Some Similarity Measures of Neutrosophic Sets Based on the Euclidean Distance and Their Application in Medical Diagnosis

Similarity measure is an important tool in multiple criteria decision-making problems, which can be used to measure the difference between the alternatives. In this paper, some new similarity measures of single-valued neutrosophic sets (SVNSs) and interval-valued neutrosophic sets (IVNSs) are defined based on the Euclidean distance measure, respectively, and the proposed similarity measures satisfy the axiom of the similarity measure. Furthermore, we apply the proposed similarity measures to medical diagnosis decision problem; the numerical example is used to illustrate the feasibility and effectiveness of the proposed similarity measures of SVNSs and IVNSs, which are then compared to other existing similarity measures.


Introduction
e concept of fuzzy set (FS) A � 〈x i , u A (x i )〉|x i ∈ X in X � x 1 , x 2 , . . . , x n was proposed by Zadeh [1], where the membership degree u A (x i ) is a single value between zero and one. e FS has been widely applied in many fields, such as medical diagnosis, image processing, supply decision-making [2][3][4], and so on. In some uncertain decisionmaking problems, the degree of membership is assumed not exactly as a numerical value but as an interval. Hence, Zadeh [5] proposed the interval-valued fuzzy set (IVFS). However, the FS and IVFS only have the membership degree, and they cannot describe the nonmembership degree of the element belonging to the set. For example, in the national entrance examination for postgraduate, a panel of ten professors evaluated the admission of a student; five professors considered the student can be accepted, three professors disapproved of his or her admission, and two professors remained neutral. In this case, the FS and IVFS cannot represent such information. In order to solve this problem, Atanassov et al. [6] proposed the intuitionistic fuzzy set (IFS) E � 〈x i , u E (x i ), v E (x i )〉|x i ∈ X , where u E (x i )(0 ≤ u E (x i ) ≤ 1) and v E (x i )(0 ≤ v E (x i ) ≤ 1) represent the membership degree and nonmembership degree, respectively, and the indeterminacy-membership degree π E (x i ) � 1 − u E (x i ) − v E (x i ). e IFS is more effective to deal with the vague information than the FS and IVFS. en, the information about the admission of the student can be represented as an IFS E � 0.5, 0.3, 0.2, where 0.5, 0.3, and 0.2 stand for the membership degree, nonmembership degree, and indeterminacy-membership degree, respectively. However, the IFS also have limitation in expressing the decision information. For example, three groups of experts evaluate the benefits of the stock, a group of experts thinks the possibility of the stock that will be profitable is 0.6, the second group of experts thinks the possibility of loss is 0.3, the third group of experts is not sure whether the stock that will be profitable is 0.4. In this case, the IFS cannot express such information because 0.6 + 0.3 + 0.4 > 1.
erefore, Wang et al. [7] proposed a single-valued neutrosophic set (SVNS) N � 〈x i , T N (x i ), I N (x i ), F N (x i )〉|x i ∈ X , where T N (x i ), I N (x i ) and F N (x i ) represent the degree of the truth-membership, indeterminacy-membership, and falsitymembership, respectively, and they belong to [0,1]. So, the information about the benefits of the stock can be represented as N � 0.6, 0.4, 0.3. However, due to the uncertainty of the decision-making environment in multiple criteria decision-making problems, the single numerical value cannot meet the needs of evaluating information. en, Wang [8] defined the interval-valued neutrosophic set (IVNS) based on the SVNS, which used the interval to describe truth membership degree, indeterminacy membership degree, and falsity membership degree, respectively. Since the neutrosophic set was proposed, there have been some researchers focusing on this subject [9][10][11][12].
On the other hand, similarity measure is an important tool in multiple criteria decision-making problems, which can be used to measure the difference between the alternatives. Many studies about the similarity measure are obtained. For example, Beg et al. [13] proposed a similarity measure of FSs based on the concept of ϵ − fuzzy transitivity and discussed the degree of transitivity of different similarity measures. Song et al. [14] considered the similarity measure of IFSs and proposed corresponding distance measure between intuitionistic fuzzy belief functions. Majumdar and Samanta [15] proposed a similarity measure between SVNSs based on the membership degree.
In addition, cosine similarity measure is also an important similarity measure, and it can be defined as the inner product of two vectors divided by the product of their lengths.
ere are some scholars who study the cosine similarity measures [16][17][18][19][20][21]. For example, Ye [16] proposed the cosine similarity measure and weighted cosine similarity measure of IVFSs with risk preference, and they were applied to the supplier selection problem. en, Ye [17] proposed the cosine similarity measure of IFSs and applied it to medical diagnosis and pattern recognition. Furthermore, Ye [18] defined the cosine similarity measure of SVNSs and IVNSs, but when the SVNSs N 1 ≠ N 2 , cos(N 1 , N 2 ) � 1 (the example can be seen in Section 3). Furthermore, Ye [19] proposed the improved cosine similarity measures of SVNSs and IVNSs based on cosine function.
In this paper, we propose a new method to construct the similarity measures of SVNSs, which is based on the existing similarity measure proposed by Majumdar and Samanta [15] and Ye [18], respectively. ey play an important role in practical application, especially in pattern recognition, medical diagnosis, and so on. Furthermore, we will propose the corresponding similarity measures of IVNSs. e rest of the paper is organized as follows. In Section 2, the basic definition and some properties about SVNS and IVNS are given. In Section 3, we proposed a method to construct the new similarity measures of SVNSs and IVNSs, respectively. In Section 4, we apply the proposed new similarity measures to medical diagnosis problems, the numerical examples are used to illustrate the feasibility and effectiveness of the proposed similarity measures, which are then compared to other existing similarity measures. Finally, the conclusions and future studies are discussed in Section 5.

Preliminaries
In this section, we give some basic knowledge about the SVNS and the IVNS. Some existing distance measures are also introduced, which will be used in the next section.

SVNS
Definition 1. Given a fixed set X � x 1 , x 2 , . . . , x n [7], the SVNS N in X is defined as follows: where the function T N (x i ) : X ⟶ [0, 1] defines the truthmembership degree, the function I N (x i ) : X ⟶ [0, 1] defines indeterminacy-membership degree, and the function F N (x i ) : X ⟶ [0, 1] defines the falsity-membership degree, respectively. For any SVNS N, it holds that the following properties are satisfied:

IVNS
Definition 2. Given a fixed set X � x 1 , x 2 , . . . , x n [8], the IVNS N ′ on X is defined as follows: where represent the truth-membership function, the indeterminacy-membership function, and the falsity-membership function, respectively. For any x i ∈ X, it holds that T N′ (x i ), the following properties are satisfied:

Existing Distance Measures between SVNSs and IVNSs
. , x n } [15]; then, the Euclidean distance between SVNSs N 1 and N 2 is defined as follows: [22]; the Euclidean distance between IVNSs N 1 ′ and N 2 ′ is defined as follows: Next, we propose a new method to construct the similarity measures of SVNSs and IVNSs based on the Euclidean distance measure.

Several New Similarity Measures
e similarity measure is a most widely used tool to evaluate the relationship between two sets. e following axiom about the similarity measure of SVNSs (or IVNSs) should be satisfied: . , x n be the universal set [18] if the similarity measure S(N 1 , N 2 ) between SVNSs (or IVNSs) N 1 and N 2 satisfies the following properties: en, the similarity measure S(N 1 , N 2 ) is a genuine similarity measure.

e New Similarity Measures between SVNSs.
To introduce the new similarity measure between SVNSs, we first review the similarity measure S 1SVNS between N 1 and N 2 defined by Majumdar et al. [15], which is given as follows: . , x n be a universal set [15], for any two SVNSs the similarity measure of SVNSs between N 1 and N 2 is defined as follows: It is already known that the similarity measure S 1SVNS defined by Majumdar et al. [15] satisfies the properties in Lemma 1. It is proposed based on the membership degree; in this section, we adopt the various methods for calculating the similarity measure between neutrosophic sets.
Firstly, we propose a new method to construct a new similarity measure of SVNSs, which is based on the similarity measure proposed by Majumdar et al. [15] and the Euclidean distance; it can be defined as follows: . , x n be a universal set, for any two SVNSs is defined as follows: e proposed similarity measure of SVNSs satisfies the following eorem 1.

Computational and Mathematical Methods in Medicine
Proof.
(1) Because D SVNS (N 1 , N 2 ) is an Euclidean distance measure, obviously, 0 ≤ D SVNS (N 1 , N 2 ) ≤ 1. Furthermore, according to Proposition 4.2.2 by Majumdar et al. [15], we know 0 ≤ S 1SVNS (N 1 , On the other hand, when N 1 � N 2 , according to formulae (3) and (5) On the other hand, cosine similarity measure is also an important similarity measure. In 2014, Ye [18] proposed a cosine similarity measure between SVNSs as follows: . , x n be a universal set [18], for any two SVNSs the cosine similarity measure between N 1 and N 2 is defined as follows: From Example 1, we know the cosine similarity measure defined by Ye [18] does not satisfy Lemma 1.

Example 1.
For two SVNSs N 1 � x, 0.4, 0.2, 0.6 and N 2 � x, 0.2, 0.1, 0.3, we can easily know N 1 ≠ N 2 . But using formula (7) to calculate the cosine similarity measure S 2SVNS (N 1 , N 2 ), we have S 2SVNS (N 1 , N 2 ) � 1. at is to say, when N 1 ≠ N 2 , S 2SVNS (N 1 , N 2 ) � 1, which means the cosine similarity measure S 2SVNS (N 1 , N 2 ) defined by Ye [18] does not satisfy the necessary condition of property 2 in Lemma 1; thus, it is not a genuine similarity measure. Furthermore, Ye [19] proposed the improved cosine similarity measures of SVNS based on the cosine similarity measure proposed by Ye [18], which overcomes its shortcoming.
In this paper, we go on proposing another new similarity measure of SVNSs based on the cosine similarity measure proposed by Ye [18] and the Euclidean distance D SVNS . It considers the similarity measure not only from the point of view of algebra but also from the point of view of geometry, which can be defined as: . , x n be a universal set, for any two SVNSs N 2 ) is defined as follows:  (N 1 , N 2 ) � 0.8920. We can see that the proposed new similarity measure S * 2SVNS (N 1 , N 2 ) overcomes the shortcoming of cosine similarity measure S 2SVNS (N 1 , N 2 ) defined by Ye [18].

Some New Similarity Measures between IVNSs.
In some situations, it is difficult to provide the truth-membership degree, false-membership degree, and indeterminate-membership degree with a precise numerical value; Wang [8] used the interval numbers to express the related membership degrees. Furthermore, Broumi et al. [22] proposed the corresponding similarity measure of IVNSs based on the similarity measure S 1SVNS proposed by Majumdar et al. [15]. Definition 9. Let X � x 1 , x 2 , . . . , x n be a universal set, for any two IVNSs [22]; the similarity measure between IVNSs N 1 ′ and N 2 ′ is defined as follows: Similarly to Section 3.1, we propose a corresponding similarity measure between IVNSs, which is based on the similarity measure S 1IVNS (N 1 ′ , N 2 ′ ) and the Euclidean distance D IVNS (N 1 ′ , N 2 ′ ) defined in Definition 4.
Definition 10. Let X � x 1 , x 2 , . . . , x n be a universal set, for any two IVNSs a new similarity measure S * 1IVNS (N 1 ′ , N 2 ′ ) is defined as follows: e proposed similarity measure also satisfies eorem 3.

Theorem 3.
e similarity measure S * 1IVNS (N 1 ′ , N 2 ′ ) satisfies the following properties: Proof. e proof is similar to eorem 1; hence, we omit it here.

Theorem 4.
e similarity measure S * 2IVNS (N 1 ′ , N 2 ′ ) satisfies the following properties: Proof. e proof is similar to eorem 2, we also omit it here.
In the next section, we will apply the proposed new similarity measures to medical diagnosis decision problem; numerical examples are also given to illustrate the application and effectiveness of the proposed new similarity measures.

e Proposed Similarity Measures between SVNSs for
Medical Diagnosis. We first give a numerical example about a medical diagnosis (adapted from Ye [19]) to illustrate the feasibility of the proposed new similarity measures S *
By applying formulae (6) and (8), we can obtain the similarity measure values S * 1SVNS (P 1 , Q i ) and S * 2SVNS (P 1 , Q i ); the results are shown in Table 2.
From the above two similarity measures S * 1SVNS and S * 2SVNS , we can conclude that the diagnoses of the patient P 1 are all malaria (Q 2 ). e proposed two similarity measures S * 1SVNS and S * 2SVNS produce the same results as Ye [19], which means the proposed similarity measures are feasible and effective.

e Proposed Similarity Measures between IVNSs for
Medical Diagnosis. We know if the doctor examines the patient two or three times a day, then the interval values of multiple inspections for the patient are obtained. In this section, we will apply the proposed similarity measures S * 1IVNS and S * 2IVNS to medical diagnosis, the example is also adapted from Ye [19].
Example 4. Let us reconsider Example 3, assume a patient P 2 has all the symptoms, which can be expressed by the following IVNS information.
e same way as Example 3 in Ye [19], the diagnosis information of SVNSs Q i with respect to symptoms S i (i � 1, 2, · · · , 5) are transformed into IVNSs, which are shown in Table 3.
By applying formulae (10) and (12), we obtain the similarity measure values S * 1IVNS (P 2 , Q i ) and S * 2IVNS (P 2 , Q i ), the results are shown in Table 4.
From the two similarity measure values in Table 4, we can see that the patient P 2 suffers from typhoid (Q 3 ); the diagnosis results are the same as shown by Ye [19].

Comparative Analyses of Existing Similarity Measures.
To illustrate the effectiveness of the proposed similarity measures for medical diagnosis, we will apply the existing similarity measures of SVNSs and IVNSs for comparative analyses.
As we can see from Table 5, the patient P 1 is still assigned to malaria (Q 2 ), and the results are same as the proposed similarity measures in this paper, which means the proposed similarity measures are feasible and effective.
Next, we introduce the existing similarity measures between IVNSs as follows: . , x n , the existing similarity measures between N 1 ′ and N 2 ′ are defined as follows: (1) Broumi et al. [23] proposed the similarity measure SM IVNS : (2) Şahin and Ahmet [24] proposed the similarity measure SD IVNS : (4) Yang et al. [25] proposed the similarity measure SY SVNS (N 1 ′ , N 2 ′ ): Example 4′. Applying formulae (9), (11), and (20)-(24) to calculate Example 6 again, the similarity measure values between P 2 and Q i (i � 1, 2, . . . , 5) are shown in Table 6. e results of Table 6 show that the patient P 2 should be assigned to typhoid (Q 3 ), they are same as the proposed similarity measures S * 1IVNS and S * 2IVNS in the paper, which means the proposed methods are feasible and effective. e proposed similarity measures in the paper have some advantages in solving multiple criteria decisionmaking problems. ey are constructed based on the existing similarity measures and Euclidean distance, which not only satisfy the axiom of the similarity measure but also consider the similarity measure from the points of view of algebra and geometry. Furthermore, they can be applied more widely in the field of decision-making problems.

Conclusions
e similarity measure is widely used in multiple criteria decision-making problems.
is paper proposed a new method to construct the similarity measures combining the existing cosine similarity measure and the Euclidean distance measure of SVNSs and IVNSs, respectively, which are based on the above existing similarity measures and the Euclidean distance measure. And, the similarity measures are proposed not only from the points of view of algebra and geometry but also satisfy the axiom of the similarity measure. Furthermore, we apply the proposed similarity measures to medical diagnosis decision problems, and the numerical example is used to illustrate the feasibility and effectiveness of the proposed similarity measure, which are then compared to other existing similarity measures. In future research, we will focus on studying the similarity measure between linguistic neutrosophic set and the application of the proposed similarity measures of neutrosophic sets, such as pattern recognition, supplier selection, and so on.

Data Availability
e data used to support the findings of this study are included within the article.