# Stochastic Differential Game in the Closed-Loop Supply Chain with Fairness Concern Retailer

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Model Setup

## 3. Benchmark: NF Model-No Fairness Concern

#### 3.1. The Feedback Equilibrium Strategies

**Proposition 1.**

**Proof.**

#### 3.2. The Evolutionary Path of the Return Rate

**Proposition 2.**

#### 3.3. The Numerical Analysis with No Fairness Concern

## 4. Equilibrium Strategies with Fairness Concern Models

#### 4.1. GF Model: Gap Fairness Concern

**Proposition 3.**

**Proposition 4.**

#### 4.2. SF Model: Self-Due Fairness Concern

**Lemma 1.**

**Proof.**

**Proposition 5.**

**Proposition 6.**

**Proposition 7.**

## 5. Numerical Analysis between Different Fairness Type

## 6. Coordinate Contract

#### 6.1. Centralized Model

**Proposition 8.**

#### 6.2. Coordinate Contract

**Proposition 9.**

**Proof.**

## 7. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

## References

- Qiang, Q.P. The closed-loop supply chain network with competition and design for remanufactureability. J. Clean. Prod.
**2015**, 105, 348–356. [Google Scholar] [CrossRef] - Savaskan, R.C.; Van Wassenhove, L.N. Reverse channel design: The case of competing retailers. Manag. Sci.
**2006**, 52, 1–14. [Google Scholar] [CrossRef][Green Version] - Qiang, Q.; Ke, K.; Anderson, T.; Dong, J. The closed-loop supply chain network with competition, distribution channel investment, and uncertainties. Omega
**2013**, 41, 186–194. [Google Scholar] [CrossRef] - Shi, J.; Zhang, G.; Sha, J. Optimal production planning for a multi-product closed loop system with uncertain demand and return. Comput. Oper. Res.
**2011**, 38, 641–650. [Google Scholar] [CrossRef][Green Version] - Tan, Y.; Guo, C. Research on Two-Way Logistics Operation with Uncertain Recycling Quality in Government Multi-Policy Environment. Sustainability
**2019**, 11, 882. [Google Scholar] [CrossRef][Green Version] - Huang, Z.; Nie, J.; Tsai, S.B. Dynamic collection strategy and coordination of a remanufacturing closed-loop supply chain under uncertainty. Sustainability
**2017**, 9, 683. [Google Scholar] [CrossRef][Green Version] - Cao, J.; Chen, X.; Zhang, X.; Gao, Y.; Zhang, X.; Kumar, S. Overview of remanufacturing industry in China: Government policies, enterprise, and public awareness. J. Clean. Prod.
**2020**, 242, 118450. [Google Scholar] [CrossRef] - Fehr, E.; Schmidt, K.M. A theory of fairness, competition, and cooperation. Q. J. Econ.
**1999**, 114, 817–868. [Google Scholar] [CrossRef] - Cui, T.H.; Raju, J.S.; Zhang, Z.J. Fairness and channel coordination. Manag. Sci.
**2007**, 53, 1303–1314. [Google Scholar] - Loch, C.H.; Wu, Y. Social preferences and supply chain performance: An experimental study. Manag. Sci.
**2008**, 54, 1835–1849. [Google Scholar] [CrossRef][Green Version] - Ho, T.H.; Su, X. Peer-induced fairness in games. Am. Econ. Rev.
**2009**, 99, 2022–2049. [Google Scholar] [CrossRef][Green Version] - Atasu, A.; Sarvary, M.; Van Wassenhove, L.N. Remanufacturing as a marketing strategy. Manag. Sci.
**2008**, 54, 1731–1746. [Google Scholar] [CrossRef][Green Version] - Govindan, K.; Soleimani, H.; Kannan, D. Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future. Eur. J. Oper. Res.
**2015**, 240, 603–626. [Google Scholar] [CrossRef][Green Version] - Govindan, K.; Soleimani, H. A review of reverse logistics and closed-loop supply chains: A Journal of Cleaner Production focus. J. Clean. Prod.
**2017**, 142, 371–384. [Google Scholar] [CrossRef] - Savaskan, R.C.; Bhattacharya, S.; Van Wassenhove, L.N. Closed-loop supply chain models with product remanufacturing. Manag. Sci.
**2004**, 50, 239–252. [Google Scholar] [CrossRef][Green Version] - Huang, M.; Song, M.; Lee, L.H.; Ching, W.K. Analysis for strategy of closed-loop supply chain with dual recycling channel. Int. J. Prod. Econ.
**2013**, 144, 510–520. [Google Scholar] [CrossRef] - De Giovanni, P.; Zaccour, G. A two-period game of a closed-loop supply chain. Eur. J. Oper. Res.
**2014**, 232, 22–40. [Google Scholar] [CrossRef] - Jacobs, B.W.; Subramanian, R. Sharing responsibility for product recovery across the supply chain. Prod. Oper. Manag.
**2012**, 21, 85–100. [Google Scholar] [CrossRef] - Subramanian, R.; Gupta, S.; Talbot, B. Product design and supply chain coordination under extended producer responsibility. Prod. Oper. Manag.
**2009**, 18, 259–277. [Google Scholar] [CrossRef] - Jena, S.K.; Sarmah, S.P. Price competition and co-operation in a duopoly closed-loop supply chain. Int. J. Prod. Econ.
**2014**, 156, 346–360. [Google Scholar] [CrossRef] - Zu-Jun, M.; Zhang, N.; Dai, Y.; Hu, S. Managing channel profits of different cooperative models in closed-loop supply chains. Omega
**2016**, 59, 251–262. [Google Scholar] [CrossRef] - Guide, V.D.R., Jr.; Van Wassenhove, L.N. Managing product returns for remanufacturing. Prod. Oper. Manag.
**2001**, 10, 142–155. [Google Scholar] [CrossRef] - Nakashima, K.; Arimitsu, H.; Nose, T.; Kuriyama, S. Optimal control of a remanufacturing system. Int. J. Prod. Res.
**2004**, 42, 3619–3625. [Google Scholar] [CrossRef][Green Version] - Fallah, H.; Eskandari, H.; Pishvaee, M.S. Competitive closed-loop supply chain network design under uncertainty. J. Manuf. Syst.
**2015**, 37, 649–661. [Google Scholar] [CrossRef] - De Giovanni, P.; Zaccour, G. Cost–Revenue Sharing in a Closed-Loop Supply Chain; Advances in Dynamic Games; Birkhäser: Boston, MA, USA, 2013; pp. 395–421. [Google Scholar]
- De Giovanni, P.; Reddy, P.V.; Zaccour, G. Incentive strategies for an optimal recovery program in a closed-loop supply chain. Eur. J. Oper. Res.
**2016**, 249, 605–617. [Google Scholar] [CrossRef] - Caliskan-Demirag, O.; Chen, Y.F.; Li, J. Channel coordination under fairness concerns and nonlinear demand. Eur. J. Oper. Res.
**2010**, 207, 1321–1326. [Google Scholar] [CrossRef] - Du, S.; Nie, T.; Chu, C.; Yu, Y. Newsvendor model for a dyadic supply chain with Nash bargaining fairness concerns. Int. J. Prod. Econ.
**2014**, 52, 5070–5085. [Google Scholar] [CrossRef] - Ho, T.H.; Su, X.; Wu, Y. Distributional and peer-induced fairness in supply chain contract design. Prod. Oper. Manag.
**2014**, 23, 161–175. [Google Scholar] [CrossRef][Green Version] - Nie, T.; Du, S. Dual-fairness supply chain with quantity discount contracts. Eur. J. Oper. Res.
**2017**, 258, 491–500. [Google Scholar] [CrossRef] - Shu, Y.; Dai, Y.; Ma, Z. Pricing Decisions in Closed-Loop Supply Chains with Peer-Induced Fairness Concerns. Sustainability
**2019**, 11, 5071. [Google Scholar] [CrossRef][Green Version] - Xiao, J.; Huang, Z. A Stochastic Differential Game in the Closed-Loop Supply Chain with Third-Party Collecting and Fairness Concerns. Sustainability
**2019**, 11, 2241. [Google Scholar] [CrossRef][Green Version] - Li, T.; Xie, J.; Zhao, X.; Tang, J. On supplier encroachment with retailer’s fairness concerns. Comput. Ind. Eng.
**2016**, 98, 499–512. [Google Scholar] [CrossRef] - Li, Q.H.; Li, B. Dual-channel supply chain equilibrium problems regarding retail services and fairness concerns. Appl. Math. Model.
**2016**, 40, 7349–7367. [Google Scholar] [CrossRef]

Notation | Definition |
---|---|

$c,{c}_{r}$ | Unit production cost of product with raw material and used products |

$\mathsf{\Delta}$ | Unit cost savings from remanufacturing |

$R(t)$ | Return rate of used products at time $t$ |

$E(t)$ | Collecting effort of the retailer at time $t$ |

$\omega (t)$, $p(t)$ | Wholesale price and retail price of product at time $t$ |

$D(t)$ | Demand rate of product |

$s$ | Unit subsidy manufacturer gives to retailer for collecting used products |

$k$ | Scaling parameter of collecting cost function |

$\theta $ | Effect of collecting efforts on return rate |

$\delta $ | Decaying rate of return rate |

$\sigma (R(t))$ | Variance of the stochastic disturbance |

$z(t)$ | Standard Wiener process |

$\zeta (t)$ | Standard normal random variables |

$a$ | Market potential of product |

$b$ | Elasticity of demand with respect to price |

$\rho $ | Discount rate for decision makers |

$\lambda ,\beta $ | Fairness concern parameter and Nash bargaining parameter of retailer |

${\pi}_{i}(t),{U}_{i}(t)$ | Profit rate and utility rate of player $i$ at time $t$, $i=m,r$ represent manufacturer and retailer |

${J}_{i}^{x}$, ${V}_{i}^{x}$ | Objective function and value function for player $i$ under the model $x$, where $i=m,r$ and $x=NF,GF,SF,C,SC.$ |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huang, Z.
Stochastic Differential Game in the Closed-Loop Supply Chain with Fairness Concern Retailer. *Sustainability* **2020**, *12*, 3289.
https://doi.org/10.3390/su12083289

**AMA Style**

Huang Z.
Stochastic Differential Game in the Closed-Loop Supply Chain with Fairness Concern Retailer. *Sustainability*. 2020; 12(8):3289.
https://doi.org/10.3390/su12083289

**Chicago/Turabian Style**

Huang, Zongsheng.
2020. "Stochastic Differential Game in the Closed-Loop Supply Chain with Fairness Concern Retailer" *Sustainability* 12, no. 8: 3289.
https://doi.org/10.3390/su12083289