*2.1. Hamacher Operations*

There is an important concept in fuzzy set theory, that is, t-norm and t-conorm, which are utilized to define a generalized intersection and union of fuzzy sets [54]. A number of t-norm and t-conorm have been proposed, including Algebraic product *TA* and Algebraic sum *SA* [1], Einstein product *TE* and Einstein sum *SE* [55], and drastic product *TD* and drastic sum *SD* [56]. Further, Hamacher [36] developed a more generalized t-norm and t-conorm, that is, the Hamacher product (Hamacher t-norm) and Hamacher sum (Hamacher t-conorm), which are calculated as follows:

$$T\_H^\upsilon(a,b) = a \otimes b = \frac{ab}{\upsilon + (1-\upsilon)(a+b-ab)}, \; \upsilon > 0$$

$$S\_H^\upsilon(a,b) = a \oplus b = \frac{a+b-ab - (1-\upsilon)ab}{1 - (1-\upsilon)ab}, \; \upsilon > 0$$

 In particular, when *υ* = 1, then the Hamacher t-norm and t-conorm are transformed into the Algebraic product *TA* and Algebraic sum *SA* [1].

 −  ·

$$T\_A(a,b) = a \cdot b$$

$$S\_A(a,b) = a + b - a \cdot b$$

When *υ* = 2, then the Hamacher t-norm and t-conorm are transformed into the Einstein product *TE* and Einstein sum *SE* [55].

 =

$$T\_E(a,b) = a \otimes b = \frac{ab}{1 + (1-a)(1-b)}$$

$$S\_E(a,b) = a \oplus b = \frac{ab}{1+ab}$$

#### *2.2. Hesitant Fuzzy Linguistic Term Set*

Motivated by the HFS and fuzzy linguistic method, Rodríguez et al. [19] introduced the notion of HFLTS.

**Definition 1.** *[19]. Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS. An HFLTS, HS, is constructed by a finite subset of the continuous linguistic terms of S.*

In order to help understand the concept of HFLTS, Liao et al. [20] gave the mathematical expression of HFLTS.

**Definition 2.** *[20]. Let X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*} *be a fixed set and S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS. An HFLST on X, HS, is defined as the following*

$$H\_{\mathbb{S}} = \{<\ge, h\_{\mathbb{S}}(\mathbf{x}\_i)>|\mathbf{x}\_i \in X\}, i = 1, 2, \dots, n. \tag{1}$$

*where hS*(*xi*) *is a collection of some linguistic terms in S and can be defined as hS*(*xi*) = {*sit*HH*sit* ∈ *S*, *i* = 1, 2, ··· , *L*} *with L being the number of linguistic term in hS*(*xi*)*. For convenience, hS*(*xi*) *is referred to as the HFLE.*

To perform the equivalent conversion between HFLE and HFE, Gou [30] defined two equivalent conversion functions.

**Definition 3.** *[30]. Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS, hS* = {*st*|*<sup>t</sup>* ∈ [−*τ*, *τ*]} *be an HFLE, and hσ* = {*σ*|*σ* ∈ [0, 1]} *be an HFE. The equivalent transformation from HFLE hS to HFE hσ is performed by the following function g*

$$\emptyset \,\, : \, [ -\tau, \tau] \to [0, 1],\\ h\_{\sigma} = \operatorname{g}(h\_{S}) = \{ \sigma = \operatorname{g}(s\_{l}) = \frac{t}{2\tau} + \frac{1}{2} \}$$

Similarly, the equivalent transformation from HFE *hσ* to HFLE *hS* is performed by the following inverse function *g*<sup>−</sup>1.

$$\mathbf{g}^{-1} : [0, 1] \to [-\tau, \tau], \ h\_{\mathbf{S}} = \mathbf{g}^{-1}(h\_{\sigma}) = \{ \mathbf{s}\_t = \mathbf{g}^{-1}(\sigma) = \mathbf{s}\_{(2r - 1)\tau} \},$$

**Definition 4.** *[57]. For any three HFLEs, hS, hS*1 *, and hS*2 *, g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion functions between HFLE and HFE, and λ* > 0*; the operational rules on HFLEs are defined as follows:*

$$(\mathcal{I})\quad h\_{\mathcal{S}\_1} \oplus h\_{\mathcal{S}\_2} = \bigcup\_{\sigma\_1 \in \mathcal{g}(h\_{\mathcal{S}\_1}), \sigma\_2 \in \mathcal{g}(h\_{\mathcal{S}\_2})} \{\mathcal{g}^{-1}(\sigma\_1 + \sigma\_2 - \sigma\_1 \sigma\_2)\};$$

$$(2)\quad h\_{\mathbb{S}\_1} \otimes h\_{\mathbb{S}\_2} = \underset{\sigma\_1 \in \mathfrak{g}(h\_{\mathbb{S}\_1}), \sigma\_2 \in \mathfrak{g}(h\_{\mathbb{S}\_2})}{\downarrow} \{\mathfrak{g}^{-1}(\sigma\_1 \sigma\_2)\};$$

$$\{3\} \quad \lambda h\_{\mathcal{S}} = \underset{\sigma \in \mathcal{S}(h\_{\mathcal{S}})}{\cup} \{\mathcal{g}^{-1}(1 - (1 - \sigma)^{\lambda})\};$$

$$(4)\quad (h\_S)^\lambda = \underset{\sigma \in \mathfrak{g}(h\_S)}{\cup} \{\mathfrak{g}^{-1}(\sigma^\lambda)\}.$$

In the following, we introduce the Hamacher t-norm and t-conorm to the HFLTS environment and define some new operational rules on HFLEs.

**Definition 5.** *For any three HFLEs, hS, hS*1 *, and hS*2 *, g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion functions between HFLE and HFE, and υ* > 0*. According to the Hamacher t-norm and t-conorm, some operational rules on HFLEs are defined as follows:*

$$(1)\qquad h\_{\mathbb{S}\_1} \oplus\_H h\_{\mathbb{S}\_2} = \bigcup\_{\sigma\_1 \in \mathcal{G}(h\_{\mathbb{S}\_1}), \sigma\_2 \in \mathcal{G}(h\_{\mathbb{S}\_2})} \left\{ \mathcal{S}^{-1} \left( \frac{\sigma\_1 + \sigma\_2 - \sigma\_1 \sigma\_2 - (1-\nu)\sigma\_1 \sigma\_2}{1 - (1-\nu)\sigma\_1 \sigma\_2} \right) \right\};$$

$$(2)\quad h\_{\mathcal{S}\_1} \otimes\_H h\_{\mathcal{S}\_2} = \underset{\substack{\sigma\_1 \in \mathcal{g}(h\mathfrak{s}\_1), \sigma\_2 \in \mathcal{g}(h\mathfrak{s}\_2)}}{\cup} \left\{ \mathcal{g}^{-1} \left( \frac{\sigma\_1 \sigma\_2}{\nu + (1 - \nu)(\sigma\_1 + \sigma\_2 - \sigma\_1 \sigma\_2)} \right) \right\} \asymp$$

$$\text{(3)} \quad \lambda h\_{\mathcal{S}} = \underset{\sigma \in \mathfrak{F}(h\_{\mathcal{S}})}{\text{cup}} \left\{ \underset{\sigma \in \mathfrak{F}(h\_{\mathcal{S}})}{\text{g}} \left\{ \underset{(1+(\upsilon-1)\sigma)^{\lambda}+(\upsilon-1)(1-\sigma)^{\lambda}}{(1+(\upsilon-1)\sigma)^{\lambda}+(\upsilon-1)(1-\sigma)^{\lambda}} \right\} \right\}, \ \lambda > 0;$$

$$(4)\quad \left(h\_{\mathcal{S}}\right)^{\lambda} = \underset{\sigma \in \mathcal{X}(h\_{\mathcal{S}})}{\cup} \left\{ \mathbf{g}^{-1} \left( \frac{\mathbf{x}\sigma^{\lambda}}{(1 + (v-1)(1-\sigma))^{\lambda} + (v-1)\sigma^{\lambda}} \right) \right\}, \ \lambda > 0.$$

**Remark 1.** *When υ* = 1*, we can see that these operations of HFLEs in Definition 5 are transformed into those in Definition 4. In other words, the operations in Definition 4 are a special case of Definition 5 by comparing Definition 4 with Definition 5.*

*In addition, when υ* = 2*, these basic operations of HFLEs in Definition 5 are transformed into the Einstein operations on HFLEs.*

$$(1)\quad h\_{S\_1} \oplus\_E h\_{S\_2} = \bigcup\_{\sigma\_1 \in \mathfrak{g}(h\_{S\_1}), \sigma\_2 \in \mathfrak{g}(h\_{S\_2})} \left\{ \mathcal{G}^{-1} \left( \frac{\sigma\_1 + \sigma\_2}{1 + \sigma\_1 \sigma\_2} \right) \right\};$$

$$\text{(2)}\quad h\_{\text{S}\_1} \otimes\_E h\_{\text{S}\_2} = \underset{\sigma\_1 \in \mathfrak{g}(h\_{\text{S}\_1}), \sigma\_2 \in \mathfrak{g}(h\_{\text{S}\_2})}{\text{U}} \left\{ \mathcal{S}^{-1} \left( \frac{\sigma\_1 \sigma\_2}{1 - (1 - \sigma\_1)(1 - \sigma\_2)} \right) \right\};$$

$$\text{(3)}\quad \lambda \text{h}\_S = \underset{\sigma \in \mathfrak{F}(\text{h}\_S)}{\text{cup}} \left\{ \text{g}^{-1} \left( \frac{(1+\sigma)^{\lambda} - (1-\sigma)^{\lambda}}{(1+\sigma)^{\lambda} + (1-\sigma)^{\lambda}} \right) \right\}, \ \lambda > 0;$$

$$(4)\quad \left(h\_S\right)^\lambda = \underset{\sigma \in \mathcal{G}(h\_S)}{\cup} \left\{ \mathcal{S}^{-1} \left(\frac{2\sigma^\lambda}{\left(2-\sigma\right)^\lambda + \sigma^\lambda}\right) \right\} , \ \lambda > 0.$$

To compare the two HFLEs, Gou [30] defined the score function of HFLE as follows.

**Definition 6.** *[30]. Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS and hS* = {*st*|*<sup>t</sup>* ∈ [−*τ*, *τ*]} *be an HFLE, then the score function of hS* is defined as the following

$$s(h\_S) = \sum\_{i=1}^{L} g(s\_i) / L \tag{2}$$

*where L is the number of the elements of hS. Therefore, the comparative relation for two HFLEs is determined as follows:*


**Definition 7.** *[58]. Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS, and hS*1 = {*sl*1*t*HHH*sl*1*t* ∈ *S*, *l* = 1, 2, ··· , *<sup>L</sup>*1} *, and hS*2 = {*sl*2*t*HHH*sl*2*t* ∈ *S*, *l* = 1, 2, ··· , *<sup>L</sup>*2} *be the two HFLEs. If L*1 = *L*2 *and λ* > 0*, then the generalized hesitant fuzzy linguistic distance between hS*1 *and hS*2 *is defined as follows*

$$d(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}) = \left(\frac{1}{L} \sum\_{i=1}^{L} \left( \left| g(s\_{1t}^i) - g(s\_{2t}^i) \right| \right)^\lambda \right)^{\frac{1}{\lambda}} \tag{3}$$

*where g is the equivalent conversion function gave in Definition 3. When λ* = 2*, <sup>d</sup>*(*hS*1 , *hS*2 ) *is called the HFL Euclidean distance between hS*1*and hS*2*.*

When applying Equation (3), if *L*1 = *L*2, then the shorter one (*L*1 < *L*2) needs to be extended by adding the linguistic terms given as *s*1 = *s*11*t* + *sL*<sup>1</sup> 1*t* /2, where *s*11*t* and *sL*<sup>1</sup> 1*t* are the smallest and biggest linguistic terms in *hS*1, respectively.

#### **3. Hesitant Fuzzy Linguistic Hamacher Aggregation Operators**

In this part, we present a hesitant fuzzy linguistic Hamacher weighted averaging (HFLHWA) and a hesitant fuzzy linguistic Hamacher weighted geometric (HFLHWG), a generalized hesitant fuzzy linguistic Hamacher weighted averaging (GHFLHWA) and a generalized hesitant fuzzy linguistic Hamacher weighted geometric (GHFLHWG) operators. Furthermore, we also discuss some special cases of these operators and explore some properties of these operators.

#### *3.1. HFLHWA and HFLHWG Operators*

**Definition 8.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and υ* > 0*. wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*)*, satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup>*wi* = 1*. If*

$$\text{HFLHWA}\_{\text{w}}^{\text{v}}(\text{h}\_{\text{S}\_{1}}, \text{h}\_{\text{S}\_{2}}, \cdot, \cdot, \text{h}\_{\text{S}\_{n}}) = w\_{1}\text{h}\_{\text{S}\_{1}} \oplus\_{H} w\_{2}\text{h}\_{\text{S}\_{2}} \oplus\_{H} \cdot \cdot \cdot \oplus\_{H} w\_{n}\text{h}\_{\text{S}\_{n}} = \bigoplus\_{i=1}^{n} \text{(w}\_{i}\text{h}\_{\text{S}\_{i}}) \tag{4}$$

*Then, HFLHWA<sup>υ</sup>w is designated as the HFL Hamacher weighted averaging (HFLHWA) operator*.

**Theorem 1.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and υ* > 0*. wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*)*, satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1*. g and g*<sup>−</sup><sup>1</sup> *are the equivalent transformation functions between HFLEs and HFEs. Then the aggregated value by the HFLHWA operator is also an HFLE, and*

$$\text{HFLHWA}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \text{g}(h\_{\text{S}\_{j}})} \left\{ \text{g}^{-1} \left( \frac{\prod\_{i=1}^{n} (1 + (\upsilon - 1)\sigma\_{i})^{w\_{i}} - \prod\_{i=1}^{n} (1 - \sigma\_{i})^{w\_{i}}}{\prod\_{i=1}^{n} (1 + (\upsilon - 1)\sigma\_{i})^{w\_{i}} + (\upsilon - 1)\prod\_{i=1}^{n} (1 - \sigma\_{i})^{w\_{i}}} \right) \right\} \tag{5}$$

**Proof.** According to mathematical induction method, Equation (5) can be proved as follows.

For *n* = 1, the result of Equation (5) clearly holds. Suppose Equation (5) hold for *n* = *k*, namely

$$\text{HFLHWA}\_{\text{w}}^{v}(h\text{s}\_{1}, h\text{s}\_{2}, \dots, h\text{s}\_{k}) = \bigcup\_{\sigma\_{i} \in \text{g}(h\text{s}\_{\text{S}\_{i}})} \left\{ \text{g}^{-1} \left( \frac{\prod\_{i=1}^{k} (1 + (v-1)\sigma\_{i})^{w\_{i}} - \prod\_{i=1}^{k} (1 - \sigma\_{i})^{w\_{i}}}{\prod\_{i=1}^{k} (1 + (v-1)\sigma\_{i})^{w\_{i}} + (v-1)\prod\_{i=1}^{k} (1 - \sigma\_{i})^{w\_{i}}} \right) \right\} \right\}$$

Then, for *n* = *k* + 1, by Equation (4), we can ge<sup>t</sup>

*HFLHWA<sup>υ</sup>w*(*hS*1 , *hS*2 , ··· , *hSk* , *hSk*+<sup>1</sup> ) = *<sup>w</sup>*1*hS*1 ⊕*H <sup>w</sup>*2*hS*2 ⊕*H* ···⊕*H wk*+<sup>1</sup>*hSk*+<sup>1</sup> = *k* ⊕*H i*=1 (*wihSi*) ⊕*H wk*+<sup>1</sup>*hSk*+<sup>1</sup> = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup>5 <sup>∏</sup>*ki*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*−<sup>∏</sup>*ki*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* <sup>∏</sup>*ki*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ki*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* 6<sup>⊕</sup>*H* ∪ *<sup>σ</sup>k*+1∈*g*(*hSk*+<sup>1</sup> ) *g*<sup>−</sup><sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>k*+<sup>1</sup>)*wk*+1−(<sup>1</sup>−*σk*+<sup>1</sup>)*wk*+<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>k*+<sup>1</sup>)*wk*+<sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σk*+<sup>1</sup>)*wk*+<sup>1</sup> 

Let ∏*ki*=<sup>1</sup> (1 + (*υ* − <sup>1</sup>)*<sup>σ</sup>i*)*wi* = *α*1, ∏*ki*=<sup>1</sup> (1 − *<sup>σ</sup>i*)*wi* = *β*1, (1 + (*υ* − <sup>1</sup>)*<sup>σ</sup>k*+<sup>1</sup>)*wk*+<sup>1</sup> = *α*2, and (1 − *<sup>σ</sup>k*+<sup>1</sup>)*wk*+<sup>1</sup> = *β*2, then

$$\bigoplus\_{i=1}^k (w\_i h\_{\mathbb{S}\_i}) = \bigcup\_{\sigma\_i \in \mathcal{G}(h\_{\mathbb{S}\_i})} \left\{ \mathcal{g}^{-1} \left( \frac{a\_1 - \beta\_1}{a\_1 + (v-1)\beta\_1} \right) \right\} \text{ and } w\_{k+1} h\_{\mathbb{S}\_{k+1}} = \bigcup\_{\sigma\_{k+1} \in \mathcal{G}(h\_{\mathbb{S}\_{k+1}})} \left\{ \mathcal{g}^{-1} \left( \frac{a\_2 - \beta\_2}{a\_2 + (v-1)\beta\_2} \right) \right\}$$

Further, the operational law (1) in Definition 5 yields

*k* ⊕*H i*=1 (*wihSi*) ⊕*H wk*+<sup>1</sup>*hSk*+<sup>1</sup> = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*∈*g*(*hSk* )*g*<sup>−</sup><sup>1</sup> *<sup>α</sup>*1−*β*<sup>1</sup> *<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*<sup>1</sup> ⊕*H* ∪ *<sup>σ</sup>k*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup> *<sup>α</sup>*2−*β*<sup>2</sup> *<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*<sup>2</sup> = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup>(*<sup>α</sup>*1−*β*1)(*<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*2)+(*<sup>α</sup>*2−*β*2)(*<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*1)−(<sup>2</sup>−*<sup>υ</sup>*)(*<sup>α</sup>*1−*β*1)(*<sup>α</sup>*2−*β*2) (*<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*1)(*<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*2)−(<sup>1</sup>−*<sup>υ</sup>*)(*<sup>α</sup>*1−*β*1)(*<sup>α</sup>*2−*β*2) = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),····,*σk*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup> *<sup>α</sup>*1*α*2−*β*1*β*<sup>2</sup> *<sup>α</sup>*1*α*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*1*β*<sup>2</sup> = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup>5 ∏*<sup>k</sup>*+<sup>1</sup> *i*=1 (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*−∏*<sup>k</sup>*+<sup>1</sup> *i*=1 (<sup>1</sup>−*σi*)*wi* ∏*<sup>k</sup>*+<sup>1</sup> *i*=1 (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*<sup>k</sup>*=<sup>1</sup> *i*=1 (<sup>1</sup>−*σi*)*wi* 6

That is, Equation (5) holds for *n* = *k* + 1. Therefore, Equation (5) holds for all *n*. -

**Remark 2.** *When υ* = 1*, then the HFLHWA operator is transformed into the following:*

$$\text{HFLMA}\_{\text{w}}(\mathsf{h}\_{\text{S}\_1}, \mathsf{h}\_{\text{S}\_2}, \dots, \mathsf{h}\_{\text{S}\_k}) = \bigoplus\_{i=1}^{n} (\mathsf{w}\_{i} \mathsf{h}\_{\text{S}\_i}) = \bigcup\_{\sigma\_{\hat{\imath}} \in \mathcal{J}(\mathsf{h}\_{\text{S}\_i})} \left\{ \mathsf{g}^{-1} \left( 1 - \prod\_{i=1}^{k} (1 - \sigma\_{\hat{\imath}})^{w\_{\hat{\imath}}} \right) \right\}.$$

*where HFLWAw is called the HFLWA operator by Zhang and Qi [29]. When υ* = 2*, the HFLHWA operator is transformed into to the following:*

$$\text{HFLEWA}\_{\text{w}}(\mathsf{h}\_{\text{S}\_{1}}, \mathsf{h}\_{\text{S}\_{2}}, \cdots, \mathsf{h}\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \mathcal{g}(\mathsf{h}\_{\text{S}\_{i}})} \left\{ \mathsf{g}^{-1} \left( \frac{\prod\_{i=1}^{n} \left(1 + \sigma\_{i}\right)^{\text{w}\_{i}} - \prod\_{i=1}^{n} \left(1 - \sigma\_{i}\right)^{\text{w}\_{i}}}{\prod\_{i=1}^{n} \left(1 + \sigma\_{i}\right)^{\text{w}\_{i}} + \prod\_{i=1}^{n} \left(1 - \sigma\_{i}\right)^{\text{w}\_{i}}} \right) \right\}$$

*Here, HFLEWAw is called the HFLEWA operator. Especially when wi* = 1/*n, then the HFLHWA operator is transformed into the hesitant fuzzy Hamacher averaging (HFLHA) operator.*

$$\text{HFLHA}\_{\text{w}}^{v}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \text{g}(h\_{\text{S}\_{i}})} \left\{ \text{g}^{-1} \left( \frac{\prod\_{i=1}^{n} \left(1 + (\upsilon - 1)\sigma\_{i}\right)^{\frac{1}{n}} - \prod\_{i=1}^{n} \left(1 - \sigma\_{i}\right)^{\frac{1}{n}}}{\prod\_{i=1}^{n} \left(1 + (\upsilon - 1)\sigma\_{i}\right)^{\frac{1}{n}} + (\upsilon - 1)\prod\_{i=1}^{n} \left(1 - \sigma\_{i}\right)^{\frac{1}{n}}} \right) \right\}$$

**Example 1.** *Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS and τ* = 3*. hS*1 = {*<sup>s</sup>*−1,*s*1} *and hS*2 = {*<sup>s</sup>*−2,*s*0} *are two HFLEs; w* = (0.4, 0.6) *are the weights of hS*1 *and hS*2 *, respectively. Then we can aggregate them by employing the HFLHWA (υ* = 3*) operator.*

*HFLHWA*3*w*(*hS*1 , *hS*2 ) = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 )*g*<sup>−</sup><sup>1</sup>5 ∏<sup>2</sup>*i*=<sup>1</sup> (1+(<sup>3</sup>−<sup>1</sup>)*<sup>σ</sup>i*)*wi*−∏<sup>2</sup>*i*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* ∏<sup>2</sup>*i*=<sup>1</sup> (1+(<sup>3</sup>−<sup>1</sup>)*<sup>σ</sup>i*)*wi*+(<sup>3</sup>−<sup>1</sup>)∏<sup>2</sup>*i*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* 6 = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ) ⎧⎪⎪⎨⎪⎪⎩*g*−1⎛⎜⎜⎝ (<sup>1</sup>+2× 13 )0.4(<sup>1</sup>+2× 16 )0.6−( 23 )0.4( 56 )0.6 (<sup>1</sup>+2× 13 )0.4(<sup>1</sup>+2× 16 )0.6+2×( 23 )0.4( 56 )0.6 , <sup>3</sup>×( 13 )0.4×( 12 )0.6 (<sup>1</sup>+2× 23 )0.4×(<sup>1</sup>+2× 12 )0.6+2×( 13 )0.4×( 12 )0.6 <sup>3</sup>×( 23 )0.4×( 16 )0.6 (<sup>1</sup>+2× 13 )0.4×(<sup>1</sup>+2× 56 )0.6+2×( 23 )0.4×( 16 )0.6 , <sup>3</sup>×( 23 )0.4×( 12 )0.6 (<sup>1</sup>+2× 13 )0.4×(<sup>1</sup>+2× 12 )0.6+2×( 23 )0.4×( 12 )0.6 ⎞⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭ = {*g*<sup>−</sup><sup>1</sup>(0.2333, 0.3862, 0.4355, 0.5716)} = {*<sup>s</sup>*−1.6004, *s*−0.6829, *s*−0.3870, *<sup>s</sup>*0.4299}

**Idempotent 1.** *Let hSi*(*i* = 1, 2, ··· , *n*) *be equal and each hSi which have only one value, namely, hSi* = *hS* = {*st*} *for any i, then*

$$\text{HFLIHM}\_{\text{uv}}^{v}(h\_{\text{S}\_1\prime}, h\_{\text{S}\_2\prime} \cdot \cdot, h\_{\text{S}\_n}) = h\_{\text{S}} \tag{6}$$

**Proof.** According to Definition 3, we have

*g* : [−*τ*, *τ*] → [0, 1], *g*(*st*) = *t*2*τ* + 12 = *σ*HHHH*t* ∈ [−*τ*, *τ*] = *hσ* Then, *HFLHWA<sup>υ</sup>w*(*hS*1 , *hS*2 , ··· , *hSn* ) = ∪ *σ*∈*g*(*hS*)*g*<sup>−</sup><sup>1</sup> ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*)*wi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*<sup>σ</sup>*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*<sup>σ</sup>*)*wi* = ∪ *σ*∈*g*(*hS*)*g*<sup>−</sup><sup>1</sup>5 (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*)<sup>∑</sup>*ni*=<sup>1</sup> *wi*−(<sup>1</sup>−*<sup>σ</sup>*)<sup>∑</sup>*ni*=<sup>1</sup> *wi* (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*)<sup>∑</sup>*ni*=<sup>1</sup> *wi*+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*<sup>σ</sup>*)<sup>∑</sup>*ni*=<sup>1</sup> *wi* 6 = ∪ *σ*∈*g*(*hS*){*g*<sup>−</sup><sup>1</sup>(*σ*)} = {*st*} = *hS*. Therefore, we have *HFLHWA<sup>υ</sup>w*(*hS*1 , *hS*2 , ··· , *hSn* ) = *hS*. -

**Remark 3.** *Note that the HFLHWA operator is not idempotent in general; the following example is provided to demonstrate this case*.

**Example 2.** *Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS, τ* = 3*, hS*1 = *hS*2 = *hS* = {*<sup>s</sup>*−1,*s*1} *and w* = (0.4, 0.6)*T. Then HFLHWA*3*w*(*hS*1 , *hS*2 ) = {0.3333, 0.4804, 0.5480, 0.6667}*, <sup>s</sup>*(*HFLHWA*3*w*(*hS*1 , *hS*2 )) = 0.5071 *and s*(*hS*) = 0.5*. Therefore, HFLHWA*3*w*(*hS*1 , *hS*2 ) > *hS.*

**Monotonic 1.** *Let haS* = {*h<sup>a</sup>*1*S* , *ha*2*S* , ··· , *hanS* } *and hbS* = {*hb*1*S* , *hb*2*S* , ··· , *hbnS* } *be two any collection of HFLEs. If for any sai t* ∈ *hai S and sbi t* ∈ *hbi S , and sai t* ≤ *sbi t for any i, then*

$$\text{HFLHWA}\_{\text{w}}^{v}(h\_{\text{S}}^{a\_1}, h\_{\text{S}}^{a\_2}, \dots, h\_{\text{S}}^{a\_n}) \leq \text{HFLHWA}\_{\text{w}}^{v}(h\_{\text{S}}^{b\_1}, h\_{\text{S}}^{b\_2}, \dots, h\_{\text{S}}^{b\_n}) \tag{7}$$

**Proof.** Let *f*(*x*) = <sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>x</sup>* 1−*<sup>x</sup>* , *x* ∈ [0, 1) and *υ* > 0. Since *f* (*x*) = *υ* (<sup>1</sup>−*<sup>x</sup>*)<sup>2</sup> > 0, *f*(*x*) is an increasing function.

According to Definition 3, we have

$$\log: [-\tau, \tau] \to [0, 1], \; \operatorname{g} \left( s\_t^{\rho\_i} \right) = \frac{t}{2\tau} + \frac{1}{2} = \sigma\_{\rho\_i \circ} \operatorname{g} \left( h\_{\mathbb{S}}^{\rho\_i} \right) = \left\{ \frac{t}{2\tau} + \frac{1}{2} = \sigma\_{\rho\_i} \, \middle| \, t \in [-\tau, \tau] \right\} = h\_{\rho\_i}$$

where *i* = 1, 2, ··· , *n* and *ρ* = *a* or *ρ* = *b*. Then for any *sai t* ≤ *sbi t* , we have *<sup>σ</sup>ai* ≤ *<sup>σ</sup>bi* , further, *f*(*<sup>σ</sup>ai*) ≤ *f*(*<sup>σ</sup>bi*).

Suppose *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) be the weight of *hSi* , satisfying *wi* ∈ [0, 1] and ∑*ni*=<sup>1</sup> *wi* = 1. Based on the above condition, we can ge<sup>t</sup>

∪ *<sup>σ</sup>ai*∈*g*(*hai S* )3*g*<sup>−</sup><sup>1</sup>5∏*ni*=<sup>1</sup><sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>ai* <sup>1</sup>−*σai wi*64 ≤ ∪ *<sup>σ</sup>bi*<sup>∈</sup>*g*(*hbi S* )3*g*<sup>−</sup><sup>1</sup>5∏*ni*=<sup>1</sup>51+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>bi* <sup>1</sup>−*<sup>σ</sup>bi* 6*wi*64 ⇒ ∪ *<sup>σ</sup>ai*∈*g*(*hai S* )3*g*<sup>−</sup><sup>1</sup>5∏*ni*=<sup>1</sup> <sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>ai* <sup>1</sup>−*σai wi* + (*υ* − 1)64 ≤ ∪ *<sup>σ</sup>bi*<sup>∈</sup>*g*(*hbi S* )3*g*<sup>−</sup><sup>1</sup>5∏*ni*=<sup>1</sup> 51+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>bi* <sup>1</sup>−*<sup>σ</sup>bi* 6*wi* + (*υ* − 1)64 ⇒ ∪ *<sup>σ</sup>ai*∈*g*(*hai S* )3*g*<sup>−</sup><sup>1</sup>5<sup>1</sup> − *<sup>υ</sup>*∏*ni*=<sup>1</sup> (<sup>1</sup>−*σai*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>ai*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σai*)*wi* 64 ≤ ∪ *<sup>σ</sup>bi*<sup>∈</sup>*g*(*hbi S* )3*g*<sup>−</sup><sup>1</sup>5<sup>1</sup> − *<sup>υ</sup>*∏*ni*=<sup>1</sup> (<sup>1</sup>−*<sup>σ</sup>bi*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>bi*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*<sup>σ</sup>bi*)*wi* 64 ⇒ ∪ *<sup>σ</sup>ai*∈*g*(*hai S* )3*g*<sup>−</sup><sup>1</sup>5 ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>ai*)*wi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*σai*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>ai*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σai*)*wi* 64 ≤ ∪ *<sup>σ</sup>bi*<sup>∈</sup>*g*(*hbi S* )3*g*<sup>−</sup><sup>1</sup>5 ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>bi*)*wi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*<sup>σ</sup>bi*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>bi*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*<sup>σ</sup>bi*)*wi* 64

Therefore, based on Theorem 1, we have *HFLHWA<sup>υ</sup>w*-*h<sup>a</sup>*1*<sup>S</sup>* , *ha*2*S* , ··· , *hanS* . ≤ *HFLHWA<sup>υ</sup>whb*1*<sup>S</sup>* , *hb*2*S* , ··· , *hbnS* . -

**Bounded 1.** *Let hSi*(*i* = 1, 2, ··· , *n*) *be a set of HFLEs, if h*+*S* = {*s*<sup>+</sup>} = max∪*sit*∈*hSi*max{*sit*}*and h*−*S* = {*s*−} = ∪*sit*∈*hSi*min{*sit*}*, then h*−*S*≤ *HFLHWA<sup>υ</sup>w*-*hS*1, *hS*2, ··· , *hSn*. ≤ *h*+*S*(8)

**Proof.** According to Definition 3, we have

$$\mathcal{g}: [-\tau, \tau] \to [0, 1], \; \mathcal{g}\left(\mathbf{s}\_t^i\right) = \frac{t}{2\tau} + \frac{1}{2} = \sigma\_i, \; \mathcal{g}\left(h\_{\mathcal{S}\_i}\right) = \left\{\frac{t}{2\tau} + \frac{1}{2} = \sigma\_i \, \middle| \, t \in [-\tau, \tau] \right\} = h\_{\mathcal{S}\_i}$$

where *i* = 1, 2, ··· , *n*. Then, *s*<sup>−</sup> ≤ *sit* ≤ *s*<sup>+</sup> for any *i*, we have *σ*<sup>−</sup> ≤ *σi* ≤ *<sup>σ</sup>*+.

Suppose *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) be the weight of *hSi* , satisfying *wi* ∈ [0, 1] and ∑*ni*=<sup>1</sup> *wi* = 1. Based on the monotonic of HFLHWA, we can ge<sup>t</sup>

*HFLHWA<sup>υ</sup>w*-*hS*1 , *hS*2 , ··· , *hSn* . = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup> ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* ≥ ∪ *σ*−∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup>5 ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*<sup>−</sup>)*wi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*σ*<sup>−</sup>)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*<sup>−</sup>)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σ*<sup>−</sup>)*wi* 6 = ∪ *σ*−∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup>5 (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*<sup>−</sup>)<sup>∑</sup>*ni*=<sup>1</sup> *wi*−(<sup>1</sup>−*σ*<sup>−</sup>)<sup>∑</sup>*ni*=<sup>1</sup> *wi* (1+(*<sup>υ</sup>*−<sup>1</sup>)*σ*<sup>−</sup>)<sup>∑</sup>*ni*=<sup>1</sup> *wi*+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σ*<sup>−</sup>)<sup>∑</sup>*ni*=<sup>1</sup> *wi* 6 = ∪ *σ*−∈*g*(*hSi*){*g*<sup>−</sup><sup>1</sup>(*σ*<sup>−</sup>)} = *h*−*S*

Similarly, *HFLHWA<sup>υ</sup>w*-*hS*1 , *hS*2 , ··· , *hSn* . ≤ *h*+*S* . Therefore, *h*−*S* ≤ *HFLHWA<sup>υ</sup>w*-*hS*1 , *hS*2 , ··· , *hSn* . ≤ *h*+*S* . -

**Commutative 1.** *Let hSi*(*i* = 1, 2, ··· , *n*) *be a set of HFLEs, and* (*hS*1 , *hS*2 , ··· , *hSn* ) *be any permutation of* (*hS*1 , *hS*2 , ··· , *hSn* )*, then*

$$\text{HFLHWA}\_{\text{w}}^{\text{v}}(\text{h}\_{\text{S}\_{1}}, \text{h}\_{\text{S}\_{2}}, \dots, \text{h}\_{\text{S}\_{n}}) = \text{HFLHWA}\_{\text{w}}^{\text{v}}(\overline{\text{h}}\_{\text{S}\_{1}}, \overline{\text{h}}\_{\text{S}\_{2}}, \dots, \overline{\text{h}}\_{\text{S}\_{n}}) \tag{9}$$

**Proof.** Equation (9) clearly holds and the proof is omitted here. -

**Lemma 1.** *[59]. Let yi* > 0 (*i* = 1, 2, ··· , *n*) *and wi be the weight of yi, satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1, *then*

$$\prod\_{i=1}^{n} \left( y\_i \right)^{w\_i} \le \sum\_{i=1}^{n} \left( w\_i y\_i \right) \tag{10}$$

*with equality if and only if y*1 = *y*2 = ··· = *yn*.

**Theorem 2.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*)*, satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1*. g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion functions between HFLEs and HFEs, and υ* > 0*. Then*

$$\text{HFLHWA}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \cdot, \cdot, h\_{\text{S}\_{n}}) \leq \text{HFLMA}\_{\text{w}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \cdot, \cdot, h\_{\text{S}\_{n}}) \tag{11}$$

**Proof.** For any *sit*∈ *hSi*, based on Definition 3, we have

$$\{g : [-\tau, \tau] \to [0, 1], \ g\left(h\_{S\_i}\right) = \left\{\frac{t}{2\tau} + \frac{1}{2} = \sigma\_i\Big| t \in [-\tau, \tau] \right\} = h\_i\}$$

Further, according to Equation (10), we have

$$\prod\_{l=1}^{\eta} \left( 1 + (\upsilon - 1)\sigma\_l \right)^{w\_l} + (\upsilon - 1)\prod\_{l=1}^{\eta} \left( 1 - \sigma\_l \right)^{w\_l} \le \sum\_{l=1}^{\eta} w\_l (1 + (\upsilon - 1)\sigma\_l) + (\upsilon - 1)\sum\_{l=1}^{\eta} w\_l (1 - \sigma\_l) = \upsilon$$

then,

*HFLHWA<sup>υ</sup>w*(*hS*1 , *hS*2 , ··· , *hSn* ) = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup> ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup><sup>1</sup> − *<sup>υ</sup>*∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi* ≤ ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup><sup>1</sup> − *<sup>υ</sup>*∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*wi υ* = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup>-<sup>1</sup> − ∏*ni*=<sup>1</sup> (1 − *<sup>σ</sup>i*)*wi* . = *HFLWAw*(*hS*1 , *hS*2 , ··· , *hSn* )

Therefore, Equation (11) holds. -

**Definition 9.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and υ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *If*

$$\mathrm{HFLHWG}\_{\mathrm{w}}^{\mathrm{w}}(\mathrm{h}\_{\mathrm{S}\_{1}},\mathrm{h}\_{\mathrm{S}\_{2}},\cdots,\mathrm{h}\_{\mathrm{S}\_{n}}) = \left(\mathrm{h}\_{\mathrm{S}\_{1}}\right)^{\mathrm{w}\_{1}} \otimes \mathrm{H}\_{\mathrm{H}}\left(\mathrm{h}\_{\mathrm{S}\_{2}}\right)^{\mathrm{w}\_{2}} \otimes \cdots \otimes \mathrm{H}\_{\mathrm{H}}\left(\mathrm{h}\_{\mathrm{S}\_{n}}\right)^{\mathrm{w}\_{n}} = \bigotimes\_{i=1}^{n} \mathrm{h}\_{\mathrm{S}\_{i}}^{\mathrm{w}\_{i}}\tag{12}$$

*then HFLHWG<sup>υ</sup>w* is designated as the HFL Hamacher weighted geometric (HFLHWG) operator.

**Theorem 3.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion*

*functions between HFLEs and HFEs, and υ* > 0. *Then the aggregated value by the HFLHWG operator is also an HFLE, and*

$$\text{HFLHWG}\_{\text{w}}^{\text{w}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \text{g} \left(h\_{\text{S}\_{i}}\right)} \left\{ \text{g}^{-1} \left( \frac{v \prod\_{i=1}^{\mathfrak{v}} \left(\sigma\right)^{w\_{i}}}{\prod\_{i=1}^{\mathfrak{v}} \left(1 + \left(v - 1\right) \left(1 - \sigma\_{i}\right)\right)^{w\_{i}} + \left(v - 1\right) \prod\_{i=1}^{\mathfrak{v}} \left(\sigma\_{i}\right)^{w\_{i}}} \right)} \right\} \tag{13}$$

**Proof.** According to the mathematical induction method, Equation (13) can be proved as follows.

For *n* = 1, the result of Equation (13) clearly holds. Suppose Equation (13) holds for *n* = *k*, namely

$$\text{HFLHWG}\_{\text{w}}^{\text{w}}(\mathsf{h}\_{\text{S}\_{1}}, \mathsf{h}\_{\text{S}\_{2}}, \dots, \mathsf{h}\_{\text{S}\_{k}}) = \bigcup\_{\sigma\_{i} \in \operatorname{\mathbb{Q}}(\mathbf{h}\_{\text{S}\_{i}})} \left\{ \mathsf{g}^{-1} \left( \frac{v \prod\_{i=1}^{k} (\sigma\_{i})^{w\_{i}}}{\prod\_{i=1}^{k} (1 + (v-1)(1-\sigma\_{i}))^{w\_{i}} + (v-1) \prod\_{i=1}^{k} (\sigma\_{i})^{w\_{i}}} \right) \right\}$$

Then, for *n* = *k* + 1, by Equation (12), we can ge<sup>t</sup>

*HFLHWG<sup>υ</sup>w*(*hS*1 , *hS*2 , ··· , *hSk* , *hSk*+<sup>1</sup> )=(*hS*1 )*<sup>w</sup>*1 ⊗*H* (*hS*2 )*<sup>w</sup>*2 ⊗*H* ···⊗*H* (*hSn* )*wn* ⊗*H* (*hSk*+<sup>1</sup> )*wk*+<sup>1</sup> = *n*⊗*H i*=1 (*hSi*)*wi* ⊗*H* (*hSk*+<sup>1</sup> )*wk*+<sup>1</sup> = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup>5 *<sup>υ</sup>*<sup>∏</sup>*ki*=<sup>1</sup> (*<sup>σ</sup>i*)*wi* <sup>∏</sup>*ki*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σi*))*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ki*=<sup>1</sup> (*<sup>σ</sup>i*)*wi* 6<sup>⊗</sup>*H* ∪ *<sup>σ</sup>k*+1∈*g*(*hSk*+<sup>1</sup> ) *g*<sup>−</sup><sup>1</sup>5 *υσwk*+<sup>1</sup> *k*+1 (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σk*+<sup>1</sup>))*wk*+<sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)*σwk*+<sup>1</sup> *k*+1 6

Let ∏*ki*=<sup>1</sup> (1 + (*υ* − 1)(1 − *<sup>σ</sup>i*))*wi* = *α*1, ∏*ki*=<sup>1</sup> (*<sup>σ</sup>i*)*wi* = *β*1, (1 + (*υ* − 1)(1 − *<sup>σ</sup>k*+<sup>1</sup>))*wk*+<sup>1</sup> = *α*2 and *σ wk*+1 *k*+1 = *β*2, then

*k* ⊗*H i*=1 (*hSi*)*wi* = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*∈*g*(*hSk* )*g*<sup>−</sup><sup>1</sup> *υβ*1 *<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*<sup>1</sup> and (*hSk*+<sup>1</sup> )*wk*+<sup>1</sup> = ∪ *<sup>σ</sup>k*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup> *υβ*2 *<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*<sup>2</sup> 

Further, the operational law (2) in Definition 5 yields

*k* ⊗*H i*=1 (*hSi*)*wi* ⊗*H* (*hSk*+<sup>1</sup> )*wk*+<sup>1</sup> = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*+1∈*g*(*hSk*+<sup>1</sup> ) ⎧⎨⎩*g*−1⎛⎝ *<sup>υ</sup>*2*β*1*β*2 (*<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*1)(*<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*2) *<sup>υ</sup>*+(<sup>1</sup>−*<sup>υ</sup>*)5 *υβ*1(*<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*2)+*υβ*2(*<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*1)−*υ*2*β*1*β*<sup>2</sup> (*<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*1)(*<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*2) 6⎞⎠⎫⎬⎭ =∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup> *υβ*1*β*2 (*<sup>α</sup>*1+(*<sup>υ</sup>*−<sup>1</sup>)*β*1)(*<sup>α</sup>*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*2)−(*<sup>υ</sup>*−<sup>1</sup>)(*<sup>α</sup>*2*β*1+*α*1*β*2+(*<sup>υ</sup>*−<sup>2</sup>)*β*1*β*2) = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup> *υβ*1*β*2 *<sup>α</sup>*1*α*2+(*<sup>υ</sup>*−<sup>1</sup>)*β*1*β*<sup>2</sup> = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ),··· ,*σk*+1∈*g*(*hSk*+<sup>1</sup> )*g*<sup>−</sup><sup>1</sup>5 *υ*∏*<sup>k</sup>*+<sup>1</sup> *i*=1 (*<sup>σ</sup>i*)*wi* ∏*<sup>k</sup>*+<sup>1</sup> *i*=1 (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σi*))*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*<sup>k</sup>*+<sup>1</sup> *i*=1 (*<sup>σ</sup>i*)*wi* 6

That is, Equation (13) holds for *n* = *k* + 1. Therefore, Equation (13) holds for all *n*. -

**Remark 4.** *When υ* = 1, *then the HFLHWG operator transforms into the following:*

$$\{\text{HFLWG}\_{\text{up}}(h\_{\text{S}\_1}, h\_{\text{S}\_2}, \dots, h\_{\text{S}\_n}) = \bigcup\_{\sigma\_i \in \text{g}(h\_{\text{S}\_i})} \left\{ \text{g}^{-1} \left( \prod\_{i=1}^n \left( \sigma\_i \right)^{w\_i} \right) \right\}.$$

*where HFLWGw is called the HFLWG operator [29]. When υ* = 2*, then the HFLHWG operator transforms into the following:*

$$\text{HFLEWG}\_{w}(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}, \dots, h\_{\mathbb{S}\_n}) = \bigcup\_{\sigma\_i \in \chi(h\_{\mathbb{S}\_i})} \left\{ \text{g}^{-1} \left( \frac{2 \prod\_{i=1}^n (\sigma\_i)^{w\_i}}{\prod\_{i=1}^n (2 - \sigma\_i)^{w\_i} + \prod\_{i=1}^n (\sigma\_i)^{w\_i}} \right) \right\}$$

*where HFLEWGw is called the HFL Einstein weighted geometric (HFLEWG) operator. Especially when wi* = 1/*n, then the HFLHWG operator is transformed into the hesitant fuzzy Hamacher geometric (HFLHG) operator.*

$$\text{HFLHG}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \mathfrak{g}/h\_{\text{S}\_{i}}} \left\{ \text{g}^{-1} \left( \frac{\upsilon \prod\_{i=1}^{n} \left( \sigma\_{i} \right)^{\frac{1}{n}}}{\prod\_{i=1}^{n} \left( 1 + (\upsilon - 1)(1 - \sigma\_{i}) \right)^{\frac{1}{n}} + (\upsilon - 1) \prod\_{i=1}^{n} \left( \sigma\_{i} \right)^{\frac{1}{n}}} \right) \right\}$$

**Example 3.** *Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS and τ* = 3. *hS*1 = {*<sup>s</sup>*−1,*s*1} *and hS*2 = {*<sup>s</sup>*−2,*s*0} *are two HFLEs, and w* = (0.4, 0.6) *is the weight of hS*1 *and hS*2 *, respectively. Then we can aggregate them by employing the HFLHWG (υ* = 3*) operator.*

*HFLHWG*3*w*(*hS*1 , *hS*2 ) = ∪ *<sup>σ</sup>*1<sup>∈</sup>*hS*1 ,*σ*2<sup>∈</sup>*hS*2*g*<sup>−</sup><sup>1</sup>5 <sup>3</sup>∏<sup>2</sup>*i*=<sup>1</sup> (*<sup>σ</sup>i*)*wi* ∏<sup>2</sup>*i*=<sup>1</sup> (1+(<sup>3</sup>−<sup>1</sup>)(<sup>1</sup>−*σi*))*wi*+(<sup>3</sup>−<sup>1</sup>)∏<sup>2</sup>*i*=<sup>1</sup> (*<sup>σ</sup>i*)*wi* 6 = ∪ *<sup>σ</sup>*1∈*g*(*hS*1 ),*σ*2∈*g*(*hS*2 ) ⎧⎪⎪⎨⎪⎪⎩*g*−1⎛⎜⎜⎝ <sup>3</sup>×( 13 )0.4×( 16 )0.6 (<sup>1</sup>+2× 23 )0.4×(<sup>1</sup>+2× 56 )0.6+2×( 13 )0.4×( 16 )0.6 , <sup>3</sup>×( 13 )0.4×( 12 )0.6 (<sup>1</sup>+2× 23 )0.4×(<sup>1</sup>+2× 12 )0.6+2×( 13 )0.4×( 12 )0.6 <sup>3</sup>×( 23 )0.4×( 16 )0.6 (<sup>1</sup>+2× 13 )0.4×(<sup>1</sup>+2× 56 )0.6+2×( 23 )0.4×( 16 )0.6 , <sup>3</sup>×( 23 )0.4×( 12 )0.6 (<sup>1</sup>+2× 13 )0.4×(<sup>1</sup>+2× 12 )0.6+2×( 23 )0.4×( 12 )0.6 ⎞⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭ = {*g*<sup>−</sup><sup>1</sup>(0.2223, 0.3120, 0.4284, 0.5645)} = {*<sup>s</sup>*−1.6662,*s*−1.1279,*s*−0.4299,*s*0.3870}

**Idempotent 2.** *Let hSi*(*i* = 1, 2, ··· , *n*) *be equal with each hSi having only one value, namely, hSi* = *hS* = {*st*} *for any i, then*

$$\text{HFLIHVG}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_1}, h\_{\text{S}\_2}, \dots, h\_{\text{S}\_n}) = h\_{\text{S}} \tag{14}$$

**Proof.** The proof of Equation (14) is similar to Equation (6) and is omitted here. -

**Remark 5.** *Note that the HFLHWG operator is not idempotent when hSi includes more than one value; the following example is provided to demonstrate this case.*

**Example 4.** *Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS*, *τ* = 3, *hS*1 = *hS*1 = *hS* = {*<sup>s</sup>*−1,*s*1} *and w* = (0.4, 0.6)*<sup>T</sup>*. *Then HFLHWG*3*w*(*hS*1 , *hS*2 ) = {0.3333, 0.4520, 0.5196, 0.6667}, *<sup>s</sup>*(*HFLHWG*3*w*(*hS*1, *hS*2 )) = 0.4929, *and s*(*hS*) = 0.5. *Therefore*, *HFLHWG*3*w*(*hS*1, *hS*2 ) < *hS*.

**Monotonic 2.** *Let haS* = {*h<sup>a</sup>*1*S* , *ha*2*S* , ··· , *hanS* } *and hbS* = {*hb*1*S* , *hb*2*S* , ··· , *hbnS* } *be two of any collection of HFLEs. If for any sai t* ∈ *hai S and sbi t* ∈ *hbi S* , *and sai t* ≤ *sbi t for any i, then*

$$\rm{HFLHVG\_w^{v}(h\_{\rm S}^{a\_1}, h\_{\rm S}^{a\_2}, \dots, h\_{\rm S}^{a\_n})} \leq \rm{HFLHVG\_w^{v}(h\_{\rm S}^{b\_1}, h\_{\rm S}^{b\_2}, \dots, h\_{\rm S}^{b\_n})} \tag{15}$$

**Proof.** Let *f*(*x*) = <sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*<sup>x</sup>*) *x* , *x* ∈ (0, 1] and *υ* > 0. Since *f* (*x*) = −*υx*<sup>2</sup> < 0, hence *f*(*x*) is a decreasing function.

According to Definition 3, we have

$$\log: [-\tau, \tau] \to [0, 1], \; \operatorname{g} \left( s\_t^{\rho\_i} \right) = \frac{t}{2\tau} + \frac{1}{2} = \sigma\_{\rho\_i}, \; \operatorname{g} \left( h\_S^{\rho\_i} \right) = \left\{ \frac{t}{2\tau} + \frac{1}{2} = \sigma\_{\rho\_i} \, \middle| \, t \in [-\tau, \tau] \right\} = h\_{\rho\_i}$$

where *i* = 1, 2, ··· , *n* and *ρ* = *a* or *ρ* = *b*. Then for any *sai t* ≤ *sbi t* , we have *<sup>σ</sup>ai* ≤ *<sup>σ</sup>bi* , further, *f*(*<sup>σ</sup>ai*) ≥ *f*(*<sup>σ</sup>bi*).

Suppose *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) is the weight of *hSi* , satisfying *wi* ∈ [0, 1] and ∑*n i*=1 *wi* = 1. Based on the above condition, we have

∪ *<sup>σ</sup>ai* <sup>∈</sup>*g*(*h ai S* ) *g*<sup>−</sup><sup>1</sup> 5 ∏*n i*=1 <sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σai* ) *<sup>σ</sup>ai wi*6 ≥ ∪ *<sup>σ</sup>bi* <sup>∈</sup>*g*(*h bi S* ) *g*<sup>−</sup><sup>1</sup> 5 ∏*n i*=1 51+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*<sup>σ</sup>bi* ) *<sup>σ</sup>bi* 6*wi*6 ⇒ ∪ *<sup>σ</sup>ai* <sup>∈</sup>*g*(*h ai S* ) *g*<sup>−</sup><sup>1</sup> 5 ∏*n i*=1 <sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σai* ) *<sup>σ</sup>ai wi* + (*υ* − 1)6 ≥ ∪ *<sup>σ</sup>bi* <sup>∈</sup>*g*(*h bi S* ) *g*<sup>−</sup><sup>1</sup> 5 ∏*n i*=1 <sup>1</sup>+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σai* ) *<sup>σ</sup>ai wi* + (*υ* − 1)6 ⇒ ∪ *<sup>σ</sup>ai* <sup>∈</sup>*g*(*h ai S* ) *g*<sup>−</sup><sup>1</sup> 5 *υ*∏*<sup>n</sup> i*=1 (*<sup>σ</sup>ai* ) *wi* ∏*n i*=1 (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σai*))*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*<sup>n</sup> i*=1 (*<sup>σ</sup>ai* ) *wi* 6 ≤ ∪ *<sup>σ</sup>bi* <sup>∈</sup>*g*(*h bi S* ) *g*<sup>−</sup><sup>1</sup> 5 *υ*∏*<sup>n</sup> i*=1 (*<sup>σ</sup>bi* ) *wi* ∏*n i*=1 (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*<sup>σ</sup>bi*))*wi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*<sup>n</sup> i*=1 (*<sup>σ</sup>bi* ) *wi* 6

Therefore, based on Theorem 3, we have *HFLHWGυ w* - *h<sup>a</sup>*1 *S* , *h<sup>a</sup>*2 *S* , ··· , *han S* . ≤ *HFLHWGυ w hb*1 *S* , *hb*2 *S* , ··· , *hbn S* . -

**Bounded 2.** *Let hSi* (*i* = 1, 2, ··· , *n*) *be a set of HFLEs, if h*<sup>+</sup> *S* = {*s*<sup>+</sup>} = max ∪ *si t*<sup>∈</sup>*hSi* max{*s<sup>i</sup> t*} *and h*− *S* = {*s*−} = ∪ *si t*<sup>∈</sup>*hSi* min{*s<sup>i</sup> t*} , *then h*− *S* ≤ *HFLHWG<sup>υ</sup> w* - *hS*1 , *hS*2 , ··· , *hSn* . ≤ *h*<sup>+</sup> *S* (16)

**Proof.** The proof of Equation (16) is similar to Equation (8) and is omitted here. -

**Commutative 2.** *Let hSi* (*i* = 1, 2, ··· , *n*) *be a collection of HFLEs, and* (*hS*1 , *hS*2 , ··· , *hSn* ) *be any permutation of* (*hS*1 , *hS*2 , ··· , *hSn* ), *then*

$$\text{HFLHWG}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \text{HFLHWG}\_{\text{w}}^{\text{v}}(\overline{h}\_{\text{S}\_{1}}, \overline{h}\_{\text{S}\_{2}}, \dots, \overline{h}\_{\text{S}\_{n}}) \tag{17}$$

**Proof.** Equation (17) clearly holds and the proof of Equation (17) is omitted here. -

**Theorem 4.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi* (*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*n i*=1 *wi* = 1. *g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion functions between HFLEs and HFEs, and υ* > 0. *Then*

$$\rm{HFLHVG}\_{w}^{v}(h\_{\rm{S}\_{1}}, h\_{\rm{S}\_{2}}, \cdots, h\_{\rm{S}\_{n}}) \ge HFLWG\_{w}(h\_{\rm{S}\_{1}}, h\_{\rm{S}\_{2}}, \cdots, h\_{\rm{S}\_{n}}) \tag{18}$$

**Proof.** For any *si t*∈ *hSi*, based on Definition 3, we have

$$\{g : [-\tau, \tau] \to [0, 1], \ g\left(h\_{S\_i}\right) = \left\{\frac{t}{2\tau} + \frac{1}{2} = \sigma\_i\Big| t \in [-\tau, \tau] \right\} = h\_i\}$$

Further, according to Equation (10), we have

$$\prod\_{i=1}^{n} \left( 1 + (\upsilon - 1)(1 - \sigma) \right)^{y\_1} + (\upsilon - 1) \prod\_{i=1}^{n} \left( \sigma\_i \right)^{y\_1} \le \sum\_{i=1}^{n} \text{w} \left( 1 + (\upsilon - 1)(1 - \sigma\_i) \right) + (\upsilon - 1) \sum\_{i=1}^{n} \text{w}\_i(\sigma\_i) = \upsilon \prod\_{i=1}^{n} \text{w}\_i(\sigma\_i)$$

then

$$\begin{split} & HFLHWG^{v}\_{\mathcal{W}}(h\_{\mathbb{S}\_{1}}, h\_{\mathbb{S}\_{2}}, \cdot, \cdot, h\_{\mathbb{S}\_{3}}) = \underset{\sigma\_{i} \in \mathcal{G}(h\_{\mathbb{S}\_{1}})}{\cup} \left\{ \boldsymbol{\mathcal{S}}^{-1} \left( \frac{v \prod\_{l=1}^{n} (\sigma\_{l})^{w\_{l}}}{\prod\_{l=1}^{n} (1 + (v-1)(1-\sigma\_{l}))^{w\_{l}} + (v-1) \prod\_{l=1}^{n} (v\_{l})^{w\_{l}}} \right) \right\} \\ & \geq \underset{\sigma\_{l} \in \mathcal{G}(h\_{\mathbb{S}\_{1}})}{\cup} \left\{ \boldsymbol{\mathcal{S}}^{-1} \left( \frac{v \prod\_{l=1}^{n} (\sigma\_{l})^{w\_{l}}}{v} \right) \right\} = \underset{\sigma\_{l} \in \mathcal{G}(h\_{\mathbb{S}\_{1}})}{\cup} \left\{ \boldsymbol{\mathcal{S}}^{-1} \left( \prod\_{l=1}^{n} (\sigma\_{l})^{w\_{l}} \right) \right\} = HFLWG\_{w}(h\_{\mathbb{S}\_{1}}, h\_{\mathbb{S}\_{2}}, \cdot, \cdot, h\_{\mathbb{S}\_{n}}) \end{split}$$

Therefore, Equation (18) holds. -

#### *3.2. GHFLHWA and GHFLHWG Operators*

**Definition 10.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs*, *υ* > 0 *and λ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *If*

$$\mathbb{E}\,\mathrm{GH}\mathrm{KL}\mathrm{W}\mathrm{A}\_{\mathrm{W}}^{\mathrm{v},\lambda}(\mathrm{h}\_{\mathrm{S}\_{1}},\mathrm{h}\_{\mathrm{S}\_{2}},\cdots,\mathrm{h}\_{\mathrm{S}\_{n}}) = \mathrm{w}\_{1}(\mathrm{h}\_{\mathrm{S}\_{1}}^{\lambda}) \oplus\_{H}\mathrm{w}\_{2}(\mathrm{h}\_{\mathrm{S}\_{2}}^{\lambda}) \oplus\_{H}\cdots \oplus\_{H}\mathrm{w}\_{n}(\mathrm{h}\_{\mathrm{S}\_{n}}^{\lambda}) = \left(\bigoplus\_{l=1}^{\mathfrak{n}}\left(\mathrm{w}\_{l}(\mathrm{h}\_{\mathrm{S}\_{l}}^{\lambda})\right)\right)^{\frac{1}{\lambda}}\tag{19}$$

*then GHFLHWA<sup>υ</sup>*,*<sup>λ</sup> w is designated as the generalized HFL Hamacher weighted averaging (GHFLHWA) operator.*

**Theorem 5.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion functions between HFLEs and HFEs, and υ* > 0. *Then the aggregated value by the GHFLHWA operator is also an HFLE and*

$$=\underset{\sigma\_{i}\in\mathcal{G}(h\_{\hat{\mathbf{s}}\_{i}})}{\operatorname{\boldsymbol{\omega}}}\left\{\operatorname{\mathcal{g}}^{-1}\left(\begin{pmatrix}\frac{\prod\_{i=1}^{n}\left(1+\frac{v(v-1)\boldsymbol{\nu}\_{i}^{\boldsymbol{\lambda}}}{(1+(v-1)(1-\boldsymbol{v}\_{i}^{\boldsymbol{\lambda}}))^{\boldsymbol{\lambda}\_{i}}+(v-1)\boldsymbol{\nu}\_{i}^{\boldsymbol{\lambda}}}\right)^{w\_{i}} - \prod\_{i=1}^{n}\left(1-\frac{nv\_{i}^{\boldsymbol{\lambda}}}{(1+(v-1)(1-\boldsymbol{v}\_{i}^{\boldsymbol{\lambda}}))^{\boldsymbol{\lambda}\_{i}}+(v-1)\boldsymbol{\nu}\_{i}^{\boldsymbol{\lambda}}}\right)^{w\_{i}}\right)\right\}\tag{20}$$

**Proof.** According to the mathematical induction method, the proof of Equation (20) is similar to that of Theorem 1 and is omitted here. -

**Remark 6.** *When λ* = 1, *the GHFLHWA operator is reduced to the HFLHWA operator; when λ* → 0, *GHFLHWA operator is reduced to the HFLHWG operator*.

*When υ* = 1*, the GHFLHWA operator is reduced to the following:*

$$\{GHFLWA\_w^{\lambda}(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}, \dots, h\_{\mathbb{S}\_n}) = \underset{\sigma\_i \in \mathfrak{g}(h\_{\mathbb{S}\_i})}{\cup} \left\{ \text{g}^{-1} \left( \left(1 - \prod\_{i=1}^n \left(1 - \sigma\_i^{\lambda} \right)^{w\_i} \right)^{\frac{1}{\lambda}} \right) \right\}$$

*where GHFLWAλw is called the generalized HFL weighted averaging (GHFLWA) operator. Particularly, when λ* = 1, *the GHFLHWA operator is further transformed into the HFLWA operator; when λ* → 0, *the GHFLHWA operator is further transformed into the HFLWG operator*.

*When υ* = 2, *the GHFLHWA operator is transformed into the following:*

$$\text{GGFLEWA}\_{\text{ulv}}^{\lambda}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{l} \in \text{g}(h\_{\text{S}\_{1}})} \left\{ \text{g}^{-1} \left( \left( \frac{\prod\_{i=1}^{n} \left(1 + \frac{2\sigma\_{l}^{\lambda}}{(2-\sigma\_{l}^{\lambda})^{\lambda} + \sigma\_{l}^{\sigma}}\right)^{w\_{l}} - \prod\_{i=1}^{n} \left(1 - \frac{2\sigma\_{l}^{\lambda}}{(2-\sigma\_{l}^{\lambda})^{\lambda} + \sigma\_{l}^{\sigma}}\right)^{w\_{l}}} \right)^{\frac{1}{\lambda}} \right) \right\}^{\frac{1}{\lambda}}$$

*where GHFLEWAλw is designated as the generalized HFL Einstein weighted averaging (GHFLEWA) operator. Particularly, when λ* = 1, *the GHFLHWA operator is further transformed into the HFLEWA operator; when λ* → 0, *GHFLHWA is further transformed into the HFLEWG operator*.

**Definition 11.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and υ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *If*

$$\text{GHFLHWG}\_{\text{IV}}^{\text{y},\lambda}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \frac{1}{\lambda} (\lambda h\_{\text{S}\_{1}})^{w\_{1}} \otimes\_{H} (\lambda h\_{\text{S}\_{2}})^{w\_{2}} \otimes\_{H} \dots \otimes\_{H} (\lambda h\_{\text{S}\_{n}})^{w\_{n}} = \frac{1}{\lambda} \left( \bigotimes\_{i=1}^{n} (\lambda h\_{\text{S}\_{i}})^{w\_{i}} \right) \tag{21}$$

*then GHFLHWG<sup>υ</sup>*,*<sup>λ</sup> w is designated as the generalized HFL Hamacher weighted geometric (GHFLHWG) operator*.

**Theorem 6.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *g and g*<sup>−</sup><sup>1</sup> *are the equivalent conversion functions between HFLEs and HFEs, and υ* > 0. *Then the aggregated value by the GHFLHWG operator is also an HFLE, and*

$$=\underset{\sigma\_{l}\in\mathcal{G}(hs\_{l})}{\text{G}}\left\{g^{-1}\left(1-\left(1-\frac{v\prod\_{i=1}^{n}\left(\frac{(1+(v-1)\rho\_{l})^{\lambda}-(1-\sigma\_{l})^{\lambda}}{(1+(v-1)\rho\_{l})^{\lambda}+(v-1)(1-\sigma\_{l})^{\lambda}}\right)^{w\_{l}}}{\prod\_{i=1}^{n}\left(\frac{v(v-1)(1-\sigma\_{l})^{\lambda}}{(1+(v-1)\rho\_{l})^{\lambda}+(v-1)(1-\sigma\_{l})^{\lambda}}\right)^{w\_{l}}}\right)^{\frac{1}{\lambda}}\right)\right\}\tag{22}$$

**Proof.** According to mathematical induction method, the proof of Equation (21) is similar to that of Theorem 3 and is omitted here. -

**Remark 7.** *When λ* = 1, *the GHFLHWG operator is transformed into the HFLHWG operator; when λ* → 0; *the GHFLHWG operator is transformed into the HFLHWA operator*.

*When υ* = 1, *GHFLHWG operator is transformed into the following:*

$$GHFLWG\_w^{\lambda} = \underset{\sigma\_i \in \mathcal{K}(h\_{S\_i})}{\cup} \left\{ \mathbf{g}^{-1} \left( 1 - \left( 1 - \prod\_{i=1}^{n} \left( 1 - \left( 1 - \sigma\_i)^{\lambda} \right)^{w\_i} \right)^{\frac{1}{\lambda}} \right) \right\} \right\}$$

*where GHFLWGλw is designated as the generalized HFL weighted geometric (GHFLWG) operator. Particularly, when λ* = 1, *the GHFLHWG operator is further transformed into the HFLWG operator; when λ* → 0, *GHFLHWG operator is further transformed into the HFLWA operator*.

*When υ* = 2, *the GHFLHWG operator is transformed into the following:*

$$GHFLEWG\_{\mathcal{U}}^{\lambda} = \underset{\sigma\_{l} \in \mathcal{g}(bs\_{j})}{\cup} \left\{ \mathbf{g}^{-1} \left( \mathbf{1} - \left( \mathbf{1} - \frac{2\prod\_{i=1}^{n} \left( \frac{(1+\sigma\_{l})^{\lambda} - (1-\sigma\_{l})^{\lambda}}{(1+\sigma\_{l})^{\lambda} + (1-\sigma\_{l})^{\lambda}} \right)^{w\_{l}}}{\prod\_{i=1}^{n} (2 - \frac{(1+\sigma\_{l})^{\lambda} - (1-\sigma\_{l})^{\lambda}}{(1+\sigma\_{l})^{\lambda} + (1-\sigma\_{l})^{\lambda}})^{w\_{l}}} + \prod\_{l=1}^{n} \frac{(1+\sigma\_{l})^{\lambda} - (1-\sigma\_{l})^{\lambda}}{(1+\sigma\_{l})^{\lambda} + (1-\sigma\_{l})^{\lambda}} \right)^{w\_{l}} \right\} \right\}^{-1}$$

*where GHFLWGλw is designated as the generalized HFL Einstein weighted geometric (GHFLEWG) operator. Particularly, when λ* = 1, *the GHFLHWG operator is transformed into the HFLEWG operator; when λ* → 0, *GHFLHWG operator is reduced to the HFLEWA operator*.

#### **4. Hesitant Fuzzy Linguistic Hamacher Power Aggregation Operators**

This section defines an HFL Hamacher power weighted averaging (HFLHPWA) operator, an HFL Hamacher power weighted geometric (HFLHPWG) operator, a generalized HFL Hamacher power weighted averaging (GHFLHPWA) operator, and a generalized HFL Hamacher power weighted geometric (GHFLHPWG) operator. In addition, we discuss some special cases withthese operators.

#### *4.1. The HFLHPWA and HFLHPWG Operators*

**Definition 12.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the hesitant fuzzy linguistic Hamacher power weighted averaging (HFLHPWA) operator is defined as follows:*

*HFLHPWAw*(*hS*1 , *hS*2 , ··· , *hSn* ) = *n* ⊕ *<sup>i</sup>*=<sup>1</sup>*wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*))*hSi*/∑*ni*=<sup>1</sup> *wi*(<sup>1</sup> + *T*(*hSi*)) (23)

*where T*(*hSi*) = <sup>∑</sup>*ni*=1,*<sup>j</sup>* =*<sup>i</sup> Sup*(*hSi* , *hSj*) *and Sup*(*hSi* , *hSj*) *expresses the support degree for hSi from hSj* , *which satisfies the following three properties*.


**Theorem 7.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and υ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the aggregated value by the HFLHPWA operator is also an HFLE, and*

$$\text{HFLHPWA}\_{\text{w}}^{\text{v}}(\text{h}\_{\text{S}\_{1}}, \text{h}\_{\text{S}\_{2}}, \dots, \text{h}\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{l} \in \text{g}(\text{h}\_{\text{S}\_{j}})} \left\{ \text{g}^{-1} \left( \frac{\prod\_{i=1}^{n} (1 + (\nu - 1)\sigma\_{l})^{p\_{i}} - \prod\_{i=1}^{n} (1 - \sigma\_{l})^{p\_{i}}}{\prod\_{i=1}^{n} (1 + (\nu - 1)\sigma\_{l})^{p\_{i}} + (\nu - 1)\prod\_{i=1}^{n} (1 - \sigma\_{l})^{p\_{i}}} \right) \right\} \tag{24}$$

*where pi* = *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*))/∑*ni*=<sup>1</sup> *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*)), *pi* ≥ 0 *and* ∑*ni*=<sup>1</sup> *pi* = 1.

**Proof.** According to mathematical induction method, the proof of Equation (24) is similar to Theorem 1 and is omitted here. -

**Remark 8.** *If Sup*(*hSi*, *hSj*) = *c*, *for all i* = *j*, *then HFLHPWA operator is transformed into the following:*

$$\text{HFLHA}^{v}(h\_{\mathbb{S}\_{1}}, h\_{\mathbb{S}\_{2}}, \dots, h\_{\mathbb{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \operatorname{g}(h\_{\mathbb{S}\_{i}})} \left\{ \mathbf{g}^{-1} \left( \frac{\prod\_{i=1}^{n} \left(1 + (v-1)\sigma\_{i}\right)^{\frac{1}{n}} - \prod\_{i=1}^{n} \left(1 - \sigma\_{i}\right)^{\frac{1}{n}}}{\prod\_{i=1}^{n} \left(1 + (v-1)\sigma\_{i}\right)^{\frac{1}{n}} + (v-1)\prod\_{i=1}^{n} \left(1 - \sigma\_{i}\right)^{\frac{1}{n}}} \right) \right\}$$

*where HFLHAυ is called the HFLHA operator*.

*When υ* = 1, *then the HFLHPWA operator is transformed into the following:*

$$\{HFLPWA\_{\mathfrak{w}}(h\_{\mathbb{S}\_1\prime}, h\_{\mathbb{S}\_2\prime}, \dots, h\_{\mathbb{S}\_n}) = \bigcup\_{\sigma\_i \in \mathfrak{g}(h\_{\mathbb{S}\_i})} \left\{ \mathfrak{g}^{-1} \left(1 - \prod\_{i=1}^n (1 - \sigma\_i)^{p\_i} \right) \right\}.$$

*where HFLPWAw is called the HFL power weighted averaging (HFLPWA) operator*.

*When υ* = 2, *then the HFLHPWA operator is transformed into the following:*

$$\text{HFLEEPWA}\_{\text{w}}(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}, \dots, h\_{\mathbb{S}\_n}) = \bigcup\_{\sigma\_i \in \mathfrak{g}(h\_{\mathbb{S}\_i})} \left\{ \text{g}^{-1} \left( \frac{\prod\_{i=1}^n (1+\sigma\_i)^{p\_i} - \prod\_{i=1}^n (1-\sigma\_i)^{p\_i}}{\prod\_{i=1}^n (1+\sigma\_i)^{p\_i} + \prod\_{i=1}^n (1-\sigma\_i)^{p\_i}} \right) \right\}$$

*where HFLEPWAw is designated as the HFL Einstein power weighted averaging (HFLEPWA) operator*.

**Remark 9.** *The HFLHPWA operator is neither idempotent, monotonic, bounded, nor commutative with regard to the input arguments, which are shown in Example 5*.

**Example 5.** *Let S* = {*st*|*<sup>t</sup>* = <sup>−</sup>*τ*, ··· , −1, 0, 1, ··· , *τ*} *be an LTS*, *τ* = 3, *hS*1 = {*<sup>s</sup>*1,*s*2}, *hS*2 = {*<sup>s</sup>*0,*s*3}, *hS*3 = {*<sup>s</sup>*0,*s*2}, *and hS*4 = {*<sup>s</sup>*0,*s*1} *be four HFLEs. Let w* = (0.3, 0.5, 0.2)*<sup>T</sup> and υ* = 3.

Based on Definition 3, according to Equation (2), we have *<sup>s</sup>*(*hS*1 ) = *<sup>s</sup>*(*hS*2 ) = 0.75, *<sup>s</sup>*(*hS*3 ) = 0.6667 and *<sup>s</sup>*(*hS*4) = 0.5833. Then, by employing HFLHPWA operator yields

> *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*1 , *hS*1 )) = 0.7572 *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*3 , *hS*4 )) = 0.6903 *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*4 , *hS*3 )) = 0.6657

$$s(HFLHPWA^3(h\_{S\_1}, h\_{S\_1}, h\_{S\_3})) = 0.7452$$

$$s(HFLHPWA^3(h\_{S\_2}, h\_{S\_2}, h\_{S\_4})) = 0.8793$$

Since *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*1 , *hS*1 )) = *<sup>s</sup>*(*hS*1 ), the HFLHPWA operator is not idempotent.

It is obvious that *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*2 , *hS*2 , *hS*4 )) > *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*1 , *hS*3 )), therefore, the HFLHPWA operator is not monotonic. On the other hand, since *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*2 , *hS*2 , *hS*4 )) > *<sup>s</sup>*(*hS*2) > *<sup>s</sup>*(*hS*4), the HFLHPWA operator is not bounded.

Furthermore, *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*3 , *hS*4 )) = *<sup>s</sup>*(*HFLHPWA*<sup>3</sup>(*hS*1 , *hS*4 , *hS*3 )), the HFLHPWA operator is not commutative.

**Theorem 8.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *g and g*<sup>−</sup><sup>1</sup> *are the equivalent transformation functions between HFLEs and HFEs, and υ* > 0. *Then*

$$\text{HFLHPWA}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_{1'}}, h\_{\text{S}\_{2'}}, \dots, h\_{\text{S}\_{n}}) \le HFLPWA\_{\text{w}}(h\_{\text{S}\_{1'}}, h\_{\text{S}\_{2'}}, \dots, h\_{\text{S}\_{n}}) \tag{25}$$

#### **Proof.** According to Equation (10), we have

∏*ni*=<sup>1</sup> (1 + (*υ* − <sup>1</sup>)*<sup>σ</sup>i*)*pi* + (*υ* − <sup>1</sup>)∏*ni*=<sup>1</sup> (1 − *<sup>σ</sup>i*)*pi* ≤ ∑*ni*=<sup>1</sup> *pi*(<sup>1</sup> + (*υ* − <sup>1</sup>)*<sup>σ</sup>i*)+(*<sup>υ</sup>* − <sup>1</sup>)∑*ni*=<sup>1</sup> *pi*(<sup>1</sup> − *<sup>σ</sup>i*) = *υ HFLHPWA<sup>υ</sup>w*(*hS*1 , *hS*2 , ··· , *hSn* ) = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup> ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*pi*−∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*pi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*pi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*pi* = ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup><sup>1</sup> − *<sup>υ</sup>*∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*pi* ∏*ni*=<sup>1</sup> (1+(*<sup>υ</sup>*−<sup>1</sup>)*<sup>σ</sup>i*)*pi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*pi* ≤= ∪ *<sup>σ</sup>i*∈*g*(*hSi*)*g*<sup>−</sup><sup>1</sup><sup>1</sup> − *<sup>υ</sup>*∏*ni*=<sup>1</sup> (<sup>1</sup>−*σi*)*pi υ* = ∪ *<sup>σ</sup>i*∈*g*(*hSi*){*g*<sup>−</sup><sup>1</sup>-<sup>1</sup> − ∏*ni*=<sup>1</sup> (1 − *<sup>σ</sup>i*)*pi* .} = *HFLPWAw*(*hS*1 , *hS*2 , ··· , *hSn* )

Therefore, Equation (25) holds. -

**Definition 13.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs, and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the HFL Hamacher power weighted geometric (HFLHPWG) operator is defined as follows:*

$$\text{HFLHPWG}\_{\text{w}}(h\_{\text{S}\_1}, h\_{\text{S}\_2}, \dots, h\_{\text{S}\_n}) = \bigcap\_{i=1}^{n} (h\_{\text{S}\_i})^{\text{w}\_i(1 + T(h\_{\text{S}\_i})) / \sum\_{i=1}^{n} \text{w}\_i(1 + T(h\_{\text{S}\_i}))} \tag{26}$$

*where T*(*hSi*) = <sup>∑</sup>*ni*=1,*<sup>j</sup>* =*<sup>i</sup> Sup*(*hSi* , *hSj*) *and Sup*(*hSi* , *hSj*) *expresses the support degree for hSi from hSj* , *which is also satisfy the three properties in Definition 12*.

**Theorem 9.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and υ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the aggregated value by the HFLHPWG operator is also an HFLE, and*

$$\text{HFLHPWG}\_{\text{w}}^{\text{v}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in h\_{\text{S}\_{j}}} \left\{ \mathcal{g}^{-1} \left( \frac{v \prod\_{i=1}^{n} (\sigma\_{i})^{p\_{i}}}{\prod\_{i=1}^{n} (1 + (v-1)(1-v\_{i}))^{p\_{i}} + (v-1) \prod\_{i=1}^{n} (\sigma\_{i})^{r\_{i}}} \right) \right\} \tag{27}$$

*where pi* = *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*))/∑*ni*=<sup>1</sup> *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*))*, pi* ≥ 0 *and* ∑*ni*=<sup>1</sup> *pi* = 1*.*

**Proof.** According to mathematical induction method, the proof of Equation (27) is similar to Theorem 3 and is omitted here. -

**Remark 10.** *If Sup*(*hSi*, *hSj*) = *c*, *for all i* = *j*, *then the HFLHPWG operator is transformed into the following:*

$$\text{HFLHG}^{\upsilon}(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}, \dots, h\_{\mathbb{S}\_n}) = \bigcup\_{\sigma\_l \in \underline{g}(l\underline{s}\_i)} \left\{ g^{-1} \left( \frac{\upsilon \prod\_{i=1}^n (\underline{\sigma}\_l)^{\frac{1}{\mathfrak{h}}}}{\prod\_{i=1}^n (1 + (\upsilon - 1)(1 - \underline{\sigma}\_l))^{\frac{1}{\mathfrak{h}}} + (\upsilon - 1)\prod\_{i=1}^n (\underline{\sigma}\_l)^{\frac{1}{\mathfrak{h}}}} \right) \right\}$$

*where HFLHGυ is called the HFL Hamacher geometric (HFLHG) operator*.

*When υ* = 1, *then the HFLHPWG operator is transformed into the following:*

$$HFLPWG\_w(h\_{S\_1}, h\_{S\_2}, \dots, h\_{S\_n}) = \bigcup\_{\sigma\_i \in \mathcal{J}(h\_{S\_i})} \left\{ \mathcal{g}^{-1} \left( \prod\_{i=1}^n \left( \sigma\_i \right)^{p\_i} \right) \right\},$$

*where HFLPWGw is called the HFL power weighted geometric (HFLPWG) operator*.

*When υ* = 2, *then the HFLHPWG operator is transformed into the following:*

$$\text{HFLEPWGw}(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}, \dots, h\_{\mathbb{S}\_n}) = \bigcup\_{\sigma\_i \in \mathcal{G}(h\_{\mathbb{S}\_i})} \left\{ \text{g}^{-1} \left( \frac{2 \prod\_{i=1}^n (\sigma\_i)^{p\_i}}{\prod\_{i=1}^n (2 - \sigma\_i)^{p\_i} + \prod\_{i=1}^n (\sigma\_i)^{p\_i}} \right) \right\}$$

*where HFLEPWGw is designated as the HFL Einstein power geometric (HFLEPWG) operator*.

**Remark 11.** *Similar to the HFLHPWA operator, the HFLHPWG operator is neither idempotent, monotonic, bounded, nor commutative with regard to the input arguments*.

**Theorem 10.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *g and g*<sup>−</sup><sup>1</sup> *are the equivalent transformation functions between HFLEs and HFEs, and υ* > 0. *Then*

$$\text{HFLHPWG}\_{\text{w}}^{v}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) \geq \text{HFLPWG}\_{\text{w}}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) \tag{28}$$

**Proof.** According to Equation (10), we have

$$\begin{split} & \| \prod\_{l=1}^{u} \left( 1 + (v-1)\sigma\_{l} \right)^{p\_{l}} + (v-1) \prod\_{l=1}^{u} \left( 1 - \sigma\_{l} \right)^{p\_{l}} \leq \sum\_{l=1}^{u} p\_{l} \left( 1 + (v-1)\sigma\_{l} \right) + (v-1) \sum\_{l=1}^{u} p\_{l} \left( 1 - \sigma\_{l} \right) = \nu \\ & \| HFLIPWG\_{w}^{u} (h\_{\mathfrak{S}\_{1}}, h\_{\mathfrak{S}\_{2}}, \dots, h\_{\mathfrak{S}\_{n}}) = \bigcup\_{\sigma\_{l} \in \mathcal{G}(h\_{\mathfrak{S}\_{1}})} \left\{ \mathcal{g}^{-1} \left( \frac{v \prod\_{l=1}^{u} (\sigma\_{l})^{p\_{l}}}{\prod\_{l=1}^{u} (1 + (v-1)(1 - \sigma\_{l}))^{p\_{l}} + (v-1) \prod\_{l=1}^{u} (\sigma\_{l})^{p\_{l}}} \right) \right\} \\ & \geq \bigcup\_{\sigma\_{l} \in \mathcal{G}(h\_{\mathfrak{S}\_{1}})} \left\{ \mathcal{g}^{-1} \left( \frac{v \prod\_{l=1}^{u} (\sigma\_{l})^{p\_{l}}}{v} \right) \right\} = \bigcup\_{\sigma\_{l} \in \mathcal{G}(h\_{\mathfrak{S}\_{1}})} \left\{ \mathcal{g}^{-1} \left( \prod\_{l=1}^{u} (\sigma\_{l})^{p\_{l}} \right) \right\} = HFLPWG\_{w} (h\_{\mathfrak{S}\_{1}}, h\_{\mathfrak{S}\_{2}}, \dots, h\_{\mathfrak{S}\_{n}}) \end{split}$$

Therefore, Equation (28) holds. -

#### *4.2. The GHFLHPWA and GHFLHPWG Operators*

**Definition 14.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the generalized hesitant fuzzy linguistic Hamacher power weighted averaging (GHFLHPWA) operator is defined as follows:*

$$\text{GHFLIHPWA}\_{\text{w}}^{\lambda}(h\_{\text{S}\_{1}}, h\_{\text{S}\_{2}}, \dots, h\_{\text{S}\_{n}}) = \left( \bigoplus\_{i=1}^{n} \left( \left( w\_{i} (1 + T(h\_{\text{S}\_{i}})) (h\_{\text{S}\_{i}})^{\lambda} \right) / \sum\_{i=1}^{n} w\_{i} (1 + T(h\_{\text{S}\_{i}})) \right) \right)^{\frac{1}{\lambda}} \tag{29}$$

*where T*(*hSi*) = <sup>∑</sup>*ni*=1,*<sup>j</sup>* =*<sup>i</sup> Sup*(*hSi* , *hSj*) *and Sup*(*hSi* , *hSj*) *expresses the support degree for hSi from hSj* , *which satisfies the three properties in Definition 12*.

**Theorem 11.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a collection of HFLEs and υ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the aggregated value by the GHFLHPWA operator is also an HFLE, and*

*GHFLHPWA<sup>υ</sup>*,*<sup>λ</sup> w* (*hS*1 , *hS*2 , ··· , *hSn* ) = ∪ *<sup>σ</sup>i*∈*g*(*hSi*) ⎧⎪⎪⎨⎪⎪⎩*g*−1⎛⎜⎜⎝⎛⎜⎝ ∏*ni*=<sup>1</sup> (<sup>1</sup>+ *<sup>υ</sup>*(*<sup>υ</sup>*−<sup>1</sup>)*σλi* (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σi*))*<sup>λ</sup>*+(*<sup>υ</sup>*−<sup>1</sup>)*σλi* )*pi*−∏*ni*=<sup>1</sup> (<sup>1</sup>− *υσλi* (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σi*))*<sup>λ</sup>*+(*<sup>υ</sup>*−<sup>1</sup>)*σλi* )*pi* ∏*ni*=<sup>1</sup> (<sup>1</sup>+ *<sup>υ</sup>*(*<sup>υ</sup>*−<sup>1</sup>)*σλi* (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σi*))*<sup>λ</sup>*+(*<sup>υ</sup>*−<sup>1</sup>)*σλi* )*pi*+(*<sup>υ</sup>*−<sup>1</sup>)∏*ni*=<sup>1</sup> (<sup>1</sup>− *υσλ* (1+(*<sup>υ</sup>*−<sup>1</sup>)(<sup>1</sup>−*σi*))*<sup>λ</sup>*+(*<sup>υ</sup>*−<sup>1</sup>)*σλi* )*pi* ⎞⎟⎠ 1*λ* ⎞⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭ (30)

*where pi* = *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*))/∑*ni*=<sup>1</sup> *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*)), *pi* ≥ 0 *and* ∑*ni*=<sup>1</sup> *pi* = 1.

**Proof.** According to the mathematical induction method, the proof of Equation (30) is similar to Theorem 1 and is omitted here. -

**Remark 12.** *Sup*(*hSi*, *hSj*) = *c*, *for all i* = *j*, *then GHFLHPWA operator is transformed into the following:*

$$=\underset{\sigma\_{i}\in\mathcal{J}(\text{bs}\_{j})}{\text{cup}}\left\{\mathcal{g}^{-1}\left(\left(\frac{\prod\_{i=1}^{n}\left(1+\frac{v(v-1)\sigma\_{i}^{\lambda}}{(1+(v-1)(1-v\_{i}))^{\lambda}+(v-1)\sigma\_{i}^{\lambda}}\right)}{\prod\_{i=1}^{n}\left(\frac{1}{1+(v-1)(1-v\_{i})}\right)^{\lambda}+(v-1)\sigma\_{i}^{\lambda}}-\Pi\_{i=1}^{n}\left(1-\frac{uv\_{i}^{\lambda}}{\left(1+(v-1)(1-v\_{i})\right)^{\lambda}+(v-1)v\_{i}^{\lambda}}\right)}{\frac{1}{1+(v-1)(1-v\_{i})}\left(1-\frac{uv\_{i}^{\lambda}}{(1+(v-1)(1-v\_{i}))^{\lambda}}+(v-1)\sigma\_{i}^{\lambda}}\right)^{\frac{1}{\lambda}}\right)\right\}$$

*where GHFLHA<sup>υ</sup>*,*<sup>λ</sup> is designated as the generalized HFL Hamacher averaging (GHFLHA) operator*. *When υ* = 1, *then the GHFLHPWA operator is transformed into the following:*

$$\operatorname{GHFLPWA}\_{\operatorname{w}}^{\lambda}(h\_{\mathbb{S}\_1}, h\_{\mathbb{S}\_2}, \dots, h\_{\mathbb{S}\_n}) = \bigcup\_{\sigma\_i \in \operatorname{g}(h\_{\mathbb{S}\_i})} \left\{ \operatorname{g}^{-1} \left( \left( 1 - \prod\_{i=1}^n \left( 1 - \sigma\_i^{\lambda} \right)^{p\_i} \right)^{\frac{1}{\lambda}} \right) \right\}$$

*where GHFLPWAλw is designated as the generalized HFL power weighted averaging (GHFLPWA) operator. Particularly, when λ* = 1, *the GHFLHPWA operator is further transformed into the HFLPWA operator; when λ* → 0, *GHFLHPWA operator is further transformed into the HFLPWG operator*.

*When υ* = 2, *then GHFLHPWA operator is transformed to the following:*

$$\{GHFLEPWA\_{\mathfrak{u}}^{\lambda}(h\_{\mathfrak{S}\_{1}},h\_{\mathfrak{S}\_{2}},\cdots,h\_{\mathfrak{S}\_{n}})=\bigcup\_{\sigma\_{i}\in\mathcal{S}\_{\mathcal{S}}(h\_{\mathfrak{S}\_{i}})}\left\{\mathcal{g}^{-1}\left(\left(\frac{\prod\_{i=1}^{n}\left(1+\frac{2\sigma\_{i}^{\lambda}}{(2-\sigma\_{i})^{\lambda}+\sigma\_{j}^{\rho}}\right)^{p\_{1}}-\prod\_{i=1}^{n}\left(1-\frac{2\sigma\_{i}^{\lambda}}{(2-\sigma\_{i})^{\lambda}+\sigma\_{j}^{\rho}}\right)^{p\_{1}}\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{\lambda}}\right\}^{\frac{1}{\lambda}}$$

*where GHFLEPWAλw is designated as the generalized HFL Einstein power weighted averaging (GHFLEPWA) operator. Particularly, when λ* = 1, *the GHFLHPWA operator is further transformed into the HFLEPWA operator; when λ* → 0, *GHFLHPWA operator is further transformed into the HFLEPWG operator*.

**Remark 13.** Similar to the HFLHPWA operator, the GHFLHPWA operator is neither idempotent, monotonic, bounded, nor commutative with regard to the input arguments.

**Definition 15.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *be the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the generalized hesitant fuzzy linguistic Hamacher power weighted geometric (GHFLHPWG) operator is defined as follows:*

$$\text{GHFLHPWG}\_{\text{w}}^{\lambda}(h\_{\text{S}\_1}, h\_{\text{S}\_2}, \dots, h\_{\text{S}\_n}) = \frac{1}{\lambda} \left( \bigotimes\_{i=1}^n (\lambda h\_{\text{S}\_i})^{\text{w}\_i(1 + T(h\_{\text{S}\_i}))/\sum\_{i=1}^n w\_i(1 + T(h\_{\text{S}\_i}))} \right) \tag{31}$$

*where T*(*hSi*) = <sup>∑</sup>*ni*=1,*<sup>j</sup>* =*<sup>i</sup> Sup*(*hSi* , *hSj*), *and Sup*(*hSi* , *hSj*) *expresses the support degree for hSi from hSj* , *which satisfies the three properties in Definition 12*.

**Theorem 12.** *Let HS* = {*hS*1 , *hS*2 , ··· , *hSn* } *be a set of HFLEs and υ* > 0. *wi*(*<sup>i</sup>* = 1, 2, ··· , *n*) *is the weight of hSi*(*i* = 1, 2, ··· , *<sup>n</sup>*), *satisfying wi* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1. *Then the aggregated value by the GHFLHPWG operator is also an HFLE, and*

$$=\bigcup\_{\sigma\_{i}\in\operatorname{rg}(h\_{\xi\_{1}})}\left\{\operatorname{g}^{-1}\left(1-\left(1-\frac{\operatorname{v}\Pi\_{i}^{\operatorname{ul}}\left(\frac{(1+(v-1)\nu\_{i})^{\operatorname{h}}-(1-\sigma\_{i})^{\operatorname{h}}}{(1+(v-1)\nu\_{i})^{\operatorname{h}}+(v-1)(1-\sigma\_{i})^{\operatorname{h}}}\right)^{p\_{i}}}{\Pi\_{i=1}^{\operatorname{v}}\left(\frac{(1+(v-1)\nu\_{i})^{\operatorname{h}}-(1-\sigma\_{i})^{\operatorname{h}}}{(1+(v-1)\nu\_{i})^{\operatorname{h}}+(v-1)(1-\sigma\_{i})^{\operatorname{h}}}\right)^{p\_{i}}}\right)^{\frac{1}{p\_{i}}}\right\}\right\}\tag{32}$$

*where pi* = *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*))/∑*ni*=<sup>1</sup> *wi*(<sup>1</sup> + *<sup>T</sup>*(*hSi*)), *pi* ≥ 0 *and* ∑*ni*=<sup>1</sup> *pi* = 1.

**Proof.** According to the mathematical induction method, the proof of Equation (32) is similar to Theorem 3 and is omitted here. -

**Remark 14.** *Sup*(*hSi*, *hSj*) = *c*, *for all i* = *j*, *then the GHFLHPWG operator is transformed into the following:*

$$=\underset{\sigma\_{i}\in\mathcal{S}(\mathbb{h}\_{\mathbb{S}\_{i}})}{\operatorname{\mathbb{L}}}\left\{\operatorname{g}^{-1}\left(1-\left(1-\frac{\mathbb{v}\prod\_{i=1}^{\mu}\left(\frac{(1+(\nu-1)\sigma\_{i})^{\lambda}-(1-\sigma\_{i})^{\lambda}}{(1+(\nu-1)\sigma\_{i})^{\lambda}+(\nu-1)(1-\sigma\_{i})^{\lambda}}\right)^{\lambda}}{\prod\_{i=1}^{\mu}\left(1+\frac{\mathbb{v}(\nu-1)(1-\sigma\_{i})^{\lambda}}{(1+(\nu-1)\sigma\_{i})^{\lambda}+(\nu-1)(1-\sigma\_{i})^{\lambda}}\right)^{\lambda}}+\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{\lambda}}\right\}$$

*where GHFLHG<sup>υ</sup>*,*<sup>λ</sup> is designated as the generalized HFL Hamacher geometric (GHFLHG) operator*.

*When υ* = 1, *then the GHFLHPWG operator is transformed into the following:*

$$\text{CHFLPWG}\_{w}^{\lambda}(\mathsf{h}\_{\text{S}\_{1}}, \mathsf{h}\_{\text{S}\_{2}}, \cdots, \mathsf{h}\_{\text{S}\_{n}}) = \bigcup\_{\sigma\_{i} \in \mathsf{g}(\mathit{h}\_{\text{S}\_{i}})} \left\{ \mathsf{g}^{-1} \left( 1 - \left( 1 - \prod\_{i=1}^{n} \left( 1 - (1 - \sigma\_{i})^{\lambda} \right)^{p\_{i}} \right)^{\frac{1}{\lambda}} \right) \right\}$$

*where GHFLPWGλw is designated as the generalized HFL power weighted geometric (GHFLPWG) operator. Particularly, when λ* = 1, *the GHFLHPWG operator is further transformed into the HFLPWG operator; when λ* → 0, *the GHFLHPWG operator is further transformed into the HFLPWA operator*.

*When υ* = 2, *then the GHFLHPWG operator is transformed into the following:*

$$=\underset{\sigma\_{i}\in\mathcal{G}(h\_{S\_{i}})}{\operatorname{\mathbb{E}}}\left\{\operatorname{\mathcal{g}}^{-1}\left(1-\left(1-\frac{2\operatorname{\mathbb{E}}\_{i=1}^{n}\left(\frac{(1+\sigma\_{i})^{\lambda}-(1-\sigma\_{i})^{\lambda}}{(1+\sigma\_{i})^{\lambda}+(1-\sigma\_{i})^{\lambda}}\right)^{\operatorname{\mathbb{P}}}}{\operatorname{\mathbb{E}}\_{i=1}^{n}\left(2-\frac{(1+\sigma\_{i})^{\lambda}-(1-\sigma\_{i})^{\lambda}}{(1+\sigma\_{i})^{\lambda}+(1-\sigma\_{i})^{\lambda}}\right)^{\operatorname{\mathbb{P}}}+(\nu-1)\operatorname{\mathbb{E}}\_{i=1}^{n}\left(\frac{(1+\sigma\_{i})^{\lambda}-(1-\sigma\_{i})^{\lambda}}{(1+\sigma\_{i})^{\lambda}+(1-\sigma\_{i})^{\lambda}}\right)^{\operatorname{\mathbb{P}}}\right)\right\}$$

*where GHFLEPWGλw is designated as the generalized HFL Einstein power weighted geometric (GHFLEPWG) operator. Particularly, when λ* = 1, *the GHFLHPWG operator is further transformed into the HFLEPWG operator; when λ* → 0, *the GHFLHPWG operator is further transformed into the HFLPWA operator*.

**Remark 15.** *Similar to the HFLHPWA operator, the GHFLHPWG operator is neither idempotent, monotonic, bounded, nor commutative with regard to the input arguments*.

#### **5. Methods for MCDM Based on the Hesitant Fuzzy Linguistic Hamacher Operators**

In this part, we develop two methods based on the presented operators to handle an MCDM problem with hesitant fuzzy linguistic information.

A general MCDM problem under the hesitant fuzzy linguistic environment can be depicted as follows.

Let *A* = {*<sup>A</sup>*1, *A*2, ··· , *Am*} be the set of *m* candidates alternatives, and *C* = {*<sup>C</sup>*1, *C*2, ··· , *Cn*} be the set of *n* evaluation criteria, which have the weight vector *w* = (*<sup>w</sup>*1, *w*2, ··· , *wn*)<sup>T</sup> satisfying *wj* ∈ [0, 1] and ∑*nj*=<sup>1</sup> *wj* = 1. Suppose that *H*ˆ *S* = (<sup>ˆ</sup>*hSij*)*m*×*n* be the hesitant fuzzy linguistic evaluation matrix, where ˆ *hSij* is an HFLE and expresses the evaluation value of alternative *Ai* with respect to the criterion *Cj*.

Generally, there are two types of criteria, the benefit criterion and cost criterion, in an MCDM problem. When all the criteria are not of the same types, the values of the cost criterion need to be transformed into the values of the benefit criterion to construct a decision-making matrix *HS* = (*hSij*)*m*×*n*by employing Equation (33).

$$h\_{\mathbb{S}\_{ij}} = \begin{cases} h\_{\mathbb{S}\_{ij'}}, \text{ for benefit criterion} \\ \left(h\_{\mathbb{S}\_{ij}}\right)^{\mathbb{C}}, \text{ for cost criterion} \end{cases}, (i = 1, 2, \dots, m; j = 1, 2, \dots, n) \tag{33}$$

In order to yield the best alternative, the GHFLHWA operator or the GHFLHWG operator, which was developed based on the Hamacher operations, is utilized for the proposed MCDM approach under the hesitant fuzzy linguistic environment. The proposed method includes the following steps.

**Method 1.** (The flowchart of Method 1 is shown in Figure 1.)


$$h\_{\mathbb{S}\_i} = \text{GHFLIWVA}(h\_{\mathbb{S}\_{i1}}, h\_{\mathbb{S}\_{i2}}, \dots, h\_{\mathbb{S}\_{in}}) \text{ or } h\_{\mathbb{S}\_i} = \text{GHFLIWVG}(h\_{\mathbb{S}\_{i1}}, h\_{\mathbb{S}\_{i2}}, \dots, h\_{\mathbb{S}\_{in}}) \tag{34}$$


To reflect the correlation between the input arguments in MCDM problem, we use the GHFLHPWA or GHFLHPWG operator for the proposed MCDM approach. The steps involved are depicted as follows.

**Method 2.** (The flowchart of Method 2 is shown in Figure 1.)

**Step 1.** Determine the linguistic term set that is applied to evaluate each alternative with respect to each criterion; then the hesitant fuzzy linguistic evaluation matrix *H* ˆ *S* = (<sup>ˆ</sup>*hSij*)*m*×*n*is obtained.

**Step 2.** Normalize the evaluation matrix *H* ˆ *S* = (<sup>ˆ</sup>*hSij*)*m*×*n*according to Equation (33).

**Step 3.** Calculate the support degree of *hSi*using the following formula.

$$T(h\_{S\_{\overline{ij}}}) = \sum\_{j=1, k \neq j}^{n} \operatorname{Sup}(h\_{S\_{\overline{ij}}}, h\_{S\_{ik}}) \tag{35}$$

$$\operatorname{Sup}(h\_{\mathbb{S}\_{ij}}, h\_{\mathbb{S}\_{ik}}) = 1 - d(h\_{\mathbb{S}\_{ij}}, h\_{\mathbb{S}\_{ik}}) \tag{36}$$

**Step 4.** Obtained the power weight vector *p* by the following formula.

$$w\_{ij} = w\_j(1 + T(h\_{S\_{ij}})) / \sum\_{j=1}^{n} w\_j(1 + T(h\_{S\_{ij}})) \tag{37}$$

**Step 5.** Aggregate the criteria values by the GHFLHPWA or GHFLHPWG operators.

$$h\_{\mathbb{S}\_i} = \text{GHFLHPWA}(h\_{\mathbb{S}\_{i1}}, h\_{\mathbb{S}\_{i2}}, \dots, h\_{\mathbb{S}\_{in}}) \text{ or } h\_{\mathbb{S}\_i} = \text{GHFLHPWG}(h\_{\mathbb{S}\_{i1}}, h\_{\mathbb{S}\_{i2}}, \dots, h\_{\mathbb{S}\_{in}}) \tag{38}$$

**Step 6.** Compute the score value of each alternative by Equation (2).

**Step 7.** Determined the priority order of alternatives by the decreasing of score value.

**Figure 1.** The flowcharts of the Method 1 and Method 2.

#### **6. An Application of the Proposed Operators to MCDM**

## *6.1. Numeric Example*

A board of directors of a venture capital company is planning to choose a suitable city to invest in a project of sharing cars in the next three years. The venture capital company determined four alternative cities *Ai*(*i* = 1, 2, 3, 4) through preliminary market research. In order to evaluate and rank these cities, four criteria (all of them are benefit criteria) are identified by the board of directors including the economic development level (*C*1), the public transportation development level (*C*2), the number of public parking lots (*C*3), and the urban road resources (*C*4). Assume that the weight vector of these criteria is *w* = (0.3, 0.1, 0.4, 0.2)T.

In what follows, we employ Method 1 to determine the most suitable city without considering the correlations of the input arguments.



$$s(h\_{\mathbb{S}\_1}) = 0.5080, s(h\_{\mathbb{S}\_2}) = 0.6534, s(h\_{\mathbb{S}\_3}) = 0.5685, s(h\_{\mathbb{S}\_4}) = 0.7340$$

**Step 5.** Based on the decreasing order of score values, we have *hS*4 > *hS*2 > *hS*3 > *hS*1 . Therefore, the best city is *A*4.


**Table 1.** The hesitant fuzzy linguistic evaluation matrix.

The parameter *υ* in the GHFLHWA operator indicates the experts' preference over the alternative with respect to each criterion. In order to explore how the different preference parameter *υ* in the GHFLHWA operator influences the score values of the alternatives, we utilized different values of *υ* ∈ (0, <sup>10</sup>], which are commonly determined by decision makers. The relative results are shown in Figure 2. It is easy to observe from Figure 2 that the score values of the alternatives become smaller with the increasing values of parameter *υ*. In addition, for the GHFLHWA operator, we can also ascertain from Figure 2 that the final ranking of alternatives for the different parameter *υ* values does not change. Therefore, the value of parameter *υ* can be chosen by the decision maker according to their preference.

If we use the GHFLHWG operator instead of the GHFLHWA operator to aggregate the criteria values, the variation of score values of the alternatives is shown in Figure 3. From Figure 3, for the GHFLHWG operator, we can see that the score values of the alternatives become greater with the increase of parameter *υ*, which is just the opposite of the GHFLHWA operator. Furthermore, the priority order of alternatives is also not influenced by the different values of parameter *υ*.

**Figure 2.** The variation of score values of alternatives with regard to *υ* in the GHFLHWA operator.

**Figure 3.** The variation of score values of alternatives with regard to *υ* in the GHFLHWG operator.

When the relationships of the input data are taken into account, we apply Method 2 to resolve the above numerical example.

The first two steps are the same as Method 1.

**Step 3.** Compute the support degree *Sup*(*hSj*, *hSk*)(*j* = 1, 2, 3, 4; *j* = *k*).

*Sup*1*j* = ⎡⎢⎢⎢⎣ 0 0.9116 0.8750 0.7500 0.9116 0 0.8024 0.8024 0.8750 0.8024 0 0.6250 0.7500 0.8024 0.6250 0 ⎤⎥⎥⎥⎦, *Sup*2*j* = ⎡⎢⎢⎢⎣ 0 0.8750 0.9116 0.8024 0.8750 0 0.9116 0.6813 0.9116 0.9116 0 0.7205 0.8024 0.6813 0.7205 0 ⎤⎥⎥⎥⎦ *Sup*3*j* = ⎡ ⎢⎢⎢⎣ 0 0.9116 0.9116 0.9116 0.9116 0 0.8750 1 0.9116 0.8750 0 0.8750 0.9116 1 0.8750 0 , ⎤ ⎥⎥⎥⎦ *Sup*4*j* = ⎡ ⎢⎢⎢⎣ 0 0.8024 0.9116 0.9116 0.8024 0 0.8750 0.8232 0.9116 0.8750 0 0.8750 0.9116 0.8232 0.8750 0 ⎤ ⎥⎥⎥⎦

then

$$T = \begin{bmatrix} 2.5366 & 2.5163 & 2.3024 & 2.1774\\ 2.5890 & 2.4679 & 2.5437 & 2.2042\\ 2.7348 & 2.7866 & 2.6616 & 2.7866\\ 2.6256 & 2.5006 & 2.6616 & 2.6908 \end{bmatrix}$$

⎤

⎥⎥⎥⎦

**Step 4.** Calculate the power weight matrix.



When *λ* = 2, let *υ* = 0.1, 0.7, 2, 5, 9, respectively. From one hand, the score values and priority orders of all alternatives determined by the GHFLHPWA operator are shown in Table 2. When the value of parameter *υ* becomes greater, we can obtain a smaller score value of the alternative. We can also see that the ranking order of alternatives is not affected by the different values of parameter *υ*.

**Table 2.** The score values and rankings of alternatives obtained by the GHFLHPWA operator.


On the other hand, if the GHFLHPWG operator is employed to replace the GHFLHPWA operator in the above calculation, Table 3 gives the score values and the final ranking of the alternatives. In Table 3, we can observe that the score values of alternatives become greater when the value of parameter *υ* increases. In addition, the priority order of alternatives does not change when the value of parameter *υ* changes. Hence, the ranking order of alternatives is robust for the parameters *υ* = 0.1, 0.7, 2, 5, 9 in this example.

**Table 3.** The score values and rankings of alternatives obtained by the GHFLHPWG operator.


Based on the above analysis, we can conclude that the priority order of alternatives obtained by the GHFLHWA and GHFLHWG operators are the same as that obtained by the GHFLHPWA and GHFLHPWG operators, that is, the ranking order of alternatives is *A*4 > *A*2 > *A*3 > *A*1. Further, the results also indicate that the correlations between the input arguments are not enough to affect the ranking order of alternatives in this example.

#### *6.2. Comparison with Existing Methods of Hesitant Fuzzy Linguistic MCDM*

In this section, we use the proposed methods comparison with the previously developed HFL MCDM approaches. The previous methods include the proposed approach with Zhang and Wu [24], where the HFL weighted averaging and HFL weighted geometric operators were employed to aggregate the HFL evaluation information, and the HFL TOPSIS method [22].

The linguistic term set in these two methods is subscript-asymmetric, however, the linguistic term set used in this paper is subscript-symmetric. Therefore, we need to transform the evaluation matrix into another form for the use of these two approaches. The transformed HFL evaluation matrix is shown in Table 4.


**Table 4.** The transformed hesitant fuzzy linguistic evaluation matrix.

In the following, we utilize the HFLWA operator [24] instead of the GHFLHWA operator in Method 1 based on the operational laws in Definition 4 to solve the numerical example. That is

$$h\_{\mathcal{S}\_i} = \text{HFLNA}(h\_{\mathcal{S}\_{i1'}}, h\_{\mathcal{S}\_{i2'}}, h\_{\mathcal{S}\_{i3'}}) \\ = \stackrel{4}{\underset{j=1}{\Leftrightarrow}} (w\_j h\_{\mathcal{S}\_{ij}}) \\ = \underset{\sigma\_{ij} \in \mathfrak{g}(h\_{\mathcal{S}\_{ij}})}{\text{cup}} \left\{ \text{g}^{-1} \left( 1 - \prod\_{j=1}^4 (1 - \sigma\_{ij})^{w\_j} \right) \right\}$$

then, we can obtain the score values of the alternatives as follows:

$$s(h\_{\mathbb{S}\_1}) = 0.5790, s(h\_{\mathbb{S}\_2}) = 0.7060, s(h\_{\mathbb{S}\_3}) = 0.6376, s(h\_{\mathbb{S}\_4}) = 0.7731$$

In this situation, the priority order of alternatives is *A*4 > *A*2 > *A*3 > *A*1, and the best city is *A*4. If we use the HFLWG operator [24] instead of the GHFLHWA operator in Method 1, we ge<sup>t</sup>

$$h\_{\mathbb{S}\_i} = \text{HFLNG}(h\_{\mathbb{S}\_{i1'}}, h\_{\mathbb{S}\_{i2'}}, h\_{\mathbb{S}\_{i3'}}, h\_{\mathbb{S}\_{i4}}) = \stackrel{\scriptstyle 4}{\odot} (h\_{\mathbb{S}\_{ij}})^{w\_j} = \underset{\sigma\_{\bar{\mathcal{V}}} \in \text{g}(h\_{\mathbb{S}\_{\bar{\mathcal{V}}}})}{\text{g}^{-1}(h\_{\bar{\mathcal{V}}})} \left\{ \text{g}^{-1} \left( \prod\_{j=1}^{n} (\sigma\_{\bar{\mathcal{V}}})^{w\_j} \right) \right\}.$$

Then, we can obtain the score values of the alternatives as follows:

$$s(h\_{\mathbb{S}\_1}) = 0.5326, s(h\_{\mathbb{S}\_2}) = 0.6749, s(h\_{\mathbb{S}\_3}) = 0.6275, s(h\_{\mathbb{S}\_4}) = 0.7094$$

In this situation, the priority order of alternatives is *A*4 > *A*2 > *A*3 > *A*1, and the best city is *A*4.

Based on the above analyses, we can see that the best city and the ranking order of alternatives obtained by the HFLWA and HFLWG operators are the same for Methods 1 and 2, which illustrate the validity of Methods 1 and 2. In addition, we should note that the GHFLHWA and GHFLHWG operators reduce to the HFLWA and HFLWG operator, respectively, when *λ* = 1 and *υ* = 1. It indicates that the method based on the GHFLHWA or GHFLHWG operators is more general and flexible than the HFLWA or HFLWG operators.

In the following, we apply the HFL TOPSIS method [22] to solve the numerical example. First, we review the HFL TOPSIS approach as follows:

**Step 1.** For an MCDM problem with HFL information, let *X* = {*<sup>x</sup>*1, *x*2, ··· , *xm*} be a collection of *m* alternatives and *C* = {*<sup>c</sup>*1, *c*2, ··· , *cn*} be a collection of *n* criteria with weight vector *w* = (*<sup>w</sup>*1, *w*2, ··· , *wn*)<sup>T</sup> satisfying *wj* ∈ [0, 1] and ∑*nj*=<sup>1</sup> *wj* = 1. Suppose *R* = (*hSij*)*m*×*n* is an HFL evaluation matrix provided by the decision makers, where *hSij*is an HFLE.

**Step 2.** Based on the evaluation matrix *R*, an HFL positive ideal solution (HFLPIS) and an HFL negative ideal solution (HFLNIS) can be determined by

$$H\_{\mathcal{S}}^{+} = (h\_{\mathcal{S}\_1}^{+}, h\_{\mathcal{S}\_2}^{+}, \cdots, h\_{\mathcal{S}\_n}^{+}) \tag{39}$$

where *h*+*Sj* = *hS*1*j* ∨ *hS*2*j* ∨···∨ *hSmj* if *cj* is a benefit criterion and *h*+*Sj* = *hS*1*j* ∧ *hS*2*j* ∧···∧ *hSmj* if *cj* is a cost criterion.

$$H\_{\mathcal{S}}^{-} = (h\_{\mathcal{S}\_1}^{-} h\_{\mathcal{S}\_2}^{-} \cdots \prime, h\_{\mathcal{S}\_n}^{-}) \tag{40}$$

where *h*−*Sj* = *hS*1*j* ∧ *hS*2*j* ∧···∧ *hSmj* if *cj* is a benefit criterion and *h*−*Sj* = *hS*1*j* ∨ *hS*2*j* ∨···∨ *hSmj* if *cj* is a cost criterion. Where ∨ and ∧ are defined by Definition 3 [22].

**Step 3.** The distance from each alternative to HFLPIS and HFLNIS are calculated as follows:

$$d\_i^+ = \sum\_{j=1}^n w\_j d(h\_{S\_{ij}} h\_{S\_j}^+) \tag{41}$$

$$d\_i^- = \sum\_{j=1}^n w\_j d(h\_{S\_{ij}} h\_{S\_j}^-) \tag{42}$$

where *d*(*hSij*, *h*+*Sj*) and *d*(*hSij*, *<sup>h</sup>*<sup>−</sup>*Sj*) are determined by Definition 7.

 **Step 4.** The closeness coefficients *di* of alternatives *xi* can be calculated by

$$cc\_i = \frac{d\_i^-}{d\_i^+ + d\_i^-} \tag{43}$$

**Step 5.** Determine the priority orders of all alternatives in the light of the decrease of the closeness coefficient *di*.

In what follows, we utilize the HFL TOPSIS approach to resolve the numerical example. The detailed steps are described as follows:

**Step 1.** The hesitant fuzzy linguistic evaluation matrix *R* is shown in Table 4.

**Step 2.** Based on the hesitant fuzzy linguistic evaluation matrix *R*, the HFLPIS and the HFLNIS are determined as

$$H\_{\mathbb{S}}^{+} = \left( \{ \mathbf{s}\_{\mathsf{6}}, \mathbf{s}\overline{\mathsf{7}} \}, \{ \mathbf{s}\_{\mathsf{6}}, \mathbf{s}\overline{\mathsf{7}} \}, \{ \mathbf{s}\_{\mathsf{6}}, \mathbf{s}\overline{\mathsf{7}} \} \right)$$

$$H\_{\mathbb{S}}^{-} = \left( \{ \mathbf{s}\_{\mathsf{4}}, \mathbf{s}\_{\mathsf{5}} \}, \{ \mathbf{s}\_{\mathsf{4}} \}, \{ \mathbf{s}\_{\mathsf{5}}, \mathbf{s}\_{\mathsf{6}} \}, \{ \mathbf{s}\_{\mathsf{2}}, \mathbf{s}\_{\mathsf{3}} \} \right)$$

**Step 3.** The distance from each alternative to HFLPIS and HFLNIS are obtained as

$$d\_1^+ = 0.2453, 
 d\_2^+ = 0.1288, 
 d\_3^+ = 0.1738, 
 d\_4^+ = 0.0551$$

$$d\_1^- = 0.0000, \, d\_2^- = 0.1443, \, d\_3^- = 0.0854, \, d\_4^- = 0.2164$$

 **Step 4.** Employ Equation (43) to compute the closeness coefficient of alternative *xi*.

$$\text{cc}\_1 = 0.0000, \text{cc}\_2 = 0.5284, \text{ cc}\_3 = 0.3293, \text{ cc}\_4 = 0.7970$$

**Step 5.** The final priority order of all alternatives obtained as follows: *A*4 > *A*2 > *A*3 > *A*1.

Based on the above calculation, we can see that the best city is *A*4.

From the obtained results above, we can ascertain that the results determined by the HFL TOPSIS are the same as that of the proposed methods, which also validates the effectiveness of the presented methods in this paper. Furthermore, the GHFLHPWA or GHFLHPWG operators in Method 2 consider the relationships between the input arguments through the weight vector determined by the support degree.

Compared with the HFLWA or HFLWG operators and the HFL TOPSIS method, the presented Methods in this paper have the following two advantages. First, decision makers can determine the parameter value *υ* in the operators of Methods1 and 2 according to their subjective preferences, which increases the flexibility of the proposed methods to handle practical decision-making problems. Second, Method 2 reduces the influences of unreasonable input arguments by using the support measure assigning a lower weight to them and reflects the correlations between the input arguments by applying the weight vector allowing the input arguments to support and reinforce each other, both of which rendering the decision result more reasonable.
