**2. Preliminaries**

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

Wei et al. [50] proposed the Pythagorean 2-tuple linguistic sets (P2TLSs) based on the PFSs [51] and 2-tuple linguistic information [52].

**Definition 1** ([50])**.** *A P2TLS A in X is given*

$$A = \left| \{ \mathbf{s}\_{\varphi(\mathbf{x})}, \rho \} , (\mathbf{u}\_A(\mathbf{x}), \mathbf{v}\_A(\mathbf{x})) , \mathbf{x} \in X \right|, \tag{1}$$

*where <sup>s</sup>*ϕ(*x*) ∈ *S,* ρ ∈ [−0.5, 0.5)*, uA*(*x*) ∈ [0, 1]*, and vA*(*x*) ∈ [0, 1]*, uA*(*x*)*, and* <sup>ν</sup>*A*(*x*) *satisfy the following condition:* 0 ≤ (*uA*(*x*))<sup>2</sup> + (*vA*(*x*))<sup>2</sup> ≤ 1*,* ∀*x* ∈ *X. The numbers uA*(*x*), <sup>ν</sup>*A*(*x*) *represent the degree of membership and degree of non-membership of the element x to linguistic variable <sup>s</sup>*ϕ(*x*), ρ *.*

*A* = + *<sup>s</sup>*ϕ, ρ ,(*uA*, *vA*) , *can be called a Pythagorean 2-tuple linguistic number (P2TLN).*

**Definition 2** ([50])**.** *Suppose that a* = + *<sup>s</sup>*ϕ, ρ ,(*ua*, *va*) , *is a P2TLN; then, the score function of P2TLN can be depicted as follows:*

$$\begin{aligned} S(a) &= \Delta \Big( \Delta^{-1} \Big( s\_{\psi(a)}, \rho \Big) \frac{1 + (u\_a)^2 - (\nu\_a)^2}{2} \Big), \\ \Delta^{-1} (S(a)) &\in [0, L]. \end{aligned} \tag{2}$$

**Definition 3** ([50])**.** *Suppose that a* = + *<sup>s</sup>*ϕ, ρ ,(*ua*, *va*) , *is a P2TLN; then, the accuracy function of P2TLN can be depicted as follows:*

$$\begin{aligned} H(a) &= \Delta \big( \Delta^{-1} \big( s\_{\varphi(a)}, \rho \big) \frac{\left( u\_d \right)^2 + \left( v\_d \right)^2}{2} \big), \\ \Delta^{-1} (H(a)) &\in [0, L]. \end{aligned} \tag{3}$$

**Definition 4** ([50])**.** *Suppose that a*1 = + *<sup>s</sup>*ϕ1 , ρ1 ,(*ua*1 , *va*1 ) , *and a*2 = + *<sup>s</sup>*ϕ2 , ρ2 ,(*ua*2 , *va*2 ) , *are two P2TLNs. Respectively, the scores of a*1 *and a*2 *are <sup>S</sup>*(*<sup>a</sup>*1) = Δ - Δ−<sup>1</sup> *<sup>s</sup>*ϕ(*<sup>a</sup>*1), ρ1 · <sup>1</sup>+(*ua*1 ) 2 <sup>−</sup>(<sup>ν</sup>*a*1 ) 2 2 .

*and <sup>S</sup>*(*<sup>a</sup>*2) = <sup>Δ</sup>-<sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>*ϕ(*<sup>a</sup>*2), ρ2 · <sup>1</sup>+(*ua*2 )<sup>2</sup>−(<sup>ν</sup>*a*2 )2 2 .*, and let <sup>H</sup>*(*<sup>a</sup>*1) = <sup>Δ</sup>-<sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>*ϕ(*<sup>a</sup>*1), ρ1 · (*ua*1 )<sup>2</sup>+(<sup>ν</sup>*a*1 )2 2 . *and* = ·(*ua*2 )<sup>2</sup>+(<sup>ν</sup>*a*2 )2 

*<sup>H</sup>*(*<sup>a</sup>*2) <sup>Δ</sup>-<sup>Δ</sup>−<sup>1</sup>*<sup>s</sup>*ϕ(*<sup>a</sup>*2), ρ2 2 .*be the accuracy degrees of a*1 *and a*2*; then, some operational laws of P2TLNs can be defined as follows:*

> (1)*if <sup>S</sup>*(*<sup>a</sup>*1) < *<sup>S</sup>*(*<sup>a</sup>*2), *a*1 < *a*2; (2)*if <sup>S</sup>*(*<sup>a</sup>*1) > *<sup>S</sup>*(*<sup>a</sup>*2), *a*1 > *a*2; (3) *if <sup>S</sup>*(*<sup>a</sup>*1) = *<sup>S</sup>*(*<sup>a</sup>*2), *<sup>H</sup>*(*<sup>a</sup>*1) < *<sup>H</sup>*(*<sup>a</sup>*2), *then a*1 < *a*2; (4) *if <sup>S</sup>*(*<sup>a</sup>*1) = *<sup>S</sup>*(*<sup>a</sup>*2), *<sup>H</sup>*(*<sup>a</sup>*1) > *<sup>H</sup>*(*<sup>a</sup>*2), *then a*1 > *a*2; (5) *if <sup>S</sup>*(*<sup>a</sup>*1) = *<sup>S</sup>*(*<sup>a</sup>*2), *<sup>H</sup>*(*<sup>a</sup>*1) = *<sup>H</sup>*(*<sup>a</sup>*2), *then a*1 = *a*2.

**Definition 5** ([50])**.** *Suppose that a*1 = +*<sup>s</sup>*ϕ1 , <sup>ρ</sup>1,(*ua*1 , *va*1 ), *and a*2 = +*<sup>s</sup>*ϕ2 , <sup>ρ</sup>2,(*ua*2 , *va*2 ), *are two P2TLNs, the normalized Hamming distance (Hd) between a*1 *and a*2 *can be depicted below:*

$$H\_d(a\_1, a\_2) = \frac{1}{2L} \left\| \begin{pmatrix} \left(1 + \left(u\_{a\_1}\right)^2 - \left(v\_{a\_1}\right)^2\right) \cdot \Delta^{-1} \left(s\_{\psi\_1}, \rho\_1\right) - \\ \left(1 + \left(u\_{a\_2}\right)^2 - \left(v\_{a\_2}\right)^2\right) \cdot \Delta^{-1} \left(s\_{\psi\_2}, \rho\_2\right) \end{pmatrix} \right\|\,\tag{4}$$

*where L represents the length of the language scale. It is a numerical value.*

**Definition 6** ([50])**.** *Suppose that a*1 = +*<sup>s</sup>*ϕ1 , <sup>ρ</sup>1,(*ua*1 , *va*1 ), *and a*2 = +*<sup>s</sup>*ϕ2 , <sup>ρ</sup>2,(*ua*2 , *va*2 ), *are two P2TLNs, then*

$$\begin{split} a\_{1}\oplus a\_{2} &= \left\langle \Delta \Big(\Delta^{-1} \Big(\mathbf{s}\_{\varphi\_{1}\prime} \rho\_{1}\Big) + \Delta^{-1} \Big(\mathbf{s}\_{\varphi\_{2}\prime} \rho\_{2}\Big)\Big), \Big(\sqrt{(u\_{a1})^{2} + (u\_{a2})^{2} - (u\_{a1})^{2}(u\_{a2})^{2}}, \nu\_{a1} \nu\_{a2}\Big)\Big)\Big\rangle; \\ a\_{1}\otimes a\_{2} &= \left\langle \Delta \Big(\Delta^{-1} \Big(\mathbf{s}\_{\varphi\_{1}\prime} \rho\_{1}\Big) \cdot \Delta^{-1} \Big(\mathbf{s}\_{\varphi\_{2}\prime} \rho\_{2}\Big)\Big), \Big(u\_{a1}u\_{a2} \cdot \sqrt{(\nu\_{a1})^{2} + (\nu\_{a2})^{2} - (\nu\_{a1})^{2}(\nu\_{a2})^{2}}\Big)\Big\rangle; \\ a\_{1}\lambda a\_{1} &= \left\langle \Delta \Big(\lambda \Delta^{-1} \Big(\mathbf{s}\_{\varphi\_{1}\prime} \rho\_{1}\Big)\Big), \Big(\sqrt{1 - \left(1 - (u\_{a1})^{2}\right)^{\lambda}}, \left(\nu\_{a1}\right)^{\lambda}\Big)\right\rangle; \\ a\_{1}(a\_{1})^{\lambda} &= \left\langle \Delta \Big(\left(\Delta^{-1} \{s\_{\varphi\_{1}\prime}, \rho\_{1}\Big)\Big)^{\lambda}\Big((u\_{a1})^{\lambda} \cdot \sqrt{1 - \left(1 - (\nu\_{a1})^{2}\right)^{\lambda}}\Big)\right). \end{split}$$

**Theorem 1** ([50])**.** *For any two P2TLNs a*1 = +*<sup>s</sup>*ϕ1 , <sup>ρ</sup>1,(*ua*1 , *va*1 ), *and a*2 = +*<sup>s</sup>*ϕ2 , <sup>ρ</sup>2,(*ua*2 , *va*2 ),*; according to Definition 6, naturally, we can get the following properties of the operation laws:*

> (1) *a*1 ⊕ *a*2 = *a*2 ⊕ *a*1 (2) *a*1 ⊗ *a*2 = *a*2 ⊗ *a*1 (3) *k*(*<sup>a</sup>*1 ⊕ *<sup>a</sup>*2) = *ka*1 ⊕ *ka*2, 0 ≤ *k* ≤ 1 (4) *k*1*a*1 ⊕ *k*2*a*1 = (*k*1 ⊕ *k*2)*<sup>a</sup>*1, 0 ≤ *k*1, *k*2, *k*1 + *k*2 ≤ 1 (5) *<sup>a</sup>*1*k*<sup>1</sup> ⊗ *<sup>a</sup>*1*k*<sup>2</sup> = (*<sup>a</sup>*1)*<sup>k</sup>*1+*k*<sup>2</sup> , 0 ≤ *k*1, *k*2, *k*1 + *k*2 ≤ 1 (6) *<sup>a</sup>*1*k*<sup>1</sup> ⊗ *<sup>a</sup>*2*k*<sup>1</sup> = (*<sup>a</sup>*1 ⊗ *<sup>a</sup>*2)*<sup>k</sup>*<sup>1</sup> , *k*1 ≥ 0 (7) (*<sup>a</sup>*1)*<sup>k</sup>*<sup>1</sup> *k*2 = (*<sup>a</sup>*1)*<sup>k</sup>*1*k*<sup>2</sup> .

#### *2.2. Pythagorean 2-Tuple Linguistic Arithmetic Aggregation Operators*

In this section, some arithmetic aggregation operators with Pythagorean 2-tuple linguistic information will be introduced, such as the P2TLWA operator and P2TLWG operator.

**Definition 7** ([50])**.** *Assume that aj* = +*<sup>s</sup>*ϕ*j* , <sup>ρ</sup>*j*,*uaj* , *vaj*,(*<sup>j</sup>* = 1, 2, ... , *n*) *is a collection of P2TLNs. The P2TLWA operator can be depicted as follows:*

$$\begin{split} & \text{P2TLWA}\_{\omega}(a\_1, a\_2, \dots, a\_n) = \bigoplus\_{j=1}^{n} \{\omega\_j a\_j\} \\ &= \left\langle \Delta \left(\sum\_{j=1}^{n} \omega\_j \Delta^{-1} \{\mathbf{s}\_{\varphi\_j}, \rho\_j\} \right) \Big| \left(\sqrt{1 - \prod\_{j=1}^{n} \left(1 - \left(\mathbf{u}\_{a\_j}\right)^2\right)^{\omega\_j}}, \prod\_{j=1}^{n} \{\mathbf{v}\_{a\_j}\}^{\omega\_j} \right) \right\rangle \end{split} \tag{5}$$

*where* ω = (<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*)*<sup>T</sup> is the weight vector of aj*(*j* = 1, 2, ... , *n*) *and* <sup>ω</sup>*j* > 0, *nj*=1 <sup>ω</sup>*j* = 1.

**Definition 8** ([50])**.** *Assume that a* = +*<sup>s</sup>*ϕ*j* , <sup>ρ</sup>*j*,*uaj* , *vaj*,(*<sup>j</sup>* = 1, 2, ... , *n*) *is a collection of P2TLNs. The P2TLWG operator can be depicted as follows:*

$$\begin{split} & \text{P2TLMNG}\_{\omega} \left( a\_1, a\_2, \dots, a\_n \right) = \stackrel{\text{in}}{\otimes} \left( \omega\_j a\_j \right) \\ &= \left\langle \Delta \left( \prod\_{j=1}^n \Delta^{-1} \left( s\_{\varphi\_j}, \rho\_j \right)^{\omega\_j} \right) \Big| \left( \prod\_{j=1}^n \left( u\_{a\_j} \right)^{\omega\_j} \sqrt{1 - \prod\_{j=1}^n \left( 1 - \left( \nu\_{a\_j} \right)^2 \right)^{\omega\_j}} \right) \right\rangle \end{split} \tag{6}$$

*where* ω = (<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*)*<sup>T</sup> is the weight vector of aj*(*j* = 1, 2, ... , *n*) *and* <sup>ω</sup>*j* > 0, *nj*=1 <sup>ω</sup>*j* = 1.

#### **3. VIKOR Method for P2TL MADM Problems**

Suppose that Φ*i* = {Φ1, Φ2, ... Φ*m*} and τ*j* = {<sup>τ</sup>1, τ2, ...τ*n*} are respectively *m* alternatives and *n* criteria. Let ω = {<sup>ω</sup>1, ω2, ...ω*n*} be the criteria's weighting vector, which satisfies <sup>ω</sup>*j* ∈ [0, 1] and *nj*=<sup>1</sup> <sup>ω</sup>*j* = 1. Let *E* = {*E*1, *E*2, ... *Ek*} be the group of DMs, *w* = {*<sup>w</sup>*1, *w*2, ... *wk*} be the weight of DMs, with *wt* ∈ [0, 1] and *kt*=<sup>1</sup> *wt* = 1. Construct a decision matrix *R*(*t*) = *r*(*t*) *ij m*×*n*, where *R*(*t*) = *r*(*t*) *ij m*×*n* = 1*s*(*t*) ϕ*ij* , ρ(*t*) *ij* ,*<sup>u</sup>*(*t*) *rij* , *v*(*t*) *rij* 2*m*×*n* means the performance of the alternative Φ*i*{*i* = 1, 2 ··· , *m*} with respect to criteria <sup>τ</sup>*jj* = 1, 2 ··· , *n* by expert *E*(*t*){*t* = 1, 2, ... *k*} using a P2TLN, 0 ≤ *u*(*t*) *rij* ≤ 1,0 ≤ *v*(*t*) *rij* ≤ 1 and 0 ≤ *u*(*t*) *rij* 2 + *v*(*t*) *rij* 2 ≤ 1, *i* = 1, 2 ··· , *m*, *j* = 1, 2 ··· , *n*, *t* = 1, 2 ··· , *k*. InviewofboththeP2TLN'stheoriesandproceduresfromtheVIKORmethod,weforwarda

 put P2TL-VIKOR method to deal with the problem of MADM effectively. The new model is shown below.

Step 1. Set up a decision-making group composed of several experts, choose the best attributes to measure alternatives, and finally ge<sup>t</sup> a series of P2TL fuzzy decision matrices *R*(*t*) = *r*(*t*) *ij m*×*n* from each decision maker.

$$\mathcal{R}^{(t)} = \begin{bmatrix} r\_{ij}^{(t)} \\ r\_{ij}^{(t)} \end{bmatrix}\_{\mathsf{m} \times \mathsf{m}} = \begin{bmatrix} r\_{11}^{(t)} & r\_{12}^{(t)} & \dots & r\_{1n}^{(t)} \\ r\_{21}^{(t)} & r\_{22}^{(t)} & \dots & r\_{2n}^{(t)} \\ \vdots & \vdots & \vdots & \vdots \\ r\_{m1}^{(t)} & r\_{m2}^{(t)} & \dots & r\_{mn}^{(t)} \end{bmatrix} \tag{7}$$

where *r*(*t*) *ij* denotes the fuzzy performance value of the *i*th alternative (*i* = 1, 2, ... , *m*) with respect to the *j*th criterion (*j* = 1, 2, ... , *n*) and *t*th decision-maker (*t* = 1, 2, ... , *k*).

Step 2. Utilize the P2TLWA operator or P2TLWG operator to the fuse assessment information; then, the group P2TL fuzzy decision matrix *R* = *rijm*×*n*can be obtained by the calculation.

$$R = \begin{bmatrix} r\_{ij} \end{bmatrix}\_{m \times n} = \begin{bmatrix} r\_{11} & r\_{12} & \dots & r\_{1n} \\ r\_{21} & r\_{22} & \dots & r\_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ r\_{m1} & r\_{m2} & \dots & r\_{mn} \end{bmatrix} \tag{8}$$

$$\begin{split} r\_{ij} &= \mathop{\frac{k}{t}}\_{t=1} r\_{ij}^{(k)} = \text{P2TLMA}(r\_{ij}^{(1)}, r\_{ij}^{(2)}, \dots, r\_{ij}^{(k)}) \\ &= \left\langle \Delta \Big( \sum\_{t=1}^{k} w\_{l} \Delta^{-1} \Big( s\_{\psi\_{ij}}^{(t)}, \rho\_{ij}^{(t)} \Big) \Big) \Big( \sqrt{1 - \prod\_{t=1}^{k} \Big( 1 - \left( u\_{r\_{ij}}^{(t)} \right)^{2}} \Big)^{w\_{l}} \Big/ \prod\_{t=1}^{k} \Big( \nu\_{r\_{ij}}^{(t)} \Big)^{w\_{l}} \right\rangle . \end{split} \tag{9}$$

Or

$$\begin{split} r\_{ij} &= \sum\_{t=1}^{k} r\_{ij}^{(k)} = \text{P2TLNG}(r\_{ij}^{(1)}, r\_{ij}^{(2)}, \dots, r\_{ij}^{(k)}) \\ &= \left\langle \Delta \left( \prod\_{t=1}^{k} \Delta^{-1} \{s\_{\varphi\_{ij}}^{(t)}, \rho\_{ij}^{(t)}\}^{w\_t} \right) \left\langle \prod\_{t=1}^{k} \left(u\_{r\_{ij}}^{(t)}\right)^{w\_t}, \sqrt{1 - \prod\_{t=1}^{k} \left(1 - \left(\nu\_{r\_{ij}}^{(t)}\right)^2\right)^{w\_t}} \right\rangle \right\rangle \end{split} \tag{10}$$

Here, *rij* means the average fuzzy performance value of the *i*th alternative relative to the *j*th criterion.

Step 3. Determine the positive ideal solutions *R*+*j*and negative ideal solutions *R*−*j*

$$\boldsymbol{R}\_{j}^{+} = \left\{ \left( \boldsymbol{\Delta}^{-1} \{ \mathbf{s}\_{\psi\_{j} \prime} \boldsymbol{\rho}\_{j} \} \right)^{+} , \left( \boldsymbol{u}\_{a\_{j} \prime}^{+} \boldsymbol{v}\_{a\_{j}}^{+} \right) \right\} , \tag{11}$$

 .

$$\mathcal{R}\_{\dot{\jmath}}^{-} = \left\{ \left( \Delta^{-1} \left( \mathbf{s}\_{\psi\_{\dot{\jmath}} \prime} \rho\_{\dot{\jmath}} \right) \right)^{-}, \left( \mathbf{u}\_{a\_{\dot{\jmath}} \prime}^{-} \mathbf{v}\_{a\_{\dot{\jmath}}}^{-} \right) \right\}. \tag{12}$$

For all the benefit criteria:

$$\left\langle \left(\Delta^{-1}\big(\mathbf{s}\_{\psi\_{\rangle^{\prime}}}\rho\_{\rangle}\right)\right\rangle^{+}, \left(\mathbf{u}\_{a\_{\rangle^{\prime}}}^{+}\mathbf{v}\_{a\_{\rangle}^{+}}^{+}\right) \right\rangle = \left\langle \max\left(\Delta^{-1}\big(\mathbf{s}\_{\psi\_{\rangle^{\prime}}}\rho\_{\rangle}\right)\right\rangle \left(\max\{\mathbf{u}\_{a\_{\rangle}}\big), \min\{\mathbf{v}\_{a\_{\rangle}}\}\right) \rangle. \tag{13}$$

$$\left\{ \left( \Delta^{-1} \left( \mathbf{s}\_{\boldsymbol{\rho}\_{j'}} \boldsymbol{\rho}\_{j} \right) \right)^{-}, \left( \mathbf{u}\_{a\_{j'}}^{-} \boldsymbol{\upsilon}\_{a\_{j}}^{-} \right) \right\} = \left\{ \min \left( \Delta^{-1} \left( \mathbf{s}\_{\boldsymbol{\rho}\_{j'}} \boldsymbol{\rho}\_{j} \right) \right), \left( \min \left( \mathbf{u}\_{a\_{j}} \right), \max \left( \mathbf{v}\_{a\_{j}} \right) \right) \right\}. \tag{14}$$

For all the cost criteria:

$$\left\{ \left( \Delta^{-1} \left( \mathbf{s}\_{\psi\_{j}}, \rho\_{j} \right) \right)^{+}, \left( \mathbf{u}\_{a\_{j}}^{+}, \mathbf{v}\_{a\_{j}}^{+} \right) \right\} = \left\{ \min \left( \Delta^{-1} \left( \mathbf{s}\_{\psi\_{j}}, \rho\_{j} \right) \right), \left( \min \left( \mathbf{u}\_{a\_{j}} \right), \max \left( \mathbf{v}\_{a\_{j}} \right) \right) \right\}, \tag{15}$$

$$\left\{ \left( \Delta^{-1} \left( \mathbf{s}\_{\psi\_{\backslash'}}, \rho\_{\rangle} \right) \right)^{-}, \left( \mathbf{u}\_{a\_{\backslash'}}^{-}, \mathbf{v}\_{a\_{\backslash}}^{-} \right) \right\} = \left\{ \max \left( \Delta^{-1} \left( \mathbf{s}\_{\psi\_{\backslash'}}, \rho\_{\rangle} \right) \right), \left( \max \left( \mathbf{u}\_{a\_{\backslash}} \right), \min \left( \mathbf{v}\_{a\_{\backslash}} \right) \right) \right\}. \tag{16}$$

Step 4. Calculate *Si* and *Pi* values using the following equations:

$$S\_{i} = \sum\_{j=1}^{n} \omega\_{j} \frac{d\left(\left(\left(\Delta^{-1}\left(\mathbf{s}\_{\psi\_{j}}, \rho\_{j}\right)\right)^{+}, \left(\mathbf{u}\_{a\_{j}^{+}}^{+}, \mathbf{v}\_{a\_{j}^{+}}^{+}\right)\right), \left(\Delta^{-1}\left(\mathbf{s}\_{\psi\_{j}}, \rho\_{j}\right), \left(\mathbf{u}\_{a\_{j}^{+}}, \mathbf{v}\_{a\_{j}^{+}}\right)\right)\right)}{d\left(\left(\left(\Delta^{-1}\left(\mathbf{s}\_{\psi\_{j}}, \rho\_{j}\right)\right)^{+}, \left(\mathbf{u}\_{a\_{j}^{+}}^{+}, \mathbf{v}\_{a\_{j}^{+}}^{+}\right)\right), \left(\Delta^{-1}\left(\mathbf{s}\_{\psi\_{j}}, \rho\_{j}\right)\right)^{-}, \left(\mathbf{u}\_{a\_{j}^{+}}^{-}, \mathbf{v}\_{a\_{j}^{+}}^{-}\right)\right)}\right)}\tag{17}$$

$$P\_{i} = \max\left\{\omega\_{j}\frac{d\left(\left(\left(\Delta^{-1}\left(\mathbf{s}\_{\overline{\boldsymbol{\rho}}\_{j}},\boldsymbol{\rho}\_{\overline{\boldsymbol{\rho}}\_{j}}\right)\right)^{+},\left(\boldsymbol{u}\_{\boldsymbol{a}\_{j}^{+}}^{+}\boldsymbol{v}\_{\overline{\boldsymbol{a}}\_{j}^{+}}^{+}\right)\right)\left(\Delta^{-1}\left(\mathbf{s}\_{\overline{\boldsymbol{\rho}}\_{j}},\boldsymbol{\rho}\_{\overline{\boldsymbol{\rho}}\_{j}}\right),\left(\boldsymbol{u}\_{\boldsymbol{a}\_{j}^{-}},\boldsymbol{v}\_{\overline{\boldsymbol{a}}\_{j}}\right)\right)\right\}}{d\left(\left(\left(\Delta^{-1}\left(\mathbf{s}\_{\overline{\boldsymbol{\rho}}\_{j}},\boldsymbol{\rho}\_{\overline{\boldsymbol{\rho}}\_{j}}\right)\right)^{+},\left(\boldsymbol{u}\_{\boldsymbol{a}\_{j}^{-}}^{-}\left(\mathbf{s}\_{\overline{\boldsymbol{\rho}}\_{j}},\boldsymbol{\rho}\_{\overline{\boldsymbol{\rho}}\_{j}}\right)\right)\right)}\right)}\right\}.\tag{18}$$

Here, *d* denotes the normalized Hamming distance and <sup>ω</sup>*j* means the weight of attributes with these conditions, 0 ≤ <sup>ω</sup>*j* ≤ 1, *n j*=1 <sup>ω</sup>*j* = 1.

Step 5: Compute *Qi* values as follows:

$$Q\_{i} = v \frac{S\_{i} - S\_{i}^{\*}}{S\_{i}^{-} - S\_{i}^{\*}} + (1 - v) \frac{P\_{i} - P\_{i}^{\*}}{P\_{i}^{-} - P\_{i}^{\*}} \, \tag{19}$$

where

$$S\_i^\* = \min\_i S\_{i\prime} S\_i^- = \max\_i S\_{i\prime} \tag{20}$$

$$P\_i^\* = \min\_i P\_{i\prime} P\_i^- = \max\_i P\_{i\prime} \tag{21}$$

where *v* can be described as the decision-making mechanism coefficient. If *v* > 0.5, it denotes "the maximum group utility"; if *v* < 0.5, it denotes "the minimum regret", and if *v* = 0.5, it denotes "equality".

Step 6: According to the *Qi* values, the optimal decision among the rank alternatives is the alternative with the minimum *Q* value.

#### **4. Numerical Example and Comparative Analysis**

#### *4.1. Numerical Example*

In this section, we shall provide a numerical example to evaluate the human factors in the process of construction project managemen<sup>t</sup> by using the P2TL-VIKOR model. Assume that five possible construction projects <sup>Φ</sup>*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) are to be selected and there are four evaluation criteria <sup>τ</sup>*j*(*j* = 1, 2, 3, 4) to evaluate these construction projects: -1 τ1 is the workers' proficiency; -2 τ2 is the workers' safety awareness; -3 τ3 is the technical workers' quality; and -4 τ4 is the workers' emergency capacity. The five possible construction projects <sup>Φ</sup>*i*(*<sup>i</sup>* = 1, 2, 3, 4, 5) are to be evaluated through using P2TLNs with the four criteria by three experts, E*k* (expert's weight *w* = (0.29, 0.38, 0.33), which have an attributes weight of ω = (0.24, 0.17, 0.31, 0.28)*<sup>T</sup>*).

In order to carry out this evaluation, decision makers use language variables to express their evaluation, and the language variables of evaluation alternatives are shown in Table 1. The following steps are used to evaluate the human factors associated with the five construction projects using the proposed P2TL-VIKOR method:


**Table 1.** Linguistic variables and their fuzzy numbers.

Step 1. Construct the evaluation matrix *R*(3) = *<sup>r</sup>*3*ij*<sup>5</sup>×<sup>4</sup>(*<sup>i</sup>* = 1, 2, ... , 5, *j* = 1, 2, ... , 4) of each decision maker as in Tables 2–4. Based on Tables 1–4 and Equation (9), the group Pythagorean 2-tuple linguistic fuzzy decision matrix is computed. Table 5 shows the results.


**Table 2.** Rating alternatives on each criterion by *E*1.

**Table 3.** Rating alternatives on each criterion by *E*2.


**Table 4.** Rating alternatives on each criterion by *E*3.


**Table 5.** The group Pythagorean 2-tuple linguistic decision matrix *R*.


Step 2. Determine the *R*+*j*and *R*−*j* by Equations (13) and (14).

$$\begin{aligned} R\_{\bar{j}}^{+} &= \left\{ \begin{array}{l} \left\langle \left( \text{s}\_{4}, 0.06 \right), \left( 0.6705, 0.3706 \right) \right\rangle \\ \left\langle \left( \text{s}\_{4}, -0.24 \right), \left( 0.5935, 0.4297 \right) \right\rangle \\ \left\langle \left( \text{s}\_{4}, 0.38 \right), \left( 0.6426, 0.3646 \right) \right\rangle \\ \left\langle \left( \text{s}\_{5}, 0.24 \right), \left( 0.743, 0.296 \right) \right\rangle \end{array} \right\}^{/2} \\ R\_{\bar{j}}^{-} &= \left\{ \begin{array}{l} \left\langle \left( \text{s}\_{1}, 0.37 \right), \left( 0.3274, 0.7651 \right) \right\rangle \\ \left\langle \left( \text{s}\_{2}, 0.27 \right), \left( 0.4786, 0.6552 \right) \right\rangle \\ \left\langle \left( \text{s}\_{1}, 0.33 \right), \left( 0.2383, 0.7378 \right) \right\rangle \\ \left\langle \left( \text{s}\_{1}, 0.14 \right), \left( 0.3095, 0.8167 \right) \right\rangle \end{array} \right\}. \end{aligned}$$

Step 3. Compute *Si* and *Pi* values by Equations (17) and (18).

$$S\_1 = 0.5917, \text{ S}\_2 = 0.7793, \text{ S}\_3 = 0.5228, \text{ S}\_4 = 0.2563, \text{ S}\_5 = 0.4590,$$

$$P\_1 = 0.2186, \ P\_2 = 0.3100, \ P\_3 = 0.2790, \ P\_4 = 0.1362, \ P\_5 = 0.2724.$$

Step 4. Calculate *Qi* values as follows (Let *v* = 0.4):

$$Q\_1 = 0.5411, \ Q\_2 = 1.0000, \ Q\_3 = 0.6968, \ Q\_4 = 0.0000, \ Q\_5 = 0.6253.$$

Step 5. According to the *Qi* values, the optimal decision among the rank alternatives is the alternative with the minimum *Q* value: Φ4 > Φ1 > Φ5 > Φ3 > Φ2. Thus, the most optimal alternative is Φ4.

#### *4.2. Comparative Analyses*

In this part, we will make some comparative analyses to compare in our proposed P2TL-VIKOR model the P2TLWA and P2TLWG operators defined by Wei, Lu, Alsaadi, Hayat and Alsaedi [50], and the P2TL-TODIM method proposed by Huang and Wei [22].

The comparison results of different methods are as follows.

It is clear from Table 6 that the results are slightly different in ranking of alternatives, but the best alternative is always Φ4 by comparing the values of our proposed P2TL-VIKOR method with the P2TLWA/P2TLWG operators and the P2TL-TODIM model. Notably, in practical MADM problems, the P2TL-VIKOR method fully considers the conflict between attributes, which is more reasonable and scientific. All these methods have their good advantages: (1) P2TLWA operators emphasize the group influences; (2) the P2TLWG operator emphasizes individual influences; (3) the P2TL-TODIM method based on the prospect theory is exactly a kind of method that considers the influence of the experts' psychological behaviors factors on the decision results; and (4) the P2TL-VIKOR method takes into account the contradictory criteria such as the objectivity of decision makers and the complexity of the decision environment, so as to obtain more useful and scientific evaluation information.


**Table 6.** Rank of alternatives by using different methods.

## **5. Conclusions**

Human factors are not only the leading factors affecting the quality of construction projects, but also the most basic and core factors in the quality assurance system; so, the evaluation of human factors in construction projects is particularly critical. Human factor evaluation in construction projects is a MADM problem, and the information available for decision making is vague or uncertain in nature. Therefore, we used language variables to express the preferences of experts. The Pythagorean 2-tuple linguistic sets (P2TLSs) can well reflect uncertain or fuzzy information and solve these kinds of problems, and the original VIKOR is characterized by handling conflicting attributes. Naturally, we combined the Pythagorean 2-tuple linguistic sets with VIKOR, and the recommended method was systematically applied to the human factor evaluation of five construction projects to find an optimal construction project. The comparative study shows that the proposed MADM algorithm is feasible. This method is very effective and useful for decision making.

The main contributions of this study is fourfold: (1) the Pythagorean 2-tuple linguistic VIKOR (P2TL-VIKOR) method is designed to tackle the Pythagorean 2-tuple linguistic MAGDM issues; (2) a

case study for evaluating human factors in construction project managemen<sup>t</sup> is designed to show the developed approach; (3) some comparative studies are provided with the existing methods to give effect to the rationality of P2TL-VIKOR; and (4) the proposed method can also successfully contribute to the selection of suitable alternatives in other selection issues.

In the future, the proposed method can be expanded to deal with other decision-making issues, such as the selection of green suppliers, the location of waste disposal station, and so on.

**Author Contributions:** T.H., G.W. (Guiwu Wei), J.L., C.W. (Cun Wei) and R.L. conceived and worked together to achieve this work, T.H. compiled the computing program by Excel and analyzed the data, T.H. and G.W. (Guiwu Wei) wrote the paper. Finally, all the authors have read and approved the final manuscript.

**Funding:** The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China (14YJCZH091). The APC was funded by Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China (14YJCZH091).

**Conflicts of Interest:** The authors declare no conflicts of interest.
