# A Quality Decision Model Considering the Delay Effects in a Dual-Channel Supply Chain

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Dual-Channel Supply Chain Study

#### 2.2. Research on Supply Chain Quality Issues

#### 2.3. Study on Delay Effects

## 3. Model Development

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

## 4. Decentralized Decision Making

**Proposition**

**1.**

**Inference**

**1.**

**Inference**

**2.**

**Inference**

**3.**

## 5. Centralized Decision-Making

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4:**

- (1)
- In the case of a delay time satisfaction relationship${d}_{s}\in RS1$, then${d}_{z}\ge 0$, the overall profit of the supply chain under centralized decision-making is not greater than that under decentralized decision-making, and decentralized decision-making is the optimal decision-making mode of the supply chain.
- (2)
- When the delay time satisfies the relationship${d}_{s}\in RS2$, if${f}_{10}\le {f}_{11}$, then${d}_{z}\in RZ1$; alternatively, when the delay time satisfies the relationship${d}_{s}\in RS2$, if${f}_{11}<{f}_{10}<{f}_{12}$, then${d}_{z}\in RZ3$; in the case of the delay time satisfaction relationship${d}_{s}\in RS2$, if${f}_{12}\le {f}_{10}$, then${d}_{z}>0$, it can be observed that the overall profit of the supply chain under decentralized decision-making is not less than that under centralized decision-making, and the supply chain should adopt decentralized decision-making mode.
- (3)
- When the delay time satisfies the relationship${d}_{s}\in RS2$, if${f}_{10}\le {f}_{11}$, then${d}_{z}\in RZ2$; alternatively, when the delay time satisfies the relationship${d}_{s}\in RS2$, if${f}_{11}<{f}_{10}<{f}_{12}$, then${d}_{z}\in RZ4$, the overall profit of the supply chain under decentralized decision-making is less than that under centralized decision-making, and the supply chain should adopt a centralized decision-making mode.
- (4)
- In the case of delay time satisfaction relationship${d}_{s}\in RS3$, if${f}_{10}\le {f}_{12}$, then${d}_{z}\in RZ5$, the overall profit of the supply chain under decentralized decision-making is not greater than that under centralized decision-making, and the supply chain should adopt a centralized decision-making mode.
- (5)
- In the case of delay time satisfaction${d}_{s}\in RS3$, if${f}_{10}\le {f}_{12}$, then${d}_{z}\in RZ6$, or in the case of delay time satisfaction${d}_{s}\in RS3$, if${f}_{10}>{f}_{12}$, then${d}_{z}>0$, the overall profit of the supply chain under decentralized decision-making is greater than that under centralized decision-making, and the supply chain should adopt a decentralized decision-making mode.

## 6. Numerical Analysis

## 7. Conclusions and Discussion

#### 7.1. Impact of Corporate Profits on Quality Decisions

#### 7.2. Impact of Delay Effects on Quality Decisions

#### 7.3. Impact of Decision-Making Approach on Quality, Profit, and Goodwill Decisions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**1.**

## Appendix B

**Proof**

**of**

**Proposition**

**2.**

## Appendix C

**Proof**

**of**

**Proposition**

**3.**

## Appendix D

**Proof**

**of**

**Proposition**

**4.**

- (1)
- If ${y}_{2}\le -\frac{{X}_{1}{}^{2}}{4{X}_{2}}$, namely, ${d}_{s}\in RS1=\{{d}_{s}|{y}_{2}+\frac{{X}_{1}{}^{2}}{4{X}_{2}}\le 0\}$. When ${d}_{z}\ge 0$, obtain ${J}_{MR}\le {J}_{M}+{J}_{R}$.
- (2)
- If $-\frac{{X}_{1}{}^{2}}{4{X}_{2}}<{y}_{2}<0$, namely, ${d}_{s}\in RS2=\{{d}_{s}|{y}_{2}+\frac{{X}_{1}{}^{2}}{4{X}_{2}}>0\}\cup \{{d}_{s}|{y}_{2}<0\}$ then, solve the equation $J({d}_{z},{d}_{s})={X}_{1}{f}_{1}({d}_{z})-{X}_{2}{f}_{1}{({d}_{z})}^{2}+{y}_{2}$ and obtain two roots of the equation, namely:$$\{\begin{array}{l}{f}_{11}=\frac{{X}_{1}-\sqrt{{X}_{1}{}^{2}+4{X}_{2}{y}_{2}}}{2{X}_{2}}\\ {f}_{12}=\frac{{X}_{1}+\sqrt{{X}_{1}{}^{2}+4{X}_{2}{y}_{2}}}{2{X}_{2}}\end{array}$$

- If ${f}_{10}\le {f}_{11}$, when $f({d}_{z})\ge {f}_{12}$, obtain ${J}_{MR}\le {J}_{M}+{J}_{R}$, which meets:$${d}_{z}\in RZ1=\{{d}_{z}|0\le {d}_{z}\le \frac{1}{\delta}\mathrm{ln}[\frac{\lambda +\delta}{b\gamma}({f}_{11}-{\theta}_{2})]\}\cup \{{d}_{z}|{d}_{z}\ge \frac{1}{\delta}\mathrm{ln}[\frac{\lambda +\delta}{b\gamma}({f}_{12}-{\theta}_{2})]\}$$

- 2.
- If ${f}_{11}<{f}_{10}<{f}_{12}$, when $f({d}_{z})\ge {f}_{12}$, obtain ${J}_{MR}\le {J}_{M}+{J}_{R}$. Obtain ${J}_{MR}\ge {J}_{M}+{J}_{R}$, namely:$${d}_{z}\in RZ3=\{{d}_{z}|{d}_{z}\ge \frac{1}{\delta}\mathrm{ln}[\frac{\lambda +\delta}{b\gamma}({f}_{1}{}_{2}-{\theta}_{2})]\}$$

- 3.
- If ${f}_{12}\le {f}_{10}$, when $f({d}_{z})\ge {f}_{10}$, namely ${d}_{z}>0$, obtain ${J}_{MR}\le {J}_{M}+{J}_{R}$.

- (3)
- If ${y}_{2}>0$, namely, ${d}_{s}\in RS3=\{{d}_{s}|{y}_{2}>0\}$.

- If ${f}_{10}\le {f}_{12}$, when ${f}_{10}\le f({d}_{z})\le {f}_{12}$, obtain ${J}_{MR}\ge {J}_{M}+{J}_{R}$, which meets:$${d}_{z}\in RZ5=\{{d}_{z}|0\le {d}_{z}\le \frac{1}{\delta}\mathrm{ln}[\frac{\lambda +\delta}{b\gamma}({f}_{12}-{\theta}_{2})]\}$$

- 2.
- If ${f}_{10}>{f}_{12}$, when ${f}_{10}<f({d}_{z})$, namely ${d}_{z}>0$, obtain ${J}_{MR}<{J}_{M}+{J}_{R}$. □

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**Figure 5.**(

**A**) Profit impact of delay time on manufacturers and retailers under decentralized decision-making. (

**B**) Profit impact of delay time on manufacturers and retailers under decentralized decision-making.

**Figure 6.**(

**A**) Impact of delay time on supply chain profit under two decisions. (

**B**) Impact of delay time on supply chain profit under two decisions.

Literature | Publish Time | Product Quality as a Decision Variable | Service Quality as a Decision Variable | Using Differential Gaming Methods | Consider the Delay Effect |
---|---|---|---|---|---|

Chen et al. [47] | 2017 | no | no | yes | yes |

Huang et al. [48] | 2018 | yes | yes | no | no |

Guo et al. [49] | 2019 | no | no | yes | yes |

Hu [43] | 2019 | no | no | yes | yes |

Malik and Kim [36] | 2020 | yes | yes | no | no |

Ahmed et al. [50] | 2021 | no | no | yes | yes |

Li and Mishra [23] | 2021 | yes | yes | no | no |

Prasenjit and Tarun [21] | 2021 | yes | no | yes | no |

Kuppusamy [41] | 2021 | yes | yes | yes | no |

Qiu et al. [25] | 2022 | yes | no | yes | no |

Lin and Wang [34] | 2022 | no | yes | no | yes |

This paper | 2022 | yes | yes | yes | yes |

Symbol | Description | Symbol | Description |
---|---|---|---|

$z(t)$ | The manufacturer’s product quality level at time t, which is the decision variable. | ${\rho}_{MF}$ | Marginal profit of offline channels for manufacturers. |

$f(t)$ | The service quality level of retailers at time t, which is a decision variable. | ${\rho}_{R}$ | Marginal profit for retailers. |

$\varphi (t)$ | The proportion of the manufacturer bearing the retailer’s service quality cost at time t, which is a decision variable. | $d(t)$ | The market demand at time t. |

$G(t)$ | The goodwill of the product at the time t. | ${a}_{i}$ | The potential sales volume in the market before the improvement of product quality and service quality, ${a}_{i}>0$. $i=1$ indicates offline channel and $i=2$ indicates online channel. |

${d}_{z}$ | Delay in manufacturer’s improvement of product quality, which in turn affects product goodwill. | $b$ | The sensitivity coefficient of product goodwill to product market demand, $b>0$. |

${d}_{s}$ | Retailers improve service quality, which in turn affects the delay of product goodwill. | ${\theta}_{1}$ | The sensitivity coefficient of the retailer’s service quality level to the product market demand, ${\theta}_{1}>0$. |

$\gamma $ | The extent to which improvements in product quality affect the rate of change in goodwill. | ${\theta}_{2}$ | The sensitivity coefficient of the manufacturer’s product quality level to the product market demand, ${\theta}_{2}>0$. |

$\beta $ | The extent to which improvements in service quality affect the rate of change in goodwill. | ${k}_{1}$ | The manufacturer’s quality cost coefficient, ${k}_{1}>0$. |

$\lambda $ | Discount rate. | ${k}_{2}$ | The retailer’s quality cost coefficient, ${k}_{2}>0$. |

${\rho}_{MO}$ | Marginal profit of online channels for manufacturers. | $\delta $ | The decay rate of product goodwill, $\delta >0$. |

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## Share and Cite

**MDPI and ACS Style**

Zhan, L.; Shu, H.; Zhou, X.; Lin, X.
A Quality Decision Model Considering the Delay Effects in a Dual-Channel Supply Chain. *Sustainability* **2022**, *14*, 6240.
https://doi.org/10.3390/su14106240

**AMA Style**

Zhan L, Shu H, Zhou X, Lin X.
A Quality Decision Model Considering the Delay Effects in a Dual-Channel Supply Chain. *Sustainability*. 2022; 14(10):6240.
https://doi.org/10.3390/su14106240

**Chicago/Turabian Style**

Zhan, Lizhen, Hui Shu, Xideng Zhou, and Xiaowei Lin.
2022. "A Quality Decision Model Considering the Delay Effects in a Dual-Channel Supply Chain" *Sustainability* 14, no. 10: 6240.
https://doi.org/10.3390/su14106240