# m-Polar Generalization of Fuzzy T-Ordering Relations: An Approach to Group Decision Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions and Properties

#### 2.1. Aggregation Functions

**A1.**- $A\left(x\right)=x$, for $n=1$ and any $x\in [0,1]$;
**A2.**- ${A}^{\left(n\right)}({x}_{1},\cdots ,{x}_{n})\le {A}^{\left(n\right)}({y}_{1},\cdots ,{y}_{n})$ if $({x}_{1},\cdots ,{x}_{n})\le ({y}_{1},\cdots ,{y}_{n})$;
**A3.**- ${A}^{\left(n\right)}(0,0,\cdots ,0)=0$ and ${A}^{\left(n\right)}(1,1,\cdots ,1)=1$.

**Definition**

**1**

**.**Let $A:{[0,1]}^{m}\to [0,1]$ and $B:{[0,1]}^{n}\to [0,1]$ be two aggregation functions where $m,n\in \mathbb{N}$. We say function A dominates function B, denoted by $A\gg B$, if

#### Triangular Norms and Conorms

**Definition**

**2**

**.**Let T be a left-continuous t-norm. The residuum or implication operation $\overrightarrow{T}:{[0,1]}^{2}\to [0,1]$ with respect to the t-norm T is defined as

**Proposition**

**1**

- 1.
- $x\le y$ if and only if $\overrightarrow{T}(x,y)=1$, so $\overrightarrow{T}(x,x)=1$;
- 2.
- $T(x,y)\le z$ if and only if $\overrightarrow{T}(y,z)\ge x$;
- 3.
- $T(\overrightarrow{T}(x,y),\overrightarrow{T}(y,z))\le \overrightarrow{T}(x,z)$;
- 4.
- $\overrightarrow{T}(1,x)=x$;
- 5.
- $T(x,\overrightarrow{T}(x,y))\le y$;
- 6.
- $y\le \overrightarrow{T}(x,T(x,y))$.

**Definition**

**3**

**.**Let T be a left-continuous t-norm. The biimplication relation of T, known as biimplication operator $\overleftrightarrow{T}:{[0,1]}^{2}\to [0,1]$, is defined by

**Proposition**

**2**

- 1.
- $x=y$ if and only if $\overleftrightarrow{T}(x,y)=1$;
- 2.
- $\overleftrightarrow{T}(x,y)=\overleftrightarrow{T}(y,x)$;
- 3.
- $\overleftrightarrow{T}(x,y)=min(\overrightarrow{T}(x,y),\overrightarrow{T}(y,x))$;
- 4.
- $\overleftrightarrow{T}(x,y)=\overrightarrow{T}(max(x,y),min(x,y))$;
- 5.
- $T(\overleftrightarrow{T}(x,y),\overleftrightarrow{T}(y,z))\le \overleftrightarrow{T}(x,z)$;
- 6.
- $\overleftrightarrow{T}:{[0,1]}^{2}\to [0,1]$ is a fuzzy T-equivalence relation.

#### 2.2. Fuzzy Orderings

- fuzzy ordering if it is reflexive;
- fuzzy preordering if it is reflexive and transitive;
- fuzzy total or linear preordering if it is strongly complete and transitive;
- fuzzy partial ordering or fuzzy weak preference ordering if it is reflexive, antisymmetric and transitive;
- fuzzy strict preference ordering if it is antisymmetric and transitive;
- similarity relation or fuzzy equivalence relation if it is reflexive, symmetric, and transitive.

**Definition**

**4.**

- α-reflexive if $\forall x\in X:R(x,x)\ge \alpha $, where $\alpha \in (0,1]$ ([12]);

**Definition**

**5**

**.**Let ∗ be a binary operation. A reflexive and ∗-transitive fuzzy relation is called fuzzy preordering w.r.t ∗ or fuzzy ∗-preordering. A fuzzy ∗-preorder relation which is also symmetric is called fuzzy ∗-equivalence relation.

**Definition**

**6**

**.**Let T be a t-norm and E be a fuzzy T-equivalence relation. The fuzzy binary relation R is called T-E-ordering if and only if it is

- 1.
- E-reflexive, i.e., $E(x,y)\le R(x,y)$ for all $x,y\in X$;
- 2.
- T-E-antisymmetric, i.e., $T\left(R\right(x,y),R(y,x\left)\right)\le E(x,y)$ for all $x,y\in X$;
- 3.
- T-transitive.

**Theorem**

**1**

**.**For an arbitrary left-continuous t-norm T, the fuzzy relation R on X is T-preorder if and only if there exists a family of ${\left\{{\mu}_{j}\right\}}_{j\in J}$ of fuzzy subsets of X such that

**Theorem**

**2**

**.**For an arbitrary left-continuous t-norm T, the fuzzy relation R on X is T-equivalence if and only if there exists a family of ${\left\{{\mu}_{j}\right\}}_{j\in J}$ of fuzzy subsets of X such that

#### m-Polar Fuzzy Relations

**Definition**

**7**

**.**An m-polar fuzzy set μ on the universe X is a mapping $\mu :X\to {[0,1]}^{m}$ such that $\mu \left(x\right)=({\pi}_{1}\circ \mu \left(x\right),\cdots ,{\pi}_{m}\circ \mu \left(x\right))$ for any $x\in X$.

- $(\mu \vee \nu )\left(x\right)=(max({\pi}_{1}\circ \mu \left(x\right),{\pi}_{1}\circ \nu \left(x\right)),\cdots ,max({\pi}_{m}\circ \mu \left(x\right),{\pi}_{m}\circ \nu \left(x\right)))$
- $(\mu \wedge \nu )\left(x\right)=(min({\pi}_{1}\circ \mu \left(x\right),{\pi}_{1}\circ \nu \left(x\right)),\cdots ,min({\pi}_{m}\circ \mu \left(x\right),{\pi}_{m}\circ \nu \left(x\right)))$

**Definition**

**8**

**.**An m-polar fuzzy relation R on the universe X is defined by a mapping $R:X\times X\to {[0,1]}^{m}$ such that $R(x,y)=({\pi}_{1}\circ R(x,y),\cdots ,{\pi}_{m}\circ R(x,y))$ for any $x,y\in X$ where for each $1\le s\le m$, the value ${\pi}_{s}\circ R(x,y)$ shows the sth degree of relationship between x and y.

## 3. Generating $\mathit{m}$-Polar $\mathit{T}$-Orders from $\mathit{m}$-Polar Fuzzy Data

**Definition**

**9.**

**Lemma**

**1.**

- 1.
- ${\pi}_{s}\circ \mu \left(x\right)\le {\pi}_{s}\circ \mu \left(y\right)\iff \overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))=1$;
- 2.
- $T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)\iff {\pi}_{s}\circ \mu \left(x\right)\le \overrightarrow{T}({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))$;
- 3.
- $T(\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right)),\overrightarrow{T}({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))\le \overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(z\right))$;
- 4.
- $\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))\le \overrightarrow{T}(T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(z\right)),T({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right)))$;
- 5.
- $\overrightarrow{T}(1,{\pi}_{s}\circ \mu \left(x\right))={\pi}_{s}\circ \mu \left(x\right)$ and $\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),1)=1$;
- 6.
- $\overrightarrow{T}(0,{\pi}_{s}\circ \mu \left(x\right))=1$ and $\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),0)=0$ if ${\pi}_{s}\circ \mu \left(x\right)\ne 0$ and T without zero divisors, and otherwise is one;
- 7.
- $T({\pi}_{s}\circ \mu \left(x\right),\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right)))\le {\pi}_{s}\circ \mu \left(y\right)$;
- 8.
- ${\pi}_{s}\circ \mu \left(y\right)\le \overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right)))$.

**Proof.**

- To prove 1, first let $\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))=1$ which means $T(\alpha ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(y\right)$ for all $\alpha <1$. Since T is a left-continuous t-norm, then ${\pi}_{s}\circ \mu \left(x\right)=T(1,{\pi}_{s}\circ \mu \left(x\right))=T(sup\{\alpha :\alpha <1\},{\pi}_{s}\circ \mu \left(x\right))=sup\{T(\alpha ,{\pi}_{s}\circ \mu \left(x\right)):\alpha <1\}\le {\pi}_{s}\circ \mu \left(y\right)$. The other side is immediately obtained because, for any $\alpha \in [0,1]$, $T(\alpha ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(x\right)\le {\pi}_{s}\circ \mu \left(y\right)$.
- For 2, first suppose that $T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)$. Thus, ${\pi}_{s}\circ \mu \left(x\right)\le sup\{\alpha \in [0,1]:T(\alpha ,{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)\}=\overrightarrow{T}({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))$. Conversely, let ${\pi}_{s}\circ \mu \left(x\right)\le \overrightarrow{T}({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))=w$. Then, by Definition 9, $T(w,{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)$. Thus, $T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))\le T(w,{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)$.
- In 3, we have: $T(\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right)),\overrightarrow{T}({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))=T(sup\{\alpha \in [0,1]:T(\alpha ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(y\right)\},sup\{\beta \in [0,1]:T(\beta ,{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)\})=sup\{T(\alpha ,\beta ):T(\alpha ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(y\right)\&T(\beta ,{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)\}\le \overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(z\right))$ since, for such $\alpha ,\beta \in [0,1]$, we have: $T(T(\alpha ,\beta ),{\pi}_{s}\circ \mu \left(x\right))=T(\beta ,T(9\alpha ,{\pi}_{s}\circ \mu \left(x\right)))\le T(\beta ,{\pi}_{s}\circ \mu \left(y\right))\le {\pi}_{s}\circ \mu \left(z\right)$ which implies $T(\alpha ,\beta )\le sup\{\gamma \in [0,1]:T(\gamma ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(z\right)\}$.
- To show 4, let $\alpha \in [0,1]$ such that $T(\alpha ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(y\right)$. Then, because of the non-decreasing and associativity of T, we have: $T(T(\alpha ,{\pi}_{s}\circ \mu \left(x\right)),{\pi}_{s}\circ \mu \left(z\right))\le T({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))$ or equivalently $T(\alpha ,T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(z\right)))\le T({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))$. Thus, $sup\{\alpha \in [0,1]:T(\alpha ,{\pi}_{s}\circ \mu \left(x\right))\le {\pi}_{s}\circ \mu \left(y\right)\}\le sup\{\beta \in [0,1]:T(\beta ,T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(z\right)))\le T({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right))\}$. Therefore, $\overrightarrow{T}({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(y\right))\le \overrightarrow{T}(T({\pi}_{s}\circ \mu \left(x\right),{\pi}_{s}\circ \mu \left(z\right)),T({\pi}_{s}\circ \mu \left(y\right),{\pi}_{s}\circ \mu \left(z\right)))$.
- The items 5, 6, 7, and 8 are clearly obtained based on Definition 9.

**Corollary**

**1.**

- 1.
- $\mu \left(x\right)\le \mu \left(y\right)\iff \overrightarrow{T}(\mu \left(x\right),\mu \left(y\right))=\mathbf{1}$;
- 2.
- $T(\mu \left(x\right),\mu \left(y\right))\le \mu \left(z\right)\iff \mu \left(x\right)\le \overrightarrow{T}(\mu \left(y\right),\mu \left(z\right))$;
- 3.
- $T(\overrightarrow{T}(\mu \left(x\right),\mu \left(y\right)),\overrightarrow{T}(\mu \left(y\right),\mu \left(z\right))\le \overrightarrow{T}(\circ \mu \left(x\right),\mu \left(z\right))$;
- 4.
- $\overrightarrow{T}(\mu \left(x\right),\mu \left(y\right))\le \overrightarrow{T}(T(\mu \left(x\right),\mu \left(z\right)),T(\mu \left(z\right),\mu \left(y\right)))$;
- 5.
- $\overrightarrow{T}(\mathbf{1},\mu \left(x\right))=\mu \left(x\right)$ and $\overrightarrow{T}(\mu \left(x\right),\mathbf{1})=\mathbf{1}$;
- 6.
- $\overrightarrow{T}(\mathbf{0},\mu \left(x\right))=\mathbf{1}$. $\overrightarrow{T}(\mu \left(x\right),\mathbf{0})=\mathbf{0}$ if $\mu \left(x\right)\ne \mathbf{0}$, otherwise it is $\mathbf{1}$;
- 7.
- $T(\mu \left(x\right),\overrightarrow{T}(\mu \left(x\right),\mu \left(y\right)))\le \mu \left(y\right)$;
- 8.
- $\mu \left(y\right)\le \overrightarrow{T}(\mu \left(x\right),T(\mu \left(x\right),\mu \left(y\right)))$.

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

- 1.
- If $T:={T}_{L}$ and $A:=WAM$ (weighted arithmetic mean) with weighting vector $w=({w}_{1},\cdots ,{w}_{K})$ where for any $1\le j\le K$: ${w}_{j}\in [0,1]$ and ${\sum}_{j=1}^{K}{w}_{j}=1$. Then, ${\pi}_{s}\circ R(x,y)={\sum}_{j=1}^{K}{w}_{j}\xb7min(1-{\pi}_{s}\circ {\mu}_{j}\left(x\right)+{\pi}_{s}\circ {\mu}_{j}\left(y\right),1)$.
- 2.
- If $T:={T}_{P}$ and $A:=WGM$ (weighted geometric mean) with weighting vector $w=({w}_{1},\cdots ,{w}_{K})$ where for any $1\le j\le K$: ${w}_{j}\in [0,1]$, and ${\sum}_{j=1}^{K}{w}_{j}=1$. Then, ${\pi}_{s}\circ R(x,y)={\prod}_{j}{\left(\frac{{\pi}_{s}\circ {\mu}_{j}\left(y\right)}{{\pi}_{s}\circ {\mu}_{j}\left(x\right)}\right)}^{{w}_{j}}$, where ${\pi}_{s}\circ {\mu}_{j}\left(x\right)\ne 0$ and ${\pi}_{s}\circ {\mu}_{j}\left(x\right)>{\pi}_{s}\circ {\mu}_{j}\left(y\right)$; otherwise, it is one.
- 3.
- If $T:={T}_{M}$ and $A:=Min$. Then, ${\pi}_{s}\circ R(x,y)=1$, if ${\pi}_{s}\circ {\mu}_{j}\left(x\right)\le {\pi}_{s}\circ {\mu}_{j}\left(y\right)$ for all $1\le j\le K$; otherwise, ${\pi}_{s}\circ R(x,y)={min}_{j}\{{\pi}_{s}\circ {\mu}_{j}\left(y\right):{\pi}_{s}\circ {\mu}_{j}\left(y\right)<{\pi}_{s}\circ {\mu}_{j}\left(x\right)\}$.

**Proposition**

**3.**

- 1.
- If ${A}_{1}\le {A}_{2}$, then ${R}_{1}\subseteq {R}_{2}$.
- 2.
- If ${T}_{1}\le {T}_{2}$, then ${R}_{1}\supseteq {R}_{2}$.
- 3.
- If for any $1\le j\le K$ and $1\le s\le m$ we have ${\pi}_{s}\circ {\mu}_{j}\left(x\right)\le {\pi}_{s}\circ {\mu}_{j}\left(y\right)$, then $R(x,y)=1$. The converse will be held if A is conjunction.
- 4.
- For any crisp point $x\in X$ such that ${\pi}_{s}\circ {\mu}_{j}\left(x\right)=1$ for any $1\le j\le K$ and $1\le s\le m$, the ${\pi}_{s}\circ R(y,x)=1$ and ${\pi}_{s}\circ R(x,y)=A({\pi}_{s}\circ {\mu}_{1}\left(y\right),\cdots ,{\pi}_{s}\circ {\mu}_{K}\left(y\right))$ for any $y\in X$.
- 5.
- If ${\pi}_{s}\circ {\mu}_{j}\left(x\right)\le {\pi}_{s}\circ {\mu}_{j}\left(y\right)$ for any $1\le j\le K$ and $1\le s\le m$, then ${\pi}_{s}\circ R(z,x)\le {\pi}_{s}\circ R(z,y)$ and ${\pi}_{s}\circ R(y,z)\le {\pi}_{s}\circ R(x,z)$ for all $z\in X$.
- 6.
- If A is disjunction, then $T\left({\pi}_{s}\circ {\mu}_{j}\left(x\right),{\pi}_{s}\circ {\mu}_{j}\left(y\right)\right)\le {\pi}_{s}\circ {\mu}_{j}\left(z\right)\Rightarrow {\pi}_{s}\circ {\mu}_{j}\left(x\right)\le {\pi}_{s}\circ R(y,z)$ for any $1\le j\le K$, $1\le s\le m$ and $x,y,z\in X$.

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Remark**

**1.**

**Theorem**

**6.**

**Proof.**

#### m-Polar T-Equivalences

**Theorem**

**7.**

**Proof.**

**Example**

**2.**

- 1.
- If $T:={T}_{L}$, $A:=WAM$ and $B:=AM$, then ${\pi}_{s}\circ E(x,y)=\frac{1}{2}{\sum}_{j=1}^{K}{w}_{j}\xb7\left[min(1-{\pi}_{s}\circ {\mu}_{j}\left(x\right)+{\pi}_{s}\circ {\mu}_{j}\left(y\right),1)+min(1-{\pi}_{s}\circ {\mu}_{j}\left(y\right)+{\pi}_{s}\circ {\mu}_{j}\left(x\right),1)\right]$ where, for $1\le j\le K$: ${w}_{j}\in [0,1]$ and ${\sum}_{j=1}^{K}{w}_{j}=1$.
- 2.
- If $T:={T}_{P}$ and $A:=WGM$ with weighting vector $w=({w}_{1},\cdots ,{w}_{m})$ where for any $1\le j\le K$: ${w}_{j}\in [0,1]$ and ${\sum}_{j=1}^{K}{w}_{j}=1$ and $B:=GM$. Then, ${\pi}_{s}\circ E(x,y)=\sqrt{{\prod}_{j}{[min(\frac{{\pi}_{s}\circ {\mu}_{j}\left(y\right)}{{\pi}_{s}\circ {\mu}_{j}\left(x\right)},1)]}^{{w}_{j}}\xb7min(\frac{{\pi}_{s}\circ {\mu}_{j}\left(x\right)}{{\pi}_{s}\circ {\mu}_{j}\left(y\right)},1){]}^{{w}_{j}}}$, where ${\pi}_{s}\circ {\mu}_{j}\left(x\right)\ne 0$ and ${\pi}_{s}\circ {\mu}_{j}\left(y\right)\ne 0$; otherwise, it is one.
- 3.
- If $T:={T}_{M}$ and $A,B:=Min$. Then, ${\pi}_{s}\circ E(x,y)={min}_{j=1}^{K}\{{\pi}_{s}\circ {\mu}_{j}\left(x\right)\}$ if ${\pi}_{s}\circ {\mu}_{j}\left(x\right)\le {\pi}_{s}\circ {\mu}_{j}\left(y\right)$ for all $1\le j\le K$; ${\pi}_{s}\circ E(x,y)={min}_{j=1}^{K}\{{\pi}_{s}\circ {\mu}_{j}\left(y\right)\}$ if ${\pi}_{s}\circ {\mu}_{j}\left(y\right)\le {\pi}_{s}\circ {\mu}_{j}\left(x\right)$ for all $1\le j\le K$; otherwise, ${\pi}_{s}\circ R(x,y)=min({min}_{j}\{{\pi}_{s}\circ {\mu}_{j}\left(y\right):{\pi}_{s}\circ {\mu}_{j}\left(y\right)<{\pi}_{s}\circ {\mu}_{j}\left(x\right)\},{min}_{j}\{{\pi}_{s}\circ {\mu}_{j}\left(x\right):{\pi}_{s}\circ {\mu}_{j}\left(x\right)<{\pi}_{s}\circ {\mu}_{j}\left(y\right)\})$.

**Proposition**

**5.**

- 1.
- If ${B}_{1}\le {B}_{2}$, then ${E}_{1}\subseteq {E}_{2}$.
- 2.
- If ${A}_{1}\le {A}_{2}$, then ${E}_{1}\subseteq {E}_{2}$.
- 3.
- If ${T}_{1}\le {T}_{2}$, then ${E}_{1}\supseteq {E}_{2}$.
- 4.
- If for any $1\le i\le K$, ${\pi}_{s}\circ {\mu}_{i}\left(x\right)={\pi}_{s}\circ {\mu}_{i}\left(y\right)$ for all $s=1,\cdots ,m$; then, ${\pi}_{s}\circ E(x,y)=1$. The converse holds if B is a conjunction.

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

## 4. Constructing Crisp Orderings of the $\mathit{m}$-Polar $\mathit{T}$-Orderings

**Theorem**

**10.**

**Proof.**

**Proposition**

**6.**

- If ${A}_{1}\le {A}_{2}$, then ${\u25c3}_{{R}_{1}}^{s}\subseteq {\u25c3}_{{R}_{2}}^{s}$ and ${\sim}_{{R}_{1}}^{s}\subseteq {\sim}_{{R}_{2}}^{s}$.
- If ${T}_{1}\le {T}_{2}$, then ${\u25c3}_{{R}_{1}}^{s}\supseteq {\u25c3}_{{R}_{2}}^{s}$ and ${\sim}_{{R}_{1}}^{s}\supseteq {\sim}_{{R}_{2}}^{s}$.
- If $\mathbf{b}\le \mathbf{c}$, then ${\u2aaf}_{R}^{s,\mathbf{b}}\supseteq {\u2aaf}_{R}^{s,\mathbf{c}}$ and ${\sim}_{R}^{s,\mathbf{b}}\supseteq {\sim}_{R}^{s,\mathbf{c}}$.

**Proof.**

**Theorem**

**11.**

- 1.
- For all $1\le s\le m$: if $x{\u25c3}_{R}^{s}y$ then $x{\u25c3}_{{R}_{F}}y$, moreover, if $x{\sim}_{R}^{s}y$ then $x{\sim}_{{R}_{F}}y$.
- 2.
- Let F be a disjunction. If for some $1\le s\le m$: $x{\u25c3}_{R}^{s}y$, then $x{\u25c3}_{{R}_{F}}y$. Similarly, if for some $1\le s\le m$: $x{\sim}_{R}^{s}y$, then $x{\sim}_{{R}_{F}}y$.
- 3.
- Let F be a conjunction. If for some $1\le s\le m$: $x{\u22ea}_{R}^{s}y$, then $x{\u22ea}_{{R}_{F}}y$. Similarly, if for some $1\le s\le m$: $x{\nsim}_{R}^{s}y$, then $x{\nsim}_{{R}_{F}}y$.
- 4.
- Let F have an annihilator element at b. If $x{\u2aaf}_{R}^{s,\mathbf{a}}y$, then $x{\u2aaf}_{{R}_{F}}^{b}y$. Similarly, if $x{\sim}_{R}^{s,\mathbf{a}}y$, then $x{\sim}_{{R}_{F}}^{b}y$.
- 5.
- Let F be a conjunction. If $min({\pi}_{1}\circ R(x,y),\cdots ,{\pi}_{m}\circ R(x,y))={b}_{\ast}$, then $x{\u2aaf}_{{R}_{F}}^{{b}_{\ast}}y$. Similarly, if $min({\pi}_{1}\circ E(x,y),\cdots ,{\pi}_{m}\circ E(x,y))={c}_{\ast}$, then $x{\sim}_{{R}_{F}}^{{c}_{\ast}}y$.
- 6.
- Let F be a disjunction. If $max({\pi}_{1}\circ R(x,y),\cdots ,{\pi}_{m}\circ R(x,y))={b}^{\ast}$, then $x{\u2aaf}_{{R}_{F}}^{{b}^{\ast}}y$. Similarly, if $max({\pi}_{1}\circ E(x,y),\cdots ,{\pi}_{m}\circ E(x,y))={c}^{\ast}$, then $x{\sim}_{{R}_{F}}^{{c}^{\ast}}y$.

**Proof.**

## 5. Application in Decision-Making

**Theorem**

**12.**

- 1.
- The score function $S(.,{\u25c3}_{{R}_{F}})$ provides a complete preference relation on X as$$\begin{array}{c}\hfill y\u2aaf\left(S\right)x\iff S(y,{\u25c3}_{{R}_{F}})\le S(x,{\u25c3}_{{R}_{F}})\end{array}$$
- 2.
- The score function $S(.,{\u2aaf}_{{R}_{F}}^{b})$ provides a complete preference relation on X defined by$$\begin{array}{c}\hfill y\u2aaf\left(S\right)x\iff S(y,{\u2aaf}_{{R}_{F}}^{b})\le S(x,{\u2aaf}_{{R}_{F}}^{b})\end{array}$$

**Remark**

**2.**

Algorithm 1: Ranking Alternatives by m-Polar Fuzzy T-Orderings |

**Theorem**

**13.**

Algorithm 2: Ranking Alternatives by m-Polar Fuzzy T-Orderings |

#### Illustrative Example

**Example**

**3.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Zadeh, L. Similarity relations and fuzzy orderings. Inf. Sci.
**1971**, 3, 177–200. [Google Scholar] [CrossRef] - Tanino, T. Fuzzy preference orderings in group decision-making. Fuzzy Sets Syst.
**1984**, 12, 117–131. [Google Scholar] [CrossRef] - Fodor, J. Strict preference relations based on weak t-norms. Fuzzy Sets Syst.
**1991**, 43, 327–336. [Google Scholar] [CrossRef] - Venugopalan, P. Fuzzy ordered sets. Fuzzy Sets Syst.
**1992**, 46, 221–226. [Google Scholar] [CrossRef] - Świtalski, Z. Transitivity of fuzzy preference relations—An empirical study. Fuzzy Sets Syst.
**2001**, 118, 503–508. [Google Scholar] [CrossRef] - Fang, J.; Qiu, Y. Fuzzy orders and fuzzifying topologies. Int. J. Approx. Reason.
**2008**, 48, 98–109. [Google Scholar] [CrossRef] [Green Version] - Xu, Y.; Wang, H.; Yu, D. Weak transitivity of interval-valued fuzzy relations. Knowl. Based Syst.
**2014**, 63, 24–32. [Google Scholar] [CrossRef] - Bentkowska, U. Aggregation of diverse types of fuzzy orders for decision-making problems. Inf. Sci.
**2018**, 424, 317–336. [Google Scholar] [CrossRef] - Fuster-Parra, P.; Martín, J.; Recasens, J.; Valero, Ó. T-Equivalences: The Metric Behavior Revisited. Mathematics
**2020**, 8, 495. [Google Scholar] [CrossRef] [Green Version] - Bodenhofer, U. A similarity-based generalization of fuzzy orderings preserving the classical axioms. Int. J. Uncertain. Fuzzy
**2000**, 8, 593–610. [Google Scholar] [CrossRef] [Green Version] - Bodenhofer, U. Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets Syst.
**2003**, 137, 113–136. [Google Scholar] [CrossRef] - Fodor, J.C.; Roubens, M.R. Fuzzy Preference Modelling and Multicriteria Decision Support; Springer Science & Business Media Dordrecht: Dordrecht, The Netherlands, 1994; eBook ISBN 978-94-017-1648-2. [Google Scholar]
- Gottwald, S. Fuzzy Sets and Fuzzy Logic: The Foundations of Application—From a Mathematical Point of View; Friedr. Vieweg & Sohn Verlagsgesellsch; Vieweg and Teubner Verlag: Braunschweig, Germany; Wiesbaden, Germany, 1993; eBook ISBN 978-3-322-86812-1. [Google Scholar]
- Valverde, L. On the structure of F-indistinguishability operators. Fuzzy Sets Syst.
**1985**, 17, 313–328. [Google Scholar] [CrossRef] [Green Version] - Zhang, W.R. Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent decision analysis. In NAFIPS/IFIS/NASA’94. Proceedings of the First, International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige; IEEE: San Antonio, TX, USA, 1994; pp. 305–309. [Google Scholar] [CrossRef]
- Chen, J.; Li, S.; Ma, S.; Wang, X. m-polar fuzzy sets: An extension of bipolar fuzzy sets. Sci. World J.
**2014**, 416530. [Google Scholar] [CrossRef] [Green Version] - Singh, P.K. m-polar fuzzy graph representation of concept lattice. Eng. Appl. Artif. Intell.
**2018**, 67, 52–62. [Google Scholar] [CrossRef] - Calvo, T.; Kolesárová, A.; Komorníková, M.; Mesiar, R. Aggregation operators: Properties, classes and construction methods. In Aggregation Operators. Studies in Fuzziness and Soft Computing; Calvo, T., Mayor, G., Mesiar, R., Eds.; Physica: Heidelberg, Germany, 2002; pp. 3–104. [Google Scholar]
- Grabisch, M.; Marichal, J.L.; Mesiar, R.; Pap, E. Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes. Inf. Sci.
**2011**, 181, 23–43. [Google Scholar] [CrossRef] [Green Version] - Zahedi Khameneh, A.; Kilicman, A. Some Construction Methods of Aggregation Operators in Decision-Making Problems: An Overview. Symmetry
**2020**, 12, 694. [Google Scholar] [CrossRef] - Saminger, S.; Mesiar, R.; Bodenhofer, U. Domination of aggregation operators and preservation of transitivity. Int. J. Uncertain. Fuzzy
**2002**, 10, 11–35. [Google Scholar] [CrossRef] - Klement, E.P.; Mesiar, R.; Pap, E. Triangular Norms; Springer Science & Business Media Dordrecht: Dordrecht, The Netherlands, 2000; eBook ISBN 978-94-015-9540-7. [Google Scholar]
- Drewniak, J.; Dudziak, U. Preservation of properties of fuzzy relations during aggregation processes. Kybernetika
**2007**, 43, 115–132. [Google Scholar] - Bentkowska, U.; Król, A. Preservation of fuzzy relation properties based on fuzzy conjunctions and disjunctions during aggregation process. Fuzzy Sets Syst.
**2016**, 291, 98–113. [Google Scholar] [CrossRef] - Mesiarova-Zemankova, A. Multi-polar t-conorms and uninorms. Inf. Sci.
**2015**, 301, 227–240. [Google Scholar] [CrossRef] - Zahedi Khameneh, A.; Kilicman, A. A fuzzy majority-based construction method for composed aggregation functions by using combination operator. Inf. Sci.
**2019**, 505, 367–387. [Google Scholar] [CrossRef] - Komorníková, M.; Mesiar, R. Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst.
**2011**, 175, 48–56. [Google Scholar] [CrossRef] - Zahedi Khameneh, A.; Kilicman, A. m-polar fuzzy soft weighted aggregation operators and their applications in group decision-making. Symmetry
**2018**, 10, 636. [Google Scholar] [CrossRef] [Green Version]

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

${h}_{1}$ | (0.7,0.58) | (0.72,0.59) | (0.74,0.64) | (0.6,0.68) | (0.84,0.6) |

${h}_{2}$ | (0.55,0.71) | (0.6,0.69) | (0.7,0.66) | (0.4,0.64) | (0.74,0.64) |

${h}_{3}$ | (0.6,0.6) | (0.73,0.57) | (0.73,0.57) | (0.8,0.62) | (0.8,0.62) |

${h}_{4}$ | (0.88,0.55) | (0.87,0.54) | (0.73,0.61) | (0.74,0.69) | (0.77,0.59) |

${h}_{5}$ | (0.64,0.69) | (0.8,0.66) | (0.69,0.65) | (0.64,0.61) | (0.76,0.61) |

${h}_{6}$ | (0.66,0.67) | (0.68,0.69) | (0.65,0.69) | (0.76,0.66) | (0.72,0.66) |

${h}_{7}$ | (0.73,0.66) | (0.8,0.66) | (0.8,0.65) | (0.53,0.66) | (0.76,0.64) |

${h}_{8}$ | (0.73,0.68) | (0.75,0.64) | (0.8,0.67) | (0.7,0.64) | (0.76,0.67) |

${h}_{9}$ | (0.8,0.64) | (0.85,0.49) | (0.75,0.57) | (0.6,0.71) | (0.69,0.44) |

${h}_{10}$ | (0.86,0.83) | (0.86,0.85) | (0.8,0.77) | (1,0.75) | (0.82,0.77) |

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

${h}_{1}$ | (1,1) | (0.8,0.96) | (0.9,0.93) | (0.93,0.95) | (0.92,0.93) | (0.88,0.98) | (0.92,0.98) | (0.92,0.96) | (0.85,0.84) | (0.98,1) |

${h}_{2}$ | (1,0.87) | (1,1) | (1,0.88) | (1,0.84) | (0.99,0.97) | (0.95,0.96) | (1,0.95) | (1,0.95) | (0.95,0.8) | (1,1) |

${h}_{3}$ | (0.8,0.98) | (0.6,1) | (1,1) | (0.94,0.95) | (0.84,0.99) | (0.92,1) | (0.73,1) | (0.9,1) | (0.8,0.82) | (1,1) |

${h}_{4}$ | (0.82,0.99) | (0.66,0.95) | (0.72,0.93) | (1,1) | (0.76,0.92) | (0.78,0.97) | (0.79,0.97) | (0.85,0.95) | (0.86,0.85) | (0.98,1) |

${h}_{5}$ | (0.92,0.89) | (0.76,1) | (0.93,0.91) | (1,0.86) | (1,1) | (0.88,0.98) | (0.89,0.97) | (0.95,0.98) | (0.93,0.83) | (1,1) |

${h}_{6}$ | (0.84,0.9) | (0.64,0.97) | (0.94,0.88) | (0.98,0.85) | (0.88,0.95) | (1,1) | (0.77,0.96) | (0.94,0.95) | (0.84,0.78) | (1,1) |

${h}_{7}$ | (0.92,0.92) | (0.8,0.98) | (0.87,0.91) | (0.93,0.88) | (0.89,0.95) | (0.85,1) | (1,1) | (0.95,0.98) | (0.93,0.8) | (1,1) |

${h}_{8}$ | (0.9,0.9) | (0.7,0.97) | (0.87,0.9) | (0.93,0.87) | (0.89,0.94) | (0.85,0.99) | (0.83,0.97) | (1,1) | (0.9,0.77) | (1,1) |

${h}_{9}$ | (0.87,0.94) | (0.75,0.93) | (0.8,0.91) | (0.98,0.91) | (0.84,0.9) | (0.83,0.95) | (0.93,0.95) | (0.9,0.93) | (1,1) | (1,1) |

${h}_{10}$ | (0.6,0.74) | (0.4,0.84) | (0.74,0.72) | (0.74,0.69) | (0.64,0.81) | (0.76,0.84) | (0.53,0.81) | (0.7,0.79) | (0.6,0.64) | (1,1) |

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

${h}_{1}$ | (1,1) | (0.8,0.87) | (0.8,0.93) | (0.82,0.95) | (0.92,0.89) | (0.84,0.9) | (0.92,0.92) | (0.9,0.9) | (0.85,0.84) | (0.6,0.74) |

${h}_{2}$ | (0.8,0.87) | (1,1) | (0.6,0.88) | (0.66,0.84) | (0.76,0.97) | (0.64,0.96) | (0.8,0.95) | (0.7,0.95) | (0.75,0.8) | (0.4,0.84) |

${h}_{3}$ | (0.8,0.93) | (0.6,0.88) | (1,1) | (0.72,0.93) | (0.84,0.91) | (0.92,0.88) | (0.73,0.91) | (0.87,0.9) | (0.8,0.82) | (0.74,0.72) |

${h}_{4}$ | (0.82,0.95) | (0.66,0.84) | (0.72,0.93) | (1,1) | (0.76,0.86) | (0.78,0.85) | (0.79,0.88) | (0.85,0.87) | (0.86,0.85) | (0.74,0.69) |

${h}_{5}$ | (0.92,0.89) | (0.76,0.97) | (0.84,0.91) | (0.76,0.86) | (1,1) | (0.88,0.95) | (0.89,0.95) | (0.89,0.94) | (0.84,0.83) | (0.64,0.81) |

${h}_{6}$ | (0.84,0.9) | (0.64,0.96) | (0.92,0.88) | (0.78,0.85) | (0.88,0.95) | (1,1) | (0.77,0.96) | (0.85,0.95) | (0.83,0.78) | (0.76,0.84) |

${h}_{7}$ | (0.92,0.92) | (0.8,0.95) | (0.73,0.91) | (0.79,0.88) | (0.89,0.95) | (0.77,0.96) | (1,1) | (0.83,0.97) | (0.93,0.8) | (0.53,0.81) |

${h}_{8}$ | (0.9,0.9) | (0.7,0.95) | (0.87,0.9) | (0.85,0.87) | (0.89,0.94) | (0.85,0.95) | (0.83,0.97) | (1,1) | (0.9,0.77) | (0.7,0.79) |

${h}_{9}$ | (0.85,0.84) | (0.75,0.8) | (0.8,0.82) | (0.86,0.85) | (0.84,0.83) | (0.83,0.78) | (0.93,0.8) | (0.9,0.77) | (1,1) | (0.6,0.64) |

${h}_{10}$ | (0.6,0.74) | (0.4,0.84) | (0.74,0.72) | (0.74,0.69) | (0.64,0.81) | (0.76,0.84) | (0.53,0.81) | (0.7,0.79) | (0.6,0.64) | (1,1) |

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

${h}_{1}$ | (1,1) | (0.4,0.64) | (0.6,0.57) | (0.73,0.54) | (0.64,0.61) | (0.65,0.66) | (0.53,0.66) | (0.76,0.64) | (0.69,0.44) | (0.82,1) |

${h}_{2}$ | (1,0.58) | (1,1) | (1,0.57) | (1,0.54) | (0.69,0.61) | (0.65,0.67) | (1,0.65) | (1,0.64) | (0.69,0.44) | (1,1) |

${h}_{3}$ | (0.6,0.58) | (0.4,1) | (1,1) | (0.74,0.54) | (0.64,0.61) | (0.65,1) | (0.53,1) | (0.7,1) | (0.6,0.44) | (1,1) |

${h}_{4}$ | (0.6,0.68) | (0.4,0.64) | (0.6,0.57) | (1,1) | (0.64,0.61) | (0.65,0.66) | (0.53,0.66) | (0.7,0.64) | (0.6,0.44) | (0.86,1) |

${h}_{5}$ | (0.6,0.58) | (0.4,1) | (0.6,0.57) | (1,0.54) | (1,1) | (0.65,0.67) | (0.53,0.66) | (0.75,0.64) | (0.60.44) | (1,1) |

${h}_{6}$ | (0.6,0.58) | (0.4,0.64) | (0.6,0.57) | (0.74,0.54) | (0.64,0.61) | (1,1) | (0.53,0.64) | (0.7,0.64) | (0.6,0.44) | (1,1) |

${h}_{7}$ | (0.7,0.58) | (0.4,0.64) | (0.6,0.57) | (0.73,0.54) | (0.64,0.61) | (0.65,1) | (1,1) | (0.75,0.64) | (0.69,0.44) | (1,1) |

${h}_{8}$ | (0.6,0.58) | (0.4,0.64) | (0.6,0.57) | (0.73,0.54) | (0.64,0.61) | (0.65,0.66) | (0.53,0.64) | (1,1) | (0.6,0.44) | (1,1) |

${h}_{9}$ | (0.7,0.58) | (0.4,0.64) | (0.6,0.6) | (0.73,0.55) | (0.64,0.61) | (0.65,0.66) | (0.53,0.66) | (0.73,0.64) | (1,1) | (1,1) |

${h}_{10}$ | (0.6,0.58) | (0.4,0.64) | (0.6,0.57) | (0.73,0.54) | (0.64,0.61) | (0.65,0.66) | (0.53,0.64) | (0.7,0.64) | (0.6,0.44) | (1,1) |

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

${h}_{1}$ | (1,1) | (0.4,0.58) | (0.6,0.57) | (0.6,0.54) | (0.6,0.58) | (0.6,0.58) | (0.53,0.58) | (0.6,0.58) | (0.69,0.44) | (0.6,0.58) |

${h}_{2}$ | (0.4,0.58) | (1,1) | (0.4,0.57) | (0.4,0.54) | (0.4,0.61) | (0.4,0.64) | (0.4,0.64) | (0.4,0.64) | (0.4,0.44) | (0.4,0.64) |

${h}_{3}$ | (0.6,0.57) | (0.4,0.57) | (1,1) | (0.6,0.54) | (0.6,0.57) | (0.6,0.57) | (0.53,0.57) | (0.6,0.57) | (0.6,0.44) | (0.6,0.57) |

${h}_{4}$ | (0.6,0.54) | (0.4,0.54) | (0.6,0.54) | (1,1) | (0.64,0.54) | (0.65,0.54) | (0.53,0.54) | (0.7,0.54) | (0.6,0.44) | (0.73,0.54) |

${h}_{5}$ | (0.6,0.58) | (0.4,0.61) | (0.6,0.57) | (0.64,0.54) | (1,1) | (0.64,0.61) | (0.53,0.61) | (0.64,0.61) | (0.6,0.44) | (0.64,0.61) |

${h}_{6}$ | (0.6,0.58) | (0.4,0.64) | (0.6,0.57) | (0.65,0.54) | (0.64,0.61) | (1,1) | (0.53,0.64) | (0.65,0.64) | (0.6,0.44) | (0.65,0.66) |

${h}_{7}$ | (0.53,0.58) | (0.4,0.64) | (0.53,0.57) | (0.53,0.54) | (0.53,0.61) | (0.53,0.64) | (1,1) | (0.53,0.64) | (0.53,0.44) | (0.53,0.64) |

${h}_{8}$ | (0.6,0.58) | (0.4,0.64) | (0.6,0.57) | (0.7,0.54) | (0.64,0.61) | (0.65,0.64) | (0.53,0.64) | (1,1) | (0.6,0.44) | (0.7,0.64) |

${h}_{9}$ | (0.69,0.44) | (0.4,0.44) | (0.6,0.44) | (0.6,0.44) | (0.6,0.44) | (0.6,0.44) | (0.53,0.44) | (0.6,0.44) | (1,1) | (0.6,0.44) |

${h}_{10}$ | (0.6,0.58) | (0.4,0.64) | (0.6,0.57) | (0.73,0.54) | (0.64,0.61) | (0.65,0.66) | (0.53,0.64) | (0.7,0.64) | (0.6,0.44) | (1,1) |

H | $\mathit{F}:=\mathit{AM}$ | $\mathit{F}:=\mathit{Min}$ | ||
---|---|---|---|---|

$\mathit{T}:={\mathit{T}}_{\mathit{L}},\mathit{A}:=\mathit{Min}$ | $\mathit{T}:={\mathit{T}}_{\mathit{M}},\mathit{A}:=\mathit{Min}$ | |||

$\mathit{S}({\mathit{x}}_{\mathit{i}},{\mathbf{\u25c3}}_{{\mathit{R}}_{\mathit{AM}}})$ | $\mathit{S}({\mathit{x}}_{\mathit{i}},{\mathbf{\u25c3}}_{{\mathit{R}}_{\mathit{Min}}})$ | $\mathit{S}({\mathit{x}}_{\mathit{i}},{\mathbf{\u2aaf}}_{{\mathit{R}}_{\mathit{Min}}}^{\mathbf{0}.\mathbf{5}})$ | $\mathit{S}({\mathit{x}}_{\mathit{i}},{\mathbf{\u2aaf}}_{{\mathit{R}}_{\mathit{Min}}}^{\mathbf{0}.\mathbf{7}})$ | |

${h}_{1}$ | 0 | 0 | 2 | −1 |

${h}_{2}$ | −1 | −1 | −8 | −1 |

${h}_{3}$ | −1 | −1 | 2 | −2 |

${h}_{4}$ | 0 | 0 | 2 | −1 |

${h}_{5}$ | −1 | −1 | 2 | −1 |

${h}_{6}$ | −1 | −1 | 2 | −1 |

${h}_{7}$ | −1 | −1 | 2 | −1 |

${h}_{8}$ | −1 | −1 | 2 | 0 |

${h}_{9}$ | −1 | −1 | −8 | −1 |

${h}_{10}$ | 7 | 7 | 2 | 9 |

Preference Order | ${h}_{2},{h}_{3},{h}_{5},{h}_{6},{h}_{7},{h}_{8},$ | ${h}_{2},{h}_{3},{h}_{5},{h}_{6},{h}_{7},{h}_{8},$ | ${h}_{2},{h}_{9}\u2aaf\left(S\right){h}_{1},{h}_{3},{h}_{4},{h}_{5},$ | ${h}_{3}\u2aaf\left(S\right){h}_{1},{h}_{2},{h}_{4},{h}_{5},{h}_{6},$ |

${h}_{9}\u2aaf\left(S\right){h}_{1},{h}_{4}\u2aaf\left(S\right){h}_{10}$ | ${h}_{9}\u2aaf\left(S\right){h}_{1},{h}_{4}\u2aaf\left(S\right){h}_{10}$ | ${h}_{6},{h}_{7},{h}_{8},{h}_{10}$ | ${h}_{7},{h}_{9}\u2aaf\left(S\right){h}_{8}\u2aaf\left(S\right){h}_{10}$ |

H | $\mathit{T}:={\mathit{T}}_{\mathit{L}},\mathit{A}:=\mathit{Min}$ | $\mathit{F}:=\mathit{AM}$ | $\mathit{T}:={\mathit{T}}_{\mathit{M}},\mathit{A}:=\mathit{Min}$ |
---|---|---|---|

$\mathit{S}({\mathit{x}}_{\mathit{i}},{\mathit{R}}_{\mathbf{1}})$ | $\mathit{S}({\mathit{x}}_{\mathit{i}},{\mathit{R}}_{\mathbf{2}})$ | ||

${h}_{1}$ | 0.479 | 0.495 | |

${h}_{2}$ | 0.44 | 0.399 | |

${h}_{3}$ | 0.487 | 0.443 | |

${h}_{4}$ | 0.512 | 0.514 | |

${h}_{5}$ | 0.483 | 0.474 | |

${h}_{6}$ | 0.508 | 0.528 | |

${h}_{7}$ | 0.485 | 0.480 | |

${h}_{8}$ | 0.511 | 0.541 | |

${h}_{9}$ | 0.462 | 0.436 | |

${h}_{10}$ | 0.634 | 0.692 | |

Preference Order | ${h}_{2}\u2aaf\left(S\right){h}_{9}\u2aaf\left(S\right){h}_{1}\u2aaf\left(S\right){h}_{5}\u2aaf\left(S\right){h}_{7}\u2aaf\left(S\right)$ | ${h}_{2}\u2aaf\left(S\right){h}_{9}\u2aaf\left(S\right){h}_{3}\u2aaf\left(S\right){h}_{5}\u2aaf\left(S\right){h}_{7}\u2aaf\left(S\right)$ | |

${h}_{3}\u2aaf\left(S\right){h}_{6}\u2aaf\left(S\right){h}_{8}\u2aaf\left(S\right){h}_{4}\u2aaf\left(S\right){h}_{10}$ | ${h}_{1}\u2aaf\left(S\right){h}_{4}\u2aaf\left(S\right){h}_{6}\u2aaf\left(S\right){h}_{8}\u2aaf\left(S\right){h}_{10}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zahedi Khameneh, A.; Kilicman, A.
*m*-Polar Generalization of Fuzzy *T*-Ordering Relations: An Approach to Group Decision Making. *Symmetry* **2021**, *13*, 51.
https://doi.org/10.3390/sym13010051

**AMA Style**

Zahedi Khameneh A, Kilicman A.
*m*-Polar Generalization of Fuzzy *T*-Ordering Relations: An Approach to Group Decision Making. *Symmetry*. 2021; 13(1):51.
https://doi.org/10.3390/sym13010051

**Chicago/Turabian Style**

Zahedi Khameneh, Azadeh, and Adem Kilicman.
2021. "*m*-Polar Generalization of Fuzzy *T*-Ordering Relations: An Approach to Group Decision Making" *Symmetry* 13, no. 1: 51.
https://doi.org/10.3390/sym13010051