**2. Preliminaries**

In this section, some basic definitions and operations related intuitionistic fuzzy sets and picture fuzzy sets are recalled briefly.

#### *2.1. Intuitionistic Fuzzy Sets*

Let *X* be universe of discourse. Then, an intuitionistic fuzzy set (IFS) [16] is defined as follows:

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

where *μ*ˆ *A* : *X* → [0, 1] and *<sup>ν</sup>*<sup>ˆ</sup>*A* : *X* → [0, 1] are called membership function and non-membership function of IFS A, respectively. Here 0 ≤ *μ*ˆ *A* + *<sup>ν</sup>*<sup>ˆ</sup>*A* ≤ 1 for all *x* ∈ *X* and *π* = 1 − (*μ*ˆ *A*(*x*) + *<sup>ν</sup>*<sup>ˆ</sup>*A*(*x*) is called degree of indeterminacy of *x* ∈ *X* in IFS *A*. The pair *μ*ˆ *A*, *<sup>ν</sup>*<sup>ˆ</sup>*A* is called intuitionistic fuzzy value (IFV) or intuitionistic fuzzy number (IFN) by Xu [31].

#### *2.2. Picture Fuzzy Sets*

Let *X* be a universe of discourse objects. A picture fuzzy set over *X*, denoted by *P*ˆ, is defined in [52,53] as follows:

$$\mathcal{P} = \{ \langle \Updownarrow \wp\_{\lozenge}(\mathtt{x}), \mathfrak{h}\_{\lozenge}(\mathtt{x}), \mathfrak{v}\_{\lozenge}(\mathtt{x}) \rangle : \mathtt{x} \in X \}, \tag{2}$$

where *μ*<sup>ˆ</sup>*P*<sup>ˆ</sup> : *X* → [0, 1], *η*ˆ*P*ˆ : *X* → [0, 1] and *ν*ˆ*P*ˆ : *X* → [0, 1] are called positive (neutral, negative)-degree of membership of picture fuzzy set *P* ˆ , respectively. Here 0 ≤ *<sup>μ</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>(*x*) + *<sup>η</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>(*x*) + *<sup>ν</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>(*x*) ≤ 1 for all *x* ∈ *X*. Besides, *<sup>π</sup>P*<sup>ˆ</sup>(*x*) denotes degree of refusal of *x* ∈ *X*, and is defined as *<sup>π</sup>P*<sup>ˆ</sup>(*x*) = 1 − (*μ*<sup>ˆ</sup>*P*<sup>ˆ</sup>(*x*) + *<sup>η</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>(*x*) + *ν* ˆ *P* <sup>ˆ</sup>(*x*)). The pair (*μ*<sup>ˆ</sup>*P*ˆ, *η*<sup>ˆ</sup>*P*ˆ, *<sup>ν</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>) is called picture fuzzy value (PFV) or picture fuzzy element (PFE).

**Definition 1** ([67])**.** *Let P* ˆ = (*μ*<sup>ˆ</sup>*P*ˆ, *η*<sup>ˆ</sup>*P*ˆ, *<sup>ν</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>) *be a PFN. Then, the score function S*ˆ *of PFN P*ˆ*, denoted by <sup>S</sup>*<sup>ˆ</sup>(*P*<sup>ˆ</sup>)*, is defined as follows:*

$$
\hat{S}(\hat{P}) = \hat{\mu}\_{\hat{P}} - \hat{\nu}\_{\hat{P}}, \ \hat{S}(\hat{P}) \in [-1, 1]. \tag{3}
$$

**Definition 2** ([67])**.** *Let P* ˆ = (*μ*<sup>ˆ</sup>*P*ˆ, *η*<sup>ˆ</sup>*P*ˆ, *<sup>ν</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>) *be a PFN. Then, the accuracy function H*ˆ *of PFN P*ˆ*, denoted by H* ˆ (*P*<sup>ˆ</sup>), *is defined as follows:*

$$
\hat{H}(\mathcal{P}) = \hat{\mu}\_{\mathcal{P}} + \hat{\eta}\_{\mathcal{P}} + \hat{\nu}\_{\mathcal{P}^\*} \hat{H}(\mathcal{P}) \in [0, 1]. \tag{4}
$$

*Here, the larger value of H*ˆ (*P*<sup>ˆ</sup>) *implies a greater degree of accuracy of the PFE P*ˆ = (*μ*<sup>ˆ</sup>*P*ˆ, *η*<sup>ˆ</sup>*P*ˆ, *<sup>ν</sup>*<sup>ˆ</sup>*P*<sup>ˆ</sup>)*.*

#### **3. Hamacher Operations (HOs) on the Picture Fuzzy Set**

## *3.1. Hamacher Operations*

The TN and TCN are useful notions in fuzzy set theory, that are used to define general union and intersection of fuzzy sets [68]. The definitions and conditions of TN and TCN are proposed by Roychowdhury and Wang [69]. The generalized union and generalized intersection of intuitionistic fuzzy sets based on TN and TCN were provided by Deschrijver and Kerre [70]. In 1978, Hamacher [33] introduced HOs known as Hamacher product () and Hamacher sum ( ), which are examples of TN and TCN, respectively. Hamacher TN and Hamacher TCN are provided in the following definition

$$T\_H(\mu, v) = \mu \bigotimes v = \frac{\mu v}{\xi + (1 - \xi)(\mu + v - \mu v)}\tag{5}$$

$$T\_H^\*(u,v) = u \bigoplus v = \frac{u+v-uv-(1-\overline{\zeta})uv}{1-(1-\overline{\zeta})uv}.\tag{6}$$

Usually, when *ξ* = 1, then Hamacher TN and TCN will reduce to the form

$$T\_H(\mu, \upsilon) = \mu \bigotimes \upsilon = \iota \upsilon \tag{7}$$

$$T\_H^\*(\iota, \upsilon) = \iota \bigoplus \upsilon = \iota + \upsilon - \iota \upsilon \tag{8}$$

which represent algebraic TN and TCN. When *ξ* = 2, then Hamacher TN and Hamacher TCN will conclude to the form

$$T\_H(u,v) = u \bigotimes v = \frac{uv}{1 + (1-u)(1-v)}\tag{9}$$

$$T\_H^\*(\mu, \upsilon) = \mu \bigoplus \upsilon = \frac{\mu + \upsilon}{1 + \mu \upsilon} \tag{10}$$

which are called Einstein TN and TCN respectively.

#### *3.2. Hamacher Operations(HOs) of Picture Fuzzy Set*

(1+(*ξ*−<sup>1</sup>)(<sup>1</sup>−*μ*ˆ1))*κ*+(*ξ*−<sup>1</sup>)(*μ*<sup>ˆ</sup>)*<sup>κ</sup>*

 ,

Here, given some Hamacher operations on PFNs which are provided by Wei [66]. Let *A* and *B* be two PFSs and *κ* > 0. Then, Hamacher product and Hamacher sum of the two PFSs *A* and *B* are denoted by (*p*ˆ1 *p*ˆ2) and (*p*ˆ1 *p*ˆ2), respectively, and defined by

$$\bullet \quad \not{p}\_1 \oplus \not{p}\_2 = \left( \frac{\not{\rho}\_1 + \not{\rho}\_2 - \not{\rho}\_1 \not{\rho}\_2 - (1 - \overline{\xi})\not{\rho}\_1 \not{\rho}\_2}{1 - (1 - \overline{\xi})\not{\rho}\_1 \not{\rho}\_2}, \frac{\not{\rho}\_1 \not{\rho}\_2}{\overline{\xi} + (1 - \overline{\xi})(\not{\rho}\_1 + \not{\rho}\_2 - \not{\rho}\_1 \not{\rho}\_2)}, \frac{\not{\rho}\_1 \not{\rho}\_2}{\overline{\xi} + (1 - \overline{\xi})(\not{\rho}\_1 + \not{\rho}\_2 - \not{\rho}\_1 \not{\rho}\_2)} \right)$$

$$\bullet \quad \not{p}\_1 \otimes \not{p}\_2 = \begin{pmatrix} \frac{\not{p}\_1 \not{p}\_2}{\xi + (1 - \xi)(\not{p}\_1 + \not{p}\_2 - \not{p}\_1 \not{p}\_2)}, \frac{\not{p}\_1 + \not{p}\_2 - \not{p}\_1 \not{p}\_2 - (1 - \xi)\not{p}\_1 \not{p}\_2}{1 - (1 - \xi)\not{p}\_1 \not{p}\_2}, \frac{\not{p}\_1 + \hat{\imath}\_2 - \hat{\imath}\_1 \hat{\imath}\_2 - (1 - \xi)\not{p}\_1 \not{p}\_2}{1 - (1 - \xi)\not{p}\_1 \not{p}\_2} \\\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \end{pmatrix}$$

$$\begin{array}{l} \bullet \quad \mathsf{x}\mathfrak{p}\_{1} = \left( \frac{(1+(\xi-1)\mathfrak{q}\_{1})^{\mathbf{x}} - (1-\mathfrak{q}\_{1})^{\mathbf{x}}}{(1+(\xi-1)\mathfrak{q}\_{1})^{\mathbf{x}} + (\xi-1)(1-\mathfrak{q}\_{1})^{\mathbf{x}}}, \frac{\mathfrak{z}(\mathfrak{q})^{\mathbf{x}}}{(1+(\xi-1)(1-\mathfrak{q}\_{1}))^{\mathbf{x}} + (\xi-1)(\mathfrak{q})^{\mathbf{x}}}, \frac{\mathfrak{z}(\mathfrak{q}\_{1})^{\mathbf{x}}}{(1+(\xi-1)(1-\mathfrak{q}\_{1}))^{\mathbf{x}} + (\xi-1)(\mathfrak{q}\_{1})^{\mathbf{x}}} \right), \mathfrak{x} > 0\\ \bullet \quad \mathfrak{p}\_{1}^{\mathbf{x}} = \left( \frac{\mathfrak{z}(\mathfrak{p})^{\mathbf{x}}}{(1+(\xi-1)(1-\mathfrak{q}\_{1}))^{\mathbf{x}} + (\xi-1)(\mathfrak{q})^{\mathbf{x}}}, \frac{(1+(\xi-1)\mathfrak{q}\_{1})^{\mathbf{x}} - (1-\mathfrak{q}\_{1})^{\mathbf{x}}}{(1+(\xi-1)\mathfrak{q}\_{1})^{\mathbf{x}} + (\xi-1)(\mathfrak{q}\_{1})^{\mathbf{x}} + (\xi-1)(\mathfrak{q}\_{1})^{\mathbf{x}}}, \frac{(1+(\xi-1)\mathfrak{q}\_{1})^{\mathbf{x}} - (1-\mathfrak{q}\_{1})^{\mathbf{x}}}{(1+(\xi-1)\mathfrak{q}\_{1})^{\mathbf{x}} + (\xi-1)(\mathfrak{q}\_{1})^{\mathbf{x}}} \right), \mathfrak{x} > 0. \end{array}$$

Now, we have drawn the attention on picture fuzzy Hamacher weighted averaing operator (PFHWA) and picture fuzzy Hamacher weighted geometric (PFHWG) operator introduced by Wei [66] that are as follows.

(1+(*ξ*−<sup>1</sup>)*η*ˆ1)*κ*+(*ξ*−<sup>1</sup>)(<sup>1</sup>−*η*ˆ1)*<sup>κ</sup>*

**Definition 3** ([66])**.** *Let p*ˆ*q* = (*μ*ˆ *q*, *η*<sup>ˆ</sup>*q*, *<sup>ν</sup>*<sup>ˆ</sup>*q*) (*q* = 1, 2, . . . ,*<sup>s</sup>*) *be several picture fuzzy number (PFNs).*

*A picture fuzzy Hamacher weighted average (PFHWA) operator is defined as a mapping from P* ˜ *s to P* ˜ *as follows:*

$$PFFHWA\_{\Psi}(\mathfrak{p}\_1, \mathfrak{p}\_2, \dots, \mathfrak{p}\_s) = \bigoplus\_{q=1}^s (\Psi\_q \mathfrak{p}\_q) \tag{11}$$

 , (1+(*ξ*−<sup>1</sup>)*ν*ˆ1)*κ*+(*ξ*−<sup>1</sup>)(<sup>1</sup>−*<sup>ν</sup>*ˆ1)*<sup>κ</sup>*

*where* Ψ = (<sup>Ψ</sup>1, Ψ2,..., <sup>Ψ</sup>*s*)*<sup>T</sup> is the weight vector of p*ˆ*q* (*q* = 1, 2, . . . ,*<sup>s</sup>*) *with* <sup>Ψ</sup>*q* > 0 *and s* ∑ *q*=1 <sup>Ψ</sup>*q* = 1*.*

Now, we considered two special cases subsequently for the PFHWA operator when the parameter *ξ* takes the values 1 or 2.

Case 1. If *ξ* = 1, then PFHWA operator will reduce to PFWA operator (Wei, 2017):

$$\begin{aligned} \langle PFWA\_{\mathbb{F}}(\mathfrak{p}\_{1}, \mathfrak{p}\_{2}, \dots, \mathfrak{p}\_{s}) &= \bigoplus\_{q=1}^{s} (\mathbb{Y}\_{q}\mathfrak{p}\_{q}) \\ &= \left(1 - \prod\_{q=1}^{s} (1 - \mathfrak{p}\_{q})^{\mathbb{Y}\_{q}} \prod\_{q=1}^{s} (\mathfrak{p}\_{q})^{\mathbb{Y}\_{q}} \prod\_{q=1}^{s} (\mathfrak{p}\_{q})^{\mathbb{Y}\_{q}}\right). \end{aligned}$$

Case 2. If *ξ* = 2, then PFHWA operator will reduce to picture fuzzy Einstein weighted averaging (PFEWA) operator:

*PFEWA*Ψ(*p*ˆ1, *p*ˆ2,..., *p*<sup>ˆ</sup>*s*) = - *s q*=1 (<sup>Ψ</sup>*q <sup>p</sup>*<sup>ˆ</sup>*q*) = *s* ∏ *q*=1 (1 + *μ*ˆ *q*)<sup>Ψ</sup>*<sup>q</sup>* − *s*∏ *q*=1 (1 − *μ*ˆ *q*)<sup>Ψ</sup> *s* ∏ *q*=1 (1 + *μ*ˆ *q*)<sup>Ψ</sup>*<sup>q</sup>* + *s*∏ *q*=1 (1 − *μ*ˆ *q*)<sup>Ψ</sup>*<sup>q</sup>* , 2 *s* ∏ *q*=1 (*η*<sup>ˆ</sup>*q*)<sup>Ψ</sup>*<sup>q</sup> s* ∏ *q*=1 (2 − *<sup>η</sup>*<sup>ˆ</sup>*q*)<sup>Ψ</sup>*<sup>q</sup>* + *s*∏ *q*=1 (*η*<sup>ˆ</sup>*q*)<sup>Ψ</sup>*<sup>q</sup>* , (12) 2 *s* ∏ *q*=1 (*ν*<sup>ˆ</sup>*q*)<sup>Ψ</sup>*<sup>q</sup> s* ∏ *q*=1 (2 − *<sup>ν</sup>*<sup>ˆ</sup>*q*)<sup>Ψ</sup>*<sup>q</sup>* + *s*∏ *q*=1 (*ν*<sup>ˆ</sup>*q*)<sup>Ψ</sup>*<sup>q</sup>* .

**Definition 4** ([66])**.** *Let p*ˆ*q* = (*μ*ˆ *q*, *η*<sup>ˆ</sup>*q*, *<sup>ν</sup>*<sup>ˆ</sup>*q*) *(q* = 1, 2, ... ,*<sup>s</sup>*) *be several PFNs. A picture fuzzy Hamacher weighted geometric (PFHWG) operator is defined as a mapping PFHWG* : *P*ˆ*s* → *P by* ˆ

$$\text{PFHWG}\_{\mathbb{F}}(\mathfrak{p}\_1, \mathfrak{p}\_2, \dots, \mathfrak{p}\_s) = \bigotimes\_{q=1}^s (\mathfrak{p}\_q)^{\mathbb{F}\_q} \tag{13}$$

*where* Ψ = (<sup>Ψ</sup>1, Ψ2,..., <sup>Ψ</sup>*s*)*<sup>T</sup> is the weight vector of p*ˆ*q* (*q* = 1, 2, . . . ,*<sup>s</sup>*) *such that* <sup>Ψ</sup>*q* > 0 *and s* ∑ *q*=1 <sup>Ψ</sup>*q* = 1*.*

Case 1. If *ξ* = 1, PFHWG operator reduces to picture fuzzy weighted geometric (PFWG) operator:

$$\begin{split} PFWG\_{\mathbb{F}}(\hat{\mathfrak{p}}\_{1}, \hat{\mathfrak{p}}\_{2}, \dots, \hat{\mathfrak{p}}\_{s}) &= \mathop{\otimes}\_{q=1}^{s} (\hat{\mathfrak{p}}\_{q})^{\mathbb{F}\_{q}} \\ &= \left( \prod\_{q=1}^{s} (\hat{\mathfrak{p}}\_{q})^{\mathbb{F}\_{q}}, 1 - \prod\_{q=1}^{s} (1 - \hat{\mathfrak{q}}\_{q})^{\mathbb{F}\_{q}}, 1 - \prod\_{q=1}^{s} (1 - \hat{\mathfrak{p}}\_{q})^{\mathbb{F}\_{q}} \right). \end{split} \tag{14}$$

Case 2. If *ξ* = 2, then PFHWG operator reduces to a picture fuzzy Einstein weighted geometric (PFEWG) operator:

$$PFEWG\_{\mathbb{F}}(\mathfrak{p}\_1, \mathfrak{p}\_{2'}, \dots, \mathfrak{p}\_s) = \bigotimes\_{q=1}^s (\mathfrak{p}\_q)^{\mathbb{F}\_q}$$

$$= \left( \frac{2\prod\_{q=1}^{s} (\boldsymbol{\hat{\mu}\_{q}})^{\mathbb{V}\_{q}}}{\prod\_{q=1}^{s} (\boldsymbol{2} - \boldsymbol{\hat{\mu}\_{q}})^{\mathbb{V}\_{q}} + \prod\_{q=1}^{s} (\boldsymbol{\hat{\mu}\_{q}})^{\mathbb{V}\_{q}}}, \frac{\prod\_{q=1}^{s} (\boldsymbol{1} + \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}} - \prod\_{q=1}^{s} (\boldsymbol{1} - \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}}}{\prod\_{q=1}^{s} (\boldsymbol{1} + \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}} + \prod\_{q=1}^{s} (\boldsymbol{1} - \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}}}, \frac{\prod\_{q=1}^{s} (\boldsymbol{1} + \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}} - \prod\_{q=1}^{s} (\boldsymbol{1} - \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}}}{\prod\_{q=1}^{s} (\boldsymbol{1} + \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}} + \prod\_{q=1}^{s} (\boldsymbol{1} - \boldsymbol{\hat{\eta}\_{q}})^{\mathbb{V}\_{q}}} \right).$$

#### **4. Model for MADM Using Picture Fuzzy Information**

To this part, multiple attribute decision making (MADM) method is proposed based on PFHA operators of which weights of attributes are real numbers and values of attributes are PFNs. To illustrate effectiveness of the proposed MADM method, an application in evaluation of enterprises performance under picture fuzzy information is given. Let *Q* = {*Q*1, *Q*2,..., *Qr*} be the discrete set of alternatives and *G* = {*<sup>G</sup>*1, *G*2,..., *Gs*} be the set of attributes.

Let Ψ = (<sup>Ψ</sup>1, Ψ2, ... , <sup>Ψ</sup>*s*) be the weight vector of the attribute such that Ψ*b* > 0 (*b* = 1, 2, ... ,*<sup>s</sup>*) and *s* ∑ *b*=1 Ψ*b* = 1, and *R* = (*μ*ˆ *ab*, *η*<sup>ˆ</sup>*ab*, *<sup>ν</sup>*<sup>ˆ</sup>*ab*)*r*×*s* be a picture fuzzy decision matrix. Here, *μ*ˆ *ab* is the degree of the positive membership for which alternative *Qa* satisfies the attribute *Gb* given by the decision makers, *η*<sup>ˆ</sup>*ab* denote the degree of neutral membership such that alternative *Qa* does not satisfy the attribute *Gb*, and *<sup>ν</sup>*<sup>ˆ</sup>*ab* provides the degree that the alternative *Qa* does not satisfy the attribute *Gb* given by the decision maker, where *μ*ˆ *ab* ⊂ [0, 1], *η*<sup>ˆ</sup>*ab* ⊂ [0, 1] and *<sup>ν</sup>*<sup>ˆ</sup>*ab* ⊂ [0, 1] such that 0 ≤ *μ*ˆ *ab* + *η*<sup>ˆ</sup>*ab* + *<sup>ν</sup>*<sup>ˆ</sup>*ab* ≤ 1, (*a* = 1, 2, . . . ,*<sup>r</sup>*) and (*b* = 1, 2, . . . ,*<sup>s</sup>*).

In the following algorithm, a MADM method using PFHWA and PFHWG operators is proposed to solve problems involving picture fuzzy information.

**Step 1.** Construction of decision matrix *R* by decision makers under PF-information:

$$R = \begin{pmatrix} \hat{\beta}\_{11} & \hat{\beta}\_{12} & \cdots & \hat{\beta}\_{1r} \\ \hat{\beta}\_{21} & \hat{\beta}\_{22} & \cdots & \hat{\beta}\_{2r} \\ \vdots & \vdots & \ddots & \vdots \\ \hat{\beta}\_{a1} & \hat{\beta}\_{s2} & \cdots & \hat{\beta}\_{ab} \end{pmatrix}$$

**Step 2.** Finding of values of *β* ˆ *a* (*a* = 1, 2, ...*<sup>r</sup>*) based on decision matrix *R*: These values are found by using PFHWA (or PFHWG) given as follow:

*β* ˆ *a* = *PFHWA*(*β*<sup>ˆ</sup> *a*1, *β*ˆ *a*2,..., *β*ˆ *ab*) = *s b*=1 (<sup>Ψ</sup>*bβ*<sup>ˆ</sup> *ab*) = *s* ∏ *b*=1 (1+(*ξ*−<sup>1</sup>)*μ*<sup>ˆ</sup> *<sup>b</sup>*)<sup>Ψ</sup>*b*− *s*∏ *b*=1 (<sup>1</sup>−*μ*<sup>ˆ</sup> *b*)<sup>Ψ</sup>*<sup>b</sup> s* ∏ *b*=1 (1+(*ξ*−<sup>1</sup>)*μ*<sup>ˆ</sup> *<sup>b</sup>*)<sup>Ψ</sup>*b*+(*ξ*−<sup>1</sup>) *s*∏ *b*=1 (<sup>1</sup>−*μ*<sup>ˆ</sup> *b*)<sup>Ψ</sup>*<sup>b</sup>* , *ξ s* ∏ *b*=1 (*η*<sup>ˆ</sup>*b*)<sup>Ψ</sup>*<sup>b</sup> s* ∏ *b*=1 (1+(*ξ*−<sup>1</sup>)(<sup>1</sup>−*η*<sup>ˆ</sup>*b*))<sup>Ψ</sup>*b*+(*ξ*−<sup>1</sup>) *s*∏ *b*=1 (*η*<sup>ˆ</sup>*b*)<sup>Ψ</sup>*<sup>b</sup>* , *ξ s* ∏ *b*=1 (*η*<sup>ˆ</sup>*b*)<sup>Ψ</sup>*<sup>b</sup> s* ∏ *b*=1 (1+(*ξ*−<sup>1</sup>)(<sup>1</sup>−*η*<sup>ˆ</sup>*b*))<sup>Ψ</sup>*b*+(*ξ*−<sup>1</sup>) *s*∏ *b*=1 (*η*<sup>ˆ</sup>*b*)<sup>Ψ</sup>*<sup>b</sup>* , (15)

$$\begin{split} \left(a = 1, 2, \ldots, r\right) \text{ or } \hat{\boldsymbol{\beta}}\_{a} &= PFHWG(\hat{\beta}\_{a1}, \hat{\beta}\_{b2}, \ldots, \hat{\beta}\_{ab}) = \underset{b=1}{\operatorname{\mathbf{\bar{S}}}} \left(\hat{\beta}\_{ab}\right)^{\mathbf{Y}\_{b}} \\ &= \left(\underset{\substack{b=1\\b=1\\b=1}}{\operatorname{\mathbf{\bar{s}}}} \left(1 + (\xi - 1)(1 - \mathfrak{f}\_{b1})\right)^{\mathbf{Y}\_{b}} + (\xi - 1)\prod\_{b=1}^{s} (\mathfrak{p}\_{b})^{\mathbf{Y}\_{b}}\right), \underset{b=1}{\operatorname{\mathbf{\bar{s}}}} \left(1 + (\xi - 1)\mathfrak{h}\_{b}\right)^{\mathbf{Y}\_{b}} - \underset{b=1}{\operatorname{\mathbf{\bar{s}}}} \left(1 - \mathfrak{h}\_{b}\right)^{\mathbf{Y}\_{b}} \\ &\overset{\mathbf{s}}{\prod}\_{q=1}^{s} (1 + (\xi - 1)\mathfrak{h}\_{b})^{\mathbf{Y}\_{b}} - \underset{b=1}{\prod} (1 - \mathfrak{h}\_{b})^{\mathbf{Y}\_{b}} \\ &\overset{\mathbf{s}}{\prod}\_{k=1}^{s} (1 + (\xi - 1)\mathfrak{h}\_{k})^{\mathbf{Y}\_{k}} + (\xi - 1)\prod\_{k=1}^{k} (1 - \mathfrak{h}\_{k})^{\mathbf{Y}\_{k}} \end{split} \tag{16}$$

(*a* = 1, 2, . . . ,*<sup>r</sup>*) to obtain the overall preference values *β*ˆ *a* (*a* = 1, 2, . . . ,*<sup>r</sup>*) of the alternative *Qr*.

**Step 3.** Calculate the score *S* <sup>ˆ</sup>(*β*<sup>ˆ</sup> *a*) (*a* = 1, 2, ... ,*<sup>r</sup>*) by using Equation (3) based on overall PF-information *β* ˆ *a* (*a* = 1, 2, ... ,*<sup>r</sup>*) in order to rank all the alternative *Qa* (*a* = 1, 2, ... ,*<sup>r</sup>*) to choose the best choice *Qa*. If score values of *S* <sup>ˆ</sup>(*β*<sup>ˆ</sup> *a*) and *S*<sup>ˆ</sup>(*β*<sup>ˆ</sup> *c*) are equal, accuracy degrees of *H*ˆ (*β*ˆ *a*) and *H*ˆ (*β*ˆ *c*) based on overall picture fuzzy information of *β* ˆ *a* and *β* ˆ *c* are calculated, and rank the alternative *Qa* depending with the accuracy of *H* ˆ (*β*ˆ *a*) and *H*ˆ (*β*ˆ *c*).

**Step 4.** To rank the alternatives *Qa* (*a* = 1, 2, ... ,*<sup>r</sup>*), choose the best one(s) in accordance with *S* <sup>ˆ</sup>(*β*<sup>ˆ</sup> *a*) (*a* = 1, 2, . . . ,*<sup>r</sup>*).

**Step 5.** Select the best alternative. **Step 6.** Stop.

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