*2.2. LIFS*

To reflect the DM's qualitative non-preference information, Zhang [12] proposes LIFS.

**Definition 2** ([12])**.** *Let X be a finite universal set and S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>2</sup>*τ*]} *be a continuous linguistic term set. Then a LIFS L in X is given as*

$$L = \{ (\mathbf{x}, \mathbf{s}\_{\theta}(\mathbf{x}), \mathbf{s}\_{\sigma}(\mathbf{x})) | \mathbf{x} \in X \}\tag{3}$$

*where sθ* (*x*),*sσ*(*x*) ∈ *S stand for the linguistic membership degree and linguistic nonmembership of the element x to L, respectively, and* 0 ≤ *θ* + *σ* ≤ 2*τ for all x* ∈ *X.*

PULTS only takes into account the DM's qualitative preference information and its probability distribution, however, in actual DM, DMs may need to give preference and non-preference information from both sides due to various uncertainties. Although LIFS takes into account the DMs' non-preference information, it requires the DMs to give only single linguistic terms as decision information, which cannot reflect the decision makers' hesitation in a complex environment. Therefore, in order to avoid the limitations mentioned above in actual DM, this paper proposes PULIFS in combination with the advantages of PULTS and LIFS.

#### **3. PULIFS and PULIFPR**
