Pythagorean fuzzy multi-criteria decision making method based on CODAS with new score function

Abstract

Pythagorean fuzzy set (PFS), as a generalization of intuitionistic fuzzy set (IFS), is more suitable to capture the indeterminacy of the experts’ decision making information. This paper is designed to build new algorithm for managing multi-criteria decision making (MCDM) issue under Pythagorean fuzzy environment. First, we initiate a novel score function based Pythagorean fuzzy number (PFN). Later, we explore an algorithm for solving MCDM problem based on CODAS (COmbinative Distance-based ASsessment). Ultimately, the availability of method is stated by some numerical examples. The dominating traits of the developed algorithm, compared to some existing Pythagorean fuzzy decision making algorithms, are (1) derive a ranking without the complex process; (2) achieve the optimal alternative without counterintuitive phenomena; (3) strong ability to differentiate the optimal alternative.

Keywords

Pythagorean fuzzy number CODAS score function MCDM

1 Introduction

Intuitionistic fuzzy set (IFS), developed by Atanassov [1], is a generalization of fuzzy set (FS) [2]. IFS is portrayed by the degrees of membership and non-membership, and so it can describe the trait of uncertain data more synthetically and detailedly. The primary trait of IFS is that it distributes to each element the membership and the nonmembership with its sum ≤ 1. Nevertheless, in certain real issues, the sum of corresponding membership degree and nonmembership degree to which a given alternative meeting a criterion offered by an expert or decision maker (DM) may be ≥ 1 while their square sum is ≤ 1.

Hence, Yager [3] explored Pythagorean fuzzy set (PFS) portrayed by the degree of membership and nonmembership, which meets the case that the square sum of corresponding nonmembership degree and membership degree is ≤1. Yager [4] provided an example to illustrate such case: a DM or expert offers his approval for membership of a given alternative is $\frac{\sqrt{3}}{2}$ and his opposition for membership is $\frac{1}{2}$ . Due to its sum is grater than 1, it isn’t impracticable for IFS while it can suit for PFS since $(\frac{\sqrt{3}}{2})^{2} + (\frac{1}{2})^{2} \leq 1$ . Apparently, PFS is more viable than IFS to simulate the indeterminacy in real multi-criteria decision making (MCDM) issues.

The PFS has been studied from different perspectives [5], including decision making technologies [6 –23], aggregation operators [24 –42], information measures [43 –47], the extensions of PFS [48 –51]. In particular, a concise literature review of the first two aspects is given as follows:

(1) Decision making technologies. Inspired by the revised TOPSIS method [52], Zhang and Xu [6] applied it to MCDM issue under Pythagorean fuzzy environment. Yager [7] proposed Pythagorean fuzzy weighted averaging operator for solving MCDM problem. Peng and Yang [8] studied their relationship, and presented multi-criteria group decision making (MCGDM) approach based on SIR (superiority and inferiority ranking). Zhang [9] presented a hierarchical QUALIFLEX (qualitative flexible) method with the closeness index-based ordering approaches for Pythagorean fuzzy decision analysis. Peng and Dai [10] introduced two Pythagorean fuzzy stochastic MCDM methods based on regret theory and prospect theory. Ren et al. [11] extended the TODIM (acronym in Portuguese for Interactive Multi-Criteria Decision Making) algorithm to deal the MCDM issues under Pythagorean fuzzy environment. Peng and Yang [12] developed a novel Pythagorean fuzzy MCGDM method based Choquet integral and MABAC (multi-attributive border approximation area comparison). Wan et al. [13] presented a mathematical programming approach for solving MCGDM issues under Pythagorean fuzzy environment.

(2) Aggregation operators. Yager [4] developed plentiful aggregation operators (AOs), and employed them in solving MCDM problems. Garg [26, 27] explored an extended Pythagorean fuzzy AOs by Einstein and geometric operations. Zeng et al. [28] explored a hybrid aggregation information for Pythagorean fuzzy information. Ma and Xu [29] explored symmetric weighted averaging/geometric operators under Pythagorean fuzzy environment. Zeng [30] proposed the Pythagorean fuzzy AOs based probabilistic information and ordered weighted averaging (OWA). Peng and Yang [31] presented some fundamental theories of interval-valued Pythagorean fuzzy AOs. Wei and Lu [32] gave Pythagorean fuzzy Maclaurin Symmetric mean operators. Zhang et al. [33] proposed some generalized Pythagorean fuzzy Bonferroni mean aggregation operators. Liu et al. [34] developed Pythagorean fuzzy interaction AOs and discussed their properties in detail.

CODAS (Combinative Distance-based Assessment), developed by Ghorabaee et al. [53], obtains the monolithic expression of one object by Taxicab distance and Euclidean distance from ideal-negative point. The CODAS employs the discriminating Euclidean distance as the all-important measure of evaluation. The Taxicab distance is employ when the Euclidean distance of two objects is quit contiguous. The closeness degree of Euclidean distance is regulatory through a threshold parameter. The Taxicab and Euclidean distances are estimated for l¹-norm and l²-norm non-differential spaces [54]. Consequently, the alternatives are firstly evaluated in an l²-norm non-differential space by the CODAS algorithm. It is time to turn l¹-norm non-differential space when the given objects are out of comparableness. To fulfill above process, each pair of objects should be compared. The CODAS method has been successfully employed in market segment assessment [55], emergency decision making [56].

Due to the deficiencies of some existing decision making methods [7 , 45] and score functions [6 , 29] for PFS, it maybe untoward for DMs to choose best or credible alternatives. Hence, the purpose of this article is to dispose the two deficiencies discussed above by presenting a MCDM approach (CODAS) to dealing decision information for PFSs, which not only can derive a ranking without the complex process, but also can obtain the best alternative without counterintuitive phenomena.

To obtain such goals, the major research innovation points can be listed as follows:

(1) The novel score function is explored and some interesting properties are proved.

(2) The Pythagorean fuzzy CODAS (PF-CODAS) is proposed for dealing some MCDM issues.

(3) The comparison with some existing Pythagorean fuzzy MCDM approaches is constructed for stating the effectiveness of the explored method.

For a better discussion, the rest of paper is listed as follows: In Sect. 2, the basic notions of PFS are retrospected in brief. In Sect. 3, the novel score function for PFN is developed and certain properties are shown. In Sect. 4, we propose a novel Pythagorean fuzzy MCDM approach based on CODAS. In Sect. 5, some numerical instances are given to illustrate the availability of developed method. Sect. 6 gives a conclusion.

2 Preliminaries

Certain basic definitions of PFS are shown in the following.

Definition 1. [3] Let X be a domain of discourse. The PFS P in X is shown as follows: $P = {< x, μ_{P} (x), ν_{P} (x) > ∣ x \in X},$ (1) where μ_P : X→[0,1] signifies membership degree and ν_P : X→[0,1] signifies nonmembership degree of the element x ∈ X to such set P, respectively. It should be fulfilled with the condition that 0 ≤ (μ_P (x)) ² + (ν_P (x)) ² ≤ 1. The hesitation degree $π_{P} (x) = \sqrt{1 - (μ_{P} (x))^{2} - (ν_{P} (x))^{2}}$ . For short, Zhang and Xu [6] regarded p = (μ_P, ν_P) as PFN (Pythagorean fuzzy number).

Definition 2. [3, 6] For three PFNs p = (μ_p, ν_p) , p₁ = (μ_{p
₁}, ν_{p
₁}) , p₂ = (μ_{p
₂}, ν_{p
₂}), the related operations are presented as follows:

(1) p₁ ∪ p₂ = (max {μ_{p
₁}, μ_{p
₂}} , min {ν_{p
₁}, ν_{p
₂}});

(2) p₁ ∩ p₂ = (min {μ_{p
₁}, μ_{p
₂}} , max {ν_{p
₁}, ν_{p
₂}});

(3) p^c = (ν_p, μ_p);

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

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

(6) $λ p = (\sqrt{1 - (1 - μ_{p}^{2})^{λ}}, ν_{p}^{λ}), λ > 0$ ;

(7) $p^{λ} = (μ_{p}^{λ}, \sqrt{1 - (1 - ν_{p}^{2})^{λ}}), λ > 0$ .

3 The novel score function for PFNs

Some existing score functions are reviewed in current section. Moreover, a novel score function, considered hesitation degree of Pythagorean fuzzy number, is explored.

3.1 Some existing pythagorean fuzzy score functions

Assume that a PFN is expressed as p =< μ, ν >, where μ and ν denote the approval and opposition, respectively. Zhang and Xu [6] developed such score function as follows: $S_{zhang} (p) = μ^{2} - ν^{2},$ (2) where S_zhang (p) ∈ [-1, 1].

Assume that p = (0.4, 0.4) and q = (0.3, 0.3). If we utilize such score function [6] to choose the biggest PFN, we have S (p) = S (q) =0. Therefore, we can’t differentiate the discrepancy, which reveals that such score function is out of achieving the biggest PFN.

Observing the drawbacks of the score function S_zhang [6], Peng and Yang [8] gave such accuracy function as follows: $H_{peng} (p) = μ^{2} + ν^{2},$ (3) where H_peng (p) ∈ [0, 1].

Moreover, Ma and Xu [29] noticed that such score function S_zhang and accuracy function H_peng were essentially capsuled after squaring difference value. Hence, they presented a revised score function in the following. $S_{ma} (p) = {\begin{matrix} \sqrt{μ^{2} - ν^{2}} & iff μ \geq ν, \\ - \sqrt{ν^{2} - μ^{2}} & iff μ \leq ν, \end{matrix}$ (4) where S_ma (p) ∈ [-1, 1]. $H_{ma} (p) = \sqrt{μ^{2} + ν^{2}},$ (5) where H_ma (p) ∈ [0, 1].

Similarly, the S_ma encounters the above mentioned case. The score function of S_zhang and S_ma can precisely classify the common PFNs. Nevertheless, the score functions (S_zhang, S_ma) and the accuracy functions (H_peng, H_ma) fail to take the effect of hesitation into consideration, which reflects the decision information loss.

Due to the drawbacks of the existing score functions [6, 29] or accuracy functions [8, 29], an efficient score function is proposed. It is not only considered the information of approval and opposition but also hesitation, which have a superior ability in distinguishing the PFNs.

3.2 A novel pythagorean fuzzy score function

In real case, a resultful score function is supposed to take the degrees of membership, non-membership and hesitation into consideration at the same time. For a PFN p = (μ, ν), we can interpret such definition using a voting model: support (μ), opposition (ν) and hesitate (π). The hesitant part might be influenced by the supporters and opponents, and then reverse to opposition or support. In other words, the degree of hesitators to objection and support is inconclusive. Nevertheless, it is easily happen such herd mentality when people make decision in hesitate environment. In other words, some DMs who hesitate is more inclined to support when μ > ν while some DMs is more inclined to oppose when μ < ν. Therefore, we should take the influence of the hesitant information for the existing score function S (p) = μ² - ν². The hesitation degree π possesses a positive influence on score function S (p) = μ² - ν² which prompts S (p) increased when μ > ν. Similarly, the hesitation degree π possesses a negative influence on score function S (p) = μ² - ν² which prompts S (p) decreased when μ < ν. Based on the above discussion, a novel score function of PFN is developed.

Definition 3. Let p = (μ, ν) be a PFN, then its score function is presented as follows: $S_{px} (p) = μ^{2} - ν^{2} + (μ^{2} - ν^{2}) π^{2} .$ (6)

Definition 4. For two PFNs p = (μ_p, ν_p) and q = (μ_q, ν_q), then

(1) If S_px (p) > S_px (q), then p > q;

(2) If S_px (p) < S_px (q), then p < q;

(3) If S_px (p) = S_px (q), then

(3.1) If π_p > π_q, then p < q;

(3.2) If π_p = π_q, then p = q.

Theorem 1. Let p = (μ, ν) be a PFN, then S_px (p) monotonically increases when μ increases and monotonically decreases when ν increases.

Proof. Based on the Equation (6), we obtain its first partial derivative of S_px (p) with μ and ν,

$\frac{\partial S_{px} (p)}{\partial μ} = 4 μ (1 - μ^{2}) \geq 0, \frac{\partial S_{px} (p)}{\partial ν} = - 4 ν (1 - ν^{2}) \leq 0$ .

Hence, we can obtain the conclusion that S_px (p) monotonically increases when μ increases and monotonically decreases when ν increases.

Theorem 2. Let p = (μ, ν) be a PFN, the new score function S_px (p) meets:

(1) -1 ≤ S_px (p) ≤1;

(2) S_px (p) =1 iff p = (1, 0); S_px (p) = -1 iff p = (0, 1).

Proof.

(1) According tp the Theorem 1, we have S_px (p) monotonically increases along with μ.

Therefore,

$S_{px} (p)_{\min} = S_{px} ((0, v)) = \min_{0 \leq ν \leq 1} {- 2 ν^{2} - ν^{4}} = - 2 ν^{2} + ν^{4} ∣_{ν = 1} = - 1$ ,

$S_{px} (p)_{\max} = S_{px} ((1, v)) = \max_{0 \leq ν \leq 1} {(1 - ν^{2})^{2}} = (1 - ν^{2})^{2} ∣_{ν = 0} = 1$ .

Furthermore, we can have -1 ≤ S_px (p) ≤1.

(2) Based on (1), S_px (p) can have the maximum value 1 when μ = 1, ν = 0; S_px (p) can have the minimum value -1 when μ = 0, ν = 1.

In other words, S_px (p) =1 iff p = (1, 0); S_px (p) = -1 iff p = (0, 1).

Theorem 3. Let p = (μ_p, ν_p) and q = (μ_q, ν_q) be two PFNs, if μ_p > μ_q and ν_p < ν_q, then S_px (p) > S_px (q).

Proof. According to Theorem 1, it can be easily seen that S_px (p) monotonically increases when μ_p increases and monotonically decreases when ν_p increases.

Therefore, if μ_p > μ_q and ν_p < ν_q, then S_px (p) > S_px (q).

Theorem 4. Let p = (μ, ν) be a PFN and p^c = (ν, μ), then S_px (p^c) = - S_px (p).

Proof. According to the definition of score function S_px (p) , we can have S_px (p) = (μ² - ν²) (2 - μ² - ν²).

S_px (p^c) = (ν² - μ²) (2 - ν² - μ²) = - (μ² - ν²) (2 - μ² - ν²) = - S_px (p) .

To verify the effectiveness of the explored score function, a comparison between the developed score function and the existing score functions (S_ma, S_zhang) is shown in Table 1.

Table 1

The comparison of score function.

Examples	The score value	Ranking
$p = (\sqrt{0.32}, 0.8), q = (\sqrt{0.17}, 0.7)$	S_px (p) = -0.3328, S_px (q) = -0.4288	p > q
	S_ma (p) = -0.5196, S_ma (q) = -0.5196	p = q
	S_zhang (p) = -0.32, S_zhang (q) = -0.32	p = q
$p = (\sqrt{0.34}, 0.7), q = (\sqrt{0.21}, 0.6)$	S_px (p) = -0.1755, S_px (q) = -0.2145	p > q
	S_ma (p) = -0.3873, S_ma (q) = -0.3873	p = q
	S_zhang (p) = -0.15, S_zhang (q) = -0.15	p = q

According to Table 1, it can be easily known that the explored score function S_px can tremendously and availably avoid the deficiencies of S_ma and S_zhang. That is to say, the explored score function can effectively distinguish PFNs while some existing score functions (S_ma, S_zhang) fail to distinguish. The conclusive decision results of using the developed score function will be same as the accuracy functions [8, 29] (H_peng, H_ma) in Table 1. It reflects that the explored score function S_peng is firsthand and effectual.

4 Approach to pythagorean fuzzy MCDM based on CODAS

In order to manage complex and real-life problem, we develop a Pythagorean fuzzy decision making method based on CODAS.

4.1 The description of pythagorean fuzzy MCDM issue

In the following, how to employ the CODAS approach in dealing with MCDM issue with Pythagorean fuzzy information is discussed. Just to make it clear, a brief description of the existing issues is presented.

The key of MCDM issue with Pythagorean fuzzy information is to achieve the optimal choice from amounts of alternatives, which are evaluated by a series of criteria, where the evaluative values are PFNs. Then such type of problem can be described by the mathematical symbols as follows:

Let A = {A₁, A₂, ⋯ , A_m} be a finite set of alternatives and C = {C₁, C₂, ⋯ , C_n} be a finite set of criteria. w = {w₁, w₂, ⋯ , w_n} is the weight vector of the criteria with w_j ∈ [0, 1] (j = 1, 2, ⋯ , n) and $\sum_{j = 1}^{n} w_{j} = 1$ . The evaluation value over the alternative A_i along with the criterion C_j is denoted as PFN p_ij = (μ_ij, ν_ij), where μ_ij denotes the degree that alternative A_i agrees the criterion C_j, and ν_ij signifies the degree that the alternative A_i disagrees the criterion C_j. Consequently, the MCDM issue can be tersely denoted in Pythagorean fuzzy decision matrix P = (p_ij) _m×n, which is presented in Table 2.

Table 2
The Pythagorean fuzzy decision matrix P = ( p_ij) _m×n

C ₁ C ₂ ⋯ C _n

A ₁ p ₁₁ p ₁₂ ⋯ p _1n

A ₂ p ₂₁ p ₂₂ ⋯ p _2n

⋮ ⋮ ⋮ ⋱ ⋮

A _m p _m1 p _m2 ⋯ p _mn

	C ₁	C ₂	⋯	C _n
A ₁	p ₁₁	p ₁₂	⋯	p _1n
A ₂	p ₂₁	p ₂₂	⋯	p _2n
⋮	⋮	⋮	⋱	⋮
A _m	p _m1	p _m2	⋯	p _mn

4.2 Pythagorean fuzzy CODAS (PF-CODAS) method

To settle MCDM issue with Pythagorean fuzzy information, we develop a revised PF-CODAS method. It is a novel and high-efficiency algorithm developed by Ghorabaee et al. [53]. The favorable alternative is calculated by two indifference spaces named l¹-norm and l²-norm for whole criteria. The combinative mode of the Euclidean distance and Taxicab distance is employed in calculating the assessment score of alternative based on above indifference spaces. Nevertheless, some existing Euclidean distances and Taxicab distances rely on the crisp or fuzzy circumstance, which fail to solve Pythagorean fuzzy issues. In order to solve such issues, we employ the fuzzy weighted Hamming distance and fuzzy weighted Euclidean distance that takes the place of the crisp distance [53]. The research emphasis is to present a revised PF-CODAS method.

At the very start, the evaluation information is normalized because there exists benefit criteria and cost criteria. Such criteria react oppositely, i.e., the bigger value reveals better performance of a benefit criterion but reflects the worse performance of a cost criterion. Therefore, in order to ensure all criteria are compatible, we continue to shift the cost criterion into benefit criterion by the following formula. $p_{ij}^{'} = (μ_{ij}^{'}, ν_{ij}^{'}) = {\begin{matrix} (μ_{ij}, ν_{ij}), C_{j} is benefit criterion, \\ (ν_{ij}, μ_{ij}), C_{j} is cost criterion . \end{matrix}$ (7) Based on the above equation, we obtain the normalized Pythagorean fuzzy matrix $P^{'} = (p_{ij}^{'})_{m \times n}$ .

Then, we compute score function t_ij (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) of $p_{ij}^{'}$ by Equation (8), $t_{ij} = {μ_{ij}^{'}}^{2} - {ν_{ij}^{'}}^{2} + ({μ_{ij}^{'}}^{2} - {ν_{ij}^{'}}^{2}) {π_{ij}^{'}}^{2} .$ (8)

For the sake of computing the weighted normalized decision matrix R = (r_ij) _m×n, we define it by weighted normalized values. $r_{ij} = w_{j} t_{ij},$ (9) where w_j (0 < w_j < 1) signifies the weight of jth criterion, and $\sum_{j = 1}^{n} w_{j} = 1$ .

The fundamental concept of the developed CODAS algorithm is negative-ideal solution (NIS). Hence, the negative-ideal solution is denoted in the following. $NIS = ({nis}_{j})_{1 \times n},$ (10) ${nis}_{j} = \min_{i} r_{ij}, i = 1, 2, \dots, m .$ (11)

Whereafter, the Euclidean distance E = (E_i) _1×m and Taxicab distance T = (T_i) _1×m of alternative A_i (i = 1, 2, ⋯ , m) from negative-ideal solution are calculated as follows: $E_{i} = \sqrt{\sum_{j = 1}^{n} (r_{ij} - {nis}_{j})^{2}},$ (12) $T_{i} = \sum_{j = 1}^{n} ∣ r_{ij} - {nis}_{j} ∣ .$ (13)

According to the Euclidean distance and Taxicab distance, the relative assessment (RA) matrix is constructed in the following. $RA = ({ra}_{ik})_{m \times m},$ (14)

$\begin{matrix} {ra}_{ik} = (E_{i} - E_{k}) + (ψ (E_{i} - E_{k}) \\ \times (T_{i} - T_{k})), i, k \in 1, 2, \dots, m \end{matrix}$ (15) where ψ signifies the threshold function to identify the equality of Euclidean distances of two different alternatives, which is denoted in the following. $ψ (x) = {\begin{matrix} 1, if ∣ x ∣ \geq Θ, \\ 0, if ∣ x ∣ < Θ . \end{matrix}$ (16)

In above function, Θ is threshold parameter that can be decided and set by expert or DM. It is advised to set such parameter at range from 0.01 to 0.05 (a value close to 0). It will continue to calculate the Taxicab distance of two alternatives when the difference of two alternatives based Euclidean distances exceeds the given Θ. In this paper, we set Θ=0.02 for the subsequent calculations.

Next, it can integrate the evaluation score of ith alternative A_i as follows: ${RA}_{i} = \sum_{k = 1}^{m} {ra}_{ik} .$ (17)

Remark 1. It is imperative to state that in Pythagorean fuzzy CODAS method, which the evaluation information is denoted by PFNs. Denoted by the degrees of membership and non-membership, the PFNs are resultful for seizing the indeterminate and inconclusive of the experts or DMs in the MCDM problem. In addition, the Pythagorean fuzzy CODAS method is an inestimable tool to manage MCDM issues with PFNs possessing a strong ability to distinguish the best alternative and obtaining the optimal alternative without counterintuitive phenomena. Nevertheless, other methods do not have this desirable characteristic when solving the decision making issues with Pythagorean fuzzy information.

Algorithm 1 :PF-CODAS

1: Construct the PF decision matrix P = (P_ij) _m×n (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n).

2: Shift the matrix P = (P_ij) _m×n into the standard matrix $P^{'} = (P_{ij}^{'})_{m \times n}$ by Equation (7)

3: Calculate the score matrix T = (t_ij) _m×n of $P^{'} = (P_{ij}^{'})_{m \times n}$ by Equation (8).

4: Compute the weighted normalized decision matrix r_ij by Equation (9).

5: Compute the NIS by Equation (10).

6: Calculate the Euclidean distance E and Taxicab distance T from the NIS by Equations (12) and (13).

7: Determine the relative evaluation matrix RA by Equation (14).

8: Compute the evaluation score of ith alternative RA_i by Equation (17).

9: Order the alternatives by evaluation score.

4.3 An illustrative example

Example 1. A college wish to buy a teaching evaluation system (TES) for assessing the teaching. The software company provides six potential TESs A_i (i = 1, 2, 3, 4, 5, 6), which maybe chose by the college. Assume that four criteria C₁(operational), C₂(functional), C₃(security), and C₄(economic), the weight vector of the criteria C_j (j = 1, 2, 3, 4) is w = (0.3, 0.3, 0.3, 0.1). Moreover, the C₁, C₂ and C₃ are benefit criteria, C₄ is cost criterion. Suppose that the TES A_i along with the criterion C_j is denoted by the PF matrix P = (p_ij) _6×4 = (μ_ij, ν_ij) _6×4, which is presented in Table 3.

Table 3
The PF decision matrix P = ( p_ij) _6×4

C ₁ C ₂ C ₃ C ₄

A ₁ (0.5, 0.2) (0.6, 0.1) (0.6, 0.4) (0.8, 0.1)

A ₂ (0.7, 0.2) (0.8, 0.1) (0.7, 0.2) (0.4, 0.5)

A ₃ (0.7, 0.4) (0.6, 0.2) (0.4, 0.7) (0.3, 0.1)

A ₄ (0.3, 0.4) (0.7, 0.1) (0.8, 0.1) (0.5, 0.3)

A ₅ (0.5, 0.2) (0.6, 0.1) (0.7, 0.1) (0.6, 0.2)

A ₆ (0.6, 0.3) (0.6, 0.1) (0.8, 0.1) (0.6, 0.2)

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.5, 0.2)	(0.6, 0.1)	(0.6, 0.4)	(0.8, 0.1)
A ₂	(0.7, 0.2)	(0.8, 0.1)	(0.7, 0.2)	(0.4, 0.5)
A ₃	(0.7, 0.4)	(0.6, 0.2)	(0.4, 0.7)	(0.3, 0.1)
A ₄	(0.3, 0.4)	(0.7, 0.1)	(0.8, 0.1)	(0.5, 0.3)
A ₅	(0.5, 0.2)	(0.6, 0.1)	(0.7, 0.1)	(0.6, 0.2)
A ₆	(0.6, 0.3)	(0.6, 0.1)	(0.8, 0.1)	(0.6, 0.2)

As we have discussed above, the PF-CODAS method is valid in dealing such MCDM issues. Next, we will take merit of the proposed method to model the MCDM process.

Step 1. Construct the PF decision matrix P = (P_ij) _6×4, shown in Table 3.

Step 2. Shift PF decision matrix P = (P_ij) _6×4 into a normalized matrix $P^{'} = (P_{ij}^{'})_{6 \times 4}$ by Equation (7), presented in Table 4.

Table 4

The normalized PF decision matrix P′

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.5, 0.2)	(0.6, 0.1)	(0.6, 0.4)	(0.1, 0.8)
A ₂	(0.7, 0.2)	(0.8, 0.1)	(0.7, 0.2)	(0.5, 0.4)
A ₃	(0.7, 0.4)	(0.6, 0.2)	(0.4, 0.7)	(0.1, 0.3)
A ₄	(0.3, 0.4)	(0.7, 0.1)	(0.8, 0.1)	(0.3, 0.5)
A ₅	(0.5, 0.2)	(0.6, 0.1)	(0.7, 0.1)	(0.2, 0.6)
A ₆	(0.6, 0.3)	(0.6, 0.1)	(0.8, 0.1)	(0.2, 0.6)

Step 3. Calculate the score matrix T = (t_ij) _6×4 of $P^{'} = (P_{ij}^{'})_{6 \times 4}$ by Equation (8), shown as follows: $T = (\begin{matrix} 0.3591 & 0.5705 & 0.2960 & - 0.8505 \\ 0.6615 & 0.8505 & 0.6615 & 0.1431 \\ 0.4455 & 0.5120 & - 0.4455 & - 0.1520 \\ - 0.1225 & 0.7200 & 0.8505 & - 0.2656 \\ 0.3591 & 0.5705 & 0.7200 & - 0.5120 \\ 0.4185 & 0.5705 & 0.8505 & - 0.5120 \end{matrix}) .$

Step 4. Compute the r_ij by Equation (9), shown as follows: $R = (\begin{matrix} 0.1077 & 0.1712 & 0.0888 & - 0.0851 \\ 0.1984 & 0.2552 & 0.1984 & 0.0143 \\ 0.1337 & 0.1536 & - 0.1337 & - 0.0152 \\ - 0.0367 & 0.2160 & 0.2552 & - 0.0266 \\ 0.1077 & 0.1712 & 0.2160 & - 0.0512 \\ 0.1256 & 0.1712 & 0.2552 & - 0.0512 \end{matrix}) .$

Step 5. Compute the NIS by Equation (10), shown as follows: $NIS = (- 0.0367, 0.1536, - 0.1337, - 0.0851) .$

Step 6. Determine the Euclidean distance E and Taxicab distance T from the NIS by Equations (12) and (13), shown as follows: $E = (0.2658, 0.4310, 0.1842, 0.3981, 0.3802, 0.4230),$ $T = (0.3845, 0.7682, 0.2403, 0.5097, 0.5455, 0.6025) .$

Step 7. Construct the relative evaluation matrix RA by Equation (14), shown as follows:

$\begin{matrix} RA = \\ (\begin{matrix} 0.0000 & - 0.4989 & 0.1810 & - 0.2130 & - 0.2350 & - 0.3263 \\ 0.4989 & 0.0000 & 0.6798 & 0.2858 & 0.2639 & 0.0068 \\ - 0.1810 & - 0.6798 & 0.0000 & - 0.3940 & - 0.4159 & - 0.5073 \\ 0.2130 & - 0.2858 & 0.3940 & 0.0000 & 0.0139 & - 0.1133 \\ 0.2350 & - 0.2639 & 0.4159 & - 0.0139 & 0.0000 & - 0.0913 \\ 0.3263 & - 0.0068 & 0.5073 & 0.1133 & 0.0913 & 0.0000 \end{matrix}) . \end{matrix}$ (18)

Step 8. Compute the assessment score of ith TES RA_i by Equation (17), presented as follows: ${RA}_{1} = - 1.0922, {RA}_{2} = 1.7353, {RA}_{3} = - 2.1781,$ ${RA}_{4} = 0.2218, {RA}_{5} = 0.2818, {RA}_{6} = 1.0314 .$

Step 9. The ordering of the TES is A₂ ≻ A₆ ≻ A₅ ≻ A₄ ≻ A₁ ≻ A₃ . That is to say, A₂ is optimal TES.

5 Comparative analysis

In the following, the TOPSIS approach [6], the TODIM approach [11], the some aggregation operators [7 , 45] with the proposed CODAS approach are compared.

5.1 Some illustrative examples

Example 2. [5] An Internet company wish to purchase an administrative management system (AMS) to assess the employees. The head of a company offers five potential administrative management systems A_i (i = 1, 2, 3, 4, 5). Supposing that four criteria C₁(operational), C₂(timeliness), C₃(functional), and C₄(economic), the weight vector of the criteria C_j (j = 1, 2, 3, 4) is w = (0.2, 0.3, 0.4, 0.1). Moreover, the C₁, C₂ and C₃ are benefit criteria, C₄ is cost criterion. Suppose that each AMS A_i along with the criterion C_j is denoted as the PF matrix P = (p_ij) _5×4 = (μ_ij, ν_ij) _5×4. The evaluation information is presented in Table 6.

Next, the comparison with some existing methods to choose the optimal AMS is presented in Table 5.

Table 5
The PF decision matrix in Example 1

C ₁ C ₂ C ₃ C ₄

A ₁ (0.7, 0.3) (0.7, 0.1) (0.6, 0.4) (0.3, 0.1)

A ₂ (0.8, 0.2) (0.9, 0.1) (0.9, 0.2) (0.4, 0.5)

A ₃ (0.6, 0.4) (0.6, 0.2) (0.4, 0.6) (0.3, 0.2)

A ₄ (0.5, 0.3) (0.7, 0.1) (0.7, 0.1) (0.4, 0.3)

A ₅ (0.4, 0.3) (0.8, 0.1) (0.8, 0.4) (0.2, 0.3)

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.7, 0.3)	(0.7, 0.1)	(0.6, 0.4)	(0.3, 0.1)
A ₂	(0.8, 0.2)	(0.9, 0.1)	(0.9, 0.2)	(0.4, 0.5)
A ₃	(0.6, 0.4)	(0.6, 0.2)	(0.4, 0.6)	(0.3, 0.2)
A ₄	(0.5, 0.3)	(0.7, 0.1)	(0.7, 0.1)	(0.4, 0.3)
A ₅	(0.4, 0.3)	(0.8, 0.1)	(0.8, 0.4)	(0.2, 0.3)

Table 6

The comparison research with some existing PF methods in Example 1

Methods	Ranking	The best AMS
Algorithm 1:proposed	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
O-PFWA [7]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
O-PFWG [7]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
TOPSIS [6]	N/A	∗
PFWA [45]	A₂ ≻ A₅ ≻ A₄ ≻ A₁ ≻ A₃	A ₂
PFWG [45]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
TODIM [11] (θ = 0.4)	A₂ ≻ A₅ ≻ A₁ ≻ A₄ ≻ A₃	A ₂
SPFWA [29]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
SPFWG [29]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
PFEWA [26]	A₂ ≻ A₅ ≻ A₄ ≻ A₁ ≻ A₃	A ₂
PFEWG [27]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
PFIWA [34]	A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃	A ₂
PFIWG [34]	A₂ ≻ A₅ ≻ A₄ ≻ A₁ ≻ A₃	A ₂

Note: N/A (division by zero problem), and ∗ (no result).

Remark 2. From the final ranking results, we can conclude that the ordering of the AMSs and the best AMS are same as references [7 , 35]. For reference [6], the eventual ordering and the best AMS can’t be obtained due to its defect (division by zero issue [5]).

Example 3. The education bureau selects four criteria to assess the five potential psycholinguistic schools (PPSs): C₁ (teaching environment), C₂ (teaching management), C₃ (curriculum design and target), C₄ (scientific study capacity). The five potential psycholinguistic schools A_i (i = 1, 2, 3, 4, 5) are evaluated by employing the PFNs under the above four criteria with weight vector w = (0.5, 0.2, 0.1, 0.2), and C₁, C₂ and C₃ are benefit criteria, and C₄ is cost criterion. Construct the matrix P = (p_ij) _5×4 = (μ_ij, ν_ij) _5×4 in Table 9.

Table 7

The PF decision matrix P = ( p_ij) _5×4

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.8, 0.2)	(0.5, 0.2)	(0.3, 0.1)	(0.1, 0.3)
A ₂	(0.6, 0.1)	(0.6, 0.3)	(0.8, 0.3)	(0.15, 0.45)
A ₃	(0.7, 0.1)	(0.7, 0.2)	(0.4, 0.4)	(0.2, 0.3)
A ₄	(0.6, 0.2)	(0.7, 0.1)	(0.7, 0.1)	(0.2, 0.4)
A ₅	(0.8, 0.2)	(0.3, 0.1)	(0.3, 0.1)	(0.2, 0.5)

Table 8

The comparison research with some existing PF methods in Example 3

Methods	Ranking	The best PPS
Algorithm 1 : proposed	A₅ ≻ A₁ ≻ A₃ ≻ A₄ ≻ A₁	A ₅
O - PFWA [7]	A₁ ∼ A₂ ∼ A₃ ∼ A₄ ∼ A₅	*
O - PFWG [7]	A₂ ≻ A₄ ≻ A₃ ≻ A₁ ∼ A₅	A ₂
TOPSIS [6]	A₁ ≻ A₅ ≻ A₃ ≻ A₄ ≻ A₂	A ₁
PFWA [45]	A₁ ∼ A₅ ≻ A₃ ≻ A₂ ≻ A₄	*
PFWG [45]	A₄ ≻ A₂ ∼ A₅ ≻ A₃ ≻ A₁	A ₄
TODIM [11] (θ = 0.4)	A₄ ≻ A₂ ≻ A₅ ≻ A₁ ≻ A₃	A ₄
TODIM [11] (θ = 100)	A₅ ≻ A₄ ≻ A₂ ≻ A₁ ≻ A₃	A ₅
SPFWA [29]	A₅ ∼ A₁ ≻ A₄ ≻ A₃ ≻ A₂	*
SPFWG [29]	A₂ ≻ A₃ ≻ A₄ ≻ A₁ ∼ A₅	A ₂
PFEWA [26]	A₅ ∼ A₁ ≻ A₃ ≻ A₂ ≻ A₄	*
PFEWG [27]	A₄ ≻ A₂ ≻ A₃ ≻ A₁ ∼ A₅	A ₄
PFIWA34	A₅ ∼ A₁ ≻ A₃ ≻ A₄ ≻ A₂	*
PFIWG [34]	A₅ ∼ A₁ ≻ A₃ ≻ A₄ ≻ A₂	*

Note: “*” denotes that there is no PPS to be chosen.

Table 9

The PF decision matrix P = ( p_ij) _5×4

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.9, 0.1)	(0.0, 1.0)	(0,2, 0.1)	(0,1, 0.9)
A ₂	(0.2, 0.1)	(0.2, 0.1)	(0.2, 0.1)	(0.05, 0.1)
A ₃	(0.1, 0.2)	(0.3, 0.4)	(0.2, 0.1)	(0.2, 0.3)
A ₄	(0.2, 0.15)	(0.2, 0.14)	(0.3, 0.2)	(0.1, 0.2)
A ₅	(0.16, 0.15)	(0.2, 0.1)	(0.3, 0.2)	(0.1, 0.3)

Next, the comparison with some existing approaches to choose the optimal psycholinguistic school is presented in Table 7.

Remark 3. According to Table 7, it can be easily found that the ordering results of five psycholinguistic schools achieved by the developed PF algorithm and some existing algorithms [6 , 45] are notably diverse, which the optimal PPS achieved by the developed algorithm is A₅ while others may be A₁ or A₂ or A₄. Actually, some existing algorithms are inconsistent in considering the experts’ psychological behaviours if the evaluations for psycholinguistic schools locate in a homologous level. Pythagorean fuzzy TOPSIS [6] can be employed in the condition that the experts are perfect rationality. Nevertheless, the experts general can’t provide the accurate preference because the incomplete preference information is filled suffused in reality. That is to say, to some extent, the experts are not rational enough. The developed PF algorithm can reasonably and effectively depict the experts’ behavior. Moreover, for Yager’s [7] O-PFWA, the ultimate decision values of proprietary psycholinguistic schools are equal, so all of them can be chose the optimal psycholinguistic school. In other words, it is unconscionable. As a matter of fact, we can select an optimal psycholinguistic school by the developed algorithm and other algorithms [6 , 45].

Example 4. The university press wish to publish one bestseller. The publishing editor provide five possible language books to choose: Java, C, PHP, Python, ASP. The university press chooses four criteria to assess the five possible language books. Let A = {A₁, A₂, A₃, A₄, A₅} be a series of books, C = {C₁, C₂, C₃, C₄} is a series of criteria (C₁ (popularity), C₂ (readability), C₃ (readability), C₄ (price)) with weight w = (0.6, 0.2, 0.1, 0.1). The criteria C₁, C₂ and C₃ are benefit criteria, and C₄ is cost criterion. Suppose that each language book A_i with respect to the criterion C_j is denoted as the PF matrix P = (p_ij) _5×4 = (μ_ij, ν_ij) _5×4. The evaluation information is displayed in Table 10.

Next, some existing algorithms [6 , 45] and proposed algorithm are employed in choosing a bestseller.

Remark 4. According to Table 10, we can see that the ranking results of five language books achieved by the developed PF algorithm and some existing algorithms (O-PFWA [7], TOPSIS [6], PFWA [45], PFEWA [26] and PFIWA [34]) are same. However, for PFWG [45], SPFWA [29], SPFWG [29], PFEWG [27] and PIFWG [34], the optimal language book is different from the proposed method and existing methods because its counterintuitive phenomena, which is discussed in [5]. Moreover, for TODIM method [11], it will produce diverse decision results when θ is diverse. In other words, the decision results are not reliable.

Table 10

The comparison research with some existing PF methods in Example 4

Methods	Ranking	The best bestseller
Algorithm 1 : proposed	A₁ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₃	A ₁
O - PFWA [7]	A₁ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₃	A ₁
O - PFWG [7]	A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁	A ₂
TOPSIS [6]	A₁ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₃	A ₁
PFWA [45]	A₁ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₃	A ₁
PFWG [45]	A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁	A ₂
TODIM [11] (θ = 0.4)	A₅ ≻ A₄ ≻ A₂ ≻ A₁ ≻ A₃	A ₅
TODIM [11] (θ = 100)	A₁ ≻ A₅ ≻ A₄ ≻ A₂ ≻ A₃	A ₁
SPFWA [29]	A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁	A ₂
SPFWG [29]	A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁	A ₂
PFEWA [26]	A₁ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₃	A ₁
PFEWG [27]	A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁	A ₂
PFIWA [34]	A₁ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₃	A ₁
PFIWG [34]	A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁	A ₂

Note: "Bold" denotes unreasonable results.

Example 5. A tourism company wants to invest a new tourism project in the best option. There are five possible tourism projects as follows: (1) A₁ is Rushan; (2) A₂ is Huangshan; (3) A₃ is Taishan; (4) A₄ is Sanqingshan; (5) A₅ is Danxiashan. The tourism company chooses four criteria to assess the five possible tourism projects: (1) C₁ is economic feasibility; (2) C₂ is technological feasibility; (3) C₃ is social feasibility; (4) C₄ is staff feasibility, and they are all benefit criteria. The DMs offers his prior weight as w = (0.6, 0.1, 0.2, 0.1). The DMs are demanded to assess the five tourism projects A_i (i = 1, 2, 3, 4, 5) under the above four criteria and the decision matrix P = (p_ij) _5×4 is shown in Table 11.

Table 11

The PF matrix given by expert team

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.1, 0.12)	(1.0, 0.0)	(0.13,0.1)	(0.2, 0.2)
A ₂	(0.6, 0.5)	(0.3, 0.1)	(0.7, 0.1)	(0.2, 0.1)
A ₃	(0.8, 0.1)	(0.9, 0.1)	(0.8, 0.2)	(0.3, 0.2)
A ₄	(0.6, 0.1)	(0.8, 0.1)	(0.4, 0.1)	(0.2, 0.15)
A ₅	(0.5, 0.1)	(0.7, 0.1)	(0.3, 0.1)	(0.3, 0.15)

Next, we employ the some existing algorithms [6 , 45] and the proposed algorithm above to choose a desirable tourism project using PF information.

Remark 5. According to Table 12, we can find that the ordering results and optimal result of the five possible tourism projects are same as the existing methods (O-PFWA [7], O-PFWG [7], TOPSIS [6], PFWG [45], SPFWA [29], PFEWG [27], TODIM [11] and PFIWA [34]). However, for PFWA [45], SPFWG [29], PFEWA [26], PIFWA [34] and PIFWG [34], the optimal tourism project cannot be obtained due to its counterintuitive phenomena discussed in [5].

Table 12

The comparison research with some existing PF methods in Example 5

Methods	The final ranking	The optimal alternative
Algorithm 1 : proposed	A₃ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₁	A ₃
O - PFWA [7]	A₃ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₁	A ₃
O - PFWG [7]	A₃ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₁	A ₃
TOPSIS [6]	A₃ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₁	A ₃
PFWA [45]	A₁ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₅	A ₁
PFWG [45]	A₃ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₁	A ₃
TODIM [11] (θ = 100)	A₃ ≻ A₂ ≻ A4_≻A₅ ≻ A₁	A ₃
SPFWA [29]	A₃ ≻ A₁ ≻ A₄ ≻ A₅ ≻ A₂	A ₃
SPFWG [29]	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂	A ₁
PFEWA [26]	A₁ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₅	A ₁
PFEWG [27]	A₃ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₁	A ₃
PFIWA [34]	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂	A ₁
PFIWG [34]	A₁ ≻ A₃ ≻ A₄ ≻ A₅ ≻ A₂	A ₁

Note: "Bold" denotes unreasonable results.

6 Conclusion

The CODAS is a quite resultful method for dealing the complicated MCDM issues, which a number of criteria are used to evaluate a lot of alternatives. In this article, we develop Pythagorean fuzzy CODAS method based novel score function for dealing decision making issues. Compared to the state-of-the-art CODAS approaches [53 –55], the prime merit of the presented approach is that it not only deal with the PF decision information but also obtain the best alternative out of counterintuitive phenomena and also has strong ability. Moreover, we develop an efficient score function, which takes the degrees of membership, non-membership and hesitation into considered simultaneously. Compared to the existing PF score functions [6 , 29], the proposed score function can differentiate the difference of PFNs when the existing score functions are out of work.

The main contributions of this article is two fold: (1) An effective ordering technology (score function) for PFNs is presented; (2) The score function based PF-CODAS method is proposed, which offers us with an effective way for dealing MCDM issues.

In the future, we will employ the excellent CODAS approach to deal with the MCDM issues in different fuzzy environment [58 –64] and gene selection [65].

Footnotes

Acknowledgments

This study was funded by National Natural Science Foundation of China (Nos. 61462019,61866011), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 18YJCZH054), Natural Science Foundation of Guangdong Province (No. 2018A030307033, 2018A0303130274), Social Science Foundation of Guangdong Province (No. GD18CFX06), Special Innovation Projects of Universities in Guangdong Province (No. KTSCX205).

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Pythagorean fuzzy multi-criteria decision making method based on CODAS with new score function

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 Some existing pythagorean fuzzy score functions

4.1 The description of pythagorean fuzzy MCDM issue

Table 2 The Pythagorean fuzzy decision matrix P = ( p ij ) m×n C 1 C 2 ⋯ C n A 1 p 11 p 12 ⋯ p 1n A 2 p 21 p 22 ⋯ p 2n ⋮ ⋮ ⋮ ⋱ ⋮ A m p m1 p m2 ⋯ p mn

5.1 Some illustrative examples

Table 5 The PF decision matrix in Example 1 C 1 C 2 C 3 C 4 A 1 (0.7, 0.3) (0.7, 0.1) (0.6, 0.4) (0.3, 0.1) A 2 (0.8, 0.2) (0.9, 0.1) (0.9, 0.2) (0.4, 0.5) A 3 (0.6, 0.4) (0.6, 0.2) (0.4, 0.6) (0.3, 0.2) A 4 (0.5, 0.3) (0.7, 0.1) (0.7, 0.1) (0.4, 0.3) A 5 (0.4, 0.3) (0.8, 0.1) (0.8, 0.4) (0.2, 0.3)

Footnotes

Acknowledgments

References

Table 2
The Pythagorean fuzzy decision matrix P = ( p_ij) _m×n

C ₁ C ₂ ⋯ C _n

A ₁ p ₁₁ p ₁₂ ⋯ p _1n

A ₂ p ₂₁ p ₂₂ ⋯ p _2n

⋮ ⋮ ⋮ ⋱ ⋮

A _m p _m1 p _m2 ⋯ p _mn