# A Novel Approach to Multi-Attribute Group Decision-Making with q-Rung Picture Linguistic Information

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. q-Rung Orthopair Fuzzy Set (q-ROFS) and q-Rung Picture Fuzzy Set (q-RPFS)

**Definition 1**

**[26]**.

**Definition 2**

**[27].**

- ${\tilde{a}}_{1}\oplus {\tilde{a}}_{2}=\left({\left({u}_{1}^{q}+{u}_{2}^{q}-{u}_{1}^{q}{u}_{2}^{q}\right)}^{1/q},{v}_{1}{v}_{2}\right)$,
- ${\tilde{a}}_{1}\otimes {\tilde{a}}_{2}=\left({u}_{1}{u}_{2},{\left({v}_{1}^{q}+{v}_{2}^{q}-{v}_{1}^{q}{v}_{2}^{q}\right)}^{1/q}\right)$,
- $\lambda {\tilde{a}}_{1}=\left({\left(1-{\left(1-{u}_{1}^{q}\right)}^{\lambda}\right)}^{1/q},{v}_{1}^{\lambda}\right)$,
- ${\tilde{a}}_{1}^{\lambda}=\left({u}_{1}^{\lambda},{\left(1-{\left(1-{v}_{1}^{q}\right)}^{\lambda}\right)}^{1/q}\right)$.

**Definition 3**

**[27].**

- If $S\left({\tilde{a}}_{1}\right)>S\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}>{\tilde{a}}_{2}$;
- If $S\left({\tilde{a}}_{1}\right)=S\left({\tilde{a}}_{2}\right)$, then
- (1)
- If $H\left({\tilde{a}}_{1}\right)>H\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}>{\tilde{a}}_{2}$;
- (2)
- If $H\left({\tilde{a}}_{1}\right)=H\left({\tilde{a}}_{2}\right)$, then ${\tilde{a}}_{1}={\tilde{a}}_{2}$.

**Definition 4**

**[29].**

**Definition**

**5.**

#### 2.2. Linguistic Term Sets and q-Rung Picture Linguistic Set (q-RPLS)

**Definition**

**6.**

**Definition**

**7.**

- ${\alpha}_{1}\oplus {\alpha}_{2}=\langle {s}_{{\theta}_{1}+{\theta}_{2}},\left({\left({u}_{1}^{q}+{u}_{2}^{q}-{u}_{1}^{q}{u}_{2}^{q}\right)}^{1/q},{\eta}_{1}{\eta}_{2},{v}_{1}{v}_{2}\right)\rangle $,
- ${\alpha}_{1}\otimes {\alpha}_{2}=\langle {s}_{{\theta}_{1}\times {\theta}_{2}},\left({u}_{1}{u}_{2},{\left({\eta}_{1}^{q}+{\eta}_{2}^{q}-{\eta}_{1}^{q}{\eta}_{2}^{q}\right)}^{1/q},{\left({v}_{1}^{q}+{v}_{2}^{q}-{v}_{1}^{q}{v}_{2}^{q}\right)}^{1/q}\right)\rangle $,
- $\lambda \alpha =\langle {s}_{\lambda \times \theta},\left({\left(1-{\left(1-{u}^{q}\right)}^{\lambda}\right)}^{1/q},{\eta}^{\lambda},{v}^{\lambda}\right)\rangle $,
- ${\alpha}^{\lambda}=\langle {s}_{{\theta}^{\lambda}},\left({u}^{\lambda},{\left(1-{\left(1-{\eta}^{q}\right)}^{\lambda}\right)}^{1/q},{\left(1-{\left(1-{v}^{q}\right)}^{\lambda}\right)}^{1/q}\right)\rangle $.

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

- if $S\left({\alpha}_{1}\right)>S\left({\alpha}_{2}\right)$, then ${\alpha}_{1}>{\alpha}_{2}$;
- if $S\left({\alpha}_{1}\right)=S\left({\alpha}_{2}\right)$, then
- (1)
- if $H\left({\alpha}_{1}\right)>H\left({\alpha}_{2}\right)$, then ${\alpha}_{1}>{\alpha}_{2}$;
- (2)
- if $H\left({\alpha}_{1}\right)>H\left({\alpha}_{2}\right)$, then ${\alpha}_{1}={\alpha}_{2}$.

#### 2.3. Heronian Mean

**Definition 11**

**[43,45].**

**Definition 12**

**[46].**

## 3. The q-Rung Picture Linguistic Heronian Mean Operators

#### 3.1. The q-Rung Picture Linguistic Heronian Mean (q-RPLHM) Operator

**Definition**

**13.**

**Theorem**

**1.**

**Proof.**

**Theorem 2**

**(Monotonicity).**

**Proof.**

**Theorem 3**

**(Idempotency)**.

**Proof.**

**Theorem 4**

**(Boundedness).**

**Proof.**

**Case 1:**When $t\to 0$, then the q-RPLHM operator reduces to the followings,

**Case 2**: When $s\to 0$, then the q-RPLHM operator reduces to the followings,

**Case 3:**When $s=t=1$, then the q-RPLHM operator reduces to the followings,

**Case 4:**When $s=t=1/2$, then the q-RPLHM operator reduces to the followings

**Case 5:**When $q=2$, then the q-RPLHM operator reduces to the followings,

**Case 6:**When $q=1$, then the q-ROLHM operator reduces to the followings,

#### 3.2. The q-Rung Picture Linguistic Weighted Heronian Mean (q-RPLWHM) Operator

**Definition**

**14.**

**Theorem**

**5.**

**Theorem**

**6 (Monotonicity).**

**Theorem 7**

**(Boundedness).**

#### 3.3. The q-Rung Picture Linguistic Geometric Heronian Mean (q-RPLGHM) Operator

**Definition**

**15.**

**Theorem**

**8.**

**Theorem 9**

**(Idempotency).**

**Theorem 10**

**(Monotonicity).**

**Theorem 11**

**(Boundedness).**

**Case 1:**When $t\to 0$, then the q-RPLGHM operator reduces to the followings,

**Case 2:**When $s\to 0$, then the q-RPLGHM operator reduces to the followings,

**Case 3:**When $s=t=1$, then the q-RPLGHM operator reduces to the followings,

**Case 4:**When $s=t=1/2$, then the q-RPLGHM operator reduces to the followings,

**Case 5:**When $q=2$, then the q-RPLGHM operator reduces to the followings

**Case 6:**When $q=1$, then the q-RPLGHM operator reduces to the followings

#### 3.4. The q-Rung Picture Linguistic Weighted Geometric Heronian Mean (q-RPLWGHM) Operator

**Definition**

**16.**

**Theorem**

**12.**

**Theorem 13**

**(Monotonicity).**

**Theorem 14**

**(Boundedness).**

## 4. A Novel Approach to MAGDM Based on the Proposed Operators

## 5. Numerical Instance

_{1}, A

_{2}, A

_{3}, A

_{4}}. Four experts ${D}_{k}(k=1,2,3,4)$ are invited to evaluate the candidates under four attributes, they are (1) technology C

_{1}; (2) strategic adaptability C

_{2}; (3) supplier’s ability C

_{3}; (4) supplier’s reputation C

_{4}. Weight vector of the four attributes is $w={\left(0.25,0.3,0.25,0.2\right)}^{T}$. The decision-makers are required to use picture fuzzy linguistic numbers (PFLNs) on the basic of the linguistic term set S = {s

_{0}= terrible, s

_{1}= bad, s

_{2}= poor, s

_{3}= neutral, s

_{4}= good, s

_{5}= well, s

_{6}= excellent} to express their preference information. Decision-makers’ weight vector is $\lambda ={\left(0.3,0.2,0.2,0.3\right)}^{T}$. After evaluation, the individual picture fuzzy linguistic decision matrix ${A}^{k}={\left({\alpha}_{ij}^{k}\right)}_{4\times 4}$ can be obtained, which are shown in Table 1, Table 2, Table 3 and Table 4.

#### 5.1. The Decision-Making Process

_{1}.

_{1}.

#### 5.2. The Influence of the Parameters on the Results

_{1}. The decision-makers can choose the appropriate parameter value q according to their preferences. From Figure 1, we can find that when $q\in \left[1,1.71\right]$ the ranking order is ${A}_{1}\succ {A}_{3}\succ {A}_{2}\succ {A}_{4}$ and when $q\in \left[1.71,10\right]$ the ranking order is ${A}_{1}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}$ by the q-RFLWHM operator. In addition, from Figure 2 we know when $q\in \left[1,4.12\right]$ the ranking order is ${A}_{1}\succ {A}_{2}\succ {A}_{4}\succ {A}_{3}$; when $q\in \left[4.12,4.34\right]$ the ranking order is ${A}_{1}\succ {A}_{4}\succ {A}_{2}\succ {A}_{3}$; when $q\in \left[4.34,4.66\right]$ the ranking order is ${A}_{1}\succ {A}_{4}\succ {A}_{3}\succ {A}_{2}$, and when $q\in \left[4.66,10\right]$ the ranking order is ${A}_{1}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}$ by the q-RFLWGHM operator.

_{1}. Especially, in the q-RPLWHM operator, the increase of the parameters s and t leads to increase of the score functions, whereas a decrease of the score functions is witnessed using q-RPLWGHM operator. Furthermore, there is a difference in the ranking orders of A

_{2}, A

_{3}and A

_{4}when $s\to 0$, $t=1$ or $s=1$, $t\to 0$ for the linear weighting by q-RPLWHM or q-RPLWGHM operator. Therefore, the parameters s and t can be also viewed a decision-makers’ optimistic or pessimistic attitude to their assessments. This demonstrates the flexibility in the aggregation processes using the proposed operators.

#### 5.3. Comparative Analysis

#### 5.3.1. Compared with the Method Proposed by Liu and Zhang [41]

#### 5.3.2. Compared with the Methods Proposed by Wang et al. [47], Liu et al. [48], and Ju et al. [49]

_{1}, (0.6, 0.4)〉, it can be transformed into a q-RPLN 〈s

_{1}, (0.6, 0, 0.4)〉. The score values and ranking results by different methods are shown in Table 10, Table 11 and Table 12.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Score values of the alternatives when $q\in \left[1,10\right],s=t=1$ based on q-RPLWHM operator.

**Figure 2.**Score values of the alternatives when $q\in \left[1,10\right],s=t=1$ using q-RPLWGHM operator.

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

A_{1} | $\langle {s}_{4},\left(0.53,0.33,0.09\right)\rangle $ | $\langle {s}_{2},\left(0.89,0.08,0.03\right)\rangle $ | $\langle {s}_{1},\left(0.42,0.35,0.18\right)\rangle $ | $\langle {s}_{3},\left(0.08,0.89,0.02\right)\rangle $ |

A_{2} | $\langle {s}_{2},\left(0.73,0.12,0.08\right)\rangle $ | $\langle {s}_{4},\left(0.13,0.64,0.21\right)\rangle $ | $\langle {s}_{2},\left(0.03,0.82,0.13\right)\rangle $ | $\langle {s}_{4},\left(0.73,0.15,0.08\right)\rangle $ |

A_{3} | $\langle {s}_{5},\left(0.91,0.03,0.02\right)\rangle $ | $\langle {s}_{1},\left(0.07,0.79,0.05\right)\rangle $ | $\langle {s}_{4},\left(0.04,0.85,0.10\right)\rangle $ | $\langle {s}_{2},\left(0.68,0.26,0.06\right)\rangle $ |

A_{4} | $\langle {s}_{5},\left(0.85,0.09,0.05\right)\rangle $ | $\langle {s}_{3},\left(0.74,0.16,0.10\right)\rangle $ | $\langle {s}_{6},\left(0.02,0.89,0.05\right)\rangle $ | $\langle {s}_{1},\left(0.08,0.84,0.06\right)\rangle $ |

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

A_{1} | $\langle {s}_{3},\left(0.53,0.33,0.09\right)\rangle $ | $\langle {s}_{3},\left(0.73,0.12,0.08\right)\rangle $ | $\langle {s}_{2},\left(0.91,0.03,0.02\right)\rangle $ | $\langle {s}_{4},\left(0.85,0.09,0.05\right)\rangle $ |

A_{2} | $\langle {s}_{1},\left(0.89,0.08,0.03\right)\rangle $ | $\langle {s}_{3},\left(0.13,0.64,0.21\right)\rangle $ | $\langle {s}_{3},\left(0.77,0.09,0.05\right)\rangle $ | $\langle {s}_{4},\left(0.74,0.16,0.10\right)\rangle $ |

A_{3} | $\langle {s}_{4},\left(0.42,0.35,0.18\right)\rangle $ | $\langle {s}_{2},\left(0.03,0.82,0.13\right)\rangle $ | $\langle {s}_{4},\left(0.04,0.85,0.10\right)\rangle $ | $\langle {s}_{3},\left(0.02,0.89,0.05\right)\rangle $ |

A_{4} | $\langle {s}_{5},\left(0.33,0.51,0.12\right)\rangle $ | $\langle {s}_{2},\left(0.53,0.31,0.16\right)\rangle $ | $\langle {s}_{6},\left(0.68,0.26,0.06\right)\rangle $ | $\langle {s}_{3},\left(0.08,0.84,0.06\right)\rangle $ |

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

A_{1} | $\langle {s}_{4},\left(0.33,0.52,0.12\right)\rangle $ | $\langle {s}_{2},\left(0.52,0.31,0.16\right)\rangle $ | $\langle {s}_{4},\left(0.31,0.39,0.25\right)\rangle $ | $\langle {s}_{5},\left(0.64,0.16,0.10\right)\rangle $ |

A_{2} | $\langle {s}_{4},\left(0.17,0.53,0.13\right)\rangle $ | $\langle {s}_{3},\left(0.51,0.24,0.21\right)\rangle $ | $\langle {s}_{4},\left(0.31,0.39,0.25\right)\rangle $ | $\langle {s}_{5},\left(0.64,0.16,0.10\right)\rangle $ |

A_{3} | $\langle {s}_{2},\left(0.90,0.05,0.02\right)\rangle $ | $\langle {s}_{1},\left(0.68,0.08,0.21\right)\rangle $ | $\langle {s}_{5},\left(0.05,0.87,0.06\right)\rangle $ | $\langle {s}_{3},\left(0.13,0.75,0.09\right)\rangle $ |

A_{4} | $\langle {s}_{3},\left(0.15,0.73,0.08\right)\rangle $ | $\langle {s}_{3},\left(0.70,0.20,0.10\right)\rangle $ | $\langle {s}_{5},\left(0.91,0.03,0.05\right)\rangle $ | $\langle {s}_{3},\left(0.18,0.64,0.06\right)\rangle $ |

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

A_{1} | $\langle {s}_{3},\left(0.90,0.05,0.02\right)\rangle $ | $\langle {s}_{1},\left(0.68,0.08,0.21\right)\rangle $ | $\langle {s}_{3},\left(0.05,0.87,0.06\right)\rangle $ | $\langle {s}_{1},\left(0.13,0.75,0.09\right)\rangle $ |

A_{2} | $\langle {s}_{6},\left(0.77,0.13,0.10\right)\rangle $ | $\langle {s}_{2},\left(0.62,0.24,0.11\right)\rangle $ | $\langle {s}_{2},\left(0.10,0.75,0.10\right)\rangle $ | $\langle {s}_{4},\left(0.64,0.16,0.10\right)\rangle $ |

A_{3} | $\langle {s}_{3},\left(0.80,0.15,0.02\right)\rangle $ | $\langle {s}_{4},\left(0.68,0.18,0.05\right)\rangle $ | $\langle {s}_{5},\left(0.05,0.87,0.06\right)\rangle $ | $\langle {s}_{1},\left(0.12,0.65,0.20\right)\rangle $ |

A_{4} | $\langle {s}_{6},\left(0.15,0.73,0.08\right)\rangle $ | $\langle {s}_{3},\left(0.61,0.25,0.10\right)\rangle $ | $\langle {s}_{5},\left(0.91,0.03,0.05\right)\rangle $ | $\langle {s}_{4},\left(0.28,0.44,0.16\right)\rangle $ |

**Table 5.**Collective picture fuzzy linguistic decision matrix (by q-rung Picture Linguistic Weighted Geometric Heronian Mean (q-RPLWHM) operator).

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

A_{1} | $\langle {s}_{3.52},\left(0.71,0.29,0.18\right)\rangle $ | $\langle {s}_{1.91},\left(0.76,0.15,0.14\right)\rangle $ | $\langle {s}_{2.44},\left(0.63,0.45,0.39\right)\rangle $ | $\langle {s}_{2.83},\left(0.57,0.61,0.44\right)\rangle $ |

A_{2} | $\langle {s}_{3.56},\left(0.75,0.19,0.15\right)\rangle $ | $\langle {s}_{3.04},\left(0.48,0.44,0.30\right)\rangle $ | $\langle {s}_{2.60}\uff0c\left(0.51,0.52,0.38\right)\rangle $ | $\langle {s}_{4.21},\left(0.69,0.17,0.14\right)\rangle $ |

A_{3} | $\langle {s}_{3.67},\left(0.83,0.13,0.11\right)\rangle $ | $\langle {s}_{2.22},\left(0.56,0.42,0.23\right)\rangle $ | $\langle {}_{}$ |