Research on Supply Chain Quality Decision Model Considering Reference Effect and Competition under Different Decision-Making Modes

Abstract

The reference effect of consumers’ product quality and service quality is an important behavioral factor that affects consumers’ purchase decisions. In this paper, considering the inherent dynamics of the operation mode, the two were combined into a product-service supply chain composed of a manufacturer and two competitive retailers, and the service reference effect was further divided into horizontal and vertical service reference effects. Differential game models between the manufacturer and the retailers were constructed. Using the continuous dynamic programming theory, the manufacturer’s quality level strategy, retailer’s service level strategy and supply chain system performance under the modes of no cost sharing decision, cost sharing decision and centralized decision were analyzed and compared. Main findings: The optimal strategy of product quality under centralized decision is greater than the corresponding value under cost sharing contract decision. Compared with decentralized decision making of no cost sharing, cost sharing contract cannot promote the manufacturer to improve product quality. Different from previous studies, the service quality under centralized decision making is not higher than other decisions. When the horizontal reference effect of service quality meets a certain condition, the supply chain profit under centralized decision making is lower than the corresponding value under decentralized decision making.

1. Introduction

For a long time, quality is the content that enterprises cannot ignore to create value for customers. With the intensification of market competition and globalization, quality management is no longer limited to the enterprise, and began to integrate into the supply chain. Therefore, only by forming a supply chain with other enterprises can any enterprise obtain the initiative of competition. On 31 August 2016, according to Reuters, Samsung Electronics said that it would postpone the delivery of the galaxy Note7 tablet phone due to additional quality control tests. South Korean local media reported that earlier, South Korean users and other market users had received Galaxy Note 7, and some users claimed that their mobile phone battery exploded. If the quality control problem was confirmed, it would be a great blow to the world’s largest smartphone manufacturer, because the company expected this new product to maintain the sales momentum of its mobile business in the second half of the year [1]. According to Agence France Presse, Tokyo, 11 October 2017, major Japanese automakers said they were trying to assess whether their vehicles using Kobe Steel products had quality problems. The steel plant admitted to tampering with product quality data in a scandal of continuous fermentation. This continuous fermenting crisis is the latest in a series of quality control and management scandals involving large Japanese enterprises in recent years, which has damaged Japan’s reputation in terms of product quality. This was another problem that gives Nissan a headache. It had announced the recall of more than 1 million cars in Japan because of a certification problem. The Kobe Steel scandal broke out on Sunday when the steel plant admitted to falsifying data related to the strength and quality of products. Internal investigations showed that between September 2016 and August 2017, Kobe Steel supplied customers with products suspected of counterfeiting, including about 19,300 tons of aluminum products, about 2200 tons of copper products and about 19,400 aluminum forgings [2]. In 2021, according to a recent report by the New York Times, Tesla “may have weakened the safety of autopilot.” According to the New York Times, Musk insisted on configuring cameras only for the autopilot system without laser radar. In this regard, the national highway traffic safety administration was launching an investigation to find out whether this decision is “part of the reason for at least 12 autopilot related traffic accidents”. In addition, Tesla was undergoing another national regulatory investigation [3]. According to Reuters, an employee of Tesla’s solar business unit accused the U.S. Securities and Exchange Commission that “the company has not properly disclosed the fire risk related to solar product defects to shareholders and the public for several years”, and the incident may affect at least 60,000 American families and 500 government and private institutions. In fact, this was not the only design defect of Tesla. Tesla had experienced many recalls in 2021. On 3 December, Tesla announced that it would recall 21,599 model Y electric vehicles produced between 4 February 2021 and 30 October 2021 because of the risk of fracture of the suspension knuckle. Tesla mentioned in the announcement that the recall was due to supplier manufacturing, and the stock price of the nominated top group fell sharply that day, with a decline of 8.86% as of the close [4].

From these quality safety events, it can be seen that the product quality defects of upstream enterprises in the supply chain are the main reasons for the occurrence of quality safety events, while the downstream core enterprises undertake the main responsibility costs. This also shows that as the main bearer of responsibility costs, the downstream core enterprises of the supply chain will have a greater willingness to require the upstream enterprises to improve the product quality level, so as to reduce the probability of quality safety accidents.

With the increasing competition in the product sales market, developing product service operation management has gradually become an important way to enhance the overall competitive advantage of the supply chain. In order to improve customer service satisfaction, retailers provide customers with real-time and convenient pre-sales and after-sales services. In this way, manufacturers focus on product quality and retailers focus on improving product service quality. The coordination and cooperation between the two can enhance the competitive advantage of products. At present, domestic and foreign retailers are changing from product marketing to service marketing, and seeking new value sources that can enhance the competitiveness and profitability of enterprises by providing customers with a series of value-added activities related to products, such as pre-sales and after-sales service support, product maintenance. The focus of competition among retailers has gradually changed from price competition to service competition.

In marketing practice, there is an important marketing phenomenon, that is, reference effect. In the past, scholars mostly studied the reference effect from the perspective of price. Recently, some researchers have extended the concept of reference point to product quality (He et al. [5], Zhou et al. [6] and Qiu et al. [7]). According to the research of Fibich et al. [8] and Zhou et al. [6], if the product quality is greater than reference quality, the market demand will increase, otherwise, the market demand will decrease.

Unlike most previous research on supply chain quality decision making, first, this paper considers the retailer service quality competition, which is more in line with practice and reality; second, the reference effect is divided into horizontal reference effect and vertical reference effect. Vertical reference effect refers to that consumers’ purchase decisions rely on the evaluation of service quality in historical purchase experience as a reference. Horizontal reference effect refers to that consumers’ purchase decisions also depend on the service quality of other retailers, and take this as a reference. The significance of this paper includes the following two aspects. Firstly, it is helpful to provide guidance for enterprise managers to make effective quality decisions; secondly, by comparing the profits of supply chain under different decision-making modes, it provides an effective way for enterprise decision makers to choose the supply chain cooperation mode.

The structure of this paper is as follows: Section 2 mainly reviews the literature from three aspects: retailer competition, supply chain quality and reference effect. Section 3 introduces basic assumptions and symbolic descriptions. Section 4 and Section 5 are the focus of this paper. Under decentralized and centralized decision making, supply chain decision-making models are constructed, and the optimal decisions are solved. Section 6 is numerical simulation analysis. Section 7 is the conclusion of this paper.

2. Literature Review

This paper mainly involves the literature review of retailer competition, reference effect, supply chain quality.

(1)

Research on Retailer competition

Lin et al. (2006) [9] considered the impact of value-added services on the optimal retail price and respective market demand. Yi (2009) [10] established a remanufacturing closed-loop supply chain game model with competitive retailers under the three power structures. Li and Wang (2010) [11] investigated the purchasing strategy of one retailer and the pricing strategy of two suppliers in the supply chain under the supply interruption environment. When suppliers are competitive, there are sufficient conditions for equilibrium price in the decentralized system, and retailers choose to purchase from multiple suppliers to disperse the risk of supply interruption. Based on the complex adaptive system, He and Wang (2013) [12] evaluated the evolutionary behavior of arm through optimization method and genetic algorithm, and found that the ability of consumers to collect pricing information has a significant impact on the competitiveness of retail chains. Glock and Kim (2015) [13] studied the supply chain situation of single supplier multi retailer, and considered three types of quantity competition, price competition and non-competition respectively. Zhou et al. (2022) [14] considered a supply chain with competitive retailers. Under government subsidies, the competitive behavior between retailers will increase the optimal retail price, which can effectively stimulate the recycling behavior under centralized decision making and promote the growth of supply chain profits under this mode. Wang et al. (2022) [15] considered the heterogeneity of consumers and built a tripartite game model for the governance of China’s Plastic straw industry. Karimi Marzieh et al. (2022) [16] investigated service, price, and inventory decisions under retailers’ competition and cooperation.

(2)

Research on reference effect

Hardie et al. (1993) [17] proposed reference quality, which added quality reference points to the prospect theory proposed by Kahneman and Tversky (1991) [18], established a reference dependence model, and confirmed that the difference between quality and reference quality would affect the purchase probability, indicating that the reference effect of quality is significant. Reference effect of quality has an impact on the demand products, and it is also affected by many factors, such as price and goodwill. Kopalle and Winer (1997) [19] first incorporated the expected quality and reference price effects into the demand model of a market monopolist. The research showed that for monopolists, maintaining periodic pricing and product quality policies is the best choice. Kihlstorm and Riordan (1984) [20] and Milgrom and Roberts (1986) [21] considered that the advertising investment of enterprises increased brand goodwill, and pointed out that brand goodwill was positively related to the product quality of enterprises to some extent. Caulkins et al. (2017) [22] interpreted market demand potential as a kind of capital inventory to measure brand goodwill, pointed out that demand is not only affected by goodwill, but also by reference quality, and studied the dynamic decision making of enterprises on price, advertising and quality investment. He et al. (2018) [5] improved the Nerlove Arrow model, incorporated the reference quality benefit and goodwill into the linear demand function. Ma and Hu (2020) [23] found that consumers’ excessive reliance on brand goodwill to judge reference quality will affect the enthusiasm of sellers’ marketing, damage the performance of the supply chain and aggravate the double marginal effect of the supply chain. Ji et al. (2021) [24] studied the impact of consumers’ online quality reference effect, offline inventory effect and consumers’ channel preference on the operational decisions of members of the dual channel supply chain system. The research shows that the reference effect of quality can motivate suppliers to improve product quality. By formulating a stylized model in an e-commerce supply chain comprising of one green manufacturer and one e-tailer, Zhou and Duan (2022) [25] investigates the choice of selling formats (reselling or agency selling) with consumer reference greenness effect and environmental awareness.

(3)

Research on supply chain quality problems

At present, academia pays more attention to the quality of supply chain, mainly involving the quality of products and services.

Boyaci and Gallego (2004) [26] studied the quality equilibrium models of three different supply chain competition situations. Gurnani and Erkoc (2010) [27] established a market demand function, in which market demand is affected by marketing input and product quality. Fan et al. (2020) [28] studied how manufacturers and retailers share the product liability costs caused by quality defects. Prasenjit and Tarun (2021) [29] explored quality decisions under product differentiation and information asymmetry. Li and Mishra (2021) [30] analyzed the impact of product quality on retailers’ after-sales service. Lejarza and Baldea (2022) [31] designed a set of optimization methods to effectively control the quality of products, so as to meet more stringent market and policy needs.

As mentioned above, there are still some deficiencies in the previous literature, such as little reference to the competitive situation of retailers and little consideration of horizontal reference and vertical effect together. The theoretical contributions of this paper include the following aspects: first, the retailer competition is considered in the supply chain quality decision-making model; second, the reference effect is divided into horizontal reference effect and vertical reference effect. It enriches the supply chain quality decision theory and reference effect theory. The actual contributions include the following aspects: first, under different cooperation modes, this paper provides guidance for enterprise managers on what factors should be considered when making quality decisions, and how to correctly determine service quality and product quality; secondly, it provides basis and judgment conditions for enterprise managers to choose the most favorable cooperation mode.

In view of this, this paper considers retailers competition and constructs demand functions including horizontal and vertical reference effect of service quality, and reference effect of product quality. Then, under different decision-making modes, the differential game models of supply chain are established, and the optimal decisions are studied. Finally, comparing the optimal decisions, we explore the application conditions of different decision-making modes.

3. Model Development

The object of this paper is a two-level supply chain, which is composed of a manufacturer and two competitive retailers, that is, the manufacturer sells products through two retailers. The symbols and relevant descriptions in this section are shown in

Table 1. Symbols and descriptions.

Symbol	Description	Symbol	Description
z(t)	The manufacturer’s product quality at time t, which is a decision variable.	β1	The consumers’ vertical reference effect of service quality, which indicates the sensitivity coefficient of the gap between the actual service quality and the reference service quality to the change in demand.
fi(t)	The retailers’ service quality at time t, which is a decision variable. i = 1, 2 indicate retailer 1 and retailer 2 respectively.	β2	The consumers’ reference effect of product quality.
rfi(t)	Consumer reference service quality of retailer i at time t. (i = 1, 2)	χ1	The consumers’ horizontal reference effect of service quality, which indicates the sensitivity coefficient of the gap between the actual service quality and the competitive retailer’s service quality to demand changes.
rz(t)	The reference product quality of consumers at time t.
${\theta }_i$	Service quality or product quality preference of consumers. (i = 1, 2)	k	The cost coefficient of providing a certain quality. k11, k12, k2, indicate retailer 1’s, retailer 2’s, and manufacturer’s cost coefficient respectively.
${\rho }_2$	Manufacturer’s marginal profit.	cm	Manufacturer’s product quality improvement cost.
${\rho }_{1i}$	Retailer i’s margin. (i = 1, 2)	cri	Retailer service quality improvement cost. (i = 1, 2)
a0i	The market potential sales volume of retailer i. (i = 1, 2)	di(t)	The market demand of retailer i at time t. (i = 1, 2)
λ	Discount rate.	${\alpha }_i$	Memory parameter of quality, ${\alpha }_i$ > 0. (i = 1, 2)

.

In the two-level supply chain system, service quality and product quality are decision variable for the retailers and the manufacturer respectively. Further, this paper proposes three assumptions.

Assumption 1.

The variation of reference product quality and vertical reference service quality with time is characterized by differential equations.

$${\dot{r}}_{fi}(t)={\alpha }_1(f_i(t)-r_{fi}(t))$$

(1)

$$r_{fi}(0)=r_{f0i}$$

$${\dot{r}}_z(t)={\alpha }_2[z(t)-r_z(t)]$$

(2)

$$r_z(0)=r_{z0}$$

where${\alpha }_1$represents memory parameter of service quality;${\alpha }_2$represents memory parameter of product quality.

Assumption 2.

Referring to reference [5,6], consumer demand is affected by z(t), r_fi(t), f(t), r_z(t) and f(t), so the consumer demand function of retailer i is

$$d_i(t)=a_{i0}+{\theta }_1{f}_i(t)+{\chi }_1[f_i(t)-f_j(t)]+{\beta }_1[f_i(t)-r_{fi}(t)]+{\theta }_{2}z(t)+{\beta }_2[z(t)-r_z(t)]$$

(3)

where${\theta }_1$service quality preference of consumers;${\theta }_2$product quality preference of consumers.

Assumption 3.

Quality improvement cost is the increasing function of quality level, which meet the conditions $c_z{}^{″}(z)>0$, $c_r{}^{″}(f_i)>0$. Their cost functions are shown in Equations (4) and (5).

$$c_{fi}=\frac{1}{2}{k}_{1i}{f_i}^2(t) (i=1,2)$$

(4)

$$c_m=\frac{1}{2}{k}_2{z}^2(t)$$

(5)

This paper assumes that manufacturer and retailers make decisions in an infinite time zone to seek their own optimal profits. To sum up, we have members’ the net discounted profits and the whole supply chain’s the net discounted profit.

$$J_{ri}={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda t}[{\rho }_{1i}{d}_i(t)}-\frac{1}{2}{k}_{1i}{f}_i{}^2(t)]dt$$

(6)

$$J_m={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda t}[{\rho }_2(d_1(t)+d_2(t))-\frac{1}{2}{k}_2{z}^2(t)}]dt$$

(7)

$$J_{mr}={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda t}[({\rho }_2+{\rho }_{11})d_1(t)+({\rho }_2+{\rho }_{12})d_2(t)-\frac{1}{2}{k}_{12}{f}_2{}^2(t)-\frac{1}{2}{k}_{11}{f}_1{}^2(t)-\frac{1}{2}{k}_2{z}^2(t)]dt}$$

(8)

The subscripts ri, m and mr represent retailer i, manufacturer and supply chain respectively.

4. Decentralized Decision Making

4.1. Decentralized Decision Making without Cost Sharing (Called DN Mode)

Manufacturer and retailers make decisions independently. Superscript n indicates DN mode. For simplicity, $f$ replaces $f(t)$ and $z$ replaces $z(t)$.

Proposition 1.

Under the DN mode, the optimal service quality and product quality are respectively

$${f_i}^{n\ast }=\frac{{\rho }_{1i}({\theta }_1\lambda +{\chi }_1\lambda +{\chi }_1{\alpha }_1+{\theta }_1{\alpha }_1+{\beta }_1\lambda )}{k_{1i}(\lambda +{\alpha }_1)}, (i=1,2),$$

$$z^{n\ast }=\frac{2{\rho }_2({\theta }_2\lambda +{\theta }_2{\alpha }_2+{\beta }_2\lambda )}{k_2(\lambda +{\alpha }_2)}.$$

Consumers’ vertical reference service quality path is given by:

$${r_{fi}}^{n\ast }(t)=(r_{f0i}-{f_i}^{n\ast })e^{-a_{1}t}+{f_i}^{n\ast }, (i=1,2).$$

And reference product quality path is given by:

$$r_z{}^{n\ast }(t)=(r_z{}_0-z^{n\ast })e^{-a_{2}t}+z^{n\ast }.$$

Proof 

(See Appendix A for the reasoning and proof process). □

According to Proposition 1, we have the following facts. (i) With time t, the reference quality of service and product will eventually be equal to the final service and product quality. Given that ${r_{fi}}^{n\ast }(t)-{f_i}^{n\ast }=(r_{f0i}-{f_i}^{n\ast })e^{-a_{1}t}$, the gap between ${r_{fi}}^{n\ast }(t)$ and ${f_i}^{n\ast }$ is directly proportional to the gap between $r_{f0i}$ and ${f_i}^{n\ast }$. Their gap decreases exponentially. For this model of ${r_z}^{n\ast }(t)-z^{n\ast }=(r_{z0}-z^{n\ast })e^{-a_{2}t}$, the conclusion is still valid. (ii) Different initial reference quality (i.e., $r_{f0i}$, $r_{z0}$) does not affect the final quality, but only the track of the reference quality changing with time. These results were also obtained by He et al. (2018). (iii) Quality is related to the marginal profit of the relevant firm. The higher the marginal profit, the greater the quality. The marginal profit of manufacturer does not affect the service quality of retailers. Similarly, the marginal profit of retailers does not affect the product quality.

According to Proposition 1, the profits of retailer i and manufacturer are:

$$\begin{array}{cc}\hfill {J_{ri}}^{n\ast }=& {\rho }_{1i}[\frac{a_0}{\lambda }+(\frac{{\theta }_1}{\lambda }+\frac{{\chi }_1}{\lambda }+\frac{{\beta }_1}{\lambda +{\alpha }_1}){f_i}^{n\ast }-\frac{{\chi }_1}{\lambda }{f_j}^{n\ast }+(\frac{{\theta }_2}{\lambda }+\frac{{\beta }_2}{\lambda +{\alpha }_2})z^{n\ast }\hfill \\ & -\frac{{\beta }_1{r}_{f0}}{\lambda +{\alpha }_1}-\frac{{\beta }_2{r}_{z0}}{\lambda +{\alpha }_2}]-\frac{1}{2\lambda }{k}_{1i}{f_i}^{n\ast 2}\hfill \end{array}, (i=1,2),$$

(9)

$$\begin{array}{l}{J_m}^{n\ast }=2\frac{{\rho }_2{a}_0}{\lambda }+{\rho }_2(\frac{{\theta }_1}{\lambda }+\frac{{\beta }_1}{\lambda +{\alpha }_1})]({f_1}^{n\ast }+{f_2}^{n\ast })+2{\rho }_2(\frac{{\theta }_2}{\lambda }+\frac{{\beta }_2}{\lambda +{\alpha }_2})z^{n\ast }-\frac{2{\rho }_2{\beta }_2{r}_{z0}^{n\ast }}{\lambda +{\alpha }_2}\\ -\frac{{\rho }_2{\beta }_1}{\lambda +{\alpha }_1}(r_{f01}^{n\ast }+r_{f02}^{n\ast })-\frac{1}{2\lambda }{\varphi }_1{k}_{11}{f_1}^{n\ast 2}(t)-\frac{1}{2\lambda }{\varphi }_2{k}_{12}{f_2}^{n\ast 2}(t)-\frac{1}{2\lambda }{k}_2{z}^{n\ast 2}\end{array}.$$

(10)

4.2. Decentralized Decision Making with Cost Sharing (Called DC Mode)

The manufacturer intends to undertake a certain proportion of the service cost, so as to promote the downstream retailer i to improve the service level and increase product sales. The superscript sm indicates the DC mode. In the DC mode, the decision sequence is described as follows: (i) the manufacturer shares a certain rate cost for the service improvement of the retailer; (ii) after observing the manufacturer’s action, the retailers determine service quality.

The profits of manufacturer and retailer i are respectively

$${J_{ri}}^s={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda t}[{\rho }_{1i}{d}_i(t)}-\frac{1}{2}(1-{\varphi }_i)k_{1i}{f}_i{}^2(t)]dt,$$

(11)

$${J_m}^s={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda t}[{\rho }_2(d_1(t)+d_2(t))-\frac{1}{2}{\varphi }_i{k}_{1i}{f}_i{}^2(t)-\frac{1}{2}{\varphi }_j{k}_{1j}{f_j}^2(t)-\frac{1}{2}{k}_2{z}^2(t)}]dt.$$

(12)

Proposition 2.

Under the DC mode, the optimal service quality of retailer i is given by:

$$f_i{}^{s\ast }=\frac{{\rho }_{1i}({\theta }_1\lambda +{\chi }_1\lambda +{\chi }_1{\alpha }_1+{\theta }_1{\alpha }_1+{\beta }_1\lambda )}{(1-{\varphi }_i)k_{1i}(\lambda +{\alpha }_1)}.$$

And the optimal product quality of manufacturer is given by:

$$z^{s\ast }=\frac{{\rho }_2({\theta }_2\lambda +{\theta }_2{\alpha }_2+{\beta }_2\lambda )}{k_2(\lambda +{\alpha }_2)}.$$

Consumers’ vertical reference service quality path is given by:

$${r_{fi}}^{s\ast }(t)=(r_{f0i}-{f_i}^{s\ast })e^{-a_{1}t}+{f_i}^{s\ast }.$$

(13)

And reference product quality path is given by:

$${r_z}^{s\ast }(t)=(r_{z0}-z^{s\ast })e^{-a_{2}t}+z^{s\ast }.$$

(14)

Proof 

(See Appendix B for the reasoning and proof process). □

Proposition 2 illustrates the following implications: (i) whatever proportion is shared by the manufacturer, the product quality remains unchanged. The higher the proportion, the higher the service quality, which leads to retailers paying more improvement costs. (ii) According to Proposition 2, retailers mainly consider following factors when determining the service quality. The first factor is the marginal profit of retailers ${\rho }_{1i}$; with the increase in marginal profit, retailers will have more enthusiasm to improve quality. The second factor is the share rate ${\varphi }_i$; the sharing rate is inversely proportional to the service quality. The third factor is the reference effect, including vertical reference effect ${\chi }_1$ and horizontal reference effect ${\beta }_1$. The higher the reference effect is, the retailer’s motivation for quality improvement will also increase. (iii) The manufacturer mainly considers two factors when determining the product quality. The first factor is the marginal profit of the manufacturer. The second factor is the reference effect of product quality.

Proposition 3.

Under the DC mode, if $2({\theta }_1\lambda +{\theta }_1{\alpha }_1+\lambda \beta ){\rho }_2>{\rho }_{1i}({\theta }_1\lambda +{\chi }_1\lambda +{\chi }_1{\alpha }_1+{\theta }_1{\alpha }_1+{\beta }_1\lambda )$, then the manufacturer’s optimal service cost share rate is

$${\varphi }_i\ast =\frac{2{\rho }_2[{\theta }_1(\lambda +{\alpha }_1)+\lambda {\beta }_1]-{\rho }_{1i}({\theta }_1\lambda +{\chi }_1\lambda +{\chi }_1{\alpha }_1+{\theta }_1{\alpha }_1+{\beta }_1\lambda )}{2{\rho }_2[{\theta }_1(\lambda +{\alpha }_1)+\lambda {\beta }_1]+{\rho }_{1i}({\theta }_1\lambda +{\chi }_1\lambda +{\chi }_1{\alpha }_1+{\theta }_1{\alpha }_1+{\beta }_1\lambda )}, (i=1,2).$$

(15)

Otherwise, ${\varphi }^{m\ast }=0$.

Proof 

(SeeAppendix C for the reasoning and proof process). □

The Proposition 2 are substituted into Equations (11) and (12), and the profits of retailer i and manufacturer is simplified as follows.

$$\begin{array}{l}{J_{ri}}^{s\ast }={\rho }_{1i}[\frac{a_0}{\lambda }+(\frac{{\theta }_1}{\lambda }+\frac{{\chi }_1}{\lambda }+\frac{{\beta }_1}{\lambda +{\alpha }_1}){f_i}^{s\ast }-\frac{{\chi }_1}{\lambda }{f_j}^{s\ast }+(\frac{{\theta }_2}{\lambda }+\frac{{\beta }_2}{\lambda +{\alpha }_2})z^{s\ast }-\frac{{\beta }_1{r}_f{}_0}{\lambda +{\alpha }_1}-\frac{{\beta }_2{r}_z{}_0}{\lambda +{\alpha }_2}]\\ -\frac{1}{2\lambda }(1-{\varphi }_i)k_{1i}{f}_i{{}^{s\ast }}^2(t)\end{array}$$

(16)

$$\begin{array}{l}{J_m}^{s\ast }=2\frac{{\rho }_2{a}_0}{\lambda }+{\rho }_2(\frac{{\theta }_1}{\lambda }+\frac{{\beta }_1}{\lambda +{\alpha }_1})]({f_1}^{s\ast }+{f_2}^{s\ast })+2{\rho }_2(\frac{{\theta }_2}{\lambda }+\frac{{\beta }_2}{\lambda +{\alpha }_2})z^{s\ast }-\frac{2{\rho }_2{\beta }_2{r}_{z0}^{s\ast }}{\lambda +{\alpha }_2}\\ -\frac{{\rho }_2{\beta }_1}{\lambda +{\alpha }_1}(r_{f01}^{s\ast }+r_{f02}^{s\ast })-\frac{1}{2\lambda }{\varphi }_1{k}_{11}{f_1}^{s\ast 2}(t)-\frac{1}{2\lambda }{\varphi }_2{k}_{12}{f_2}^{s\ast 2}(t)-\frac{1}{2\lambda }{k}_2{z}^{s\ast 2}\end{array}$$

(17)

Corollary 1.

Under the DC mode, the share rate is positively correlated with the marginal profit of upstream manufacturer ($d{\varphi }^{m\ast }/d{\rho }_2>0$). This is because the higher the marginal profit of the manufacturer, in order to obtain more revenue, the manufacturer encourages the retailer i to improve the service quality by undertaking more service costs of retailers. Although the manufacturer will undertake more service costs, by weighing the benefits and costs, the income of the manufacturer will still increase. On the contrary, the share rate is negatively correlated with the marginal profits of downstream retailers ($d{\varphi }^{m\ast }/d{\rho }_{1i}<0$). The higher the marginal profits of retailers, the higher the service quality costs.

5. Centralized Decision Making (Called CD Mode)

Under the CD mode, the manufacturer and the two retailers make decisions as a whole, and maximize the profits of the supply chain by designing the optimal quality. Superscript c represents the CD mode.

Proposition 4.

Under the CD mode, the optimal service quality and product quality are respectively

$$f_i{}^{c\ast }=\frac{({\rho }_2+{\rho }_{1i})[{\theta }_1(\lambda +{\alpha }_1)+{\beta }_1\lambda ]}{k_{1i}(\lambda +{\alpha }_1)},$$

$$z^{c\ast }=\frac{(2{\rho }_2+{\rho }_{11}+{\rho }_{12})[{\theta }_2(\lambda +{\alpha }_2)+{\beta }_2\lambda ]}{k_2(\lambda +{\alpha }_2)}.$$

Consumers’ vertical reference service quality path is given by:

$${r_{fi}}^{c\ast }(t)=(r_{f0i}-{f_i}^{c\ast })e^{-a_{1}t}+{f_i}^{c\ast }.$$

And reference product quality path is given by:

$${r_z}^{c\ast }(t)=(r_{z0}-z^{c\ast })e^{-a_{2}t}+z^{c\ast }.$$

Proof 

(See Appendix D for the reasoning and proof process). □

Substituting Proposition 4 into Equation (8), we obtain the profit of the supply chain, as shown below.

$$\begin{array}{l}{J_m}^{c\ast }=2\frac{{\rho }_2{a}_0}{\lambda }+{\rho }_2(\frac{{\theta }_1}{\lambda }+\frac{{\beta }_1}{\lambda +{\alpha }_1})]({f_1}^{c\ast }+{f_2}^{c\ast })+2{\rho }_2(\frac{{\theta }_2}{\lambda }+\frac{{\beta }_2}{\lambda +{\alpha }_2})z^{c\ast }-\frac{2{\rho }_2{\beta }_2{r}_{z0}^{c\ast }}{\lambda +{\alpha }_2}\\ -\frac{{\rho }_2{\beta }_1}{\lambda +{\alpha }_1}(r_{f01}^{c\ast }+r_{f02}^{c\ast })-\frac{1}{2\lambda }{\varphi }_1{k}_{11}{f_1}^{c\ast 2}(t)-\frac{1}{2\lambda }{\varphi }_2{k}_{12}{f_2}^{c\ast 2}(t)-\frac{1}{2\lambda }{k}_2{z}^{c\ast 2}\end{array}.$$

(18)

By comparing and analyzing the optimal decisions, the following Corollary 2 is obtained.

Corollary 2.

(1)

$z^{c\ast }>z^{s\ast }=z^{n\ast }$;

(2)

When a certain condition is met, then ${f_i}^{c\ast }>{f_i}^{s\ast }$, otherwise ${f_i}^{s\ast }\le {f_i}^{n\ast }$;

(3)

When a certain condition is met, then ${f_i}^{s\ast }>{f_i}^{n\ast }$, otherwise ${f_i}^{s\ast }={f_i}^{n\ast }$;

(4)

When a certain condition is met, then ${f_i}^{c\ast }>{f_i}^{n\ast }$, otherwise ${f_i}^{c\ast }\le {f_i}^{n\ast }$.

Proof 

(See Appendix E for the reasoning and proof process). □

Comparing the optimal strategies under the three decision-making modes, we can find that the optimal strategy of product quality under the CD mode is greater than that under the DC mode. Compared with the DN mode, DC mode cannot promote the manufacturer to improve product quality. Different from previous studies, the service quality under the CD mode is not the best, and the service quality under the DC mode is not the lowest. Under certain conditions, the service quality under the CD mode may be higher than that under the DC mode, or lower than that under the DN mode.

By comparing and analyzing the optimal profits, the following Corollary 3 is obtained.

Corollary 3.

The supply chain profits under different decision-making modes have the following relationships:

• If ${\Delta }_1+{\Delta }_2>0$, then ${J_{mr}}^c>{J_{mr}}^s$, otherwise ${J_{mr}}^c\le {J_{mr}}^s$.
• If ${\Delta }_3+{\Delta }_4>0$, then ${J_{mr}}^s>{J_{mr}}^n$, otherwise ${J_{mr}}^s\le {J_{mr}}^n$.
• If ${\Delta }_5+{\Delta }_6>0$, then ${J_{mr}}^c>{J_{mr}}^n$, otherwise ${J_{mr}}^c\le {J_{mr}}^n$.

Proof 

(See Appendix F for the reasoning and proof process). □

According to Corollary 3, we can find that the supply chain profit under the DC mode is not the highest, and the supply chain profit under the DN mode is not the lowest. Under certain conditions, the profit of supply chain under the DC mode may be higher than or lower than that under the DC and DC mode.

Corollary 4.

Under the three different decision-making modes, the greater the quality preference, the better the retailer i and manufacturer can improve quality; the smaller the memory parameter, that is, the longer the consumer’s memory of quality tends to be. In addition, the retailers i and the manufacturer are more motivated to improve quality; the larger the consumers’ reference effect of quality is, the retailer i and manufacturer have more incentive to improve quality.

6. Simulation Analysis

Through simulation, the difference of supply chain profits under different modes is analyzed to verify the relevant conclusions of this paper. We assume that the parameters in the models are $\lambda =0.1$, ${\rho }_2=5$, ${\rho }_{11}=1$, ${\rho }_{12}=2$, ${\alpha }_1=0.1$, ${\alpha }_2=10$, ${\beta }_1=0.2$, ${\beta }_2=0.01$, ${\theta }_1=0.2$, ${\theta }_2=0.01$, ${\chi }_1=0.1$, $k_{11}=k_{12}=1$, $k_2=10$, $r_{z0}=3$, $r_{f10}=r_{f20}=2$, $a_{10}=a_{20}=2$.

Figure 1 shows that when the initial service reference level is low, that is, $r_{f01}=0.1$, the profits of supply chain are positively correlated with the vertical reference effect under the CD and DC modes. This is because when the initial service reference level is low, the retailer i’s service strategy is higher than consumers’ expectations, which has a positive impact on demand and increases the profits of the supply chain; under the DN mode the supply chain’s profit first decreases and then increases with increases in the vertical reference effect ${\beta }_1$. This is because when the vertical reference effect ${\beta }_1$ is low, the retailer i’s service level is also low, resulting in a negative effect on demand and reducing supply chain profits. However, with the increase in ${\beta }_1$, the service level set by retailer i is higher, which increases the overall profit of the supply chain.

When the initial service reference level is high, that is, $r_{f01}=2$, the supply chain profit under the CD and DC modes decreases first and then increases with the increase in the vertical reference effect ${\beta }_1$, because when the vertical reference effect ${\beta }_1$ is low, the service level set by retailer i is also low, which results in a negative effect on demand and reducing the supply chain profit. However, with the increase in ${\beta }_1$, the service level set by retailer i has also been improved, resulting in a positive effect on demand and increasing the supply chain profit. Under the DN mode, supply chain profit is negatively correlated with the vertical reference effect ${\beta }_1$. This is because when the initial service reference level is high, consumers expect the initial reference service level of retailer i to be high. Even if the vertical reference effect ${\beta }_1$ increases, the service level set by retailer i is difficult to meet the requirements of consumers. In addition, it can be seen from Figure 1 that when the initial service quality reference level is low, the supply chain profits are greater than those when the initial service quality reference level is high.

Figure 2 shows that when the initial product reference level is low, that is, $r_{z0}=0.1$, the supply chain profits are less sensitive to the changes in the reference effect ${\beta }_2$. When the initial product quality reference level is high, that is, $r_{z0}=3$, the supply chain profits are negatively correlated with the reference effect of product quality. This is because when the initial reference quality of product is high, the product quality set by the manufacturer can hardly meet the requirements of consumers. Even if the product quality is improved, it still fails to meet the expectations of consumers, it is not conducive to improving the supply chain profits.

Figure 3a–c show the changes in supply chain profit with horizontal reference effect under any two decision-making modes, respectively. Based on the benchmark data, Figure 3a shows that when the horizontal reference effect ${\chi }_1$ is within the range of [0,0.3], the supply chain profit under the DC mode is less than or equal to the supply chain profit under the CD mode. When the horizontal reference effect ${\chi }_1$ is within the range of (0.3, 1), the supply chain profit under the DC mode is greater than that under the CD mode. It can be seen from Figure 3b that when the horizontal reference effect ${\chi }_1$ is within the range of (0,0.65), the supply chain profit under the DC mode is greater than or equal to the supply chain profit under the DN mode. When the horizontal reference effect ${\chi }_1$ is within the range of (0.65, 1), the supply chain profit under the DN mode is greater than that under the DC mode. Reset ${\rho }_2=1$,${\rho }_{11}=3$, and other data are consistent with the benchmark data. It can be seen from Figure 3c that when the horizontal reference effect ${\chi }_1$ is within the range of [0,0.065] and [0.135,1], the supply chain profit under the DN mode is less than or equal to the supply chain profit under the DC mode. When the horizontal reference effect ${\chi }_1$ is within the range of (0.065, 0.135), the supply chain profit under the DN mode is greater than that under the CD mode.

Figure 4 shows the changes in supply chain profits with preference ${\theta }_1$ for service quality under different decision-making modes. It can be seen that the supply chain profits increase with the increase in service quality preference ${\theta }_1$. This is because the higher the preference coefficient, the better the service quality of retailer i, and then increase the profits of the supply chain.

Figure 5 shows the changes in supply chain profits with consumers’ preference ${\theta }_2$ under different decision-making modes. It can be seen that the supply chain profits increase with the increase in preference ${\theta }_2$. This is because the higher the preference coefficient, the better the quality, and this increases the supply chain profits.

Finally, the changes in the reference quality with time are plotted, as shown in Figure 6.

It can be seen from Figure 6 that under the DN mode, DC mode and CD mode, when the initial reference level is high, the reference service quality decreases with time and tends to a stable value. When the initial reference level is low, the reference service quality increases with time and tends to a stable value. The stable value of the reference service quality is equal to the optimal service quality. In the same decision-making mode, the difference of the initial reference service quality does not affect the reference service quality, but only affects the trajectory of the reference service quality level with time.

7. Summary

In the competitive environment of retailers, consumers’ reference effects are considered together, in which consumer reference effect of service quality is divided into horizontal and vertical reference effect. By comparing the optimal decisions and the supply chain profits under the three decision-making modes, the following conclusions are obtained.

(1)

According to the above, retailers mainly consider following factors when making decisions. The first factor is the marginal profit of retailers. The second factor is the share rate; the sharing rate is inversely proportional to the service quality. The third factor is the reference effect, including vertical reference effect and horizontal reference effect. The higher the reference effect is, the retailer’s motivation for quality improvement will also increase. The fourth factor is initial reference quality. Different initial reference quality will affect the relationship between reference effect and supply chain profit. Manufacturers mainly consider three factors when making decisions. That is, the marginal profit of manufacturers, the reference effect of product quality and the initial reference quality.

(2)

Comparing the quality under different modes, we draw the following implications: (i) the optimal product quality under the CD mode is greater than that under the DC mode. (ii) DC mode cannot promote the manufacturer to improve product quality. (iii) Different from previous studies, the service quality under the CD mode is not higher than other decisions. (iv) Under certain conditions, the service quality under the CD mode may be higher than or lower than that under the DC mode and DN mode. The above management enlightenment helps enterprise managers have a deeper understanding of quality decision making under different modes, and adjust quality strategy when certain conditions change (such as consumer sensitivity coefficient, sharing proportion and marginal profit).

(3)

By comparing the profits of supply chain under different modes, enterprises can gain the following insights: (i) The profits of supply chain under the CD mode are not the highest, which is contrary to the traditional view; the profit of supply chain under the DN mode is not the lowest. (ii) When the reference effect of service quality level meets certain conditions, the supply chain profit under the CD mode is lower than that under the DN mode and DC mode. (iii) The share rate is positively correlated with the manufacturer’s marginal profit; on the contrary, the share rate is negatively correlated with the marginal profit of retailer i. The above insights will help managers better choose their own cooperation mode under the given conditions.

From the above conclusions, we can draw some research implications.

(1)

Enterprises should realize that the way to increase profits is not limited to reducing costs, and vigorously improving product quality and service quality is also a more effective way, such as enhancing the reliability and performance of products, extending the service life of products, creating an experiential shopping environment and improving the service ability of sellers.

(2)

When making quality decisions, enterprises should not only collect their own data, but also obtain the reference behavior data of consumers through market questionnaires, especially the horizontal reference effect data in the case of retailer competition, so that enterprises can make more accurate quality decisions. It is also necessary to fully investigate the demand characteristics of consumers for high-quality products and high-quality service levels. When consumers have strong product reference effect or service reference effect (vertical), the more enterprises need to increase their investment in product quality and service quality.

(3)

Generally speaking, alliance is a way for enterprises to achieve resource complementarity and gain greater competitive advantages. However, whether it is established in the case of retailer competition depends on the strength of consumers’ horizontal reference effect. Therefore, before making decisions, supply chain enterprises need to conduct in-depth research on market consumers in order to select a more appropriate cooperation mode.

(4)

Under certain conditions, suppliers can share part of the service cost with retailers to reduce the economic pressure of retailers, so that retailers have enough power to improve service quality to expand market demand.

This paper considers the situation of retailers’ competition. Future research could consider manufacturers’ competition, and explore the impact of horizontal reference effect and vertical reference effect on decision making and profits.