**3. PHFS**

According to the PFS and UTHFS, we can define the PHFS that is composed by four membership functions, namely, positive, neutral, negative, and refusal membership functions. The four membership degrees are denoted by several values belonging to [0, 1], respectively, which can convey the hesitancy of decision makers.

**Definition 10.** *Let X be a non-empty and finite set, a PHFS N on X is defined by*

$$N = \{ \langle \mathbf{x}, \widetilde{\mu}(\mathbf{x}), \widetilde{\eta}(\mathbf{x}), \widetilde{v}(\mathbf{x}) \rangle | \mathbf{x} \in X \}, \tag{13}$$

*where μ*(*x*) = {*α*|*α* ∈ *μ*(*x*)}*, η*(*x*) = {*β*|*β* ∈ *η*(*x*)}*, and v*(*x*) = {*γ*|*γ* ∈ *v*(*x*)} *are three sets of several values in* [0, 1]*, representing the potential positive, neutral, and negative membership degrees. The degrees above satisfy the condition of* 0 ≤ *α*<sup>+</sup> + *β*<sup>+</sup> + *γ*<sup>+</sup> ≤ 1*, where α*<sup>+</sup> = <sup>∪</sup>*α*∈*μ*(*x*)max{*α*}*, β*<sup>+</sup> = <sup>∪</sup>*β*<sup>∈</sup>*η*(*x*)max{*β*}*, and γ*<sup>+</sup> = <sup>∪</sup>*γ*∈*v*(*x*)max{*γ*}*. For convenience, we call n* = {*μ*(*x*), *η*(*x*), *v*(*x*)} *is a PHFE, denoted by n* = {*μ*, *η*, *<sup>v</sup>*}.

During the process of applying the PHFEs to the practical MCDM problems, it is necessary to rank the PHFEs; thus, we develop the score and accuracy functions of PHFEs.

**Definition 11.** *Let n* = {*μ*, *η*, *v*} *be a PHFE, the numbers of values in μ*, *η*, *v are l*, *p*, *q, respectively. Thus, the score function is defined as*

$$s(\widehat{n}) = \left(1 + \frac{1}{l} \sum\_{i=1}^{l} \alpha\_i - \frac{1}{p} \sum\_{i=1}^{p} \beta\_i - \frac{1}{q} \sum\_{i=1}^{q} \gamma\_i\right) / 2, s(\widehat{n}) \in [0, 1]. \tag{14}$$

*the accuracy function is expressed as*

$$h(\boldsymbol{\tilde{n}}) = \frac{1}{l} \sum\_{i=1}^{l} \boldsymbol{a}\_{i} + \frac{1}{p} \sum\_{i=1}^{p} \beta\_{i} + \frac{1}{q} \sum\_{i=1}^{q} \gamma\_{i\prime} h(\boldsymbol{\tilde{n}}) \in [0, 1]. \tag{15}$$

Based on the score and accuracy values of PHFEs, we can determine the order relations between two PHFEs as in the following.

**Definition 12.** *Let n*1 = {*μ*1, *η*1, *<sup>v</sup>*1} *and n*2 = {*μ*2, *η*2, *<sup>v</sup>*2} *be two PHFEs, then*

*(1) If <sup>s</sup>*(*n*1) > *<sup>s</sup>*(*n*2)*, then n*1 > *<sup>n</sup>*2*;*

*(2) If <sup>s</sup>*(*n*1) = *<sup>s</sup>*(*n*2)*, then*


For example, let *n*1 = {{0.3, 0.4}, {0.2}, {0.2, 0.3}}± and *n*2 = {{0.3}, {0.2, 0.3}, {0.1, 0.2}} be two PHFEs, according to the Definition 11, we have *<sup>s</sup>*(*n*1) = *<sup>s</sup>*(*n*2) = 0.45, *h*(*n*1) = 0.4, and *h*(*n*2) = 0.35, then *n*1 > *<sup>n</sup>*2.

Inspired by the operational laws of PFNs and UTHFEs, i.e., the Definition 3 and 8, we propose the operational laws of PHFEs as follows.

**Definition 13.** *Let n* = {*μ*, *η*, *<sup>v</sup>*}*, n*1 = {*μ*1, *η*1, *<sup>v</sup>*1}*, and n*2 = {*μ*2, *η*2, *<sup>v</sup>*2} *be three PHFEs, λ* > 0*, and n<sup>c</sup> is the complementary set of n, and the operations of PHFEs are represented as* 

$$
\hat{n}^c = \underset{a \in \overline{\mu}, \beta \in \overline{\eta}, \gamma \in \overline{v}}{\cup} \{ \{\gamma\}, \{\beta\}, \{\alpha\} \}; \tag{16}
$$

$$\begin{aligned} \overline{u}\_1 \oplus \overline{v}\_2 = \left\{ \overline{\mu}\_1 \oplus \overline{\mu}\_2, \overline{\eta}\_1 \otimes \overline{\eta}\_2, \overline{v}\_1 \otimes \overline{v}\_2 \right\} = \underset{a\_1 \in \overline{\mu}\_1, \theta\_1 \in \overline{\eta}\_1, \gamma\_1 \in \overline{\eta}\_2, \theta\_2 \in \overline{\eta}\_2, \gamma\_2 \in \overline{\nu}\_2}{\left\{ \overline{\mu}\_1, \overline{\mu}\_1 \} \in \overline{\eta}\_1, \overline{\mu}\_1 \in \overline{\eta}\_2, \overline{\mu}\_2 \in \overline{\eta}\_2, \gamma\_2 \in \overline{\nu}\_2} \; \left\{ \left. \left\{ \overline{\mu}\_1, \overline{\mu}\_2 \right\} \right|, \left\{ \overline{\mu}\_1, \overline{\mu}\_2 \right\} \right\} \; \left\{ \left. \overline{\mu}\_2, \overline{\mu}\_1 \right\} \right\} \end{aligned} \tag{17}$$

$$\begin{aligned} \overline{n}\_1 \otimes \overline{n}\_2 = \{ \overline{\mu}\_1 \otimes \overline{\mu}\_2, \overline{\eta}\_1 \oplus \overline{\eta}\_2, \overline{\upsilon}\_1 \oplus \overline{\upsilon}\_2 \} = \underline{\lim}\_{\underline{a}\_1 \in \overline{\mu}\_1, \underline{b}\_1 \in \overline{\eta}\_1, \underline{\eta}\_1 \in \overline{\eta}\_2, \underline{b} \in \overline{\eta}\_2, \underline{\eta}\_2 \in \overline{\upsilon}\_2} \{ \{ \underline{a}\_1 \underline{a}\_2 \}, \{ \underline{\beta}\_1 + \underline{\beta}\_2 - \underline{\beta}\_1 \underline{\beta}\_2 \}, \{ \gamma\_1 + \gamma\_2 - \gamma\_1 \gamma\_2 \} \}; \tag{18} \end{aligned} \tag{19}$$

$$
\lambda \widetilde{\mathfrak{n}} =\_{a \in \overline{\mu}, \overline{\beta} \in \overline{\eta}, \gamma \in \overline{v}} \left\{ \left\{ 1 - \left( 1 - a \right)^{\lambda} \right\} , \left\{ \beta^{\lambda} \right\} , \left\{ \gamma^{\lambda} \right\} \right\} ; \tag{19}
$$

$$\hat{m}^{\lambda} = \underset{a \in \overline{\mu}, \beta \in \overline{\eta}, \gamma \in \overline{\nu}}{\text{cup}} \left\{ \left\{ a^{\lambda} \right\} / \left\{ 1 - (1 - \beta)^{\lambda} \right\} / \left\{ 1 - (1 - \gamma)^{\lambda} \right\} \right\}. \tag{20}$$

*For example, let n*1 = {{0.3, 0.4}, {0.2}, {0.2, 0.3}} *and n*2 = {{0.3}, {0.2, 0.3}, {0.1, 0.2}} *be two PHFEs, λ* = 2*, then*


Obviously, the following theorem can be obtained based on the Definition 13.

**Theorem 1.** *Let n* = {*μ*, *η*, *<sup>v</sup>*}*, n*1 = {*μ*1, *η*1, *<sup>v</sup>*1}*, and n*2 = {*μ*2, *η*2, *<sup>v</sup>*2} *be three PHFEs, λ*, *λ*1, *λ*2 > 0*, then*


$$(\mathcal{T}) \quad \left(\widetilde{\boldsymbol{\pi}}^{\lambda\_1}\right)^{\lambda\_2} = \widetilde{\boldsymbol{\pi}}^{\lambda\_1 \lambda\_2}.$$

#### **4. Generalized Picture Hesitant Fuzzy Aggregation Operators**

Combined with the concept and operations of PHFS, the GPHFWA, GPHFWG, GPHFPWA, and GPHFPWG operators are developed. Then, several properties of them are discussed, and some other aggregation operators under PHF environment that reduced by the proposed operators are presented.

#### *4.1. The GPHFWA Operator*

**Definition 14.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, the GPHFWA operator is a mapping* Ω*n* → Ω *as*

$$\text{GPHFWA}\_{\lambda}(\widetilde{n}\_{1}, \widetilde{n}\_{2}, \dots, \widetilde{n}\_{n}) = \left(w\_{1}\widetilde{n}\_{1}^{\lambda} \oplus w\_{2}\widetilde{n}\_{2}^{\lambda} \oplus \dots \oplus w\_{n}\widetilde{n}\_{n}^{\lambda}\right)^{1/\lambda} = \underset{j=1}{\underset{j=1}{\oplus}} \left(w\_{j}\widetilde{n}\_{j}^{\lambda}\right)^{1/\lambda}.\tag{21}$$

*where w* = (*<sup>w</sup>*1, *w*2,..., *wn*) *is the weight vector of PHFEs nj, and satisfies the conditions of wj* > 0 *and* ∑*nj*=<sup>1</sup> *wj* = 1.

Based on the Definition 13, we can obtain the theorems as follows.

**Theorem 2.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, then their aggregated value by using the GPHFWA operator is also a PHFE, and*

*GPHFWAλ*(*n*1, *<sup>n</sup>*2,..., *<sup>n</sup>n*) = ∪ *<sup>α</sup>*1∈*μ* 1,*α*2∈*μ* 2,...,*αn*∈*μ <sup>n</sup>*,*β*1∈*η*1,*β*2∈*η*2,...,*βn*∈*ηn*,*γ*1∈*v*1,*γ*2∈*v*2,...,*γn*∈*v<sup>n</sup>* ⎧⎨⎩⎧⎨⎩1 − *n*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *<sup>α</sup>j<sup>λ</sup>*.*wj*1/*λ*⎫⎬⎭ , ⎧⎨⎩1 − 1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>β</sup>j*.*<sup>λ</sup>wj*1/*λ*⎫⎬⎭, ⎧⎨⎩1 − 1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>γ</sup>j*.*<sup>λ</sup>wj*1/*λ*⎫⎬⎭⎫⎬⎭. (22)

**Proof.** See Appendix A. -

**Theorem 3.** *(Idempotency) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if all the PHFEs are equal, i.e., n j* = *n* = {*μ*, *η*, *<sup>v</sup>*}*, μ* = *α*, *η* = *β*, *v* = *γ, then*

$$
\mathbb{G}PHFWA\_{\lambda}(\tilde{n}\_1, \tilde{n}\_2, \dots, \tilde{n}\_n) = \tilde{n} = \{\tilde{\mu}, \tilde{\eta}, \tilde{\upsilon}\}.\tag{23}
$$

**Proof.** See Appendix B. -

**Theorem 4.** *(Boundedness) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if n*<sup>−</sup> = {{*α*−}, {*β*<sup>+</sup>}, {*γ*<sup>+</sup>}} *and n*<sup>+</sup> = {{*α*<sup>+</sup>}, {*β*−}, {*γ*−}}*, where α*<sup>−</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*min,*<sup>α</sup>j*/*, β*− = <sup>∪</sup>*βj*∈*ηj*min,*βj*/*, γ* − = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, α*<sup>+</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*max,*<sup>α</sup>j*/*, β*<sup>+</sup> = <sup>∪</sup>*βj*∈*ηj*max,*βj*/*, and γ*<sup>−</sup> = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, thus*

$$
\hat{\boldsymbol{n}}^{-} \le \text{GPHFVM}\_{\lambda}(\hat{\boldsymbol{n}}\_{1}, \hat{\boldsymbol{n}}\_{2}, \dots, \hat{\boldsymbol{n}}\_{n}) \le \hat{\boldsymbol{n}}^{+}.\tag{24}
$$

**Proof.** See Appendix C. -

**Theorem 5.** *(Monotonicity) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>n</sup>j*<sup>∗</sup>(*j* = 1, 2, . . . , *n*) *be two collections of PHFEs, if nj* ≤ *<sup>n</sup>j*<sup>∗</sup>*, then*

$$GPHFWA\_{\lambda}(\left[\vec{n}\_{1}, \vec{n}\_{2}, \dots, \vec{n}\_{n}\right] \leq GPHFWA\_{\lambda}(\left[\vec{n}\_{1} \; ^{\*}\right], \vec{n}\_{2} \; ^{\*}, \dots, \vec{n}\_{n} \; ^{\*}).\tag{25}$$

**Proof.** Theorem 5 can be obtained by the Theorem 4. -

Under some specific situations, we can obtain the reduced operators of the GPHFWA operator.

**Case 1.** If *λ* = 1, then the GPHFWA operator is reduced to the picture hesitant fuzzy weighted averaging (PHFWA) operator

$$PHFWA(\tilde{\mathfrak{n}}\_1, \tilde{\mathfrak{n}}\_2, \dots, \tilde{\mathfrak{n}}\_n) = \left(w\_1 \tilde{\mathfrak{n}}\_1 \oplus w\_2 \tilde{\mathfrak{n}}\_2 \oplus \dots \oplus w\_n \tilde{\mathfrak{n}}\_n\right) = \underset{j=1}{\stackrel{\text{\textquotedbl{}}}{\stackrel{\text{\textquotedbl{}}}{\stackrel{\text{\textquotedbl{}}}{\cdot}}} \left(w\_j \tilde{\mathfrak{n}}\_j\right). \tag{26}$$

**Case 2.** If *λ* = 1 and *w* = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, . . . , 1/*n*), then the GPHFWA operator is reduced to the picture hesitant fuzzy arithmetic averaging (PHFAA) operator

$$PHFAA(\widetilde{n}\_1, \widetilde{n}\_2, \dots, \widetilde{n}\_n) = \left(\frac{1}{n}\widetilde{n}\_1 \oplus \frac{1}{n}\widetilde{n}\_2 \oplus \dots \oplus \frac{1}{n}\widetilde{n}\_n\right). \tag{27}$$

#### *4.2. The GPHFWG Operator*

Similarly, the GPHFWG operator can be defined as in the following.

**Definition 15.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, the GPHFWG operator is a mapping* Ω*n* → Ω *as*

$$\text{GPHFWG}\_{\lambda}(\tilde{\mathbf{n}}\_{1}, \tilde{\mathbf{n}}\_{2}, \dots, \tilde{\mathbf{n}}\_{n}) = \frac{1}{\lambda} (\lambda \tilde{\mathbf{n}}\_{1}^{w\_{1}} \otimes \lambda \tilde{\mathbf{n}}\_{2}^{w\_{2}} \otimes \dots \otimes \lambda \tilde{\mathbf{n}}\_{n}^{w\_{n}}) = \frac{1}{\lambda} \underset{j=1}{\underset{j=1}{\odot}} \left(\lambda \tilde{\mathbf{n}}\_{j}^{w\_{j}}\right). \tag{28}$$

*where w* = (*<sup>w</sup>*1, *w*2,..., *wn*) *is the weight vector of PHFEs nj, and satisfies the conditions of wj* > 0 *and* ∑*nj*=<sup>1</sup> *wj* = 1.

According to the operational laws of PHFEs, the theorem can be obtained as follows.

**Theorem 6.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, then their aggregated value by using the GPHFWG operator is also a PHFE, and*

$$\begin{split} GPHFG\_{\Lambda}(\hat{\boldsymbol{\mu}}\_{1},\hat{\boldsymbol{\mu}}\_{2},\ldots,\hat{\boldsymbol{\mu}}\_{N}) &= \\ \max\_{\boldsymbol{\mu}\_{1}\in\{\boldsymbol{\mu}\_{2},\boldsymbol{\mu}\_{3}\in\{\hat{\boldsymbol{\mu}}\_{1},\boldsymbol{\mu}\_{2}\},\boldsymbol{\mu}\_{3}\in\{\hat{\boldsymbol{\mu}}\_{2},\boldsymbol{\mu}\_{3}\in\{\hat{\boldsymbol{\mu}}\_{3},\boldsymbol{\mu}\_{2}\in\{\hat{\boldsymbol{\mu}}\_{2},\boldsymbol{\mu}\_{3}\in\{\hat{\boldsymbol{\mu}}\_{3},\boldsymbol{\mu}\_{2}\}\}} \left\{ \left\{ 1 - \left( 1 - \prod\_{j=1}^{n} \left( 1 - (1 - a\_{j})^{\lambda} \right)^{\boldsymbol{\mu}\_{j}} \right)^{1/\lambda} \right\}, \left\{ \left( 1 - \prod\_{j=1}^{n} \left( 1 - \boldsymbol{\beta}\_{j}^{\lambda} \right)^{\boldsymbol{\mu}\_{j}} \right)^{1/\lambda} \right\}. \end{split} \tag{29}$$

It can be proven by the same process as Theorem 3–5 that the GPHFWG operator also has several properties.

**Theorem 7.** *(Idempotency) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if all the PHFEs are equal, i.e., nj* = *n* = {*μ*, *η*, *<sup>v</sup>*}*, μ* = *α*, *η* = *β*, *v* = *γ, then*

$$
\delta P HFWG\_{\lambda}(\tilde{\mu}\_1, \tilde{\mu}\_2, \dots, \tilde{\mu}\_n) = \tilde{\mu} = \{\tilde{\mu}, \tilde{\eta}, \tilde{\upsilon}\}.\tag{30}
$$

**Theorem 8.** *(Boundedness) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if n*<sup>−</sup> = {{*α*−}, {*β*<sup>+</sup>}, {*γ*<sup>+</sup>}} *and n*<sup>+</sup> = {{*α*<sup>+</sup>}, {*β*−}, {*γ*−}}*, where α*<sup>−</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*min,*<sup>α</sup>j*/*, β*− = <sup>∪</sup>*βj*∈*ηj*min,*βj*/*, γ* − = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, α*<sup>+</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*max,*<sup>α</sup>j*/*, β*<sup>+</sup> = <sup>∪</sup>*βj*∈*ηj*max,*βj*/*, and γ*<sup>−</sup> = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, thus*

$$
\widetilde{n}^- \le GPHFWG\_\lambda(\widetilde{n}\_1, \widetilde{n}\_2, \dots, \widetilde{n}\_n) \le \widetilde{n}^+.\tag{31}
$$

**Theorem 9.** *(Monotonicity) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>n</sup>j*<sup>∗</sup>(*j* = 1, 2, . . . , *n*) *be two collections of PHFEs, if nj* ≤ *<sup>n</sup>j*<sup>∗</sup>*, then*

$$GPHFWG\_{\lambda}(\tilde{n}\_1, \tilde{n}\_2, \dots, \tilde{n}\_n) \le GPHFWG\_{\lambda}(\tilde{n}\_1 \, ^\*, \tilde{n}\_2 \, ^\*, \dots, \tilde{n}\_n \, ^\*). \tag{32}$$

Several reduced operators of the GPHFWG operator are presented as: **Case 3.** If *λ* = 1, then the GPHFWG operator is reduced to the picture hesitant fuzzy weighted geometric (PHFWG) operator

$$PHFWG(\widetilde{\mathfrak{n}}\_1, \widetilde{\mathfrak{n}}\_2, \dots, \widetilde{\mathfrak{n}}\_n) = (\widetilde{\mathfrak{n}}\_1^{w\_1} \otimes \widetilde{\mathfrak{n}}\_2^{w\_2} \otimes \dots \otimes \widetilde{\mathfrak{n}}\_n^{w\_n}) = \bigotimes\_{j=1}^n (\widetilde{\mathfrak{n}}\_j^{w\_j}).\tag{33}$$

**Case 4.** If *λ* = 1 and *w* = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, . . . , 1/*n*), then the GPHFWG operator is reduced to the picture hesitant fuzzy geometric averaging (PHFGA) operator

$$PHFGA(\widetilde{n}\_1, \widetilde{n}\_2, \dots, \widetilde{n}\_n) = \left(\widetilde{n}\_1 \otimes \widetilde{n}\_2 \otimes \dots \otimes \widetilde{n}\_n\right)^{1/n}.\tag{34}$$

#### *4.3. The GPHFPWA Operator*

In real life, the criteria sometimes have different priority levels. For example, safety has a higher priority than price when a couple chooses a toy for their child. Obviously, the GPHFWA and GPHFWG operators cannot deal with this situation; then, the GPHFPWA and GPHFPWG operators are developed according to the PA operator proposed by Yager [31].

**Definition 16.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, the GPHFPWA operator is a mapping* Ω*n* → Ω *as*

$$\text{GPHFPWA}\_{\Lambda}(\widetilde{n}\_{1}, \widetilde{n}\_{2}, \dots, \widetilde{n}\_{n}) = \left(\frac{T\_{1}}{\sum\_{j=1}^{n} T\_{j}} \widetilde{n}\_{1} \, ^{\lambda} \oplus \frac{T\_{2}}{\sum\_{j=1}^{n} T\_{j}} \widetilde{n}\_{2} \, ^{\lambda} \oplus \dots \oplus \frac{T\_{n}}{\sum\_{j=1}^{n} T\_{j}} \widetilde{n}\_{n} \, ^{\lambda}\right)^{1/\lambda}.\tag{35}$$

*where Tj* = ∏*j*−<sup>1</sup> *k*=1 *<sup>s</sup>*(*nk*)(*j* = 2, . . . , *<sup>n</sup>*)*, T*1 = 1*, and <sup>s</sup>*(*nk*) *is the score value of PHFE <sup>n</sup>k*.

Similarly, the following theorem can be put forward.

**Theorem 10.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, then their aggregated value by using the GPHFPWA operator is also a PHFE, and*

$$\begin{split} \; \text{GPHF}[\text{F}\&\text{A}\_{\lambda}\left(\hat{\boldsymbol{\Pi}}\_{1},\hat{\boldsymbol{\Pi}}\_{2},\ldots,\hat{\boldsymbol{\Pi}}\_{N}\right) &= \\ \; \left. \; \text{G}\left[\text{F}\left(\boldsymbol{\Pi}\_{1},\ldots,\boldsymbol{\Pi}\_{N}\right)\right] \; \text{G}\left(\boldsymbol{\Pi}\_{2},\ldots,\boldsymbol{\Pi}\_{N}\right) \; \text{G}\left(\boldsymbol{\Pi}\_{2},\ldots,\boldsymbol{\Pi}\_{N}\right) \; \text{G}\left(\boldsymbol{\Pi}\_{N},\ldots,\boldsymbol{\Pi}\_{N}\right) \\ & \quad \; \left\{ \left\{ 1-\prod\_{j=1}^{n}\left(1-\lambda\_{j}\right)^{\frac{\mathsf{T}\_{j}}{\mathsf{T}\_{j-1}\mathsf{T}\_{j}}} \right\}^{1/\lambda} \right\} \; \left\{ \left\{ 1-\left(1-\prod\_{j=1}^{n}\left(1-\left(1-\beta\_{j}\right)^{\lambda}\right)^{\frac{\mathsf{T}\_{j}}{\mathsf{T}\_{j-1}\mathsf{T}\_{j}}}\right)^{1/\lambda} \right\}^{1/\lambda} \\ & \quad \; \left\{ 1-\left(1-\prod\_{j=1}^{n}\left(1-\left(1-\gamma\_{j}\right)^{\lambda}\right)^{\frac{\mathsf{T}\_{j}}{\mathsf{T}\_{j-1}\mathsf{T}\_{j}}}\right)^{1/\lambda} \right\}. \end{split} \tag{36}$$

The GPHFPWA operator also has the properties as follows.

**Theorem 12.** *(Idempotency) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if all the PHFEs are equal, i.e., n j* = *n* = {*μ*, *η*, *<sup>v</sup>*}*, μ* = *α*, *η* = *β*, *v* = *γ, then*

$$
\Gamma^{\rm PHFPWA}\_{\lambda}(\tilde{\boldsymbol{n}}\_1, \tilde{\boldsymbol{n}}\_2, \dots, \tilde{\boldsymbol{n}}\_n) = \tilde{\boldsymbol{n}} = \{\tilde{\boldsymbol{\mu}}, \tilde{\boldsymbol{\eta}}, \tilde{\boldsymbol{v}}\}.\tag{37}$$

**Theorem 13.** *(Boundedness) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, n*<sup>−</sup> = {{*α*−}, {*β*<sup>+</sup>}, {*γ*<sup>+</sup>}} *and n* + = {{*α*<sup>+</sup>}, {*β*−}, {*γ*−}}*, where α*<sup>−</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*min,*<sup>α</sup>j*/*, β*− = <sup>∪</sup>*βj*∈*ηj*min,*βj*/*, γ*<sup>−</sup> = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, α*<sup>+</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*max,*<sup>α</sup>j*/*, β*<sup>+</sup> = <sup>∪</sup>*βj*∈*ηj*max,*βj*/*, and γ*<sup>−</sup> = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, thus*

$$
\hat{n}^- \le \text{GPHFPVA}\_{\lambda}(\hat{n}\_1, \hat{n}\_2, \dots, \hat{n}\_n) \le \hat{n}^+.\tag{38}
$$

**Theorem 14.** *(Monotonicity) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>n</sup>j*<sup>∗</sup>(*j* = 1, 2, . . . , *n*) *be two collections of PHFEs, if nj* ≤ *<sup>n</sup>j*<sup>∗</sup>*, then*

$$GPHPVM\_{\lambda}(\tilde{\boldsymbol{n}}\_{1}, \tilde{\boldsymbol{n}}\_{2}, \dots, \tilde{\boldsymbol{n}}\_{n}) \leq GPHPWA\_{\lambda}(\tilde{\boldsymbol{n}}\_{1}^{\*}, \tilde{\boldsymbol{n}}\_{2}^{\*}, \dots, \tilde{\boldsymbol{n}}\_{n}^{\*}).\tag{39}$$

Then, the reduced operators of the GPHFPWA operator can be obtained.

**Case 5.** If *λ* = 1, then the GPHFPWA operator is reduced to the picture hesitant fuzzy prioritized weighted averaging (PHFPWA) operator

$$PHFPWA(\tilde{n}\_1, \tilde{n}\_2, \dots, \tilde{n}\_n) = \left(\frac{T\_1}{\sum\_{j=1}^n T\_j} \tilde{n}\_1 \oplus \frac{T\_2}{\sum\_{j=1}^n T\_j} \tilde{n}\_2 \oplus \dots \oplus \frac{T\_n}{\sum\_{j=1}^n T\_j} \tilde{n}\_n\right). \tag{40}$$

**Case 6.** If *λ* = 1 and the criteria are at the same priority, then the GPHFPWA operator is reduced to the PHFWA operator

$$PHFWA(\tilde{\mathfrak{n}}\_1, \tilde{\mathfrak{n}}\_2, \dots, \tilde{\mathfrak{n}}\_n) = (w\_1 \tilde{\mathfrak{n}}\_1 \oplus w\_2 \tilde{\mathfrak{n}}\_2 \oplus \dots \oplus w\_n \tilde{\mathfrak{n}}\_n) = \bigoplus\_{j=1}^{\mathfrak{N}} (w\_j \tilde{\mathfrak{n}}\_j). \tag{41}$$

**Case 7.** If *λ* = 1, *w* = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, . . . , 1/*n*), and the criteria are at the same priority, then the GPHFPWA operator is reduced to the PHFAA operator

$$PHFAA(\left|\vec{n}\_1, \vec{n}\_2, \dots, \vec{n}\_n\right\rangle = \left(\frac{1}{n}\vec{n}\_1 \oplus \frac{1}{n}\vec{n}\_2 \oplus \dots \oplus \frac{1}{n}\vec{n}\_n\right). \tag{42}$$

#### *4.4. The GPHFPWG Operator*

Similarly, the GPHFPWG operator is constructed as below.

**Definition 17.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, the GPHFPWG operator is a mapping* Ω*n* → Ω *as*

$$\text{GPHFPWG}\_{\Lambda}(\tilde{n}\_1, \tilde{n}\_2, \dots, \tilde{n}\_n) = \frac{1}{\Lambda} \left( (\lambda \tilde{n}\_1)^{\frac{\sum\_{j=1}^{T\_1} T\_j}{\sum\_{j=1}^{T\_1} \xi\_j}} \otimes (\lambda \tilde{n}\_2)^{\frac{T\_2}{\sum\_{j=1}^{T\_1} T\_j}} \otimes \dots \otimes (\lambda \tilde{n}\_n)^{\frac{T\_n}{\sum\_{j=1}^{T\_1} T\_j}} \right). \tag{43}$$

*where Tj* = ∏*j*−<sup>1</sup> *k*=1 *<sup>s</sup>*(*nk*)(*j* = 2, . . . , *<sup>n</sup>*)*, T*1 = 1*, and <sup>s</sup>*(*nk*) *is the score value of PHFE <sup>n</sup>k*.

Combined with the operations of PHFEs, the following theorems are obtained. **Theorem 15.** *Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, then their aggregated value by using the GPHFPWG operator is also a PHFE, and*

$$\begin{split} \; \text{GPHF}[\text{WG}\_{\lambda}(\hat{\mathbf{f}}\_{1}, \hat{\mathbf{f}}\_{2}, \dots, \hat{\mathbf{f}}\_{n}) &= \\ \; \text{Pr}\_{\lambda} \left\{ \text{Pr}\_{\lambda} \text{Pr}\_{\lambda} \left[ \text{Pr}\_{\lambda} \left\| \mathbf{f}\_{1} - \mathbf{f}\_{\lambda} \right\|\_{2} \left\| \mathbf{f}\_{1} - \mathbf{f}\_{\lambda} \right\|\_{2} \left\| \mathbf{f}\_{2} - \mathbf{f}\_{\lambda} \right\|\_{2} \left\| \mathbf{f}\_{2} - \mathbf{f}\_{\lambda} \right\|\_{2} \left\| \mathbf{f}\_{2} - \mathbf{f}\_{\lambda} \right\|\_{2} \left\| \mathbf{f}\_{1} - \mathbf{f}\_{\lambda} \right\|\_{2} \left\| \mathbf{f}\_{2} - \mathbf{f}\_{\lambda} \right\|\_{2} \\ & \left\{ \left( 1 - \prod\_{j=1}^{n} \left( 1 - \gamma\_{j}^{\lambda} \right)^{\frac{\eta\_{j}}{\eta\_{j} - 1}} \right)^{1/\lambda} \right\} \end{split} \tag{44}$$

**Theorem 16.** *(Idempotency) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if all the PHFEs are equal, i.e., nj* = *n* = {*μ*, *η*, *<sup>v</sup>*}*, μ* = *α*, *η* = *β*, *v* = *γ, then*

$$
\vec{\bf G}PHFPWG\_{\lambda}(\vec{n}\_1, \vec{n}\_2, \dots, \vec{n}\_n) = \vec{n} = \{\vec{\mu}, \vec{\eta}, \vec{v}\}.\tag{45}
$$

**Theorem 17.** *(Boundedness) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *be a collection of PHFEs, if n*<sup>−</sup> = {{*α*−}, {*β*<sup>+</sup>}, {*γ*<sup>+</sup>}} *and n*<sup>+</sup> = {{*α*<sup>+</sup>}, {*β*−}, {*γ*−}}*, where α*<sup>−</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*min,*<sup>α</sup>j*/*, β*− = <sup>∪</sup>*βj*∈*ηj*min,*βj*/*, γ* − = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, α*<sup>+</sup> = <sup>∪</sup>*<sup>α</sup>j*∈*μj*max,*<sup>α</sup>j*/*, β*<sup>+</sup> = <sup>∪</sup>*βj*∈*ηj*max,*βj*/*, and γ*<sup>−</sup> = <sup>∪</sup>*γj*∈*vj*min,*<sup>γ</sup>j*/*, thus*

$$
\hat{n}^- \le \text{GPHFPPWG}\_{\lambda}(\hat{n}\_1, \hat{n}\_2, \dots, \hat{n}\_n) \le \hat{n}^+.\tag{46}
$$

**Theorem 18.** *(Monotonicity) Let <sup>n</sup>j*(*j* = 1, 2, . . . , *n*) *and <sup>n</sup>j*<sup>∗</sup>(*j* = 1, 2, . . . , *n*) *be two collections of PHFEs, if nj* ≤ *<sup>n</sup>j*<sup>∗</sup>*, then*

$$\text{GPPFFWG}\_{\lambda}(\tilde{\boldsymbol{n}}\_{1}, \tilde{\boldsymbol{n}}\_{2}, \dots, \tilde{\boldsymbol{n}}\_{n}) \leq \text{GPPFFWG}\_{\lambda}(\tilde{\boldsymbol{n}}\_{1}^{\*}, \tilde{\boldsymbol{n}}\_{2}^{\*}, \dots, \tilde{\boldsymbol{n}}\_{n}^{\*}).\tag{47}$$

Several reduced operators of the GPHFPWG operator are presented as below:

**Case 8.** If *λ* = 1, then the GPHFPWG operator is reduced to the picture hesitant fuzzy prioritized weighted geometric (PHFPWG) operator

$$PHFP\mathsf{WG}(\widetilde{\boldsymbol{n}}\_{1}, \widetilde{\boldsymbol{n}}\_{2}, \dots, \widetilde{\boldsymbol{n}}\_{n}) = \left( \left( \widetilde{\boldsymbol{n}}\_{1} \right)^{\frac{\widetilde{\boldsymbol{n}}\_{1}}{\sum\_{j=1}^{n} \widetilde{\boldsymbol{T}}\_{j}}} \otimes \left( \widetilde{\boldsymbol{n}}\_{2} \right)^{\frac{\widetilde{\boldsymbol{n}}\_{2}}{\sum\_{j=1}^{n} \widetilde{\boldsymbol{T}}\_{j}}} \otimes \dots \otimes \left( \widetilde{\boldsymbol{n}}\_{n} \right)^{\frac{\widetilde{\boldsymbol{n}}\_{n}}{\sum\_{j=1}^{n} \widetilde{\boldsymbol{T}}\_{j}}} \right). \tag{48}$$

**Case 9.** If *λ* = 1 and the criteria are at the same priority, then the GPHFPWG operator is reduced to the PHFWG operator

$$PHFWWG(\widetilde{\mathfrak{n}}\_1, \widetilde{\mathfrak{n}}\_2, \dots, \widetilde{\mathfrak{n}}\_n) = \left(\widetilde{\mathfrak{n}}\_1^{w\_1} \otimes \widetilde{\mathfrak{n}}\_2^{w\_2} \otimes \dots \otimes \widetilde{\mathfrak{n}}\_n^{w\_n}\right) = \bigotimes\_{j=1}^n \left(\widetilde{\mathfrak{n}}\_j^{w\_j}\right). \tag{49}$$

**Case 10.** If *λ* = 1, *w* = (1/*<sup>n</sup>*, 1/*<sup>n</sup>*, . . . , 1/*n*) and the criteria are at the same priority, then the GPHFPWG operator is reduced to the PHFGA operator

$$PHFGA(\widetilde{n}\_1, \widetilde{n}\_2, \dots, \widetilde{n}\_n) = \left(\widetilde{n}\_1 \otimes \widetilde{n}\_2 \otimes \dots \otimes \widetilde{n}\_n\right)^{1/n}.\tag{50}$$

#### **5. MCDM Methods under PHF Environment**

We utilize the proposed operators to deal the different MCDM problems under PHF environment in this section. Let *A* = {*<sup>A</sup>*1, *A*2,..., *Am*} be a collection of alternatives and *C* = {*<sup>C</sup>*1, *C*2,..., *Cn*} be a set of criteria; decision maker evaluates the *m* alternatives concerning the *n* criteria by using the PHFEs. Thus, suppose that *N* = -*nij*.(*<sup>i</sup>* = 1, 2, . . . , *m*; *j* = 1, 2, . . . , *n*) is the PHF evaluation matrix, and *nij* = ,*μij*, *<sup>η</sup>ij*, *<sup>v</sup>ij*/ is the evaluation information when the alternative *Ai* is evaluated concerning the criteria *Cj*. In general, the criteria can be divided into two types in practice, namely, the cost criteria and benefit criteria; therefore, the evaluation information concerning the cost criteria should be transformed into the evaluation information concerning the benefit criteria to obtain the standardized PHF evaluation matrix *N* = -*nij*. as

$$
\overline{n}\_{\overline{i}\overline{j}} = \begin{cases}
\quad \widetilde{n}\_{\overline{i}j\prime} & \text{for the benefit criteria;} \\
\quad \left(\widetilde{n}\_{\overline{i}j}\right)^{c} & \text{for the cost criteria.}
\end{cases}
\tag{51}
$$

According to the aforementioned assumptions, when the criteria of a specific MCDM problem are in same priority level, and let *w* = (*<sup>w</sup>*1, *w*2,..., *wn*) be the weight vector of the criteria. We can construct a novel approach, i.e., Algorithm 1 to solve it based on the GPHFWA or the GPHFWG operator. The flow diagram of the Algorithm 1 is presented in Figure 1, and the ranking result can be obtained by the following steps.

**Figure 1.** Flow diagram of the Algorithm 1.

**Algorithm 1.** MCDM method based on the GPHFWA or the GPHFWG operator.

1: Normalize the PHF evaluation matrix *N* to obtain the standardized PHF evaluation matrix *N* combined with Equation (51).

2: Utilize the GPHFWA operator

*GPHFWAλ*(*ni*1, *ni*2,..., *nin*) = *<sup>w</sup>*1*ni*1*<sup>λ</sup>* ⊕ *<sup>w</sup>*2*ni*2*<sup>λ</sup>* ⊕···⊕ *wnnin<sup>λ</sup>*1/*<sup>λ</sup>* = *n*⊕*<sup>j</sup>*=<sup>1</sup>*wjnij<sup>λ</sup>*1/*<sup>λ</sup>* = ∪ *<sup>α</sup>i*1∈*μ i*1,*αi*2∈*μ i*2,...,*αin*∈*μ in*,*β<sup>i</sup>*1∈*ηi*1,*β<sup>i</sup>*2∈*ηi*2,...,*βin*∈*ηin*,*γi*1∈*vi*1,*γi*2∈*vi*2,...,*γin*∈*vin* ⎧⎨⎩⎧⎨⎩1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *<sup>α</sup>ij<sup>λ</sup>wj*1/*λ*⎫⎬⎭,⎧⎨⎩<sup>1</sup> − 1 − *n*∏*j*=151 − 1 − *<sup>β</sup>ij<sup>λ</sup>*6*wj*1/*λ*⎫⎬⎭, ⎧⎨⎩1 − 1 − *n*∏*j*=151 − 1 − *<sup>γ</sup>ij<sup>λ</sup>*6*wj*1/*λ*⎫⎬⎭⎫⎬⎭ (52)

or the GPHFWG operator

$$\begin{split}GPHWG\_{\lambda}\left(\mathfrak{R}\_{1},\mathfrak{R}\_{12},\ldots,\mathfrak{R}\_{10}\right) &= \frac{1}{\lambda}\left(\lambda\mathfrak{R}\_{11}^{un}\otimes\lambda\mathfrak{R}\_{22}^{un}\otimes\cdots\otimes\lambda\mathfrak{R}\_{10}^{un}\right) - \frac{1}{\lambda}\left(\lambda\mathfrak{R}\_{1j}^{un}\right) - \\ \mathfrak{R}\_{1i}\mathfrak{R}\_{1i}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\left\{\mathfrak{R}\_{11}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\left\{\mathfrak{R}\_{12}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\mathfrak{R}\_{2}\right\}\right\}\left\{\left\{1-\left(1-\prod\_{j=1}^{n}\left(1-\left(1-\overline{\pi}\_{ij}\right)^{1}\right)^{w\_{j}}\right\}^{1/\lambda}\right\},\left\{\left(1-\prod\_{j=1}^{n}\left(1-\overline{\pi}\_{ij}\right)^{w\_{j}}\right)^{1/\lambda}\right\},\end{split}\tag{53}$$

$$\left\{\left(1-\prod\_{j=1}^{n}\left(1-\overline{\pi}\_{ij}^{\lambda}\right)^{w}\right)^{1/\lambda}\right\}$$

to aggregate the standardized PHF evaluation matrix *N* to obtain the collective evaluation information of each alternative, i.e., *ni* = {*μi*, *ηi*, *<sup>v</sup>i*}.

3: Compute the score and accuracy values of each alternative using Equation (14) and (15). 4: Based on the comparison method of PHFEs, rank the alternatives.

When the criteria are in different priorities, we can solve the MCDM problem combined with the Algorithm 2 based on the GPHFPWA or the GPHFPWG operator. The flow diagram of Algorithm 2 is presented in Figure 2, and the ranking result can be obtained by the following steps.

**Figure 2.** Flow diagram of the Algorithm 2.

**Algorithm 2.** MCDM method based on the GPHFPWA or the GPHFPWG operator.

1: Normalize the PHF evaluation matrix *N* to obtain the standardized PHF evaluation matrix combined with Equation (51).

2: Compute the values of *Tij* using the equations as

$$T\_{lj} = \prod\_{k=1}^{j-1} s(\overline{u}\_{lk}), \\ T\_{l1} = 1. \tag{54}$$

3: Utilize the GPHFPWA operator

*GPHFPWAλ*(*ni*1, *ni*2,..., *nin*) = 5 *Ti*1 ∑*nj*=<sup>1</sup> *Tij ni*1*<sup>λ</sup>* ⊕ *Ti*2 ∑*nj*=<sup>1</sup> *Tij ni*2*<sup>λ</sup>* ⊕···⊕ *Tin* ∑*nj*=<sup>1</sup> *Tij nin<sup>λ</sup>*61/*<sup>λ</sup>* = ∪ *<sup>α</sup>i*1∈*μ i*1,*αi*2∈*μ i*2,...,*αin*∈*μ in*,*β<sup>i</sup>*1∈*ηi*1,*β<sup>i</sup>*2∈*ηi*2,...,*βin*∈*ηin*,*γi*1∈*vi*1,*γi*2∈*vi*2,...,*γin*∈*vin* ⎧⎨⎩⎧⎨⎩1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *<sup>α</sup>ij<sup>λ</sup> Tij* <sup>∑</sup>*nj*=<sup>1</sup> *Tij* 1/*<sup>λ</sup>*⎫⎬⎭,⎧⎪⎨⎪⎩<sup>1</sup> − ⎛⎝1 − *n*∏*j*=151 − 1 − *<sup>β</sup>ij<sup>λ</sup>*6 *Tij* <sup>∑</sup>*nj*=<sup>1</sup> *Tij* ⎞⎠1/*λ*⎫⎪⎬⎪⎭, ⎧⎪⎨⎪⎩1 − ⎛⎝1 − *n*∏*j*=151 − 1 − *<sup>γ</sup>ij<sup>λ</sup>*6 *Tij* <sup>∑</sup>*nj*=<sup>1</sup> *Tij* ⎞⎠1/*λ*⎫⎪⎬⎪⎭⎫⎪⎬⎪⎭ (55)

or the GPHFPWG operator

$$\begin{split} GPHPWG\_{\mathsf{k}}(\overline{\mathsf{a}}\_{1},\overline{\mathsf{a}}\_{2},\ldots,\overline{\mathsf{a}}\_{n}) &= \frac{1}{\mathsf{a}} \Bigg\{ (\mathsf{A}\mathsf{\mathsf{T}}\_{\Pi})^{\frac{\mathsf{T}\_{\Pi}}{\mathsf{a}\_{1}+1}} \otimes (\mathsf{A}\mathsf{\mathsf{T}}\_{\Pi})^{\frac{\mathsf{T}\_{\Pi}}{\mathsf{a}\_{1}+1}} \otimes \cdots \otimes (\mathsf{A}\mathsf{\mathsf{T}}\_{\Pi})^{\frac{\mathsf{T}\_{\Pi}}{\mathsf{a}\_{1}+1}} \} \\ \times \, e^{\mathsf{T}\_{\Pi}\overline{\mathsf{a}}\_{1}\cdot\mathsf{a}\_{2}\underline{\mathsf{a}}\_{2}\dots\mathsf{a}\_{n}\underline{\mathsf{a}}\_{n}} \, e^{\mathsf{T}\_{\Pi}\underline{\mathsf{a}}\_{1}\cdot\mathsf{a}\_{2}\underline{\mathsf{a}}\_{2}\cdot\mathsf{a}\_{3}\underline{\mathsf{a}}\_{3}\cdot\mathsf{a}\_{1}\underline{\mathsf{a}}\_{2}\cdot\mathsf{a}\_{3}\underline{\mathsf{a}}\_{3}\cdot\mathsf{a}\_{1}\underline{\mathsf{a}}\_{3}\cdot\mathsf{a}\_{1})} \, e^{\frac{\mathsf{T}\_{\Pi}}{\mathsf{a}\_{1}}} \left\{ \left\{ 1 - \prod\_{j=1}^{n} \left( 1 - \left( 1 - \mathsf{T}\_{\Pi} \right)^{\lambda} \right)^{\frac{\mathsf{T}\_{\Pi}}{\mathsf{a}\_{1}+1}} \right\}^{1/\lambda} \right\} \, \left\{ \left\{ 1 - \prod\_{j=1}^{n} \left( 1 - \overline{\mathsf{P}}\_{\Pi} \right)^{\frac{\mathsf{T}\_{\Pi}}{\mathsf{a}\_{1}+1}} \right\}^{1/\lambda} \right\} \, \left\{ \left\{$$

to aggregate the standardized PHF evaluation matrix *N* to obtain the collective evaluation information of each alternative, i.e., *ni* = {*μi*, *ηi*, *<sup>v</sup>i*}.

4: Compute the score and accuracy values of each alternative using Equation (14) and (15).

5: Based on the comparison method of PHFEs, rank the alternatives.

## **6. Numerical Examples**

We adopt two numerical examples of MCDM problems from the study of [25] and [34] and an application of web service selection [37] to show the feasibility and advantages of the proposed methods.
