*3.1. PULIFS*

**Definition 3.** *Let S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>2</sup>*τ*]} *be a continuous LTS, then a PULIFS on S is expressed by a mathematical symbol:*

$$\mathcal{U}(p) = \{ \langle ([s\_{\underline{u}^k}, s\_{\overline{\pi}^k}], [s\_{\underline{x}^k}, s\_{\overline{\pi}^k}]), p^k \rangle \, | \, p^k \ge 0, k = 1, 2 \cdots, \#\mathcal{U}(p), \sum\_{k=1}^{\#\mathcal{U}(p)} p^k \le 1 \},\tag{4}$$

*where* ([*suk* ,*suk* ], [*svk* ,*svk* ]), *p<sup>k</sup> is a PULIF element (PULIFE), which denotes the k-th uncertain linguistic intuitionistic variable (ULIV)* ([*suk* ,*suk* ], [*svk* ,*svk* ]) *associated with its probability pk in <sup>U</sup>*(*p*)*, and* [*suk* ,*suk* ] ⊆ [*<sup>s</sup>*0,*s*2*τ*] *,* [*svk* ,*svk* ] ⊆ [*<sup>s</sup>*0,*s*2*τ*] *represent non-vagueness qualitative uncertain preference and non-preference information respectively. suk* ,*suk* ,*svk* ,*svk* ∈ *S, are the linguistic terms, suk* ≤ *suk* ,*svk* ≤ *svk* , *uk* + *vk* ≤ 2*τ, and* #*<sup>U</sup>*(*p*) *is the cardinality of <sup>U</sup>*(*p*)*. Similarly, the uncertain linguistic variable <sup>s</sup>π<sup>k</sup>* = [*<sup>s</sup>π<sup>k</sup>* ,*sπ<sup>k</sup>* ] *represent vagueness uncertain information, where π<sup>k</sup>* = 2*τ* − *uk* − *<sup>v</sup>k*, *π<sup>k</sup>* = 2*τ* − *uk* − *vk.*

In actual DM, DMs tend to compare two alternatives and give preference information instead of directly giving evaluation information to one alternative. Therefore, we further give the concept of PULIFPR based on PULIFS. For convenience, we use *u*(*p*) = {([*suk* ,*suk* ], [*svk* ,*svk* ]), *p<sup>k</sup>*} to represent the PULIFS, where *k* = 1, 2, ··· , #*u*(*p*), and #*u*(*p*) is the number of PULIFE in *<sup>u</sup>*(*p*).
