# Research on Quality Decisions and Coordination with Reference Effect in Dual-Channel Supply Chain

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Development

_{f}(t) is the consumer’s reference service quality level over time t, f(t) is the retailer’s actual service quality level, r

_{z}(t) is the consumer’s reference product quality level over time t, and z(t) is the manufacturer’s actual product quality level. α

_{1}and α

_{2}indicate service quality memory parameter and product quality memory parameter, respectively, α

_{1}> 0, α

_{2}> 0.

#### 3.1. Model D

**Proposition**

**1.**

#### 3.2. Model DC

**Proposition**

**2.**

**Proposition**

**3.**

**Property 1.**

**Property 2.**

#### 3.3. Model C

**Proposition**

**4.**

## 4. Comparative Analysis

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

## 5. Supply Chain Coordination Mechanism (Model CS)

**Proposition**

**5.**

## 6. Numerical Analysis

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**1.**

## Appendix B

**Proof**

**of**

**Proposition**

**3.**

## Appendix C

#### Appendix C.1

**Proof**

**of**

**Corollary**

**1.**

#### Appendix C.2

**Proof**

**of**

**Corollary**

**2.**

#### Appendix C.3

**Proof**

**of**

**Corollary**

**3.**

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**Figure 3.**The impact of effectiveness parameters on profits. (

**a**) The impact of ${\theta}_{ft}$ on profits (

**b**) The impact of ${\theta}_{zt}$ on profits (

**c**) The impact of ${\theta}_{ze}$ on profits.

**Figure 4.**The impact of initial reference quality on profits. (

**a**) The impact of ${r}_{f0}$ on profits. (

**b**) The impact of ${r}_{z0}$ on profits.

**Figure 5.**Impact of reference effect on profits. (

**a**) The impact of ${\beta}_{ft}$ on profits. (

**b**) The impact of ${\beta}_{fe}$ on profits. (

**c**) The impact of ${\beta}_{ze}$ on profits. (

**d**) The impact of ${\beta}_{zt}$ on profits.

**Figure 6.**The change of reference quality with time. (

**a**) The change of ${r}_{f}$ with time. (

**b**) The change of ${r}_{z}$ with time.

Notation | Description |
---|---|

f(t) | Retailer’s service quality at time t, which is a decision variable. |

z(t) | Manufacturer’s product quality at time t, which is a decision variable. |

${\theta}_{ft}$ | Service quality’s effectiveness on market demand in traditional channel. |

${\theta}_{zt}$ | Product quality’s effectiveness on market demand in traditional channel. |

${\theta}_{ze}$ | Product quality’s effectiveness on market demand in online channel. |

${\rho}_{1}$ | Retailer’s marginal profit in traditional channel. |

${\rho}_{2}$ | Manufacturer’s marginal profit in traditional channel. |

${\rho}_{3}$ | Manufacturer’s marginal profit in online channel. |

k_{1} | Cost parameter associated with service quality improvement by the retailer. |

k_{2} | Cost parameter associated with product quality improvement by the manufacturer. |

λ | Discount rate. λ > 0. |

${\varphi}^{d}$ | Cost-sharing rate provided under decentralized decision making. |

${\varphi}^{cs}$ | Cost-sharing rate provided under coordinating mechanism. |

Model D | Model DC | Model C | |
---|---|---|---|

f | ${f}^{n\ast}=\frac{{\rho}_{1}({\theta}_{ft}+{\beta}_{ft})(\lambda +{\alpha}_{1})-{\rho}_{1}{\beta}_{ft}\text{}{\alpha}_{1}}{(\lambda +{\alpha}_{1}){k}_{1}}$ | ${f}^{d\ast}=\frac{{\rho}_{1}({\theta}_{ft}+{\beta}_{ft})(\lambda +{\alpha}_{1})-{\rho}_{1}{\beta}_{ft}\text{}{\alpha}_{1}}{(1-{\varphi}^{d})(\lambda +{\alpha}_{1}){k}_{1}}$ | ${f}^{c\ast}=\frac{({\rho}_{2}+{\rho}_{1})({\theta}_{ft}+{\beta}_{ft})(\lambda +{\alpha}_{1})+{\rho}_{3}{\beta}_{fe}\lambda -({\rho}_{2}+{\rho}_{1}){\beta}_{ft}{\alpha}_{1}}{(\lambda +{\alpha}_{1}){k}_{1}}$ |

z | ${z}^{n\ast}=\frac{\left\{\begin{array}{l}{\rho}_{2}[({\alpha}_{2}+\lambda ){\theta}_{zt}+\lambda {\beta}_{zt}]\\ +{\rho}_{3}[({\alpha}_{2}+\lambda ){\theta}_{ze}+\lambda {\beta}_{ze}]\end{array}\right\}}{(\lambda +{\alpha}_{2}){k}_{2}}$ | ${z}^{d\ast}=\frac{\left\{\begin{array}{l}{\rho}_{2}[({\alpha}_{2}+\lambda ){\theta}_{zt}+\lambda {\beta}_{zt}]\\ +{\rho}_{3}[({\alpha}_{2}+\lambda ){\theta}_{ze}+\lambda {\beta}_{ze}]\end{array}\right\}}{(\lambda +{\alpha}_{2}){k}_{2}}$ | ${z}^{c\ast}=\frac{\begin{array}{l}[({\rho}_{2}+{\rho}_{1})({\theta}_{zt}+{\beta}_{zt})(\lambda +{\alpha}_{2})+{\rho}_{3}({\theta}_{ze}+\\ {\beta}_{ze})(\lambda +{\alpha}_{2})-({\rho}_{2}+{\rho}_{1}){\beta}_{zt}{\alpha}_{2}-{\rho}_{3}{\beta}_{ze}{\alpha}_{2}]\end{array}}{(\lambda +{\alpha}_{2}){k}_{2}}$ |

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**MDPI and ACS Style**

Xideng, Z.; Bing, X.; Fei, X.; Yu, L.
Research on Quality Decisions and Coordination with Reference Effect in Dual-Channel Supply Chain. *Sustainability* **2020**, *12*, 2296.
https://doi.org/10.3390/su12062296

**AMA Style**

Xideng Z, Bing X, Fei X, Yu L.
Research on Quality Decisions and Coordination with Reference Effect in Dual-Channel Supply Chain. *Sustainability*. 2020; 12(6):2296.
https://doi.org/10.3390/su12062296

**Chicago/Turabian Style**

Xideng, Zhou, Xu Bing, Xie Fei, and Li Yu.
2020. "Research on Quality Decisions and Coordination with Reference Effect in Dual-Channel Supply Chain" *Sustainability* 12, no. 6: 2296.
https://doi.org/10.3390/su12062296