left(1+\gamma \right){{\rm E}}_1\right\}}$$

(45)

$${\overline{{\eta }_1}}^{\ast }=\frac{\left[\begin{array}{c}\left\{\begin{array}{c}{\lambda }_1{\left(1+\gamma \right)}^2{{\rm E}}_1+{\lambda }_2{\left(1-\gamma \right)}^2{{\rm E}}_1\\ -[{\epsilon }_1\left(1-\gamma \right)+2{\epsilon }_2\left(1+\gamma \right)]{\lambda }_1{\lambda }_2\end{array}\right\}w\\ +\left(1-\gamma \right){{\lambda }_1}^2{\lambda }_2\left(1-\rho \right)D_0\left(1+\gamma \right)\end{array}\right]\sqrt{{{\rm E}}_1}}{{{\lambda }_1}^2\left(1+\gamma \right)\{2{\epsilon }_2{\lambda }_2{(q_0{k}_2)}^2-{\lambda }_1\left(1+\gamma \right){{\rm E}}_1\}}$$

(46)

$${\overline{{\eta }_2}}^{\ast }=\frac{\left\{\left[-{\lambda }_1{\epsilon }_1+\left(1-\gamma \right){{\rm E}}_1\right]w+{\lambda }_1{\lambda }_1\left(1-\rho \right)D_0\left(1+\gamma \right)\right\}\sqrt{{{\rm E}}_2}}{2{\epsilon }_2{\lambda }_1{\lambda }_2-{\lambda }_1\left(1+\gamma \right){{\rm E}}_1}$$

(47)

Under the revenue sharing contract, to provide the supply chain with Pareto improvement, ${\overline{p}}^{\ast }$, ${\overline{{\eta }_1}}^{\ast }$, and ${\overline{{\eta }_2}}^{\ast }$ must be equal to ${\tilde{p}}^{\ast },{\widetilde{{\eta }_1}}^{\ast },\mathrm{and}{\widetilde{{\eta }_2}}^{\ast }$, respectively, and can be obtained simultaneously:

$$\gamma =\frac{{\lambda }_2{\widetilde{{\eta }_2}}^{\ast }-k_2\left(\theta +\Delta \theta \right){\tilde{p}}^{\ast }}{k_2\left(\theta +\Delta \theta \right){\tilde{p}}^{\ast }}$$

(48)

$$w=\frac{2\left(1-p\right)D_0{{\lambda }_1}^2{E}_2{\tilde{p}}^{\ast }+{\lambda }_1{\lambda }_2{E}_1{{\widetilde{{\eta }_2}}^{\ast }}^2-\left[\left(1-p\right)D_0{\lambda }_1+2{\epsilon }_2{\tilde{p}}^{\ast }\right]{\lambda }_1{\lambda }_2{\widetilde{{\eta }_2}}^{\ast }\sqrt{E_2}}{{\lambda }_2{k_1}^2{E_1}^{2/3}{\widetilde{{\eta }_2}}^{\ast }-{\epsilon }_1{\lambda }_1{E}_1{\tilde{p}}^{\ast }}$$

(49)

If the designed contract is to be accepted by manufacturers and retailers, both individual rationality and collective rationality must be satisfied. This is to ensure that the profits of the manufacturer and the retailer under the revenue-sharing contract reach Pareto improvement, and so we provide Proposition 5.

Proposition 5.

Under the benefit-sharing contract coordination, there exists${\gamma }_1,{\gamma }_2\in \left(0,1\right)$, such that${\widetilde{{\pi }_M}}^{\ast }={\overline{{\pi }_M}}^{\ast },{\widetilde{{\pi }_R}}^{\ast }={\overline{{\pi }_R}}^{\ast }$. If${\gamma }_1<{\gamma }_2$, then$\gamma \in \left[{\gamma }_1,{\gamma }_2\right]$, and all members of the dual-channel SC can achieve Pareto improvement under the revenue sharing contract.

The proof is presented in Appendix A.

Proposition 5 shows that, as long as the profits of the manufacturer and the retailer are not less than their respective profits under decentralized decision making, the dual-channel supply chain can be coordinated under a revenue-sharing contract.

8. Numerical Simulation

To further verify the validity of the model, this paper will use numerical examples for analysis. The parameter setting is combined with the actual investigation of fresh food sales in e-commerce [36,40] and studies on the quality effort and loss of fresh food [25,38]. These parameters express that the current offline sales channels have a slight advantage in channel selection, and consumers are not sensitive to offline prices. According to the constraints of Lemma 1 and Lemma 3, the specific parameter settings are set as shown in

Table 4. Numerical assumptions.

${\mathit{D}}_0$	$\mathit{\rho }$	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	$\mathit{\theta }$	${\mathit{q}}_0$	${\mathit{k}}_1$	${\mathit{k}}_2$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$
100	0.6	0.6	0.8	0.6	1	0.85	0.9	1.8	1.2

.

(1)

Divided into two cases according to $\Delta \theta >0$ and $\Delta \theta <0$, with $\left(\Delta \theta ,{\mu }_{M,R}\right)$ as the independent variable, then $\Delta \theta \in \left[-0.6,0.6\right]$, ${\mu }_M^{+}\in \left[0,6\right]$, ${\mu }_R^{+}\in \left[0,9\right]$, ${\mu }_M^{-}\in \left[0,15\right]$, ${\mu }_R^{-}\in \left[0,18\right]$, the trend curve of $\widetilde{{\pi }_M}$ and $\widetilde{{\pi }_R}$ versus $\left(\Delta \theta ,{\mu }_{M,R}\right)$ is shown in Figure 2.

Figure 2 illustrates that regardless of whether it is a manufacturer or a retailer in the decentralized decision model, as long as the disruption of quality preference is considered, it will increase their profit. Among them, when $\Delta \theta >0$, the improvement in $\Delta \theta$ under the original decision does not cause an increase in the profit for the manufacturer, missing the opportunity to improve the quality profit. When $\Delta \theta <0$, the original decision is more affected by $\Delta \theta$, and the profit of manufacturers considering disruptions will be more stable. This proves that it is necessary to consider the disruption of quality preference, which helps to stabilize the profit and avoid demand disruption.

However, it is worth noting that the Stackelberg game is more beneficial to manufacturers, and retailers will be at a massive disadvantage. Retailer profits plummeted as demand in offline channels plummeted due to disruptions in quality preferences. This may be because offline channel consumers have a more direct perception of the quality of fresh food, and the occurrence of disruptions will cause consumers to stop their consumption behavior. Therefore, retailers should pay more attention to the disruption of quality preferences. While striving to improve quality, new technologies such as BCT are used to improve the traceability and information sharing of fresh products, providing consumers with a tool to easily explore the source of fresh products on their own to create a trusted consumption environment.

(2)

In the centralized decision model, when ${\mu }_C^{+},{\mu }_C^{-}\in \left[0,40\right]$, the “resilience range” of $\sqrt{1-{\rm T}{\mu }_C^{-}}-\theta <\Delta \theta <\sqrt{{\rm T}{\mu }_C^{+}+1}-\theta$ is present, as shown in Figure 3. The “resilience range” refers to the range of resilience in decision-making when $\Delta \theta$ changes with ${\mu }_C^{+}$, ${\mu }_C^{-}$. From Figure 3, we can find that the upper and lower limits of the resilience range are mainly affected by ${\mu }_C^{+}$ and ${\mu }_C^{-}$. The increase in the resilience range with ${\mu }_C^{+}$ is obviously greater than the increase with ${\mu }_C^{-}$. Therefore, decision makers need to analyze the cost of disruptions. The resilience range is mainly above the original value of $\theta$, indicating that when $\Delta \theta$ rises, production decisions are more likely to be resilient. Therefore, it is necessary to enhance consumers’ preference for the quality of fresh food through advertising, publicity posters, etc., so as to make the SC income more stable.

When $\theta$ is in the resilience range $\left({\mu }_C^{+}=10,{\mu }_C^{-}=30\right)$, the change curve of each decision variable is shown in Figure 4. When the disruption is within the resilience range, price, quality, and quantity do not need to change. However, due to the high stock quantity decisions out costing ${\mu }_C^{+}$, the profit of the SC will decline. It will also be significantly higher than the profit under the original decision. When the disruption is lower than the resilience range, the price should be lowered to promote sales. In contrast, the quantity needs to be appropriately increased due to the low price advantage. When the disruption is higher than the resilience range, more extraordinary quality efforts are required, but the benefits are far less than the resilience interval decision-making.

In either case, offline channels have to make more quality efforts than online channels to form SC coordination. This shows that in a centralized decision model, the quality of fresh food that finally reaches consumers through dual channels needs to be differentiated, and appropriate environmental conditions must be maintained in retail stores. For fresh food that needs to be refrigerated, the freshness-keeping temperature must be kept stable to avoid the deterioration of quality due to accidental opening of the freezer, power failure, or other reasons.

(3)

In the decentralized decision model, there is also a resilience range similar to Lemma 2, and the solutions of each decision variable are:

$${\tilde{w}}^{\ast }=-\frac{{{\lambda }_1}^2\rho D_0{{\rm B}}_2+{\lambda }_1{\lambda }_2\left(1-\rho \right)D_0{A}_1}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(50)

$${\tilde{p}}^{\ast }=\frac{{\lambda }_1{\lambda }_2\left[\rho D_0{A}_1+2\left(1-\rho \right)D_0{{\rm E}}_1\right]}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(51)

$${\widetilde{{\eta }_2}}^{\ast }=\frac{{\lambda }_1\left[\rho D_0{A}_1+2\left(1-\rho \right)D_0{{\rm E}}_1\right]\sqrt{{{\rm E}}_2}}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(52)

$${\widetilde{{\eta }_1}}^{\ast }=\frac{\left\{{\lambda }_2\left[2\left(1-\rho \right)D_0{{\rm E}}_1+A_1{D}_0\right]+{\lambda }_1\rho D_0{{\rm B}}_2\right\}{{\rm E}}_1}{2{\lambda }_1{{\rm E}}_1{{\rm B}}_2-{\lambda }_2{A_1}^2}$$

(53)

We compare the change curves of the decision variables in the centralized and decentralized model and draw Figure 5. Figure 5 illustrates that when the disruption is in the resilience range, the price of the centralized decision model will be higher than that of the decentralized decision model, and the total profit in the centralized decision model is much higher. This shows that a centralized decision model can effectively alleviate channel conflicts and avoid "price wars" between channels. The higher the consumer’s preference for quality, the greater the quality effort required by retailers. Moreover, prices under a centralized decision model are more stable, reducing the adverse effects of significant price changes on consumers. In addition, the total demand in the centralized decision model is less affected by the disruption. The original decision making is not changed much and can be maintained to reduce decision frequency.

(4)

It can be seen from Figure 6 that when the revenue-sharing contract parameter $\gamma$ gradually increases, the manufacturer’s profits decrease, and the retailer’s profits gradually increase. Figure 6 reveals the effect of $\gamma$ on the profits of manufacturers and retailers before and after the SC reaches coordination when in a resilience range of $\theta +\Delta \theta$. It is calculated that, after contract coordination, when $\gamma \in \left[0.1423,0.5389\right]$, the manufacturer and retailer achieve Pareto improvement.

9. Conclusions

This paper conducts modeling on the disruption management problem of the dual-channel FSC based on the perspective of consumer quality preference disruption. First, in view of the baseline case, it studies the price, quality, and production decision-making of the centralized and decentralized decision models. Second, in the case of quality preference disruptions, it divides situations into three scenarios and compared them with the decision-making in the baseline case. We propose a revenue sharing contract to achieve supply chain coordination. Finally, the numerical experiment is used to analyze the profit of SC members under the two decisions and the resilience range generated by the disruption of quality preference.

Based on the above model analysis and numerical simulations, the management insights from this work are summarized as follows:

• When consumer quality preference is slightly disturbed, the FSC’s pricing, quality, and quantity decisions have a range of resilience. The resilience range is mainly affected by disruption cost. Production decisions that establish a resiliency strategy are most effective in mitigating disruption risks. When making decisions, adequate market research should be conducted, not just of production but also quality preferences.
• In response to demand disruptions, a centralized decision model can achieve higher SC profits and quality levels, effectively avoiding channel conflicts and improving the quality of fresh food. The use of revenue sharing contracts can solve the problem of profit disruption in the case of centralized decision-making. However, it is worth noting that the Stackelberg game is more beneficial to manufacturers. Retailers will be at a massive disadvantage and should pay more attention to the disruption of quality preference.
• Regardless of whether it is a centralized or decentralized decision model, considering quality preference disruptions when formulating plans will significantly increase the profit of each member of the SC. Therefore, decision makers must always pay attention to market changes and could actively use big data analysis and other means to identify consumer quality preference.
• In either case, offline channels have to make more quality efforts than online channels to form SC coordination. For fresh food that needs to be refrigerated, the freshness-keeping temperature needs to be kept stable to avoid the deterioration of quality due to accidental opening of the freezer, power failure, or other reasons. While striving to improve quality, new technologies such as BCT are used to improve the traceability and information sharing of fresh food, providing consumers with a tool to easily explore the source of fresh products on their own to create a trusted consumption environment.

It is worth exploring the following points further: (1) the research model in this paper only considers the simple dual-channel approach of a single product, and the SC will be more complicated in actual market conditions. (2) In the disruption management for quality preference, we could consider repurchase contracts, quantity discount contracts, etc., to coordinate the SC. (3) The description of the resilience of the SC is still in the model stage, and its identification, prediction, intervention and other means need to continue to be studied.