# An Improved Shapley Value Method for a Green Supply Chain Income Distribution Mechanism

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## Abstract

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## 1. Introduction

- (1)
- We incorporate and quantify carbon emissions reductions and environmental remediation in a resource-based supply chain that includes a resource developer and a resource processor in order to determine the costs and contributions of each enterprise with regard to green development.
- (2)
- We include ecological factors, risk factors, and other efforts to update the traditional Shapley value method in an attempt to achieve fair and effective income distribution between members, and achieve sustainable resource-based supply chain development.

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. “Low-Carbon and Environmental Remediation” Cost to the Enterprise

#### 3.2. Market Demand that Considers the Consumers’ Low-Carbon Preferences

#### 3.3. Decentralized Decision and Centralized Decision Situations

- (1)
- Upstream resource developers mainly provide resource commodities through mining and primary processing, and sell them to downstream resource processors with an optimal price; downstream resource processors mainly manufacture resource commodities provided by resource developers, and then sell to the market.
- (2)
- Shortage and inventory costs are not considered when modeling; furthermore, resource recycling is not considered.
- (3)
- When it comes to calculating the cost for resource developers, three costs are considered: resource exploration costs, refinement costs, and ecological restoration cost. For resource processors, costs comprise resource transportation cost, emission reduction cost, and other additional costs.

#### 3.3.1. Decentralized Decision Situation

- (1)
- A downstream-dominated Stackelberg game;
- (2)
- An upstream-dominated Stackelberg game;
- (3)
- An upstream-and-downstream Nash game.

#### 3.3.2. Cooperative Centralized Decision Situation

#### 3.4. The Income Distribution Mechanism

#### 3.4.1. Income Distribution Principles

#### Symmetry between Risk and Return

#### Symmetry between Low-Carbon Input and Return

#### Symmetry between Efforts and Return

#### 3.4.2. Quantification of Correction Factors

#### Risk Factor Quantification

#### Ecological Factor Quantification

#### Technological Level Quantification

#### 3.4.3. Improved Shapley Value Method

## 4. Numeric Study

#### 4.1. Experimental Solution Process

#### 4.1.1. Determination of Correction Factors

#### 4.1.2. Weight Determination for the Correction Factors

- Construct the hierarchy model: the hierarchy is divided into three layers—an uppermost goal layer, a criterion layer, and a program layer (in which the index exists)
- Establish the judgment matrix: after experiments and a pairwise comparison of the significant magnitudes of two factors, Saaty and other scholars found that a nine-level ratio scale was most suitable; that is, frequently used numerical judgments (1, 3, 5, 7, 9 and 2, 4, 6, 8 there-between) that correspond to written narrative evaluations
- Hierarchical single arrangement and the corresponding consistency test: we perform a consistency test to obtain reasonable factor weights using the same test approach as hierarchical single arrangements
- Hierarchical overall arrangement and the corresponding consistency test and the specific construction of the hierarchical model

#### 4.1.3. Determination of Correction Factors

#### 4.1.4. Income Distribution Plans

#### 4.2. Testing for Robustness

#### 4.3. Managerial Insights

#### 4.3.1. Cooperative Centralized Decision-Making Is Beneficial to the Enterprises in the Resource-Based Supply Chain

#### 4.3.2. The Income Distribution Mechanism Should Be Closely Related to the Participators’ Efforts

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

${p}_{m}$ | price of a processed resource good unit |

${p}_{r}$ | price of a resource good unit |

${c}_{r}$ | development costs for a resource goods unit |

${c}_{e}$ | cost of exploration for the allocation of a resource goods unit |

$b$ | additional cost of processed resource goods minus the cost of transfer and emissions reduction in the production process |

$t$ | profit of a product unit by resource developers |

$\tau $ | profit of a product unit by resource processors |

$U$ | upstream dominance |

$D$ | downstream dominance |

$B$ | upstream-and-downstream equilibrium |

$T$ | cooperative centralized decision |

$E(\pi )$ | profit utility function for a resource-based supply chain system as well as the expected return in the paper. This paper assumes $E({\pi}_{i}{}^{\ast})=E({\pi}_{r(i)}^{\ast})+E({\pi}_{m(i)}^{\ast}),i=U,D,B,T$ |

R.I. | mean random consistency index |

CI and CR | consistency text index |

${\phi}_{i}$ | return given to i enterprise in a cooperative resource-based supply chain |

${\delta}_{i}$ | risk taken by i enterprise in the cooperative resource-based supply chain |

$V(S)$ | total benefit of set S |

$V(S\backslash i)$ | total benefit of set S minus participator i |

$W\left(\left|S\right|\right)$ | weighting factor, representing the contribution of each participator in set S |

## Appendix B

#### Appendix B.1. Upstream-Dominated Stackelberg Game

#### Appendix B.2. Upstream-and-Downstream Nash Equilibrium

Participators | Resource Developer | Resource Processor | Total Resource-Based Supply Chain Benefit |
---|---|---|---|

Downstream leading market | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{16\beta}\\ -\frac{1}{2}m{\gamma}^{2}-\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}{}_{}\end{array}$ | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{8\beta}\\ -\frac{1}{2}m{\xi}^{2}-\frac{1}{2}A{\delta}_{\epsilon}{}^{2}\end{array}$ | $\begin{array}{l}\frac{3{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{16\beta}-\frac{1}{2}m{\gamma}^{2}-\frac{1}{2}m{\xi}^{2}\\ -\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}-\frac{1}{2}A{\delta}_{\epsilon}{}^{2}\end{array}$ |

Upstream leading market | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{8\beta}\\ -\frac{1}{2}m{\gamma}^{2}-\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}\end{array}$ | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{16\beta}\\ -\frac{1}{2}m{\xi}^{2}-\frac{1}{2}A{\delta}_{\epsilon}{}^{2}\end{array}$ | $\begin{array}{l}\frac{3{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{16\beta}-\frac{1}{2}m{\xi}^{2}-\frac{1}{2}m{\gamma}^{2}\\ -\frac{1}{2}A{\delta}_{\epsilon}{}^{2}-\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}\end{array}$ |

Equilibrium market | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{9\beta}\\ -\frac{1}{2}m{\gamma}^{2}-\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}\end{array}$ | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{9\beta}\\ -\frac{1}{2}m{\xi}^{2}-\frac{1}{2}A{\delta}_{\epsilon}{}^{2}\end{array}$ | $\begin{array}{l}\frac{2{[\alpha -\beta a({c}_{r}+{c}_{e})-\beta b+\beta \xi +s\theta ]}^{2}}{9\beta}-\frac{1}{2}m{\xi}^{2}-\frac{1}{2}m{\gamma}^{2}\\ -\frac{1}{2}A{\delta}_{\epsilon}{}^{2}-\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}\end{array}$ |

Centralized decision market | $\begin{array}{l}\frac{{[\alpha -\beta a({c}_{e}+{c}_{r})-\beta b+\beta \xi +s\theta ]}^{2}}{4\beta}-\frac{1}{2}m{\xi}^{2}-\frac{1}{2}m{\gamma}^{2}\\ -\frac{1}{2}A{\delta}_{\epsilon}{}^{2}-\frac{1}{2}A{(a{\delta}_{\epsilon})}^{2}\end{array}$ |

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Factors | Risk Factor | Ecological Factor | Technological Level | $\mathbf{Weight}\text{}{\mathit{w}}_{\mathit{i}}\text{}$ |
---|---|---|---|---|

Risk factor | 1 | 1/3 | 1/5 | 0.1095 |

Ecological factor | 3 | 1 | 1/2 | 0.3090 |

Technological level | 5 | 2 | 1 | 0.5816 |

Matrix order (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

RI | 0.00 | 0.00 | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 |

$\mathbf{S}$ | $\mathit{E}({\mathit{\pi}}_{\mathit{r}})\text{}$ | $\mathit{E}({\mathit{\pi}}_{\mathit{m}})\text{}$ | $\mathit{V}\left\{\mathit{r},\mathit{m}\right\}\text{}$ |
---|---|---|---|

$V(\mathrm{S})$ | 14.296 | 9.0575 | 70.146 |

$\mathbf{S}$ | $\mathit{V}\left\{\mathit{r}\right\}\text{}$ | $\mathit{V}\left\{\mathit{r},\mathit{m}\right\}\text{}$ |
---|---|---|

$V(\mathrm{S})$ | 14.296 | 70.146 |

$V(\mathrm{S}\backslash \mathrm{r})$ | 0 | 9.0575 |

$V(\mathrm{S})-V(\mathrm{S}\backslash \mathrm{r})$ | 14.296 | 61.0885 |

$\left|\mathrm{S}\right|$ | 1 | 2 |

$W(\left|\mathrm{S}\right|)$ | 0.5 | 0.5 |

$W(\left|\mathrm{S}\right|)\left(V(\mathrm{S})-V(\mathrm{S}\backslash \mathrm{r})\right)$ | 7.148 | 30.544 |

$\mathbf{S}$ | $\mathit{V}\left\{\mathit{m}\right\}\text{}$ | $\mathit{V}\left\{\mathit{r},\mathit{m}\right\}\text{}$ |
---|---|---|

$V(\mathrm{S})$ | 9.0575 | 70.146 |

$V(\mathrm{S}\backslash \mathrm{m})$ | 0 | 14.296 |

$V(\mathrm{S})-V(\mathrm{S}\backslash \mathrm{r})$ | 9.0575 | 55.850 |

$\left|\mathrm{S}\right|$ | 1 | 2 |

$W(\left|\mathrm{S}\right|)$ | 0.5 | 0.5 |

$W(\left|\mathrm{S}\right|)\left(V(\mathrm{S})-V(\mathrm{S}\backslash \mathrm{r})\right)$ | 4.529 | 27.925 |

Ecological Remediation Effort | $\mathit{V}\left\{\mathit{r},\mathit{m}\right\}\text{}$ | Improved Income Distribution Vector |
---|---|---|

$\gamma =0.5$ | 94.416 | ${\Phi}^{\prime}=(\begin{array}{cc}40.672& 53.744\end{array})$ |

$\gamma =0.6$ | 88.584 | ${\Phi}^{\prime}=(\begin{array}{cc}40.479& 48.068\end{array})$ |

$\gamma =0.8$ | 76.866 | ${\Phi}^{\prime}=(\begin{array}{cc}37.140& 39.726\end{array})$ |

Ecological Remediation Effort | $\mathit{V}\left\{\mathit{r},\mathit{m}\right\}\text{}$ | Improved Income Distribution Vector |
---|---|---|

$\xi =0.4$ | 87.178 | ${\Phi}^{\prime}=(\begin{array}{cc}42.731& 44.448\end{array})$ |

$\xi =0.5$ | 88.584 | ${\Phi}^{\prime}=(\begin{array}{cc}40.479& 48.068\end{array})$ |

$\xi =0.6$ | 91.788 | ${\Phi}^{\prime}=(\begin{array}{cc}39.533& 52.245\end{array})$ |

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## Share and Cite

**MDPI and ACS Style**

Xu, Z.; Peng, Z.; Yang, L.; Chen, X.
An Improved Shapley Value Method for a Green Supply Chain Income Distribution Mechanism. *Int. J. Environ. Res. Public Health* **2018**, *15*, 1976.
https://doi.org/10.3390/ijerph15091976

**AMA Style**

Xu Z, Peng Z, Yang L, Chen X.
An Improved Shapley Value Method for a Green Supply Chain Income Distribution Mechanism. *International Journal of Environmental Research and Public Health*. 2018; 15(9):1976.
https://doi.org/10.3390/ijerph15091976

**Chicago/Turabian Style**

Xu, Zhongwen, Zixuan Peng, Ling Yang, and Xudong Chen.
2018. "An Improved Shapley Value Method for a Green Supply Chain Income Distribution Mechanism" *International Journal of Environmental Research and Public Health* 15, no. 9: 1976.
https://doi.org/10.3390/ijerph15091976