Pythagorean fuzzy power Muirhead mean operators with their application to multi-attribute decision making

Abstract

The sum of membership and non-membership degrees of the Pythagorean fuzzy set is greater than one with their square sum less than or equal to one. Thus, as an extension of intuitionistic fuzzy set, the Pythagorean fuzzy set is a powerful tool to describe fuzziness and uncertainty. The aim of this study is to introduce some new operators for aggregating Pythagorean fuzzy information and apply them to multi-attribute decision making. Considering the advantages of the power average operator and Muirhead mean, we introduce the power Muirhead mean operator and investigate it under Pythagorean fuzzy environment. Thus, some new Pythagorean fuzzy aggregation operators, such as the Pythagorean fuzzy power Muirhead mean and the weighted Pythagorean fuzzy power Muirhead mean are developed. The prominent advantage of these proposed operators is that they consider the relationships between fused data and the interrelationships between all aggregated values, thereby obtaining more information in the process of multi-attribute decision making. Furthermore, we introduce a novel approach to multi-attribute decision-making problems based on the proposed operators. Finally, we provide a numerical example to illustrate the validity of the proposed approach.

Keywords

Pythagorean fuzzy set multi-attribute decision making aggregation operators power Muirhead mean

1 Introduction

One of the difficulties in practical multi-attribute decision making (MADM) problems is representing attribute values in fuzzy and vague decision-making environments. Fuzzy set (FS) theory, which was originally introduced by Zadeh [1], is a powerful tool for depicting and expressing impreciseness and uncertainty. Since its introduction, FS has received a substantial amount of attention and has been studied by thousands of scientists around the world in theoretical and practical aspects [2]. Subsequently, Atanassov [3] introduced the concept of intuitionistic fuzzy set (IFS), which is characterized by a membership degree and a non-membership degree. Therefore, IFS can describe fuzziness and uncertainty more comprehensively and detailedly than FS. Since its appearance, IFS has drawn much attention among scholars and has been applied to medical diagnosis [4, 5], pattern recognition [6, 7], data mining [8, 9], and MADM [10 –13].

IFSs must meet the condition that the sum of the membership degree and membership degree is equal to or less than 1. However, in certain situations, the sum of membership and non-membership degrees that decision makers provide is greater than one. For instance, the membership degree and non-membership degree provided by a decision maker are 0.7 and 0.6 respectively. Evidently, the pair (0.7, 0.6) is not valid for intuitionistic fuzzy numbers (IFNs). To address these situations, Yager [14, 15] introduced the concept of Pythagorean fuzzy set (PFS), which can be considered as a generalization of IFS. In other words, all intuitionistic fuzzy degrees are part of the Pythagorean fuzzy degrees. Therefore, PFS has a higher capability than IFS to model vagueness in real decision-making situations. A significant step in selecting the best alternative in a Pythagorean fuzzy environment is to aggregate Pythagorean fuzzy numbers (PFNs). In this process, information aggregation operators are important tools. In fact, several aggregation operators have been proposed [16 –21]. Additional achievements can be found in the literature [22]. To aggregate PFNs, a number of Pythagorean fuzzy aggregation operators have been developed in the past years. Peng and Yuan [23] proposed the generalized Pythagorean fuzzy weighted averaging (GPFWA) operator and some generalized Pythagorean fuzzy point weighted averaging (GPFPWA) operators. Zeng et al. [24] introduced the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator, based on which a hybrid method to MADM was also proposed. Ma and Xu [25] developed some Pythagorean fuzzy symmetric aggregation operators, such as the symmetric Pythagorean fuzzy weighted geometric (SPFWG) operator and the symmetric Pythagorean fuzzy weighted averaging (SPFWA) operator. Based on the Einstein operations for PFNs, Garg [26, 27] and Rahman et al. [28] proposed some Pythagorean fuzzy Einstein arithmetic and geometric aggregation operators. After considering that the aforementioned operators cannot capture the relationship between PFNs, Peng and Yuan [29] proposed the Choquet integral operator to fuse Pythagorean fuzzy information. To capture the interrelationships among fused PFNs, Liang et al. [30] proposed some Pythagorean fuzzy geometric Bonferroni mean operators and discussed their desirable properties. Wei and Lu [31] also developed a family of Pythagorean fuzzy Maclaurin symmetric mean operators. Zhang et al. [32] investigated the generalized Bonferroni mean in the Pythagorean fuzzy environment and proposed a family of generalized Pythagorean fuzzy Bonferroni mean operators.

Due to the increased complexity in practical decision-making problems, we have to consider the following questions when determining the best alternative. (1) In some cases, attribute values that decision makers provide may be unduly high or unduly low, negatively affecting the final ranking results. The power averaging (PA) operator that was originally proposed by Yager [33] is a useful aggregation technology that allows the evaluated values to support and reinforce each other. Therefore, we may utilize the PA operator to reduce such bad influence by assigning the different weights produced by the support measure. (2) In some real decision-making problems, the attribute values are dependent. Thus, the interrelationship between the attribute values should be considered. The Bonferroni mean (BM) [34], Heronian mean (HM) [35], and Muirhead mean (MM) [36] can complete this function. However, Liu and Li [37] pointed out that MM has some advantages over BM and HM. Some existing aggregation operators, such as BM and the Maclaurin symmetric mean (MSM) [38], are special cases of MM. Moreover, MM has a parameter vector, which can increase the flexibility in aggregation processes. Thus, we extend MM to PFSs so that the interrelationship among aggregated PFNs can be captured.

Therefore, we combine PA with MM and propose the power Muirhead mean (PMM) operator that takes the advantages of PA and MM. Furthermore, we apply the PMM to aggregating Pythagorean fuzzy information, so that some new Pythagorean fuzzy aggregation operators are developed. The proposed operators can not only relieve the bad influence of unduly high or low arguments on the results but also consider the interrelationship between arguments. Finally, we apply the proposed operators to solving MADM problems. The rest of the paper is organized as follows. Section 2 introduces relevant concepts. Section 3 proposes several Pythagorean fuzzy power Muirhead mean operators and discusses their desirable properties and special cases. Section 4 introduces a novel method to MADM based on the proposed aggregation operators. Finally, Section 5 provides an instance to illustrate the validity of the proposed method.

2 Basic concepts

This section briefly recalls some basic concepts about PFS, PA, and MM.

2.1 Pythagorean fuzzy set

Definition 1. [14, 15] Let X be an ordinary fixed set. A PFS P defined on X is expressed as follows: $P = {〈 x, μ_{P} (x), v_{P} (x) 〉 | x \in X},$ (1) where μ_P (x) and v_P (x) denote membership and non-membership degrees, respectively, thereby satisfying μ_P (x), v_P (x) ∈ [0, 1] and (μ_P (x)) ² + (v_P (x)) ² ≤ 1. The indeterminacy degree of P is defined as $π_{P} (x) = \sqrt{1 - {(μ_{P} (x))}^{2} - {(v_{P} (x))}^{2}}$ . Zhang and Xu [39] called (μ_P (x), v_p (x)) as a PFN, which can be denoted by p = (μ_p, v_p).

Zhang and Xu [39] also proposed some operations for PFNs.

Definition 2. [39] Let p = (μ, v), p₁ = (μ₁, v₁), and p₂ = (μ₂, v₂) be any three PFNs, and let λ be a positive real number. Then,

$p_{1} \oplus p_{2} = (\sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}}, v_{1} v_{2})$ ,

$p_{1} \otimes p_{2} = (μ_{1} μ_{2}, \sqrt{v_{1}^{2} + v_{2}^{2} - v_{1}^{2} v_{2}^{2}})$ ,

$λ p = (\sqrt{1 - {(1 - μ^{2})}^{λ}}, v^{λ})$ ,

$p^{λ} = (μ^{λ}, \sqrt{1 - {(1 - v^{2})}^{λ}})$ .

To compare two PFNs, Peng and Yuan [23] introduced a comparison law for PFNs.

Definition 3. [23] For any PFN p = (μ, v), the score function of p is defined as s (p) = μ² - v². For any two PFNs, such as p₁ = (μ₁, v₁) and p₂ = (μ₂, v₂), if s (p₁) > s (p₂), then p₁ > p₂; if s (p₁) = s (p₂), then p₁ = p₂.

Zhang and Xu [39] also proposed a Pythagorean fuzzy distance measure for PFNs.

Definition 4. [39] Let p₁ = (μ₁, v₁) and p₂ = (μ₂, v₂) be two PFNs. Then, the distance between p₁ and p₂ is defined as follows: $\begin{matrix} d (p_{1}, p_{2}) \\ = \frac{1}{2} (| μ_{1}^{2} - μ_{2}^{2} | + | v_{1}^{2} - v_{2}^{2} | + | π_{1}^{2} - π_{2}^{2} |) . \end{matrix}$ (2)

2.2 Power average operator

Yager [33] first proposed the PA operator for crisp numbers. The most important advantage of the PA operator is its capability to reduce the bad effect of unduly high and low arguments on the finalresults.

Definition 5. [33] Let a_i (i = 1, 2, …, n) be a collection of crisp numbers. The PA operator is then expressed as follows: $\begin{matrix} PA (a_{1}, a_{2}, \dots, a_{n}) \\ = \sum_{i = 1}^{n} ((1 + T (a_{i})) a_{i} / - \sum_{j = 1}^{n} (1 + T (a_{j}))), \end{matrix}$ (3) where $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (a_{i}, a_{j})$ and Sup (a_i, a_j) is the support for a_i and a_j. The support satisfies the following properties:

Sup (a_i, a_j) ∈ [0, 1],

Sup (a_i, a_j) = Sup (a_j, a_i),

If d (a_i, a_j) < d (a_l, a_k), then Sup (a_i, a_j) > Sup (a_l, a_k), where d (a_i, a_j) is the distance between a_i and a_j. For convenience, we call the aforementioned properties Property 1.

2.3 Muirhead mean

Muirhead [36] proposed MM for crisp numbers. The advantage of MM is its capability to consider the interrelationship between all aggregated arguments.

Definition 6. [36] Let a_i (i = 1, 2, …, n) be a collection of crisp numbers and P = (p₁, p₂, …, p_n) ∈ Rⁿ be a vector of parameters. Then, the MM operator is defined as $\begin{matrix} {MM}^{P} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{j = 1}^{n} a_{ϑ (j)}^{P_{j}})}^{\frac{1}{\sum_{j = 1}^{n} P_{j}}}, \end{matrix}$ (4) where ϑ (j) (j = 1, 2, …, n) is any permutation of (1, 2, …, n), and S_n is the collection of all permutations of (1, 2, …, n).

We can obtain some special cases of MM with respect to the parameter vector P, which are shown as follows:

If P = (1, 0, …, 0), then MM is reduced to ${MM}^{(1, 0, \dots, 0)} (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{n} \sum_{j = 1}^{n} a_{j},$ (5) which is the arithmetic averaging operator.

If P = (1/ - n, 1/ - n, …, 1/ - n), then MM is reduced to ${MM}^{(1 / - n, 1 / - n, \dots, 1 / - n)} (a_{1}, a_{2}, \dots, a_{n}) = \prod_{j = 1}^{n} a_{j}^{1 / - n},$ (6) which is the geometric averaging operator.

If P = (1, 1, 0, 0, …, 0), then MM is reduced to $\begin{array}{l} M M^{(1, 1, 0, 0, \dots, 0)} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n + 1)} \sum_{_{i \neq j}^{i, j = 1}}^{n} a_{i} a_{j})}^{\frac{1}{2}}, \end{array}$ (7) $\begin{array}{l} M M \overset{k}{\overset{︷}{(1, 1, \dots, 1}} \overset{n - k}{\overset{︷}{0, 0, \dots, 0)_{(a_{1}, a_{2}, \dots, a_{n})}}} \\ = {(\frac{1 \leq i_{1} \leq \sum_{\dots} \leq i_{i} \leq n \prod_{j = 1}^{k} a i_{j}}{c_{n}^{k}})}^{1 / k} \end{array}$ (8) which is the BM [34] operator.

If $P = {\overset{︷}{(1, 1, \dots, 1}}^{k}, {\overset{︷}{0, 0, \dots, 0)}}^{n - k}$ , then MM is reduced to $\begin{matrix} M M^{{\overset{︷}{(1, 1, \dots, 1}}^{k}, {\overset{︷}{0, 0, \dots, 0)}}^{n - k}} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1 \leq i_{1} \leq \dots_{\sum^{}}^{} \leq i_{i} \leq n \prod_{j = 1}^{k} a_{i_{j}}}{C_{n}^{k}})}^{1 / - k}, \end{matrix}$ (9) which is the MSM [38] operator.

3 Power Muirhead mean for Pythagorean fuzzy information

In this section, we first propose the PMM by combining PA with MM. Then, we extend PMM to aggregate Pythagorean fuzzy information.

Definition 7. Let a_i (i = 1, 2, …, n) be a collection of crisp numbers and P = (p₁, p₂, …, p_n) ∈ Rⁿ be a vector of parameters. Then, PMM is defined as follows: $\begin{matrix} {PMM}^{P} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(\frac{n (1 + T (a_{ϑ (i)})) a_{ϑ (i)}}{\sum_{j = 1}^{n} (1 + T (a_{j}))})}^{p_{i}})}^{\frac{1}{\sum_{i = 1}^{n} P_{i}}}, \end{matrix}$ (10) where $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (a_{i}, a_{j})$ and Sup (a_i, a_j) is the support for a_i and a_j, satisfying Property 1.

3.1 Pythagorean fuzzy power Muirhead mean operator

Definition 8. Let p_i (i = 1, 2, …, n) be a collection of PFNs and R = (r₁, r₂, …, r_n) ∈ Rⁿ be a vector of parameters. Then, the Pythagorean fuzzy power Muirhead mean (PFPMM) is defined as follows: $\begin{matrix} {PFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(\frac{n (1 + T (p_{ϑ (i)}))}{\sum_{j = 1}^{n} (1 + T (p_{j}))} p_{ϑ (i)})}^{r_{i}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \end{matrix}$ (11) where $T (p_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (p_{i}, p_{j})$ and Sup (p_i, p_j) is the support for p_i and p_j, satisfying the following properties:

Sup (p_i, p_j) ∈ [0, 1],

Sup (p_i, p_j) = Sup (p_j, p_i),

If d (p_i, p_j) < d (p_l, p_k), then Sup (p_i, p_j) > Sup (p_l, p_k), where d (p_i, p_j) is the distance between p_i and p_j. For convenience, we call the above properties Property 2.

To simplify Equation (10), we can define it as $ω_{i} = \frac{(1 + T (p_{i}))}{\sum_{j = 1}^{n} (1 + T (p_{j}))} .$ (12)

For convenience, we can call (ω₁, ω₂, …, ω_n) the power weighting vector (PWV), which clearly satisfies ω_i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1$ . Thereafter, Equation (10) can be denoted by $\begin{matrix} {PFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}} . \end{matrix}$ (13)

According to the operations FOR PFNs in Definition 2, the following theorem can be obtained:

Theorem 1. Let p_i = (μ_i, v_i) (i = 1, 2, …, n) be a collection of PFNs and R = (r₁, r₂, …, r_n) ∈ Rⁿ be a vector of parameters. Then, the aggregated value by PFPMM is still a PFN and $\begin{matrix} {PFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) = ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \\ \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) . \end{matrix}$ (14)

Proof. According to Definition 2, we have $n ω_{i} p_{ϑ (i)} = (\sqrt{1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}}}, v_{ϑ (i)}^{n ω_{i}}) .$ (15)

Therefore, ${(n ω_{i} p_{ϑ (i)})}^{r_{i}} = ({(\sqrt{1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}}})}^{r_{i}}, \sqrt{1 - {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}}) .$ (16)

Thus, we can obtain $\prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}} = (\prod_{i = 1}^{n} {(\sqrt{1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}}})}^{r_{i}}, \sqrt{1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}})$ (17) and $\begin{matrix} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}} \\ = (\sqrt{1 - \prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}})}^{r_{i}})}, \prod_{ϑ \in S_{n}} (\sqrt{1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}})) . \end{matrix}$ (18)

Thus, $\begin{matrix} \frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}} \\ = (\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}}, {(\prod_{ϑ \in S_{n}} (\sqrt{1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}}))}^{\frac{1}{n!}}) . \end{matrix}$ (19)

Therefore, $\begin{matrix} {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}} = ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \\ \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) . \end{matrix}$ (20)

In the following equations, we determine PWV. To calculate the PWV ω, we first have to determine the support degree Sup (p_i, p_j). According to the distance between any two PFNs shown in Definition 4, we can obtain Sup (p_i, p_j) using $Sup (p_{i}, p_{j}) = 1 - d (p_{i}, p_{j}) .$ (21)

Thereafter, we utilize the equation $T (p_{i}) = \sum_{substack i = 1 i \neq j}^{n} Sup (p_{i}, p_{j})$ (22) to calculate T (p_i) (i = 1, 2, …, n). Then, according to Equation (11), we can obtain PWV. We provide an example to improve the illustration of the calculation process.

Example. Let p₁ = (0.9, 0.3), p₂ = (0.7, 0.6), and p₃ = (0.5, 0.8) be three PFNs. We then utilize the PFPMM operator to aggregate them. The steps are as follows:

Step 1. We calculate the supports Sup (p_i, p_j), where i, j = 1, 2, 3, according to Equations (2 and 20). Thus, we have $\begin{matrix} Sup (p_{1}, p_{2}) = 0.68, Sup (p_{1}, p_{3}) = 0.44, \\ Sup (p_{2}, p_{1}) = 0.68, Sup (p_{2}, p_{3}) = 0.72, \\ Sup (p_{3}, p_{1}) = 0.44, Sup (p_{3}, p_{2}) = 0.72 . \end{matrix}$

Step 2. We calculate the PWV according to Equation (11) to obtain $\begin{matrix} T (p_{1}) & = & Sup (p_{1}, p_{2}) + Sup (p_{1}, p_{3}) \\ = & 0.68 + 0.44 = 1.12; \\ T (p_{2}) & = & Sup (p_{2}, p_{1}) + Sup (p_{2}, p_{3}) \\ = & 0.68 + 0.72 = 1.4; \\ T (p_{3}) & = & Sup (p_{3}, p_{1}) + Sup (p_{3}, p_{2}) \\ = & 0.44 + 0.72 = 1.16 . \end{matrix}$

Therefore, $\begin{matrix} ω_{1} & = & \frac{1 + T (p_{1})}{(1 + T (p_{1})) + (1 + T (p_{2})) + (1 + T (p_{3}))} \\ = & \frac{1 + 1.12}{(1 + 1.12) + (1 + 1.4) + (1 + 1.16)} \\ = & 0.3174 . \end{matrix}$

Similarly, we calculate ω₂ and ω₃ in the same method to obtain ω₂ = 0.3593 and ω₃ = 0.3233.

Step 3. We utilize the PFPMM operator to aggregate the three PFNs. For convenience, let the comprehensive value be p = (μ, v). In this example, we suppose R = (0.1, 0.2, 0.3). The calculation process is shown at the top of the next page. $\begin{matrix} μ & = & {(\sqrt{1 - {(\prod_{ϑ \in S_{3}} (1 - \prod_{i = 1}^{3} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{3 ω_{i}})}^{r_{i}}))}^{\frac{1}{3!}}})}^{\frac{1}{0.1 + 0.2 + 0.3}} \\ = & {(1 - (\begin{matrix} (1 - {(1 - {(1 - 0 . 9^{2})}^{3 ω_{1}})}^{0.1} \times {(1 - {(1 - 0 . 7^{2})}^{3 ω_{2}})}^{0.2} \times {(1 - {(1 - 0 . 5^{2})}^{3 ω_{3}})}^{0.3}) \times \\ (1 - {(1 - {(1 - 0 . 9^{2})}^{3 ω_{1}})}^{0.1} \times {(1 - {(1 - 0 . 5^{2})}^{3 ω_{3}})}^{0.2} \times {(1 - {(1 - 0 . 7^{2})}^{3 ω_{2}})}^{0.3}) \times \\ (1 - {(1 - {(1 - 0 . 7^{2})}^{3 ω_{2}})}^{0.1} \times {(1 - {(1 - 0 . 9^{2})}^{3 ω_{1}})}^{0.2} \times {(1 - {(1 - 0 . 5^{2})}^{3 ω_{3}})}^{0.3}) \times \\ (1 - {(1 - {(1 - 0 . 7^{2})}^{3 ω_{2}})}^{0.1} \times {(1 - {(1 - 0 . 5^{2})}^{3 ω_{3}})}^{0.2} \times {(1 - {(1 - 0 . 9^{2})}^{3 ω_{1}})}^{0.3}) \times \\ (1 - {(1 - {(1 - 0 . 5^{2})}^{3 ω_{3}})}^{0.1} \times {(1 - {(1 - 0 . 7^{2})}^{3 ω_{2}})}^{0.2} \times {(1 - {(1 - 0 . 9^{2})}^{3 ω_{1}})}^{0.3}) \times \\ (1 - {(1 - {(1 - 0 . 5^{2})}^{3 ω_{3}})}^{0.1} \times {(1 - {(1 - 0 . 9^{2})}^{3 ω_{1}})}^{0.2} \times {(1 - {(1 - 0 . 7^{2})}^{3 ω_{2}})}^{0.3}) \end{matrix}))}^{\frac{1}{0.1 + 0.2 + 0.3}} \\ = & 0.9496 \end{matrix}$ Similarly, we can obtain v = 0.2458. Therefore, the comprehensive value is p = (0.9496, 0.2458).

In the following, we present and discuss some desirable properties of the PFPMM operator.

Theorem 2. (Idempotency) Let p_i = (μ_i, v_i) (i = 1, 2, …, n) be a collection of PFNs, and p_i = p = (μ, v) for i = 1, 2, …, n. Then, ${PFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) = p .$ (23)

Proof. As p_i = p = (μ, v) holds for all i, we have Sup (p_i, p_j) = 1 for i, j = 1, 2, …, n. Therefore, we can obtain ω_i = 1/ - n for all i. Furthermore, $\begin{matrix} {PFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) = {PFPMM}^{R} (p, p, \dots, p) \\ = ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ^{2})}^{n \frac{1}{n}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - v^{2 n \frac{1}{n}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) \\ = ({(\sqrt{1 - {({(1 - {(1 - (1 - μ^{2}))}^{\sum_{i = 1}^{n} r_{i}})}^{n!})}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \end{matrix} \sqrt{1 - {(1 - (1 - {(1 - v^{2})}^{\sum_{i = 1}^{n} r_{i}}))}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) & & = (\sqrt{μ^{2}}, \sqrt{v^{2}}) = (μ, v) = p$ which completes the proof of Theorem 2.

Theorem 3. (Boundedness) Let p_i = (μ_i, v_i) (i = 1, 2, …, n) be a collection of PFNs, p⁻ = min(p₁, p₂, …, p_n) = (a, b) and p⁺ = max(p₁, p₂, …, p_n) = (c, d). Then, $x \leq {PFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) \leq y,$ (24) where $\begin{matrix} x & = & ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - a^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - b^{2 n ω_{i}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}), \end{matrix}$ and $\begin{matrix} y & = & ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - c^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - d^{2 n ω_{i}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) . \end{matrix}$

Proof. $\begin{matrix} n ω_{i} p_{ϑ (i)} \\ = (\sqrt{1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}}}, v_{ϑ (i)}^{n ω_{i}}) \\ \geq (\sqrt{1 - {(1 - a^{2})}^{n ω_{i}}}, b^{n ω_{i}}) . \end{matrix}$

Thus, $\begin{matrix} {(n ω_{i} p_{ϑ (i)})}^{r_{i}} \\ = ({(\sqrt{1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}}})}^{r_{i}}, \sqrt{1 - {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}}) \\ \geq ({(\sqrt{1 - {(1 - a^{2})}^{n ω_{i}}})}^{r_{i}}, \sqrt{1 - {(1 - b^{2 n ω_{i}})}^{r_{i}}}) . \end{matrix}$

Therefore, $\begin{matrix} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}} = (\prod_{i = 1}^{n} {(\sqrt{1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}}})}^{r_{i}}, \sqrt{1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}}) \\ \geq (\prod_{i = 1}^{n} {(\sqrt{1 - {(1 - a^{2})}^{n ω_{i}}})}^{r_{i}}, \sqrt{1 - \prod_{i = 1}^{n} {(1 - b^{2 n ω_{i}})}^{r_{i}}}) \end{matrix}$ and $\begin{matrix} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}} = (\sqrt{1 - \prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}})}^{r_{i}})}, \\ \prod_{ϑ \in S_{n}} (\sqrt{1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}}})) \\ \geq (\sqrt{1 - \prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - a^{2})}^{n ω_{i}})}^{r_{i}})}, \prod_{ϑ \in S_{n}} (\sqrt{1 - \prod_{i = 1}^{n} {(1 - b^{2 n ω_{i}})}^{r_{i}}})) . \end{matrix}$

Thus, $\begin{matrix} {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(n ω_{i} p_{ϑ (i)})}^{r_{i}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}} = ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \\ \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - v_{ϑ (i)}^{2 n ω_{i}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) \\ \geq ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - a^{2})}^{n ω_{i}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \\ \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - b^{2 n ω_{i}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) = x \end{matrix}$ which means that x ≤ PFPMM^R (p₁, p₂, …, p_n).

Similarly, we can also prove that PFPMM^R (p₁, p₂, …, p_n) ≤ y, which completes the proof of Theorem 3.

Evidently, PFPMM does not have the property of monotonicity.

One of the prominent advantage of PFPMM is its capability to capture the interrelationship between PFNs. Moreover, PFPMM has a parameter vector leading to a flexible aggregation process. By assigning different values to the parameter vector, some special cases can be obtained.

Case 1. If R = (1, 0, …, 0), then PFPMM is reduced to $\begin{matrix} {PFPMM}^{(1, 0, \dots, 0)} (p_{1}, p_{2}, \dots, p_{n}) \\ = \sum_{i = 1}^{n} ((1 + T (p_{i})) p_{i} / - \sum_{j = 1}^{n} (1 + T (p_{j}))), \end{matrix}$ (25) which is the Pythagorean fuzzy power averaging operator.

Case 2. If R = (1/ - n, 1/ - n, …, 1/ - n), then PFPMM is reduced to $\begin{matrix} {PFPMM}^{(1, 0, \dots, 0)} (p_{1}, p_{2}, \dots, p_{n}) \\ = \prod_{i = 1}^{n} p_{i}^{\frac{1 + T (p_{i})}{\sum_{j = 1}^{n} (1 + T (p_{j}))}}, \end{matrix}$ (26) which is Pythagorean fuzzy power geometric operator.

Case 3. If R = (1, 1, 0, 0, …, 0), then PFPMM is reduced to Equation (26), which is shown at the top of this page and is the Pythagorean fuzzy power Bonferroni mean operator. $\begin{array}{l} P F P M M \overset{k}{\overset{︷}{(1, 1, \dots, 0}} {\overset{k}{\overset{︷}{0, 0, \dots, 0}}}_{(p_{1}, p_{2}, \dots, p_{n})} \\ {({(1 - \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - {(1 - μ_{i_{j}}^{2})}^{ω i_{j}})))}^{\frac{1}{c_{n}^{k}}})}^{\frac{1}{2 k}}, \\ \sqrt{1 - {(1 - \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} {(1 - \prod_{j = 1}^{k} (1 - υ_{i_{j}}^{2 ω i_{j}}))}^{\frac{1}{c_{n}^{k}}})}^{\frac{1}{k}}}) . \end{array}$ (27) $\begin{array}{l} W P F P M M^{R} (p_{1}, p_{2}, \dots p_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} (\frac{n ω θ (i) w ϑ (i)}{\sum_{j = 1}^{n} ω_{j} w_{j}} p ϑ (i)))}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \end{array}$ (28) $\begin{array}{l} W P F P M M^{R} (p_{1}, p_{2}, \dots p_{n}) \\ = {({(\sqrt{1 - (\prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - μ_{ϑ (i)}^{2})}^{^{\frac{n ω θ (i) w ϑ (i)}{\sum_{j = 1}^{n} {\bar{ω}}_{j} w_{j}}}})}^{r_{i}})})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \\ = \sqrt{1 - {(1 - {\prod_{ϑ \in S_{n}} (1 - {\prod_{i = 1}^{n} (1 - {(υ_{ϑ (j)})}^{^{\frac{2 n n ω θ (i) w ϑ (i)}{\sum_{j = 1}^{n} ω_{j} w_{j}}}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}} . \end{array}$ (29)

Case 4. If $R = {\overset{︷}{(1, 1, \dots, 1}}^{k}, {\overset{︷}{0, 0, \dots, 0)}}^{n - k}$ , then PFPMM is reduced to Equation (27), which is shown at the top of this page and is the Pythagorean fuzzy power Maclaurin symmetric meanoperator. $\begin{array}{l} P F P M M \overset{k}{\overset{︷}{(1, 1, \dots, 1}} {\overset{k}{\overset{︷}{0, 0, \dots, 0)}}}_{(p_{1}, p_{2}, \dots, p_{n})} \\ = {((1 - \prod_{1 \leq i_{1} < \dots < i_{k} \leq n} (1 - \prod_{j = 1}^{k} (1 - μ_{i_{j}}^{2}) ω_{i j})) \frac{1}{c_{n}^{k}})}^{\frac{1}{2 k}}, \\ \sqrt{1 - {(1 - \prod_{1 \leq i_{1} < \dots < i_{k} \leq n}^{k} (1 - \prod_{j = 1}^{k} (1 - υ_{i_{j}}^{2 ω i_{j}})) \frac{1}{C_{n}^{k}})}^{\frac{1}{k}}}) . \end{array}$ (30)

3.2 Weighted Pythagorean fuzzy power Muirhead mean

The PFPMM does not consider the importance of the aggregated PFNs. In this subsection, we propose the weighted Pythagorean fuzzy power Muirhead mean (WPFPMM) operator, which is capable of considering the weights of PFNs.

Definition 9. Let p_i (i = 1, 2, …, n) be a collection of PFNs and R = (r₁, r₂, …, r_n) ∈ Rⁿ be a vector of parameters. Then, the WPFPMM is defined as follows: $\begin{matrix} {WPFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) \\ = {(\frac{1}{n!} \sum_{ϑ \in S_{n}} \prod_{i = 1}^{n} {(\frac{n ω_{θ (i)} w_{ϑ (i)}}{\sum_{j = 1}^{n} ω_{j} w_{j}} p_{ϑ (i)})}^{r_{i}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \end{matrix}$ (31) where w = (w₁, w₂, …, w_n) ^T is the weight vector of p_i (i = 1, 2, …, n) satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , and ω_j is the PWV, satisfying $ω_{j} = (1 + T (p_{j})) / - (\sum_{j = 1}^{n} (1 + T (p_{j})))$ ; $\sum_{j = 1}^{n} ω_{j} = 1$ , $T (p_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (p_{i}, p_{j})$ , and Sup (p_i, p_j) is the support for p_i and p_j, satisfying Property 2.

Similarly, we can obtain the following theorem according to the operations for PFNs provided in Definition 2.

Theorem 5. Let p_i (i = 1, 2, …, n) be a collection of PFNs and R = (r₁, r₂, …, r_n) ∈ Rⁿ be a vector of parameters. Then, the aggregated value by WPFPMM is also a PFN, and $\begin{matrix} {WPFPMM}^{R} (p_{1}, p_{2}, \dots, p_{n}) \\ = ({(\sqrt{1 - {(\prod_{ϑ \in S_{n}} (1 - \prod_{i = 1}^{n} {(1 - {(1 - μ_{ϑ (i)}^{2})}^{\frac{n ω_{θ (i)} w_{ϑ (i)}}{\sum_{j = 1}^{n} ϖ_{j} w_{j}}})}^{r_{i}}))}^{\frac{1}{n!}}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}, \\ \sqrt{1 - {(1 - \prod_{ϑ \in S_{n}} {(1 - \prod_{i = 1}^{n} {(1 - {(v_{ϑ (j)})}^{\frac{2 n ω_{θ (i)} w_{ϑ (i)}}{\sum_{j = 1}^{n} ω_{j} w_{j}}})}^{r_{i}})}^{\frac{1}{n!}})}^{\frac{1}{\sum_{i = 1}^{n} r_{i}}}}) . \end{matrix}$ (32)

4 An approach to MADM with Pythagorean fuzzy information

In this section, we present a novel method to MADM in which attribute values take the form of PFNs. Consider a typical MADM problem in which X ={ x₁, x₂, …, x_m } is a series of alternatives and G ={ G₁, G₂, …, G_n } is a set of attributes with the weight vector w = (w₁, w₂, …, w_n) ^T, thereby satisfying $\sum_{i = 1}^{n} w_{i} = 1$ and w_i ∈ [0, 1]. For the attribute G_j (j = 1, 2, …, n) of alternative x_i (i = 1, 2, …, m), decision makers are required to use a PFN p_ij = (μ_ij, v_ij) where i = 1, 2, …, m and j = 1, 2, …, n to express their preference information. Therefore, a Pythagorean fuzzy decision matrix can be obtained. In the following, we utilize the proposed Pythagorean fuzzy aggregation operators to solve this problem.

Step 1. Standardize the original decision matrix. Generally, attributes can be categorized into two types: benefit and cost attributes. Thus, the original decision matrix should be standardized as follows: $p_{ij} = {\begin{matrix} (μ_{ij}, v_{ij}) & G_{j} \in I_{1} \\ (v_{ij}, μ_{ij}) & G_{j} \in I_{2} \end{matrix},$ (33) where I₁ and I₂ represent the benefit type and cost type attributes, respectively.

Step 2. The supports are calculated as follows: $Sup (p_{ik}, p_{il}) = 1 - d (p_{ik}, p_{il}),$ (34) where i = 1, 2, …, m, k, l = 1, 2, …, n and k ≠ l.

Step 3. T (p_ij) is calculated as follows: $T (p_{ij}) = \sum_{substack l = 1 l \neq k}^{n} Sup (p_{ik}, p_{il}),$ (35) where i = 1, 2, …, m, k, l = 1, 2, …, n and k ≠ l.

Step 4. The weight vector ω_ij of the power operator of the associated PFN p_ij is calculated as follows: $ω_{ij} = \frac{(1 + T (p_{ij}))}{\sum_{k = 1}^{n} (1 + T (p_{ik}))},$ (36) where i = 1, 2, …, m, k, l = 1, 2, …, n and k ≠ l.

Step 5. For each alternative, utilize WPFPMM $p_{i} = {WPFPMM}^{R} (p_{i 1}, p_{i 2}, \dots, p_{in}),$ (37) to aggregate the corresponding evaluation information. Thus, a series of overall values can be obtained.

Step 6. The scores of the overall values are calculated and ranked.

Step 7. The alternatives are ranked according to the rank of the corresponding overall values, and the best alternative is selected.

5 Numerical example

This section provides a numerical example adopted from Ma and Xu [25] to illustrate the validity of the proposed method.

The civil aviation administration of Taiwan hopes to identify the best Taiwanese airline to allow others to learn from it. Several decision makers comprise the committee that will decide which among the four domestic airlines is the best. The airlines are UNI Air (x₁), TransAsia (x₂), Mandarin (x₃), and Daily Air (x₄). To select the best airline, the four alternatives are evaluated from several aspects: booking and ticketing service (G₁), check-in and boarding process (G₂), cabin service (G₃), and responsiveness (G₄). The weight vector of the attributes is w = (0.15, 0.25, 0.35, 0.25) ^T. For attribute G_j (j = 1, 2, 3, 4) of alternative x_i (i = 1, 2, 3, 4), decision makers are required to utilize PFNs to represent their preference. Therefore, a Pythagorean fuzzy decision making matrix can be obtained and is shown as follows: $\begin{matrix} P_{4 \times 4} (x_{ij}) \\ = [\begin{matrix} (0.9, 0.3) (0.7, 0.6) (0.5, 0.8) (0.6, 0.3) \\ (0.4, 0.7) (0.9, 0.2) (0.8, 0.1) (0.5, 0.3) \\ (0.8, 0.4) (0.7, 0.5) (0.6, 0.2) (0.7, 0.4) \\ (0.7, 0.2) (0.8, 0.2) (0.8, 0.4) (0.6, 0.6) \end{matrix}] . \end{matrix}$

5.1 Decision-making process

In the following steps, we utilize the proposed method to solve the problem.

Step 1. Evidently, all the attributes are of the benefit type. Thus, they do not have to be standardized.

Step 2. The supports Sup (p_ik, p_il) are calculated using Equation (31), where i, k, l = 1, 2, 3, 4 and k ≠ l.

When i = 1, $\begin{matrix} Sup (p_{11}, p_{12}) = 0.68, & Sup (p_{11}, p_{13}) = 0.44, \\ Sup (p_{11}, p_{14}) = 0.55, & Sup (p_{12}, p_{11}) = 0.68, \\ Sup (p_{12}, p_{13}) = 0.72, & Sup (p_{12}, p_{14}) = 0.60, \\ Sup (p_{13}, p_{11}) = 0.44, & Sup (p_{13}, p_{12}) = 0.72, \\ Sup (p_{13}, p_{14}) = 0.45, & Sup (p_{14}, p_{11}) = 0.55, \\ Sup (p_{14}, p_{12}) = 0.60, & Sup (p_{14}, p_{13}) = 0.45 . \end{matrix}$

When i = 2, $\begin{matrix} Sup (p_{21}, p_{22}) = 0.35, & Sup (p_{21}, p_{23}) = 0.52, \\ Sup (p_{21}, p_{24}) = 0.60, & Sup (p_{22}, p_{21}) = 0.35, \\ Sup (p_{22}, p_{23}) = 0.80, & Sup (p_{22}, p_{24}) = 0.44, \\ Sup (p_{23}, p_{21}) = 0.52, & Sup (p_{23}, p_{22}) = 0.80, \\ Sup (p_{23}, p_{24}) = 0.61, & Sup (p_{24}, p_{21}) = 0.60, \\ Sup (p_{24}, p_{22}) = 0.44, & Sup (p_{24}, p_{23}) = 0.61 . \end{matrix}$

When i = 3, $\begin{matrix} Sup (p_{31}, p_{32}) = 0.85, & Sup (p_{31}, p_{33}) = 0.60, \\ Sup (p_{31}, p_{34}) = 0.85, & Sup (p_{32}, p_{31}) = 0.85, \\ Sup (p_{32}, p_{33}) = 0.66, & Sup (p_{32}, p_{34}) = 0.91, \\ Sup (p_{33}, p_{31}) = 0.60, & Sup (p_{33}, p_{32}) = 0.66, \\ Sup (p_{33}, p_{34}) = 0.75, & Sup (p_{34}, p_{31}) = 0.85, \\ Sup (p_{34}, p_{32}) = 0.91, & Sup (p_{34}, p_{33}) = 0.75 . \end{matrix}$

When i = 4, $\begin{matrix} Sup (p_{41}, p_{42}) = 0.85, & Sup (p_{41}, p_{43}) = 0.73, \\ Sup (p_{41}, p_{44}) = 0.68, & Sup (p_{42}, p_{41}) = 0.85, \\ Sup (p_{42}, p_{43}) = 0.88, & Sup (p_{42}, p_{44}) = 0.68, \\ Sup (p_{43}, p_{41}) = 0.73, & Sup (p_{43}, p_{42}) = 0.88, \\ Sup (p_{43}, p_{44}) = 0.72, & Sup (p_{44}, p_{41}) = 0.68, \\ Sup (p_{44}, p_{42}) = 0.68, & Sup (p_{44}, p_{43}) = 0.72 . \end{matrix}$

Step 3. In accordance with Equation (32), T (p_ij) is calculated where i, j = 1, 2, 3, 4: $\begin{matrix} T (p_{11}) = 1.67, & T (p_{12}) = 2 . 0, & T (p_{13}) = 1.61, \\ T (p_{14}) = 1.6, & T (p_{21}) = 1.47, & T (p_{22}) = 1.59, \\ T (p_{23}) = 1.93, & T (p_{24}) = 1.65, & T (p_{31}) = 2.30, \\ T (p_{32}) = 2.42, & T (p_{33}) = 2.01, & T (p_{34}) = 2.51, \\ T (p_{41}) = 2.26, & T (p_{42}) = 2.41, & T (p_{43}) = 2.33, \\ T (p_{44}) = 2.08 . \end{matrix}$

Step 4. According to Equation (33), the PWVs can obtained, shown as follows: $\begin{matrix} ω_{11} = 0.2454, & ω_{12} = 0.2757, & ω_{13} = 0.2399, \\ ω_{14} = 0.2390, & ω_{21} = 0.2321, & ω_{22} = 0.2434, \\ ω_{23} = 0.2754, & ω_{24} = 0.2491, & ω_{31} = 0.2492, \\ ω_{32} = 0.2583, & ω_{33} = 0.2273, & ω_{34} = 0.2652, \\ ω_{41} = 0.2492, & ω_{42} = 0.2607, & ω_{43} = 0.2546, \\ ω_{44} = 0.2355 . \end{matrix}$

Step 5. For each alternative, Equation (34) is utilized to aggregate the assessments. In this step, we assume that R = (1, 1, 1, 1). Therefore, we can obtain $\begin{matrix} p_{1} = (0.8814, 0.3439), & p_{2} = (0.8769, 0.2476), \\ p_{3} = (0.8757, 0.2849), & p_{4} = (0.8850, 0.2634) . \end{matrix}$

Step 6. The scores of the overall values are calculated according to Definition 3. Thus, we have $\begin{matrix} s (p_{1}) = 0.6585, & s (p_{2}) = 0.7078, & s (p_{3}) = 0.6858, \\ s (p_{4}) = 0.7138 . \end{matrix}$

Therefore, the rank of the overall values is p₄ > p₂ > p₃ > p₁.

Step 7. According to the rank of the overall values, we can obtain the rank of the alternatives, that is, x₄ ≻ x₂ ≻ x₃ ≻ x₁. Therefore, Daily Air is the best airline.

According to Ma and Xu [25], the ranking result is also x₄ ≻ x₂ ≻ x₃ ≻ x₁ when the Pythagorean fuzzy weighted geometric operator is used to aggregate the assessments. This result proves the validity of the proposed method.

5.2 Influence of parameter vector R on ranking results

The proposed method to MADM problems has two prominent advantages. First, it can reduce the bad effects of the unduly high and low arguments on the final results. Second, it can capture the interrelationship between Pythagorean fuzzy attribute values. Moreover, both the proposed aggregation operators have a parameter vector that leads to a flexible and nimble aggregation process. In other words, when different parameters are assigned to the PFPMM and WPFPMM operators, distinct overall values can be obtained, resulting in varying scores and ranking results. To illustrate the influence of the parameter vector R on the ranking results, we set different parameter vectors R in the WPFWMM operator and discuss the ranking results. The details are in Table 1.

Table 1
Ranking results obtained by utilizing different parameter vectors R in WPFPMM operator

Parameter vector R Score functions s (p_i) Ranking results

R = (1, 0, 0, 0) s (p₁) = -0.0315 s (p₂) = 0.1263 s (p₃) = 0.0587 s (p₄) = 0.1262 x₂ ≻ x₄ ≻ x₃ ≻ x₁

R = (1, 1, 0, 0) s (p₁) = 0.3808 s (p₂) = 0.4726 s (p₃) = 0.4324 s (p₄) = 0.4794 x₄ ≻ x₂ ≻ x₃ ≻ x₁

R = (1, 1, 1, 0) s (p₁) = 0.5596 s (p₂) = 0.6237 s (p₃) = 0.5953 s (p₄) = 0.6304 x₄ ≻ x₂ ≻ x₃ ≻ x₁

R = (1, 1, 1, 1) s (p₁) = 0.6585 s (p₂) = 0.7078 s (p₃) = 0.6858 s (p₄) = 0.7138 x₄ ≻ x₂ ≻ x₃ ≻ x₁

R = (2, 0, 0, 0) s (p₁) = 0.0247 s (p₂) = 0.1834 s (p₃) = 0.0707 s (p₄) = 0.1429 x₂ ≻ x₄ ≻ x₃ ≻ x₁

R = (3, 0, 0, 0) s (p₁) = 0.0763 s (p₂) = 0.2280 s (p₃) = 0.0837 s (p₄) = 0.1590 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Parameter vector R	Score functions s (p_i)	Ranking results
R = (1, 0, 0, 0)	s (p₁) = -0.0315 s (p₂) = 0.1263 s (p₃) = 0.0587 s (p₄) = 0.1262	x₂ ≻ x₄ ≻ x₃ ≻ x₁
R = (1, 1, 0, 0)	s (p₁) = 0.3808 s (p₂) = 0.4726 s (p₃) = 0.4324 s (p₄) = 0.4794	x₄ ≻ x₂ ≻ x₃ ≻ x₁
R = (1, 1, 1, 0)	s (p₁) = 0.5596 s (p₂) = 0.6237 s (p₃) = 0.5953 s (p₄) = 0.6304	x₄ ≻ x₂ ≻ x₃ ≻ x₁
R = (1, 1, 1, 1)	s (p₁) = 0.6585 s (p₂) = 0.7078 s (p₃) = 0.6858 s (p₄) = 0.7138	x₄ ≻ x₂ ≻ x₃ ≻ x₁
R = (2, 0, 0, 0)	s (p₁) = 0.0247 s (p₂) = 0.1834 s (p₃) = 0.0707 s (p₄) = 0.1429	x₂ ≻ x₄ ≻ x₃ ≻ x₁
R = (3, 0, 0, 0)	s (p₁) = 0.0763 s (p₂) = 0.2280 s (p₃) = 0.0837 s (p₄) = 0.1590	x₂ ≻ x₄ ≻ x₃ ≻ x₁

Table 1 shows that by using different parameter vectors R, the scores of the overall values vary and different ranking results are obtained. Moreover, as we increase the number of attribute interrelationships being considered, the values of score functions also increase.

5.3 Comparative analysis

To illustrate the effectiveness and advantages of our proposed method, we conduct comparative analysis. We use some existing methods to solve the above example and analyze the results. In this section, we compare our method with that proposed by Ma and Xu [25] based on the Pythagorean fuzzy weighted averaging (PFWA) operator, that proposed by Wei and Lu [40] based on the Pythagorean fuzzy power weighted averaging (PFPWA) operator, that proposed by Liang et al. [30] based on the weighted Pythagorean fuzzy geometric Bonferroni mean (WPFGBM) operator, that proposed by Wei and Lu [31] based on the Pythagorean fuzzy weighted Maclaurin symmetric mean (PFWMSM) operator, and that proposed by Zhang et al. [32] based on the dual generalized Pythagorean fuzzy weighted Bonferroni geometric mean (DGPFWBGM) operator. The ranking results obtained by the six methods are shown in Table 2.

Table 2
Ranking results obtained by using different methods

Methods Score functions s (p_i)(i = 1, 2, 3, 4) Ranking results

Method proposed by Ma and Xu [25] based on PFWA s (p₁) = -0.0315, s (p₂) = 0.1263 s (p₃) = 0.0587, s (p₄) = 0.1262 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Method proposed by Wei and Lu [40] based on PFPWA s (p₁) = 0.1763, s (p₂) = 0.4864 s (p₃) = 0.2398, s (p₄) = 0.3587 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Method proposed by Liang et al. [30] based on WPFGBM (s = t = 1) s (p₁) = 0.3173, s (p₂) = 0.5969 s (p₃) = 0.4892, s (p₄) = 0.5563 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Method proposed by Wei and Lu [31] based on PFWMSM operator (k = 2) s (p₁) = 0.4127, s (p₂) = 0.6947 s (p₃) = 0.5698, s (p₄) = 0.6048 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Method proposed by Zhang et al. [32] based on DGPFWBGM operator (R = (1, 1, 1, 1)) s (p₁) = 0.5150, s (p₂) = 0.7103 s (p₃) = 0.5869, s (p₄) = 0.7032 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Method proposed in this study (R = (1, 1, 1, 1)) s (p₁) = 0.6585, s (p₂) = 0.7078 s (p₃) = 0.6858, s (p₄) = 0.7138 x₄ ≻ x₂ ≻ x₃ ≻ x₁

Methods	Score functions s (p_i)(i = 1, 2, 3, 4)	Ranking results
Method proposed by Ma and Xu [25] based on PFWA	s (p₁) = -0.0315, s (p₂) = 0.1263 s (p₃) = 0.0587, s (p₄) = 0.1262	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Method proposed by Wei and Lu [40] based on PFPWA	s (p₁) = 0.1763, s (p₂) = 0.4864 s (p₃) = 0.2398, s (p₄) = 0.3587	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Method proposed by Liang et al. [30] based on WPFGBM (s = t = 1)	s (p₁) = 0.3173, s (p₂) = 0.5969 s (p₃) = 0.4892, s (p₄) = 0.5563	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Method proposed by Wei and Lu [31] based on PFWMSM operator (k = 2)	s (p₁) = 0.4127, s (p₂) = 0.6947 s (p₃) = 0.5698, s (p₄) = 0.6048	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Method proposed by Zhang et al. [32] based on DGPFWBGM operator (R = (1, 1, 1, 1))	s (p₁) = 0.5150, s (p₂) = 0.7103 s (p₃) = 0.5869, s (p₄) = 0.7032	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Method proposed in this study (R = (1, 1, 1, 1))	s (p₁) = 0.6585, s (p₂) = 0.7078 s (p₃) = 0.6858, s (p₄) = 0.7138	x₄ ≻ x₂ ≻ x₃ ≻ x₁

The Ma and Xu’s [25] method is based on the basic weighted average operator, which cannot consider the interrelationship between PFNs. In addition, this method cannot consider power weighting, which means that the bad influences of unduly high and low arguments on the final ranking cannot be reduced or eliminated. Our method is based on the WPFPMM operator, which can capture the interrelationship between attribute values. Moreover, our method considers power weighting, enabling the elimination of the influences of the unreasonable data.

The Wei and Lu’s [40] method and proposed method in this study are based on PA, which can reduce the bad influence of unduly high or low arguments on the final ranking results. The difference between the two methods is that the proposed method in this study can also consider the interrelationship between the aggregated arguments, whereas the Wei and Lu’s [40] method cannot. Consequently, the ranking results produced by the two methods are different. In addition, if we set R = (1, 0, 0, 0) in the proposed method, which means that the interrelationship between the aggregated arguments is not considered, we obtain the ranking order x₂ ≻ x₄ ≻ x₃ ≻ x₁ which is the same as the result obtained by Wei and Lu’s [40] method. In other words, if we do not consider interrelationships among attribute values, then the same ranking result can be obtained as the Wei and Lu’s [40] method.

The Liang et al.’s [30] method is based on the WPFGBM operator, which can consider the interrelationship between arguments. The distinction between our method and that of Liang et al. [30] is that ours can also consider power weighting to eliminate the bad effects of unreasonable arguments. In addition, the Liang et al.’s [30] method is based on BM, which can only consider the interrelationship between any two arguments, whereas ours is based on MM which can consider the interrelationship between all the fused arguments. In practical decision-making problems, relationships commonly exist among all attributes for MADM problems. Moreover, BM is only a special case of MM. Due to the above reasons, our method is more reasonable than that of Liang et al. [30].

The Wei and Lu’s [31] method is based on the PFWMSM operator. Similar to the Liang et al.’ [30] method, Wei and Lu’s [31] can consider the interrelationship among attributes but cannot eliminate the effects of unreasonable data. Moreover, the Wei and Lu’s [31] method is based on MSM, which is a special case of MM. Thus, our method is better and more flexible than that of Wei and Lu [30].

The Zhang et al.’s [32] method is based on the DGPFWBGM operator. Although this method is better than those of Liang et al. [30] and Wei and Lu [31], it cannot consider power weighting. In other words, the Zhang et al.’s [32] method can consider the interrelationships among all arguments but cannot eliminate the effects of unreasonable data on the ranking results. Our method not only captures the interrelationship between aggregated attribute values but also eliminates the influence of unduly high or low arguments on the rankingresults.

Moreover, our method is more reasonable than other existing Pythagorean fuzzy MADM approaches. We conducted comparative analyses from a qualitative angle, the details of which are shown in Table 3.

Table 3

Comparison of different approaches and aggregation operators

Approaches	Whether or not the interrelationship of two attributes is captured	Whether or not the interrelationship of three attributes is captured	Whether or not the interrelationship of multiple attributes is captured	Whether or not the bad effects of unduly high and unduly low arguments can be reduced	Whether or not the parameter vector increases the flexibility of the method
GPFWA [23]	No	No	No	No	Yes
GPFPWA [23]	No	No	No	No	Yes
PFOWAWAD [24]	No	No	No	No	No
SPFWG [25]	No	No	No	No	No
SPFWA [25]	No	No	No	No	No
PFEWA [26]	No	No	No	No	No
GPFEWA [26]	No	No	No	No	Yes
PFEWG [27, 28]	No	No	No	No	No
GPFEWG [17, 28]	No	No	No	No	Yes
WPFPMM	Yes	Yes	Yes	Yes	Yes

As mentioned, interrelationships usually exist between attributes in real decision-making problems. Sometimes, decision makers provide unduly high or low arguments over alternatives, thereby negatively affecting the final ranking results. Our method is based on the WPFPMM operator, which can take full advantage of PA and MM operators. In other words, our proposed method can consider the interrelationship between attribute values and eliminate the bad effects of unreasonable arguments on the ranking results. In addition, our method is based on WPFPMM, which has a parameter vector, thereby increasing the flexibility of the aggregation operator. Some existing operators are merely special cases of it. Therefore, our proposed method is more reasonable for real applications than the other methods.

6 Conclusions

In this paper, we proposed several aggregation operators for aggregating PFNs. We combined the PA and MM operators and proposed the PMM operator. Furthermore, we extended the PMM operator to PFSs and proposed the PFPMM and WPFPMM operators. The proposed aggregation operators take full advantages of the PA and MM operators. In other words, the proposed operator not only captures interrelationships among all the aggregated values but also reduces the bad influence of unduly high or unduly low arguments on the final results. In addition, we investigated and discussed some desirable properties and special cases of the proposed operators. We also introduced a novel approach to MADM with Pythagorean fuzzy information based on the proposed operators. Finally, we provided a numerical instance to illustrate the effectiveness and superiority of the proposed method. In our future research, we will extend the PMM operator to different fuzzy environments, such as IFSs [3], hesitant fuzzy sets [41], dual hesitant fuzzy sets [42], and q-rung orthopair fuzzy sets [43].

Footnotes

Acknowledgments

This work was partially supported by a key program of the National Natural Science Foundation of China (Grant No. 71532002) and the Fundamental Research Funds for the Central Universities (Grant No. 2017YJS075).

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

Merigó

J.M.

, Gil-Lafuente

A.M.

, Yager

R.R.

, An overview of fuzzy research with bibliometric indicators, Applied Soft Computing 27 (2015), 420–433.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989), 37–46.

Deng

, Deng

W.K.

, Sun

X.P.

, Ye

C.H.

, Zhou

, Adaptive intuitionistic fuzzy enhancement of brain tumor MR images, Scientific Reports 6 (2016).

Maheshwari

, Srivastava

, Study on divergence measures for intuitionistic fuzzy sets and its application in medical diagnosis, Journal of Applied Analysis and Computation 6 (2016), 772–789.

Chou

W.S.

, New algorithm of similarity measures for pattern recognition problems, Journal of Testing and Evaluation 44 (2015), 1473–1484.

Chen

S.M.

, Cheng

S.H.

, Lan

T.C.

, A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition, Information Sciences 343 (2016), 15–40.

Wang

, Xu

Z.S.

, Liu

S.S.

, Yao

Z.Q.

, Direct clustering analysis based on intuitionistic fuzzy implication, Applied Soft Computing 23 (2014), 1–8.

Wang

, Xu

Z.S.

, Liu

S.S.

, Tang

, A netting clustering analysis method under intuitionistic fuzzy environment, Applied Soft Computing 11 (2011), 5558–5564.

10.

Zhang

Z.H.

, Hu

, Ma

, Xu

J.H.

, Yuan

S.G.

, Chen

, Incentive-punitive risk function with interval valued intuitionistic fuzzy information for outsourced software project risk assessment, Fuzzy Systems 32 (2017), 3749–3760.

11.

Xie

N.X.

, Li

Z.W.

, Zhang

G.Q.

, An intuitionistic fuzzy soft set method for stochastic decision-making applying prospect theory and grey relational analysis, Fuzzy Systems 33 (2017), 15–25.

12.

G.H.

, Qu

W.H.

, Zhang

Z.H.

, Wang

J.X.

, Choquet integral correlation coefficient of intuitionistic fuzzy sets and its applications, Fuzzy Systems 33 (2017), 543–553.

13.

Zeng

S.Z.

, Merigó

J.M.

, Palacios

M.D.

, Jin

H.H.

, Gu

F.J.

, Intuitionistic fuzzy induced ordered weighted averaging distance operator and its application to decision making, Fuzzy Systems 32 (2017), 11–22.

14.

Yager

R.R.

, Abbasov

A.M.

, Pythagorean membership grades, complex numbers, and decision making, International Journal of Intelligent Systems 28 (2013), 439–452.

15.

Yager

R.R.

, Pythagorean membership grades in multi-criteria decision making, IEEE Transactions on Fuzzy Systems 22 (2014), 958–965.

16.

Yager

R.R.

, On ordered weighted averaging aggregation operators in multicriteria decision-making, IEEE Transactions on Systems, Man, and Cybernetics 18 (1988), 183–190.

17.

Yager

R.R.

, On generalized Bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning 50 (2009), 1279–1286.

18.

Blanco-Mesa

, Merigó

J.M.

, Kacprzyk

, Bonferroni means with distance measures and the adequacy coefficient in entrepreneurial group theory, Knowledge-Based Systems 111 (2016), 217–227.

19.

Liu

J.P.

, Chen

H.Y.

, Xu

, Zhou

L.G.

, Tao

Z.F.

, Generalized ordered modular averaging operator and its application to group decision making, Fuzzy Sets and Systems 299 (2016), 1–25.

20.

Liu

J.P.

, Chen

H.Y.

, Zhou

L.G.

, Tao

Z.F.

, He

Y.D.

, On the properties of the generalized OWHA operators and their application to group decision making, Fuzzy Systems 4 (2014), 2077–2089.

21.

Liu

J.P.

, Chen

H.Y.

, Zhou

L.G.

, Tao

Z.F.

, Generalized linguistic ordered weighted hybrid logarithm averaging operators and applications to group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 3 (2015), 421–442.

22.

Blanco-Mesa

, Merigó

J.M.

, Gil-Lafuente

A.M.

, Fuzzy decision making: A bibliometric-based review, Fuzzy Systems 32 (2017), 2033–2050.

23.

Peng

X.D.

, Yuan

H.H.

, Fundamental properties of Pythagorean fuzzy aggregation operators, Fundamenta Informaticae 147 (2016), 415–446.

24.

Zeng

S.Z.

, Chen

J.P.

, Li

X.S.

, A hybrid method for Pythagorean fuzzy multiple-criteria decision making, Decision Making 15 (2016), 403–422.

25.

Z.M.

, Xu

Z.S.

, Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multi-criteria decision making problems, International Journal of Intelligent Systems 31 (2016), 1198–1219.

26.

Garg

, A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making, International Journal of Intelligent Systems 31 (2016), 886–920.

27.

Garg

, Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multi-criteria decision making process, International Journal of Intelligent Systems 32 (2017), 597–630.

28.

Rahman

, Abdullah

, Ahmed

, Ullah

, Pythagorean fuzzy Einstein weighted geometric aggregation operator and their application to multiple attribute group decision making, Fuzzy Systems 33 (2017), 635–647.

29.

Peng

X.D.

, Yang

, Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making, International Journal of Intelligent Systems 31 (2016), 989–1020.

30.

Liang

D.C.

, Xu

Z.S.

, Darko

A.P.

, Projection model for fusing the information of Pythagorean fuzzy multi-criteria group decision making based on geometric Bonferroni mean, International Journal of Intelligent Systems 32 (2017), 966–987.

31.

Wei

G.W.

, Lu

, Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making, International Journal of Intelligent Systems, published online.

32.

Zhang

R.T.

, Wang

, Zhu

X.M.

, Xia

M.M.

, Yu

, Some generalized Pythagorean fuzzy Bonferroni mean aggregation operators with their application to multi-attribute group decision making, Complexity (2017), Article ID 5937376.

33.

Yager

R.R.

, The power average operator, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 31 (2001), 724–731.

34.

Bonferroni

, Sulle medie multiple di potenze, Bollettino dell’Unione Matematica Italiana 5 (1950), 267–270.

35.

Sykora

, Mathematical means and averages: Generalized Heronian means, Sykora S Stan’s, Library, 2009.

36.

Muirhead

R.F.

, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proceedings of the Edinburgh Mathematical Society 21 (1902), 144–162.

37.

Liu

P.D.

, Li

D.F.

, Some Muirhead mean operators for intuitionistic fuzzy numbers and their applications to group decision making, Plos One 12, Article ID e 0168767.

38.

Maclaurin

, A second letter to Martin Folkes, Esq.; concerning the roots of equations, with demonstration of other rules of algebra, Philos Trans Roy Soc London Ser A 36 (1729), 59–96.

39.

Zhang

X.L.

, Xu

Z.S.

, Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, International Journal of Intelligent Systems 29 (2014), 1061–1078.

40.

Wei

G.W.

, Lu

, Pythagorean fuzzy power aggregation operators in multiple attribute decision making, International Journal of Intelligent Systems 33 (2018), 169–186.

41.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

42.

Zhu

, Xu

Z.S.

, Xia

M.M.

, Dual hesitant fuzzy sets, Journal of Applied Mathematics 2012 (2012), Article ID 879629.

43.

Yager

R.R.

, Generalized orthopair fuzzy sets, IEEE Transactions on Fuzzy Systems 25 (5), 1222–1230.