# A Green Supply Chain Member Selection Method Considering Green Innovation Capability in a Hesitant Fuzzy Environment

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Hesitation Fuzzy Sets Theory

**Definition**

**1**

**Definition**

**2**

**.**Let the score function and variance function of the hesitant fuzzy element${h}_{1}\left(x\right)$be$s\left({h}_{1}\right)$,$v\left({h}_{1}\right)$, and the score function and variance function of${h}_{2}\left(x\right)$be$s\left({h}_{2}\right)$,$v\left({h}_{2}\right)$, respectively, then:

- (1)
- If$s\left({h}_{1}\right)s\left({h}_{2}\right)$, then${h}_{1}{h}_{2}$
- (2)
- If $s\left({h}_{1}\right)=s\left({h}_{2}\right)$, then:
- If$v\left({h}_{1}\right)=v\left({h}_{2}\right)$, then ${h}_{1}={h}_{2}$;
- If$v\left({h}_{1}\right)v\left({h}_{2}\right)$, then ${h}_{1}<{h}_{2}$;
- If$v\left({h}_{1}\right)v\left({h}_{2}\right)$, then ${h}_{1}>{h}_{2}$.

**Definition**

**3**

**.**Let the hesitant fuzzy element$h(y)=\left\{{{\displaystyle \vartheta}}_{j}\right|j=1,2,\dots ,{l}_{h}\}$, where${h}^{+}$and${h}^{-}$denote the maximum and minimum values in $h(y)$, respectively, i.e.,

#### 3.2. Three-Point Estimation Method

## 4. Problem Description

## 5. Decision-Making Method for Green Supply Chain Member Selection

#### 5.1. Weight Determination Method

- (1)
- When $\alpha =1,\beta =\gamma =0$, the decision maker focuses on the program’s own characteristics.
- (2)
- When $\beta =1,\alpha =\gamma =0$, the decision maker focuses on the attribute characteristics.
- (3)
- When $\gamma =1,\alpha =\beta =0$, the decision maker focuses on the interrelationship between attributes.
- (4)
- When $\beta \cdot \gamma \cdot \alpha \ne 0$, the decision maker integrates the program and attribute characteristics, makes full use of the decision information, and makes a more comprehensive, objective, and reasonable evaluation of the evaluation object.

#### 5.2. Decision-Making Steps

- Step 1:
- The green supply chain member selection decision maker based on the green innovation capability for the set of attributes $M=\left\{{M}_{1},{M}_{2},{M}_{3},\dots \dots ,{M}_{t}\right\}$ in the solution set $H=\left\{{H}_{1},{H}_{2},{H}_{3},\dots \dots ,{H}_{i}\right\}$ is measured according to each attribute, the hesitant fuzzy decision matrix $D=\left\{{h}_{i\times t}\right\}$ for this green supply chain member selection is obtained, and the positive ideal solution is calculated according to Equation (4).
- Step 2:
- Calculate ${\alpha}^{\prime}$, ${\beta}^{\prime}$, and ${\gamma}^{\prime}$ according to Equations (5)–(7), correct ${\alpha}^{\prime}$, ${\beta}^{\prime}$, and ${\gamma}^{\prime}$ according to Equations (8)–(10), and determine the values of $\alpha $, $\beta $, and $\gamma $.
- Step 3:
- The attribute weights of the decision attributes of each decision option in the decision problem of selecting green supply chain members based on the green innovation capability perspective are determined according to Equation (11).
- Step 4:
- According to Formulas (2) and (3), calculate the similarity ${S}_{O}\left({H}_{i}\right)$, ${S}_{m}\left({H}_{i}\right)$, and ${S}_{p}\left({H}_{i}\right)$ of each decision-making scheme and the positive ideal point in the decision-making problem of selecting green supply chain members from the perspective of green innovation capability
- Step 5:
- According to the three-point estimation method, the decision-making schemes in the decision-making problem of selecting green supply chain members from the perspective of green innovation ability are ranked, and the optimal green supply chain member selection scheme is determined.

## 6. Numerical Example

## 7. Comparative Analysis

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**Green supplier decision matrix $D={\left({h}_{ij}\right)}_{5\times 5}$ for new energy vehicle manufacturers.

Program | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} |
---|---|---|---|---|---|

${H}_{1}$ | {0.2, 0.4, 0.7, 0.9} | {0.1, 0.2, 0.5, 0.7} | {0.2, 0.3, 0.5, 0.7, 0.8} | {0.1, 0.4, 0.6} | {0.3, 0.5, 0.7, 0.8} |

${H}_{2}$ | {0.4, 0.6, 0.7, 0.8} | {0.1, 0.2, 0, 4, 0.6} | {0.3, 0.4, 0.7, 0.8, 0.9} | {0.1, 0.2, 0.4} | {0.4, 0.5, 0.7, 0.9} |

${H}_{3}$ | {0.2, 0.3, 0.6, 0.7} | {0.3, 0.4, 0.5, 0.7} | {0.2, 0.4, 0.6, 0.7, 0.8} | {0.3, 0.4, 0.8} | {0.2, 0.6, 0.7, 0.8} |

${H}_{4}$ | {0.2, 0.3, 0.5, 0.8} | {0.2, 0.3, 0.5, 0.7} | {0.4, 0.6, 0.7, 0.8, 0.9} | {0.1, 0.2, 0.7} | {0.3, 0.5, 0.8, 0.9} |

${H}_{5}$ | {0.2, 0.3, 0.4, 0.7} | {0.3, 0.4, 0.6, 0.7} | {0.1, 0.5, 0.6, 0.8, 0.9} | {0.2, 0.5, 0.6} | {0.2, 0.4, 0.6, 0.9} |

**Table 2.**Similarity of ${S}_{O}\left({H}_{i}\right)$, ${S}_{m}\left({H}_{i}\right)$, and ${S}_{p}\left({H}_{i}\right)$.

Program | ${\mathit{S}}_{\mathit{O}}\left({\mathit{H}}_{\mathit{i}}\right)$ | ${\mathit{S}}_{\mathit{m}}\left({\mathit{H}}_{\mathit{i}}\right)$ | ${\mathit{S}}_{\mathit{p}}\left({\mathit{H}}_{\mathit{i}}\right)$ |
---|---|---|---|

${H}_{1}$ | 0.657 | 0.732 | 0.788 |

${H}_{2}$ | 0.692 | 0.736 | 0.774 |

${H}_{3}$ | 0.717 | 0.78 | 0.828 |

${H}_{4}$ | 0.660 | 0.74 | 0.796 |

${H}_{5}$ | 0.670 | 0.734 | 0.785 |

Program | ${\mathit{S}}_{\mathit{e}}\left({\mathit{H}}_{\mathit{i}}\right)$ | Three-Point Estimation Method | Program Ranking (Excellent → Poor) |
---|---|---|---|

${H}_{1}$ | 0.729 | ${S}_{e}({H}_{i})=\frac{{S}_{o}\left({H}_{i}\right)+4{S}_{m}\left({H}_{i}\right)+{S}_{p}\left({H}_{i}\right)}{6}$ | ${H}_{3}$ |

${H}_{2}$ | 0.735 | ${H}_{4}$ | |

${H}_{3}$ | 0.777 | ${H}_{2}$ | |

${H}_{4}$ | 0.736 | ${H}_{5}$ | |

${H}_{5}$ | 0.732 | ${H}_{1}$ |

**Table 4.**Comparison with the method in the literature [29].

Decision-Making Methods | Balance Factor | Attribute Weights | Program Sorting |
---|---|---|---|

Program Sorting | $\alpha =1$ $\beta =0$ $\gamma =0$ | ${(0.200,0.194,0.176,0.210,0.210)}^{T}$ | ${H}_{3}\succ {H}_{2}\succ {H}_{5}\succ {H}_{4}\succ {H}_{1}(p)$ |

${H}_{3}\succ {H}_{4}\succ {H}_{5}\succ {H}_{1}\succ {H}_{2}(m)$ | |||

${H}_{3}\succ {H}_{4}\succ {H}_{1}\succ {H}_{5}\succ {H}_{2}(o)$ | |||

$\alpha =0$ $\beta =1$ $\gamma =0$ | ${(0.256,0.141,0.163,0.335,0.105)}^{T}$ | ${H}_{5}\succ {H}_{2}\succ {H}_{3}\succ {H}_{1}\succ {H}_{4}(p)$ | |

${H}_{5}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}\succ {H}_{4}(m)$ | |||

${H}_{5}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}\succ {H}_{4}(o)$ | |||

$\alpha =0$ $\beta =0$ $\gamma =1$ | ${(0.122,0.261,0.165,0.287,0.168)}^{T}$ | ${H}_{1}\succ {H}_{3}\succ {H}_{5}\succ {H}_{2}\succ {H}_{4}(p)$ | |

${(0.109,0.250,0.176,0.265,0.173)}^{T}$ | ${H}_{1}\succ {H}_{3}\succ {H}_{5}\succ {H}_{4}\succ {H}_{2}(m)$ | ||

${(0.106,0.192,0.226,0.245,0.212)}^{T}$ | ${H}_{1}\succ {H}_{3}\succ {H}_{5}\succ {H}_{4}\succ {H}_{2}(o)$ | ||

$\alpha =1/3$ $\beta =1/3$ $\gamma =1/3$ | ${(0.200,0.194,0.173,0.222,0.201)}^{T}$ | ${H}_{3}\succ {H}_{2}\succ {H}_{5}\succ {H}_{1}\succ {H}_{4}(p)$ | |

${(0.200,0.193,0.174,0.220,0.202)}^{T}$ | ${H}_{3}\succ {H}_{5}\succ {H}_{2}\succ {H}_{1}\succ {H}_{4}(m)$ | ||

${(0.200,0.191,0.176,0.219,0.203)}^{T}$ | ${H}_{3}\succ {H}_{5}\succ {H}_{1}\succ {H}_{4}\succ {H}_{2}(o)$ | ||

Program Sequencing for this Article | $\alpha =0.621$ $\beta =0.245$ $\gamma =0.134$ | ${(0.201,0.193,0.175,0.214,0.207)}^{T}$ | ${H}_{3}\succ {H}_{4}\succ {H}_{2}\succ {H}_{5}\succ {H}_{1}$ |

${\mathit{H}}_{1}$ | ${\mathit{H}}_{2}$ | ${\mathit{H}}_{3}$ | ${\mathit{H}}_{4}$ | ${\mathit{H}}_{5}$ | |
---|---|---|---|---|---|

${H}_{1}$ | 0 | −0.203 | 0.146 | −0.154 | 0.012 |

${H}_{2}$ | −0.428 | 0 | −0.198 | −0.037 | −0.410 |

${H}_{3}$ | −0.805 | −0.756 | 0 | −0.348 | −0.314 |

${H}_{4}$ | −0.443 | −0.699 | −0.268 | 0 | −0.315 |

${H}_{5}$ | −0.703 | 0.516 | −0.125 | −0.244 | 0 |

**Table 6.**Global dominance degree and comparison with the method in the literature [30].

${\mathit{H}}_{1}$ | ${\mathit{H}}_{2}$ | ${\mathit{H}}_{3}$ | ${\mathit{H}}_{4}$ | ${\mathit{H}}_{5}$ | |
---|---|---|---|---|---|

Global Dominance Degree | 0 | 0.106 | 1 | 0.826 | 0.699 |

Program Sorting | ${H}_{3}\succ {H}_{4}\succ {H}_{5}\succ {H}_{2}\succ {H}_{1}$ | ||||

Program Sequencing for this Article | ${H}_{3}\succ {H}_{4}\succ {H}_{2}\succ {H}_{5}\succ {H}_{1}$ |

$\mathit{C}\left({\mathit{H}}_{1},{\mathit{H}}_{0}\right)$ | $\mathit{C}\left({\mathit{H}}_{2},{\mathit{H}}_{0}\right)$ | $\mathit{C}\left({\mathit{H}}_{3},{\mathit{H}}_{0}\right)$ | $\mathit{C}\left({\mathit{H}}_{4},{\mathit{H}}_{0}\right)$ | $\mathit{C}\left({\mathit{H}}_{5},{\mathit{H}}_{0}\right)$ | |
---|---|---|---|---|---|

${{\displaystyle C|}}_{\lambda =1}$ | 0.759 | 0.515 | 0.903 | 0.731 | 0.838 |

${{\displaystyle C|}}_{\lambda =2}$ | 0.939 | 0.744 | 0.989 | 0.916 | 0.973 |

Program Sorting | ${H}_{3}\succ {H}_{5}\succ {H}_{1}\succ {H}_{4}\succ {H}_{2}$ | ||||

Program Sequencing for this Article | ${H}_{3}\succ {H}_{4}\succ {H}_{2}\succ {H}_{5}\succ {H}_{1}$ |

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## Share and Cite

**MDPI and ACS Style**

Su, J.; Xu, B.; Li, L.; Wang, D.; Zhang, F.
A Green Supply Chain Member Selection Method Considering Green Innovation Capability in a Hesitant Fuzzy Environment. *Axioms* **2023**, *12*, 188.
https://doi.org/10.3390/axioms12020188

**AMA Style**

Su J, Xu B, Li L, Wang D, Zhang F.
A Green Supply Chain Member Selection Method Considering Green Innovation Capability in a Hesitant Fuzzy Environment. *Axioms*. 2023; 12(2):188.
https://doi.org/10.3390/axioms12020188

**Chicago/Turabian Style**

Su, Jiafu, Baojian Xu, Lvcheng Li, Dan Wang, and Fengting Zhang.
2023. "A Green Supply Chain Member Selection Method Considering Green Innovation Capability in a Hesitant Fuzzy Environment" *Axioms* 12, no. 2: 188.
https://doi.org/10.3390/axioms12020188