Multiple criteria decision making method based on the new similarity measures of Pythagorean fuzzy set

Abstract

As an extension of the intuitionistic fuzzy set, the Pythagorean fuzzy set can depict uncertain information more effectively, so it has been well applied in multiple criteria decision making problems. At present, the multiple criteria decision making methods using the Pythagorean fuzzy set are generally ranked based on the aggregation operator or the distance measure, ignoring the important tool of the similarity measure. Therefore, this paper proposes several new similarity measures of the Pythagorean fuzzy set and applies them to multiple criteria decision making problems. Firstly, several new similarity measures of the Pythagorean fuzzy set are proposed, and their properties are discussed. Then, based on the weighted similarity measures, the multiple criteria decision making method is proposed. Finally, the accuracy and reliability of the new similarity measures and the proposed multiple criteria decision making method are verified by the simulation cases.

Keywords

Intuitionistic fuzzy set Pythagorean fuzzy set similarity measure multiple criteria decision making

1 Introduction

The decision making (DM) problem is widely existed in various aspects of life [1]. In the actual DM process, the decision-maker will have to take into account many factors for making scientific and reasonable results [2]. These different factors are usually referred to as criteria, and the process in which decision makers rank a number of schemes according to multiple criteria is referred to as multiple criteria decision making (MCDM). MCDM can solve problems in many fields such as military, economic and management, for example, weapon threat assessment [3], commercial project selection [4], medical diagnosis [5] and supplier selection [6]. Therefore, the method of MCDM is a research hotspot.

With the deepening of the research of MCDM, the practical problems to be dealt with become more and more complex and changeable, and human thinking itself is fuzzy and uncertain, so how to depict the uncertainty in MCDM problem has become the mainstream of research. In 1965, Zadeh [7] put forward the fuzzy set (FS), which provides a new method for people to deal with fuzzy information. On the basis of the FS, Atanassov [8] puts forward the intuitionistic fuzzy set (IFS). The IFS characterize the FS by the membership function and the non-membership function, and the sum of the membership degree (MD) and the non-membership degree (ND) is less than or equal to 1. Yager [9] proposes the Pythagorean fuzzy set (PFS) by specifying the square sum of the MD and the ND is less than or equal to 1.

The PFS is well applied to MCDM problems. Yager [9] presented several kinds of aggregation operators of the PFS and applied them to MCDM problems. Grag [10] proposed an idea related to weighted aggregated operators of the PFS and applied these operators to DM problem. Wei and Lu [11] developed some Pythagorean fuzzy power aggregation operators and utilized these operators to develop some approaches to solve the Pythagorean fuzzy MCDM problems. Zeng et al. [12] developed a new method for Pythagorean fuzzy MCDM problems with aggregation operators and distance measures. Li et al. [13] proposed the Hamming distance measure, the Euclidean distance measure and the Minkowski distance measure between the PFSs, and then presented a MCDM method in the Pythagorean fuzzy environment based on the proposed distance measures. Zhang and Xu [14] defined a distance measure to calculate the distances between each alternative and the Pythagorean fuzzy ideal solution, and then proposed a novel DM method based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to deal with the MCDM problem.

It can be seen from the above that the MCDM methods of the PFS can be divided into two categories: one ranking based on the aggregation operator; and one ranking based on the distance measure. There is also an important measure of information in the FS: similarity measure. Similarity measure is used as a tool to measure the similarity degree between two objects, and is widely used in many fields, such as pattern recognition, medical diagnosis, machine learning and image processing.

At present, there are many studies on the similarity measure of the IFS. Gong [15] proposed a new similarity measure of the IFS and applied it to pattern recognition. Li and Cheng [16] proposed some new similarity measures of the IFS and applied them to pattern recognition. Baccour et al. [17] presented several studies on similarity measures between the IFSs. Ye [18] proposed a cosine similarity measure and a weighted cosine similarity measure between the IFSs based on the concept of the cosine similarity measure of the FS. In order to improve the efficiency of the Ye’s cosine similarity measure, Hwang and Yang [19] modified the cosine similarity measure between the IFSs. Based on the cosine function and the information in the IFS, Ye [20] proposed two new cosine similarity measures. Muthukumar and Krishnan [21] proposed a new similarity measure and a weighted similarity measure on intuitionistic fuzzy soft set. Szmidt and Kacprzyk [22] proposed a new similarity measure of the IFS to analyze the agreement extent in a group of experts. Hung and Yang [23] presented a new similarity measure between the IFSs based on the Hausdorff distance concept. Comparatively speaking, there are few studies on the similarity measure of the PFS. Peng et al. [24] proposed several similarity measures of the PFS, and applied them to solve some practical problems. By considering the MD, the ND and the hesitation degree (HD) in the PFS, Wei and Wei [25] presented several similarity measures between the PFSs based on the cosine function, and applied them to solve the problems of pattern recognition and medical diagnosis. In order to handle the problems of multiple criteria group decision making (MCGDM) within Pythagorean fuzzy environment, Zhang [26] developed a new decision method based on the similarity measure. Zeng et al. [27] presented some similarity measures of the PFS based on a variety of distance measures, and then proposed a new Pythagorean fuzzy MCGDM approach based on the similarity measures.

Considering that the MCDM methods based on the similarity measures of the PFS are less studied. This paper proposes several new similarity measures of the PFS, and proposes a new MCDM method based on the similarity measures of the PFS. The rest of the paper is organized as follows: In section 2, the basic concepts of the IFS and the PFS are introduced. In section 3, six existing similarity measures of the PFS are introduced, and four new similarity measures of the PFS are proposed, and then the weighted similarity measures are defined. In section 4, a new MCDM method is proposed. In section 5, the new method is applied through some simulation cases. In section 6, the paper is summarized.

2 Preliminaries

2.1 Intuitionistic fuzzy set

Definition 1. [28] An IFS on the domain X is given by $I = {〈 x, μ_{I} (x), v_{I} (x) 〉 | x \in X}$ (1) where μ_I (x) : X → [0, 1] and ν_I (x) : X → [0, 1]. For any x ∈ X, μ_I (x) + ν_I (x) ≤ 1, then μ_I (x) and ν_I (x) are referred to as the MD and the ND of x to I, respectively. π_I (x) = 1 - μ_I (x) - ν_I (x) is the HD of x to I.

The intuitionistic fuzzy number (IFN) is represented by i = (μ_i, ν_i).

Definition 2. [27] Let i = (μ_i, ν_i) be an IFN, the score function and the exact function of can be defined as $\begin{matrix} S (i) = μ_{i} - ν_{i} \\ E (i) = μ_{i} + ν_{i} \end{matrix}$ (2)

For two IFNs i₁ and i₂, the following rules can be obtained as

if S (i₁) > S (i₂), then i₁ > i₂

if S (i₁) = S (i₂), then

if E (i₁) > E (i₂), then i₁ > i₂

if E (i₁) = E (i₂), then i₁ = i₂

2.2 Pythagorean fuzzy set

Definition 3. [29] A PFS on the domain X is given by $P = {〈 x, μ_{P} (x), v_{P} (x) 〉 | x \in X}$ (3) where μ_P (x) : X → [0, 1] and ν_P (x) : X → [0, 1]. For any $x \in X, μ_{P}^{2} (x) + ν_{P}^{2} (x) \leq 1$ . μ_P (x) and ν_P (x) are referred to as the MD and the ND of x to P, respectively. $π_{P} (x) = \sqrt{1 - μ_{P}^{2} (x) - ν_{P}^{2} (x)}$ is the HD of x to P.

The Pythagorean fuzzy number (PFN) is represented by p = (μ_p, ν_p) [14].

Definition 4. [30] Let p₁, p₂, …, p_n be n PFNs, where p_i = (μ_{p
_i}, ν_{p
_i}) (i = 1, 2, ⋯ , n). w₁, w₂, ⋯ , w_n are the weights with respect to the PFNs, satisfying 0 ≤ w_i ≤ 1 and $\sum_{i = 1}^{n} w_{i} = 1$ , then Pythagorean fuzzy weighted average (PFWA) operator can be defined as

$\begin{matrix} PFWA (p_{1}, p_{2}, \dots, p_{n}) \\ = (\sum_{i = 1}^{n} w_{i} μ_{p_{i}}, \sum_{i = 1}^{n} w_{i} ν_{p_{i}}) \end{matrix}$ (4)

Next, two ranking methods of the PFNs are defined as follow

Definition 5. [31] Let p = (μ_p, ν_p) be a PFN, the score function and the exact function of p can be defined as $\begin{matrix} S (p) = μ_{p}^{2} - ν_{p}^{2} \\ E (p) = μ_{p}^{2} + ν_{p}^{2} \end{matrix}$ (5)

For two PFNs p₁ and p₂, the ranking rules can be obtained as

if S (p₁) > S (p₂), then p₁ > p₂

if S (p₁) = S (p₂), then

if E (p₁) > E (p₂), then p₁ > p₂

if E (p₁) = E (p₂), then p₁ = p₂

Definition 6. [9] Let p = (r_p, d_p) be a PFN, the rank function of p can be defined as $R (p) = \frac{1}{2} + r_{p} (d_{p} - \frac{1}{2}) = \frac{1}{2} + r_{p} (\frac{1}{2} - \frac{2 θ_{p}}{π})$ (6) where $r_{P} = \sqrt{μ_{P}^{2} + ν_{P}^{2}}$ (7) $d_{P} = 1 - 2 θ_{p} / π$ (8) $θ_{P} = arctan (ν_{P} / μ_{P})$ (9)

For two PFNs p₁ and p₂, the ranking rules can be obtained as

if R (p₁) > R (p₂), then R (p₁) > R (p₂)

if R (p₁) = R (p₂), then R (p₁) = R (p₂)

Next, an example is given to compare the two ranking methods.

Example 1. Let α₁ = (0.6, 0.4) and α₂ = (0.5, 0.25). α₁ and α₂ can be regarded not only as IFNs, but also as PFNs. When they are regarded as IFNs, we can get S (α₁) = 0.2 and S (α₂) = 0.25 by using Equation (2). According to the ranking rules in Definition 2, we can get α₁ < α₂. When they are regarded as PFNs, we can get S (α₁) = 0.2 and S (α₂) = 0.1875 by using Equation (5). According to the ranking rules in Definition 5, we can get α₁ > α₂. It can be seen that when α₁ and α₂ are regarded as different types of fuzzy numbers, the ranking results obtained by using the corresponding ranking methods are opposite, which shows that the ranking method based on the score function and the exact function is unreliable in some cases. We can get R (α₁) = 0.5906 and R (α₂) = 0.6145 by using Equation (6), and can get α₁ < α₂ according to the ranking rules in Definition 6. Since the ranking method in Definition 6 takes into account the direction of r_P (x), the ranking result is more reliable than that obtained by the ranking method based on the score function and the exact function.

3 Similarity measures of pythagorean fuzzy set

3.1 Similarity measures

In this paper, six typical cosine similarity measures are selected from the reference [25] and four new similarity measures based on exponential function are proposed, which are defined as follows

Definition 7. [25] Let A = {〈x_j, μ_A (x_j) , ν_A (x_j) 〉|x_j ∈ X} and B = {〈x_j, μ_B (x_j) , ν_B (x_j) 〉|x_j ∈ X} be two PFNs on the domain X, then the similarity measures between A and B can be calculated as

$\begin{matrix} s m^{1} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} \\ \frac{μ_{A}^{2} (x_{j}) μ_{B}^{2} (x_{j}) + ν_{A}^{2} (x_{j}) ν_{B}^{2} (x_{j})}{\sqrt{μ_{A}^{4} (x_{j}) + ν_{A}^{4} (x_{j})} \sqrt{μ_{B}^{4} (x_{j}) + ν_{B}^{4} (x_{j})}} \end{matrix}$ (10)

$s m^{2} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} \frac{μ_{A}^{2} (x_{j}) μ_{B}^{2} (x_{j}) + ν_{A}^{2} (x_{j}) ν_{B}^{2} (x_{j}) + π_{A}^{2} (x_{j}) π_{B}^{2} (x_{j})}{\sqrt{μ_{A}^{4} (x_{j}) + ν_{A}^{4} (x_{j}) + π_{A}^{4} (x_{j})} \sqrt{μ_{B}^{4} (x_{j}) + ν_{B}^{4} (x_{j}) + π_{B}^{4} (x_{j})}}$ (11)

$\begin{matrix} s m^{3} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} cos [\frac{π}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \\ \lor | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |)] \end{matrix}$ (12)

$\begin{matrix} s m^{4} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} cos [\frac{π}{4} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \\ + | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |)] \end{matrix}$ (13)

$\begin{matrix} s m^{5} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} cos [\frac{π}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \lor \\ | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) | \lor | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |)] \end{matrix}$ (14)

$\begin{matrix} s m^{6} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} cos [\frac{π}{4} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | + | \\ ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) | + | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |)] \end{matrix}$ (15)

$\begin{matrix} s m^{7} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} \\ [2^{1 - (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \lor | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |)} - 1] \end{matrix}$ (16)

$\begin{matrix} s m^{8} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} \\ [2^{1 - \frac{1}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | + | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |)} - 1] \end{matrix}$ (17)

$\begin{matrix} s m^{9} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} [2^{1 - (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \lor | ν_{A}^{2} (x_{j})} \\ - ν_{B}^{2} (x_{j}) | \lor | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |) - 1] \end{matrix}$ (18)

$\begin{matrix} s m^{10} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} [2^{1 - \frac{1}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | + | ν_{A}^{2} (x_{j})} \\ - ν_{B}^{2} (x_{j}) | + | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |) - 1] \end{matrix}$ (19) where “∨ ” indicates the maximum operation.

The above similarity measures of the PFS are represented by sm^α ( ) (α = 1, 2, …, 10). When α = 1, 2, …, 6, sm^α ( ) is the existing similarity measures, and when α = 7, 8, 9, 10, sm^α ( ) is the new similarity measures.

Property 1. The similarity measures between two PFSs A and B should satisfy the following properties

0 ≤ sm^α (A, B) ≤ 1

sm^α (A, B) = 1 if and only if A = B

sm^α (A, B) = sm^α (B, A)

If C is a PFS and A ⊆ B ⊆ C, then sm^α(A, C) ≤ sm^α (A, B), sm^α (A, C) ≤ sm^α (B, C).

Next, an example is given to verify the superiority of the new similarity measures compared with the existing similarity measures.

Example 2. Let the feature space be X = {x₁, x₂, x₃}, A₁ = {(x₁, 0.5, 0.3) , (x₂, 0.6, 0.2) , (x₃, 0.8, 0.1)} and A₂ = {(x₁, 0.5, 0.3) , (x₂, 0.6, 0.2) , (x₃, 0.8, 0.13)} are two PFSs on X, respectively. A₁ and A₂ are highly similar but not exactly the same. Next, the existing similarity measures and the new similarity measures between A₁ and A₂ are calculated, respectively. The calculation results are as follows: $\begin{matrix} s m^{1} (A_{1}, A_{2}) = 1 & s m^{2} (A_{1}, A_{2}) = 1 \end{matrix}$ $\begin{matrix} s m^{3} (A_{1}, A_{2}) = 1 & s m^{4} (A_{1}, A_{2}) = 1 \end{matrix}$ $\begin{matrix} s m^{5} (A_{1}, A_{2}) = 1 & s m^{6} (A_{1}, A_{2}) = 1 \end{matrix}$ $\begin{matrix} s m^{7} (A_{1}, A_{2}) = 0.9968 & s m^{8} (A_{1}, A_{2}) = 0.9984 \end{matrix}$ $\begin{matrix} s m^{9} (A_{1}, A_{2}) = 0.9968 & s m^{10} (A_{1}, A_{2}) = 0.9968 \end{matrix}$

It can be seen from the calculation results that the existing similarity measures cannot distinguish A₁ and A₂ from each other, mistakenly thinking that the two are the same. The existing similarity measures are unreasonable in this case. The new similarity measures can distinguish the highly similar but inconsistent PFSs. In summary, the new similarity measures have better distinguish ability, and can be applied to solve a wider range of problems.

3.2 Weighted similarity measures

In MCDM problems, the decision criteria will be given different weights according to importance degree. Thus, the weighted similarity measures of the PFS are defined as follows

Definition 8. Let A = {〈x_j, μ_A (x_j) , ν_A (x_j) 〉|x_j ∈ X} and B = {〈x_j, μ_B (x_j) , ν_B (x_j) 〉|x_j ∈ X} be two PFNs on the domain X, w = (w₁, w₂, …, w_n) ^T be the weights with respect to the decision criteria. Then the weighted similarity measures between A and B can be calculated as

$\begin{matrix} {sm}_{w}^{1} (A, B) = \sum_{j = 1}^{n} w_{j} \\ \frac{μ_{A}^{2} (x_{j}) μ_{B}^{2} (x_{j}) + ν_{A}^{2} (x_{j}) ν_{B}^{2} (x_{j})}{\sqrt{μ_{A}^{4} (x_{j}) + ν_{A}^{4} (x_{j})} \sqrt{μ_{B}^{4} (x_{j}) + ν_{B}^{4} (x_{j})}} \end{matrix}$ (20)

${sm}_{w}^{2} (A, B) = \sum_{j = 1}^{n} w_{j} \frac{μ_{A}^{2} (x_{j}) μ_{B}^{2} (x_{j}) + ν_{A}^{2} (x_{j}) ν_{B}^{2} (x_{j}) + π_{A}^{2} (x_{j}) π_{B}^{2} (x_{j})}{\sqrt{μ_{A}^{4} (x_{j}) + ν_{A}^{4} (x_{j}) + π_{A}^{4} (x_{j})} \sqrt{μ_{B}^{4} (x_{j}) + ν_{B}^{4} (x_{j}) + π_{B}^{4} (x_{j})}}$ (21)

$\begin{matrix} {sm}_{w}^{3} (A, B) = \sum_{j = 1}^{n} w_{j} cos [\frac{π}{2} (| μ_{A}^{2} (x_{j}) \\ - μ_{B}^{2} (x_{j}) | \lor | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |)] \end{matrix}$ (22)

$\begin{matrix} {sm}_{w}^{4} (A, B) = \sum_{j = 1}^{n} w_{j} cos [\frac{π}{4} (| μ_{A}^{2} (x_{j}) \\ - μ_{B}^{2} (x_{j}) | + | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |)] \end{matrix}$ (23)

$\begin{matrix} {sm}_{w}^{5} (A, B) = \sum_{j = 1}^{n} w_{j} cos [\frac{π}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \\ \lor | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) | \lor | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |)] \end{matrix}$ (24)

$\begin{matrix} {sm}_{w}^{6} (A, B) = \sum_{j = 1}^{n} w_{j} cos [\frac{π}{4} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) | \\ + | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) | + | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |)] \end{matrix}$ (25)

$\begin{matrix} {sm}_{w}^{7} (A, B) = \sum_{j = 1}^{n} w_{j} [2^{1 - (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) |} \\ \lor | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |) - 1] \end{matrix}$ (26)

$\begin{matrix} {sm}_{w}^{8} (A, B) = \sum_{j = 1}^{n} w_{j} [2^{1 - \frac{1}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) |} \\ + | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) |) - 1] \end{matrix}$ (27)

$\begin{matrix} {sm}_{w}^{9} (A, B) = \sum_{j = 1}^{n} w_{j} [2^{1 - (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) |} \\ \lor | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) | \lor | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |) - 1] \end{matrix}$ (28)

$\begin{matrix} {sm}_{w}^{10} (A, B) = \sum_{j = 1}^{n} w_{j} [2^{1 - \frac{1}{2} (| μ_{A}^{2} (x_{j}) - μ_{B}^{2} (x_{j}) |} \\ + | ν_{A}^{2} (x_{j}) - ν_{B}^{2} (x_{j}) | + | π_{A}^{2} (x_{j}) - π_{B}^{2} (x_{j}) |) - 1] \end{matrix}$ (29) Where 0 ≤ w_j ≤ 1 and $\sum_{j = 1}^{n} w_{j} = 1$ .

The above weighted similarity measures of the PFS are represented by ${sm}_{w}^{β} () (β = 1, 2, \dots, 10)$ . When β = 1, 2, …, 6, ${sm}_{w}^{β} ()$ is the existing weighted similarity measure (hereinafter referred to as the existing measure), and when α = 7, 8, 9, 10, ${sm}_{w}^{β} ()$ is the new weighted similarity measure (hereinafter referred to as the new measure).

4 Multiple criteria decision making method

The process of MCDM can be simply described as: first, suppose A = {A₁, A₂, …, A_m} (i = 1, 2, …, m) is m decision schemes, C = {C₁, C₂, …, C_n} (j = 1, 2, …, n) is n decision criteria, E = {e₁, e₂, …, e_l} (k = 1, 2, …, l) is l decision makers; then, each decision maker judges m schemes according to n criteria, so as to obtain the decision values; finally, decision method is used to rank all schemes and select an optimal scheme.

Assuming that w = (w₁, w₂, …, w_n) ^T is the weight corresponding to the decision criterion, satisfying 0 ≤ w_j ≤ 1 and $\sum_{j = 1}^{n} w_{j} = 1$ , and v = (v₁, v₂, …, v_l) ^T is the weight corresponding to the decision maker, satisfying 0 ≤ v_k ≤ 1 and $\sum_{k = 1}^{l} v_{k} = 1$ . Assuming that $p_{ij}^{(k)} = (μ_{ij}^{k}, ν_{ij}^{k})$ is the decision value given by k-th decision maker to the j-th criterion of the i-th scheme, which is represented by the PFN. The Pythagorean fuzzy decision matrix $P^{(k)} = {(p_{ij}^{(k)})}_{m \times n}^{T}$ of decision maker e_k can be expressed as $P^{(k)} = \begin{matrix} A_{1} & A_{2} \dots & A_{m} \\ \begin{matrix} C_{1} \\ C_{2} \\ ⋮ \\ C_{n} \end{matrix} & [\begin{matrix} (μ_{11}^{k}, ν_{11}^{k}) \\ (μ_{21}^{k}, ν_{21}^{k}) \\ \begin{matrix} ⋮ \end{matrix} \\ (μ_{n 1}^{k}, ν_{n 1}^{k}) \end{matrix} & \begin{matrix} (μ_{12}^{k}, ν_{12}^{k}) \begin{matrix} \begin{matrix} \end{matrix} \dots \end{matrix} \\ (μ_{22}^{k}, ν_{22}^{k}) \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \\ \begin{matrix} ⋮ \end{matrix} \begin{matrix} ⋱ \end{matrix} \\ (μ_{n 2}^{k}, ν_{n 2}^{k}) \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \end{matrix} & \begin{matrix} (μ_{1 m}^{k}, ν_{1 m}^{k}) \\ (μ_{2 m}^{k}, ν_{2 m}^{k}) \\ \begin{matrix} ⋮ \end{matrix} \\ (μ_{nm}^{k}, ν_{nm}^{k}) \end{matrix}] \end{matrix}$ (30)

Next, the specific calculation process of the MCDM method based on the similarity measures of the PFS is as follows

Step 1. Using Equation (4) to aggregate the Pythagorean fuzzy decision matrix P^(k) = ${(p_{ij}^{(k)})}_{m \times n}^{T}$ of each decision maker to obtain a comprehensive Pythagorean fuzzy decision matrix P = (p_ij) _m×n^T

$\begin{matrix} p_{ij} = PFWA (p_{ij}^{(1)}, p_{ij}^{(2)}, \dots, p_{ij}^{(l)}) \\ = (\sum_{k = 1}^{l} v_{k} μ_{ij}^{k}, \sum_{k = 1}^{l} v_{k} ν_{ij}^{k}) \end{matrix}$ (31)

If there is only one decision maker, this step can be ignored.

Step 2. For P = (p_ij) _m×n^T, the ranking function value R (p_ij) is calculated by using Equation (6), and the Pythagorean fuzzy ranking matrix R (P) is obtained. The ranking function value R (p_ij) of all schemes with respect to the same criterion C_j is compared according to the ranking rules in Definition 6. Then the optimal value $p_{j}^{o} = max {(μ_{ij}, ν_{ij})}$ and the worst value $p_{j}^{w} = min {(μ_{ij}, ν_{ij})}$ . under criterion C_j are determined, and then the positive ideal criterion value p⁺ and the negative ideal criterion value p^- can be obtained as $\begin{matrix} p^{+} = (p_{1}^{+}, p_{2}^{+}, \dots, p_{j}^{+}, \dots, p_{n}^{+}) \\ p_{j}^{+} = {\begin{matrix} p_{j}^{o} \begin{matrix} j \in J^{+} \end{matrix} \\ p_{j}^{w} \begin{matrix} j \in J^{-} \end{matrix} \end{matrix} \end{matrix}$ (32) $\begin{matrix} p^{-} = (p_{1}^{-}, p_{2}^{-}, \dots, p_{j}^{-}, \dots, p_{n}^{-}) \\ p_{j}^{-} = {\begin{matrix} p_{j}^{w} \begin{matrix} j \in J^{+} \end{matrix} \\ p_{j}^{o} \begin{matrix} j \in J^{-} \end{matrix} \end{matrix} \end{matrix}$ (33) where, J⁺ and J^- are benefit criterion set and cost criterion set, respectively.

Step 3. The weighted similarity measures ${sm}_{w}^{β} (p_{i}, p^{+})$ between the criterion value vector p_i = (p_i1, p_i2, …, p_in) (i = 1, 2, …, m) of each scheme and the positive ideal criterion value are calculated. Similarly, the weighted similarity measures ${sm}_{w}^{β} (p_{i}, p^{-})$ between the criterion value vector and the negative ideal criterion value are also calculated.

Step 4. The relative closeness of each scheme is calculate as $λ_{i}^{β} = \frac{{sm}_{w}^{β} (p_{i}, p^{-})}{{sm}_{w}^{β} (p_{i}, p^{+}) + {sm}_{w}^{β} (p_{i}, p^{-})} (i = 1, 2, \dots, m)$ (34)

According to the size of $λ_{i}^{β}$ , the schemes are ranked, and the smaller the value of $λ_{i}^{β}$ , the better the scheme.

5 Simulation case

5.1 A case of multiple criteria decision making for single decision maker

The simulation case selects the company investment decision problem in the reference [10].

An investor plans to invest in a company. Five companies have been selected as candidates, namely, computer company A₁, furniture company A₂, automobile company A₃, chemical company A₄, food company A₅. The investor considers the following six criteria: technical strength C₁, expected profit C₂, market competitiveness C₃, anti-risk ability C₄, management ability C₅ and corporate culture C₆. The investor assesses the value of company A_i (i = 1, 2, …, 5) under the corresponding criterion C_j (j = 1, 2, …, 6). The scheme set is A ={ A₁, A₂, …, A₅ }, and the criterion set is C ={ C₁, C₂, …, C₆ }, the weight of the criterion set is w = (0.2, 0.1, 0.3, 0.15, 0.15, 0.1). Through market research, the investor has determined the decision values of six criteria of five companies, which are given in the form of the PFNs. Then the Pythagorean fuzzy decision matrix P = (p_ij) _5×6^T is obtained, as shown in Table 1.

Table 1
The Pythagorean fuzzy decision matrix

A ₁ A ₂ A ₃ A ₄ A ₅

C ₁ (0.2, 0.5) (0.3, 0.5) (0.4, 0.3) (0.4, 0.7) (0.6, 0.8)

C ₂ (0.3, 0.8) (0.8, 0.5) (0.7, 0.7) (0.6, 0.8) (0.4, 0.7)

C ₃ (0.6, 0.8) (0.5, 0.6) (0.6, 0.7) (0.4, 0.7) (0.7, 0.4)

C ₄ (0.3, 0.7) (0.5, 0.6) (0.5, 0.8) (0.7, 0.5) (0.3, 0.4)

C ₅ (0.2, 0.4) (0.4, 0.7) (0.5, 0.8) (0.5, 0.7) (0.5, 0.6)

C ₆ (0.8, 0.4) (0.6, 0.5) (0.7, 0.4) (0.5, 0.8) (0.4, 0.7)

	A ₁	A ₂	A ₃	A ₄	A ₅
C ₁	(0.2, 0.5)	(0.3, 0.5)	(0.4, 0.3)	(0.4, 0.7)	(0.6, 0.8)
C ₂	(0.3, 0.8)	(0.8, 0.5)	(0.7, 0.7)	(0.6, 0.8)	(0.4, 0.7)
C ₃	(0.6, 0.8)	(0.5, 0.6)	(0.6, 0.7)	(0.4, 0.7)	(0.7, 0.4)
C ₄	(0.3, 0.7)	(0.5, 0.6)	(0.5, 0.8)	(0.7, 0.5)	(0.3, 0.4)
C ₅	(0.2, 0.4)	(0.4, 0.7)	(0.5, 0.8)	(0.5, 0.7)	(0.5, 0.6)
C ₆	(0.8, 0.4)	(0.6, 0.5)	(0.7, 0.4)	(0.5, 0.8)	(0.4, 0.7)

The ranking function value is calculated and the Pythagorean fuzzy ranking matrix R (P) is obtained, as shown in Table 2.

Table 2

The Pythagorean fuzzy ranking matrix

	A ₁	A ₂	A ₃	A ₄	A ₅
C ₁	0.3612	0.4091	0.5452	0.3633	0.4097
C ₂	0.2679	0.6362	0.5000	0.4097	0.3633
C ₃	0.4097	0.4549	0.4549	0.3633	0.6367
C ₄	0.3155	0.4549	0.3638	0.5904	0.4548
C ₅	0.4084	0.3633	0.3638	0.4096	0.4549
C ₆	0.6832	0.5451	0.6367	0.3638	0.3633

After determining the optimal value $p_{j}^{o}$ and the worst value $p_{j}^{w}$ under criterion C_j, $p_{j}^{+}$ and $p_{j}^{-}$ are obtained (six criteria are all benefit index) $p_{j}^{+} = {\begin{matrix} (0.4, 0.3), (0.8, 0.5), (0.7, 0.4), \\ (0.7, 0.5), (0.5, 0.6), (0.8, 0.4) \end{matrix}}$ $p_{j}^{-} = {\begin{matrix} (0.2, 0.5), (0.3, 0.8), (0.4, 0.7), \\ (0.3, 0.7), (0.4, 0.7), (0.4, 0.7) \end{matrix}}$

Next, the weighted similarity measures between the criterion value of each scheme and $p_{j}^{+}$ are calculated, and the calculated results are shown in Table 3. The weighted similarity measures between the criterion value and $p_{j}^{-}$ are shown in Table 4.

Table 3

The calculated results of ${sm}_{w}^{β} (p_{A_{i}}, p^{+})$

	A ₁	A ₂	A ₃	A ₄	A ₅
${sm}_{w}^{1} (p_{A_{i}}, p^{+})$	0.7258	0.8599	0.8932	0.7583	0.8625
${sm}_{w}^{2} (p_{A_{i}}, p^{+})$	0.7496	0.9163	0.8560	0.7826	0.6989
${sm}_{w}^{3} (p_{A_{i}}, p^{+})$	0.8406	0.9495	0.9093	0.8740	0.8470
${sm}_{w}^{4} (p_{A_{i}}, p^{+})$	0.9091	0.9670	0.9537	0.9140	0.9202
${sm}_{w}^{5} (p_{A_{i}}, p^{+})$	0.8187	0.9495	0.9093	0.8740	0.7801
${sm}_{w}^{6} (p_{A_{i}}, p^{+})$	0.8187	0.9495	0.9093	0.8740	0.7801
${sm}_{w}^{7} (p_{A_{i}}, p^{+})$	0.6115	0.7589	0.7029	0.6506	0.6874
${sm}_{w}^{8} (p_{A_{i}}, p^{+})$	0.6936	0.8041	0.7896	0.7188	0.7685
${sm}_{w}^{9} (p_{A_{i}}, p^{+})$	0.5780	0.7589	0.7029	0.6506	0.6383
${sm}_{w}^{10} (p_{A_{i}}, p^{+})$	0.5780	0.7589	0.7029	0.6506	0.6383

Table 4

The calculated results of ${sm}_{w}^{β} (p_{A_{i}}, p^{-})$

	A ₁	A ₂	A ₃	A ₄	A ₅
${sm}_{w}^{1} (p_{A_{i}}, p^{-})$	0.9470	0.8992	0.8452	0.9299	0.8487
${sm}_{w}^{2} (p_{A_{i}}, p^{-})$	0.8593	0.9187	0.8710	0.8883	0.7402
${sm}_{w}^{3} (p_{A_{i}}, p^{-})$	0.9385	0.9464	0.9379	0.9442	0.8986
${sm}_{w}^{4} (p_{A_{i}}, p^{-})$	0.9599	0.9596	0.9621	0.9691	0.9216
${sm}_{w}^{5} (p_{A_{i}}, p^{-})$	0.8928	0.9464	0.9188	0.9228	0.8230
${sm}_{w}^{6} (p_{A_{i}}, p^{-})$	0.8928	0.9464	0.9188	0.9228	0.8230
${sm}_{w}^{7} (p_{A_{i}}, p^{-})$	0.8044	0.8091	0.7299	0.7940	0.6756
${sm}_{w}^{8} (p_{A_{i}}, p^{-})$	0.8392	0.8365	0.7927	0.8504	0.7209
${sm}_{w}^{9} (p_{A_{i}}, p^{-})$	0.7338	0.8091	0.6870	0.7561	0.6149
${sm}_{w}^{10} (p_{A_{i}}, p^{-})$	0.7338	0.8091	0.6870	0.7561	0.6149

Next, the relative closeness of each scheme is calculated, as shown in Table 5.

Table 5

The calculated results of the relative closeness $λ_{A_{i}}^{β}$

	A ₁	A ₂	A ₃	A ₄	A ₅
$λ_{A_{i}}^{1}$	0.9470	0.8992	0.8452	0.9299	0.8487
$λ_{A_{i}}^{2}$	0.8593	0.9187	0.8710	0.8883	0.7402
$λ_{A_{i}}^{3}$	0.9385	0.9464	0.9379	0.9442	0.8986
$λ_{A_{i}}^{4}$	0.9599	0.9596	0.9621	0.9691	0.9216
$λ_{A_{i}}^{5}$	0.8928	0.9464	0.9188	0.9228	0.8230
$λ_{A_{i}}^{6}$	0.8928	0.9464	0.9188	0.9228	0.8230
$λ_{A_{i}}^{7}$	0.8044	0.8091	0.7299	0.7940	0.6756
$λ_{A_{i}}^{8}$	0.8392	0.8365	0.7927	0.8504	0.7209
$λ_{A_{i}}^{9}$	0.7338	0.8091	0.6870	0.7561	0.6149
$λ_{A_{i}}^{10}$	0.7338	0.8091	0.6870	0.7561	0.6149

From Table 5, the ranking results of each scheme are shown in Table 6.

Table 6

The ranking results of each scheme

	Ranking results
$λ_{A_{i}}^{1}$	A₃ > A₅ > A₂ > A₄ > A₁
$λ_{A_{i}}^{2}$	A₂ > A₃ > A₅ > A₄ > A₁
$λ_{A_{i}}^{3}$	A₂ > A₃ > A₅ > A₄ > A₁
$λ_{A_{i}}^{4}$	A₂ > A₅ > A₃ > A₁ > A₄
$λ_{A_{i}}^{5}$	A₂ > A₃ > A₅ > A₄ > A₁
$λ_{A_{i}}^{6}$	A₂ > A₃ > A₅ > A₄ > A₁
$λ_{A_{i}}^{7}$	A₅ > A₃ > A₂ > A₄ > A₁
$λ_{A_{i}}^{8}$	A₅ > A₃ > A₂ > A₄ > A₁
$λ_{A_{i}}^{9}$	A₅ > A₃ > A₂ > A₄ > A₁
$λ_{A_{i}}^{10}$	A₅ > A₃ > A₂ > A₄ > A₁

It can be seen that the ranking results obtained by the new measures all are A₅ > A₃ > A₂ > A₄ > A₁, and the ranking results obtained by the existing measures are A₂ > A₃ > A₅ > A₄ > A₁ (except ${sm}_{w}^{1} ()$ and ${sm}_{w}^{4} ()$ ).

The ranking results obtained by the new measures and the existing measures are different. In order to verify the accuracy and reliability of the new measures compared with the existing measures, the PFWA operator in Definition 5 and the ranking method in Definition 6 (hereinafter referred to as the verification method) are used to rank all schemes.

First, the comprehensive decision value of each scheme is calculated by Equation (4). $PFWA (A_{1}) = (0.4050, 0.6250)$ $PFWA (A_{2}) = (0.4850, 0.5750)$ $PFWA (A_{3}) = (0.5500, 0.6200)$ $PFWA (A_{4}) = (0.4900, 0.6900)$ $PFWA (A_{5}) = (0.5300, 0.5700)$

Then, the ranking function value of the comprehensive decision value of each scheme is calculated by Equation (6). $\begin{matrix} R (A_{1}) = 0.4002 & R (A_{2}) = 0.4594 & R (A_{3}) = 0.4685 \\ R (A_{4}) = 0.4095 & R (A_{5}) = 0.4820 \end{matrix}$

According to the ranking rules in Definition 6, the ranking result of schemes is A₅ > A₃ > A₂ > A₄ > A₁, which is completely consistent with the ranking results obtained by the new measures. Therefore, the accuracy and reliability of the new measures are verified.

5.2 A case of multiple criteria decision making for multiple decision makers

The simulation case selects the stock investment decision problem in the reference [30].

An investment bank wants to make a big profit in Internet stocks, so they invite four experts as decision makers to assess the potential value of stocks to be invested. Four stocks with high returns have been selected as candidates, namely, SINA A₁, BIDU A₂, NETS A₃, BABA A₄. The investment bank considers the following three criteria: market trend C₁, policy guide C₂, financial performance C₃. The experts e_k (k = 1, 2, 3, 4) assess the value of stock A_i (i = 1, 2, 3, 4) under the criterion C_j (j = 1, 2, 3). The scheme set is A ={ A₁, A₂, A₃, A₄ }, the criterion set is C ={ C₁, C₂, C₃ }, the expert set is E ={ e₁, e₂, e₃, e₄ }. The weight of the criterion set is w = (0.35, 0.35, 0.3), the weight of the expert set is v = (0.3, 0.2, 0.25, 0.25).

Next, each expert’s Pythagorean fuzzy decision matrix $P^{(k)} = {(p_{ij}^{(k)})}_{4 \times 3}^{T}$ is constructed, as shown in Tables 7 –10, respectively.

Table 7
The Pythagorean fuzzy ranking matrix constructed by expert e₁

A ₁ A ₂ A ₃ A ₄

C ₁ (0.9, 0.3) (0.4, 0.7) (0.8, 0.4) (0.7, 0.2)

C ₂ (0.7, 0.6) (0.9, 0.2) (0.7, 0.5) (0.8, 0.2)

C ₃ (0.5, 0.8) (0.8, 0.1) (0.6, 0.2) (0.8, 0.4)

	A ₁	A ₂	A ₃	A ₄
C ₁	(0.9, 0.3)	(0.4, 0.7)	(0.8, 0.4)	(0.7, 0.2)
C ₂	(0.7, 0.6)	(0.9, 0.2)	(0.7, 0.5)	(0.8, 0.2)
C ₃	(0.5, 0.8)	(0.8, 0.1)	(0.6, 0.2)	(0.8, 0.4)

Table 8

The Pythagorean fuzzy ranking matrix constructed by expert e₂

	A ₁	A ₂	A ₃	A ₄
C ₁	(0.7, 0.2)	(0.6, 0.7)	(0.7, 0.1)	(0.6, 0.6)
C ₂	(0.9, 0.2)	(0.5, 0.6)	(0.6, 0.5)	(0.5, 0.8)
C ₃	(0.7, 0.2)	(0.6, 0.2)	(0.8, 0.4)	(0.6, 0.4)

Table 9

The Pythagorean fuzzy ranking matrix constructed by expert e₃

	A ₁	A ₂	A ₃	A ₄
C ₁	(0.8, 0.1)	(0.7, 0.6)	(0.9, 0.4)	(0.8, 0.6)
C ₂	(0.9, 0.2)	(0.6, 0.4)	(0.8, 0.6)	(0.7, 0.5)
C ₃	(0.8, 0.1)	(0.7, 0.2)	(0.7, 0.4)	(0.6, 0.4)

Table 10

The Pythagorean fuzzy ranking matrix constructed by expert e₄

	A ₁	A ₂	A ₃	A ₄
C ₁	(0.6, 0.3)	(0.5, 0.7)	(0.6, 0.7)	(0.5, 0.8)
C ₂	(0.6, 0.2)	(0.9, 0.3)	(0.9, 0.2)	(0.7, 0.5)
C ₃	(0.7, 0.4)	(0.5, 0.2)	(0.8, 0.3)	(0.9, 0.2)

The Equation (4) is used to aggregate the Pythagorean fuzzy decision matrix $P^{(k)} = {(p_{ij}^{(k)})}_{4 \times 3} T$ of each decision maker to obtain the comprehensive Pythagorean fuzzy decision matrix P = (p_ij) _4×3^T, as shown in Table 11.

Table 11

The comprehensive Pythagorean fuzzy decision matrix

	A ₁	A ₂	A ₃	A ₄
C ₁	(0.760, 0.230)	(0.540, 0.675)	(0.755, 0.415)	(0.655, 0.530)
C ₂	(0.765, 0.320)	(0.745, 0.355)	(0.755, 0.450)	(0.690, 0.470)
C ₃	(0.665, 0.405)	(0.660, 0.170)	(0.715, 0.315)	(0.735, 0.350)

The ranking function value is calculated and the Pythagorean fuzzy ranking matrix R (P). is obtained, as shown in Table 12.

Table 12

The Pythagorean fuzzy ranking matrix

	A ₁	A ₂	A ₃	A ₄
C ₁	0.7485	0.4391	0.6551	0.5564
C ₂	0.7055	0.6790	0.6387	0.5996
C ₃	0.6182	0.7314	0.6842	0.6767

After determining the optimal value $p_{j}^{o}$ and the worst value $p_{j}^{w}$ under criterion C_j, $p_{j}^{+}$ and $p_{j}^{-}$ are obtained (three criteria are all benefit index) $p_{j}^{+} = {\begin{matrix} (0.7600, 0.2300), (0.7650, 0.3200), \\ (0.6600, 0.1700) \end{matrix}}$ $p_{j}^{-} = {\begin{matrix} (0.5400, 0.6750), (0.6900, 0.4700), \\ (0.6650, 0.4050) \end{matrix}}$

Next, the weighted similarity measures between the criterion value of each scheme and $p_{j}^{+}$ are calculated, the calculated results are shown in Table 13.

Table 13

The calculated results of ${sm}_{w}^{β} (p_{A_{i}}, p^{+})$

	A ₁	A ₂	A ₃	A ₄
${sm}_{w}^{1} (p_{A_{i}}, p^{+})$	0.9876	0.8643	0.9856	0.9435
${sm}_{w}^{2} (p_{A_{i}}, p^{+})$	0.9885	0.8933	0.9732	0.9410
${sm}_{w}^{3} (p_{A_{i}}, p^{+})$	0.9933	0.9319	0.9874	0.9677
${sm}_{w}^{4} (p_{A_{i}}, p^{+})$	0.9981	0.9497	0.9949	0.9756
${sm}_{w}^{5} (p_{A_{i}}, p^{+})$	0.9926	0.9319	0.9817	0.9573
${sm}_{w}^{6} (p_{A_{i}}, p^{+})$	0.9926	0.9319	0.9817	0.9573
${sm}_{w}^{7} (p_{A_{i}}, p^{+})$	0.9464	0.8150	0.8669	0.8005
${sm}_{w}^{8} (p_{A_{i}}, p^{+})$	0.9712	0.8384	0.9129	0.8214
${sm}_{w}^{9} (p_{A_{i}}, p^{+})$	0.9438	0.8150	0.8398	0.7655
${sm}_{w}^{10} (p_{A_{i}}, p^{+})$	0.9438	0.8150	0.8398	0.7655

The calculated results of the weighted similarity measures between the criterion value of each scheme and $p_{j}^{-}$ are shown in Table 14.

Table 14

The calculated results of ${sm}_{w}^{β} (p_{A_{i}}, p^{-})$

	A ₁	A ₂	A ₃	A ₄
${sm}_{w}^{1} (p_{A_{i}}, p^{-})$	0.8529	0.9798	0.9104	0.9667
${sm}_{w}^{2} (p_{A_{i}}, p^{-})$	0.8843	0.9826	0.9211	0.9697
${sm}_{w}^{3} (p_{A_{i}}, p^{-})$	0.9262	0.9894	0.9603	0.9834
${sm}_{w}^{4} (p_{A_{i}}, p^{-})$	0.9445	0.9949	0.9635	0.9877
${sm}_{w}^{5} (p_{A_{i}}, p^{-})$	0.9262	0.9887	0.9603	0.9834
${sm}_{w}^{6} (p_{A_{i}}, p^{-})$	0.9262	0.9887	0.9603	0.9834
${sm}_{w}^{7} (p_{A_{i}}, p^{-})$	0.7743	0.9018	0.8030	0.8808
${sm}_{w}^{8} (p_{A_{i}}, p^{-})$	0.7983	0.9303	0.8222	0.8999
${sm}_{w}^{9} (p_{A_{i}}, p^{-})$	0.7743	0.8993	0.8030	0.8808
${sm}_{w}^{10} (p_{A_{i}}, p^{-})$	0.7743	0.8993	0.8030	0.8808

Next, the relative closeness of each scheme is calculated, as shown in Table 15.

Table 15

The calculated results of the relative closeness $λ_{A_{i}}^{β}$

	A ₁	A ₂	A ₃	A ₄
$λ_{A_{i}}^{1}$	0.4634	0.5313	0.4802	0.5061
$λ_{A_{i}}^{2}$	0.4722	0.5238	0.4862	0.5075
$λ_{A_{i}}^{3}$	0.4825	0.5150	0.4930	0.5040
$λ_{A_{i}}^{4}$	0.4862	0.5116	0.4920	0.5031
$λ_{A_{i}}^{5}$	0.4827	0.5148	0.4945	0.5067
$λ_{A_{i}}^{6}$	0.4827	0.5148	0.4945	0.5067
$λ_{A_{i}}^{7}$	0.4500	0.5253	0.4809	0.5239
$λ_{A_{i}}^{8}$	0.4511	0.5260	0.4739	0.5228
$λ_{A_{i}}^{9}$	0.4507	0.5246	0.4888	0.5350
$λ_{A_{i}}^{10}$	0.4507	0.5246	0.4888	0.5350

From Table 15, the ranking results of each scheme are as shown in Table 16.

Table 16

The ranking results of each scheme

	Ranking results
$λ_{A_{i}}^{1}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{2}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{3}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{4}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{5}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{6}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{7}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{8}$	A₁ > A₃ > A₄ > A₂
$λ_{A_{i}}^{9}$	A₁ > A₃ > A₂ > A₄
$λ_{A_{i}}^{10}$	A₁ > A₃ > A₂ > A₄

It can be seen that the ranking results obtained by the existing measures all are A₁ > A₃ > A₄ > A₂, and the ranking results obtained by the new measures are A₁ > A₃ > A₂ > A₄ (except ${sm}_{w}^{7} ()$ and ${sm}_{w}^{8} ()$ ).

The verification method is also used to verify the accuracy and reliability of the new measures compared with the existing measures.

First, the comprehensive decision value of each scheme is calculated by Equation (4). $PFWA (A_{1}) = (0.7332, 0.3140)$ . $PFWA (A_{2}) = (0.6478, 0.4115)$ $PFWA (A_{3}) = (0.7430, 0.3972)$ $PFWA (A_{4}) = (0.6912, 0.4550)$

Then, the ranking function value of each scheme is calculated by Equation (6). $\begin{matrix} R (A_{1}) = 0.6933 & R (A_{2}) = 0.6072 \end{matrix}$ $\begin{matrix} R (A_{3}) = 0.6579 & R (A_{4}) = 0.6071 \end{matrix}$

According to the ranking rules, the ranking result of all schemes is A₁ > A₃ > A₂ > A₄, which is completely consistent with the ranking results obtained by the new measures (except ${sm}_{w}^{7} ()$ and ${sm}_{w}^{8} ()$ ). Since ${sm}_{w}^{7} ()$ and ${sm}_{w}^{8} ()$ do not consider the HD when calculating the similarity measures, the calculation result may have a slight error, resulting in inaccurate ranking, but the error has been greatly reduced compared with the existing measures.

5.3 A case in which the new measures are compared with the verification method

In the previous two cases, the new measures are compared with the existing measures, and the ranking results are verified by the verification method. Verification method is a MCDM method based on the PFWA operator. Therefore, it is necessary to compare the new measures with the PFWA operator to reflect the superiority of the new measures.

The simulation case setting is basically consistent with Section 5.2, with the only difference being that only one expert assesses two Internet stocks. Thus, the scheme set is A ={ A₁, A₂ }, the criterion set is C = { C₁, C₂, C₃ }. The weight of the criterion set is w = (0.35, 0.35, 0.3). Next, the expert’s Pythagorean fuzzy decision matrix $P^{(k)} = {(p_{ij}^{(k)})}_{2 \times 3}^{T}$ is constructed, as shown in Table 17.

Table 17
The expert’s Pythagorean fuzzy decision matrix

A ₁ A ₂

C ₁ (0.63, 0.48) (0.75, 0.58)

C ₂ (0.53, 0.42) (0.61, 0.52)

C ₃ (0.38, 0.35) (0.48, 0.45)

	A ₁	A ₂
C ₁	(0.63, 0.48)	(0.75, 0.58)
C ₂	(0.53, 0.42)	(0.61, 0.52)
C ₃	(0.38, 0.35)	(0.48, 0.45)

First, the comprehensive decision value of each scheme is calculated by Equation (4). $PFWA (A_{1}) = (0.52, 0.42)$ $PFWA (A_{2}) = (0.62, 0.52)$

Then, the ranking function value of each scheme is calculated by Equation (6). $R (A_{1}) = 0.5451 R (A_{2}) = 0.5451$

According to the ranking rules in Definition 6, the ranking result of all schemes is A₁ = A₂. It can be seen from Table 17 that the decision vectors of A₁ and A₂ are different, but the ranking results calculated by the verification method are equal. Thus, the verification method cannot effectively rank all schemes.

Next, the MCDM method based on similarity measures of the PFS is used to ranking all schemes.

The ranking function value is calculated and the Pythagorean fuzzy ranking matrix R (P). is obtained, as shown in Table 18.

Table 18

The Pythagorean fuzzy ranking matrix

	A ₁	A ₂
C ₁	0.5677	0.5767
C ₂	0.5496	0.5406
C ₃	0.5135	0.5135

The positive ideal criterion value and the negative ideal criterion value are obtained, respectively (three criteria are all benefit index) $p_{j}^{+} = {(0.75, 0.58), (0.53, 0.42), (0.38, 0.35)}$ $p_{j}^{-} = {(0.63, 0.48), (0.61, 0.52), (0.48, 0.45)}$

Next, the weighted similarity measures between the criterion value of each scheme and $p_{j}^{+}$ are calculated, and the calculated results are shown in Table 19. The calculated results of the weighted similarity measures between the criterion value of each scheme and $p_{j}^{-}$ are shown in Table 20.

Table 19

The calculated results of ${sm}_{w}^{β} (p_{A_{i}}, p^{+})$

	A ₁	A ₂
${sm}_{w}^{1} (p_{A_{i}}, p^{+})$	1.0000	0.9992
${sm}_{w}^{2} (p_{A_{i}}, p^{+})$	0.9521	0.9682
${sm}_{w}^{3} (p_{A_{i}}, p^{+})$	0.9882	0.9935
${sm}_{w}^{4} (p_{A_{i}}, p^{+})$	0.9921	0.9938
${sm}_{w}^{5} (p_{A_{i}}, p^{+})$	0.9686	0.9752
${sm}_{w}^{6} (p_{A_{i}}, p^{+})$	0.9686	0.9752
${sm}_{w}^{7} (p_{A_{i}}, p^{+})$	0.9241	0.9211
${sm}_{w}^{8} (p_{A_{i}}, p^{+})$	0.9371	0.9229
${sm}_{w}^{9} (p_{A_{i}}, p^{+})$	0.8799	0.8505
${sm}_{w}^{10} (p_{A_{i}}, p^{+})$	0.8799	0.8505

Table 20

The calculated results of ${sm}_{w}^{β} (p_{A_{i}}, p^{-})$

	A ₁	A ₂
${sm}_{w}^{1} (p_{A_{i}}, p^{-})$	0.9992	1.0000
${sm}_{w}^{2} (p_{A_{i}}, p^{-})$	0.9682	0.9521
${sm}_{w}^{3} (p_{A_{i}}, p^{-})$	0.9935	0.9882
${sm}_{w}^{4} (p_{A_{i}}, p^{-})$	0.9938	0.9921
${sm}_{w}^{5} (p_{A_{i}}, p^{-})$	0.9752	0.9686
${sm}_{w}^{6} (p_{A_{i}}, p^{-})$	0.9752	0.9686
${sm}_{w}^{7} (p_{A_{i}}, p^{-})$	0.9211	0.9241
${sm}_{w}^{8} (p_{A_{i}}, p^{-})$	0.9229	0.9371
${sm}_{w}^{9} (p_{A_{i}}, p^{-})$	0.8505	0.8799
${sm}_{w}^{10} (p_{A_{i}}, p^{-})$	0.8505	0.8799

Next, the relative closeness of each scheme is calculated, as shown in Table 21.

Table 21

The calculated results of the relative closeness $λ_{A_{i}}^{β}$

	A ₁	A ₂
$λ_{A_{i}}^{1}$	0.4634	0.5313
$λ_{A_{i}}^{2}$	0.4722	0.5238
$λ_{A_{i}}^{3}$	0.4825	0.5150
$λ_{A_{i}}^{4}$	0.4862	0.5116
$λ_{A_{i}}^{5}$	0.4827	0.5148
$λ_{A_{i}}^{6}$	0.4827	0.5148
$λ_{A_{i}}^{7}$	0.4500	0.5253
$λ_{A_{i}}^{8}$	0.4511	0.5260
$λ_{A_{i}}^{9}$	0.4507	0.5246
$λ_{A_{i}}^{10}$	0.4507	0.5246

From Table 21, the ranking results of each scheme are as shown in Table 22. It can be seen that the ranking results obtained by the new measures all are A₁ > A₂, and the ranking results obtained by the existing measures are A₂ > A₁ (except ${sm}_{w}^{1} ()$ ).

Table 22

The ranking results of each scheme

	Ranking results
$λ_{A_{i}}^{1}$	A₁ > A₂
$λ_{A_{i}}^{2}$	A₂ > A₁
$λ_{A_{i}}^{3}$	A₂ > A₁
$λ_{A_{i}}^{4}$	A₂ > A₁
$λ_{A_{i}}^{5}$	A₂ > A₁
$λ_{A_{i}}^{6}$	A₂ > A₁
$λ_{A_{i}}^{7}$	A₁ > A₂
$λ_{A_{i}}^{8}$	A₁ > A₂
$λ_{A_{i}}^{9}$	A₁ > A₂
$λ_{A_{i}}^{10}$	A₁ > A₂

When the previous verification method is used to rank, if the ranking function value of each scheme is retained in the 9 digits after the decimal points, then $R (A_{1}) = 0.545100201$ $R (A_{2}) = 0.545073303$

At this point, it can be seen that the schemes are ranked as A₁ > A₂, which is consistent with the ranking results obtained by the new measures. Thus, when the verification method is adopted, the calculation results must be strictly distinguished to obtain the accurate ranking results, which makes the calculation results more difficult and error-prone.

In summary, on the one hand, the case verifies the accuracy and reliability of the new measures compared with the existing measures. On the other hand, the case verifies the simplicity and operability of the new measures compared with the PFWA operator.

6 Conclusions

In this paper, several new similarity measures of the PFS are proposed, and a MCDM method based on the similarity measures is proposed. In the simulation case, the proposed method is applied to solve different MCDM problems. The main conclusions are as follows:

For the highly similar but inconsistent PFSs, the existing similarity measures can not distinguish them, while the new similarity measure can effectively distinguish. Therefore, the new similarity measures have better distinguish ability and wider application range.

In the case of single decision maker and multiple decision makers, it is verified that the MCDM method based on the new measures is more accurate and reliable than the MCDM method based on the existing measures.

In the case of comparison with the verification method, on the one hand, it is verified that the new measures are more accurate and reliable than the existing measures. On the other hand, it is verified that the new measures are simpler and easier to operate than the PFWA operator.

References

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem[J], Computers & Mathematics with Applications 44(8–9) (2002), 1077–1083.

Z.S.

and Yager

R.R.

, Dynamic intuitionistic fuzzy multi-criterion decision making[J], International Journal of Approximate Reasoning 48(1) (2008), 246–262.

Zhang

, Hu

J.H.

, Feng

J.F.

and Liu

, Air multi-target threat assessment method based on improved GGIFSS, Journal of Intelligent & Fuzzy Systems 36(5) (2019), 1–13.

Liang

D.C.

and Xu

Z.S.

, The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets[J], Applied Soft Computing 60 (2017), 167–179.

Ngan

R.T.

, Son

L.H.

and Cuong

B.C.

, et al. H-max distance measure of intuitionistic fuzzy sets in decision making [J], Applied Soft Computing 69 (2018), 393–425.

Yue

Z.L.

and Jia

Y.Y.

, A method to aggregate crisp values into interval-valued intuitionistic fuzzy information for group decision making[J], Applied Soft Computing 13(5) (2013), 2304–2317.

Zadeh

L.A.

, Fuzzy Sets[J], Information and Control 8(3) (1965), 338–353.

Atanassov

K.T.

, Intuitionistic fuzzy sets[J], Fuzzy Sets & Systems 20(1) (1986), 87–96.

Yager

R.R.

, Pythagorean membership grades in multi-criteria decision making[J], IEEE Transactions on Fuzzy Systems 22(4) (2014), 958–965.

10.

Garg

, A new generalized Pythagorean fuzzy information aggregation using einstein operations and its application to decision making[J], International Journal of Intelligent Systems 31(9) (2016), 886–920.

11.

Wei

G.W.

and Lu

, Pythagorean fuzzy power aggregation operators in multiple criterion decision making[J], International Journal of Intelligent Systems 33(1) (2018), 169–186.

12.

Zeng

S.Z.

, Chen

J.P.

and Li

X.S.

, A hybrid method for Pythagorean fuzzy multiple-criteria decision making[J], International Journal of Information Technology & Decision Making 14 (2015), 1–20.

13.

D.Q.

, Zeng

W.Y.

and Yin

, Distance measures of Pythagorean fuzzy sets and their applications multi-criterion decision making[J], Control and Decision 32(10) (2017), 1817–1823.

14.

Zhang

X.L.

and Xu

Z.S.

, Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets[J], International Journal of Intelligent Systems 29(12) (2014), 1061–1078.

15.

Gong

Y.B.

, A new similarity measures of intuitionistic fuzzy sets and application to pattern recognitions[J], Advanced Materials Research 219–220 (2011), 160–164.

16.

D.F.

and Cheng

C.T.

, New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions[J], Pattern Recognition Letters 23(1–3) (2002), 221–225.

17.

Baccour

, Alimi

A.M.

and John

R.I.

, Similarity measures for intuitionistic fuzzy sets: State of the art[J], Journal of Intelligent & Fuzzy Systems 24(1) (2013), 37–49.

18.

, Cosine similarity measures for intuitionistic fuzzy sets and their applications[J], Mathematical and Computer Modelling 53 (2011), 91–97.

19.

Hwang

C.M.

and Yang

M.S.

, Modified cosine similarity measure between Intuitionistic fuzzy sets[C]// Proceedings of the 4th international conference on Artificial Intelligence and Computational Intelligence. Springer, Berlin, Heidelberg (2012).

20.

, Similarity measures of intuitionistic fuzzy sets based on cosine function for the decision making of mechanical design schemes, Journal of Intelligent & Fuzzy Systems 30(1) (2016), 151–158.

21.

Muthukumar

and Krishnan

G.S.S.

, A similarity measure of Intuitionistic fuzzy soft sets and its application in medical diagnosis[J], Applied Soft Computing 41 (2016), 148–156.

22.

Szmidt

and Kacprzyk

, A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making[J], Lecture Notes in Computer Science 3558 (2005), 272–282.

23.

Hung

W.L.

and Yang

M.S.

, Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance[J], Pattern Recognition Letters 25(14) (2004), 1603–1611.

24.

Peng

X.D.

, Yuan

H.Y.

and Yang

, Pythagorean fuzzy information measures and their applications, International Journal of Intelligent Systems 32(10) (2017), 991–1029.

25.

Wei

G.W.

and Wei

, Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications[J], International Journal of Intelligent Systems 33(3) (2018), 634–652.

26.

Zhang

X.L.

, A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making[J], International Journal of Intelligent Systems 31(6) (2016), 593–611.

27.

Zeng

W.Y.

, Li

D.Q.

and Yin

, Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making, International Journal of Intelligent Systems 11(33) (2018).

28.

Atanassov

K.T.

, More on intuitionistic fuzzy sets[J], Fuzzy Sets & Systems 33(1) (1989), 37–45.

29.

Z.S.

, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15(6) (2007), 1179–1187.

30.

Peng

X.D.

and Yang

, Some Results for Pythagorean Fuzzy Sets[J], International Journal of Intelligent Systems 30(11) (2015), 1133–1160.

31.

Ren

P.J.

, Xu

Z.S.

and Gou

X.J.

, Pythagorean fuzzy TODIM approach to multi-criteria decision making[J], Applied Soft Computing 42 (2016), 246–259.

Multiple criteria decision making method based on the new similarity measures of Pythagorean fuzzy set

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Intuitionistic fuzzy set

3.1 Similarity measures

5.1 A case of multiple criteria decision making for single decision maker

Table 7 The Pythagorean fuzzy ranking matrix constructed by expert e1 A 1 A 2 A 3 A 4 C 1 (0.9, 0.3) (0.4, 0.7) (0.8, 0.4) (0.7, 0.2) C 2 (0.7, 0.6) (0.9, 0.2) (0.7, 0.5) (0.8, 0.2) C 3 (0.5, 0.8) (0.8, 0.1) (0.6, 0.2) (0.8, 0.4)

Table 17 The expert’s Pythagorean fuzzy decision matrix A 1 A 2 C 1 (0.63, 0.48) (0.75, 0.58) C 2 (0.53, 0.42) (0.61, 0.52) C 3 (0.38, 0.35) (0.48, 0.45)

References

Table 7
The Pythagorean fuzzy ranking matrix constructed by expert e₁

A ₁ A ₂ A ₃ A ₄

C ₁ (0.9, 0.3) (0.4, 0.7) (0.8, 0.4) (0.7, 0.2)

C ₂ (0.7, 0.6) (0.9, 0.2) (0.7, 0.5) (0.8, 0.2)

C ₃ (0.5, 0.8) (0.8, 0.1) (0.6, 0.2) (0.8, 0.4)

Table 17
The expert’s Pythagorean fuzzy decision matrix

A ₁ A ₂

C ₁ (0.63, 0.48) (0.75, 0.58)

C ₂ (0.53, 0.42) (0.61, 0.52)

C ₃ (0.38, 0.35) (0.48, 0.45)