# m-Polar Fuzzy Soft Weighted Aggregation Operators and Their Applications in Group Decision-Making

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Weighted Aggregation Operator

#### 2.2. Fuzzy Sets

**Theorem**

**1**(Multiplicative transitivity)

**.**

**Definition**

**1**(m-polar fuzzy set)

**.**

- If ${\mu}_{i}^{s}(u)\le {\mu}_{j}^{s}(u)$ for all $s=1,\cdots ,m$, then ${\mu}_{i}\le {\mu}_{j}$.
- $({\bigvee}_{k}{\mu}_{k})(u)={sup}_{k}\left\{{\mu}_{k}(u)\right\}=({sup}_{k}\left\{{\mu}_{k}^{1}(u)\right\},\cdots ,{sup}_{k}\left\{{\mu}_{k}^{m}(u)\right\})$.
- $({\bigwedge}_{k}{\mu}_{k})(u)={inf}_{k}\left\{{\mu}_{k}(u)\right\}=({inf}_{k}\left\{{\mu}_{k}^{1}(u)\right\},\cdots ,{inf}_{k}\left\{{\mu}_{k}^{m}(u)\right\})$.

#### 2.3. Fuzzy Soft Sets

**Definition**

**2**(Fuzzy soft set)

**.**

- If ${P}_{i}\subset {P}_{j}$ and ${f}_{i}(p)(u)\le {f}_{j}(p)(u)$ for all $p\in {P}_{i}$, then $({f}_{i},{P}_{i})\tilde{\le}({f}_{j},{P}_{j})$.
- $({\tilde{\vee}}_{k}{f}_{k})(p)(u)={sup}_{k}\left\{{f}_{k}(p)(u)\right\}$ for all $p\in {\cup}_{k}{P}_{k}$.
- $({\tilde{\wedge}}_{k}{f}_{k})(p)(u)={inf}_{k}\left\{{f}_{k}(p)(u)\right\}$ for all $p\in {\cap}_{k}{P}_{i}$.

## 3. A New Weighted Aggregation Operator for M-pFSSs

#### 3.1. m-Polar Fuzzy Soft Sets

**Definition**

**3**(m-polar fuzzy soft set)

**.**

**Example**

**1.**

**Definition**

**4.**

- 1.
- $({f}_{i},{P}_{j})\tilde{\le}({f}_{j},{P}_{j})$ if ${P}_{i}\subseteq {P}_{j}$ and ${f}_{i}^{s}(a)(u)\le {f}_{j}^{s}(a)(u)$ for all $a\in {P}_{i}$ and $s=1,2,\cdots ,m$.
- 2.
- $({\tilde{\vee}}_{k}{f}_{k})(a)(u)={sup}_{k}\left\{{f}_{k}(a)(u)\right\}=({sup}_{k}\left\{{f}_{k}^{1}(a)(u)\right\},\cdots ,{sup}_{k}\left\{{f}_{k}^{m}(a)(u)\right\})$, for all $a\in {\bigcup}_{k\in K}{P}_{k}$.
- 3.
- $({\tilde{\wedge}}_{k}{f}_{k})(a)(u)={inf}_{k}\left\{{f}_{k}(a)(u)\right\}=({inf}_{k}\left\{{f}_{k}^{1}(a)(u)\right\},\cdots ,{inf}_{k}\left\{{f}_{k}^{m}(a)(u)\right\})$, for all $a\in {\bigcap}_{k\in K}{P}_{k}$.

**Proposition**

**1.**

- 1.
- $(f,P)\tilde{\vee}\tilde{0}=(f,P)$, $(f,P)\tilde{\wedge}\tilde{0}=\tilde{0}$ and $(f,P)\tilde{\vee}\tilde{1}=\tilde{1}$ and $(f,P)\tilde{\wedge}\tilde{1}=(f,P)$.
- 2.
- (Idempotent) $(f,P)\tilde{\vee}(f,P)=(f,P)$ and $(f,P)\tilde{\wedge}(f,P)=(f,P)$.
- 3.
- (Commutative) $(f,P)\tilde{\vee}(g,Q)=(g,Q)\tilde{\vee}(f,P)$ and $(f,P)\tilde{\wedge}(g,Q)=(g,Q)\tilde{\wedge}(f,P)$.
- 4.
- (Associative) $(f,P)\tilde{\vee}\left[(g,Q)\tilde{\vee}(h,E)\right]=\left[(f,P)\tilde{\vee}(g,Q)\right]\tilde{\vee}(h,E)$ and$(f,P)\tilde{\wedge}\left[(g,Q)\tilde{\wedge}(h,E)\right]=\left[(f,P)\tilde{\wedge}(g,Q)\right]\tilde{\wedge}(h,E)$.
- 5.
- (Distributive) $(f,P)\tilde{\vee}\left[(g,Q)\tilde{\wedge}(h,E)\right]=\left[(f,P)\tilde{\vee}(g,Q)\right]\tilde{\wedge}\left[(f,P)\tilde{\vee}(h,E)\right]$ and$(f,P)\tilde{\wedge}\left[(g,Q)\tilde{\vee}(h,E)\right]=\left[(f,P)\tilde{\wedge}(g,Q)\right]\tilde{\vee}\left[(f,P)\tilde{\wedge}(h,E)\right]$.

**Proof.**

**Proposition**

**2**(De Morgan Law)

**.**

- 1.
- ${\left[(f,P)\tilde{\vee}(g,Q)\right]}^{c}=({f}^{c},P)\tilde{\wedge}({g}^{c},Q)$.
- 2.
- ${\left[(f,P)\tilde{\wedge}(g,Q)\right]}^{c}=({f}^{c},P)\tilde{\vee}({g}^{c},Q)$.

**Proof.**

#### 3.2. The M-pFSMWM Operator

**Definition**

**5**(M-pFSMWM Operator)

**.**

**Theorem**

**2.**

- 1.
- (Idempotency) If $({f}_{k},P)=(f,P)$ for all k, then$$M-pFSMW{M}^{(K,\alpha ,m)}\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle =(f,P)$$
- 2.
- (Boundary Conditions)$$M-pFSMW{M}^{(K,\alpha ,m)}\langle \tilde{\mathbf{0}},\cdots ,\tilde{\mathbf{0}}\rangle =\tilde{\mathbf{0}}$$$$M-pFSMW{M}^{(K,\alpha ,m)}\langle \tilde{\mathbf{1}},\cdots ,\tilde{\mathbf{1}}\rangle =\tilde{\mathbf{1}}$$
- 3.
- (Monotonicity) If $({f}_{k},P)\tilde{\le}({g}_{k},P)$ for all k, then$$M-pFSMW{M}^{(K,\alpha ,m)}\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle \tilde{\le}M-pFSMW{M}^{(K,\alpha ,m)}\langle ({g}_{1},P),\cdots ,({g}_{K},P)\rangle $$
- 4.
- (Boundedness)$$\underset{k}{min}{\left\{({f}_{k},P)\right\}}_{k=1}^{K}\tilde{\le}M-pFSMW{M}^{(K,\alpha ,m)}\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle \tilde{\le}\underset{k}{max}{\left\{({f}_{k},P)\right\}}_{k=1}^{K}$$
- 5.
- (Commutativity or Symmetry)$$M-pFSMW{M}^{(K,\alpha ,m)}\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle =M-pFSMW{M}^{(K,\alpha ,m)}\langle ({f}_{\sigma (1)},P),\cdots ,({f}_{\sigma (K)},P)\rangle $$

**Proof.**

- Let for all k: $({f}_{k},P)=(f,P)$. Thus, it is clear that the distinct α-combinations of K objects is reduced to the trivial case K-combination of K with ${C}_{K,K}=1$ and ${\omega}_{k}=\frac{1}{K}$ for all k, i.e., the unweighted case. Thus,$$\begin{array}{cc}\hfill M-pFSMW{M}^{(K,\alpha ,m)}\u2329{f}_{1},\cdots ,{f}_{K}\u232a(p)(u)& =M-pFSMW{M}^{(K,K,m)}\u2329f,\cdots ,f\u232a(p)(u)\hfill \\ & =({f}^{1}(p)(u),\cdots ,{f}^{m}(p)(u))=f(p)(u)\hfill \end{array}$$
- First, assume for all k: $({f}_{k},P)=\tilde{\mathbf{0}}$. Then, by Property 1 of Theorem 2, we have $M-pFSMW{M}^{(K,\alpha ,m)}\langle \tilde{\mathbf{0}},\cdots ,\tilde{\mathbf{0}}\rangle =\tilde{\mathbf{0}}$. The property $M-pFSMW{M}^{(K,\alpha ,m)}\langle \tilde{\mathbf{1}},\cdots ,\tilde{\mathbf{1}}\rangle =\tilde{\mathbf{1}}$ follows the similar way for $({f}_{k},P)=\tilde{\mathbf{1}}$, $\forall k$.
- Let $({f}_{k},P)\tilde{\le}({g}_{k},P)$ for all k. Then, for each $s=1,2,\cdots ,m$: ${f}_{k}^{s}(p)(u)\le {g}_{k}^{s}(p)(u)$. Thus, the condition is hold since $min{\{{f}_{\sigma ({\delta}_{1}(l))}^{s}(p)(u),\cdots ,{f}_{\sigma ({\delta}_{k}(l))}^{s}(p)(u)\}}_{{\delta}_{k}(l)\in {\Delta}_{K,\alpha}(l)}\le min{\{{g}_{\sigma ({\delta}_{1}(l))}^{s}(p)(u),\cdots ,{g}_{\sigma ({\delta}_{k}(l))}^{s}(p(u))\}}_{{\delta}_{k}(l)\in {\Delta}_{K,\alpha}(l)}$: $s=1,2,\cdots ,m$, for any l of the ${C}_{K,\alpha}$ possible choices of K.
- Let for $p\in P$ and $u\in U$: ${min}_{k}{f}_{k}^{s}(p)(u)={{B}_{\ast}}^{s}$ and ${max}_{k}{f}_{k}^{s}(p)(u)={{B}^{\ast}}^{s}$ for each $s=1,2,\cdots ,m$. Then, for all k: ${\mathit{B}}_{\ast}\le {f}_{k}(p)(u)\le {\mathit{B}}^{\ast}$ where ${\mathit{B}}_{\ast}=({B}_{\ast}^{1},\cdots ,{B}_{\ast}^{m})$ and ${\mathit{B}}^{\ast}=({{B}^{\ast}}^{1},\cdots ,{{B}^{\ast}}^{m})$. Hence, by Properties 1 and 3 of Theorem 2, the inequality holds.
- It is trivial from Definition 5.

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Theorem**

**4.**

- 1.
- If $\mathit{\omega}={(0,\cdots ,\underset{\underset{j-th}{\ufe38}}{1},\cdots ,0)}^{T}$ i.e., ${\omega}_{j}=1$ for $k=j$ and ${\omega}_{k}=0$ for $k\ne j$, then$$M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1},\cdots ,{f}_{K}\rangle (p)(u)={f}_{j}(p)(u)$$
- 2.
- When $K=\alpha $, we have$$M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1}\cdots ,{f}_{K}\rangle (p)(u)=\langle \underset{{\omega}_{1},\cdots ,{\omega}_{K}}{\overset{\otimes}{min}}{\left\{{f}_{k}^{1}(p)(u)\right\}}_{k=1}^{K},\cdots ,\underset{{\omega}_{1},\cdots ,{\omega}_{K}}{\overset{\otimes}{min}}{\left\{{f}_{k}^{m}(p)(u)\right\}}_{k=1}^{K}\rangle $$
- 3.
- When $K=\alpha $: if $\omega ={(\frac{1}{K},\frac{1}{K},\cdots ,\frac{1}{K})}^{T}$, then$$M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1}\cdots ,{f}_{K}\rangle (p)(u)=\langle \underset{k}{min}{\left\{{f}_{k}^{1}(p)(u)\right\}}_{k=1}^{K},\cdots ,\underset{k}{min}{\left\{{f}_{k}^{m}(p)(u)\right\}}_{k=1}^{K}\rangle $$
- 4.
- When $K=\alpha $: if ${f}_{\sigma (1)}\tilde{\ge}\cdots \tilde{\ge}{f}_{\sigma (K)}$; and ${\omega}_{\sigma (1)}=1$ and ${\omega}_{\sigma (k)}=0$ for all $k\ne 1$, then$$M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1}\cdots ,{f}_{K}\rangle (p)(u)=\langle \underset{k}{max}{\left\{{f}_{k}^{1}(p)(u)\right\}}_{k=1}^{K},\cdots ,\underset{k}{max}{\left\{{f}_{k}^{m}(p)(u)\right\}}_{k=1}^{K}\rangle $$
- 5.
- When $K=\alpha $: If ${f}_{\sigma (1)}\tilde{\ge}\cdots \tilde{\ge}{f}_{\sigma (K)}$; and ${\omega}_{\sigma (K)}=1$ and ${\omega}_{\sigma (k)}=0$ for all $k\ne K$, then$$M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1}\cdots ,{f}_{K}\rangle (p)(u)=\langle \underset{k}{min}{\left\{{f}_{k}^{1}(p)(u)\right\}}_{k=1}^{K},\cdots ,\underset{k}{min}{\left\{{f}_{k}^{m}(p)(u)\right\}}_{k=1}^{K}\rangle $$

**Proof.**

- Let $\mathit{\omega}={({\omega}_{1},\cdots ,1,\cdots ,{\omega}_{K})}^{T}$, then in any lth α-combination of K objects involving jth element, the value of ${f}_{j}(p)(u)$ for $p\in P$ and $u\in U$ is interpreted as the first object, i.e., ${f}_{\sigma ({\delta}_{1}(l))}(p)(u)$, where ${\omega}_{\sigma ({\delta}_{1}(l))}={\omega}_{j}=1$ and for rest ${\omega}_{k}=0$. Thus, by using Equation (5): $M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1},\cdots ,{f}_{K}\rangle (p)(u)=({f}_{j}^{1}(p)(u),\cdots ,{f}_{j}^{m}(p)(u))={f}_{j}(p)(u)$.
- Let $K=\alpha $. Then, we have ${C}_{K,K}=1$ (only one trivial combination) and thus ${\Delta}_{K,K}(1)=\{1,\cdots ,K\}$. Hence, by Equation (5):$$\begin{array}{cc}\hfill M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1},\cdots ,{f}_{K}\rangle (p)(u)& =M-pFSMW{M}^{(K,K,m)}\langle {f}_{1},\cdots ,{f}_{K}\rangle (p)(u)\hfill \\ & =\langle \underset{{\omega}_{1},\cdots ,{\omega}_{K}}{\overset{\otimes}{min}}\{{f}_{1}^{1}(p)(u),\cdots ,{f}_{K}^{1}(p)(u)\},\cdots ,\hfill \\ & \underset{{\omega}_{1},\cdots ,{\omega}_{K}}{\overset{\otimes}{min}}\{{f}_{1}^{m}(p)(u),\cdots ,{f}_{K}^{m}(p)(u)\}\rangle \hfill \end{array}$$
- When $\mathit{\omega}=(\frac{1}{K},\frac{1}{K},\cdots ,\frac{1}{K})$, then the resultant weighted minimum in Part 2 of Theorem 4 acts as the standard (unweighted) minimum operator. Thus, the $M-pFMWM$ operator is derived by the minimum operator, easily.
- When ${\omega}_{\sigma (1)}=1$ and ${\omega}_{\sigma (k)}=0$ for all $k\ne 1$, then by Part 1 of Theorem 4 we have $M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1},\cdots ,{f}_{K}\rangle (p)(u)={f}_{\sigma (1)}(p)(u)$ that is the largest argument since ${f}_{\sigma (1)}\tilde{\ge}\cdots \tilde{\ge}{f}_{\sigma (K)}$, i.e., the $M-pFMWM$ operator is derived by the maximum operator.
- When ${\omega}_{\sigma (K)}=1$ and ${\omega}_{\sigma (k)}=0$ for all $k\ne K$, then by Part 1 of Theorem 4 we have $M-pFSMW{M}^{(K,\alpha ,m)}\langle {f}_{1},\cdots ,{f}_{K}\rangle (p)(u)={f}_{\sigma (K)}(p)(u)$ that is the lowest argument since ${f}_{\sigma (1)}\tilde{\ge}\cdots \tilde{\ge}{f}_{\sigma (K)}$, i.e., the $M-pFMWM$ operator is derived by the minimum operator.

## 4. Application of M-pFSMWM Operator in Group Decision-Making

#### 4.1. A Fuzzy Soft Preference Relationship

**Definition**

**6.**

**Proposition**

**3.**

- 1.
- ${\tilde{r}}_{ij}({p}_{y})+{\tilde{r}}_{ji}({p}_{y})=1$
- 2.
- $(\frac{{\tilde{r}}_{ji}({p}_{y})}{{\tilde{r}}_{ij}({p}_{y})})(\frac{{\tilde{r}}_{kj}({p}_{y})}{{\tilde{r}}_{jk}({p}_{y})})=\frac{{\tilde{r}}_{ki}({p}_{y})}{{\tilde{r}}_{ik}({p}_{y})}$
- 3.
- If ${\tilde{r}}_{ij}({p}_{y})\ge 0.5$ and ${\tilde{r}}_{jk}({p}_{y})\ge 0.5$, then ${\tilde{r}}_{ik}({p}_{y})\ge max\{{\tilde{r}}_{ij}({p}_{y}),{\tilde{r}}_{jk}({p}_{y})\}$.

**Proof.**

**Definition**

**7.**

**Proposition**

**4.**

- 1.
- ${\tilde{\mathit{a}}}_{ij}+{\tilde{\mathit{a}}}_{ji}=\underset{\underset{M-times}{\ufe38}}{(1,1,\cdots ,1)}$.
- 2.
- If ${\tilde{\mathit{a}}}_{ij}\xb7{\tilde{\mathit{a}}}_{jk}=({\tilde{a}}_{ij}^{1},{\tilde{a}}_{ij}^{2},\cdots ,{\tilde{a}}_{ij}^{M})\xb7({\tilde{a}}_{jk}^{1},{\tilde{a}}_{jk}^{2},\cdots ,{\tilde{a}}_{jk}^{M})=({\tilde{a}}_{ij}^{1}\xb7{\tilde{a}}_{jk}^{1},{\tilde{a}}_{ij}^{2}\xb7{\tilde{a}}_{jk}^{2},\cdots ,{\tilde{a}}_{ij}^{M}\xb7{\tilde{a}}_{jk}^{M})$, then ${\tilde{\mathit{a}}}_{ij}\xb7{\tilde{\mathit{a}}}_{jk}={\tilde{\mathit{a}}}_{ik}$.
- 3.
- If ${\tilde{\mathit{a}}}_{ij}=\underset{\underset{M-times}{\ufe38}}{(1,1,\cdots ,1)}$ and ${\tilde{\mathit{a}}}_{jk}=\underset{\underset{M-times}{\ufe38}}{(1,1,\cdots ,1)}$, then ${\tilde{\mathit{a}}}_{ik}=\underset{\underset{M-times}{\ufe38}}{(1,1,\cdots ,1)}$.

**Proof.**

#### 4.2. An Approach to Group Decision-Making Based on M-pFSMWM Operator

**Definition**

**8.**

Algorithm 1. Finding the optimum solution in MAGDM problems based on M-pFSMWM operator for M-pFSSs. |

**Remark**

**2**(Analysing Algorithm 1)

**.**

## 5. The M-pFSIOWA and M-pFSIOWG Operators

**Definition**

**9.**

- 1.
- The M-pFSIOWA operator of dimension k is the mapping $M-pFSIOWA:{\bigcup}_{K\in \mathbb{N}}{(mfs(U))}^{K}\to mfs(U)$ such that for an associated weighting vector $\mathit{w}={({w}_{1},{w}_{2},\cdots ,{w}_{K})}^{T}$, where ${w}_{j}\in [0,1]$ and ${\sum}_{j=1}^{K}{w}_{j}=1$, is defined as below:$$\begin{array}{c}\hfill M-pFSIOWA\u2329{f}_{1},\cdots ,{f}_{K}\u232a({p}_{y})({u}_{i})=\u2329\sum _{j=1}^{K}{w}_{j}\xb7{F}_{jyi}^{1},\cdots ,\sum _{j=1}^{K}{w}_{j}\xb7{F}_{jyi}^{m}\u232a\end{array}$$
- 2.
- The M-pFSIOWG operator of dimension k is the mapping $M-pFSIOWG:{\bigcup}_{K\in \mathbb{N}}{(mfs(U))}^{K}\to mfs(U)$ such that for an associated weighting vector $\mathit{w}={({w}_{1},{w}_{2},\cdots ,{w}_{K})}^{T}$, where ${w}_{j}\in [0,1]$ and ${\sum}_{j=1}^{K}{w}_{j}=1$, can be defined by$$\begin{array}{c}\hfill M-pFSIOWG\u2329{f}_{1},\cdots ,{f}_{K}\u232a({p}_{y})({u}_{i})=\u2329\prod _{j=1}^{K}{({F}_{jyi}^{1})}^{{w}_{j}},\cdots ,\prod _{j=1}^{K}{({F}_{jyi}^{m})}^{{w}_{j}}\u232a\end{array}$$

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

- 1.
- (Idempotency) If $({f}_{k},P)=(f,P)$ $\forall k$, then$$M-pFSIOWA\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle =(f,P)$$$$M-pFSIOWG\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle =(f,P)$$
- 2.
- (Monotonicity) If $({f}_{k},P)\tilde{\le}({g}_{k},P)$ $\forall k$, then$$M-pFSIOWA\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle \tilde{\le}M-pFSIOWA\langle ({g}_{1},P),\cdots ,({g}_{K},P)\rangle $$$$M-pFSIOWG\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle \tilde{\le}M-pFSIOWG\langle ({g}_{1},P),\cdots ,({g}_{K},P)\rangle $$
- 3.
- (Boundedness)$$\underset{k}{min}\left\{({f}_{k},P)\right\}\tilde{\le}M-pFSIOWA\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle \tilde{\le}\underset{k}{max}\left\{({f}_{k},P)\right\}$$$$\underset{k}{min}\left\{({f}_{k},P)\right\}\tilde{\le}M-pFSIOWG\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle \tilde{\le}\underset{k}{max}\left\{({f}_{k},P)\right\}$$
- 4.
- (Commutativity or Symmetry)$$M-pFSIOWA\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle =M-pFSIOWA\langle ({f}_{\sigma (1)},P),\cdots ,({f}_{\sigma (K)},P)\rangle $$$$M-pFSIOWG\langle ({f}_{1},P),\cdots ,({f}_{K},P)\rangle =M-pFSIOWG\langle ({f}_{\sigma (1)},P),\cdots ,({f}_{\sigma (K)},P)\rangle $$

**Theorem**

**7.**

- 1.
- If $\mathit{w}={(\frac{1}{K},\cdots ,\frac{1}{K})}^{T}$, then$$M-pFSIOWA\langle {f}_{1},\cdots ,{f}_{K}\rangle ({p}_{y})({u}_{i})=\langle \frac{1}{K}\sum _{j=1}^{K}{F}_{jyi}^{1},\cdots ,\frac{1}{K}\sum _{j=1}^{K}{F}_{jyi}^{m}\rangle $$$$M-pFSIOWG\langle {f}_{1},\cdots ,{f}_{K}\rangle ({p}_{y})({u}_{i})=\langle \prod _{j=1}^{K}{({F}_{jyi}^{1})}^{\frac{1}{K}},\cdots ,\prod _{j=1}^{K}{({F}_{jyi}^{m})}^{\frac{1}{K}}\rangle $$
- 2.
- If $\mathit{w}={(1,0,\cdots ,0)}^{T}$, then$$M-pFSIOWA\langle {f}_{1},\cdots ,{f}_{K}\rangle ({p}_{y})({u}_{i})=\langle \underset{j}{max}{\left\{{F}_{jyi}^{1}\right\}}_{j=1}^{K},\cdots ,\underset{j}{max}{\left\{{F}_{jyi}^{m}\right\}}_{j=1}^{K}\rangle $$$$M-pFSIOWG\langle {f}_{1},\cdots ,{f}_{K}\rangle ({p}_{y})({u}_{i})=\langle \underset{j}{max}{\left\{{F}_{jyi}^{1}\right\}}_{j=1}^{K},\cdots ,\underset{j}{max}{\left\{{F}_{jyi}^{m}\right\}}_{j=1}^{K}\rangle $$
- 3.
- If $\mathit{w}={(0,\cdots ,0,1)}^{T}$, then$$M-pFSIOWA\langle {f}_{1},\cdots ,{f}_{K}\rangle ({p}_{y})({u}_{i})=\langle \underset{j}{min}{\left\{{F}_{jyi}^{1}\right\}}_{j=1}^{K},\cdots ,\underset{j}{min}{\left\{{F}_{jyi}^{m}\right\}}_{j=1}^{K}\rangle $$$$M-pFSIOWG\langle {f}_{1},\cdots ,{f}_{K}\rangle ({p}_{y})({u}_{i})=\langle \underset{j}{min}{\left\{{F}_{jyi}^{1}\right\}}_{j=1}^{K},\cdots ,\underset{j}{min}{\left\{{F}_{jyi}^{m}\right\}}_{j=1}^{K}\rangle $$

#### Application of M-pFSIOWA and M-pFSIOWG Operators in Group Decision-Making

Algorithm 2. Finding the optimum solution in MAGDM problems based on M-pFSIOWA or M-pFSIOWG operators for M-pFSSs. |

**Remark**

**3.**

## 6. Illustrative Example

#### Hotel Booking Problem

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The effect of consensus degree on scores of alternatives based on the M-pFSMWM aggregation method.

**Figure 3.**The effect of consensus degree on scores of alternatives based on different M-pFS-based aggregation methods.

**Figure 4.**The effect of different methods on scores of alternatives for different consensus degrees.

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

${p}_{1}$: | |||||

${p}_{1}^{1}$ | Close | Far | Very Far | Very Close | |

${p}_{1}^{2}$ | No | No | Yes | Yes | |

${p}_{1}^{3}$ | Yes | Yes | No | No | |

${p}_{2}$: | |||||

${p}_{2}^{1}$ | Very Good | Medium | Medium Poor | Good | |

${p}_{2}^{2}$ | Very Poor | Good | Medium | Good | |

${p}_{2}^{3}$ | Medium | Very Good | Poor | Medium | |

${p}_{3}$: | |||||

${p}_{3}^{1}$ | Yes | Yes | Yes | No | |

${p}_{3}^{2}$ | No | Yes | Yes | Yes | |

${p}_{3}^{3}$ | Medium Good | Medium | Poor | Good |

U | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ |
---|---|---|---|

${x}_{1}$ | (0.9,0,1) | (1,0,0.5) | (1,0,0.7) |

${x}_{2}$ | (0.1,0,1) | (0.5,0.9,1) | (1,1,0.5) |

${x}_{3}$ | (0,1,0) | (0.3,0.5,0.1) | (1,1,0.1) |

${x}_{4}$ | (1,1,0) | (0.9,0.9,0.5) | (0,1,0.9) |

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

${h}_{1}$ | (0.87,0.86) | (0.7,0.86) | (0.86,0.67) | (0.58,0.81) |

${h}_{2}$ | (0.7,0.7) | (0.55,0.7) | (0.74,0.73) | (0.71,0.7) |

${h}_{3}$ | (0.82,0.86) | (0.6,0.86) | (0.79,0.58) | (0.6,0.8) |

${h}_{4}$ | (0.83,0.81) | (0.88,0.81) | (0.84,0.78) | (0.55,0.81) |

${h}_{5}$ | (0.89,0.81) | (0.64,0.81) | (0.82,0.71) | (0.69,0.84) |

${h}_{6}$ | (0.68,0.69) | (0.66,0.69) | (0.82,0.77) | (0.67,0.74) |

${h}_{7}$ | (0.82,0.78) | (0.73,0.78) | (0.77,0.83) | (0.66,0.8) |

${h}_{8}$ | (0.78,0.8) | (0.73,0.8) | (0.74,0.8) | (0.68,0.79) |

${h}_{9}$ | (0.82,0.71) | (0.8,0.71) | (0.69,0.84) | (0.64,0.76) |

${h}_{10}$ | (0.88,0.89) | (0.86,0.89) | (0.7,0.8) | (0.83,0.85) |

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

${h}_{1}$ | (0.89,0.79) | (0.72,0.84) | (0.84,0.7) | (0.59,0.8) |

${h}_{2}$ | (0.74,0.76) | (0.6,0.76) | (0.77,0.75) | (0.69,0.71) |

${h}_{3}$ | (0.77,0.68) | (0.73,0.77) | (0.71,0.6) | (0.57,0.69) |

${h}_{4}$ | (0.78,0.72) | (0.87,0.79) | (0.85,0.65) | (0.54,0.78) |

${h}_{5}$ | (0.91,0.83) | (0.8,0.91) | (0.81,0.71) | (0.66,0.88) |

${h}_{6}$ | (0.69,0.7) | (0.68,0.71) | (0.82,0.75) | (0.69,0.75) |

${h}_{7}$ | (0.86,0.78) | (0.8,0.86) | (0.74,0.8) | (0.66,0.8) |

${h}_{8}$ | (0.78,0.78) | (0.75,0.8) | (0.74,0.77) | (0.64,0.8) |

${h}_{9}$ | (0.71,0.75) | (0.85,0.69) | (0.8,0.84) | (0.49,0.72) |

${h}_{10}$ | (0.89,0.94) | (0.86,0.89) | (0.82,0.86) | (0.85,0.92) |

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

${h}_{1}$ | (0.85,0.79) | (0.74,0.85) | (0.84,0.8) | (0.64,0.81) |

${h}_{2}$ | (0.71,0.69) | (0.7,0.71) | (0.74,0.66) | (0.66,0.69) |

${h}_{3}$ | (0.77,0.78) | (0.73,0.8) | (0.73,0.6) | (0.57,0.73) |

${h}_{4}$ | (0.83,0.76) | (0.73,0.84) | (0.82,0.7) | (0.61,0.8) |

${h}_{5}$ | (0.83,0.75) | (0.69,0.82) | (0.7,0.69) | (0.65,0.79) |

${h}_{6}$ | (0.69,0.7) | (0.65,0.7) | (0.8,0.76) | (0.69,0.73) |

${h}_{7}$ | (0.81,0.74) | (0.8,0.82) | (0.75,0.59) | (0.65,0.78) |

${h}_{8}$ | (0.76,0.73) | (0.8,0.76) | (0.71,0.6) | (0.67,0.75) |

${h}_{9}$ | (0.78,0.67) | (0.75,0.8) | (0.77,0.56) | (0.57,0.73) |

${h}_{10}$ | (0.88,0.88) | (0.8,0.86) | (0.68,0.77) | (0.77,0.84) |

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

${h}_{1}$ | (0.83,0.82) | (0.6,0.83) | (0.75,0.7) | (0.68,0.83) |

${h}_{2}$ | (0.72,0.72) | (0.4,0.71) | (0.69,0.71) | (0.64,0.7) |

${h}_{3}$ | (0.74,0.76) | (0.8,0.76) | (0.75,0.59) | (0.62,0.68) |

${h}_{4}$ | (0.88,0.84) | (0.74,0.89) | (0.83,0.77) | (0.69,0.84) |

${h}_{5}$ | (0.79,0.79) | (0.64,0.83) | (0.75,0.76) | (0.61,0.82) |

${h}_{6}$ | (0.69,0.69) | (0.76,0.68) | (0.79,0.81) | (0.66,0.72) |

${h}_{7}$ | (0.81,0.76) | (0.53,0.81) | (0.73,0.84) | (0.66,0.78) |

${h}_{8}$ | (0.73,0.72) | (0.7,0.73) | (0.68,0.79) | (0.64,0.72) |

${h}_{9}$ | (0.82,0.81) | (0.6,0.89) | (0.79,0.87) | (0.71,0.79) |

${h}_{10}$ | (0.87,0.87) | (1,0.87) | (0.7,0.76) | (0.75,0.83) |

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

${h}_{1}$ | (0.87,0.6) | (0.84,0.87) | (0.93,0.84) | (0.6,0.9) |

${h}_{2}$ | (0.74,0.8) | (0.74,0.74) | (0.8,0.74) | (0.64,0.74) |

${h}_{3}$ | (0.85,0.8) | (0.8,0.8) | (0.81,0.8) | (0.62,0.8) |

${h}_{4}$ | (0.81,0.84) | (0.77,0.76) | (0.77,0.76) | (0.59,0.77) |

${h}_{5}$ | (0.8,0.8) | (0.76,0.77) | (0.84,0.77) | (0.61,0.76) |

${h}_{6}$ | (0.67,0.8) | (0.72,0.78) | (0.67,0.78) | (0.66,0.72) |

${h}_{7}$ | (0.78,0.69) | (0.76,0.74) | (0.78,0.74) | (0.64,0.76) |

${h}_{8}$ | (0.77,0.72) | (0.76,0.72) | (0.79,0.72) | (0.67,0.76) |

${h}_{9}$ | (0.65,0.6) | (0.69,0.76) | (0.78,0.76) | (0.44,0.69) |

${h}_{10}$ | (0.88,0.86) | (0.82,0.66) | (0.88,0.66) | (0.77,0.82) |

Aggregation Methods | $\mathit{\alpha}$ | Number of Computational Steps | Scores of Alternatives (${\mathit{S}}_{\mathit{i}}$) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{h}}_{1}$ | ${\mathit{h}}_{2}$ | ${\mathit{h}}_{3}$ | ${\mathit{h}}_{4}$ | ${\mathit{h}}_{5}$ | ${\mathit{h}}_{6}$ | ${\mathit{h}}_{7}$ | ${\mathit{h}}_{8}$ | ${\mathit{h}}_{9}$ | ${\mathit{h}}_{10}$ | |||

M-pFSMWM | $\alpha =3$ | 18 | 6.95 | 2.15 | 1.25 | 5.85 | 6 | 4.1 | 5.8 | 3.65 | 3.05 | 6.2 |

M-pFSMWM | $\alpha =5$ | 9 | 6.65 | 2.45 | 2.15 | 6.55 | 7.1 | 4.1 | 4.95 | 3.55 | 0.6 | 6.9 |

M-pFSIOWA | $\alpha =3$ | 9 | 6.6 | 1.8 | 1.45 | 5.65 | 5.825 | 3.825 | 5.15 | 3.375 | 3.375 | 7.95 |

M-pFSIOWA | $\alpha =5$ | 9 | 7.65 | 3.875 | 7 | 3.8 | 5.9 | 1.025 | 2.85 | 3.6 | 1.4 | 7.9 |

M-pFSIOWG | $\alpha =3$ | 9 | 6.75 | 1.8 | 1.45 | 5.65 | 5.85 | 3.9 | 5.15 | 3.3 | 3.2 | 7.95 |

M-pFSIOWG | $\alpha =5$ | 9 | 7.65 | 3.875 | 7 | 3.8 | 5.9 | 1.025 | 2.85 | 3.6 | 1.4 | 7.9 |

Operators | Idempotency | Boundedness | Monotonicity | Symmetry | Flexible by Consensus Degree α | Flexible for Different Choices of K |
---|---|---|---|---|---|---|

M-pFSMWM | Yes | Yes | Yes | Yes | Yes | Yes |

M-pFSIOWA | Yes | Yes | Yes | Yes | No | No |

M-pFSIOWG | Yes | Yes | Yes | Yes | No | No |

Methods | Consensus Degree α | Preference Order |
---|---|---|

M-pFSMWM | 3 | ${h}_{1}\succ {h}_{10}\succ {h}_{5}\succ {h}_{4}\succ {h}_{7}\succ {h}_{6}\succ {h}_{8}\succ {h}_{9}\succ {h}_{2}\succ {h}_{3}$ |

M-pFSMWM | 5 | ${h}_{5}\succ {h}_{10}\succ {h}_{1}\succ {h}_{4}\succ {h}_{7}\succ {h}_{6}\succ {h}_{8}\succ {h}_{2}\succ {h}_{3}\succ {h}_{9}$ |

M-pFSIOWA | 3 | ${h}_{10}\succ {h}_{1}\succ {h}_{5}\succ {h}_{4}\succ {h}_{7}\succ {h}_{6}\succ {h}_{8}\succ {h}_{9}\succ {h}_{2}\succ {h}_{3}$ |

M-pFSIOWA | 5 | ${h}_{10}\succ {h}_{1}\succ {h}_{3}\succ {h}_{5}\succ {h}_{2}\succ {h}_{4}\succ {h}_{8}\succ {h}_{7}\succ {h}_{9}\succ {h}_{6}$ |

M-pFSIOWG | 3 | ${h}_{10}\succ {h}_{1}\succ {h}_{5}\succ {h}_{4}\succ {h}_{7}\succ {h}_{6}\succ {h}_{8}\succ {h}_{9}\succ {h}_{2}\succ {h}_{3}$ |

M-pFSIOWG | 5 | ${h}_{10}\succ {h}_{1}\succ {h}_{3}\succ {h}_{5}\succ {h}_{2}\succ {h}_{4}\succ {h}_{8}\succ {h}_{7}\succ {h}_{9}\succ {h}_{6}$ |

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**MDPI and ACS Style**

Zahedi Khameneh, A.; Kiliçman, A.
m-Polar Fuzzy Soft Weighted Aggregation Operators and Their Applications in Group Decision-Making. *Symmetry* **2018**, *10*, 636.
https://doi.org/10.3390/sym10110636

**AMA Style**

Zahedi Khameneh A, Kiliçman A.
m-Polar Fuzzy Soft Weighted Aggregation Operators and Their Applications in Group Decision-Making. *Symmetry*. 2018; 10(11):636.
https://doi.org/10.3390/sym10110636

**Chicago/Turabian Style**

Zahedi Khameneh, Azadeh, and Adem Kiliçman.
2018. "m-Polar Fuzzy Soft Weighted Aggregation Operators and Their Applications in Group Decision-Making" *Symmetry* 10, no. 11: 636.
https://doi.org/10.3390/sym10110636