# Multi-Attribute Group Decision-Making Methods Based on Entropy Weights with q-Rung Picture Uncertain Linguistic Fuzzy Information

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**One can describe the representation of a PFS ${C}_{HT}$ in a nonempty fixed set X as follows:

**Definition**

**2**

**.**Let us define a q-ROFS on a nonempty fixed set X as follows:

**Definition**

**3**

**.**We can define a q-RPFS on set of positive real numbers X in the following manner:

**Definition**

**4**

**.**We can define a q-RPLS ${C}_{HT}$ on an ordinary fixed set X:

**Definition**

**5**

**.**Linguistic term set $S=\{{s}_{0},{s}_{1},\dots ,{s}_{t-1}\}$ is composed of k elements, where k is an odd number. Generally, t = 3, 5, 7, 9. For example, t equals 5, $S=\{{s}_{0},{s}_{1},{s}_{2},{s}_{3},{s}_{4}\}=\{verypoor,poor,good,verygood,excellent\}.$

**Definition**

**6**

**.**We can define a q-ROULS ${C}_{HT}$ on an ordinary fixed set X:

**Definition**

**7.**

**Definition**

**8.**

- 1.
- ${\alpha}_{1}\oplus {\alpha}_{2}$ = $<[{s}_{{\varphi}_{1}+{\varphi}_{2}},{s}_{{\psi}_{1}+{\psi}_{2}}],({({u}_{1}^{q}+{u}_{2}^{q}-{u}_{1}^{q}{u}_{2}^{q})}^{\frac{1}{q}},{\eta}_{1}{\eta}_{2},{v}_{1}{v}_{2})>$;
- 2.
- ${\alpha}_{1}\otimes {\alpha}_{2}$= $<[{s}_{{\varphi}_{1}{\varphi}_{2}},{s}_{{\psi}_{1}{\psi}_{2}}],({u}_{1}{u}_{2},{({\eta}_{1}^{q}+{\eta}_{2}^{q}-{\eta}_{1}^{q}{\eta}_{2}^{q})}^{\frac{1}{q}},{({v}_{1}^{q}+{v}_{2}^{q}-{v}_{1}^{q}{v}_{2}^{q})}^{\frac{1}{q}})>$;
- 3.
- $m{\alpha}_{1}=<[{s}_{m{\varphi}_{1}},{s}_{m{\psi}_{1}}],({(1-{(1-{u}_{1}^{q})}^{m})}^{\frac{1}{q}},{\eta}_{1}^{m},{v}_{1}^{m})>$;
- 4.
- ${\alpha}_{1}^{m}=<[{s}_{{\varphi}_{1}^{m}},{s}_{{\psi}_{1}^{m}}],({u}_{1}^{m},{(1-{(1-{\eta}_{1}^{q})}^{m})}^{\frac{1}{q}},{(1-{(1-{v}_{1}^{q})}^{m})}^{\frac{1}{q}})>$.

**Theorem**

**1.**

- 1.
- ${\Gamma}_{1}\oplus {\Gamma}_{2}={\Gamma}_{2}\oplus {\Gamma}_{1}$;
- 2.
- ${\Gamma}_{1}\otimes {\Gamma}_{2}={\Gamma}_{2}\otimes {\Gamma}_{1}$;
- 3.
- $\lambda ({\Gamma}_{1}\oplus {\Gamma}_{2})=\lambda {\Gamma}_{1}\oplus \lambda {\Gamma}_{2}$
- 4.
- ${({\Gamma}_{1}\otimes {\Gamma}_{2})}^{\lambda}={\Gamma}_{1}^{\lambda}\otimes {\Gamma}_{2}^{\lambda}$
- 5.
- ${\lambda}_{1}\Gamma \oplus {\lambda}_{2}\Gamma =({\lambda}_{1}+{\lambda}_{2})\Gamma $
- 6.
- ${\Gamma}^{{\lambda}_{1}}\otimes {\Gamma}^{{\lambda}_{2}}={\Gamma}^{({\lambda}_{1}+{\lambda}_{2})}$
- 7.
- ${({\Gamma}^{{\lambda}_{1}})}^{{\lambda}_{2}}={\Gamma}^{{\lambda}_{1}{\lambda}_{2}}$

**Definition**

**9.**

**Definition**

**10**

**.**Let ${\Gamma}_{1}=<[{s}_{{\varphi}_{1}},{s}_{{\psi}_{1}}],({u}_{1},{\eta}_{1},{v}_{1})>$ and ${\Gamma}_{2}=<[{s}_{{\varphi}_{2}},{s}_{{\psi}_{2}}],({u}_{2},{\eta}_{2},{v}_{2})>$ be two q-RPULSs; then:

- If $({\Gamma}_{1})<S({\Gamma}_{2})$, then ${\Gamma}_{1}<{\Gamma}_{2}$;
- If $({\Gamma}_{1})>S({\Gamma}_{2})$, then ${\Gamma}_{1}>{\Gamma}_{2}$;
- If $({\Gamma}_{1})=S({\Gamma}_{2})$, then
- 1.
- If $A({\Gamma}_{1})<A({\Gamma}_{2})$, then ${\Gamma}_{1}<{\Gamma}_{2}$;
- 2.
- If $A({\Gamma}_{1})>A({\Gamma}_{2})$, then ${\Gamma}_{1}>{\Gamma}_{2}$;
- 3.
- If $A({\Gamma}_{1})=A({\Gamma}_{2})$, then ${\Gamma}_{1}={\Gamma}_{2}$.

**Definition**

**11**

**.**Let ${\Gamma}_{1}=<[{s}_{{\varphi}_{1}},{s}_{{\psi}_{1}}],({u}_{1},{\eta}_{1},{v}_{1})>$ and ${\Gamma}_{2}=<[{s}_{{\varphi}_{2}},{s}_{{\psi}_{2}}],({u}_{2},{\eta}_{2},{v}_{2})>$ be two q-RPULSs, and $q\ge 1$; then, the q-RPUL distance measure between them is defined as:

**Definition**

**12**

**.**Suppose we have a collection of crisp numbers represented as ${\xi}_{i}$ (where i = 1, 2, …, n), then the $MS{M}^{(k)}$ operator is the following:

**Theorem**

**2.**

- 1.
- Idempotency. If ${\sigma}_{i}(i=1,2,\dots ,n)$ = σ for all i, then $MS{M}^{(k)}=(\sigma ,\sigma ,\dots ,\sigma )=\sigma $;
- 2.
- Monotonicity. If ${\sigma}_{i}\le {\zeta}_{i}$ for all i, then $MS{M}^{(k)}({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n})\le MS{M}^{(k)}({\zeta}_{1},{\zeta}_{2},\dots ,{\zeta}_{n})$;
- 3.
- Boundedness. $min({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n})\le MS{M}^{(k)}({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n})\le max({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n})$.

## 3. The q-Rung Picture Uncertain Linguistic Set Aggregation Operators

#### 3.1. The q-Rung Picture Uncertain Linguistic Set Aggregation Maclaurin Symmetric Mean Operators

**Definition**

**13.**

**Proof.**

**Theorem**

**3.**

- 1.
- Idempotency. If all ${h}_{l}=h=<[{s}_{\varphi},{s}_{\psi}],(u,\eta ,v)>$ are equal for all l, then $q-RPULMS{M}^{(k)}$ = $(h,h,\dots ,h)=h$.
- 2.
- Monotonicity. Let ${h}_{l}=<[{s}_{{\varphi}_{l}},{s}_{{\psi}_{l}}],({u}_{l},{\eta}_{l},{v}_{l})>$ and ${g}_{j}=<[{s}_{{\varphi}_{j}},{s}_{{\psi}_{j}}],({u}_{j},{\eta}_{j},{v}_{j})>(l,j=1,2,\dots ,n)$ be two sets of q-RPULSs; if ${s}_{{\varphi}_{l}}\le {s}_{{\varphi}_{j}},{s}_{{\psi}_{l}}\ge {s}_{{\psi}_{j}},{u}_{i}\le {u}_{j},{\eta}_{l}\ge {\eta}_{j},{v}_{l}\ge {v}_{j}$, then$q-RPULMS{M}^{(k)}({h}_{1},{h}_{2},\dots ,{h}_{n})\le q-RPULMS{M}^{(k)}({g}_{1},{g}_{2},\dots ,{g}_{n})$.
- 3.
- Boundedness. Let ${h}_{i}=<[{s}_{{\varphi}_{i}},{s}_{{\psi}_{i}}],({u}_{i},{\eta}_{i},{v}_{i})>(i=1,2,\dots ,n)$ be a set of q-RPULNs and${h}^{+}=([max({s}_{{\varphi}_{i}}),min({s}_{{\psi}_{i}})],max({u}_{i}),min({\eta}_{i}),min({v}_{i}))$,${h}^{-}=([min({s}_{{\varphi}_{i}}),max({s}_{{\psi}_{i}})],min({u}_{i}),max({\eta}_{i}),max({v}_{i}))$; then,${h}^{-}\le q-RPULMS{M}^{(k)}({a}_{1},{a}_{2},\dots ,{a}_{n})\le {h}^{+}$.
- 4.
- Permutation Invariance. Let ${h}_{i}$ be a permutation of ${h}_{i}^{*}$ for all i; therefore,$q-RPULMS{M}^{(k)}({h}_{1},{h}_{2},\dots ,{h}_{n})=q-RPULMS{M}^{(k)}({h}_{1}^{*},{h}_{2}^{*},\dots ,{h}_{n}^{*})$

**Definition**

**14.**

**Theorem**

**4.**

- 1.
- Idempotency. If all ${h}_{i}=h=<[{s}_{\varphi},{s}_{\psi}],(u,\eta ,v)>$ are equal for all i, then $q-RPULWMS{M}^{(k)}$ = $(h,h,\dots ,h)=h$;
- 2.
- Monotonicity. Let ${h}_{i}=<[{s}_{{\varphi}_{i}},{s}_{{\psi}_{i}}],({u}_{i},{\eta}_{i},{v}_{i})>$ and ${g}_{j}=<[{s}_{{\varphi}_{j}},{s}_{{\psi}_{j}}],({u}_{j},{\eta}_{j},{v}_{j})>(i,j=1,2,\dots ,n)$ be two sets of q-RPULSs; if ${s}_{{\varphi}_{i}}\le {s}_{{\varphi}_{j}},{s}_{{\psi}_{i}}\ge {s}_{{\psi}_{j}},{u}_{i}\le {u}_{j},{\eta}_{i}\ge {\eta}_{j},{v}_{i}\ge {v}_{j}$, then$q-RPULWMS{M}^{(k)}({h}_{1},{h}_{2},\dots ,{h}_{n})\le q-RPULWMS{M}^{(k)}({g}_{1},{g}_{2},\dots ,{g}_{n})$;
- 3.
- Boundedness. Let ${h}_{i}=<[{s}_{{\varphi}_{i}},{s}_{{\psi}_{i}}],({u}_{i},{\eta}_{i},{v}_{i})>(i=1,2,\dots ,n)$ be a set of q-RPULNs and${h}^{+}=([max({s}_{{\varphi}_{i}}),min({s}_{{\psi}_{i}})],max({u}_{i}),min({\eta}_{i}),min({v}_{i}))$,${h}^{-}=([min({s}_{{\varphi}_{i}}),max({s}_{{\psi}_{i}})],min({u}_{i}),max({\eta}_{i}),max({v}_{i}))$; then,${h}^{-}\le q-RPULWMS{M}^{(k)}({a}_{1},{a}_{2},\dots ,{a}_{n})\le {h}^{+}$.
- 4.
- Commutativity. Let ${h}_{i}$ be a permutation of ${h}_{i}^{*}$ for all i; therefore,$q-RPULWMS{M}^{(k)}({h}_{1},{h}_{2},\dots ,{h}_{n})=q-RPULWMS{M}^{(k)}({h}_{1}^{*},{h}_{2}^{*},\dots ,{h}_{n}^{*})$

#### 3.2. A Method to Determine the Attribute Weights Based on Entropy

**Definition**

**15.**

- 1.
- G(${a}_{1}$) = 0, if and only if ${a}_{1}$ is a crisp set;
- 2.
- G(${a}_{1}$) = 1 if and only if ${s}_{{\varphi}_{1}}={s}_{{\psi}_{1}}$, ${u}_{1}={\eta}_{1}={v}_{1}$;
- 3.
- G(${a}_{1}$) ≤ G(${a}_{2}$), if and only if ${u}_{1}\le {u}_{2}$, ${\eta}_{1}\ge {\eta}_{2}$ and ${v}_{1}\ge {v}_{2}$;
- 4.
- G(${a}_{1}$) = G(${a}_{1}^{c}$).

## 4. MAGDM Methods Based on q-RPULMSM Operator

## 5. Numerical Example and Comparative Analysis

#### 5.1. Evaluation Steps for the q-RPULWMSM Operator

#### 5.2. Comparative Analysis and Discussion

- Q-rung picture uncertain linguistic sets (q-RPULSs) incorporate qualitative and quantitative aspects of decision-making, while also utilizing linguistic terms that are easily comprehensible and relatable to people’s perception.
- The MSM operator provides a powerful tool to account for the interdependence of multiple input parameters, resulting in the improved accuracy and reliability of evaluation results.
- A novel solution is put forward for the issue of MAGDM with unknown attribute weights.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Abbreviation | Full Name |
---|---|

MAGDM | multi-attribute group decision-making |

FSs | fuzzy sets |

IFSs | intuitionistic fuzzy sets |

PyFSs | Pythagorean fuzzy sets |

PFSs | picture fuzzy sets |

q-ROFs | q-rung orthopair fuzzy sets |

q-ROULSs | q-rung orthopair uncertain linguistic sets |

q-RPULSs | q-rung picture uncertain linguistic sets |

q-RPFSs | q-rung picture fuzzy sets |

ULVs | uncertain linguistic variables |

MSM | Maclaurin symmetric mean |

q-RPULMSM | q-rung picture uncertain linguistic Maclaurin symmetric mean |

q-RPULWMSM | q-rung picture uncertain linguistic weighted Maclaurin symmetric mean |

2TLCq-RPFSs | 2-tuple linguistic complex q-rung picture fuzzy sets |

LDULFPEWG | linear diophantine uncertain linguistic fuzzy power Einstein-weighted geometric |

TSULWA | T-spherical uncertain linguistic weighted averaging |

T-SFWGMSM | T-spherical fuzzy weighted geometric Maclaurin symmetric mean |

q-RPFDHM | q-rung picture fuzzy dual Heronian mean |

q-RPULA | q-rung picture uncertain linguistic averaging |

PULMSM | picture uncertain linguistic Maclaurin symmetric mean |

q-RPULBM | q-rung picture uncertain linguistic Bonferroni mean |

SPULMSM | spherical picture uncertain linguistic Maclaurin symmetric mean |

RAM | random access memory |

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

${A}_{1}$ | <[s${}_{5}$, ${s}_{5}$], (0.5, 0.4, 0.1)> | <[s${}_{2}$, ${s}_{3}$], (0.8, 0.1, 0.1)> | <[s${}_{5}$, ${s}_{6}$], (0.4, 0.3, 0.2)> | <[s${}_{3}$, ${s}_{4}$], (0.1, 0.8, 0.1)> |

${A}_{2}$ | <[s${}_{4}$, ${s}_{5}$], (0.7, 0.1, 0.1)> | <[s${}_{5}$, ${s}_{5}$], (0.1, 0.7, 0.2)> | <[s${}_{3}$, ${s}_{4}$], (0.1, 0.7, 0.2)> | <[s${}_{4}$, ${s}_{4}$], (0.7, 0.1, 0.1)> |

${A}_{3}$ | <[s${}_{3}$, ${s}_{4}$], (0.8, 0.1, 0.1)> | <[s${}_{4}$, ${s}_{4}$], (0.1, 0.8, 0.1)> | <[s${}_{4}$, ${s}_{5}$], (0.1, 0.8, 0.1)> | <[s${}_{4}$, ${s}_{5}$], (0.6, 0.2, 0.1)> |

${A}_{4}$ | <[s${}_{6}$, ${s}_{6}$], (0.7, 0.1, 0.1)> | <[s${}_{2}$, ${s}_{3}$], (0.7, 0.2, 0.1)> | <[s${}_{3}$, ${s}_{4}$], (0.1, 0.7, 0.1)> | <[s${}_{3}$, ${s}_{3}$], (0.1, 0.8, 0.1)> |

Methods | Score Values | Ranking |
---|---|---|

q-RPULWMSM | $S({a}_{1})$ = 0.4916, $S({a}_{2})$ = 0.5171, $S({a}_{3})$ = 0.4590, $S({a}_{4})$ = 0.4387. | ${A}_{2}>{A}_{1}>{A}_{3}>{A}_{4}$ |

TSULWA [36] | $S({a}_{1})$ = 0.4416, $S({a}_{2})$ = 0.4671, $S({a}_{3})$ = 0.4590, $S({a}_{4})$ = 0.4364. | ${A}_{2}>{A}_{3}>{A}_{1}>{A}_{4}$ |

T-SFWGMSM [40] | $S({a}_{1})$ = 0.4334, $S({a}_{2})$ = 0.4571, $S({a}_{3})$ = 0.4532, $S({a}_{4})$ = 0.4236. | ${A}_{2}>{A}_{3}>{A}_{1}>{A}_{4}$ |

q-RPFDWHM [41] | $S({a}_{1})$ = 0.4853, $S({a}_{2})$ = 0.4951, $S({a}_{3})$ = 0.4590, $S({a}_{4})$ = 0.4596. | ${A}_{2}>{A}_{1}>{A}_{4}>{A}_{3}$ |

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**MDPI and ACS Style**

Sun, M.; Geng, Y.; Zhao, J.
Multi-Attribute Group Decision-Making Methods Based on Entropy Weights with q-Rung Picture Uncertain Linguistic Fuzzy Information. *Symmetry* **2023**, *15*, 2027.
https://doi.org/10.3390/sym15112027

**AMA Style**

Sun M, Geng Y, Zhao J.
Multi-Attribute Group Decision-Making Methods Based on Entropy Weights with q-Rung Picture Uncertain Linguistic Fuzzy Information. *Symmetry*. 2023; 15(11):2027.
https://doi.org/10.3390/sym15112027

**Chicago/Turabian Style**

Sun, Mengran, Yushui Geng, and Jing Zhao.
2023. "Multi-Attribute Group Decision-Making Methods Based on Entropy Weights with q-Rung Picture Uncertain Linguistic Fuzzy Information" *Symmetry* 15, no. 11: 2027.
https://doi.org/10.3390/sym15112027