# Approach to Multi-Criteria Group Decision-Making Problems Based on the Best-Worst-Method and ELECTRE Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (1)
- $0\le {d}_{IMN}\le 1$;
- (2)
- ${d}_{IMN}({\mathsf{\alpha}}_{ij},{\mathsf{\beta}}_{ij})={d}_{IMN}({\mathsf{\beta}}_{ij},{\mathsf{\alpha}}_{ij})$;
- (3)
- ${d}_{IMN}({\mathsf{\alpha}}_{ij},{\mathsf{\beta}}_{ij})=0$, iff ${\rho}_{{\mathsf{\alpha}}_{ij}}={\mathsf{\rho}}_{{\mathsf{\beta}}_{ij}}$, ${\mathsf{\sigma}}_{{\mathsf{\alpha}}_{ij}}={\mathsf{\sigma}}_{{\mathsf{\beta}}_{ij}}$.

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**1.**

**Proof.**

#### 2.1. The Best-Worst Multi-Criteria Decision-Making Method

#### 2.2. ELECTRE III Method

## 3. Framework of the Proposed Novel Method

## 4. A Numerical Example

## 5. Discussions

- (1)
- If $s({\mathsf{\alpha}}_{1})>s({\mathsf{\alpha}}_{2})$, then ${\mathsf{\alpha}}_{1}>{\mathsf{\alpha}}_{2}$,
- (2)
- If $s({\mathsf{\alpha}}_{1})=s({\mathsf{\alpha}}_{2})$ and if $h({\mathsf{\alpha}}_{1})>h({\mathsf{\alpha}}_{2})$, then ${\mathsf{\alpha}}_{1}>{\mathsf{\alpha}}_{2}$; and if $h({\mathsf{\alpha}}_{1})=h({\mathsf{\alpha}}_{2})$, then ${\mathsf{\alpha}}_{1}={\mathsf{\alpha}}_{2}$.

- (a)
- The proposed method of this paper is based on BWM. It can decrease the comparison times to $2n-3$, compared with at least $(n-1)n/2$ times (AHP, TOPSIS). We increase the method’s efficiency in this way. Besides, comparison results are expressed by intuitionistic multiplicative preference relations, which is different from the original BWM.
- (b)
- We study the ELECTRE III method with intuitionistic multiplicative preference relations. Additionally, we introduce a distance formula, which can measure the distance between two IMNs. This distance formula is also suitable for interval-valued fuzzy preference relations. The research scope would be expended with this formula.
- (c)
- Comparing with the general ELECTRE method [3,37,41], based on BWM, we give the outranking functions with two kinds of matrices: the best-to-others, which is about the comparison results of the best alternative over the others; the others-to-worst, which is about the comparison result of the other alternatives over the worst one. That is consistent with the practical situation and improves the rationality of the final ranking result.
- (d)
- Our method can change some improper elements from the decision matrices automatically. This means that within a limited condition, our method would readjust itself by preference relations from the given matrices, if the decision results do not meet the requirements.

- (a)
- Our decision matrices are obtained based on BWM, including two parts: the matrix about best-to-others and the matrix about others-to-worst. Each kind of matrix does not have the problems of consistency. However, there exists a consistency issue between comparisons about best-to-others and others-to-worst, which have not been discussed in this paper.
- (b)
- The proposed method is based on the intuitionistic preference relations. However, in some more complicated conditions, this tool may still be beyond expression. Therefore, this research should be discussed further with more practical tools, such as interval-valued intuitionistic preference relations.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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${\mathit{q}}_{\mathit{l}}$ | ${\mathit{p}}_{\mathit{l}}$ | ${\mathit{v}}_{\mathit{l}}$ | |
---|---|---|---|

$D{M}_{1}$ | 0.04 | 0.14 | 0.30 |

$D{M}_{2}$ | 0.06 | 0.16 | 0.36 |

$D{M}_{3}$ | 0.10 | 0.20 | 0.35 |

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

${\overline{\mathrm{\Phi}}}_{1}$ | 0.8005 | −0.2158 | 0.1949 | 0.1171 | −0.8966 |

${\overline{\mathrm{\Phi}}}_{2}$ | 0.7777 | −0.6771 | 0.4672 | −0.1521 | −0.4157 |

${\overline{\mathrm{\Phi}}}_{3}$ | 0.8149 | −0.0957 | 0.9315 | −0.7321 | −0.9187 |

${\overline{\mathrm{\Phi}}}_{G}$ | 0.7977 | −0.3295 | 0.5312 | −0.2557 | −0.7437 |

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

${\tilde{\mathrm{\Phi}}}_{1}$ | −1.6060 | −0.2130 | 0.3546 | 0.4152 | 1.0492 |

${\tilde{\mathrm{\Phi}}}_{2}$ | −0.9758 | 0.1201 | 0.4721 | −0.8090 | 1.1926 |

${\tilde{\mathrm{\Phi}}}_{3}$ | −0.9711 | 0.0699 | −0.3674 | −0.2863 | 1.5549 |

${\tilde{\mathrm{\Phi}}}_{G}$ | −1.1843 | −0.0077 | −0.1531 | −0.2267 | 1.2655 |

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

${\overline{U}}_{1}$ | 5 | 2 | 4 | 3 | 1 |

${\overline{U}}_{2}$ | 5 | 1 | 4 | 3 | 2 |

${\overline{U}}_{3}$ | 4 | 3 | 5 | 2 | 1 |

${\overline{U}}_{G}$ | 5 | 2 | 4 | 3 | 1 |

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

${\tilde{U}}_{1}$ | 5 | 4 | 3 | 2 | 1 |

${\tilde{U}}_{2}$ | 5 | 3 | 2 | 4 | 1 |

${\tilde{U}}_{3}$ | 5 | 2 | 4 | 3 | 1 |

${\tilde{U}}_{G}$ | 5 | 3 | 2 | 4 | 1 |

${\mathsf{\phi}}_{(1)}^{0}$ | ${\mathsf{\phi}}_{(2)}^{0}$ | ${\mathsf{\phi}}_{(3)}^{0}$ | ${\mathsf{\phi}}_{(\mathit{G})}^{0}$ | |
---|---|---|---|---|

Best-to-Others | 0.9 | 0.7 | 0.6 | 0.7 |

Others-to-Worst | 0.7 | 0.6 | 0.9 | 0.7 |

**Table 7.**The first roundadjusted group outranking flow indexes of every alternative about the worst-to-others aspect.

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

${\tilde{\mathrm{\Phi}}}_{1}$ | −1.2744 | −0.2130 | 0.3546 | 0.0836 | 1.0492 |

${\tilde{\mathrm{\Phi}}}_{2}$ | −0.9758 | 0.1201 | 0.4721 | −0.8090 | 1.1926 |

${\tilde{\mathrm{\Phi}}}_{3}$ | −0.9711 | 0.0699 | −0.3674 | −0.2863 | 1.5549 |

${\tilde{\mathrm{\Phi}}}_{G}$ | −1.0737 | −0.0077 | 0.1531 | −0.3372 | 1.2655 |

**Table 8.**The first roundadjusted group outranking flow orders of every alternative about the worst aspect.

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

${\tilde{U}}_{1}$ | 5 | 4 | 2 | 3 | 1 |

${\tilde{U}}_{2}$ | 5 | 3 | 2 | 4 | 1 |

${\tilde{U}}_{3}$ | 5 | 2 | 4 | 3 | 1 |

${\tilde{U}}_{G}$ | 5 | 3 | 2 | 4 | 1 |

${\mathsf{\phi}}_{(1)}^{1}$ | ${\mathsf{\phi}}_{(2)}^{1}$ | ${\mathsf{\phi}}_{(3)}^{1}$ | ${\mathsf{\phi}}_{(\mathit{G})}^{1}$ | |
---|---|---|---|---|

Others-to-Worst | 0.90 | 1.00 | 0.70 | 0.87 |

Best-to-Others | Others-to-Worst | |
---|---|---|

M.M. Xia et al. | ${x}_{5}\succ {x}_{1}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}$ | ${x}_{5}\succ {x}_{1}\succ {x}_{3}\succ {x}_{4}\succ {x}_{2}$ |

Z.S. Xu | ${x}_{5}\succ {x}_{3}\succ {x}_{2}\succ {x}_{1}\succ {x}_{4}$ | ${x}_{5}\succ {x}_{3}\succ {x}_{2}\succ {x}_{4}\succ {x}_{1}$ |

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**MDPI and ACS Style**

You, X.; Chen, T.; Yang, Q.
Approach to Multi-Criteria Group Decision-Making Problems Based on the Best-Worst-Method and ELECTRE Method. *Symmetry* **2016**, *8*, 95.
https://doi.org/10.3390/sym8090095

**AMA Style**

You X, Chen T, Yang Q.
Approach to Multi-Criteria Group Decision-Making Problems Based on the Best-Worst-Method and ELECTRE Method. *Symmetry*. 2016; 8(9):95.
https://doi.org/10.3390/sym8090095

**Chicago/Turabian Style**

You, Xinshang, Tong Chen, and Qing Yang.
2016. "Approach to Multi-Criteria Group Decision-Making Problems Based on the Best-Worst-Method and ELECTRE Method" *Symmetry* 8, no. 9: 95.
https://doi.org/10.3390/sym8090095