New correlation coefficients of Pythagorean fuzzy set and its application to extended TODIM method

Abstract

With the increasing complexity of decision making (DM) problems, powerful mathematical tools are needed to represent and process fuzzy and uncertain DM information, and Pythagorean fuzzy set (PFS) is such a mathematical tool. PFS has been successfully applied in the field of fuzzy multiple criteria decision making (MCDM). Correlation coefficient is an information measure of PFS, and plays an important role in the application of PFS. At present, there is a problem that the existing correlation coefficients cannot moderately measure the correlation degree between PFSs, so this paper proposes the new correlation coefficients of PFS. The TODIM method has been proved to be effective in dealing with MCDM problems that consider the psychological behavior of decision makers. This paper extends the TODIM method with the new correlation coefficients of PFS, and the extended TODIM method is called Pythagorean fuzzy CC-TODIM method. By numerical examples, it is verified that the new correlation coefficients of PFS are more reasonable and valid. By case analysis, it is verified that the Pythagorean fuzzy CC-TODIM method can effectively solve the MCDM problems, and the Pythagorean fuzzy CC-TODIM method based on the new correlation coefficients is more accurate and reliable.

Keywords

Pythagorean fuzzy set correlation coefficient TODIM method multiple criteria decision making

1 Introduction

Decision making (DM) problems exist in all aspects of life [1]. In simple terms, DM is to select the best one from many alternatives. When making decisions, it is usually necessary to consider a variety of influencing factors, which can be regarded as criteria. For specific DM problems, the process of selecting the best one from many alternatives based on several criteria is called multiple criteria decision making (MCDM). MCDM problems widely exist in many fields such as economy, management and military [2 –4]. Many researchers have proposed a variety of MCDM methods, such as ELECTRE method [5], TOPSIS method [6], PROMETHEE method [7] and TODIM method [8]. These classic MCDM methods can solve many types of MCDM problems, and are constantly being extended and improved.

Due to the fuzziness of human cognition and the uncertainty of the decision problem itself, there is usually great fuzziness and uncertainty in decision information [9]. Fuzzy set (FS) is a powerful tool to deal with fuzzy and uncertain information [10]. Since Zadeh proposed FS, various extension forms of FS have emerged in endlessly. For example, interval-valued fuzzy set [11], hesitant fuzzy set [12], intuitionistic fuzzy set (IFS) [13], Pythagorean fuzzy set (PFS) [14] and so on. PFS is a new extended form of FS. Similar to IFS, PFS use membership degree (MD) and non-membership degree (ND) to depict FS, but the restrictive condition is more relaxed than IFS, only specifies that the square sum of MD and ND is less than or equal to 1. Compared with FS and IFS, PFS can more effectively represent and process fuzzy and uncertain information, so it has been successfully applied in the field of DM [15 –17].

Correlation coefficient is a statistical indicator that reflects the correlation degree between variables. It is often used in statistics, economics, and engineering. With the cross-development of various disciplines and the advancement of FS theory, correlation coefficient has also been introduced into FS theory. As an important information measure of IFS and PFS, the correlation coefficient has attracted the attention of many researchers. In general, there are relatively more studies on the correlation coefficients of IFS. Gerstenkorn and Manko [18] defined the correlation coefficient function of IFS. Furthermore, Bustince and Burillo [19] proposed the correlation coefficient of interval-valued IFS. Hong and Hwang [20] defined the correlation coefficient concept of IFS in probability space. Hung and Wu [21] calculated the correlation coefficient of IFS by centroid method. Hung [22] studied the correlation coefficient of IFS from the statistical perspective. Xu [23] presented a new correlation coefficient calculate method of IFS, and extended it to the interval-valued IFS theory. At present, there are few studies on the correlation coefficient of PFS. Grag [24] first proposed several new correlation coefficients to measure the relationship between two PFSs, and used the correlation coefficients to solve the DM problems. Ejegwa [25] improved the correlation coefficient proposed by Grag, and proposed generalized triparametric correlation coefficient of PFS. Singh and Ganie [26] proposed several new correlation coefficients of PFS by considering the element weights. Thao [27] introduced a new PFS correlation coefficient from a statistical point of view, and the correlation coefficient value is in the range [–1, 1].

The study of correlation coefficients can enrich the information measurement types and expand the application range of PFS. The correlation coefficients proposed by Grag extend the correlation coefficient theory from IFS to PFS. However, Grag’s correlation coefficients still have some shortcomings. For example, for the same two PFSs, the calculation results of Grag’s correlation coefficients differ greatly, indicating that the correlation degree between PFSs is not moderately measured. So, it is necessary to study the new correlation coefficient of PFS.

The TODIM method is effective in solving the MCDM problem that considers the psychological behavior of decision makers. Gao et al. [28] applied the TODIM method to assess the threat of warning detection. Sen et al. [29] used the TODIM method to select the industrial robot. Zhang and Fan [30] applied the TODIM method to solve the linguistic MCDM problem. Hua et al. [31] adopted the TODIM method to select the medical treatment. In recent years, many researchers have extended TODIM method by using FS theory. For example, fuzzy TODIM method [32, 33], intuitionistic fuzzy TODIM method [34, 35] and Pythagorean fuzzy TODIM method [36]. So far, there is no study on the Pythagorean fuzzy TODIM method based on the correlation coefficient in the existing literature. Therefore, it is of great significance to extend the TODIM method with the correlation coefficient of PFS.

In this paper, we first propose the new correlation coefficients of PFS, and then extend the classic TODIM method with the new correlation coefficients of PFS. The expanded TODIM method is called Pythagorean fuzzy CC-TODIM method. The rest of the paper is organized as follows: Section 2 introduces the basic theory of PFS and the calculation process of the classic TODIM method. Section 3 first introduces several existing correlation coefficients of PFS, and then proposes the new correlation coefficients of PFS. Section 4 proposes the Pythagorean fuzzy CC-TODIM method based on the new correlation coefficients of PFS. Section 5 compares the different correlation coefficients by three examples. Section 6 verifies and analyzes the performance of the Pythagorean fuzzy CC-TODIM method by two cases of MCDM.

2 Preliminaries

In this section, we first introduce several concepts of PFS, and then introduce the TODIM method.

2.1 Pythagorean fuzzy set

The definition of PFS, the operation rule and ranking method of Pythagorean fuzzy number (PFN), Pythagorean fuzzy weighted average (PFWA) operator, distance measure and similarity measure of PFS are introduced in turn. These preliminaries will be used in other sections of this paper.

Definition 1. [37] A PFS in the universe of discourse X is expressed as

$φ = {〈 x, ξ_{φ} (x), ς_{φ} (x) 〉 | x \in X}$ (1) where ξ_φ (x) , ς_φ (x) : X → [0, 1]. ξ_φ (x) and ς_φ (x) are MD and ND of x to φ. For any x ∈ X, $ξ_{φ}^{2} (x) +$ $ς_{φ}^{2} (x) ⩽ 1$ . The hesitation degree (HD) of x to φ is $ɛ_{φ} (x) = \sqrt{1 - ξ_{φ}^{2} (x) - ς_{φ}^{2} (x)}$ . In summary, it can be seen that the IFS must be a PFS, but the PFS is not necessarily an IFS, that is, the IFS is a special PFS.

In this paper, PFN is expressed as κ = (ξ_κ, ς_κ).

Definition 2. [2, 38] Let κ_i = (ξ_{κ_i}, ς_{κ_i}) (i = 1, 2) be PFN, then

$κ_{1}^{c} = (ς_{κ_{1}}, ξ_{κ_{1}})$

$κ_{1} \oplus κ_{2} = (\sqrt{ξ_{κ_{1}}^{2} + ξ_{κ_{2}}^{2} - ξ_{κ_{1}}^{2} ξ_{κ_{2}}^{2}}, ς_{κ_{1}} ς_{κ_{2}})$

$κ_{1} \otimes κ_{2} = (ξ_{κ_{1}} ξ_{κ_{2}}, \sqrt{ς_{κ_{1}}^{2} + ς_{κ_{2}}^{2} - ς_{κ_{1}}^{2} ς_{κ_{2}}^{2}})$

$λ κ_{1} = (\sqrt{1 - (1 - ξ_{κ_{1}}^{2})^{λ}}, (ς_{κ_{1}})^{λ}), λ > 0$

$κ_{1}^{λ} = ((ξ_{κ_{1}})^{λ}, \sqrt{1 - (1 - ς_{κ_{1}}^{2})^{λ}}), λ > 0$

Definition 3. [39] Let κ_i = (ξ_{κ_i}, ς_{κ_i}) (i = 1, 2, ⋯ , m) be PFN, ω_i is the weight of κ_i, satisfying ω_i ⩾ 0 and $\sum_{i = 1}^{m} ω_{i} = 1$ , then the PFWA operator can be expressed as

$PFWA (κ_{1}, κ_{2}, \dots, κ_{m}) = (\sum_{i = 1}^{m} ω_{i} ξ_{κ_{i}}, \sum_{i = 1}^{m} ω_{i} ς_{κ_{i}})$ (2)

Definition 4. [3, 40] Let φ = { 〈 x_i, ξ_φ (x_i) , ς_φ (x_i) 〉 |x_i∈ X } and γ ={ 〈 x_i, ξ_γ (x_i) , ς_γ (x_i) 〉 |x_i ∈ X } be two PFSs in X, then the Euclidean distance between φ and γ can be calculated as

$D (φ, γ) = \sqrt{\frac{1}{2} \sum_{i = 1}^{m} ((ξ_{φ}^{2} (x_{i}) - ξ_{γ}^{2} (x_{i}))^{2} + (ς_{φ}^{2} (x_{i}) - ς_{γ}^{2} (x_{i}))^{2} + (ɛ_{φ}^{2} (x_{i}) - ɛ_{γ}^{2} (x_{i}))^{2})}$ (3)

The similarity measure between φ and γ can be calculated as

$\begin{matrix} S (φ, γ) = \frac{1}{n} \sum_{i = 1}^{m} \\ 2^{1 - \frac{1}{2} (| ξ_{φ}^{2} (x_{i}) - ξ_{γ}^{2} (x_{i}) | + | ς_{φ}^{2} (x_{i}) - ς_{γ}^{2} (x_{i}) | + | ɛ_{φ}^{2} (x_{i}) - ɛ_{γ}^{2} (x_{i}) |)} - 1 \end{matrix}$ (4)

2.2 The TODIM method

Let A_i (i = 1, 2, ⋯ , m) be m schemes, C_j (j = 1, 2, ⋯ , n) be n criteria, ω_j is the weight with respect to the criteria C_j, satisfying 0 ⩽ ω_j ⩽ 1 and $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose the decision matrix is A = (a_ij) _m×n, then the calculation process of the TODIM method is as follows:

Step 1. Determine the normalized decision matrix R = (r_ij) _m×n, and

$r_{ij} = \frac{a_{ij}}{\sqrt{\sum_{i = 1}^{m} a_{ij}^{2}}} \begin{matrix} j \in J^{+} \end{matrix}$ (5)

$r_{ij} = \frac{1 / a_{ij}}{\sqrt{\sum_{i = 1}^{m} (1 / a_{ij})^{2}}} \begin{matrix} j \in J^{-} \end{matrix}$ (6)

Step 2. Calculate the relative weight ω_jr of each criterion C_j

$ω_{jr} = \frac{ω_{j}}{ω_{r}} = \frac{ω_{j}}{max {ω_{j}}}$ (7) where ω_j and ω_r are the weight of criterion C_j and reference criterion C_r, respectively.

Step 3. Calculate the overall dominance degree π (A_i, A_t) of the scheme A_i over each scheme A_t (t = 1, 2, ⋯ , m) about the criterion C_j.

$π (A_{i}, A_{t}) = \sum_{j = 1}^{n} φ_{j} (A_{i}, A_{t})$ (8)

$φ_{j} (A_{i}, A_{t}) = {\begin{matrix} \sqrt{\frac{ω_{jr} (r_{ij} - r_{tj})}{\sum_{j = 1}^{n} ω_{jr}}} \begin{matrix} r_{ij} > r_{tj} \end{matrix} \\ 0 \begin{matrix} r_{ij} = r_{tj} \end{matrix} \\ - \frac{1}{θ} \sqrt{\frac{\sum_{j = 1}^{n} ω_{jr} (r_{tj} - r_{ij})}{ω_{jr}}} \begin{matrix} r_{ij} < r_{tj} \end{matrix} \end{matrix}$ (9) where θ indicates the loss attenuation factor.

Step 4. Calculate the overall value ζ (A_i) of the scheme A_i

$ζ (A_{i}) = \frac{\sum_{t = 1}^{m} π (A_{i}, A_{t}) - min_{i} {\sum_{t = 1}^{m} π (A_{i}, A_{t})}}{max_{i} {\sum_{t = 1}^{m} π (A_{i}, A_{t})} - min_{i} {\sum_{t = 1}^{m} π (A_{i}, A_{t})}}$ (10)

where ζ (A_i) is larger, the scheme A_i is better.

Step 5. Rank schemes A_i according to ζ (A_i), and determine the optimal scheme.

3 Correlation coefficients of PFS

In this section, we first introduce the existing correlation coefficients of PFS, and then propose the new correlation coefficients based on the analysis of the shortcomings of the existing correlation coefficients.

3.1 Existing correlation coefficients

Inspired by the correlation coefficients of IFS proposed by Ye [41] and Xu [42], Grag proposed several correlation coefficients of PFS, which are uniformly defined as follows

Definition 5. [24 , 42] Let ς_φ (x_i)〉 |x_i ∈ X } and γ ={ 〈 x_i, ξ_γ (x_i) , ς_γ (x_i) 〉 |x_i ∈ X } be two PFSs in the discourse universe X = {x₁, x₂, ⋯, x_m}, then the correlation coefficients between φ and γ are

$\begin{matrix} K_{1}^{E} (φ, γ) = \frac{C (φ, γ)}{\sqrt{C (φ, φ) \cdot C (γ, γ)}} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})}{\sqrt{\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i})} \sqrt{\sum_{i = 1}^{m} ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i})}} \end{matrix}$ (11)

$\begin{matrix} K_{2}^{E} (φ, γ) = \frac{C (φ, γ)}{max {C (φ, φ), C (γ, γ)}} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})}{max {\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}), \sum_{i = 1}^{m} ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i})}} \end{matrix}$ (12)

If MD, ND and HD of φ and γ are all considered, then the correlation coefficients between φ and γ are

$\begin{matrix} K_{3}^{E} (φ, γ) = \frac{C (φ, γ)}{\sqrt{C (φ, φ) \cdot C (γ, γ)}} = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{\sqrt{\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i})} \sqrt{\sum_{i = 1}^{m} ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i})}} \end{matrix}$ (13)

$\begin{matrix} K_{4}^{E} (φ, γ) = \frac{C (φ, γ)}{max {C (φ, φ), C (γ, γ)}} = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{max {\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}), \sum_{i = 1}^{m} ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i})}} \end{matrix}$ (14)

The above existing correlation coefficients of PFS can be expressed as $K_{α}^{E} (α = 1, 2, 3, 4)$ .

Example 1. Two PFSs in in the discourse universe X ={ x₁, x₂ } are $\begin{matrix} \begin{matrix} φ = {(x_{1}, 0.7, 0.2), (x_{2}, 0.6, 0.4)} \\ γ = {(x_{1}, 0.9, 0.4), (x_{2}, 0.9, 0.3)} \end{matrix} \end{matrix}$

Then, $K_{α}^{E} (φ, γ)$ can be obtained as $\begin{matrix} \begin{matrix} K_{1}^{E} (φ, γ) = 0.9705 K_{2}^{E} (φ, γ) = 0.5270 \\ K_{3}^{E} (φ, γ) = 0.7191 K_{4}^{E} (φ, γ) = 0.5685 \end{matrix} \end{matrix}$

Let ω_i be the weight of x_i, then the weighted correlation coefficients between two PFSs φ and γ are

$\begin{matrix} {\tilde{K}}_{1}^{E} (φ, γ) = \frac{\tilde{C} (φ, γ)}{\sqrt{\tilde{C} (φ, φ) \cdot \tilde{C} (γ, γ)}} \\ = \frac{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}))}{\sqrt{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}))} \sqrt{\sum_{i = 1}^{m} ω_{i} (ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}))}} \end{matrix}$ (15)

$\begin{matrix} {\tilde{K}}_{2}^{E} (φ, γ) = \frac{\tilde{C} (φ, γ)}{max {\tilde{C} (φ, φ), \tilde{C} (γ, γ)}} \\ = \frac{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}))}{max {\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i})), \sum_{i = 1}^{m} ω_{i} (ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}))}} \end{matrix}$ (16)

$\begin{matrix} {\tilde{K}}_{3}^{E} (φ, γ) = \frac{\tilde{C} (φ, γ)}{\sqrt{\tilde{C} (φ, φ) \cdot \tilde{C} (γ, γ)}} \\ = \frac{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i}))}{\sqrt{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}))} \sqrt{\sum_{i = 1}^{m} ω_{i} (ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i}))}} \end{matrix}$ (17)

$\begin{matrix} {\tilde{K}}_{4}^{E} (φ, γ) = \frac{\tilde{C} (φ, γ)}{max {\tilde{C} (φ, φ), \tilde{C} (γ, γ)}} \\ = \frac{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i}))}{max {\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i})), \sum_{i = 1}^{m} ω_{i} (ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i}))}} \end{matrix}$ (18)

The above existing weighted correlation coefficients of PFS can be expressed as ${\tilde{K}}_{α}^{E} (α = 1, 2, 3, 4)$ .

3.2 New correlation coefficients

From Example 1, it can be seen that for two PFSs φ and γ, the correlation coefficients $K_{1}^{E} (φ, γ)$ and $K_{2}^{E} (φ, γ)$ have a large difference (so are $K_{3}^{E} (φ, γ)$ and $K_{4}^{E} (φ, γ)$ ). For the same two PFSs, it is unreasonable that the correlation coefficient is too large or too small. Therefore, it is necessary to moderately measure the correlation degree between PFSs.

From the previous section, $K_{α}^{E} (α = 1, 2, 3, 4)$ can be classified as follows $\begin{matrix} \begin{matrix} K_{α}^{E} (φ, γ) = \frac{C (φ, γ)}{\sqrt{C (φ, φ) \cdot C (γ, γ)}} (α = 1, 3) \\ K_{α}^{E} (φ, γ) = \frac{C (φ, γ)}{max {C (φ, φ) \cdot C (γ, γ)}} (α = 2, 4) \end{matrix} \end{matrix}$

It can be concluded that the reason why $K_{α}^{E} (φ, γ)$ (α = 1, 3) is too large or $K_{α}^{E} (φ, γ) (α = 2, 4)$ is too small is $\begin{matrix} max {C (φ, φ), C (γ, γ)} ⩾ \sqrt{C (φ, φ) \cdot C (γ, γ)} \end{matrix}$

Thus, in order to measure the correlation degree between PFSs moderately, a function formula with the value between max{ C (φ, φ) , C (γ, γ) } and $\sqrt{C (φ, φ) \cdot C (γ, γ)}$ is needed.

Let C (φ, φ) = δ, C (γ, γ) = σ, then $\begin{matrix} \begin{matrix} max {C (φ, φ), C (γ, γ)} = max {δ, σ} \\ \sqrt{C (φ, φ) \cdot C (γ, γ)} = \sqrt{δ σ} \end{matrix} \end{matrix}$

If δ ⩾ σ, then $\begin{matrix} \frac{1}{2} (δ + σ) ⩽ δ \end{matrix}$

If δ ⩽ σ, then $\begin{matrix} \frac{1}{2} (δ + σ) ⩽ σ \end{matrix}$

Thus $\begin{matrix} \frac{1}{2} (δ + σ) ⩽ max (δ, σ) \end{matrix}$

Moreover, as a fundamental inequality, $\sqrt{δ σ} ⩽$ $\frac{1}{2} (δ + σ)$ is obviously true.

In summary, we can get $\begin{matrix} \sqrt{δ σ} ⩽ \frac{1}{2} (δ + σ) ⩽ max (δ, σ) \end{matrix}$

That is $\begin{matrix} \sqrt{C (φ, φ) \cdot C (γ, γ)} ⩽ \frac{1}{2} (C (φ, φ) + C (γ, γ)) \\ ⩽ max {C (φ, φ), C (γ, γ)} \end{matrix}$

So, $\frac{1}{2} (C (φ, φ) + C (γ, γ))$ is the function formula which we need.

Next, by using $\frac{1}{2} (C (φ, φ) + C (γ, γ))$ to improve the existing correlation coefficients, we propose the new correlation coefficients of PFS.

Definition 6. Let φ ={ 〈 x_i, ξ_φ (x_i) , ς_φ (x_i) 〉 |x_i ∈ X } and γ ={ 〈 x_i, ξ_γ (x_i) , ς_γ (x_i) 〉 |x_i ∈ X } be two PFSs in X, then the new correlation coefficient between φ and γ is

$\begin{matrix} K_{1}^{N} (φ, γ) = \frac{C (φ, γ)}{\frac{1}{2} (C (φ, φ) + C (γ, γ))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})}{\frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}))} \end{matrix}$ (19)

Similarly, if MD, ND and HD of φ and γ are all considered, then the new correlation coefficient between φ and γ is

$\begin{matrix} K_{2}^{N} (φ, γ) = \frac{C (φ, γ)}{\frac{1}{2} (C (φ, φ) + C (γ, γ))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{\frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i}))} \end{matrix}$ (20)

The above new correlation coefficients of PFS can be expressed as $K_{β}^{N} (β = 1, 2)$ .

The correlation coefficients $K_{β}^{N} (φ, γ) (β = 1, 2)$ should satisfy the following properties:

$0 ⩽ K_{β}^{N} (φ, γ) ⩽ 1$

$K_{β}^{N} (φ, γ) = K_{β}^{N} (γ, φ)$

$K_{β}^{N} (φ, γ) = 1$ if and only if φ = γ

Proof.

(1) For $K_{1}^{N}$ , first

$\begin{matrix} \begin{matrix} K_{1}^{N} (φ, γ) = \frac{C (φ, γ)}{\frac{1}{2} (C (φ, φ) + C (γ, γ))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})}{\frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})}{\sum_{i = 1}^{m} \frac{1}{2} (ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i})) + \frac{1}{2} (ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}))} \\ ⩽ \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})}{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i})} = 1 \end{matrix} \end{matrix}$

Then, it can be obtained that $K_{1}^{N} (φ, γ) ⩾ 0$ is obviously true, so $0 ⩽ K_{1}^{N} (φ, γ) ⩽ 1$ is true.

For $K_{2}^{N}$ , first

$\begin{matrix} \begin{matrix} K_{2}^{N} (φ, γ) = \frac{C (φ, γ)}{\frac{1}{2} (C (φ, φ) + C (γ, γ))} = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{\frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i}))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{\sum_{i = 1}^{m} \frac{1}{2} (ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i})) + \frac{1}{2} (ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i})) + \frac{1}{2} (ɛ_{φ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i}))} \\ ⩽ \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})} = 1 \end{matrix} \end{matrix}$

Then, it can be obtained that $K_{2}^{N} (φ, γ) ⩾ 0$ is obviously true, so $0 ⩽ K_{2}^{N} (φ, γ) ⩽ 1$ is true.

(2) Take $K_{2}^{N}$ as an example. $\begin{matrix} \begin{matrix} K_{2}^{N} (φ, γ) = \frac{C (φ, γ)}{\frac{1}{2} (C (φ, φ) + C (γ, γ))} \\ K_{2}^{N} (γ, φ) = \frac{C (γ, φ)}{\frac{1}{2} (C (γ, γ) + C (φ, φ))} \end{matrix} \end{matrix}$

Obviously $\begin{matrix} \begin{matrix} C (φ, γ) = C (γ, φ) \\ C (φ, φ) + C (γ, γ) = C (γ, γ) + C (φ, φ) \end{matrix} \end{matrix}$

So, $K_{β}^{N} (φ, γ) = K_{β}^{N} (γ, φ)$ is true.

(3) Take $K_{2}^{N}$ as an example.

If φ = γ, then

$\begin{matrix} \begin{matrix} K_{2}^{N} (φ, γ) = \frac{C (φ, γ)}{\frac{1}{2} (C (φ, φ) + C (γ, γ))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i})}{\frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i}))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{2} (x_{i}) \cdot ξ_{φ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{φ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{φ}^{2} (x_{i})}{\frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}))} \\ = \frac{\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i})}{\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i})} = 1 \end{matrix} \end{matrix}$

If $K_{2}^{N} (φ, γ) = 1$ , let τ be a nonzero positive real number, then $\begin{matrix} \begin{matrix} ξ_{φ} (x_{i}) = τ \cdot ξ_{γ} (x_{i}) \\ ς_{φ} (x_{i}) = τ \cdot ς_{γ} (x_{i}) \\ ɛ_{φ} (x_{i}) = τ \cdot ɛ_{γ} (x_{i}) \\ C (φ, γ) = \sum_{i = 1}^{n} ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) \\ + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i}) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} = \sum_{i = 1}^{n} τ^{2} ξ_{γ}^{4} (x_{i}) + τ^{2} ς_{γ}^{4} (x_{i}) + τ^{2} ɛ_{γ}^{4} (x_{i}) \\ \frac{1}{2} (C (φ, φ) + C (γ, γ)) \\ = \frac{1}{2} (\sum_{i = 1}^{m} ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) \\ + ς_{γ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i})) \\ = \frac{1}{2} (\sum_{i = 1}^{m} (1 + τ^{4}) ξ_{γ}^{4} (x_{i}) + (1 + τ^{4}) ς_{γ}^{4} (x_{i}) \\ + (1 + τ^{4}) ɛ_{γ}^{4} (x_{i})) \end{matrix} \end{matrix}$

Next,

$\begin{matrix} K_{2}^{N} (φ, γ) = \frac{τ^{2} \cdot \sum_{i = 1}^{m} ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i})}{\frac{1}{2} (1 + τ^{4}) \cdot \sum_{i = 1}^{m} ξ_{γ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i})} \end{matrix}$

Considering that $K_{2}^{N} (φ, γ) = 1$ , then $\begin{matrix} τ^{2} = \frac{1}{2} (1 + τ^{4}) \end{matrix}$

τ = 1 can be obtained, then we can get $\begin{matrix} \begin{matrix} ξ_{φ} (x_{i}) = ξ_{γ} (x_{i}) \\ ς_{φ} (x_{i}) = ς_{γ} (x_{i}) \\ ɛ_{φ} (x_{i}) = ɛ_{γ} (x_{i}) \end{matrix} \end{matrix}$

So, φ = γ.

The proof is over.

Example 2. Two PFSs in in the discourse universe X ={ x₁, x₂ } are $\begin{matrix} \begin{matrix} φ = {(x_{1}, 0.7, 0.2), (x_{2}, 0.6, 0.4)} \\ γ = {(x_{1}, 0.9, 0.4), (x_{2}, 0.9, 0.3)} \end{matrix} \end{matrix}$

Then, $K_{β}^{N} (φ, γ)$ can be obtained as $\begin{matrix} K_{1}^{N} (φ, γ) = 0.8140 K_{2}^{N} (φ, γ) = 0.6997 \end{matrix}$

From Example 1 and Example 2, we can get $\begin{matrix} \begin{matrix} K_{1}^{E} (φ, γ) > K_{1}^{N} (φ, γ) > K_{2}^{E} (φ, γ) \\ K_{3}^{E} (φ, γ) > K_{2}^{N} (φ, γ) > K_{4}^{E} (φ, γ) \end{matrix} \end{matrix}$

Thus, for the same two PFSs φ and γ, the new correlation coefficients are more moderate than the existing correlation coefficient.

Let ω_i be the weight of x_i, then the new weighted correlation coefficients between φ and γ are

$\begin{matrix} {\tilde{K}}_{1}^{N} (φ, γ) = \frac{\tilde{C} (φ, γ)}{\frac{1}{2} (\tilde{C} (φ, φ) + \tilde{C} (γ, γ))} \\ = \frac{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}))}{\frac{1}{2} (\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i})))} \end{matrix}$ (21)

$\begin{matrix} {\tilde{K}}_{2}^{N} (φ, γ) = \frac{\tilde{C} (φ, γ)}{\frac{1}{2} (\tilde{C} (φ, φ) + \tilde{C} (γ, γ))} \\ = \frac{\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{2} (x_{i}) \cdot ξ_{γ}^{2} (x_{i}) + ς_{φ}^{2} (x_{i}) \cdot ς_{γ}^{2} (x_{i}) + ɛ_{φ}^{2} (x_{i}) \cdot ɛ_{γ}^{2} (x_{i}))}{\frac{1}{2} (\sum_{i = 1}^{m} ω_{i} (ξ_{φ}^{4} (x_{i}) + ξ_{γ}^{4} (x_{i}) + ς_{φ}^{4} (x_{i}) + ς_{γ}^{4} (x_{i}) + ɛ_{φ}^{4} (x_{i}) + ɛ_{γ}^{4} (x_{i})))} \end{matrix}$ (22)

The above new weighted correlation coefficients of PFS can be expressed as ${\tilde{K}}_{β}^{N} (β = 1, 2)$ .

4 Pythagorean fuzzy CC-TODIM method

In this section, we use the correlation coefficient of PFS to expand the TODIM method, and thus propose the Pythagorean fuzzy CC-TODIM method.

In general, MCDM cases can be set as: Let A = { A₁, A₂, ⋯ , A_m } (i = 1, 2, ⋯ , m) be decision scheme set, C = { C₁, C₂, ⋯ , C_n } (j = 1, 2, ⋯ , n) be decision criterion set, E = { e₁, e₂, ⋯ , e_l } (k = 1, 2, ⋯ , l) be decision maker set. ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weight of criterion set, satisfying 0 ⩽ ω_j ⩽ 1 and $\sum_{j = 1}^{n} ω_{j} = 1$ ; υ = (υ₁, υ₂, ⋯ , υ_l) ^T is the weight of decision maker set, satisfying 0 ⩽ υ_k ⩽ 1 and $\sum_{k = 1}^{l} υ_{k} = 1$ .

The decision matrix $P^{k} = (κ_{ij}^{k})_{m \times n}^{T}$ of decision maker e_k can be expressed as

$\begin{matrix} P^{k} = (κ_{ij}^{k})_{m \times n}^{T} \\ = \begin{matrix} A_{1} & A_{2} \dots & A_{m} \\ \begin{matrix} C_{1} \\ C_{2} \\ ⋮ \\ C_{n} \end{matrix} & [\begin{matrix} (ξ_{11}^{k}, ς_{11}^{k}) \\ (ξ_{21}^{k}, ς_{21}^{k}) \\ \begin{matrix} ⋮ \end{matrix} \\ (ξ_{n 1}^{k}, ς_{n 1}^{k}) \end{matrix} & \begin{matrix} (ξ_{12}^{k}, ς_{12}^{k}) \begin{matrix} \begin{matrix} \end{matrix} \dots \end{matrix} \\ (ξ_{22}^{k}, ς_{22}^{k}) \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \\ \begin{matrix} ⋮ \end{matrix} \begin{matrix} ⋱ \end{matrix} \\ (ξ_{n 2}^{k}, ς_{n 2}^{k}) \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \end{matrix} & \begin{matrix} (ξ_{1 m}^{k}, ς_{1 m}^{k}) \\ (ξ_{2 m}^{k}, ς_{2 m}^{k}) \\ \begin{matrix} ⋮ \end{matrix} \\ (ξ_{nm}^{k}, ς_{nm}^{k}) \end{matrix}] \end{matrix} \end{matrix}$ (23) where $κ_{ij}^{k} = (ξ_{ij}^{k}, ς_{ij}^{k})$ is the decision value of the j-th criterion about the i-th scheme given by the k-th decision maker.

Next, the calculation process of the Pythagorean fuzzy CC-TODIM method is as follows

Step 1. Aggregate each decision maker’s decision matrix $P^{k} = (κ_{ij}^{k})_{m \times n}^{T}$ by the PFWA operator, then the comprehensive decision matrix is

$P = (κ_{ij})_{m \times n}^{T} = {(\sum_{k = 1}^{l} υ_{k} ξ_{ij}^{k}, \sum_{k = 1}^{l} υ_{k} ς_{ij}^{k})}_{m \times n}^{T}$ (24)

Step 2. Determine the normalized decision matrix $\bar{P} = ({\bar{κ}}_{ij})_{m \times n}^{T}$ , and

${\bar{κ}}_{ij} = {\begin{matrix} κ_{ij} = (\sum_{k = 1}^{l} υ_{k} ξ_{ij}^{k}, \sum_{k = 1}^{l} υ_{k} ς_{ij}^{k}) \begin{matrix} j \in J^{+} \end{matrix} \\ κ_{ij}^{c} = (\sum_{k = 1}^{l} υ_{k} ς_{ij}^{k}, \sum_{k = 1}^{l} υ_{k} ξ_{ij}^{k}) \begin{matrix} j \in J^{-} \end{matrix} \end{matrix}$ (25) where J⁺ is the benefit criterion set and J^- is the cost criterion set.

Step 3. Calculate the relative weight ω_jr of each criterion C_j

$ω_{jr} = \frac{ω_{j}}{ω_{r}} = \frac{ω_{j}}{max {ω_{j}}}$ (26)

Step 4. Calculate the dominance degree φ_j (A_i, A_t) of the scheme A_i over each scheme A_t (t = 1, 2, ⋯ , m) about the criterion C_j

$\begin{matrix} φ_{j} (A_{i}, A_{t}) \\ = {\begin{matrix} \sqrt{\frac{ω_{jr} (1 - K_{β}^{N} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj}))}{\sum_{j = 1}^{n} ω_{jr}}} \begin{matrix} {\bar{κ}}_{ij} > {\bar{κ}}_{tj} \end{matrix} \\ 0 \begin{matrix} {\bar{κ}}_{ij} = {\bar{κ}}_{tj} \end{matrix} \\ - \frac{1}{θ} \sqrt{\frac{\sum_{j = 1}^{n} ω_{jr} (1 - K_{β}^{N} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj}))}{ω_{jr}}} \begin{matrix} {\bar{κ}}_{ij} < {\bar{κ}}_{tj} \end{matrix} \end{matrix} \end{matrix}$ (27) where $K_{β}^{N} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj}) (β = 1, 2)$ is the new correlation coefficient between $\begin{matrix} {\bar{κ}}_{ij} \end{matrix}$ and ${\bar{κ}}_{tj}$ , θ is the loss attenuation factor and θ > 0.

Then, the dominance matrix φ_j can be obtained as

$\begin{matrix} φ_{j} = {[φ_{j} (A_{i}, A_{t})]}_{m \times m} \\ = \begin{matrix} A_{1} & \begin{matrix} A_{2} & \dots \end{matrix} & A_{m} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} & [\begin{matrix} 0 \\ φ_{j} (A_{2}, A_{1}) \\ \begin{matrix} ⋮ \end{matrix} \\ φ_{j} (A_{m}, A_{1}) \end{matrix} & \begin{matrix} φ_{j} (A_{1}, A_{2}) \begin{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \dots \end{matrix} \\ 0 \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \\ \begin{matrix} ⋮ \end{matrix} \begin{matrix} ⋱ \end{matrix} \\ φ_{j} (A_{m}, A_{2}) \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \end{matrix} & \begin{matrix} φ_{j} (A_{1}, A_{m}) \\ φ_{j} (A_{2}, A_{m}) \\ \begin{matrix} ⋮ \end{matrix} \\ 0 \end{matrix}] \end{matrix} \end{matrix}$ (28)

Step 5. Calculate the overall dominance degree π (A_i, A_t) of the scheme A_i over each scheme A_t about the criterion C_j

$π (A_{i}, A_{t}) = \sum_{j = 1}^{n} φ_{j} (A_{i}, A_{t})$ (29)

Then, the dominance matrix π can be obtained as

$\begin{matrix} π = {[π (A_{i}, A_{t})]}_{m \times m} \\ = \begin{matrix} A_{1} & \begin{matrix} A_{2} & \dots \end{matrix} & A_{m} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} & [\begin{matrix} 0 \\ π (A_{2}, A_{1}) \\ \begin{matrix} ⋮ \end{matrix} \\ π (A_{m}, A_{1}) \end{matrix} & \begin{matrix} π (A_{1}, A_{2}) \begin{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \dots \end{matrix} \\ 0 \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \\ \begin{matrix} ⋮ \end{matrix} \begin{matrix} ⋱ \end{matrix} \\ π (A_{m}, A_{2}) \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \dots \end{matrix} \end{matrix} & \begin{matrix} π (A_{1}, A_{m}) \\ π (A_{2}, A_{m}) \\ \begin{matrix} ⋮ \end{matrix} \\ 0 \end{matrix}] \end{matrix} \end{matrix}$ (30)

Step 6. Calculate the overall value ζ (A_i) of the scheme A_i

where ζ (A_i) is larger, the scheme A_i is better.

Step 7. Rank all schemes A_i according to the size of the overall value ζ (A_i), and then determine the optimal scheme.

5 Example comparative

In this section, in order to reflect the innovation and effectiveness of the new correlation coefficients, the new correlation coefficients are compared with the existing correlation coefficients, Ejegwa’s correlation coefficients, Singh and Ganie’s correlation coefficients, Thao’s correlation coefficient by several examples.

Example 3. This example compares the new correlation coefficients with the existing correlation coefficients.

Three PFSs in the discourse universe X ={ x₁, x₂ } are $\begin{matrix} \begin{matrix} A = {(x_{1}, 0.5, 0.2), (x_{2}, 0.4, 0.2)} \\ A_{1} = {(x_{1}, 0.7, 0.2), (x_{2}, 0.6, 0.3)} \\ A_{2} = {(x_{1}, 0.4, 0.1), (x_{2}, 0.5, 0.3)} \end{matrix} \end{matrix}$

The weights of x₁ and x₂ are 0.65 and 0.35, then ${\tilde{K}}_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix}) (α = 1, 2, 3, 4)$ and ${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix}) (β = 1, 2)$ are shown in Table 1.

Table 1
Calculation results of ${\tilde{K}}_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$ and ${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$

A ₁ A ₂

${\tilde{K}}_{1}^{E} (A, A_{i})$ 0.9959 0.9018

${\tilde{K}}_{2}^{E} (A, A_{i})$ 0.4972 0.8112

${\tilde{K}}_{3}^{E} (A, A_{i})$ 0.9034 0.9798

${\tilde{K}}_{4}^{E} (A, A_{i})$ 0.7846 0.9497

${\tilde{K}}_{1}^{N} (A, A_{i})$ 0.7960 0.8968

${\tilde{K}}_{2}^{N} (A, A_{i})$ 0.8945 0.9793

	A ₁	A ₂
${\tilde{K}}_{1}^{E} (A, A_{i})$	0.9959	0.9018
${\tilde{K}}_{2}^{E} (A, A_{i})$	0.4972	0.8112
${\tilde{K}}_{3}^{E} (A, A_{i})$	0.9034	0.9798
${\tilde{K}}_{4}^{E} (A, A_{i})$	0.7846	0.9497
${\tilde{K}}_{1}^{N} (A, A_{i})$	0.7960	0.8968
${\tilde{K}}_{2}^{N} (A, A_{i})$	0.8945	0.9793

By analyzing Table 1, we can get: on the one hand, for two PFSs A and A_i, the calculation results of ${\tilde{K}}_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$ are still quite different, and ${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$ is more moderate than ${\tilde{K}}_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$ ; on the other hand, ${\tilde{K}}_{1}^{E} (\begin{matrix} A, A_{1} \end{matrix}) > {\tilde{K}}_{1}^{E} (\begin{matrix} A, A_{2} \end{matrix})$ , while other correlation coefficients between A and A₁ is smaller than that between A and A₂. It shows that ${\tilde{K}}_{1}^{E}$ sometimes cannot accurately measure the correlation degree between PFSs. It can be obtained that, compared with ${\tilde{K}}_{β}^{E}$ , ${\tilde{K}}_{β}^{N}$ can more moderately and reasonably measure the correlation degree between PFSs

Example 4. This example compares the new correlation coefficients with the Ejegwa’s correlation coefficients, Singh and Ganie’s correlation coefficients, Thao’s correlation coefficient. For convenience, Ejegwa’s correlation coefficients are expressed as ${\tilde{K}}_{k}^{Ee} (k = 1, 2, 3, 4)$ , Singh and Ganie’s correlation coefficients are expressed as ${\tilde{K}}_{r}^{Esg} (r = 1, 2)$ , Thao’s correlation coefficient is expressed as ${\tilde{K}}^{Et}$ .

Three PFSs in the discourse universe X ={ x₁, x₂ } are $\begin{matrix} \begin{matrix} A = {(x_{1}, 0.5, 0.2), (x_{2}, 0.4, 0.2)} \\ A_{1} = {(x_{1}, 0.5, 0.2), (x_{2}, 0.4, 0.3)} \\ A_{2} = {(x_{1}, 0.5, 0.2), (x_{2}, 0.4, 0.1)} \end{matrix} \end{matrix}$

The weights of x₁ and x₂ are equal, then calculation results and raking results of ${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix}) (β = 1, 2)$ , ${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix}) (k = 1, 2, 3, 4)$ , ${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix}) (r = 1, 2)$ and ${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$ are shown in Table 2.

Table 2

Calculation results and raking results of ${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$ ${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$ , ${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$ and ${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$

		A ₁	A ₂	Ranking results
${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$	β = 1	0.9868	0.9950	A₁ < A₂
	β = 2	0.9979	0.9993	A₁ < A₂
${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$	k = 1	0.9868	0.9834	A₁ > A₂
	k = 2	0.9973	0.9973	A₁ = A₂
	k = 3	0.9821	0.9879	A₁ < A₂
	k = 4	0.9692	0.9808	A₁ < A₂
${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$	r = 1	0.9835	0.9901	A₁ < A₂
	r = 2	0.9780	0.9867	A₁ < A₂
${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$		0.9985	0.9985	A₁ = A₂

As shown in Table 2, the ranking results of ${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$ and ${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$ are all A₁ < A₂, the ranking results of ${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$ is A₁ = A₂, and there are three kinds of ranking results of ${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$ . It can be obtained that, compared with ${\tilde{K}}^{Et}$ and ${\tilde{K}}_{k}^{Ee}$ , ${\tilde{K}}_{β}^{N}$ and ${\tilde{K}}_{r}^{Esg}$ can accurately measure the correlation degree between PFSs.

Example 5. This example compares the new correlation coefficients with all other correlation coefficients mentioned in the paper.

Three known patterns A_i (i = 1, 2, 3) and an unknown pattern A are represented by PFSs in the discourse universe X ={ x₁, x₂, x₃ } as $\begin{matrix} \begin{matrix} A_{1} = {(x_{1}, 0.9, 0.1), (x_{2}, 0.7, 0.2), (x_{3}, 0.7, 0.22)} \\ A_{2} = {(x_{1}, 0.8, 0.1), (x_{2}, 0.7, 0.2), (x_{3}, 0.8, 0.23)} \\ A_{3} = {(x_{1}, 0.7, 0.3), (x_{2}, 0.8, 0.2), (x_{3}, 0.7, 0.11)} \\ A = {(x_{1}, 0.8, 0.1), (x_{2}, 0.8, 0.2), (x_{3}, 0.8, 0.3)} \end{matrix} \end{matrix}$

In order to recognize which pattern A belongs to, the correlation coefficients between A and A_i are calculated, as shown in Table 3.

Table 3

Calculation results of $K_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$ , $K_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$ , ${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$ , ${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$ and ${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$

		A ₁	A ₂	A ₃
${\tilde{K}}_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$	α = 1	0.9690	0.9925	0.9856
	α = 2	0.9298	0.9198	0.8400
	α = 3	0.9481	0.9843	0.9621
	α = 4	0.9272	0.9731	0.9270
${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$	β = 1	0.9682	0.9896	0.9731
	β = 2	0.9479	0.9843	0.9614
${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$	k = 1	0.9878	0.9968	0.9815
	k = 2	0.9826	0.9949	0.9777
	k = 3	0.9558	0.9871	0.9569
	k = 4	0.9272	0.9731	0.9270
${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$	r = 1	0.9115	0.9594	0.9005
	r = 2	0.8820	0.9457	0.8989
${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$		0.7746	0.0872	–0.2363

The above calculation results are ranked as shown in Table 4.

Table 4

Ranking results of $K_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$ , $K_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$ , ${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$ , ${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$ and ${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$

		Ranking results
${\tilde{K}}_{α}^{E} (\begin{matrix} A, A_{i} \end{matrix})$	α = 1	A₂ > A₃ > A₁
	α = 2	A₁ > A₂ > A₃
	α = 3	A₂ > A₃ > A₁
	α = 4	A₂ > A₁ > A₃
${\tilde{K}}_{β}^{N} (\begin{matrix} A, A_{i} \end{matrix})$	β = 1	A₂ > A₃ > A₁
	β = 2	A₂ > A₃ > A₁
${\tilde{K}}_{k}^{Ee} (\begin{matrix} A, A_{i} \end{matrix})$	k = 1	A₂ > A₁ > A₃
	k = 2	A₂ > A₁ > A₃
	k = 3	A₂ > A₃ > A₁
	k = 4	A₂ > A₁ > A₃
${\tilde{K}}_{r}^{Esg} (\begin{matrix} A, A_{i} \end{matrix})$	r = 1	A₂ > A₁ > A₃
	r = 2	A₂ > A₁ > A₃
${\tilde{K}}^{Et} (\begin{matrix} A, A_{i} \end{matrix})$		A₁ > A₂ > A₃

As shown in Table 4, the ranking results of all correlation coefficients are not consistent, so distance measure and similarity measure of PFS are used for pattern recognition of A and A_i to verify the accuracy of the correlation coefficients, respectively.

(1) Pattern recognition by distance measure

The Euclidean distance between A and A_i are calculated as $\begin{matrix} \begin{matrix} D (\begin{matrix} A, A_{1} \end{matrix}) = 0.2861 \\ D (\begin{matrix} A, A_{2} \end{matrix}) = 0.1545 \\ D (\begin{matrix} A, A_{3} \end{matrix}) = 0.2391 \end{matrix} \end{matrix}$

According to the calculation results of $D (\begin{matrix} A, A_{i} \end{matrix})$ , we can get the distance between A₂ and A is the closest, so A belongs to the pattern A₂. In addition, the ranking result of $D (\begin{matrix} A, A_{i} \end{matrix})$ is A₁ > A₃ > A₂, this means that the similarity degree between A and A_i is ranked as A₂ > A₃ > A₁.

(2) Pattern recognition by similarity measure

The similarity measure between A and A_i are calculated as $\begin{matrix} \begin{matrix} S (\begin{matrix} A, A_{1} \end{matrix}) = 0.7771 \\ S (\begin{matrix} A, A_{2} \end{matrix}) = 0.9172 \\ S (\begin{matrix} A, A_{3} \end{matrix}) = 0.8368 \end{matrix} \end{matrix}$

According to the calculation results of $S (\begin{matrix} A, A_{i} \end{matrix})$ , we can get the similarity measure between A₂ and A is the largest, so A belongs to the pattern A₂. In addition, the ranking result of $S (\begin{matrix} A, A_{i} \end{matrix})$ is A₂ > A₃ > A₁.

We can see that the recognition results obtained by using the distance measure and similarity measure are completely consistent with those obtained by the new correlation coefficients, but are not completely the same as the recognition results obtained by all other correlation coefficients. Therefore, the performance of the new correlation coefficients is better than all other correlation coefficients in the field of pattern recognition.

In summary, the new correlation coefficients can more moderately and accurately measure the correlation degree between PFSs, and it is more reasonable in handling pattern recognition problem.

6 Case analysis

In this section, the performance of the Pythagorean fuzzy CC-TODIM method is analyzed by two cases of MCDM.

6.1 Case 1

This section verifies the performance of the Pythagorean fuzzy CC-TODIM method in dealing with a typical MCDM problem.

The environmental protection organization invites three experts to assess the effect of urban smog control, and selects three heavily polluted cities, namely Bei A₁, Tian A₂, and Shi A₃. The organization mainly considers the following four aspects: air quality index C₁, air quality level C₂, particle matter 2.5 concentration C₃, particle matter 10 concentration C₄. Thus, the scheme set is A = { A₁, A₂, A₃ } , the criterion set is C = { C₁, C₂, C₃, C₄ } , and the decision maker set is E = { E₁, E₂, E₃ } . Each expert E_k (k = 1, 2, 3), as a decision maker, assesses the effect of smog control of each city A_i (i = 1, 2, 3) according to the criterion C_j (j = 1, 2, 3, 4). The weight of the criterion set is ω = (0 . 2861, 0 . 2200, 0 . 2787, 0 . 2152), the weight of the decision maker set is υ = (0 . 3361, 0 . 3305, 0 . 3334).

The decision matrix $P^{k} = (κ_{ij}^{k})_{3 \times 4}^{T}$ of three experts is shown in Tables 5 7.

Table 5
Decision matrix $P^{1} = (κ_{ij}^{1})_{3 \times 4}^{T}$

A ₁ A ₂ A ₃

C ₁ (0.2, 0.7) (0.2,0.8) (0.1,0.8)

C ₂ (0.6,0.2) (0.7,0.3) (0.3,0.4)

C ₃ (0.3,0.7) (0.5,0.7) (0.5,0.4)

C ₄ (0.2,0.5) (0.9,0.4) (0.3,0.7)

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

Table 6

Decision matrix $P^{2} = (κ_{ij}^{2})_{3 \times 4}^{T}$

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

Table 7

Decision matrix $P^{3} = (κ_{ij}^{3})_{3 \times 4}^{T}$

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

The PFWA operator is used to aggregate $P^{k} = (κ_{ij}^{k})_{3 \times 4}^{T}$ to obtain the comprehensive decision matrix $P = (κ_{ij})_{3 \times 4}^{T}$ , as shown in Table 8.

Table 8

Comprehensive decision matrix $P = (κ_{ij})_{3 \times 4}^{T}$

	A ₁	A ₂	A ₃
C ₁	(0.4983,0.5005)	(0.4319,0.6348)	(0.3319,0.6339)
C ₂	(0.5339,0.3992)	(0.5669,0.3997)	(0.,4995,0.5325)
C ₃	(0.4328,0.7330)	(0.4997,0.6339)	(0.4994,0.4000)
C ₄	(0.4328,0.4997)	(0.6347,0.4661)	(0.6322,0.3680)

Considering that C₂ is benefit criteria and the rest are cost criteria, the normalized decision matrix $\bar{P} = ({\bar{κ}}_{ij})_{3 \times 4}^{T}$ can be obtained, as shown in Table 9.

Table 9

Normalized decision matrix $\bar{P} = ({\bar{κ}}_{ij})_{3 \times 4}^{T}$

	A ₁	A ₂	A ₃
C ₁	(0.5005,0.4983)	(0.6348,0.4319)	(0.6339,0.3319)
C ₂	(0.5339,0.3992)	(0.5669,0.3997)	(0.4995,0.5325)
C ₃	(0.7330,0.4328)	(0.6339,0.4997)	(0.4000,0.4994)
C ₄	(0.4997,0.4328)	(0.4661,0.6347)	(0.3680,0.6322)

The reference criterion is C₁ and its weight is $\begin{matrix} ω_{1} = max {ω_{j}} = 0.2861 \end{matrix}$

The relative weight ω_j1 is calculated as $\begin{matrix} ω_{j 1} = (1.0000, 0.7690, 0.9741, 0.7522) \end{matrix}$

Let θ = 2.5, then the overall dominance matrix $π_{α}^{E} (α = 1, 2, 3, 4)$ and $π_{β}^{N} (β = 1, 2)$ determined by $K_{α}^{E} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj})$ and $K_{β}^{N} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj}) (i, t = 1, 2, 3)$ are as follows $\begin{matrix} \begin{matrix} \begin{matrix} π_{1}^{E} = [\begin{matrix} 0 & 0.0105 & 0.2812 \\ - 0.2326 & 0 & 0.2600 \\ - 0.7221 & - 0.5084 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{2}^{E} = [\begin{matrix} 0 & - 0.1181 & 0.5136 \\ - 0.4505 & 0 & 0.4464 \\ - 1.2128 & 1.2094 & 0 \end{matrix}] \end{matrix} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} \begin{matrix} π_{3}^{E} = [\begin{matrix} 0 & 0.0360 & 0.3287 \\ - 0.2740 & 0 & 0.2546 \\ - 0.6824 & - 0.4846 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{4}^{E} = [\begin{matrix} 0 & 0.0447 & 0.3658 \\ - 0.3949 & 0 & 0.3105 \\ - 0.8104 & - 0.6896 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{1}^{N} = [\begin{matrix} 0 & 0.0081 & 0.3683 \\ - 0.3017 & 0 & 0.3249 \\ - 0.8676 & - 0.6090 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{2}^{N} = [\begin{matrix} 0 & 0.0251 & 0.3672 \\ - 0.2735 & 0 & 0.3043 \\ - 0.7571 & - 0.5793 & 0 \end{matrix}] \end{matrix} \end{matrix} \end{matrix}$

The overall values $ζ_{α}^{E} (A_{i})$ and $ζ_{β}^{N} (A_{i})$ are calculated and ranked, as shown in Table 10.

Table 10

Calculation results and ranking results of $ζ_{α}^{E} (A_{i})$ and $ζ_{β}^{N} (A_{i})$

	A ₁	A ₂	A ₃	Ranking results
$ζ_{1}^{E} (A_{i})$	1	0.8264	0	A₁ > A₂ > A₃
$ζ_{2}^{E} (A_{i})$	1	0.8582	0	A₁ > A₂ > A₃
$ζ_{3}^{E} (A_{i})$	1	0.7492	0	A₁ > A₂ > A₃
$ζ_{4}^{E} (A_{i})$	1	0.7410	0	A₁ > A₂ > A₃
$ζ_{1}^{N} (A_{i})$	1	0.8094	0	A₁ > A₂ > A₃
$ζ_{2}^{N} (A_{i})$	1	0.7909	0	A₁ > A₂ > A₃

As shown in Table 10, all ranking results of the overall value are A₁ > A₂ > A₃, so the city with the best smog control effect is A₁, that is, Bei. The results obtained in this section are consistent with those in the reference [43], verifying that Pythagorean fuzzy CC-TODIM method can accurately and effectively solve MCDM problem.

6.2 Case 2

This section compares the performance of the Pythagorean fuzzy CC-TODIM method based on different correlation coefficients $K_{α}^{E} (α = 1, 2, 3, 4)$ and $K_{β}^{N} (β = 1, 2)$ .

A company is preparing to select the best employee of the year. After recommendation, there are two candidates, Wang A₁ and Zhang A₂. The company considers two aspects: sales performance C₁ and daily performance C₂. Thus, the scheme set is A = { A₁, A₂ } , the criterion set is C = { C₁, C₂ } . The company, as the decision maker, assesses each candidate A_i (i = 1, 2) according to the criterion C_j (j = 1, 2). The weight of the criterion set is ω = (0 . 6, 0 . 4).

The decision matrix $P = (κ_{ij})_{2 \times 2}^{T}$ is shown in Table 11.

Table 11
Decision matrix $P = (κ_{ij})_{2 \times 2}^{T}$

A ₁ A ₂

C ₁ (0.8,0.3) (0.8,0.1)

C ₂ (0.7,0.2) (0.6,0.3)

	A ₁	A ₂
C ₁	(0.8,0.3)	(0.8,0.1)
C ₂	(0.7,0.2)	(0.6,0.3)

Considering that C₁ and C₂ are the benefit criteria, the normalized decision matrix $\bar{P} = ({\bar{κ}}_{ij})_{2 \times 2}^{T} = (κ_{ij})_{2 \times 2}^{T}$ .

The reference criterion is C₁ and its weight is $\begin{matrix} ω_{1} = max {ω_{j}} = 0.6 \end{matrix}$

The relative weight ω_j1 of C_j is calculated as $\begin{matrix} ω_{j 1} = (1.0000, 0.6667) \end{matrix}$

Let θ = 2.5, then the overall dominance matrix $π_{α}^{E} (α = 1, 2, 3, 4)$ and $π_{β}^{N} (β = 1, 2)$ determined by $K_{α}^{E} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj})$ and $K_{β}^{N} (\begin{matrix} {\bar{κ}}_{ij} \end{matrix}, {\bar{κ}}_{tj}) (i, t = 1, 2, 3)$ are as follows $\begin{matrix} \begin{matrix} \begin{matrix} π_{1}^{E} = [\begin{matrix} 0 & 0.0276 \\ - 0.0049 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{2}^{E} = [\begin{matrix} 0 & 0.2519 \\ - 0.2180 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{3}^{E} = [\begin{matrix} 0 & 0.0505 \\ - 0.0226 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{4}^{E} = [\begin{matrix} 0 & 0.0278 \\ 0.0306 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{1}^{N} = [\begin{matrix} 0 & 0.0977 \\ - 0.0750 & 0 \end{matrix}] \end{matrix} \\ \begin{matrix} π_{2}^{N} = [\begin{matrix} 0 & 0.0493 \\ - 0.0204 & 0 \end{matrix}] \end{matrix} \end{matrix} \end{matrix}$

The overall values $ζ_{α}^{E} (A_{i})$ and $ζ_{β}^{N} (A_{i})$ are calculated and ranked, as shown in Table 12.

Table 12

Calculation results and ranking results of $ζ_{α}^{E} (A_{i})$ and $ζ_{β}^{N} (A_{i})$

	A ₁	A ₂	Ranking results
$ζ_{1}^{E} (A_{i})$	1	0	A₁ > A₂
$ζ_{2}^{E} (A_{i})$	1	0	A₁ > A₂
$ζ_{3}^{E} (A_{i})$	1	0	A₁ > A₂
$ζ_{4}^{E} (A_{i})$	0	1	A₂ > A₁
$ζ_{1}^{N} (A_{i})$	1	0	A₁ > A₂
$ζ_{2}^{N} (A_{i})$	1	0	A₁ > A₂

As shown in Table 12, all ranking results are A₁ > A₂ (except $ζ_{4}^{E} (A_{i})$ ), so the best employee of the year is Wang. The ranking result of $ζ_{4}^{E} (A_{i})$ is A₂ > A₁, which indicates that the Pythagorean fuzzy CC-TODIM method based on the existing correlation coefficient $K_{4}^{E}$ is prone to errors in the application.

In summary, compared with $K_{α}^{E}$ , the Pythagorean fuzzy CC-TODIM method based on $K_{β}^{N}$ is more effective and reliable.

7 Conclusions

In this paper, first, the new correlation coefficients of PFS are proposed, then the classic TODIM method is extended and the Pythagorean fuzzy CC-TODIM method is proposed based on the new correlation coefficients of PFS. The performance of the new correlation coefficients and the Pythagorean fuzzy CC-TODIM method is verified by examples and cases. In summary, the main conclusions are as follows

Compared with the existing correlation coefficients, the new correlation coefficients can more moderately and reasonably measure the correlation degree between PFSs, and can accurately and reliably handle the pattern recognition problems.

The Pythagorean fuzzy CC-TODIM method can accurately and effectively solve MCDM problems. Compared with the existing correlation coefficients, the method based on the new correlation coefficients is more effective and reliable in dealing with MCDM problems.

In the future, we except to further expand the application range of the new correlation coefficients of PFS, and continue to test the performance of the new correlation coefficients. Meanwhile, the TODIM method can continue to be improved from other aspects, such as combining with other types of fuzzy sets, or using other information measures, so that it can solve different DM problems more effectively.

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New correlation coefficients of Pythagorean fuzzy set and its application to extended TODIM method

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Pythagorean fuzzy set

3.1 Existing correlation coefficients

Table 1 Calculation results of K ˜ α E ( A , A i ) and K ˜ β N ( A , A i ) A 1 A 2 K ˜ 1 E ( A , A i ) 0.9959 0.9018 K ˜ 2 E ( A , A i ) 0.4972 0.8112 K ˜ 3 E ( A , A i ) 0.9034 0.9798 K ˜ 4 E ( A , A i ) 0.7846 0.9497 K ˜ 1 N ( A , A i ) 0.7960 0.8968 K ˜ 2 N ( A , A i ) 0.8945 0.9793

6.1 Case 1

Table 5 Decision matrix P 1 = ( κ ij 1 ) 3 × 4 T A 1 A 2 A 3 C 1 (0.2, 0.7) (0.2,0.8) (0.1,0.8) C 2 (0.6,0.2) (0.7,0.3) (0.3,0.4) C 3 (0.3,0.7) (0.5,0.7) (0.5,0.4) C 4 (0.2,0.5) (0.9,0.4) (0.3,0.7)

Table 11 Decision matrix P = ( κ ij ) 2 × 2 T A 1 A 2 C 1 (0.8,0.3) (0.8,0.1) C 2 (0.7,0.2) (0.6,0.3)

References

Table 5
Decision matrix $P^{1} = (κ_{ij}^{1})_{3 \times 4}^{T}$

A ₁ A ₂ A ₃

C ₁ (0.2, 0.7) (0.2,0.8) (0.1,0.8)

C ₂ (0.6,0.2) (0.7,0.3) (0.3,0.4)

C ₃ (0.3,0.7) (0.5,0.7) (0.5,0.4)

C ₄ (0.2,0.5) (0.9,0.4) (0.3,0.7)

Table 11
Decision matrix $P = (κ_{ij})_{2 \times 2}^{T}$

A ₁ A ₂

C ₁ (0.8,0.3) (0.8,0.1)

C ₂ (0.7,0.2) (0.6,0.3)