Some new hybrid hesitant fuzzy weighted aggregation operators based on Archimedean and Dombi operations for multi-attribute decision making

Abstract

Hesitant fuzzy set (HFS) is more flexible and general tool in comparison to fuzzy set theory. There is not yet reported an aggregation operator (AO) which can provide desirable generality, flexibility and compatibility in adjusting risk preferences while aggregating attribute values under hesitant fuzzy (HF) environment, although based on algebraic t-norm and t-conorm, Einstein t-norm and t-conorm, Hammacher t-norm and t-conorm, Dombi t-norm and t-conorm and Frank t-norm and t-conorm; weighted AOs have been developed earlier to attempt to meet above such eventualities. So, the primary objective of this paper is to develop some general, flexible as well as compatible AOs that can be exploited to solve MADM problems with HF information. From this perspective, at the very beginning, we develop new operations between HFEs by uniting the features of Archimedean and Dombi operations. Next, based on these operations, we develop HF Archimedean-Dombi weighted arithmetic and geometric AOs, HF Archimedean-Dombi ordered weighted arithmetic and geometric AOs and HF Archimedean-Dombi hybrid arithmetic and geometric AOs. We have shown that the existing HF-algebraic weighted AOs, HF-Einstein weighted AOs and HF-Hammacher weighted AOs are special cases of our developed AOs. We discuss in detail some intriguing properties of the proposed AOs. Next, we establish a procedure of MADM endowed by the proposed operators under HF environment. Then, we present a practical example concerning the personnel selection to gloss the decision steps of the proposed method. We also conduct a validity test to show that our proposed AOs are authentic and legal. Moreover, we exhibit a sensitivity investigation with diverse criteria weight sets to examine the stability of our proposed intriguing approach. Also, we draw attention toward a comparison between the existing decision-making methods with the proposed method to prove the superiority of our model.

Appendices

Appendix 1: list of abbreviations

Abbreviation	Full name
HFS	Hesitant fuzzy set
HFE	Hesitant fuzzy element
HF	Hesitant fuzzy
AO	Aggregation operator
MADM	Multi-attribute decision making
HFADWAA	Hesitant fuzzy Archimedean–Dombi weighted arithmetic aggregation operator
HFADOWAA	Hesitant fuzzy Archimedean–Dombi ordered weighted arithmetic aggregation operator
HFADHAA	Hesitant fuzzy Archimedean–Dombi hybrid arithmetic aggregation operator
HFADWGA	Hesitant fuzzy Archimedean–Dombi weighted geometric aggregation operator
HFADOWGA	Hesitant fuzzy Archimedean–Dombi ordered weighted geometric aggregation operator
HFADHGA	Hesitant fuzzy Archimedean–Dombi hybrid geometric aggregation operator
HFWA	Hesitant fuzzy weighted arithmetic aggregation operator
HFWG	Hesitant fuzzy weighted geometric aggregation operator
HFFWA	Hesitant fuzzy Frank weighted averaging operator
HFFWG	Hesitant fuzzy Frank weighted geometric operator
HFHWA	Hesitant fuzzy Hamachar weighted arithmetic aggregation operator
HFHWG	Hesitant fuzzy Hamachar weighted geometric aggregation operator
HFHOWA	Hesitant fuzzy Hamachar ordered weighted arithmetic aggregation operator
HFHOWG	Hesitant fuzzy Hamachar ordered weighted geometric aggregation operator
HFDWA	Hesitant fuzzy Dombi weighted arithmetic aggregation operator
HFDWG	Hesitant fuzzy Dombi weighted geometric aggregation operator
HFDOWA	Hesitant fuzzy Dombi ordered weighted arithmetic aggregation operator
HFDOWG	Hesitant fuzzy Dombi ordered weighted geometric aggregation operator
HFEWA	Hesitant fuzzy Einstein weighted arithmetic aggregation operator
HFEWG	Hesitant fuzzy Einstein weighted geometric aggregation operator
HFEOWA	Hesitant fuzzy Einstein ordered weighted arithmetic aggregation operator
HFEOWG	Hesitant fuzzy Einstein ordered weighted geometric aggregation operator

Appendix 2: hesitant fuzzy sets

Definition 8

[1] Let U be a finite universe of discourse. Then a hesitant fuzzy set $\xi$ on U is defined as:

$$ \xi = \{ < x,\mu_{\xi }^{h} (x) > :x \in U\} $$

where $\mu_{\xi }^{h} :U \to 2^{[0,1]}$ is a mapping, and $\mu_{\xi }^{h} (x)$ denotes the set (finite) of possible membership degrees of the element $x \in U$.

Given $x \in U$, $\mu_{\xi }^{h}$ is termed as a hesitant fuzzy element (HFE) [20]. For sake of simplicity, the hesitant fuzzy set $\xi$ is represented by $\left\langle {\mu_{\xi }^{h} } \right\rangle$. The set of all HFEs on U is denoted by ${\text{HFE}}^{U}$.

Definition 9

[20] For two HFEs $\mu_{{\xi_{1} }}^{h} \,\,{\text{and}}\,\,\,\mu_{{\xi_{2} }}^{h}$ on U, some basic operations between them can be described as:

• (1) $\mu_{{\xi_{1} }}^{h} \cup \,\mu_{{\xi_{2} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} ,\alpha_{2} \in \mu_{{\xi_{2} }}^{h} }} max\{ \alpha_{1} ,\alpha_{2} \}$

• (2) $\mu_{{\xi_{1} }}^{h} \cap \,\mu_{{\xi_{2} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} ,\alpha_{2} \in \mu_{{\xi_{2} }}^{h} }} min\{ \alpha_{1} ,\alpha_{2} \}$

• (3) $\left( {\mu_{{\xi_{1} }}^{h} } \right)^{c} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} }} \{ 1 - \alpha_{1} \}$

• (4) $\mu_{{\xi_{1} }}^{h} \oplus \,\mu_{{\xi_{2} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} ,\alpha_{2} \in \mu_{{\xi_{2} }}^{h} }} \{ \alpha_{1} + \alpha_{2} - \alpha_{1} \alpha_{2} \}$

• (5) $\mu_{{\xi_{1} }}^{h} \otimes \,\mu_{{\xi_{2} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} ,\alpha_{2} \in \mu_{{\xi_{2} }}^{h} }} \{ \alpha_{1} \alpha_{2} \}$

• (6) $\lambda \mu_{{\xi_{1} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} }} \{ 1 - (1 - \alpha_{1} )^{\lambda } \} \,\,\,\,\,\,(\lambda > 0)$

• (7) $\left( {\mu_{{\xi_{1} }}^{h} } \right)^{\lambda } = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} }} \{ (\alpha_{1} )^{\lambda } \} \,\,\,\,\,\,(\lambda > 0)$

Theorem 24

[20] Let $\mu_{{\xi_{1} }}^{h} \,\,{\text{and}}\,\,\,\mu_{{\xi_{2} }}^{h}$ be two HFEs defined on U and $\lambda ,\lambda_{1} ,\,\lambda_{2} > 0$. Then

• (i) $\mu_{{\xi_{1} }}^{h} \oplus \mu_{{\xi_{2} }}^{h} = \mu_{{\xi_{2} }}^{h} \oplus \mu_{{\xi_{1} }}^{h}$

• (ii) $\mu_{{\xi_{1} }}^{h} \otimes \mu_{{\xi_{2} }}^{h} = \mu_{{\xi_{2} }}^{h} \otimes \mu_{{\xi_{1} }}^{h}$

• (iii) $\lambda (\mu_{{\xi_{1} }}^{h} \oplus \mu_{{\xi_{2} }}^{h} ) = (\lambda \mu_{{\xi_{1} }}^{h} ) \oplus (\lambda \mu_{{\xi_{2} }}^{h} )$

• (iv) $(\mu_{{\xi_{1} }}^{h} \otimes \mu_{{\xi_{2} }}^{h} )^{\lambda } = (\mu_{{\xi_{1} }}^{h} )^{\lambda } \otimes (\mu_{{\xi_{2} }}^{h} )^{\lambda }$

• (v) $(\lambda_{1} + \,\lambda_{2} )\mu_{{\xi_{1} }}^{h} = (\lambda_{1} \mu_{{\xi_{1} }}^{h} ) \oplus (\lambda_{2} \mu_{{\xi_{1} }}^{h} )$

• (vi) $(\mu_{{\xi_{1} }}^{h} )^{{\lambda_{1} + \,\lambda_{2} }} = (\mu_{{\xi_{1} }}^{h} )^{{\lambda_{1} }} \otimes (\mu_{{\xi_{1} }}^{h} )^{{\lambda_{2} }}$

Definition 10

[20] Let $\mu_{\xi }^{h}$ be a HFE on U. Then the score value of $\mu_{\xi }^{h}$ is defined as:

$$ S(\mu_{\xi }^{h} ) = \frac{1}{{\# \mu_{\xi }^{h} }}\sum\nolimits_{{\alpha \in \mu_{\xi }^{h} }} \alpha $$

where $\# \mu_{\xi }^{h}$ denotes the number of elements in $\mu_{\xi }^{h}$.

Based on the score values of HFEs, a comparison method of HFEs is described below:

Definition 11

[20] Suppose $\mu_{{\xi_{1} }}^{h} \,\,{\text{and}}\,\,\,\mu_{{\xi_{2} }}^{h}$ be two HFEs on U. Then

• (1) If $S(\mu_{{\xi_{1} }}^{h} \,) > S(\mu_{{\xi_{2} }}^{h} )$, then $\mu_{{\xi_{1} }}^{h} \succ \,\mu_{{\xi_{2} }}^{h}$

• (2) If $S(\mu_{{\xi_{1} }}^{h} \,) < S(\mu_{{\xi_{2} }}^{h} )$, then $\mu_{{\xi_{1} }}^{h} \prec \,\mu_{{\xi_{2} }}^{h}$

• (3) If $S(\mu_{{\xi_{1} }}^{h} \,) = S(\mu_{{\xi_{2} }}^{h} )$, then $\mu_{{\xi_{1} }}^{h} = \,\mu_{{\xi_{2} }}^{h}$

Appendix 3: Archimedean t-norm and t-conorm

Definition 12

[39, 40] A fuzzy t-norm $f:[0,1] \times [0,1] \to [0,1]$ is a function which satisfies the following axioms:

• (i) $f(x,1) = x\,\,{\text{for}}\,\,x \in [0,1]$

• (ii) $f(x,y) \le f(x^{\prime},y^{\prime})\,\,provided\,\,\,x \le x^{\prime},\,y \le y^{\prime}\,\,{\text{for}}\,\,x,x^{\prime},\,y,y^{\prime} \in [0,1]$

• (iii) $f(x,y) = f(y,x)\,\,{\text{for}}\,\,\,x,\,y, \in [0,1]$

• (iv) $f(x,f(y,z)) = f(f(x,y),z)\,\,{\text{for}}\,\,\,x,\,y, \in [0,1]$

Definition 13

[39, 40] A fuzzy t-conorm $g:[0,1] \times [0,1] \to [0,1]$ is a function which satisfies the following axioms:

• (i) $g(x,0) = x\,\,{\text{for}}\,\,x \in [0,1]$

• (ii) $g(x,y) \le g(x^{\prime},y^{\prime})\,\,provided\,\,\,x \le x^{\prime},\,y \le y^{\prime}\,\,{\text{for}}\,\,x,x^{\prime},\,y,y^{\prime} \in [0,1]$

• (iii) $g(x,y) = g(y,x)\,\,{\text{for}}\,\,\,x,\,y, \in [0,1]$

• (iv) $g(x,g(y,z)) = g(g(x,y),z)\,\,{\text{for}}\,\,\,x,\,y, \in [0,1]$

Definition 14

[39, 40] A t-norm function $f(x,y)$ is called a strictly Archimedean t-norm if it is continuous, $f(x,x) < x\,\,\,\forall \,x \in (0,1)$ and strictly increasing for $x,y \in (0,1).$

Definition 15

[39, 40] A t-conorm function $g(x,y)$ is called a strictly Archimedean t-conorm if it is continuous, $g(x,x) > x\,\,\,\forall \,x \in (0,1)$ and strictly increasing for $x,y \in (0,1)$.

Definition 16

[41] Suppose $\theta :(0,1] \to R$ is a continuous function such that $\theta$ is strictly decreasing. Then a strictly Archimedean t-norm is expressed by:

$$ \delta (x,y) = \theta^{ - 1} (\theta (x) + \theta (y))\,\,\,for\,x,y \in (0,1] $$

Definition 17

[41] Suppose $\psi :[0,1) \to R$ is a continuous function such that $\psi (l) = \theta (1 - l),\,l \in [0,1)$, and $\psi$ is strictly increasing. Then a strictly Archimedean t-conorm is expressed by:

$$ \rho (x,y) = \psi^{ - 1} (\psi (x) + \psi (y))\,\,\,for\,x,y \in [0,1) $$

Appendix 4: Dombi operations

The operations of t-norm and t-conorm, developed by Dombi [38] are generally known as Dombi operations described below:

Definition 18

[38] For any two real numbers x and y in [0, 1], the Dombi t-norm and Dombi t-conorm can be defined as follows:

$$ {\text{Dom}}(x,y) = \frac{1}{{1 + \left\{ {\left( {\frac{1 - x}{x}} \right)^{k} + \left( {\frac{1 - y}{y}} \right)^{k} } \right\}^{{\frac{1}{k}}} }},\,\,\,\,\,{\text{Dom}}^{c} (x,y) = 1 - \frac{1}{{1 + \left\{ {\left( {\frac{x}{1 - x}} \right)^{k} + \left( {\frac{y}{1 - y}} \right)^{k} } \right\}^{{\frac{1}{k}}} }}\,\,(k \ge 1) $$

Dombi operations have good precedence of change w.r.t values of the parameter ‘k’. Based on the Dombi operations, He [32] developed a few operations between HFEs given below:

Definition 19

[32] Let $\mu_{{\xi_{1} }}^{h} \,\,{\text{and}}\,\,\,\mu_{{\xi_{2} }}^{h}$ be two HFEs on U.

• (i) $\mu_{{\xi_{1} }}^{h} \oplus_{D} \mu_{{\xi_{2} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} ,\,\,\alpha_{2} \in \mu_{{\xi_{2} }}^{h} }} \left\{ {1 - \left( {1 + \left\{ {\left( {\frac{{\alpha_{1} }}{{1 - \alpha_{1} }}} \right)^{k} + \left( {\frac{{\alpha_{2} }}{{1 - \alpha_{2} }}} \right)^{k} } \right\}^{{\frac{1}{k}}} } \right)^{ - 1} } \right\}$

• (ii) $\mu_{{\xi_{1} }}^{h} \otimes_{D} \mu_{{\xi_{2} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} ,\,\,\alpha_{2} \in \mu_{{\xi_{2} }}^{h} }} \left\{ {\left( {1 + \left\{ {\left( {\frac{{1 - \alpha_{1} }}{{\alpha_{1} }}} \right)^{k} + \left( {\frac{{1 - \alpha_{2} }}{{\alpha_{2} }}} \right)^{k} } \right\}^{{\frac{1}{k}}} } \right)^{ - 1} } \right\}$

• (iii) $\lambda *_{D} \mu_{{\xi_{1} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} }} \left\{ {1 - \left( {1 + \left\{ {\lambda \left( {\frac{{\alpha_{1} }}{{1 - \alpha_{1} }}} \right)^{k} } \right\}^{{\frac{1}{k}}} } \right)^{ - 1} } \right\}\,\,\,\,\,\,\,(\lambda > 0)$

• (iv) $\lambda \circ_{D} \mu_{{\xi_{1} }}^{h} = \mathop \cup \limits_{{\alpha_{1} \in \mu_{{\xi_{1} }}^{h} }} \left\{ {\left( {1 + \left\{ {\lambda \left( {\frac{{1 - \alpha_{1} }}{{\alpha_{1} }}} \right)^{k} } \right\}^{{\frac{1}{k}}} } \right)^{ - 1} } \right\}\,\,\,\,\,\,\,(\lambda > 0)$