**2. Preliminaries**

In the following, we introduced some basic concepts related to 2-tuple linguistic term sets and PFSs.

### *2.1. 2-Tuple Linguistic Term Sets*

Let *S* = {*si* |*i* = 1, 2, ··· , *t* } be a linguistic term set with odd cardinality. *si* represents a possible value for a linguistic variable, and it should satisfy the following characteristics [61]:

(1) The set is ordered: *si* > *sj*, if *i* > *j*; (2) Max operator: max*si*,*sj* = *si*, if *si* ≥ *sj*; (3) Min operator: min*si*,*sj* = *si*, if *si* ≤ *sj*. For example, *S* can be defined as

*S* = {*<sup>s</sup>*1 = extremely poor,*<sup>s</sup>*2 = very poor,*<sup>s</sup>*3 = poor,*<sup>s</sup>*4 = medium, *s*5 = good,*<sup>s</sup>*6 = very good,*<sup>s</sup>*7 = extremely good}.

Herrera and Martinez [60] defined the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is utilized for depicting the linguistic information with a 2-tuple (*si*, *<sup>α</sup>i*), where *si* is a linguistic label from predefined linguistic term set *S*, and *αi* is the value of symbolic translation, and *αi* ∈ [−0.5, 0.5).

*2.2. Picture Fuzzy Sets (PFSs)*

**Definition 1** ([17])**.** *A PFS on the universe. X is an object of the form*

$$A = \{ \langle \mathbf{x}, \mu\_A(\mathbf{x}), \eta\_A(\mathbf{x}), \nu\_A(\mathbf{x}) \rangle | \mathbf{x} \in X \},\tag{1}$$

*where μA*(*x*) ∈ [0, 1] *is called the "degree of positive membership of A", ηA*(*x*) ∈ [0, 1] *is defined as the "degree of neutral membership of A", and <sup>ν</sup>A*(*x*) ∈ [0, 1] *is defined as the "degree of negative membership of A", and μA*(*x*)*, ηA*(*x*)*, <sup>ν</sup>A*(*x*) *satisfy the following condition:* 0 ≤ *μA*(*x*) + *ηA*(*x*) + *<sup>ν</sup>A*(*x*) ≤ 1*,* ∀ *x* ∈ *X. Then, for x* ∈ *X, <sup>π</sup>A*(*x*) = 1 − (*μA*(*x*) + *ηA*(*x*) + *<sup>ν</sup>A*(*x*)) *could be defined as the degree of refusal membership of x in A*.

**Definition 2** ([17])**.** *Let α* = (*μα*, *ηα*, *να*) *and β* = *μβ*, *ηβ*, *νβ be two PFNs, the operation formula of them can be given:*

$$(1)\quad \kappa \oplus \beta = (\mu\_{\alpha} + \mu\_{\beta} - \mu\_{\alpha}\mu\_{\beta}, \eta\_{\alpha}\eta\_{\beta}, \upsilon\_{\alpha}\upsilon\_{\beta}) \text{ y}$$

(2) *α* ⊗ *β* = *μαμβ*, *ηα* + *ηβ* − *ηαηβ*, *να* + *νβ* − *νανβ*;

$$\text{(3)}\quad \lambda\mathfrak{a} = \left(1 - (1 - \mu\_{\mathfrak{a}})^{\lambda}, \eta\_{\mathfrak{a}}^{\lambda}, \nu\_{\mathfrak{a}}^{\lambda}\right), \lambda > 0;$$

$$\text{(4)}\quad \pi^{\lambda} = \left(\mu\_{a\prime}^{\lambda}, 1 - \left(1 - \eta\_{a}\right)^{\lambda}, 1 - \left(1 - \nu\_{a}\right)^{\lambda}\right), \lambda > 0.$$

According to Definition 2, the operation laws have the following properties [17].

$$
\mathfrak{a} \oplus \mathfrak{b} = \mathfrak{b} \oplus \mathfrak{a}, \mathfrak{a} \otimes \mathfrak{b} = \mathfrak{b} \otimes \mathfrak{a}, \left( \left( a \right)^{\lambda\_1} \right)^{\lambda\_2} = \left( a \right)^{\lambda\_1 \lambda\_2}; \tag{2}
$$

$$
\lambda \left( a \oplus \beta \right) = \lambda a \oplus \lambda \beta,\ (a \otimes \beta)^{\lambda} = (a)^{\lambda} \otimes (\beta)^{\lambda};\tag{3}
$$

$$
\lambda\_1 \mathfrak{a} \oplus \lambda\_2 \mathfrak{a} = (\lambda\_1 + \lambda\_2) \mathfrak{a}, \ (\mathfrak{a})^{\lambda\_1} \otimes (\mathfrak{a})^{\lambda\_2} = (\mathfrak{a})^{(\lambda\_1 + \lambda\_2)}.\tag{4}$$

### *2.3. Picture 2-Tuple Linguistic Sets (P2TLSs)*

In the following, we introduce the concepts and basic operations of the P2TLSs based on the PFSs [17] and 2-tuple linguistic information model [60].

**Definition 3** ([44,45])**.** *A P2TLS A in X* is given

$$A = \left\{ \left( \mathbf{s}\_{\theta(x)}, \rho \right), \left( \mu\_A(\mathbf{x}), \eta\_A(\mathbf{x}), \nu\_A(\mathbf{x}) \right), \mathbf{x} \in X \right\},\tag{5}$$

*where <sup>s</sup>θ*(*x*), *ρ* ∈ *S, ρ* ∈ [−0.5, 0.5)*, uA*(*x*) ∈ [0, 1], *ηA*(*x*) ∈ [0, 1]*, and vA*(*x*) ∈ [0, 1]*, with the condition* 0 ≤ *uA*(*x*) + *ηA*(*x*) + *vA*(*x*) ≤ 1*,* ∀*x* ∈ *X, <sup>s</sup>θ*(*a*) ∈ *S, and ρ* ∈ [−0.5, 0.5)*. The numbers μA*(*x*), *ηA*(*x*), *<sup>ν</sup>A*(*x*) *represent, respectively, the degree of positive membership, degree of negative membership, and degree of negative membership of the element x to 2-tuple linguistic variable <sup>s</sup>θ*(*x*), *ρ* .

For convenience, we call +*α* =< *<sup>s</sup>θ*(*a*), *<sup>ρ</sup>* ,(*u*(*a*), *η*(*a*), *v*(*a*)) > a P2TLN, where *μα* ∈ [0, 1], *ηα* ∈ [0, 1], *να* ∈ [0, 1], *μα* + *ηα* + *να* ≤ 1, *<sup>s</sup>θ*(*a*) ∈ *S* and *ρ* ∈ [−0.5, 0.5).

**Definition 4** ([44])**.** *Let* +*a* =< *<sup>s</sup>θ*(*a*), *<sup>ρ</sup>* ,(*u*(*a*), *η*(*a*), *v*(*a*)) > *be a P2TLN, and a score function* +*a can be defined as follows:*

$$S(\tilde{\boldsymbol{\mu}}) = \Delta \left( \boldsymbol{\Delta}^{-1} \left( \boldsymbol{s}\_{\theta(\boldsymbol{a})^{\prime}} \boldsymbol{\rho} \right) \cdot \frac{1 + \mu\_{\boldsymbol{a}} - \nu\_{\boldsymbol{a}}}{2} \right), \ \boldsymbol{\Delta}^{-1} (\mathcal{S}(\tilde{\boldsymbol{a}})) \in [1, t]. \tag{6}$$

**Definition 5** ([44])**.** *Let* +*a* =< *<sup>s</sup>θ*(*a*), *<sup>ρ</sup>* ,(*u*(*a*), *η*(*a*), *v*(*a*)) > *be a P2TLN, and the accuracy function can be defined as follows:*

$$H(\hat{\mathfrak{a}}) = \Delta \left( \Delta^{-1} \left( \mathfrak{s}\_{\theta(a)}, \rho \right) \frac{\mu\_{\mathfrak{a}} + \eta\_{\mathfrak{a}} + \nu\_{\mathfrak{a}}}{2} \right), \Delta^{-1} \left( H(\hat{\mathfrak{a}}) \right) \in [1, t]. \tag{7}$$

**Definition 6** ([44])**.** *Let* +*a*1 =< *<sup>s</sup>θ*(*<sup>a</sup>*1), *<sup>ρ</sup>*<sup>1</sup> ,(*u*(*<sup>a</sup>*1), *η*(*<sup>a</sup>*1), *<sup>v</sup>*(*<sup>a</sup>*1)) > *and* +*a*2 =< *<sup>s</sup>θ*(*<sup>a</sup>*2), *ρ*2 , (*u*(*<sup>a</sup>*2), *η*(*<sup>a</sup>*2), *<sup>v</sup>*(*<sup>a</sup>*2)) > *be two P2TLNs, <sup>S</sup>*(<sup>+</sup>*a*1) = <sup>Δ</sup><sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>θ*(*<sup>a</sup>*1), *ρ*1 · <sup>1</sup>+*μα*1<sup>−</sup>*να*1 2 *and <sup>S</sup>*(<sup>+</sup>*a*2) = <sup>Δ</sup><sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>θ*(*<sup>a</sup>*2), *ρ*2 · <sup>1</sup>+*μα*2<sup>−</sup>*να*2 2 *be the scores of* +*a*1 *and* +*a*2*, respectively, and let <sup>H</sup>*(<sup>+</sup>*a*1) = <sup>Δ</sup><sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>θ*(*<sup>a</sup>*1), *ρ*1 · *μα*1+*ηα*1+*να*1 2 *and <sup>H</sup>*(<sup>+</sup>*a*2) = <sup>Δ</sup><sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>θ*(*<sup>a</sup>*2), *ρ*2 · *μα*2+*ηα*2+*να*2 2 *be the accuracy degrees of* +*a*1 *and* +*a*2*, respectively, then if <sup>S</sup>*(<sup>+</sup>*a*1) < *<sup>S</sup>*(<sup>+</sup>*a*2)*,* +*a*1 < +*a*2*; if <sup>S</sup>*(<sup>+</sup>*a*1) = *<sup>S</sup>*(<sup>+</sup>*a*2)*, then* (1) *if <sup>H</sup>*(<sup>+</sup>*a*1) = *<sup>H</sup>*(<sup>+</sup>*a*2)*, then* + *a*1 = + *a*2*;* (2) *if <sup>H</sup>*(<sup>+</sup>*a*1) < *<sup>H</sup>*(<sup>+</sup>*a*2)*, then,* +*a*1 < +*a*2.

Some operational laws of P2TLNs are defined as follows: **Definition 7** ([44])**.** *Let* +*a*1 = < *<sup>s</sup>θ*(*<sup>a</sup>*1), *ρ*1 ,(*u*(*<sup>a</sup>*1), *η*(*<sup>a</sup>*1), *<sup>v</sup>*(*<sup>a</sup>*1)) > *and* +*a*2 = < *<sup>s</sup>θ*(*<sup>a</sup>*2), *ρ*2 ,(*u*(*<sup>a</sup>*2), *η*(*<sup>a</sup>*2), *<sup>v</sup>*(*<sup>a</sup>*2)) > *be two P2TLNs, then*

$$\begin{array}{lcl}\widetilde{a}\_{1}\oplus\widetilde{a}\_{2} &=& \left\langle \Delta\left(\Delta^{-1}\big(s\_{\theta(a\_{1})},\rho\_{1}\right)+\Delta^{-1}\big(s\_{\theta(a\_{2})},\rho\_{2}\right)\right\rangle, \\ & (u(a\_{1})+u(a\_{2})-u(a\_{1})u(a\_{2}),\eta(a\_{1})\eta(a\_{2}),\nu(a\_{1})\nu(a\_{2}))\right\rangle; \\ \widetilde{a}\_{1}\odot\widetilde{a}\_{2} &=& \left\langle \Delta\left(\Delta^{-1}\big(s\_{\theta(a\_{1})},\rho\_{1}\right)\cdot\Delta^{-1}\big(s\_{\theta(a\_{2})},\rho\_{2}\big)\right\rangle, \\ & (u(a\_{1})u(a\_{2}),\eta(a\_{1})+\eta(a\_{2})-\eta(a\_{1})\eta(a\_{2}),\nu(a\_{1})+\nu(a\_{2})-\nu(a\_{1})\nu(a\_{2}))\right\rangle; \\ \lambda\widetilde{a}\_{1} &=& \left\langle \Delta\left(\lambda\Delta^{-1}\big(s\_{\theta(a\_{1})},\rho\_{1}\right)\right), \left(1-(1-u(a\_{1}))^{\lambda},\eta(a\_{1})^{\lambda},\nu(a\_{1})^{\lambda}\right)\rangle; \\ (\widetilde{a}\_{1})^{\lambda} &=& \left\langle \Delta\left(\left(\Delta^{-1}\big(s\_{\theta(a\_{1})},\rho\_{1}\right)\right)^{\lambda}\right), \left(u(a\_{1})^{\lambda},1-(1-\eta(a\_{1}))^{\lambda},1-(1-\nu(a\_{1}))^{\lambda}\right)\right\rangle. \end{array}$$

### **3. Picture 2-Tuple Linguistic Aggregation Operators**

In this section, we propose some aggregation operators with P2TLNs, such as the P2TLWA operator and the P2TLWG operator.

**Definition 8.** *Let* +*αj* = /*sj*, *ρj* , - *μj*, *ηj*, *<sup>ν</sup>j*0(*j* = 1, 2, ··· , *n*) *be a collection of P2TLNs, and the P2TLWA operator can be represented as*

$$\text{P2TLWA}\_{\omega}(\tilde{\mathfrak{a}}\_{1}, \tilde{\mathfrak{a}}\_{2}, \cdots, \tilde{\mathfrak{a}}\_{n}) = \underset{j=1}{\stackrel{n}{\leftrightarrow}} (\omega\_{j} \tilde{\mathfrak{a}}\_{j})\_{\prime} \tag{8}$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*) *T is the weight vector of* +*<sup>α</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>ω</sup>j* > 0, *n* ∑ *j*=1 *<sup>ω</sup>j* = 1.

Based on the Definition 8, we can ge<sup>t</sup> the following result:

**Theorem 1.** *The aggregated value by using the P2TLWA operator is also a P2TLN, where*

$$\begin{split} & \text{P2TINA}\_{\omega} \left( \tilde{\mathbf{a}}\_{1}, \tilde{\mathbf{a}}\_{2}, \cdots, \tilde{\mathbf{a}}\_{n} \right) = \bigoplus\_{j=1}^{n} \left( \omega\_{j} \tilde{\mathbf{a}}\_{j} \right) \\ &= \left\langle \Delta \left( \sum\_{j=1}^{n} \omega\_{j} \mathbf{A}^{-1} \{ s\_{j}, \rho\_{j} \} \right), \left( \mathbf{1} - \prod\_{j=1}^{n} \left( 1 - \mu\_{j} \right)^{\omega\_{j}}, \prod\_{j=1}^{n} \{ \eta\_{j} \}^{\omega\_{j}}, \prod\_{j=1}^{n} \left( \nu\_{j} + \eta\_{j} \right)^{\omega\_{j}} - \prod\_{j=1}^{n} \{ \eta\_{j} \}^{\omega\_{j}} \right) \right\rangle \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*) *T is the weight vector of* +*<sup>α</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>ω</sup>j* > 0, ∑ *j*=1 *<sup>ω</sup>j* = 1.

**Definition 9.** *Let* +*αj* = /*sj*, *ρj* , - *μj*, *ηj*, *<sup>ν</sup>j*0(*j* = 1, 2, ··· , *n*) *be a collection of P2TLNs, the P2TLWG operator can be represented as*

$$\text{P2TLWG}\_{\omega}(\tilde{\mathfrak{a}}\_1, \tilde{\mathfrak{a}}\_2, \dots, \tilde{\mathfrak{a}}\_n) = \bigotimes\_{j=1}^n \left(\tilde{\mathfrak{a}}\_j\right)^{\omega\_j} \tag{10}$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*) *T is the weight vector of* +*<sup>α</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>ω</sup>j* > 0, *n* ∑ *j*=1 *<sup>ω</sup>j* = 1.

Based on Definition 9, we can ge<sup>t</sup> the following result: **Theorem 2.** *The aggregated value by using the P2TLWG operator is also a P2TLN, where*

$$\begin{split} & \text{P2TLW}\_{\omega}(\overline{a}\_{1}, \overline{a}\_{2}, \dots, \overline{a}\_{n}) = \prod\_{j=1}^{n} \left( \overline{a}\_{j} \right)^{\omega\_{j}} \\ &= \left\langle \Delta \left( \prod\_{j=1}^{n} \left( \Delta^{-1} \left( s\_{j}, \rho\_{j} \right)^{\omega\_{j}} \right) \right), \left( \prod\_{j=1}^{n} \left( \mu\_{a\_{j}} + \eta\_{a\_{j}} \right)^{\omega\_{j}} - \prod\_{j=1}^{n} \left( \eta\_{a\_{j}} \right)^{\omega\_{j}}, \prod\_{j=1}^{n} \left( \eta\_{a\_{j}} \right)^{\omega\_{j}}, 1 - \prod\_{j=1}^{n} \left( 1 - \nu\_{a\_{j}} \right)^{\omega\_{j}} \right) \right\rangle \end{split} \tag{11}$$
  $where \ \omega = \left( \omega\_{1}, \omega\_{2}, \dots, \omega\_{n} \right)^{T}$  is the weight vector of  $a\_{j}(j = 1, 2, \dots, n)$  and  $\omega\_{j} > 0$ ,  $\sum\_{j=1}^{n} \omega\_{j} = 1$ .

### **4. The EDAS Model with P2TLNs**

The traditional EDAS method [1], which can consider the conflicting attributes, has been studied in many MCDM problems. By computing the average solution (AV), this model can describe the difference between all the alternatives and the AV based on two distance measures which are namely PDA (positive distance from average) and NDA (negative distance from average); the alternative with higher values of PDA and lower values of PDA is the best choice. To combine the EDAS model with P2TLNs, we construct the EDAS model so the evaluation values are presented by P2TLNs. The computing steps of our proposed model can be established as follows.

Suppose there are *m* alternatives {*<sup>δ</sup>*1, *δ*2,... *<sup>δ</sup>m*}, *n* attributes {*<sup>G</sup>*1, *G*2,... *Gn*}, and *r* experts {*<sup>a</sup>*1, *a*2,... *ar*}, let {*<sup>ω</sup>*1, *ω*2,... *<sup>ω</sup>n*} and {*<sup>θ</sup>*1, *θ*2,... *<sup>θ</sup>r*} be the attribute's weighting vector and expert's weighting vector which satisfy *ωi* ∈ [0, 1], *θi* ∈ [0, 1] and ∑*ni*=<sup>1</sup>*ωi* = 1, ∑*ti*=<sup>1</sup>*θi* = 1. Then:

**Step 1.** Construct the picture 2-tuple linguistic decision matrix *R* = -+*rijm*×*n* = /*sij*, *<sup>ρ</sup>ij*, *μij*, *ηij*, *<sup>ν</sup>ij*0*m*×*n*, *i* = 1, 2, . . . , *m*, *j* = 1, 2, . . . , *n*, which can be depicted as follows.

$$
\widetilde{R} = \left( \widetilde{r}\_{\widetilde{i}\widetilde{j}} \right)\_{m \times n} = \begin{bmatrix}
\widetilde{r}\_{11} & \widetilde{r}\_{12} & \dots & \widetilde{r}\_{1n} \\
\widetilde{r}\_{21} & \widetilde{r}\_{22} & \dots & \widetilde{r}\_{2n} \\
\vdots & \vdots & \vdots & \vdots \\
\widetilde{r}\_{m1} & \widetilde{r}\_{m2} & \dots & \widetilde{r}\_{mn}
\end{bmatrix} \tag{12}
$$

+

where +*rij*denotes the P2TLNs of alternative *ϑi* on attribute *Uj* by expert *qr*.

**Step 2.** Normalize the evaluation matrix *R* + = -+*rijm*×*n* to *R*+ = +*<sup>r</sup> ij m*×*n*. For benefit attributes:

$$\tilde{\mathbf{r}}\_{\text{ij}}^{\prime} = \tilde{\mathbf{r}}\_{\text{ij}} = \langle \left( \mathbf{s}\_{\text{ij}}, \rho\_{\text{ij}} \right), \left( \mu\_{\text{ij}}, \eta\_{\text{ij}}, \nu\_{\text{ij}} \right) \rangle, \mathbf{i} = 1, 2, \dots, m, \mathbf{j} = 1, 2, \dots, n \tag{13}$$

For cost attributes:

$$\left(\tilde{r}\_{\vec{i}\vec{j}}^{\prime} = \left(\tilde{r}\_{\vec{i}\vec{j}}\right)^{\mathcal{L}} = \left\langle \Delta \left(T - \Delta^{-1} \left(s\_{\vec{i}\vec{j}\cdot\vec{r}} \rho\_{\vec{i}\vec{j}}\right)\right), \left(v\_{\vec{i}\vec{j}\cdot\vec{r}} \eta\_{\vec{i}\vec{j}\cdot\vec{r}}\right)\right\rangle, \mathfrak{i} = 1, 2, \dots, m, \mathfrak{j} = 1, 2, \dots, n. \tag{14}$$

**Step 3.** According to the decision making matrix *R* + = +*<sup>r</sup> ij m*×*n* and expert's weighting vector {*<sup>δ</sup>*1, *δ*2,... *<sup>δ</sup>r*}, we can utilize overall *r* + *ij* to *r ij* by using P2TLWA or P2TLWG aggregation operators, and the computing results can be presented as follows.

$$R = \begin{bmatrix} r'\_{ij} \end{bmatrix}\_{m \times n} = \begin{bmatrix} r'\_{11} & r'\_{12} & \dots & r'\_{1n} \\ r'\_{21} & r'\_{22} & \dots & r'\_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ r'\_{m1} & r'\_{m2} & \dots & r'\_{mn} \end{bmatrix} \tag{15}$$

**Step 4.** Compute the value of AV based on all proposed attributes;

$$\text{AV} = \begin{bmatrix} \text{AV}\_{\dot{f}} \end{bmatrix}\_{1 \times n} = \left[ \frac{\sum\_{i=1}^{m} r'\_{\dot{ij}}}{m} \right]\_{1 \times n} . \tag{16}$$

Based on Definition 8,

$$\sum\_{i=1}^{m} r'\_{\,\,ij} = \left\langle \Delta \left( \sum\_{i=1}^{m} \Delta^{-1} (s\_{i\,\,p} \rho\_i) \right), \left( 1 - \prod\_{i=1}^{m} \left( 1 - \mu'\_{\,\,ij} \right) \prod\_{i=1}^{m} \eta'\_{\,\,ij'} \prod\_{i=1}^{m} \left( \nu'\_{\,\,ij} + \eta'\_{\,\,ij} \right) - \prod\_{i=1}^{m} \eta'\_{\,\,ij} \right) \right\rangle \tag{17}$$

$$\begin{split} \text{AV} &= \begin{bmatrix} \text{AV}\_{l} \end{bmatrix}\_{1 \times n} = \begin{bmatrix} \frac{\sum\_{i=1}^{n} r\_{il}'}{m} \end{bmatrix}\_{1 \times n} \\ &= \left\langle \Delta \left( \sum\_{i=1}^{m} \frac{1}{m} \Delta^{-1} (s\_{i\cdot} \rho\_{l}) \right), \left( 1 - \prod\_{i=1}^{m} \left( 1 - \mu\_{il}' \right)^{\frac{1}{m}}, \prod\_{i=1}^{m} \left( \eta\_{\,i\'\_{l}}' \right)^{\frac{1}{m}}, \prod\_{i=1}^{m} \left( \nu\_{\,i\'\_{l}}' + \eta\_{\,i\'\_{l}}' \right)^{\frac{1}{m}} - \prod\_{i=1}^{m} \left( \eta\_{\,i\'\_{l}}' \right)^{\frac{1}{m}} \right) \end{split} \tag{18}$$

**Step 5.** According to the results of AV, we can compute the PDA and NDA by using the following formula:

$$\text{PDA}\_{i\bar{j}} = \left[ \text{PDA}\_{i\bar{j}} \right]\_{m \times n} = \frac{\max \left( 0, \left( r'\_{i\bar{j}} - \text{AV}\_{\bar{j}} \right) \right)}{\text{AV}\_{\bar{j}}},\tag{19}$$

$$\text{NDA}\_{ij} = \left[\text{NDA}\_{ij}\right]\_{m \times n} = \frac{\max\left(0, \left(\text{AV}\_j - r\_{ij}'\right)\right)}{\text{AV}\_j}.\tag{20}$$

For convenience, we can use the score function of P2TLNs presented in Definition 4 to determine the results of PDA and NDA as follows.

$$\text{PDA}\_{ij} = \begin{bmatrix} \text{PDA}\_{ij} \end{bmatrix}\_{m \times n} = \frac{\max \left( 0, \left( s \left( r'\_{i\bar{j}} \right) - s \left( \text{AV}\_{\bar{j}} \right) \right) \right)}{s \left( \text{AV}\_{\bar{j}} \right)} \tag{21}$$

$$\text{NDA}\_{i\bar{j}} = \left[ \text{NDA}\_{i\bar{j}} \right]\_{m \times n} = \frac{\max \left( 0, \left( s \left( \text{AV}\_{\bar{j}} \right) - s \left( r'\_{i\bar{j}} \right) \right) \right)}{s \left( \text{AV}\_{\bar{j}} \right)} \tag{22}$$

**Step 6.** Calculate the values of SP*i* and SN*i* which denotes the weighted sum of PDA and NDA, the computing formula are provided as follows.

$$\text{SP}\_{i} = \sum\_{j=1}^{n} w\_{j} PDA\_{i\bar{j}\bar{\prime}} \text{ SN}\_{i} = \sum\_{j=1}^{n} w\_{j} NDA\_{i\bar{j}} \tag{23}$$

**Step 7.** The results of Equation (23) can be normalized as

$$\text{NSP}\_{i} = \frac{\text{SP}\_{i}}{\max(\text{SP}\_{i})}, \text{ NSN}\_{i} = 1 - \frac{\text{SN}\_{i}}{\max(\text{SN}\_{i})}.\tag{24}$$

**Step 8.** Compute the values of appraisal score (AS) based on each alternative's NSP*i* and NSN*i*.

$$\text{AS}\_{i} = \frac{1}{2}(\text{NSP}\_{i} + \text{NSN}\_{i}) \tag{25}$$

**Step 9.** According to the calculating results of the AS, we can rank all the alternatives; the bigger the value of AS is, the better the selected alternative will be.

### **5. The Numerical Example**

### *5.1. Numerical for MCGDM Problems with PFNs*

In this section, we provide a numerical example for green supplier selection by using EDAS models with P2TLNs. Assuming that five possible green suppliers *<sup>ϑ</sup>i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) are to be selected and there are four criteria to assess these green suppliers: -1 U1 is the price factor; -2 U2 is the delivery factor; -3 U3 is the environmental factors; -4 U4 is the product quality factor. The five possible green suppliers *<sup>ϑ</sup>i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) are to be evaluated with P2TLNs with the four criteria by three experts, *a<sup>r</sup>* (attributes weight *ω* = (0.22, 0.36, 0.28, 0.14), expert's weight *δ* = (0.24, 0.45, 0.31).).

**Step 1.** Construct the evaluation matrix *R* + = -+*rijm*×*n*, *i* = 1, 2, ... , *m*, *j* = 1, 2, ... , *n* for each of the three experts, which are listed in Tables 1–3.


**Table 1.** Picture 2-tuple linguistic evaluation information by *q*1.

**Table 2.** Picture 2-tuple linguistic evaluation information by *q*2.


**Table 3.** Picture 2-tuple linguistic evaluation information by *q*3.


**Step 2.** Normalize the evaluation matrix *R* + = %+*rij*&*m*×*n* to *R*+ = %+*r ij* &*m*×*n*; if all the attributes are benefitted, then normalization is not needed.

**Step 3.** According to the decision making matrix *R* + = %+*r ij* &*m*×*n* and expert's weighting vector {*<sup>δ</sup>*1, *δ*2,... *<sup>δ</sup>r*}, utilize overall +*r ij* to *r ij* by using the P2TLWA aggregation operator, and the computing results can be presented as follows in Table 4.


**Table 4.** The fused values by using picture 2-tuple linguistic weighted averaging (P2TLWA) operator.

**Step 4.** According to Table 4, we can obtain the value of the AV based on all proposed attributes by Formula (16), which is listed in Table 5.

**Table 5.** The value of the average solution (AV).


**Step 5.** According to the results of the AV, we can compute the PDA and the NDA by using the Formulas (19) and (20), which are listed in Tables 6–8.


**Table 6.** The score values of *<sup>ϑ</sup> ij* and AV*j*.

**Table 7.** The results of PDA*ij*.



**Table 8.** The results of NDA*ij*.

**Step 6.** By calculating the values of SP*i* and SN*i* by Equation (23) and the attributes weighting vector *ω* = (0.22, 0.36, 0.28, 0.14), we can obtain the results as

> SP1 = 0.0000, SP2 = 0.9218, SP3 = 0.0000, SP4 = 0.0117, SP5 = 0.0348 SN1 = 0.1520, SN2 = 0.0000, SN3 = 0.2752, SN4 = 0.0907, SN5 = 0.1889

**Step 7.** The results of Step 6 can be normalized by Formula (24) and are listed as

NSP1 = 0.0000, NSP2 = 1.0000, NSP3 = 0.0000, NSP4 = 0.0127, NSP5 = 0.0378 NSN1 = 0.4475, NSN2 = 1.0000, NSN3 = 0.0000, NSN4 = 0.6705, NSN5 = 0.3135

**Step 8.** Based on each alternative's NSP*i* and NSN*i*, compute the values of AS;

> AS1 = 0.2238, AS2 = 1.0000, AS3 = 0.0000, AS4 = 0.3416, AS5 = 0.1756.

**Step 9.** According to the calculated results of AS, we can rank all the alternatives; the bigger the value of AS is, the better the selected alternative will be. Clearly, the rank of all alternatives is *ϑ*2 > *ϑ*4 > *ϑ*1 > *ϑ*5 > *ϑ*3, and *ϑ*2 is the best green supplier.

### *5.2. Compare P2TLNs EDAS Method with Some Aggregation Operators with P2TLNs*

In this section, we compare our proposed picture 2-tuple linguistic EDAS method when using either the P2TLWA operator or the P2TLWG operator. According to the results of Table 4 and attributes weighting vector *ω* = (0.22, 0.36, 0.28, 0.14), we can utilize overall *r ij* to *r i* by using the P2TLWA and P2TLWG operators, which is listed in Table 9.

**Table 9.** The fused values by using some picture 2-tuple linguistic number (P2TLN) aggregation operators.


According to the score function of P2TLNs, we can obtain the alternative score results which are shown in Table 10.

The ranking of alternatives by some P2TLN aggregation operators are listed in Table 11.

Comparing the results of the picture 2-tuple linguistic EDAS model using either P2TLWA or P2TLWG operators, the aggregation results are slightly different in the ranking of alternatives, and the best alternatives are the same. However, the picture 2-tuple linguistic EDAS model has the valuable characteristic of considering the conflicting attributes, and can be more accurate and effective in the application of MCGDM problems.


**Table 10.** Score results of alternatives *ϑi*.

**Table 11.** Rank of alternatives by some P2TLN aggregation operators.

