*2.2. HFS*

Due to the complexity of the evaluated object in practice, decision makers may have difficulty determining an accurate value of the membership level. To deal with this situation, Torra [8] developed the HFS theory in which the membership degree is expressed by several possible values.

**Definition 4.** *Let* ℘([0, 1]) *be the set of all subsets of the unitary interval and X be a non- empty set. Let hA* : *X* → ℘([0, <sup>1</sup>])*, then an HFS A on X is defined by*

$$A = \{ \langle \mathbf{x}, h\_A(\mathbf{x}) \rangle | \mathbf{x} \in X \}. \tag{7}$$

**Definition 5.** *A hesitant fuzzy element (HFE) is a non-empty and finite subset of* [0, 1] *[27].*

Although a HFE can be given by any subset of [0, 1], in practice, HFS is commonly restricted to finite set in the MCDM problems [27]. Therefore, Bedregal at al. [35] proposed the typical hesitant fuzzy set (THFS), which is the finite and non-empty HFS. Later, Alcantud and Torra [36] defined the uniformly typical hesitant fuzzy set (UTHFS) that can simplify many theoretical and practical arguments, which is a generalized form of THFS. In this paper, the evaluation information of decision makers is expressed by UTHFS during the MCDM processes under hesitant fuzzy environment.

**Definition 6.** *Let* H ⊆ ℘([0, 1]) *be the set of all finite and non-empty subsets of* [0, 1]*, and let X be a nonempty set. Then, a THFS A on X is defined by Equation (7), where hA* : *X* → H*. Each h* ∈ H *is called a typical hesitant fuzzy element (THFE) [35].*

**Definition 7.** *Let A be a THFS on X, if there is N such that the cardinality of the THFS lA*(*x*) ≤ *N for each x* ∈ *X. Then, the THFS A is an UTHFS. Each h* ∈ H *is called an uniformly typical hesitant fuzzy element (UTHFE) [36].*

To aggregate the hesitant fuzzy evaluation information, Xia and Xu [27] investigated the operations of HFEs, which is also valid for fusing UTHFEs.

**Definition 8.** *Let h, h*1*, and h*2 *be three UTHFEs, λ* > 0*, then*

$$h\_1 \oplus h\_2 = \underset{\gamma\_1 \in h\_1, \gamma\_2 \in h\_2}{\cup} \{\gamma\_1 + \gamma\_2 - \gamma\_1 \gamma\_2\};\tag{8}$$

$$h\_1 \otimes h\_2 = \underset{\gamma\_1 \in h\_1, \gamma\_2 \in h\_2}{\cup} \{\gamma\_1 \gamma\_2\};\tag{9}$$

$$\lambda h = \underset{\gamma \in h}{\cup} \left\{ 1 - (1 - \gamma)^{\lambda} \right\};\tag{10}$$

$$h^{\lambda} = \underset{\gamma \in \mathcal{h}}{\cup} \{\gamma^{\lambda}\}. \tag{11}$$

#### *2.3. The PA Operator*

Aggregation operator plays a crucial role in the process of information fusion. Sometimes, the criteria have different priorities according to their important degree; thus, Yager [31] constructed the PA operator to address these situations.

**Definition 9.** *Let C* = {*<sup>C</sup>*1, *C*2,..., *Cn*} *be a set of criteria, which are divided into several priority levels, i.e., the priority of Cp is higher than Cq when p* < *q. The Cj*(*x*) ∈ [0, 1] *is the evaluation value of the alternative x concerning the criteria Cj. Thus, the PA operator is expressed by*

$$PA(\mathbb{C}\_1(\mathbf{x}), \mathbb{C}\_2(\mathbf{x}), \dots, \mathbb{C}\_n(\mathbf{x})) = \sum\_{j=1}^n w\_j \mathbb{C}\_j(\mathbf{x}).\tag{12}$$

*where wj* = *Tj*/∑*n j*=1 *Tj, Tj* = ∏*j*−<sup>1</sup> *k*=1 *Ck*(*x*)*, and T*1 = 1.
