**2. Preliminaries**

In this part, we review some basic concepts of LIFS, PLTS and PULTS, and point out the main disadvantage of these fuzzy sets.

#### *2.1. PLTS and PULTS*

For convenience, all the LST mentioned in this article are represented by *S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>2</sup>*τ*]} except for special explanations. In order to present the probability distribution information of the HFLTS, Pang et al. [14] proposed PLTS.

**Definition 1** ([14])**.** *Let S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>2</sup>*τ*]} *be a continuous LTS, then a PLTS is defined as*

$$L(p) = \{ L^{(k)}(p^{(k)}) | L^{(k)} \in \mathcal{S}, p^{(k)} \ge 0, k = 1, 2, \dots, \#L(p), \sum\_{k=1}^{\#L(p)} p^{(k)} \le 1 \},\tag{1}$$

*where <sup>L</sup>*(*k*)(*p*(*k*)) *is the linguistic term L*(*k*) *associated with the probability <sup>p</sup>*(*k*)*, and* #*<sup>L</sup>*(*p*) *is the number of all different linguistic terms in <sup>L</sup>*(*p*)*. To further reflect the hesitation of DMs, Lin et al. [15] expanded PLTS into PULTS:*

$$\mathcal{S}(p) = \{ \langle [L^k, \mathcal{U}^k], p^k \rangle | p^k \ge 0, k = 1, 2 \cdots, \#\mathcal{S}(p), \sum\_{k=1}^{\#\mathcal{S}(p)} p^k \le 1 \},\tag{2}$$

*where* [*Lk*, *<sup>U</sup><sup>k</sup>*], *p<sup>k</sup> represents the uncertain linguistic variable* [*Lk*, *U<sup>k</sup>*] *associated with its probability pk. <sup>L</sup>k*, *U<sup>k</sup>* ∈ *S are the linguistic terms, Lk* ≤ *Uk, and* #*<sup>S</sup>*(*p*) *is the cardinality of <sup>S</sup>*(*p*)*.*
