# Probabilistic Interval-Valued Fermatean Hesitant Fuzzy Set and Its Application to Multi-Attribute Decision Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- This paper adds corresponding probability information for each membership degree and innovatively proposes the concept of PIVFHFSs.
- (2)
- This paper defines two new operators for PIVFHFSs combined with the Hamacher operation, including the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher weighted averaging (PIVFHFHWA) operator and geometric (PIVFHFHWG) operator.
- (3)
- Considering the correlation between different attributes, this paper further proposes the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher Choquet integral averaging (PIVFHFHCIA) operator and geometric (PIVFHFHCIG) operator based on Choquet integral.
- (4)
- Based on the PIVFHFHCIG operator, a MADM model is constructed to solve the carbon emission reduction decision-making problem of manufacturers in the supply chain.

## 2. Preliminaries

#### 2.1. Fermatean Fuzzy Sets

**Definition**

**1 [10].**

**Definition**

**2 [19].**

**Definition**

**3 [28].**

**Definition**

**4 [34].**

#### 2.2. Hamacher t-Norm and t-Conorm

**Definition**

**5 [54].**

#### 2.3. Choquet Integral and Fuzzy Measure

**Definition**

**6 [49].**

- (1)
- $\kappa (X)=1$, $\kappa (\xd8)=0$;
- (2)
- $\forall \alpha ,\beta \subseteq P(X)$, if $\alpha \subseteq \beta $, then $\kappa (\alpha )\le \kappa (\beta )$, where $P(X)$ is a power set of $X$;
- (3)
- $\kappa (\alpha \cap \beta )=\kappa (\alpha )+\kappa (\beta )+\gamma \kappa (\alpha )\kappa (\beta )$ for all $\alpha ,\beta \in X$ and $\alpha \cap \beta =\xd8$, where $\gamma $ denotes the interaction of indicators with $\gamma >-1$.

**Definition**

**7 [50].**

## 3. Probabilistic Interval-Valued Fermatean Hesitant Fuzzy Set

#### 3.1. The Probabilistic Interval-Valued Fermatean Hesitant Fuzzy Set

**Definition**

**8.**

**Example**

**1.**

**Definition**

**9.**

**Definition**

**10.**

- (1)
- If $S({M}_{1})>S({M}_{2})$, then ${M}_{1}\succ {M}_{2}$;
- (2)
- If $S({M}_{1})<S({M}_{2})$, then ${M}_{1}\prec {M}_{2}$;
- (3)
- If $S({M}_{1})=S({M}_{2})$, then
- (i)
- If $E({M}_{1})>E({M}_{2})$, then ${M}_{1}\succ {M}_{2}$;
- (ii)
- If $E({M}_{1})=E({M}_{2})$, then ${M}_{1}\sim {M}_{2}$;
- (iii)
- If $E({M}_{1})<E({M}_{2})$, then ${M}_{1}\prec {M}_{2}$.

#### 3.2. Hamacher Operations on PIVFHFEs

**Definition**

**11.**

**Theorem**

**1.**

- (1)
- ${M}_{1}\otimes {M}_{2}={M}_{2}\otimes {M}_{1}$;
- (2)
- ${M}_{1}\oplus {M}_{2}={M}_{2}\oplus {M}_{1}$;
- (3)
- $\lambda ({M}_{1}\oplus {M}_{2})=\lambda {M}_{1}\oplus \lambda {M}_{2},\lambda >0$;
- (4)
- ${({M}_{1}\otimes {M}_{2})}^{\lambda}={M}_{2}{}^{\lambda}\otimes {M}_{1}{}^{\lambda},\lambda >0$;
- (5)
- $({\lambda}_{1}+{\lambda}_{2})M={\lambda}_{1}M\oplus {\lambda}_{2}M,\forall {\lambda}_{1},{\lambda}_{2}>0$;
- (6)
- ${M}^{{\lambda}_{1}}\otimes {M}^{{\lambda}_{2}}={M}^{{\lambda}_{1}+{\lambda}_{2}},\forall {\lambda}_{1},{\lambda}_{2}>0$.

## 4. Probabilistic Interval-Valued Fermatean Hesitant Fuzzy Hamacher Aggregation Operators

**Definition**

**12.**

**Theorem**

**2.**

**Proof of Theorem**

**2.**

**Example**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Definition**

**13.**

**Theorem**

**6.**

## 5. Hamacher Choquet Integral Aggregation Operators of PIVFHFSs

**Definition**

**14.**

**Theorem**

**7.**

**Proof of Theorem**

**7.**