# An Interactive Consensus Model in Group Decision Making with Heterogeneous Hesitant Preference Relations

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Two Tuple Linguistic Expressive Model

**Definition**

**1**

**([36]).**

**Definition**

**2**

**([36]).**

#### 2.2. Heterogeneous Hesitant Preference Relations and Their Consistencies

**Definition**

**3**

**([23]).**

**Definition**

**4**

**([23]).**

**Definition**

**5**

**([24]).**

**Definition**

**6**

**([37]).**

**Definition**

**7.**

**Definition**

**8.**

## 3. The Framework of Consensus Reaching Process with HHPRs

#### 3.1. The Selection Phase

#### 3.1.1. Obtaining the Individual Priority Weight Vector

- (1)
- ${e}_{k}\in {E}^{H}$

- (2)
- For ${e}_{k}\in {E}^{A}$, let $\delta \left({a}_{ij}^{\sigma \left(l\right)}\right)={a}_{ij}^{\left(1\right)}or{a}_{ij}^{\left(2\right)}or\cdots or{a}_{ij}^{\left(\#{a}_{ij}\right)}$, then $\frac{{w}_{i}}{{w}_{j}}={a}_{ij}^{\left(1\right)}or{a}_{ij}^{\left(2\right)}or\cdots or{a}_{ij}^{\left(\#{a}_{ij}\right)}\iff \frac{{w}_{i}}{{w}_{j}}=\delta \left({a}_{ij}^{\sigma \left(l\right)}\right)\iff {w}_{i}-{w}_{j}\cdot \delta \left({a}_{ij}^{\sigma \left(l\right)}\right)=0$, in order to minimize the total deviation $\left|{w}_{i}-{w}_{j}\cdot \delta \left({a}_{ij}^{\sigma \left(l\right)}\right)\right|$, we obtain the following optimization model$$\begin{array}{l}\mathrm{min}F={\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^{n}{d}_{ij,k}^{+}+{d}_{ij,k}^{-}}}\\ s.t.\{\begin{array}{l}{w}_{i}^{k}-\left({\displaystyle \sum _{l=1}^{\#{p}_{ij,k}}{z}_{ij,k}^{\sigma \left(l\right)}}{a}_{ij,k}^{\sigma \left(l\right)}\right){w}_{j}^{k}-{d}_{ij,k}^{+}+{d}_{ij,k}^{-}=0\\ {\displaystyle \sum _{i=1}^{n}{w}_{i}^{k}}=1,{w}_{i}^{k}\ge 0,i=1,2,\cdots ,n,k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\\ {\displaystyle \sum _{l=1}^{\#{p}_{ij,k}}{z}_{ij,k}^{\sigma \left(l\right)}}=1,i,j=1,2,\cdots ,n,ji,k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\\ {z}_{ij,k}^{\sigma \left(l\right)}=0\mathrm{or}1,i,j=1,2,\cdots ,n,ji,k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\\ {d}_{ij,k}^{+},{d}_{ij,k}^{-}\ge 0,i,j=1,2,\cdots ,n,ji,k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\end{array}\end{array}$$

- (3)
- For ${e}_{k}\in {E}^{T}$, let $\delta \left({\mathrm{\Delta}}^{-1}({t}_{ij}^{\sigma \left(l\right)})\right)={\mathrm{\Delta}}^{-1}({t}_{ij}^{\left(1\right)})or{\mathrm{\Delta}}^{-1}({t}_{ij}^{\left(2\right)})or\cdots or{\mathrm{\Delta}}^{-1}({t}_{ij}^{\left(\#{t}_{ij}\right)})$, then $\frac{gn}{2}\left({w}_{i}-{w}_{j}\right)+\frac{g}{2}=\delta \left({\mathrm{\Delta}}^{-1}({t}_{ij}^{\sigma \left(l\right)})\right)\iff \frac{gn}{2}\left({w}_{i}-{w}_{j}\right)+\frac{g}{2}-\delta \left({\mathrm{\Delta}}^{-1}({t}_{ij}^{\sigma \left(l\right)})\right)=0$, in order to minimize total deviation $\left|\frac{gn}{2}\left({w}_{i}-{w}_{j}\right)+\frac{g}{2}-\delta \left({\mathrm{\Delta}}^{-1}({t}_{ij}^{\sigma \left(l\right)})\right)\right|$, the following mathematical programming is established:$$\begin{array}{l}\mathrm{min}F={\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=i+1}^{n}{d}_{ij,k}^{+}+{d}_{ij,k}^{-}}}\\ s.t.\{\begin{array}{l}\frac{gn}{2}\left({w}_{i}^{k}-{w}_{j}^{k}\right)+\frac{g}{2}-\left({\displaystyle \sum _{l=1}^{\#{t}_{ij,k}}{z}_{ij,k}^{\sigma \left(l\right)}}{\mathrm{\Delta}}^{-1}\left({t}_{ij,k}^{\sigma \left(l\right)}\right)\right)-{d}_{ij,k}^{+}+{d}_{ij,k}^{-}=0\\ {\displaystyle \sum _{i=1}^{n}{w}_{i}^{k}}=1,{w}_{i}^{k}\ge 0,i=1,2,\cdots ,n,k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\\ {\displaystyle \sum _{l=1}^{\#{t}_{ij,k}}{z}_{ij,k}^{\sigma \left(l\right)}}=1,i,j=1,2,\cdots ,n,ji,k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\\ {z}_{ij,k}^{\sigma \left(l\right)}=0\mathrm{or}1,i,j=1,2,\cdots ,n,ji,k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\\ {d}_{ij,k}^{+},{d}_{ij,k}^{-}\ge 0,i,j=1,2,\cdots ,n,ji,k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\end{array}\end{array}$$

#### 3.1.2. Obtaining the Group Priority Weight Vector

#### 3.2. Consensus Reaching Process

#### 3.2.1. Group Consensus Degree

**Definition**

**9.**

#### 3.2.2. Feedback Adjustment

- (1)
- for ${e}_{k}\in {E}^{H}\left(k=1,2,\cdots ,{m}_{1}\right)$,$$\{\begin{array}{cc}\overline{{h}_{ij,k}^{\sigma \left(l\right)}}\in \left[\mathrm{min}\left\{{h}_{ij,k}^{\sigma \left(l\right)},{h}_{ij,c}\right\},\mathrm{max}\left\{{h}_{ij,k}^{\sigma \left(l\right)},{h}_{ij,c}\right\}\right],& i<j\\ \overline{{h}_{ij,k}^{\sigma \left(l\right)}}=1,& i=j\\ \overline{{h}_{ij,k}^{\sigma \left(l\right)}}=1-\overline{{h}_{ji,k}^{\sigma \left(l\right)}}& i>j\end{array}$$
- (2)
- for ${e}_{k}\in {E}^{A}\left(k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\right)$,$$\{\begin{array}{cc}\overline{{a}_{ij,k}^{\sigma \left(l\right)}}\in \left[\mathrm{min}\left\{{a}_{ij,k}^{\sigma \left(l\right)},{a}_{ij,c}\right\},\mathrm{max}\left\{{a}_{ij,k}^{\sigma \left(l\right)},{a}_{ij,c}\right\}\right],& i<j\\ \overline{{a}_{ij,k}^{\sigma \left(l\right)}}=1,& i=j\\ \overline{{a}_{ij,k}^{\sigma \left(l\right)}}=\frac{1}{\overline{{a}_{ji,k}^{\sigma \left(l\right)}}}& i>j\end{array}$$
- (3)
- for ${e}_{k}\in {E}^{T}\left(k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\right)$,$$\{\begin{array}{cc}\overline{{t}_{ij,k}^{\sigma \left(l\right)}}\in \left[\mathrm{min}\left\{{t}_{ij,k}^{\sigma \left(l\right)},{t}_{ij,c}\right\},\mathrm{max}\left\{{t}_{ij,k}^{\sigma \left(l\right)},{t}_{ij,c}\right\}\right],& i<j\\ \overline{{t}_{ij,k}^{\sigma \left(l\right)}}=1,& i=j\\ \overline{{t}_{ij,k}^{\sigma \left(l\right)}}=1-\overline{{t}_{ji,k}^{\sigma \left(l\right)}}& i>j\end{array}$$

Algorithm 1: A consensus reaching process in GDM with HHPRs |

Input: Initial HHPRs and preset threshold of GCD ($\overline{GCD}$). |

Output: The adjusted HHPRs and final group priority weight vector. |

Step 1: Set $z=0$, ${H}_{k}^{\left(z\right)}={\left({h}_{ij,k}^{\left(z\right)}\right)}_{n\times n}\left(k=1,2,\cdots ,{m}_{1}\right)$, ${A}_{k}^{\left(z\right)}={\left({a}_{ij,k}^{\left(z\right)}\right)}_{n\times n}\left(k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\right)$, ${T}_{k}^{\left(z\right)}={\left({t}_{ij,k}^{\left(z\right)}\right)}_{n\times n}\left(k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\right)$. |

Step 2: The individual priority weight vector ${w}_{k}^{\left(z\right)}=\left({w}_{1,k}^{\left(z\right)},{w}_{2,k}^{\left(z\right)},\cdots ,{w}_{n,k}^{\left(z\right)}\right)\left(k=1,2,\cdots ,m\right)$ and consistent level $c{l}_{k}^{\left(z\right)}\left(k=1,2,\cdots ,m\right)$ are obtained according to the Equations (16)–(21). |

Step 3: According to Equation (22), group priority weight vector ${w}_{c}^{\left(z\right)}=\left({w}_{1,c}^{\left(z\right)},{w}_{2,c}^{\left(z\right)},\cdots ,{w}_{n,c}^{\left(z\right)}\right)$ is obtained by IOWA operator. |

Step 4: Obtaining $GCD\left\{{e}_{1},{e}_{2},\cdots ,{e}_{m}\right\}$ according to Equation (25), if $GCD\left\{{e}_{1},{e}_{2},\cdots ,{e}_{m}\right\}\ge \overline{GCD}$, go directly to Step 6; Otherwise, proceed to the next step. |

Step 5: The decision makers adjust their preferences according to Equations (26)–(28), respectively. Then, set $z=z+1$, go back to step 2. |

Step 6: Suppose $\overline{{H}^{\left(k\right)}}={H}_{z}^{\left(k\right)}\left(k=1,2,\cdots ,{m}_{1}\right)$, $\overline{{\mathrm{A}}^{\left(k\right)}}{=\mathrm{A}}_{z}^{\left(k\right)}\left(k={m}_{1}+1,{m}_{1}+2,\cdots ,{m}_{2}\right)$, $\overline{{T}^{\left(k\right)}}={T}_{z}^{\left(k\right)}\left(k={m}_{2}+1,{m}_{2}+2,\cdots ,{m}_{3}\right)$, and ${w}^{*}={w}_{c}^{\left(z\right)}$. |

## 4. Example Application: Emergency Plan Selection

- (a)
- Mining rescue channel and escape passage in the style of roadway drivage underground (${x}_{1}$);
- (b)
- Dispatching large mechanical equipment and deep-hole drilling machines above mine (${x}_{2}$);
- (c)
- Repair the wellbore and take mine cars down into the mine (${x}_{3}$);
- (d)
- Using partial blasting and arranging mining machines (${x}_{4}$).

- (1)
- Obtaining the priority weight vector of individual

- (2)
- Obtaining the group priority weight vector

- (2)
- Consensus reaching process

## 5. Discussion

#### 5.1. Comparison with Aggregation Operators

#### 5.2. Comparison with the Related Studies

#### 5.3. Managerial Implications

#### 5.4. Contributions and Importance of This Study

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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x_{1} | x_{2} | x_{3} | x_{4} | |
---|---|---|---|---|

x_{1} | {0.5} | {0.3, 0.4} | {0.4} | {0.6} |

x_{2} | {0.7, 0.6} | {0.5} | {0.5} | {0.5, 0.6} |

x_{3} | {0.6} | {0.5} | {0.5} | {0.6}} |

x_{4} | {0.3} | {0.5, 0.4} | {0.3} | {0.5} |

x_{1} | x_{2} | x_{3} | x_{4} | |
---|---|---|---|---|

x_{1} | {1} | {1/3} | {3} | {3, 5} |

x_{2} | {3} | {1} | {3} | {5} |

x_{3} | {1/3} | {0.5} | {1} | {3, 4}} |

x_{4} | {1/3, 1/5} | {1/5} | {1/3, 1/4} | {1} |

x_{1} | x_{2} | x_{3} | x_{4} | |
---|---|---|---|---|

x_{1} | {s_{3}} | {s_{3}} | {s_{3}, s_{4}} | {s_{4}} |

x_{2} | {s_{5}} | {s_{3}} | {s_{5}} | {s_{5}, s_{6}} |

x_{3} | {s_{5}, s_{4}} | {s_{3}} | {s_{3}} | {s_{5}}} |

x_{4} | {s_{4}} | {s_{3}, s_{2}} | {s_{3}} | {s_{3}} |

Priority Vectors | Ranking Results | |
---|---|---|

WA operator | ${\left(0.2697,0.3546,0.2486,0.1271\right)}^{T}$ | ${x}_{2}\succ {x}_{1}\succ {x}_{3}\succ {x}_{4}$ |

IOWA operator | ${\left(0.2619,0.3551,0.255,0.128\right)}^{T}$ | ${x}_{2}\succ {x}_{1}\succ {x}_{3}\succ {x}_{4}$ |

Our study | ${\left(0.3337,0.3911,0.2059,0.07\right)}^{T}$ | ${x}_{2}\succ {x}_{1}\succ {x}_{3}\succ {x}_{4}$ |

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**MDPI and ACS Style**

Song, Y.
An Interactive Consensus Model in Group Decision Making with Heterogeneous Hesitant Preference Relations. *Axioms* **2022**, *11*, 517.
https://doi.org/10.3390/axioms11100517

**AMA Style**

Song Y.
An Interactive Consensus Model in Group Decision Making with Heterogeneous Hesitant Preference Relations. *Axioms*. 2022; 11(10):517.
https://doi.org/10.3390/axioms11100517

**Chicago/Turabian Style**

Song, Yongming.
2022. "An Interactive Consensus Model in Group Decision Making with Heterogeneous Hesitant Preference Relations" *Axioms* 11, no. 10: 517.
https://doi.org/10.3390/axioms11100517