# Supply Chain Strategies for Quality Inspection under a Customer Return Policy: A Game Theoretical Approach

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## Abstract

**:**

## 1. Introduction

## 2. Model Framework

#### 2.1. Assumptions

- All the actors are rational.
- All parameters are deterministic and known in advance.
- Demand and market price functions $\left({I}_{1},{I}_{2},{I}_{3}\right)$ are constant. Although the demand and product quality are related to each other but we assume that retailer provides enough compensation to satisfy the consumer when the consumer reports a faulty/non-conforming product. Following Ref. [17], we assume ${I}_{1}>{I}_{2}>{I}_{3}$.
- Production quality, random quality inspection level and traceability level are controllable and follow their respective cost functions (i.e., ${C}_{p}(P)$ and ${C}_{Qs}(Q,s)$) (also used in the literature, cf. [3]).
- Return of the product is applicable to the faulty products, which were sold as a good product by the retailer. The return rate i.e., probability of a product return is an exogenous function.
- The retailer can accept a faulty product but cannot reject a non-faulty/conforming product to the textile manufacturer.
- Both actors should individually make a positive profit to remain in the game.

#### 2.2. Textile Manufacturer–Retailer Interaction

#### 2.3. Textile Manufacturers Model Formulation

#### 2.4. Retailer’s Model Formulation

## 3. Formulation of Game Scenarios

#### 3.1. Non-Cooperative Games

#### 3.1.1. The Stackelberg’s Textile Manufacturer Game

#### 3.1.2. The Stackelberg’s Retailer Game

#### 3.1.3. The Nash Game

#### 3.2. Cooperative Game

**Proposition**

**1.**

**Proof.**

## 4. Bargaining Feasibility and Game Change Scenarios

#### 4.1. Game Change and Bargaining

#### 4.2. Competition among Non-Cooperative Games

#### 4.3. Forced Game Change and Cooperation

## 5. Results and Discussion

^{TM}) whereas it remains lowest for the retailer-dominated game (P

^{R}) (Figure 2a). The retailer’s quality inspection Q remains at the lowest level for r < 0.4 for all the games except textile manufacturer-dominated (Q

^{TM}) game (Figure 2b). Traceability remains at the lowest level for the games where there is no single supply chain captain (i.e., either both are cooperative (s

^{C}) or both are non-cooperative (s

^{N})), whereas the traceability level in the manufacturer-dominant game (s

^{TM}) exceeds that in a retailer-dominatant game (s

^{R}) (Figure 2c). Nevertheless, non-zero traceability confirms that companies should invest on traceability when the two actors are non-cooperating in the supply chain. Figure 2d shows the change in internal transfer payment I for different game scenarios with varying return rate (r).

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Variation of textile manufacturer and retailer controlled variables with variations in return rate (r) in different game scenarios. Here the superscripts R, TM, N and C represent the parameters pertaining to Stackelberg’s retailer game, Stackelberg’s textile manufacturer game, Nash game and cooperative game models, respectively. (

**a**) shows the parameter P; (

**b**) shows the parameter Q; (

**c**) shows the parameter s; (

**d**) shows the parameter I; (

**e**) shows the parameter $\Pi $; (

**f**) shows the parameter ${\Pi}_{R}$; (

**g**) shows the parameter ${\Pi}_{TM}$.

**Figure 3.**Competition among non-cooperative games (

**a**) the Stackelberg’s retailer game and Nash game; (

**b**) the Stackelberg’s textile manufacturer game and Nash game. The shaded region shows the favourable region for the Nash game over the respective Stackelberg’s games.

**Figure 4.**Competition among (

**a**) the Stackelberg’s retailer game and cooperative game; (

**b**) the Stackelberg’s textile manufacturer game and cooperative game. Region I represents the favourable region for the leader in the game to go in cooperation with the follower and II represents the favourable region for leader to go on follower’s position.

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

A, B | Penalty costs to the textile manufacturer for the non-conforming products identified in random quality inspection and traceability respectively |

${C}_{p}(P),{C}_{Qs}(Q,s)$ | Cost functions for implementing product quality (P), and quality inspection techniques (Q, s) respectively |

I | The price per product charged by the manufacturer to the retailer |

${I}_{1}$ | Selling price charged by the retailer to the consumer |

${I}_{2}$ | Selling price of non-conforming product identified by the retailer |

${I}_{3}$ | Net selling price of the non-conforming/defected product identified by the consumer |

K | Coefficient deciding the profit proportion of the follower |

P | Product quality rate, as proportion of product that succeed to meet the retailer’s quality requirement or conformation, $0\le P\le 1$ |

Q | Random quality inspection rate, as the proportion of nonconforming products detected in quality inspection, $0\le Q\le 1$ |

r | Return rate, as the proportion of the non-conforming product returned by the consumer. $0\le r\le 1$ |

s | Traceability success rate, as the proportion of the return successfully identified by the retailer to the manufacturer. $0\le s\le 1$ |

${\alpha}_{P},{\beta}_{p}$ | Coefficients related to cost function ${C}_{p}(P)$ |

${\alpha}_{Q},{\alpha}_{s},{\beta}_{Q},{\beta}_{s}$ | Coefficients related to cost function ${C}_{Qs}(Q,s)$ |

$\gamma $ | Factor related to relative bargaining power of the textile manufacturer and the retailer |

${\Pi}_{TM}$ | Manufacturer’s profit per product |

${\Pi}_{R}$ | Retailer’s profit per product |

$\Pi $ | Total supply chain profit, $\Pi ={\Pi}_{TM}+{\Pi}_{R}$ |

Sr. No. | Description | Proportion |
---|---|---|

1 | Proportion of good products among total produced products | P |

2 | Proportion of non-conforming products among identified in retailer’s random quality inspection | $\left(1-P\right)Q$ |

3 | Proportion of traced non-conforming products which were not identified in retailer’s random quality inspection but returned by the consumer | $\left(1-P\right)\left(1-Q\right)sr$ |

5 | Proportion of non-conforming products which were not identified in retailer’s random quality inspection and also did not return by the consumer | $(1-P)(1-Q)(1-r)$ |

6 | Proportion of non-traceable non-conforming products which were not identified in retailer’s random quality inspection but returned by the consumer | $(1-P)(1-Q)(1-s)r$ |

Total | 1.0 |

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Kumar, V.; Ekwall, D.; Wang, L. Supply Chain Strategies for Quality Inspection under a Customer Return Policy: A Game Theoretical Approach. *Entropy* **2016**, *18*, 440.
https://doi.org/10.3390/e18120440

**AMA Style**

Kumar V, Ekwall D, Wang L. Supply Chain Strategies for Quality Inspection under a Customer Return Policy: A Game Theoretical Approach. *Entropy*. 2016; 18(12):440.
https://doi.org/10.3390/e18120440

**Chicago/Turabian Style**

Kumar, Vijay, Daniel Ekwall, and Lichuan Wang. 2016. "Supply Chain Strategies for Quality Inspection under a Customer Return Policy: A Game Theoretical Approach" *Entropy* 18, no. 12: 440.
https://doi.org/10.3390/e18120440