# Centroid Transformations of Intuitionistic Fuzzy Values Based on Aggregation Operators

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Sets

- $(\mu \cap \nu )\left(x\right)=min\left\{\mu \right(x),\nu (x\left)\right\}$,
- $(\mu \cup \nu )\left(x\right)=max\left\{\mu \right(x),\nu (x\left)\right\}$,
- ${\mu}^{c}\left(x\right)=1-\mu \left(x\right)$,

#### 2.2. Intuitionistic Fuzzy Sets

**Definition**

**1.**

- $A\bigsqcup B=\{(x,max\{{t}_{A}\left(x\right),{t}_{B}\left(x\right)\},min\{{f}_{A}\left(x\right),{f}_{B}\left(x\right)\})\mid x\in U\}$;
- $A\sqcap B=\{(x,min\{{t}_{A}\left(x\right),{t}_{B}\left(x\right)\},max\{{f}_{A}\left(x\right),{f}_{B}\left(x\right)\})\mid x\in U\}$;
- $A\u2291B$ if and only if ${t}_{A}\left(x\right)\le {t}_{B}\left(x\right)$ and ${f}_{A}\left(x\right)\ge {f}_{B}\left(x\right)$ for all $x\in U$.

## 3. Operations and an Admissible Order of Intuitionistic Fuzzy Values

**Definition**

**2.**

- $A\oplus B=({t}_{A}+{t}_{B}-{t}_{A}\xb7{t}_{B},{f}_{A}\xb7{f}_{B})$;
- $A\otimes B=({t}_{A}\xb7{t}_{B},{f}_{A}+{f}_{B}-{f}_{A}\xb7{f}_{B})$;
- $\lambda A=(1-{(1-{t}_{A})}^{\lambda},{f}_{A}^{\lambda})$.

**Theorem**

**1.**

- (1)
- $A\oplus B=B\oplus A$;
- (2)
- $\lambda (A\oplus B)=\lambda A\oplus \lambda B$;
- (3)
- $({\lambda}_{1}+{\lambda}_{2})A={\lambda}_{1}A\oplus {\lambda}_{2}A$.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Proposition**

**1.**

- (1)
- $\delta (0,1)=0$;
- (2)
- $\delta (1,0)=1$;
- (3)
- $\delta ({t}_{A},{f}_{A})$ is increasing with respect to ${t}_{A}$;
- (4)
- $\delta ({t}_{A},{f}_{A})$ is deceasing with respect to ${f}_{A}$.

**Definition**

**6.**

- (1)
- ⪯ is a linear order on ${L}^{*}$;
- (2)
- For all $A,B\in {L}^{*}$, $A{\u2a7d}_{{L}^{*}}B$ implies $A\u2aafB$.

**Definition**

**7.**

- if ${s}_{A}<{s}_{B}$, A is smaller than B and denoted by $A<B$;
- if ${s}_{A}={s}_{B}$, then we have:
- (1)
- if ${h}_{A}={h}_{B}$, A is equivalent to B and denoted by $A=B$;
- (2)
- if ${h}_{A}<{h}_{B}$, A is smaller than B and denoted by $A<B$;
- (3)
- if ${h}_{A}>{h}_{B}$, A is greater than B and denoted by $A>B$.

## 4. Centroid Transformations of Intuitionistic Fuzzy Values

**Definition**

**8.**

**Definition**

**9.**

**Proposition**

**2.**

- (1)
- ${h}_{A}={t}_{A}+{f}_{A}=1$;
- (2)
- ${\pi}_{A}=1-{h}_{A}=0$;
- (3)
- $\overline{A}=A=\underline{A}$;
- (4)
- ${\u25b5}_{A}=\left\{A\right\}$.

**Proof.**

**Definition**

**10**

**([6]).**Let ${\alpha}_{i}$ ($i=1,2,\cdots ,n$) be IFVs in ${L}^{*}$. The intuitionistic fuzzy weighted averaging (IFWA) operator of dimension n is a mapping ${\mathsf{\Xi}}_{w}:{\left({L}^{*}\right)}^{n}\to {L}^{*}$ given by

**Theorem**

**2.**

**Definition**

**11.**

**Example**

**1.**

**Example**

**2.**

**Definition**

**12.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Definition**

**13.**

**Proposition**

**5.**

**Proof.**

## 5. Limits of Simple Centroid Transformations Sequences

**Definition**

**14.**

**Proposition**

**6.**

**Proof.**

**Example**

**3.**

**Proposition**

**7.**

- (1)
- ${s}_{A}={t}_{A}-{f}_{A}={t}_{B}-{f}_{B}={s}_{B}$;
- (2)
- ${\delta}_{A}=({s}_{A}+1)/2=({s}_{B}+1)/2={\delta}_{B}$;
- (3)
- ${h}_{B}={h}_{A}+2{\pi}_{A}/3=(2+{h}_{A})/3$;
- (4)
- ${\pi}_{B}=(1-{h}_{A})/3={\pi}_{A}/3$.

**Proof.**

**Proposition**

**8.**

**Proof.**

**Definition**

**15.**

**Proposition**

**9.**

- (1)
- $(0,1){\le}_{(s,h)}{A}_{n}{\le}_{(s,h)}(1,0)$;
- (2)
- ${A}_{n}{\le}_{(s,h)}{A}_{n+1}$,

**Proof.**

**Definition**

**16.**

**Proposition**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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

${p}_{1}$ | $(0.8,0.1)$ | $(0.6,0.2)$ | $(0.4,0.4)$ | $(0.7,0.2)$ |

${p}_{2}$ | $(0.6,0.2)$ | $(0.7,0.2)$ | $(0.4,0.5)$ | $(0.5,0.1)$ |

${p}_{3}$ | $(0.7,0.3)$ | $(0.8,0.1)$ | $(0.6,0.3)$ | $(0.7,0.1)$ |

${p}_{4}$ | $(0.8,0.1)$ | $(0.9,0.1)$ | $(0.6,0.2)$ | $(0.5,0.3)$ |

${p}_{5}$ | $(0.5,0.4)$ | $(0.6,0.3)$ | $(0.7,0.2)$ | $(0.2,0.6)$ |

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**MDPI and ACS Style**

Liu, X.; Kim, H.S.; Feng, F.; Alcantud, J.C.R.
Centroid Transformations of Intuitionistic Fuzzy Values Based on Aggregation Operators. *Mathematics* **2018**, *6*, 215.
https://doi.org/10.3390/math6110215

**AMA Style**

Liu X, Kim HS, Feng F, Alcantud JCR.
Centroid Transformations of Intuitionistic Fuzzy Values Based on Aggregation Operators. *Mathematics*. 2018; 6(11):215.
https://doi.org/10.3390/math6110215

**Chicago/Turabian Style**

Liu, Xiaoyan, Hee Sik Kim, Feng Feng, and José Carlos R. Alcantud.
2018. "Centroid Transformations of Intuitionistic Fuzzy Values Based on Aggregation Operators" *Mathematics* 6, no. 11: 215.
https://doi.org/10.3390/math6110215