A new multi-criteria decision-making method utilizing power heronian operators with picture hesitant fuzzy information

Abstract

In practice, picture hesitant fuzzy sets (PHFSs) combining the picture fuzzy sets (PFSs) and hesitant fuzzy sets (HFSs) are suitable to represent more complex multi-criteria decision-making (MCDM) information. The power heronian (PH) operators, which have the merits of power average (PA) and heronian mean (HM) operators, are extended to the environment of PHFSs in this article. First, some algebraic operations of picture hesitant fuzzy numbers (PHFNs), comparative functions and distance measure are introduced. Second, two novel operators, called as picture hesitant fuzzy weighted power heronian (PHFWPH) operator and picture hesitant fuzzy weighted geometric power heronian (PHFWGPH) operator, are defined. Meanwhile, some desirable characteristics and special instances of two operators are investigated as well. Third, a novel MCDM approach applying the proposed PH operators to handle PHFNs is explored. Lastly, to indicate the effectiveness of this novel method, an example regarding MCDM problem is conducted, as well as sensitivity and comparison analysis.

Keywords

Multi-criteria decision-making power heronian operators picture hesitant fuzzy sets

1 Introduction

In recent years, picture fuzzy sets (PFSs) initially conceived by Cuong [1], as an expansion of Atanassov’s intuitionistic fuzzy sets (IFSs) [2], have become an important research hotspot on multi-criteria decision-making (MCDM). PFSs can simultaneously depict different preferences of decision makers (DMs), including the membership degrees of positive, neutral, negative and refusal. Therefore, many achievements on PFSs [3 –10] have been conducted, such as the aggregation operators (AOs), distance measures, similarity coefficient, and so on.

In some cases, people may be hesitant when describing their evaluation results. Thus, Torra introduced hesitant fuzzy sets (HFSs) [11], which apply multiple values to convey the membership degree. As the increasing complex decision-making process, under PFSs environment, DMs may select several numbers to represent their possible opinions regarding a criterion. For instance, DMs may prefer to use 0.5 or 0.6 to express the positive degree, the negative degree is 0.2 or 0.3 and the neural degree is 0.1. This circumstance is beyond the expression of PFSs. Therefore, Wang and Li [12] initially defined the notion of picture hesitant fuzzy sets (PHFSs), and extended prioritized weighted AOs to PHFSs. Subsequently, Yang et al. [13] built up a group MCDM method to solve end-of-life vehicle problem under picture hesitant fuzzy environment. Ullah et al. [14] constructed a MCDM model using picture hesitant fuzzy information where the criteria weights are unknown. Mathew et al. [15] conceived of multi-granulation picture hesitant fuzzy rough sets by combing the multi-granulation rough sets and PHFSs, and also explored the correlations of them. Yang and Li [16] firstly proposed the definition of multiple-valued picture fuzzy linguistic set by considering the advantages of PHFSs and linguistic term set, and utilized generalized heronian mean operators to cope with MCDM problem.

In MCDM process, AOs, which can merge a series of assessment values into a global value, have become an important tool for information fusion. For example, Liu and Shi [17] explored heronian mean operators to handle MCDM under neutrosophic uncertain linguistic information. Ye [18] extended weighted arithmetic average and weighted geometric average operators to interval neutrosophic uncertain linguistic set. Li et al. [19] proposed normalized weighted Bonferroni mean (NWBM) operator according to Hamacher operation and applied to MCDM with multivalued neutrosophic linguistic information. Yang and Li [20] developed NWBM operator according to Dombi operations to multiple-valued neutrosophic uncertain linguistic sets. Besides AOs mentioned above, there are diverse operators to manage different fuzzy information [21 –29]. For PFSs, some AOs have also been applied to aggregate picture fuzzy numbers, including Dombi Heronian mean [30], Hamacher [31], Muirhead mean [32], Dombi [33], Bonferroni mean [34], Interaction Partitioned Heronian [35], and so on.

Power average (PA) operator, firstly conceived by Yager [36], is able to eliminate the impact of irrational asessment results given by DMs. Heronian mean (HM) operator introduced by Beliakov et al. [37] can reflect the correlations of assessment data and avoid the redundant computation compared with Bonferroni mean [38] operator. Inspired by the advantages of these two operators, Liu introduced the Power Heronian (PH) AOs by combining PA operator with HM operator to manage interval-valued intuitionistic fuzzy information [39], linguistic neutrosophic information [40] and [41]. Liu et al. [42] established PH AOs based on Dombi operations to cope with 2-tuple linguistic neutrosophic set. Shi et al. [43] explored power geometric heronian AOs to IFSs. Zhao et al. [44] investigated PH AOs to single-valued neutrosophic environment. Ju and Wang [45] extended power generalized Heronian AOs to hesitant fuzzy linguistic environment. Wang et al. [46] expanded PH operators to q-rung orthopair hesitant fuzzy information. Jiang et al. [47] utilized PH AOs to dispose of MCDM problem with interval-valued dual hesitant fuzzy numbers. Furthermore, Dordevic et al. [48] combined the full consistency approach with the rough PH AOs to select criteria of service quality in rail transport. Zhong et al. [49] applied PH AOs based on Dombi operations to q-rung orthopair fuzzy numbers. Bai et al. [50] extended power partitioned Heronian AOs to q-rung orthopair uncertain linguistic numbers.

It can be seen from the above discussion that PH AOs simultaneously considering the correlations among attributes and reducing the impact of irrational data have gained more attentions day by day. Nevertheless, to date, there is no achievement about expanding PH AOs to MCDM problem with picture hesitant fuzzy numbers (PHFNs) information, which is more appropriate for describing the hesitancy attitude of DMs than PFSs. Consequently, it is significant to explore PH AOs to manage MCDM problem with PHFNs information. Thus, the key goal of this paper is to provide a novel MCDM approach under PHFNs environment and expand PH AOs to PHFSs.

Thus, the remainder section of this article is arranged as below. In Section 2, the concept of PHFSs is recalled, and the comparison functions and distance measure of PHFNs are proposed. In Section 3, two novel PH AOs are established to fuse PHFNs, and their special properties and particular cases are investigated as well. In Section 4, a novel MCDM approach is developed, which is based on the PH AOs defined above. In Section 5, an instance is performed, along with the sensitivity analysis and comparison analysis with other studies. Eventually, some remarks are sum up In Section 6.

2 Picture hesitant fuzzy sets

Here, PHFSs and their algebraic operations are introduced, and the comparative method and distance measure are investigated as well.

2.1 PHFSs and operations

Definition 1. [12] A PHFSs B in Y is shown as $B = {〈 y, {\tilde{μ}}_{B} (y), {\tilde{η}}_{B} (y), {\tilde{ν}}_{B} (y) 〉 | y \in Y} .$ where ${\tilde{μ}}_{B} (y)$ , ${\tilde{η}}_{B} (y)$ and ${\tilde{ν}}_{B} (y)$ respectively represent three sets of possible positive, neutral and negative membership degrees, and each of them may contain multiple values in [0, 1], satisfying the condition below, 0 ⩽ μ⁺ + η⁺ + ν⁺ ⩽ 1. Where $μ^{+} = max {μ | μ \in {\tilde{μ}}_{B} (y)}, η^{+} = max {η | η \in {\tilde{η}}_{B} (y)}$ , and $ν^{+} = max {ν | ν \in {\tilde{ν}}_{B} (y)}$ .

If there is only one point in Y, then B is expressed as $B = 〈 {\tilde{μ}}_{B}, {\tilde{η}}_{B}, \tilde{$} C = {\tilde{μ}}_{c}, {\tilde{η}}_{c}, {\tilde{ν}}_{c} {nu}_{B} 〉$ , called as a picture hesitant fuzzy number (PHFN).

Definition 2. [12] Let $B = 〈 {\tilde{μ}}_{B}, {\tilde{η}}_{B}, {\tilde{ν}}_{B} 〉$ and $C = 〈 {\tilde{μ}}_{C}, {\tilde{η}}_{C}, {\tilde{ν}}_{C} 〉$ be two PHFNs, and γ > 0, and the algebraic operations of PHFNs are given by:

$(1) B \oplus C = 〈 \begin{matrix} ⋃_{\begin{matrix} μ_{B} \in {\tilde{μ}}_{B} \\ μ_{C} \in {\tilde{μ}}_{C} \end{matrix}} {μ_{B} + μ_{C} - μ_{B} μ_{C}}, \\ ⋃_{\begin{matrix} η_{B} \in {\tilde{η}}_{B} \\ η_{C} \in {\tilde{η}}_{C} \end{matrix}} {η_{B} η_{C}}, \\ ⋃_{\begin{matrix} ν_{B} \in {\tilde{ν}}_{B} \\ ν_{C} \in {\tilde{ν}}_{C} \end{matrix}} {ν_{B} ν_{C}} \end{matrix} 〉;$

$(2) B \otimes C = 〈 \begin{matrix} ⋃_{\begin{matrix} μ_{B} \in {\tilde{μ}}_{B} \\ μ_{C} \in {\tilde{μ}}_{C} \end{matrix}} {μ_{B} μ_{C}}, \\ ⋃_{\begin{matrix} η_{B} \in {\tilde{η}}_{B} \\ η_{C} \in {\tilde{η}}_{C} \end{matrix}} {η_{B} + η_{C} - η_{B} η_{C}}, \\ ⋃_{\begin{matrix} ν_{B} \in {\tilde{ν}}_{B} \\ ν_{C} \in {\tilde{ν}}_{C} \end{matrix}} {ν_{B} + ν_{C} - ν_{B} ν_{C}} \end{matrix} 〉;$

$(3) γ B = 〈 \begin{matrix} ⋃_{μ_{B} \in {\tilde{μ}}_{B}} {1 - (1 - μ_{B})^{γ}}, \\ ⋃_{η_{B} \in {\tilde{η}}_{B}} {η_{B}^{γ}}, \\ ⋃_{ν_{B} \in {\tilde{ν}}_{B}} {ν_{B}^{γ}} \end{matrix} 〉;$

$(4) B^{γ} = 〈 \begin{matrix} ⋃_{μ_{B} \in {\tilde{μ}}_{B}} {μ_{B}^{γ}}, \\ ⋃_{η_{B} \in {\tilde{η}}_{B}} {1 - (1 - η_{B})^{γ}}, \\ ⋃_{ν_{B} \in {\tilde{ν}}_{B}} {1 - (1 - ν_{B})^{γ}} \end{matrix} 〉$ .

2.2 Comparative method

Definition 3. Let $B = 〈 {\tilde{μ}}_{B}, {\tilde{η}}_{B}, {\tilde{ν}}_{B} 〉$ be a PHFN, then the score and accuracy functions, represented by s (B) and a (B), are listed below.

$(1) s (B) = \frac{1}{ℓ_{{\tilde{μ}}_{B}}} \sum_{μ_{B} \in {\tilde{μ}}_{B}} μ_{B} + \frac{1}{ℓ_{{\tilde{η}}_{B}}} \sum_{η_{B} \in {\tilde{η}}_{B}} (1 - η_{B}) + \frac{1}{ℓ_{{\tilde{ν}}_{B}}} \sum_{ν_{B} \in {\tilde{ν}}_{B}} (1 - ν_{B})$

$(2) a (B) = \frac{1}{ℓ_{{\tilde{μ}}_{B}}} \sum_{μ_{B} \in {\tilde{μ}}_{B}} μ_{B} + \frac{1}{ℓ_{{\tilde{η}}_{B}}} \sum_{η_{B} \in {\tilde{η}}_{B}} η_{B} + \frac{1}{ℓ_{{\tilde{ν}}_{B}}} \sum_{ν_{B} \in {\tilde{ν}}_{B}} ν_{B}$

Where $ℓ_{{\tilde{μ}}_{B}}$ , $ℓ_{{\tilde{η}}_{B}}$ and $ℓ_{{\tilde{ν}}_{B}}$ are the numbers of object in ${\tilde{μ}}_{B}$ , ${\tilde{η}}_{B}$ and ${\tilde{ν}}_{B}$ .

Theorem 1. Let $B = 〈 {\tilde{μ}}_{B}, {\tilde{η}}_{B}, {\tilde{ν}}_{B} 〉$ and $C = 〈 {\tilde{μ}}_{c}, {\tilde{η}}_{c}, {\tilde{ν}}_{c} 〉$ b e two PHFNs, then

(1) if s (B) > s (C), then B > C;

(2) if s (B) = s (C) and a (B) = a (C), then B = C;

(3) if s (B) > s (C) and a (B) = a (C) then B > C.

2.3 Distance measure

Definition 4. Let $B = 〈 {\tilde{μ}}_{B}, {\tilde{η}}_{B}, {\tilde{ν}}_{B} 〉$ and $C = 〈 {\tilde{μ}}_{c}, {\tilde{η}}_{c}, {\tilde{ν}}_{c} 〉$ be any two PHFNs, then the Hamming distance between B and C is provided. $d (B, C) = \frac{1}{2} (| \frac{1}{ℓ_{{\tilde{μ}}_{B}}} \sum_{μ_{B} \in {\tilde{μ}}_{B}} μ_{B} - \frac{1}{ℓ_{{\tilde{μ}}_{C}}} \sum_{μ_{C} \in {\tilde{μ}}_{C}} μ_{C} |$ $+ | \frac{1}{ℓ_{{\tilde{η}}_{B}}} \sum_{η_{B} \in {\tilde{η}}_{B}} η_{B} - \frac{1}{ℓ_{{\tilde{η}}_{C}}} \sum_{η_{C} \in {\tilde{η}}_{C}} η_{C} |$ $+ | \frac{1}{ℓ_{{\tilde{ν}}_{B}}} \sum_{ν_{B} \in {\tilde{ν}}_{B}} ν_{B} - \frac{1}{ℓ_{{\tilde{ν}}_{C}}} \sum_{ν_{C} \in {\tilde{ν}}_{C}} ν_{C} |)$

It can be easily proved that the Hamming distance mentioned above for PHFNs A, B and C meets characteristics below.

(1) d (B, B) =0;

(2) d (B, C) = d (C, B) , d (B, C)∈ [0, 1] ;

(3) d (A, B) + d (B, C) ⩾ d (A, C).

3 Novel power heronian operators

In the following, corresponding power heronian AOs for PNs are elored, and some interesting properties and particular instances are also presented.

3.1 Power and heronian operators

Definition 5. [36] Let β_g (g = 1, 2, …, h) be many real numbers, the form of PA operator is shown below: $PA (β_{1}, β_{2}, \dots, β_{h}) = \frac{\sum_{g = 1}^{h} (1 + ψ (β_{g})) β_{g}}{\sum_{f = 1}^{h} (1 + ψ (β_{f}))}$

Where $ψ (β_{g}) = \sum_{\begin{matrix} f = 1 \\ f \neq g \end{matrix}}^{h} sup (β_{g}, β_{f})$ , and sup(β_g, β_f) is the support for β_g from β_f. Satisfying the conditions below:

(1) sup(β_g, β_f)∈ [0, 1] ;

(2) sup(β_g, β_f) = sup(β_f, β_g) ;

(3) sup(β_g, β_f) ⩾ sup(β_m, β_n), if d (β_g, β_f) ⩽ d (β_m, β_n).

Definition 6. Let s, t ⩾ 0, and β_g (g = 1, 2, ⋯ , h) be many real numbers, HM operator [51] and geometric HM (GHM) operator [51] are shown below respectively: $\begin{matrix} {HM}^{s, t} (β_{1}, β_{2}, \dots, β_{h}) \\ = {(\frac{2}{h (h + 1)} \sum_{g = 1}^{h} \sum_{f = g}^{h} β_{g}^{s} β_{f}^{t})}^{\frac{1}{s + t}}; \end{matrix}$ $\begin{matrix} {GHM}^{s, t} (β_{1}, β_{2}, \dots, β_{h}) \\ = \frac{1}{s + t} \prod_{g = 1}^{h} \prod_{f = g}^{h} (s β_{g} + t β_{f})^{\frac{2}{h (h + 1)}} . \end{matrix}$

3.2 Picture hesitant fuzzy weighted power heronian operator

Definition 7. Let s, t ⩾ 0, and $B_{g} = 〈 {\tilde{μ}}_{B_{g}}, {\tilde{η}}_{B_{g}},$ ${\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and the importance of B_g (g = 1, 2, ⋯ , h) is denoted by ω_g. Satisfying ω_g ⩾ 0 (g = 1, 2, ⋯ , h) and the sum of ω_g is equal to 1, namely, $\sum_{g = 1}^{h} ω_{g} = 1$ . The picture hesitant fuzzy weighted power heronian (PHFWPH) operator is expressed as follows: $\begin{matrix} {PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) = \\ (\frac{2}{h (h + 1)} \oplus_{g = 1}^{h} \oplus_{f = g}^{h} {(\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))} B_{g})}^{s} \end{matrix}$ $\begin{matrix} {\otimes {(\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))} B_{f})}^{t})}^{\frac{1}{s + t}} \end{matrix}$ (1) where $ψ (B_{g}) = \sum_{f = 1 f \neq g}^{h} sup (B_{g}, B_{f})$ and sup(B_g, B_f) =1 - d (B_g, B_f).

Theorem 2. Let s, t ⩾ 0, and $B_{g} = 〈 {\tilde{μ}}_{B_{g}},$ ${\tilde{η}}_{B_{g}}, {\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and the importance of B_g (g = 1, 2, ⋯ , h) is denoted by ω_g. Satisfying ω_g ⩾ 0 (g = 1, 2, ⋯ , h) and $\sum_{g = 1}^{h} ω_{g} = 1$ . The result utilizing PHFWPH operator is still a PHFN, which is shown as below. ${PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 \cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - μ_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - {(1 - μ_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - η_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - ν_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$

Proof. Let $ς_{g} = \frac{ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}$ , then the formula (1) is transformed into the following form. ${PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) =$ ${(\frac{2}{h (h + 1)} \overset{\underset{h}{g = 1}}{\oplus} \overset{\underset{h}{f = g}}{\oplus} {(h ς_{g} B_{g})}^{s} \otimes {(h ς_{f} B_{f})}^{t})}^{\frac{1}{s + t}}$

According to algebraic orational rules of PHFNs provided in Definition 2, we get $h ς_{g} B_{g} = \begin{matrix} 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}} {1 - (1 - μ_{B_{g}})^{h ς_{g}}}, \\ ⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}}} {η_{B_{g}}^{h ς_{g}}}, \\ ⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}} {ν_{B_{g}}^{h ς_{g}}} 〉; \end{matrix} .$ $h ς_{f} B_{f} = \begin{matrix} 〈 ⋃_{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - (1 - μ_{B_{f}})^{h ς_{f}}}, \\ ⋃_{η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {η_{B_{f}}^{h ς_{f}}}, \\ ⋃_{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {ν_{B_{f}}^{h ς_{f}}} 〉; \end{matrix} .$ ${(h ς_{g} B_{g})}^{s} = \begin{matrix} 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}} {{(1 - (1 - μ_{B_{g}})^{h ς_{g}})}^{s}}, \\ ⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}}} {1 - (1 - η_{B_{g}}^{h ς_{g}})^{s}}, \\ ⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}} {1 - (1 - ν_{B_{g}}^{h ς_{g}})^{s}} 〉; \end{matrix}$ ${(h ς_{f} B_{f})}^{t} = \begin{matrix} 〈 ⋃_{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - (1 - μ_{B_{f}})^{h ς_{f}})}^{t}}, \\ ⋃_{η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - (1 - η_{B_{f}}^{h ς_{f}})^{t}}, \\ ⋃_{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - (1 - ν_{B_{f}}^{h ς_{f}})^{t}} 〉 \end{matrix}$ Then, ${(h ς_{g} B_{g})}^{s} \otimes {(h ς_{f} B_{f})}^{t} =$ Thus, $〈 \cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {{(1 - {(1 - μ_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - μ_{B_{f}})}^{h ς_{f}})}^{t}},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {\begin{matrix} (1 - {(1 - η_{B_{g}}^{h ς_{g}})}^{s}) + (1 - {(1 - η_{B_{f}}^{h ς_{f}})}^{t}) \\ - (1 - {(1 - η_{B_{g}}^{h ς_{g}})}^{s}) (1 - {(1 - η_{B_{f}}^{h ς_{f}})}^{t}) \end{matrix}},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {\begin{matrix} (1 - {(1 - ν_{B_{g}}^{h ς_{g}})}^{s}) + (1 - {(1 - ν_{B_{f}}^{h ς_{f}})}^{t}) \\ - (1 - {(1 - ν_{B_{g}}^{h ς_{g}})}^{s}) (1 - {(1 - ν_{B_{f}}^{h ς_{f}})}^{t}) \end{matrix}} 〉$ $= \cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {{(1 - {(1 - μ_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - μ_{B_{f}})}^{h ς_{f}})}^{t}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - η_{B_{g}}^{h ς_{g}})}^{s} {(1 - η_{B_{g}}^{h ς_{f}})}^{t}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - ν_{B_{g}}^{h ς_{g}})}^{s} {(1 - ν_{B_{f}}^{h ς_{f}})}^{t}} 〉$ $\overset{\underset{h}{g = 1}}{\oplus} \overset{\underset{h}{f = g}}{\oplus} {(h ς_{g} B_{g})}^{s} \otimes {(h ς_{f} B_{f})}^{t} =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - \prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - μ_{B_{g}})^{h ς_{g}})}^{s} {(1 - (1 - μ_{B_{f}})^{h ς_{f}})}^{t})},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{h ς_{g}})}^{s} {(1 - η_{B_{f}}^{h ς_{f}})}^{t})},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{h ς_{g}})}^{s} {(1 - ν_{B_{f}}^{h ς_{f}})}^{t})} 〉$ Then, $\frac{2}{h (h + 1)} \overset{\underset{h}{g = 1}}{\oplus} \overset{\underset{h}{f = g}}{\oplus} {(h ς_{g} B_{g})}^{s} \otimes {(h ς_{f} B_{f})}^{t}$ $= 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - μ_{B_{g}})^{h ς_{g}})}^{s} {(1 - (1 - μ_{B_{f}})^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {{(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{h ς_{g}})}^{s} {(1 - η_{B_{f}}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {{(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{h ς_{g}})}^{s} {(1 - ν_{B_{f}}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}}} 〉$ So, ${PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h})$ $= {(\frac{2}{h (h + 1)} \overset{\underset{h}{g = 1}}{\oplus} \overset{\underset{h}{f = g}}{\oplus} {(h ς_{g} B_{g})}^{s} \otimes {(h ς_{f} B_{f})}^{t})}^{\frac{1}{s + t}}$ $= 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - μ_{B_{g}})^{h ς_{g}})}^{s} {(1 - (1 - μ_{B_{f}})^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{h ς_{g}})}^{s} {(1 - η_{B_{f}}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{h ς_{g}})}^{s} {(1 - ν_{B_{f}}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$ $= \cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - μ_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - {(1 - μ_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - η_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - ν_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$

Consequently, Theorem 2 is provided.

Theorem 3. (Idempotency) Let $B_{g} = 〈 {\tilde{μ}}_{B_{g}}, {\tilde{η}}_{B_{g}},$ ${\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and ω = (ω₂, …, ω_h) ^T, satisfying $ω_{g} = \frac{1}{h} (g = 1, 2, \dots, h)$ and $\sum_{g = 1}^{h} ω_{g} = 1$ . IfB₁ = B₂ = ⋯ = B_g = B, then PHFWPH operator is degenerated to PHFPH operator, which has the property of idempotency. Thus, ${PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) = B .$

Proof. ${PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h})$ $= {PHFPH}^{s, t} (B, B, \dots, B)$ $\begin{matrix} = (\frac{2}{h (h + 1)} \overset{\underset{h}{g = 1}}{\oplus} \overset{\underset{h}{f = g}}{\oplus} {(h \frac{ω_{g} h}{\sum_{e = 1}^{h} ω_{e} h} B)}^{s} \end{matrix}$ $\begin{matrix} {\otimes {(h \frac{ω_{f} h}{\sum_{e = 1}^{h} ω_{e} h} B)}^{t})}^{\frac{1}{s + t}} \\ = {(\frac{2}{h (h + 1)} \overset{\underset{h}{g = 1}}{\oplus} \overset{\underset{h}{f = g}}{\oplus} B^{{s +}^{t}})}^{\frac{1}{s + t}} = B . \end{matrix}$

Theorem 4. (Boundedness) Let $B_{g} = 〈 {\tilde{μ}}_{B_{g}}, {\tilde{η}}_{B_{g}},$ ${\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and ω = (ω₁, ω₂, …, ω_h) ^T, satisfying ω_g = $\frac{1}{h} (g = 1, 2, \dots, h)$ and $\sum_{g = 1}^{h} ω_{g} = 1$ . B^- = min ${{\tilde{μ}}_{B_{g}}}, max {{\tilde{η}}_{B_{g}}}, max {{\tilde{ν}}_{B_{g}}}$ and $B^{+} = max {{\tilde{μ}}_{B_{g}}}, min {{\tilde{η}}_{B_{g}}}, min {{\tilde{ν}}_{B_{g}}}$ . Then, $B^{-} ⩽ {PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) ⩽ B^{+} .$

Proof. Since $ω_{g} = \frac{1}{h} (g = 1, 2, \dots, h)$ , then $ψ (B^{-}) = \sum_{\begin{matrix} f = 1 \\ f \neq g \end{matrix}}^{h} sup (B^{-}, B^{-}) = h - 1, ψ (B^{+}) = h - 1$ .

Since $ς_{g} = \frac{ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}$ , then, $ς_{B^{-}} = ς_{B^{+}} = \frac{1}{h}$ , hς_{B
^-} = hς_{B
⁺} = 1,

Thus, $B^{-} = {PHFWPH}^{s, t} (B^{-}, B^{-}, \dots, B^{-})$ $= 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - μ_{B_{g}}^{s} μ_{B_{f}}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}})}^{s} {(1 - η_{B_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}})}^{s} {(1 - ν_{B_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$ $= 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - μ_{B^{-^{s + t}}}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B^{-}})}^{s +^{t}}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B^{-}})}^{s +^{t}}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$ $⩽ {PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h})$ $B^{+} = {PHFWPH}^{s, t} (B^{+}, B^{+}, \dots, B^{+})$ $= 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - μ_{B^{+^{s + t}}}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B^{+}})}^{s +^{t}}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B^{+}})}^{s +^{t}}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$ $⩾ {PHFWPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h})$

Then, B^- ⩽ PHFWPH^s,t (B₁, B₂, ⋯ , B_h) ⩽ B⁺.

Thus, the proof of Theorem 4 is obtained.

Next, some particular instances of PHFWPH operator are explored.

(1) If s = 0, then the PHFWPH operator in Theorem 2 is degenerated to the following forms. ${PHFWPH}^{0, t} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - μ_{B_{f}})^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{t}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{t}}} 〉$

(2) If t = 0, then the PHFWPH operator in Theorem 2 is degenerated to the following forms. ${PHFWPH}^{s, 0} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - μ_{B_{g}})^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s}}},$

$⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s}}} 〉$

(3) If $s = t = \frac{1}{2}$ , then the PHFWPH operator in Theorem 2 is degenerated to the following forms. ${PHFWPH}^{\frac{1}{2}, \frac{1}{2}} (B_{1}, B_{2}, \dots, B_{h}) =$ $\cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - μ_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} {(1 - {(1 - μ_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}}))}^{\frac{2}{h (h + 1)}})},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {1 - (1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} {(1 - η_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}}))}^{\frac{2}{h (h + 1)}})},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {1 - (1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} {(1 - ν_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}}))}^{\frac{2}{h (h + 1)}})} 〉$

(4) If s = t = 1, then the PHFWPH operator in Theorem 2 is degenerated to the following forms. ${PHFWPH}^{1, 1} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 \cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - (1 - {(1 - μ_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}}) (1 - {(1 - μ_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{2}}},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - (1 - η_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}}) (1 - η_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{2}}},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - (1 - ν_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}}) (1 - ν_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{2}}} 〉$

3.3 Picture hesitant fuzzy weighted geometric power heronian operator

Definition 8. Let s, t ⩾ 0, and $B_{g} = 〈 {\tilde{μ}}_{B_{g}}, {\tilde{η}}_{B_{g}},$ ${\tilde{ν}}_{B_{g}} 〈 (g = 1, 2, \dots, h)$ be a group of PHFNs, and the importance of B_g (g = 1, 2, ⋯ , h) is denoted by ω_g. Satisfying ω_g ⩾ 0 (g = 1, 2, ⋯ , h) and the sum of ω_g is equal to 1, namely, $\sum_{g = 1}^{h} ω_{g} = 1$ . The picture hesitant fuzzy weighted geometric power heronian (PHFWGPH) operator is expressed as follows: ${PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) =$ $\frac{1}{s + t} \overset{\underset{h}{g = 1}}{\otimes} \overset{\underset{h}{f = g}}{\otimes} {(s_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}} \oplus t_{B_{f}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{2}{h (h + 1)}}$ (2)

where $ψ (B_{g}) = \sum_{\begin{matrix} f = 1 \\ f \neq g \end{matrix}}^{h} sup (B_{g}, B_{f})$ , and sup(B_g, B_f) = 1 - d (B_g, B_f).

Theorem 5. Let s, t ⩾ 0, and $B_{g} = 〈 {\tilde{μ}}_{B_{g}},$ ${\tilde{η}}_{B_{g}}, {\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and the importance of B_g (g = 1, 2, ⋯ , h) is denoted by ω_g. Satisfying ω_g ⩾ 0 (g = 1, 2, ⋯ , h) and $\sum_{g = 1}^{h} ω_{g} = 1$ . The global result aggregated by PHFWGPH operator is still a PHFN, wch is shown as below. ${PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) =$ $\cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - μ_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - η_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - {(1 - η_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - ν_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s} {(1 - {(1 - ν_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$

Proof. Let $ς_{g} = \frac{ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}$ , then the formula (2) is transformed into the following form. ${PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) =$ $\frac{1}{s + t} \overset{\underset{h}{g = 1}}{\otimes} \overset{\underset{h}{f = g}}{\otimes} {({sB}_{g}^{h ς_{g}} \oplus {tB}_{f}^{h ς_{f}})}^{\frac{2}{h (h + 1)}}$

According to algebraic operational rules of PHFNs provided in Definition 2, we get. $B_{g}^{h ς_{g}} = \begin{matrix} 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}} {μ_{B_{g}}^{h ς_{g}}}, \\ ⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}}} {1 - (1 - η_{B_{g}})^{h ς_{g}}}, \\ ⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}} {1 - (1 - ν_{B_{g}})^{h ς_{g}}} 〉; \end{matrix}$ $B_{f}^{h ς_{f}} = \begin{matrix} 〈 ⋃_{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {μ_{B_{f}}^{h ς_{f}}}, \\ ⋃_{η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - (1 - η_{B_{f}})^{h ς_{f}}}, \\ ⋃_{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - (1 - ν_{B_{f}})^{h ς_{f}}} 〉; \end{matrix}$

And, ${sB}_{g}^{h ς_{g}} = \begin{matrix} 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}} {1 - {(1 - μ_{B_{g}}^{h ς_{g}})}^{s}}, \\ ⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}}} {{(1 - (1 - η_{B_{g}})^{h ς_{g}})}^{s}}, \\ ⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}} {{(1 - (1 - ν_{B_{g}})^{h ς_{g}})}^{s}} 〉; \end{matrix}$ ${tB}_{f}^{h ς_{f}} = \begin{matrix} 〈 ⋃_{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - {(1 - μ_{B_{f}}^{h ς_{f}})}^{t}}, \\ ⋃_{η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {{(1 - (1 - η_{B_{f}})^{h ς_{f}})}^{t}}, \\ ⋃_{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {{(1 - (1 - ν_{B_{f}})^{h ς_{f}})}^{t}} 〉 . \end{matrix}$

Then, ${sB}_{g}^{h ς_{g}} \oplus {tB}_{f}^{h ς_{f}} =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - (1 - μ_{B_{g}}^{h ς_{g}})^{s} (1 - μ_{B_{f}}^{h ς_{f}})^{t}},$ $〈 \cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {{(1 - {(1 - η_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - η_{B_{f}})}^{h ς_{f}})}^{t}},$ $\cup_{\underset{ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}}{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}}}} {{(1 - {(1 - ν_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - ν_{B_{f}})}^{h ς_{f}})}^{t}} 〉$

Thus, $\overset{\underset{h}{g = 1}}{\otimes} \overset{\underset{h}{f = g}}{\otimes} ({sB}_{g}^{h ς_{g}} \otimes {tB}_{f}^{h ς_{f}}) =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{g}}^{h ς_{g}})}^{s} {(1 - μ_{B_{f}}^{h ς_{f}})}^{t})},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - \prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - η_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - η_{B_{f}})}^{h ς_{f}})}^{t})},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - \prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - ν_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - ν_{B_{f}})}^{h ς_{f}})}^{t})} 〉$

Then, $\overset{\underset{h}{g = 1}}{\otimes} \overset{\underset{h}{f = g}}{\otimes} {({sB}_{g}^{h ς_{g}} \otimes {tB}_{f}^{h ς_{f}})}^{\frac{2}{h (h + 1)}} =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{g}}^{h ς_{g}})}^{s} {(1 - μ_{B_{f}}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - η_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - η_{B_{f}})}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - ν_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - ν_{B_{f}})}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}}} 〉$

So, ${PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) = \frac{1}{s + t} \overset{\underset{h}{g = 1}}{\otimes} \overset{\underset{h}{f = g}}{\otimes} {({sB}_{g}^{h ς_{g}} \oplus {tB}_{f}^{h ς_{f}})}^{\frac{2}{h (h + 1)}}$ $= 〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{g}}^{h ς_{g}})}^{s} {(1 - μ_{B_{f}}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - η_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - η_{B_{f}})}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - ν_{B_{g}})}^{h ς_{g}})}^{s} {(1 - {(1 - ν_{B_{f}})}^{h ς_{f}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$

Consequently, Theorem 5 is provided.

Analogously, PHFGPH operator h the characteristics of idempotency and boundedness when all weights of criteria are equal.

Theorem 6. (Idempotency) Let $B_{g} = 〈 {\tilde{μ}}_{B_{g}}, {\tilde{η}}_{B_{g}},$ ${\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and ω = (ω₁, ω₂, …, ω_h) ^T, satisfying ω_g = $\frac{1}{h} (g = 1, 2, \dots, h)$ and $\sum_{g = 1}^{h} ω_{g} = 1$ . If B₁ =B₂ = ⋯ = B_g = B, then PHFWGPH operator is degenerated to PHFGPH operator, which has the property of idempotency. Thus, ${PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) = B .$

Proof. ${PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h})$ $= {PHFGPH}^{s, t} (B, B, \dots, B)$ $= \frac{1}{s + t} \overset{\overset{g = 1}{h}}{\otimes} \overset{\overset{f = g}{h}}{\otimes} {(s B^{h \frac{ω_{g} h}{\sum_{e = 1}^{h} ω_{e} h}} \oplus t B^{h \frac{ω_{f} h}{\sum_{e = 1}^{h} ω_{e} h}})}^{\frac{2}{h (h + 1)}}$ $= \frac{1}{s + t} \overset{\underset{h}{g = 1}}{\otimes} \overset{\underset{h}{f = g}}{\otimes} {(sB \oplus tB)}^{\frac{2}{h (h + 1)}} = B .$

Theorem 7. (Boundedness) Let $B_{g} = 〈 {\tilde{μ}}_{B_{g}}, {\tilde{η}}_{B_{g}},$ ${\tilde{ν}}_{B_{g}} 〉 (g = 1, 2, \dots, h)$ be a group of PHFNs, and ω = (ω₁, ω₂, …, ω_h) ^T, satisfying $ω_{g} = \frac{1}{h} (g = 1, 2, \dots, h)$ and $\sum_{g = 1}^{h} ω_{g} = 1$ . $B^{-} = 〈 min {{\tilde{μ}}_{B_{g}}}, max {{\tilde{η}}_{B_{g}}}, max {{\tilde{ν}}_{B_{g}}} 〉$ and $B^{+} = 〈 max {{\tilde{μ}}_{B_{g}}}, min {{\tilde{η}}_{B_{g}}}, min {{\tilde{ν}}_{B_{g}}} 〉$ then, $B^{-} ⩽ {PHFWGPH}^{s, t} (B_{1}, B_{2}, \dots, B_{h}) ⩽ B^{+} .$

Next, some particular instances of PHFWGPH operator are explored.

(1) If s = 0, then the PHFWGPH operator in Theorem 5 is degenerated to the following form of formula (3).

(2) If t = 0, then the PHFWGPH operator in Theorem 5 is degenerated to the following form of formula (4).

(3) If $s = t = \frac{1}{2}$ , then the PHFWGPH operator in Theorem 5 is degenerated to the following form of formula (5).

(4) If s = t = 1, then the PHFWGPH operator in Theorem 5 is degenerated to the following form of formula (6). ${PHFWGPH}^{0, t} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{t}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - η_{B_{f^{}}})^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{t}}},$

$⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - ν_{B_{f}})^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{t}}} 〉$ (3) ${PHFWGPH}^{s, 0} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - η_{B_{g}})^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s}}},$

$⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - ν_{B_{g}})^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{s}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s}}} 〉$ (4) ${PHFWGPH}^{\frac{1}{2}, \frac{1}{2}} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 ⋃_{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}} μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}} {{(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} {(1 - μ_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}}))}^{\frac{2}{h (h + 1)}}},$ $⋃_{η_{B_{g}} \in {\tilde{η}}_{B_{g}} η_{B_{f}} \in {\tilde{η}}_{B_{f}}} {1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (\begin{matrix} 1 - {(1 - (1 - η_{B_{g}})^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} \\ {(1 - (1 - η_{B_{f}})^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} \end{matrix}))}^{\frac{2}{h (h + 1)}}},$

$⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (\begin{matrix} 1 - {(1 - (1 - ν_{B_{g}})^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} \\ {(1 - (1 - ν_{B_{f}})^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})}^{\frac{1}{2}} \end{matrix}))}^{\frac{2}{h (h + 1)}}} 〉$ (5) ${PHFWGPH}^{1, 1} (B_{1}, B_{2}, \dots, B_{h}) =$ $〈 \cup_{\underset{μ_{B_{f}} \in {\tilde{μ}}_{B_{f}}}{μ_{B_{g}} \in {\tilde{μ}}_{B_{g}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - (1 - μ_{B_{g}}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}}) (1 - μ_{B_{f}}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{2}}},$ $\cup_{\underset{η_{B_{f}} \in {\tilde{η}}_{B_{f}}}{η_{B_{g}} \in {\tilde{η}}_{B_{g}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - (1 - {(1 - η_{B_{g}})}^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}}) (1 - {(1 - η_{B_{f}})}^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{2}}},$ $⋃_{ν_{B_{g}} \in {\tilde{ν}}_{B_{g}} ν_{B_{f}} \in {\tilde{ν}}_{B_{f}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - (1 - (1 - ν_{B_{g}})^{\frac{h ω_{g} (1 + ψ (B_{g}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}}) (1 - (1 - ν_{B_{f}})^{\frac{h ω_{f} (1 + ψ (B_{f}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ (B_{e}))}})))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{2}}} 〉$ (6)

4 Novel MCDM method based on PH operators

In this part, the proposed PH operators will be applied to solve MCDM problem with picture hesitant fuzzy information. Suppose X ={ X₁, X₂, ⋯ , X_p } be p alternatives and Y ={ Y₁, Y₂, ⋯ , Y_h } be h criteria. The importance of each criterion is represented by ω_g (g = 1, 2, ⋯ , h), satisfying $\sum_{g = 1}^{h} ω_{g} = 1$ and ω_g ⩾ 0. DMs provide the assessment values $β_{mg} = 〈 {\tilde{μ}}_{β_{mg}}, {\tilde{η}}_{β_{mg}}, {\tilde{ν}}_{β_{mg}} 〉$ (m = 1, 2, ⋯ , p ; g = 1, 2, ⋯ , h) of alternative X_m regarding criterion Y_g, which take the form of PHFN. The initial matrix is denoted as D = [β_mg] _p×h (m = 1, 2, ⋯ , p ; g = 1, 2, ⋯ , h).

The novel MCDM procedure using the PH operators is illustrated in Fig. 1.

Step 1. Transform the initial matrix.

In MCDM problem, the set of attributes commonly contain the benefit and cost criteria, and the cost criteria need to be converted into the benefit criteria. Thus, the initial matrix D = [β_mg] _p×h is transformed into the normalized picture hesitant fuzzy matrix $\tilde{D} = {[{\tilde{β}}_{mg}]}_{p \times h}$ based on the formula below. ${\tilde{β}}_{mg} = {\begin{matrix} 〈 {\tilde{μ}}_{β_{mg}}, {\tilde{η}}_{β_{mg}}, {\tilde{ν}}_{β_{mg}} 〉 for benefit criteria \\ 〈 {\tilde{ν}}_{β_{mg}}, {\tilde{η}}_{β_{mg}}, {\tilde{μ}}_{β_{mg}} 〉 for cost criteria \end{matrix}$ (7)

Here m = 1, 2, ⋯ , p ; g = 1, 2, ⋯ , h.

Step 2. Compute the supports $sup ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ . $sup ({\tilde{β}}_{mg}, {\tilde{β}}_{mf}) = 1 - d ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ (8)

Where m = 1, 2, ⋯ , p ; g, f = 1, 2, ⋯ , h, and $d ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ is the distance between ${\tilde{β}}_{mg}$ and ${\tilde{β}}_{mf}$ as given in Definition 4.

Fig. 1

The procedure of proposed algorithm.

Step 3. Compute the weigths ς_mg relating to the PHFN ${\tilde{β}}_{mg}$ . $ς_{mg} = \frac{ω_{g} (1 + ψ ({\tilde{β}}_{mg}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{me}))}$ (9)

Where ς_mg ⩾ 0, $\sum_{g = 1}^{h} ς_{mg} = 1$ , and $ψ ({\tilde{β}}_{mg}) = \sum_{\begin{matrix} f = 1 \\ f \neq g \end{matrix}}^{h} sup ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ , m = 1, 2, ⋯ , p ; g, f = 1, 2, ⋯ , h.

Step 4. Obtain the overall value ${\tilde{β}}_{m}$ of each alternative.

Aggregate each assessment value ${\tilde{β}}_{mg}$ of alternative X_m (m = 1, 2, ⋯ , p) associated with criterion Y_g (g = 1, 2, ⋯ , h) using PHFWPH or PHFWGPH operator, and the overall value ${\tilde{β}}_{m}$ is gained. The values are calculated based on formular (10) and formular (11) in the following.

Step 5. Compute the score value $s ({\tilde{β}}_{m})$ and accuracy value $a ({\tilde{β}}_{m})$ .

According to the formulas in Definition 3, the comparative function values of alternative X_m (m = 1, 2, ⋯ , p) are derived.

Step 6. Rank and choose alternative(s).

We can rank all alternative in terms of their function values. The alternative providing the maximize score value is the best one. ${\tilde{β}}_{m} = {PHFWPH}^{s, t} ({\tilde{β}}_{m_{1}}, {\tilde{β}}_{m_{2}}, \dots, {\tilde{β}}_{m_{h}}) =$ $\cup_{\underset{μ_{{\tilde{β}}_{m_{f}}} \in {\tilde{μ}}_{{\tilde{β}}_{m_{f}}}}{μ_{{\tilde{β}}_{m_{g}}} \in {\tilde{μ}}_{{\tilde{β}}_{m_{g}}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - μ_{{\tilde{β}}_{m_{g}}})}^{\frac{h ω_{g} (1 + ψ ({\tilde{β}}_{m_{g}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{m_{e}}))}})}^{s} {(1 - {(1 - μ_{{\tilde{β}}_{m_{f}}})}^{\frac{h ω_{f} (1 + ψ ({\tilde{β}}_{m_{f}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{m_{e}}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{η_{{\tilde{β}}_{m_{f}}} \in {\tilde{η}}_{{\tilde{β}}_{m_{f}}}}{η_{{\tilde{β}}_{m_{g}}} \in {\tilde{η}}_{{\tilde{β}}_{m_{g}}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - η_{{\tilde{β}}_{m_{g}}}^{\frac{h ω_{g} (1 + ψ ({\tilde{β}}_{m_{g}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{m_{e}}))}})}^{s} {(1 - η_{{\tilde{β}}_{m_{f}}}^{\frac{h ω_{f} (1 + ψ ({\tilde{β}}_{m_{f}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{m_{e}}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $⋃_{ν_{{\tilde{β}}_{m_{g}}} \in {\tilde{ν}}_{{\tilde{β}}_{m_{g}}} ν_{{\tilde{β}}_{m_{f}}} \in {\tilde{ν}}_{{\tilde{β}}_{m_{f}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - ν_{{\tilde{β}}_{m_{g}}}^{\frac{h ω_{g} (1 + ψ ({\tilde{β}}_{m_{g}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{m_{e}}))}})}^{s} {(1 - ν_{{\tilde{β}}_{m_{f}}}^{\frac{h ω_{f} (1 + ψ ({\tilde{β}}_{m_{f}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{β}}_{m_{e}}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉,$ (10) or ${\tilde{β}}_{m} = {PHFWGPH}^{s, t} ({\tilde{β}}_{m_{1}}, {\tilde{β}}_{m_{2}}, \dots, {\tilde{β}}_{m_{h}}) =$ $\cup_{\underset{μ_{{\tilde{β}}_{m_{f}}} \in {\tilde{μ}}_{{\tilde{β}}_{m_{f}}}}{μ_{{\tilde{β}}_{m_{g}}} \in {\tilde{μ}}_{{\tilde{β}}_{m_{g}}}}} {1 - {(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - μ_{{\tilde{β}}_{m_{g}}}^{\frac{h ω_{g} (1 + ψ ({\tilde{μ}}_{m_{g}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{μ}}_{m_{e}}))}})}^{s} {(1 - μ_{{\tilde{β}}_{m_{f}}}^{\frac{h ω_{f} (1 + ψ ({\tilde{μ}}_{m_{f}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{μ}}_{m_{e}}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}},$ $\cup_{\underset{η_{{\tilde{β}}_{m_{f}}} \in {\tilde{η}}_{{\tilde{β}}_{m_{f}}}}{η_{{\tilde{β}}_{m_{g}}} \in {\tilde{η}}_{{\tilde{β}}_{m_{g}}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - {(1 - η_{{\tilde{β}}_{m_{g}}})}^{\frac{h ω_{g} (1 + ψ ({\tilde{η}}_{m_{g}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{η}}_{m_{e}}))}})}^{s} {(1 - {(1 - η_{{\tilde{β}}_{m_{f}}})}^{\frac{h ω_{f} (1 + ψ ({\tilde{η}}_{m_{f}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{η}}_{m_{e}}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}}$ $⋃_{ν_{{\tilde{β}}_{m_{g}}} \in {\tilde{ν}}_{{\tilde{β}}_{m_{g}}} ν_{{\tilde{β}}_{m_{f}}} \in {\tilde{ν}}_{{\tilde{β}}_{m_{f}}}} {{(1 - {(\prod_{g = 1}^{h} \prod_{f = g}^{h} (1 - {(1 - (1 - ν_{{\tilde{β}}_{m_{g}}})^{\frac{h ω_{g} (1 + ψ ({\tilde{ν}}_{m_{g}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{ν}}_{m_{e}}))}})}^{s} {(1 - (1 - ν_{{\tilde{β}}_{m_{f}}})^{\frac{h ω_{f} (1 + ψ ({\tilde{ν}}_{m_{f}}))}{\sum_{e = 1}^{h} ω_{e} (1 + ψ ({\tilde{ν}}_{m_{e}}))}})}^{t}))}^{\frac{2}{h (h + 1)}})}^{\frac{1}{s + t}}} 〉$ (11)

5 Instance and analysis

In this part, an instance is explored to verify the practicability of the MCDM approach mentioned above. Meanwhile, we provide sensitivity and comparison analysis to show the effectiveness of the novel method.

5.1 Example

A company wants to choose the most suitable investment project from four alternatives, which are expressed as X_i (i = 1, 2, 3, 4). There are three criteria, namely, risk (Y₁), growth (Y₂) and environmental impact (Y₃), and the corresponding weight of each criterion is ω = (0.35, 0.25, 0.4). The assessment values decided by DMs take the form of PHFN, which is represented as $β_{mg} = 〈 {\tilde{μ}}_{β_{mg}}, {\tilde{η}}_{β_{mg}}, {\tilde{ν}}_{β_{mg}} 〉 (m = 1, 2, 3, 4; g = 1, 2, 3)$ . The initial decision matrix D = [β_mg] _4×3 is provided as follows. $\begin{matrix} D = \\ [\begin{matrix} 〈 (\begin{matrix} {0.4, 0.5}, \\ {0.2}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.4}, \\ {0.2, 0.3}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.2}, \\ {0.2}, \\ {0.5} \end{matrix}) 〉 \\ 〈 (\begin{matrix} {0.6}, \\ {0.1, 0.2}, \\ {0.2} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.6}, \\ {0.1}, \\ {0.2} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.5} \\ {0.2}, \\ {0.1, 0.2} \end{matrix}) 〉 \\ 〈 (\begin{matrix} {0.3, 0.4}, \\ {0.2}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.5}, \\ {0.2}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.5}, \\ {0.2, 0.3}, \\ {0.2} \end{matrix}) 〉 \\ 〈 (\begin{matrix} {0.7}, \\ {0.1, 0.2}, \\ {0.1} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.6}, \\ {0.1}, \\ {0.2} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.4}, \\ {0.3}, \\ {0.2} \end{matrix}) 〉 \end{matrix}] \end{matrix}$ Then, we will utilize the novel MCDM approach proposed above to solve this actual example. For simplicity, let s = t = 1.

Step 1. Transform the initial matrix.

Because the criterion Y₃ is the cost type, the normalized evaluation PHFN matrix $\tilde{D} = {[{\tilde{β}}_{mg}]}_{4 \times 3}$ is gained based on the formula (3). $\begin{matrix} \tilde{D} = \\ [\begin{matrix} 〈 (\begin{matrix} {0.4, 0.5}, \\ {0.2}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.4}, \\ {0.2, 0.3}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.5}, \\ {0.2}, \\ {0.2} \end{matrix}) 〉 \\ 〈 (\begin{matrix} {0.6}, \\ {0.1, 0.2}, \\ {0.2} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.6}, \\ {0.1}, \\ {0.2} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.1, 0.2} \\ {0.2}, \\ {0.5} \end{matrix}) 〉 \\ 〈 (\begin{matrix} {0.3, 0.4}, \\ {0.2}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.5}, \\ {0.2}, \\ {0.3} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.2}, \\ {0.2, 0.3}, \\ {0.5} \end{matrix}) 〉 \\ 〈 (\begin{matrix} {0.7}, \\ {0.1, 0.2}, \\ {0.1} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.6}, \\ {0.1}, \\ {0.2} \end{matrix}) 〉 & 〈 (\begin{matrix} {0.2}, \\ {0.3}, \\ {0.4} \end{matrix}) 〉 \end{matrix}] \end{matrix}$

Step 2. Compute the supports $sup ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ .

Based on the formula (4), the support $sup ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ for PHFN ${\tilde{β}}_{mg}$ from ${\tilde{β}}_{mf}$ is computed. $\begin{matrix} sup ({\tilde{β}}_{11}, {\tilde{β}}_{12}) = sup ({\tilde{β}}_{12}, {\tilde{β}}_{11}) = 0.9667; \\ sup ({\tilde{β}}_{11}, {\tilde{β}}_{13}) = sup ({\tilde{β}}_{13}, {\tilde{β}}_{11}) = 0.9500; \\ sup ({\tilde{β}}_{12}, {\tilde{β}}_{13}) = sup ({\tilde{β}}_{13}, {\tilde{β}}_{12}) = 0.9167; \\ sup ({\tilde{β}}_{21}, {\tilde{β}}_{22}) = sup ({\tilde{β}}_{22}, {\tilde{β}}_{21}) = 0.9833; \\ sup ({\tilde{β}}_{21}, {\tilde{β}}_{23}) = sup ({\tilde{β}}_{23}, {\tilde{β}}_{21}) = 0.7333; \\ sup ({\tilde{β}}_{22}, {\tilde{β}}_{23}) = sup ({\tilde{β}}_{23}, {\tilde{β}}_{22}) = 0.7167; \\ sup ({\tilde{β}}_{31}, {\tilde{β}}_{32}) = sup ({\tilde{β}}_{32}, {\tilde{β}}_{31}) = 0.9500; \\ sup ({\tilde{β}}_{31}, {\tilde{β}}_{33}) = sup ({\tilde{β}}_{33}, {\tilde{β}}_{31}) = 0.8667; \\ sup ({\tilde{β}}_{32}, {\tilde{β}}_{33}) = sup ({\tilde{β}}_{33}, {\tilde{β}}_{32}) = 0.8167; \\ sup ({\tilde{β}}_{41}, {\tilde{β}}_{42}) = sup ({\tilde{β}}_{42}, {\tilde{β}}_{41}) = 0.9167; \\ sup ({\tilde{β}}_{41}, {\tilde{β}}_{43}) = sup ({\tilde{β}}_{43}, {\tilde{β}}_{41}) = 0.6833; \\ sup ({\tilde{β}}_{42}, {\tilde{β}}_{43}) = sup ({\tilde{β}}_{43}, {\tilde{β}}_{42}) = 0.7333; \end{matrix}$

Step 3. Compute the weigths ς_mg.

Based on the formula $ψ ({\tilde{β}}_{mg}) = \sum_{\begin{matrix} f = 1 \\ f \neq g \end{matrix}}^{h}$ $sup ({\tilde{β}}_{mg}, {\tilde{β}}_{mf})$ , we can get ${[ψ ({\tilde{β}}_{mg})]}_{4 \times 3} = [\begin{matrix} 1.9167 & 1.8833 & 1.8667 \\ 1.7167 & 1.7000 & 1.4500 \\ 1.8167 & 1.7667 & 1.6833 \\ 1.6000 & 1.6500 & 1.4167 \end{matrix}]$

Thus, based on the formula (5), the weigths ς_mg are obtained. ${[ς_{mg}]}_{4 \times 3} = [\begin{matrix} 0.3534 & 0.2496 & 0.3970 \\ 0.3649 & 0.2590 & 0.3761 \\ 0.3584 & 0.2514 & 0.3902 \\ 0.3584 & 0.2609 & 0.3807 \end{matrix}]$

Step 4. Compute the global values ${\tilde{β}}_{m}$ of all alternatives.

According to the PHFWPH operator defined in Theorem 2, the global value ${\tilde{β}}_{m}$ of X_m (m = 1, 2, 3, 4) can be obtained. $\begin{matrix} {\tilde{β}}_{1} = {0.43920.4738}, {0.20330.2290}, {0.2636}; \\ {\tilde{β}}_{2} = {0.4485, 0.4738}, {0.1326, 0.1671}, {0.2905}; \\ {\tilde{β}}_{3} = {0.3218, 0.3577}, {0.2031, 0.2363}, {0.3691}; \\ {\tilde{β}}_{4} = {0.5174}, {0.1577, 0.1961}, {0.2175} . \end{matrix}$

Similarly, the global value ${\tilde{β}}_{m}$ of X_m (m = 1, 2, 3, 4) using PHFWGPH operator defined in Theorem 5 can be obtained as well. $\begin{matrix} {\tilde{β}}_{1} = {0.4381, 0.4732}, {0.3930, 0.4784}, {0.4809}; \\ {\tilde{β}}_{2} = {0.3513, 0.4220}, {0.2593, 0.2849}, {0.4830}; \\ {\tilde{β}}_{3} = {0.3068, 0.3404}, {0.3401, 0.3119}, {0.5148}; \\ {\tilde{β}}_{4} = {0.4463}, {0.2743, 0.5449}, {0.4569} . \end{matrix}$

Step 5. Compute the score value $s ({\tilde{β}}_{m})$ and accuracy value $a ({\tilde{β}}_{m})$ .

Based on the formulas in Definition 3, the following results using PHFWPH and PHFWGPH operator can be obtained, respectively.

For PHFWPH operator, $s ({\tilde{β}}_{1}) = 1.9767; s ({\tilde{β}}_{2}) = 2.0208;$ $s ({\tilde{β}}_{3}) = 1.7510; s ({\tilde{β}}_{4}) = 2.1231 .$ For PHFWGPH operator, $s ({\tilde{β}}_{1}) = 1.5391; s ({\tilde{β}}_{2}) = 1.6315;$ $s ({\tilde{β}}_{3}) = 1.4828; s ({\tilde{β}}_{4}) = 1.5798;$

Step 6. Rank and choose alternative(s).

The final ranking using PHFWPH operator is X₄ ≻ X₂ ≻ X₁ ≻ X₃, the most suitable solution is X₄, while X₃ is the worst one. Meanwhile, the final ranking using PHFWGPH operator is X₂ ≻ X₄ ≻ X₁ ≻ X₃, then the most suitable solution is X₂, while X₃ is the worst one.

5.2 Sensitivity analysis

In this part, the influences of different parameters s and t regarding the scores and ranking results of four alternatives are shown in the following Figs. 2 –13. When parameter s is equal to 1 and parameter t is varying from 0 to 20, the changing trend of scores regarding four alternatives utilizing PHFWPH operator and PHFWGPH operator are illustrated in Figs. 2 3, respectively. Similarly, when parameter t is equal to 1 and parameter s is varying from 0 to 20, the changing trend of scores regarding four alternatives utilizing PHFWPH operator and PHFWGPH operator are obtained in Figs. 4 5, respectively. The scores of alternative X_i (i = 1, 2, 3, 4) utilizing PHFWPH operator are presented in Figs. 6 –9 when parameters s and t are varying from 0.1 to 20. Similarly, the scores of alternative X_i (i = 1, 2, 3, 4) utilizing PHFWGPH operator are presented in Figs. 10 –13 when parameters s and t are varying from 0.1 to 20.

Fig. 2

The score values using PHFWPH operator when s = 1, t ∈ [0, 20].

Fig. 3

The score values using PHFWGPH operator when s = 1, t ∈ [0, 20].

Fig. 4

The score values using PHFWPH operator when t = 1, s ∈ [0, 20].

Fig. 5

The score values using PHFWGPH operator when t = 1, s ∈ [0, 20].

Fig. 6

The score values of X₁ using PHFWPH operator when s, t ∈ [0 . 1, 20].

Fig. 7

The score values of X₂ using PHFWPH operator when s, t ∈ [0 . 1, 20].

Fig. 8

The score values of X₃ using PHFWPH operator when s, t ∈ [0 . 1, 20].

Fig. 9

The score values of X₄ using PHFWPH operator when s, t ∈ [0 . 1, 20].

Fig. 10

The score values of X₁ using PHFWGPH operator when s, t ∈ [0 . 1, 20].

Fig. 11

The score values of X₂ using PHFWGPH operator when s, t ∈ [0 . 1, 20].

Fig. 12

The score values of X₃ using PHFWGPH operator when s, t ∈ [0 . 1, 20].

Fig. 13

The score values of X₄ using PWGPH operator when s, t ∈ [0 . 1, 20].

In Fig. 2, let s = 1, it is clearly that the optimal alternative is always X₄ and the worst alternative is always X₃, no matter what the value of parameter t is. This illustrates the robustness of the PHFWPH operator.

From Fig. 3, we can obatin that the scores of four alternatives utilzing PHFWGPH operator become smaller as the parameter t becomes bigger, and the optimal alternative changes from X₂ to X₁ as the parameter t becomes bigger.

In Fig. 4, it is clearly that the scores of four alternatives utilzing PHFWPH operator become bigger as the parameter s becomes bigger and s is greater than 1, and the optimal alternative changes from X₁ to X₄ with the increasing of parameter s, while the worst alternative ialways X₃.

In Fig. 5, the optimal alternative is X₁ or X₂ as parameter s changes, and the worst alternative is X₃ or X₄. Since the deloping trend of Fig. 5 differs from Fig. 3, then the parameters s and t are asymmetric.

From Figs. 6 –9, we can observe that the score values of alternative X_i (i = 1, 2, 3, 4) utilizing PHFWPH operator become smaller as parameters sand t become smaller. As a contrast, the scores of alternative X_i (i = 1, 2, 3, 4) utilizing PHFWGPH operator become bigger as parameters sand t become smaller in Figs. 10 –13.

To sum up, different parameters sand t may derive different scores and affect the final selection. DMs can decide the values of parameters sand t in terms of their preference. Therefore, the proposed MCDM method is flexible and practical.

5.3 Comparison analysis

In this subsection, the novel approach proposed in this paper with existing picture fuzzy MCDM approach [12, 52] is compared.

Some characteristics comparison of our method with these methods mentioned above are presented in Table 1.

Table 1
Characteristics comparison of different method

Characteristic Wei [52] Wang [12] Proposed method

Consider the correlations of criteria No No Yes

Reduce the impact of irrational data No No Yes

Types of Fuzzy set PFS PFS PHFS PFS PHFS

Numbers of parameter none λ s, t

Characteristic	Wei [52]	Wang [12]	Proposed method
Consider the correlations of criteria	No	No	Yes
Reduce the impact of irrational data	No	No	Yes
Types of Fuzzy set	PFS	PFS PHFS	PFS PHFS
Numbers of parameter	none	λ	s, t

Wei [52] applied weighted average (WA), weighted geometric (WG), ordered weighted average (OWA), and ordered weighted geometric (OWG) operators to solve MCDM problem with picture fuzzy information. One hand, the PH AOs proposed in this paper are more powerful than those proposed by Wei [52], which ignores the influence of irrational data and correlations between criteria. On the other hand, picture hesitant fuzzy information is more practical than picture fuzzy information [52], which is unsuitable to situations where DMs feel hesitant to be difficult to decide a fixed membership degree value.

Wang [12] proposed the definition of PHFS and explored prioritized weighted AOs to MCDM problem. The picture hesitant fuzzy prioritized weighted (PHFPW) AOs investigated by Wang [12] consider different priorities of criteria but cannot capture the interrelationships of criteria and handle the impact of unreasonable data. Thus, the PH AOs proposed in this paper have more advantages than that of Wang [12].

In sum, fuzzy set is only a special case of PHFS. In contrast, our method not only consider the correlations of criteria, but also reduce the impact of irrational data. Moreover, it has two parameters s and t. These characteristics make the proposed method more general and flexible.

6 Conclusion and future directions

PHFSs can better express the hesitancy of DMs in actual situation with the increasing complicated decision-making environment. Nevertheless, the study on AOs applying to PHFN environment is inadequate. Therefore, we propose two novel AOs, namely, PHFWPH operator and PHFWGPH operator. A novel MCDM approach based on the proposed PH AOs is also investigated under PHFN environment.

In summary, the main characteristics of our work are listed as below: (1) PHFSs are more appropriate to convey uncertain and hesitant information in MCDM problems. (2) The novel MCDM approach is more practical for utilizing PH AOs, which can not only decrease the impact of irrational data but also capture the correlations among criteria. (3) The PH AOs including PHFWPH operator and PHFWGPH operator proposed in this paper are more powerful for fusing multiple values, which provide flexible parameters s and t by DMs according to their actual requirements.

In future, we will explore to apply the proposed PH AOs to other fuzzy sets, such as pythagorean fuzzy set [53], multiple-valued picture fuzzy linguistic set [16], multiple-valued neutrosophic uncertain linguistic set [20], and so on. Meanwhile, we will consider the interaction between membership and non-membership, and further explore the MCDM problems under PHFSs environment where the criteria weights are unknown.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper.

This work was supported by the Social Science Foundation of Hubei Province (No. 20ZD065) and the industry-university cooperation collaborative education project provided by the Ministry of Education (No. 201802234027).

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A new multi-criteria decision-making method utilizing power heronian operators with picture hesitant fuzzy information

Abstract

Keywords

1 Introduction

2 Picture hesitant fuzzy sets

2.1 PHFSs and operations

2.2 Comparative method

2.3 Distance measure

3 Novel power heronian operators

3.1 Power and heronian operators

3.2 Picture hesitant fuzzy weighted power heronian operator

5.1 Example

5.2 Sensitivity analysis

Table 1 Characteristics comparison of different method Characteristic Wei [52] Wang [12] Proposed method Consider the correlations of criteria No No Yes Reduce the impact of irrational data No No Yes Types of Fuzzy set PFS PFS PHFS PFS PHFS Numbers of parameter none λ s, t

Conflicts of interest

Acknowledgments

References

Table 1
Characteristics comparison of different method

Characteristic Wei [52] Wang [12] Proposed method

Consider the correlations of criteria No No Yes

Reduce the impact of irrational data No No Yes

Types of Fuzzy set PFS PFS PHFS PFS PHFS

Numbers of parameter none λ s, t