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
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.
Preliminaries
Intuitionistic fuzzy set
The intuitionistic fuzzy number (IFN) is represented by i = (μ i , ν i ).
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
The Pythagorean fuzzy number (PFN) is represented by p = (μ p , ν p ) [14].
Next, two ranking methods of the PFNs are defined as follow
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
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.
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
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.
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.
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.
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
The above weighted similarity measures of the PFS are represented by
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
Next, the specific calculation process of the MCDM method based on the similarity measures of the PFS is as follows
If there is only one decision maker, this step can be ignored.
According to the size of
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.
The Pythagorean fuzzy decision matrix
The Pythagorean fuzzy decision matrix
The ranking function value is calculated and the Pythagorean fuzzy ranking matrix R (P) is obtained, as shown in Table 2.
The Pythagorean fuzzy ranking matrix
After determining the optimal value
Next, the weighted similarity measures between the criterion value of each scheme and
The calculated results of
The calculated results of
Next, the relative closeness of each scheme is calculated, as shown in Table 5.
The calculated results of the relative closeness
From Table 5, the ranking results of each scheme are shown in Table 6.
The ranking results of each scheme
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
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).
Then, the ranking function value of the comprehensive decision value of each scheme is calculated by Equation (6).
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.
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
The Pythagorean fuzzy ranking matrix constructed by expert e1
The Pythagorean fuzzy ranking matrix constructed by expert e1
The Pythagorean fuzzy ranking matrix constructed by expert e2
The Pythagorean fuzzy ranking matrix constructed by expert e3
The Pythagorean fuzzy ranking matrix constructed by expert e4
The Equation (4) is used to aggregate the Pythagorean fuzzy decision matrix
The comprehensive Pythagorean fuzzy decision matrix
The ranking function value is calculated and the Pythagorean fuzzy ranking matrix R (P). is obtained, as shown in Table 12.
The Pythagorean fuzzy ranking matrix
After determining the optimal value
Next, the weighted similarity measures between the criterion value of each scheme and
The calculated results of
The calculated results of the weighted similarity measures between the criterion value of each scheme and
The calculated results of
Next, the relative closeness of each scheme is calculated, as shown in Table 15.
The calculated results of the relative closeness
From Table 15, the ranking results of each scheme are as shown in Table 16.
The ranking results of each scheme
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
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).
Then, the ranking function value of each scheme is calculated by Equation (6).
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
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
The expert’s Pythagorean fuzzy decision matrix
The expert’s Pythagorean fuzzy decision matrix
First, the comprehensive decision value of each scheme is calculated by Equation (4).
Then, the ranking function value of each scheme is calculated by Equation (6).
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.
The Pythagorean fuzzy ranking matrix
The positive ideal criterion value and the negative ideal criterion value are obtained, respectively (three criteria are all benefit index)
Next, the weighted similarity measures between the criterion value of each scheme and
The calculated results of
The calculated results of
Next, the relative closeness of each scheme is calculated, as shown in Table 21.
The calculated results of the relative closeness
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
The ranking results of each scheme
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
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.
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.
