*2.1. PFS*

Atanassov [5] applied the non-membership degree to extend FS; however, expressing the evaluation information depend on IFS is unreasonable in practice, at times. Therefore, Cuong [13] proposed the PFS theory based on FS and IFS, which can represent more information of decision makers, including yes, abstain, no, and refusal.

**Definition 1.** *Let X be a non-empty and finite set, a PFS P on X* is defined by

$$P = \{ \langle \mathbf{x}, \mu\_P(\mathbf{x}), \eta\_P(\mathbf{x}), \upsilon\_P(\mathbf{x}) \rangle | \mathbf{x} \in X \}, \tag{1}$$

*where μP*(*x*)*, ηP*(*x*)*, and vP*(*x*) *are the positive, neutral, and negative membership functions that are belonging to* [0, 1]*, respectively, and they meet the condition of* 0 ≤ *μP*(*x*) + *ηP*(*x*) + *vP*(*x*) ≤ 1*. Furthermore, <sup>π</sup>P*(*x*) = 1 − *μP*(*x*) − *ηP*(*x*) − *vP*(*x*) *is the refusal membership function.*

**Definition 2.** *A picture fuzzy number (PFN) is represented by a* = (*μ<sup>a</sup>*, *η<sup>a</sup>*, *va*)*, where μa* ∈ [0, 1]*, ηa* ∈ [0, 1]*, va* ∈ [0, 1]*, and μa* + *ηa* + *va* ≤ 1 *[25].*

Wei [25] proposed the operations of PFNs based on the operations of IFNs in [21].

**Definition 3.** *Let a*1 = (*μ*1, *η*1, *<sup>v</sup>*1)*, a*2 = (*μ*2, *η*2, *<sup>v</sup>*2)*, and a* = (*μ*, *η*, *v*) *be three PFNs, λ* > 0*, and a<sup>c</sup> is the complementary set of a, then [25]*

$$a^c = (v, \eta, \mu);\tag{2}$$

$$a\_1 \oplus a\_2 = (\mu\_1 + \mu\_2 - \mu\_1 \mu\_2, \eta\_1 \eta\_2, \upsilon\_1 \upsilon\_2);\tag{3}$$

$$a\_1 \otimes a\_2 = (\mu\_1 \mu\_2, \eta\_1 + \eta\_2 - \eta\_1 \eta\_2, \upsilon\_1 + \upsilon\_2 - \upsilon\_1 \upsilon\_2);\tag{4}$$

$$
\lambda a = \left(1 - (1 - \mu)^{\lambda}, \eta^{\lambda}, \upsilon^{\lambda}\right);
\tag{5}
$$

$$a^{\lambda} = \left(\mu^{\lambda}, 1 - (1 - \eta)^{\lambda}, 1 - (1 - v)^{\lambda}\right). \tag{6}$$
