Apply new entropy based similarity measures of single valued neutrosophic sets to select supplier material

Abstract

The single-valued neutrosophic set (SVNS) is an extension of the fuzzy set and intuitionistic fuzzy set. This is a useful tool to deal with uncertain and inconsistent information. In the information theory, the distance measure, entropy measure and similarity measures have an important role. Several entropy measures of SVNSs have been proposed and applied in many real problems. But they have some restriction in practice and in the academic study. The similarity measures induced from entropy were studied and gave interesting results. In this paper, we introduce a new entropy measure concept based on the SVNS, which overcomes the restriction of existing entropy measures. At the same time, we also investigate some similarity measures which are induced from new entropy measures and apply them to propose the multi-criteria decision making (MCDM) model in selecting the supplier.

Keywords

Entropy of SVNS similarity measure of SVNS MCDM

1 Introduction

Neutrosophic set [21], proposed by Smarandache in 1998, is an extension of fuzzy set [47], intuitionistic fuzzy sets [2] and interval valued intuitionistic fuzzy set [3]. A SVNS A in a universal set X has three membership functions: a truth-membership function T_A, an indeterminacy-membership function I_A and a falsity-membership function F_A. For each element x of X its neutrosophic component functions T_A (x) , I_A (x) , F_A (x) are real standard or nonstandard subsets of ] ^-0, 1⁺ [. This makes them difficult to use in real problems. To overcome this difficult, the single valued SVNS set (SVNS) is introduced by Smarandache and Wang et al. [32]. In the context of SVNS, three memberships T_A (x) , I_A (x) , F_A (x) are real standard subsets of [0, 1] for each element x of X. The SVSN is a generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set, q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical Fuzzy Set [22]. SVNS is designed to deal with uncertainty, incomplete and inconsistent information. Today, SVNS is studied in theoretical approach and applied in many real problems [1 , 18]. Broumi and Smarandache [4] proposed some similarity measures based on the Hausdorff distance measure and were applied in MCDM. Huang [11] gave new distance measures of SVNS and applied them in clustering analysis and MCDM. Kharal [13] constructed the score functions for each SVNS to use in MCDM problems. Wang et al. [33] investigated a wide range of generalized Maclaurin symmetric mean (MSM) aggregation operators to solve MCDM problem. Liu et al. [15] combined Hamacher operations and extended the aggregation operators to NS, and proposed several new aggregation operators and applied them to MCDM. Ye investigated some similarity measures of SVNSs that he applied in MCDM, clustering analysis and engineering fields [37 –46]. Thao and Smarandache [26] study the divergence measure of SVSN and apply them to the medical diagnosis and the classification problems. Most of the above similarity measures are either directly determined by explicit formulas or generated from distance measures. Three vector similarity measures proposed by Ye [37, 38] have some restrictions when evaluating the similarity of two single valued SVNS sets (see Example 4 in Section 4).

In terms of information evaluation, beside the important measures of distance measure and similarity measure, an entropy measure is also a useful tool for the processing of uncertainty, inconsistency. There are many fuzzy entropies [10 , 36], intuitionistic fuzzy entropies [5,23 , 31], Pythagorean fuzzy entropies [19, 28], and Picture fuzzy entropies [30], that are studied and applied in real problems. The relationship between distance, similarity, and fuzzy entropy is also investigated by many researchers [7 , 36]. So far, only a few studies have been done on the entropy measure of the SVNS set. In 2014, Ye [40] proposed a cross-entropy of SVNS as an extension of a cross-entropy of a fuzzy set in each membership function of the SVNSs and applied it in MCDM. Ye [41, 42] improved cross entropy measure of SVNSs and investigated its properties, and then extended it to a cross-entropy measure between interval SVNS and applied them in MCDM. Majumdar and Samanta [16] proposed the similarity measure based on the distance measure of SVNS. They also investigated the entropy measure of SVNS and give a formula of entropy measure based on their entropy of SVNS. But this has certain limitations, and may not even be feasible. This is shown in Example 1 in Section 2. In 2018, Wu et al. [34] introduced the entropy that overcomes the limitations of the entropy concept by Majumdar and Samanta in 2014 [16]. In this entropy of Wu et al. [34] it is used the condition $E (\bar{A}) = E (A)$ where $\bar{A} = {(1 - T_{A} (x), 1 - I_{A} (x), 1 - F_{A} (x)) | x \in X}$ and called $\bar{A}$ is the complement of SVNS A ={ (x, T_A (x) , I_A (x) , F_A (x)) |x ∈ X }. A better concept of complement of SVNS is A^C ={ (F_A (x) , 1 - I_A (x) , T_A (x)) |x ∈ X }. It also gives rise to the new similarity measures that the concept of similarity measures need to redefine. It causes difficulties in the practical applications of inherent similar measures. This also is not a natural extension of the knowledge concept of fuzzy entropy and intuitionistic fuzzy entropy, picture fuzzy entropy. Moreover, some extensions of the entropy measures on fuzzy sets or intuitionistic sets do not satisfy this axiom (see Example 2, in Section 3). These are the reasons that motivate us to study a more appropriate concept of entropy measures for neutrosophic sets and for finding their applications.

In this paper, we introduce a new concept of entropy measure of SVNS. This entropy will overcome the limitation of the entropy of Majumdar and Samanta [16] (see Example 1 in Section 2). This concept is also the natural extension of the concept of entropy measure of fuzzy sets and intuitionistic fuzzy sets. Afterwards, we investigated the similarity measure of SVNS which were generated by our proposed entropies. These similarity measures overcome some drawbacks of three vector similarity measures by Ye [37–39 , 46]. In this paper we also use the new similarity measures in the MCDM in order to choose the supplier problems. To do so, the rest of this paper is organized as follows: Several needed concepts and definitions are called in Section 2. In Section 3, we propose a new concept of entropy of SVNS and give its formula. The similarity measures derived from our proposed entropies will be discussed in Section 4. Finally, the paper will give some applications of the newly introduced similarity measures in MCDM in Section 5.

2 Preliminaries

In this section, we recall the concept of SVNS. Also we remind the concept of similarity measure and entropy of SVNS in the contexts.

Definition 1 [32]. Let X be a universal set. A SVNS A in X is characterized by a truth-membership function T_A, an indeterminacy-membership function I_A and a falsity-membership function F_A. For each element x of X its neutrosophic components T_A (x) , I_A (x) , F_A (x) are real standard subsets of [0, 1] that satisfy the inequality T_A (x) + I_A (x) + F_A (x) ⩽3 for all x ∈ X.

A SVNS A can be denoted by the following symbol: $A = {(x, T_{A} (x), I_{A} (x), F_{A} (x)) | x \in X}$

The collection of SVNS of X is denoted by SVNS (X).

For two sets A, B ∈ SVNS (X) we have several operators as follows:

Complement of a SVNS A: $A^{C} = {(x, F_{A} (x), 1 - I_{A} (x), T_{A} (x)) | x \in X}$

Inclusion

A ⊆ B if only if T_A (x) ⩽ T_B (x), I_A (x) ⩾ I_B (x) and F_A (x) ⩾ F_B (x) for all x ∈ X.

Equality A = B if only if A ⊆ B and B ⊆ A.

Union A∪ B = { (x, T_A (x) ∨ T_B (x) , I_A (x) ∧ I_B (x) , F_A (x) ∧ F_B (x)) |x ∈ X }.

Intersection A∩ B = { (x, T_A (x) ∧ T_B (x) , I_A (x) ∨ I_B (x) , F_A (x) ∨ F_B (x)) |x ∈ X }.

Where ∧ is the fuzzy t-norm and ∨ is the fuzzy t-conorm.

For a set A ∈ SVNS (X), the triple (T_A (x) , I_A (x) , F_A (x)) is called a single valued neutrosophic number (SVNN) and denoted by a = (T_A, I_A, F_A).

- For λ positive real number we define the operational laws $\begin{matrix} λ a = (1 - (1 - T_{A})^{λ}, I_{A}^{λ}, F_{A}^{λ}) \\ a^{λ} = (T_{A}^{λ}, 1 - (1 - I_{A})^{λ}, 1 - (1 - F_{A})^{λ}) . \end{matrix}$ (1)

In the rest of this paper, we always consider X be the discrete set X ={ x₁, x₂, . . . , x_n }.

For two sets A, B ∈ SVNS (X) we define the similarity measure as follows:

Definition 2 [4]. A real–valued function S : SVNS (X) × SVNS (X) → [0, 1] is called a similarity measure on SVNS (X), if it satisfies the following axiomatic:

(S1) 0 ⩽ S (A, B) ⩽1,

(S2) S (A, B) = S (B, A),

(S3) S (A, B) =1 if only if A = B,

(S4) If A ⊂ B ⊂ C then S (A, C) ⩽ S (A, B) and S (A, C) ⩽ S (B, C).

The entropy measure determines the uncertainty and ambiguity of an object. It is very important to determine the uncertainty and ambiguity for a neutrosophic set. Now, we recall the entropy concept of SVNS by Majumdar and Samanta [16].

Definition 3 [16]. An entropy on SVNS (X) is a function EN : SVNS (X) → [0, 1], satisfying all following conditions

(E1) E (A) =0 if A is a crisp set,

(E2) E (A) =1 if A ={ (x_i, 0.5, 0.5, 0.5) |x_i ∈ X },

(E3) E (A) ⩾ E (B) for all A, B ∈ SVNS (X) satisfy T_A (x_i) + F_A (x_i) ⩽ T_B (x_i) + F_B (x_i) , and |I_A (x_i) - I_{A
^C} (x_i) | ⩽ |I_B (x_i) - I_{B
^C} (x_i) | for all x_i ∈ X,

(E4) E (A) = E (A^C), for all A ∈ SVNS (X).

This entropy is questionable in condition (E3). See the below example.

Example 1. For two SVNS A ={ (x, 0, 0, 1) } and B ={ (x, 0.8, 0, 0.7) } on X ={ x }, we have T_A (x) + F_A (x) =1 < T_B (x) + F_B (x) =1.5 and |I_A (x) - I_{A
^C} (x) | = 1 = |I_B (x) - I_{B
^C} (x) |. It means that E (B) ⩽ E (A) =0. It limits the use of this entropy concept.

In 2018, Wu et al. [34] investigated the following concept of SVNS entropy.

Definition 4 [34]. (Wu et al. 2018) An entropy on SVNS (X) is a function EN : SVNS (X) → [0, 1], satisfying all following conditions

(E1) E (A) =0 if only if T_A (x_i) , I_A (x_i) , F_A (x_i)∈ { 0, 1 } for all x_i ∈ X.

(E2) E (A) =1 if A ={ (x_i, 0.5, 0.5, 0.5) |x_i ∈ X }.

(E3) $E (A) = E (\bar{A})$ , where $\bar{A} = {(x_{i}, 1 - T_{A} (x_{i}), 1 - I_{A} (x_{i}), 1 - F_{A} (x_{i})) | x_{i} \in X}$ for all A ∈ SVNS (X).

(E4) E (A) ⩽ E (B) if A is more uncertain than B i.e. T_A (x_i) ⩽ T_B (x_i) , I_A (x_i) ⩽ I_B (x_i), F_A (x_i) ⩽ F_B (x_i) when $T_{B} (x_{i}) - T_{\bar{B}} (x_{i}) ⩽ 0$ , $I_{B} (x_{i}) - I_{\bar{B}} (x_{i}) ⩽ 0$ , $F_{B} (x_{i}) - F_{\bar{B}} (x_{i}) ⩽ 0$ , or T_A (x_i) ⩾ T_B (x_i) , I_A (x_i) ⩾ I_B (x_i), F_A (x_i) ⩾ F_B (x_i) when $T_{B} (x_{i}) - T_{\bar{B}} (x_{i}) ⩾ 0$ , $I_{B} (x_{i}) - I_{\bar{B}} (x_{i}) ⩾ 0$ , $F_{B} (x_{i}) - F_{\bar{B}} (x_{i}) ⩾ 0$ , where $T_{\bar{B}} (x_{i}) = 1 - T_{B} (x_{i})$ , $I_{\bar{B}} (x_{i}) = 1 - I_{B} (x_{i})$ , $F_{\bar{B}} (x_{i}) = 1 - F_{B} (x_{i})$ for all x_i ∈ X.

This entropy of Wu et al. [34] has the disadvantages mentioned in the introduction of this article. That is in third condition (E3) $E (A) = E (\bar{A})$ , where $\bar{A} = {(x_{i}, 1 - T_{A} (x_{i}), 1 - I_{A} (x_{i}), 1 - F_{A} (x_{i})) | x_{i} \in X}$ for all A ∈ SVNS (X). It is not an accurate extension of the concept of entropy of fuzzy sets and intuitionistic fuzzy sets. Moreover, some extensions of the entropy measures on fuzzy sets or intuitionistic sets do not satisfy this axiom (see Example 2, in Section 3). These are the motivations that made us to study a more appropriate concept of entropy measures for neutrosophic sets and find their applications. Therefore, the definition of entropy for SVNSs needs to be improved to it as a natural extension of fuzzy sets and intuitionistic fuzzy sets. We do this work in Section 3.

3 The new entropy of single valued SVNS sets

In this section we will introduce the new concept of a SVNS entropy. This concept is an extension of the concept of an intuitionistic fuzzy set entropy which was known in the literature. The entropy of a SVNS is a useful tool to measure the uncertain of information. Unlike the entropy of fuzzy sets, the entropy of the SVNS must show the effect of the indeterminacy in the SVNS.

Definition 5. An entropy on SVNS (X) is a function E : SVNS (X) → [0, 1] satisfying all following conditions

(E1) E (A) =0 if A is a crisp set, i.e. A_i = (T_A (x_i) , I_A (x_i) , F_A (x_i)) = (1, 0, 0) or A_i = (T_A (x_i) , I_A (x_i) , F_A (x_i)) = (0, 0, 1) for all x_i ∈ X,

(E2) E (A) =1 if A ={ (x_i, 0.5, 0.5, 0.5) |x_i ∈ X },

(E3) E (A) = E (A^C), for all A ∈ SVNS (X),

(E4) E (A) ⩽ E (B) if either T_A (x_i) ⩽ T_B (x_i) , I_A (x_i) ⩽ I_B (x_i) , F_A (x_i) ⩽ F_B (x_i) when max { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩽ 0.5 or T_A (x_i) ⩾ T_B (x_i) , I_A (x_i) ⩾ I_B (x_i) , F_A (x_i) ⩾ F_B (x_i) when min { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩾ 0.5.

Lemma 1. If we have either T_A (x_i) ⩽ T_B (x_i) , I_A (x_i) ⩽ I_B (x_i) , F_A (x_i) ⩽ F_B (x_i) when max { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩽ 0.5 or T_A (x_i) ⩾ T_B (x_i) , I_A (x_i) ⩾ I_B (x_i) , F_A (x_i) ⩾ F_B (x_i) when min { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩾ 0.5 then we have $\begin{matrix} | T_{B} (x_{i}) - 0.5 | & ⩽ | T_{A} (x_{i}) - 0.5 |, | I_{B} (x_{i}) - 0.5 | \\ ⩽ | I_{A} (x_{i}) - 0.5 |, | T_{B} (x_{i}) - 0.5 | \\ ⩽ | T_{A} (x_{i}) - 0.5 |, \end{matrix}$ and $| I_{B^{C}} (x_{i}) - 0.5 | ⩽ | I_{A^{C}} (x_{i}) - 0.5 |$ for all x_i ∈ X.

Now, we give several formulas to determine entropy measure of a single valued neutrosophic set on X.

Definition 6. Let A ={ (x_i, T_A (x_i) , I_A (x_i) , F_A (x_i)) |x_i ∈ X } be a SVNS set on X. We define $E_{T} (A) = 1 - \frac{1}{n} \sum_{i = 1}^{n} \frac{| T_{A} (x_{i}) - 0.5 | + | F_{A} (x_{i}) - 0.5 | + | I_{A} (x_{i}) - 0.5 | + | I_{A^{C}} (x_{i}) - 0.5 |}{2}$ (2)

Theorem 2. The function defined by Equation (2) is an entropy measure of a SVNS set.

Proof.

(E1) If A is a crisp set then for all x_i ∈ X, we have $E_{T}^{i} (A) = \frac{| T_{A} (x_{i}) - 0.5 | + | F_{A} (x_{i}) - 0.5 | + | I_{A} (x_{i}) - 0.5 | + | I_{A^{C}} (x_{i}) - 0.5 |}{2} = 1 .$

It imply that E_T (A) =0.

(E2) If A ={ (x_i, 0.5, 0.5, 0.5) |x_i ∈ X } then $E_{T}^{i} (A) = 0$ . It implies that E_T (A) =1.

(E3) It is easy to verify that E (A) = E (A^C) for all A ∈ SVNS (X).

(E4) From Lemma 1, we have E_T (A) ⩽ E_T (B) if either T_A (x_i) ⩽ T_B (x_i) , I_A (x_i) ⩽ I_B (x_i) , F_A (x_i) ⩽ F_B (x_i) when max { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩽ 0.5 or T_A (x_i) ⩾ T_B (x_i) , I_A (x_i) ⩾ I_B (x_i) , F_A (x_i) ⩾ F_B (x_i) when min { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩾ 0.5.□

By proving similar to theorem 1 and theorem 2, we also have the following entropy measures.

Definition 7. Let A ={ (x_i, T_A (x_i) , I_A (x_i) , F_A (x_i)) |x_i ∈ X } be a SVNS set on X. We define

$\begin{matrix} E_{MT} (A) \\ = \frac{1}{n} \sum_{i = 1}^{n} \frac{min {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}}{\max {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}} \end{matrix}$ (3)

Theorem 3. The function defined by Equation (3) is an entropy of a SVNS set.

Proof. $\begin{matrix} We put E_{MT}^{i} (A) \\ = \frac{min {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}}{\max {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}} . \end{matrix}$

(E1). If T_A (x_i) , I_A (x_i) , F_A (x_i)∈ { 0, 1 } for all i = 1, 2, . . . , n then min { T_A (x_i) , F_A (x_i) , I_A (x_i) , I_{A
^C} (x_i) } = 0,

max{ T_A (x_i) , F_A (x_i) , I_A (x_i) , I_{A
^C} (x_i) } = max { T_A (x_i) , F_A (x_i) , I_A (x_i) , 1 - I_A (x_i) } = 1.

It implies that $E_{MT}^{i} (A) = 0$ for all i = 1, 2, . . . , n. So that $E_{MT} (A) = \frac{1}{n} \sum_{i = 1}^{n} E_{MT}^{i} (A) = 0 .$

(E2). If A ={ (x_i, 0.5, 0.5, 0.5) |x_i ∈ X } then $E_{MT}^{i} (A) = 1$ . It implies that $E_{MT} (A) = \frac{1}{n} \sum_{i = 1}^{n} E_{MT}^{i} (A) = 1 .$

(E3) It is easy to verify that E_MT (A^C) = E_MT (A).

(E4) For all i = 1, 2, . . . , n, we have:

+If T_A (x_i) ⩽ T_B (x_i) , I_A (x_i) ⩽ I_B (x_i) , F_A (x_i) ⩽ F_B (x_i) when max { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩽ 0.5, then I_{A
^C} (x_i) ⩾ I_{B
^C} (x_i) ⩾0.5. So that min{ T_A (x_i) , F_A (x_i) , I_A (x_i) , I_{A
^C} (x_i) } ⩽ min{ T_B (x_i) , F_B (x_i) , I_B (x_i) , I_{B
^C} (x_i) } and max{ T_A (x_i) , F_A (x_i) , I_A (x_i) , I_{A
^C} (x_i) } ⩾max{ T_B (x_i) , F_B (x_i) , I_B (x_i) , I_{B
^C} (x_i) }. We get E_MT (A) ⩽ E_MT (B).

+If T_A (x_i) ⩾ T_B (x_i) , I_A (x_i) ⩾ I_B (x_i) , F_A (x_i) ⩾ F_B (x_i) when min { T_B (x_i) , I_B (x_i) , F_B (x_i) } ⩾ 0.5, then I_{A
^C} (x_i) ⩽ I_{B
^C} (x_i) ⩽0.5. So that min{ T_A (x_i) , F_A (x_i) , I_A (x_i) , I_{A
^C} (x_i) } ⩽ min{ T_B (x_i) , F_B (x_i) , I_B (x_i) , I_{B
^C} (x_i) } and max{ T_A (x_i) , F_A (x_i) , I_A (x_i) , I_{A
^C} (x_i) } ⩾ max{ T_B (x_i) , F_B (x_i) , I_B (x_i) , I_{B
^C} (x_i) }. We get E_MT (A) ⩽ E_MT (B).□

3.1 Compare to the existing entropy measures

Now, we compare the proposed entropy measures E_T and E_MT with some existing entropy measures.

- With entropy measure of Majumdar and Samanta [16]. $\begin{matrix} E_{MM} (A) = & 1 - \frac{1}{n} \sum_{i = 1}^{n} [T_{A} (x_{i}) + F_{A} (x_{i})] \\ | I_{A} (x_{i}) - I_{A^{C}} (x_{i}) | \end{matrix}$

This entropy measure satisfies the concept of entropy measure in Definition 3. It is maximize equal 1 when either T_A (x_i) = F_A (x_i) =0 or I_A (x_i) = I_{A
^C} (x_i) =0.5. But it is not well as shown in the example 1, i.e. with a SVNS B ={ (x, 0.8, 0, 0.7) } on X ={ x } then $E_{MM} (B) = 1 - \frac{1}{1} (0.8 + 0.7) | 0 - 1 | = - 0.5 \notin [0, 1]$ . This contradicts the value of the entropy in [0,1]. If we use new entropy measures, we have $E_{T} (A) = 1 - \frac{| 0.8 - 0.5 | + | 0 - 0.5 | + | 1 - 0.5 | + | 0.7 - 0.5 |}{2} = 0.25$ and E_MT (A) =0.

- With entropy measure of Wu et al. [34]: $E_{W} (A) = \frac{1}{n} \sum_{i = 1}^{n} E_{W}^{i} (A)$ where $\begin{matrix} E_{W}^{i} (A) = \frac{1}{3 (\sqrt{2} - 1)} {\sqrt{2} \cos \frac{2 T_{A} (x_{i}) - 1}{4} π \\ + \sqrt{2} \cos \frac{2 I_{A} (x_{i}) - 1}{4} π + \sqrt{2} \cos \frac{2 F_{A} (x_{i}) - 1}{4} π - 3} \end{matrix}$

for all i = 1, 2, . . . , n.

- With entropy measure of Cui and Ye [7, 8]: $\begin{matrix} E_{CY 1} (A) = \frac{1}{3 n} \sum_{i = 1}^{n} [3 - 2 \times (| T_{A} (x_{i}) - 0.5 | + | I_{A} (x_{i}) - 0.5 | + | F_{A} (x_{i}) - 0.5 |)] \\ E_{CY 2} (A) = 1 - \frac{1}{n} \sum_{i = 1}^{n} {[\frac{4 \times ({| T_{A} (x_{i}) - 0.5 |}^{2} + {| I_{A} (x_{i}) - 0.5 |}^{2} + {| F_{A} (x_{i}) - 0.5 |}^{2})}{3}]}^{\frac{1}{3}} \end{matrix}$

$\begin{matrix} E_{sin} (A) = \frac{1}{3 n} \sum_{i = 1}^{n} [sin (T_{A} (x_{i}) π) + sin (I_{A} (x_{i}) π) + sin (F_{A} (x_{i}) π)] \\ E_{exp} (A) = \frac{1}{3 n (\sqrt{e} - 1)} \sum_{i = 1}^{n} [\begin{matrix} T_{A} (x_{i}) e^{1 - T_{A} (x_{i})} + (1 - T_{A} (x_{i})) e^{T_{A} (x_{i})} - 1 \\ + I_{A} (x_{i}) e^{1 - I_{A} (x_{i})} + (1 - I_{A} (x_{i})) e^{I_{A} (x_{i})} - 1 \\ + F_{A} (x_{i}) e^{1 - F_{A} (x_{i})} + (1 - F_{A} (x_{i})) e^{F_{A} (x_{i})} - 1 \end{matrix}] \end{matrix}$ - With entropy measure of Ye [46]: $\begin{matrix} E_{Y 1} (A) = 1 - 2 {\frac{1}{3 n} \sum_{i = 1}^{n} ({| T_{A} (x_{i}) - 0.5 |}^{2} + {| I_{A} (x_{i}) - 0.5 |}^{2} + {| F_{A} (x_{i}) - 0.5 |}^{2})}^{\frac{1}{2}} \\ E_{Y 2} (A) = 1 - \frac{2}{n} \sum_{i = 1}^{n} [\max (| T_{A} (x_{i}) - 0.5 |, | I_{A} (x_{i}) - 0.5 |, | F_{A} (x_{i}) - 0.5 |)] \end{matrix}$

It is easy to verify that E_W (A) is an entropy measure based on the new concept of our proposed entropy measure.

3.2 Here we come to the question: does the concept of entropy measure in this paper coincide with the concept of Wu’s entropy measure?

The answer is no.

Indeed, we shall show that E_MT is not the Wu’s entropy measure through the following example.

Example 2. Let A ={ (x, 0.8, 0.4, 0.7) } be a SVNS on X ={ x }. We have A^C ={ (x, 0.7, 0.6, 0.8) } and $\bar{A} = {(x, 1 - 0.8, 1 - 0.4, 1 - 0.7)} = {(x, 0.2, 0.6,$ 0.3)}. Then $E_{MT} (A^{C}) = E_{MT} (A) = \frac{min {0.8, 0.4, 0.7, 0.6}}{max {0.8, 0.4, 0.7, 0.6}} = 0.5 \neq E_{MT} (\bar{A}) = \frac{min {0.2, 0.4, 0.3, 0.6}}{max {0.2, 0.4, 0.3, 0.6}} = \frac{1}{3}$ . It means that the third condition (E3) $E (A) = E (\bar{A})$ of Definition 4 is not satisfied. This implies that E_MT is not the Wu’s entropy measure, but E_MT is the entropy measure based on our proposed concept of entropy measure in Definition 5.

We conclude this comparison with an example to compare the effectiveness of the proposed entropy measures E_T, E_MT and E_TM with some of the existing entropy measures as follows.

Example 3. Let X ={ x₁, x₂, x₃, x₄ } be a universal set and A = { (x₁, 0 . 9, 0 . 2, 0 . 5) , (x₂, 0 . 8, 0 . 8, 0 . 6) , (x₃, 0 . 6, 0 . 8, 0 . 7) , (x₄, 0 . 5, 0 . 6, 0 . 8) } be a SVNS on X. In the characterizations of the linguistic variables, we regard A as “Large” then according to the operator law in the Equation (1) we have

$A^{\frac{1}{2}}$ may be regarded as “Quite Large”,

A² may be regarded as “Very Large”,

A³ may be regarded as “Quite Very Large”,

A⁴ may be regarded as “Very Very Large”,

in which

$\begin{matrix} A^{\frac{1}{2}} = {(x_{1}, 0 . 9487, 0 . 1056, 0 . 2929), (x_{2}, 0 . 8944, 0 . 5528, 0 . 2929), \\ (x_{3}, 0 . 7746, 0 . 5528, 0 . 4523), (x_{4}, 0 . 7071, 0 . 3675, 0 . 5528)} \\ A^{2} = {(x_{1}, 0 . 81, 0 . 36, 0 . 75), (x_{2}, 0 . 64, 0 . 96, 0 . 75), \\ (x_{3}, 0 . 36, 0 . 96, 0 . 91), (x_{4}, 0 . 25, 0 . 84, 0 . 96)} \\ A^{3} = {(x_{1}, 0 . 729, 0 . 488, 0 . 875), (x_{2}, 0 . 512, 0 . 992, 0 . 875), \\ (x_{3}, 0 . 126, 0 . 992, 0 . 973), (x_{4}, 0 . 125, 0 . 936, 0 . 992)} \end{matrix}$ $A^{4} = {\begin{matrix} (x_{1}, 0 . 6561, 0 . 5904, 0 . 9375), (x_{2}, 0 . 4096, 0 . 9984, 0 . 9375), \\ (x_{3}, 0 . 1296, 0 . 9984, 0 . 9919), (x_{4}, 0 . 0625, 0 . 9744, 0 . 9984) \end{matrix}}$

The results of new entropy measures are shown in the Table 1.

Table 1
Comparison of the SVNS with various entropies of SVNSs

E _T E _MT E _MM E _W E _CY1 E _CY2 E _sin E _exp E _Y1 E _Y2

$A^{\frac{1}{2}}$ 0.6119 0.3840 0.6079 0.7457 0.5880 0.4076 0.7260 0.7572 0.5017 0.3376

A 0.5875 0.3056 0.3350 0.7697 0.6167 0.4007 0.7466 0.7829 0.5274 0.35

A² 0.3738 0.1736 0.0733 0.5715 0.3983 0.2586 0.5446 0.5871 0.3517 0.155

A³ 0.3151 0.1596 0.1132 0.4284 0.3255 0.1877 0.4072 0.4408 0.2483 0.0745

A⁴ 0.2446 0.1164 0.0611 0.3405 0.2531 0.1466 0.3253 0.3495 0.1908 0.0337

	E _T	E _MT	E _MM	E _W	E _CY1	E _CY2	E _sin	E _exp	E _Y1	E _Y2
$A^{\frac{1}{2}}$	0.6119	0.3840	0.6079	0.7457	0.5880	0.4076	0.7260	0.7572	0.5017	0.3376
A	0.5875	0.3056	0.3350	0.7697	0.6167	0.4007	0.7466	0.7829	0.5274	0.35
A²	0.3738	0.1736	0.0733	0.5715	0.3983	0.2586	0.5446	0.5871	0.3517	0.155
A³	0.3151	0.1596	0.1132	0.4284	0.3255	0.1877	0.4072	0.4408	0.2483	0.0745
A⁴	0.2446	0.1164	0.0611	0.3405	0.2531	0.1466	0.3253	0.3495	0.1908	0.0337

Based on the mathematical view and intuitive of man, the entropy measure E must be satisfied $E (A^{\frac{1}{2}}) > E (A) > E (A^{2}) > E (A^{3}) > E (A^{4})$ . According to Table 1, we find that the three measures of entropy satisfies this include of two proposed measures and E_CY2, but others are not well. Meanwhile, all entropy measures satisfying Definition 5. One more time, we find the benefit and validity of the new concept of entropy measure proposed in this paper.

In applications, we often consider the weight of elements to be considered. So here we also consider determining the entropy over the weights assigned to the components of the universal set.

Let X ={ x₁, x₂, . . . , x_n } be a universal set. In which, each element x_i in X will be assigned with a the weight ω_i ∈ [0, 1] for all i = 1, 2, . . . , n and $\sum_{i = 1}^{n} ω_{i} = 1$ , ω = (ω₁, ω₂, . . . , ω_n) is called the vector weight on X. We have the entropies which generated from Definition 6 as follow:

Definition 8. Let A ={ (x_i, T_A (x_i) , I_A (x_i) , F_A (x_i)) |x_i ∈ X } be a SVNS set on X. We define

$E_{T}^{ω} (A) = \sum_{i = 1}^{n} ω_{i} \times {1 - \frac{| T_{A} (x_{i}) - 0.5 | + | F_{A} (x_{i}) - 0.5 | + | I_{A} (x_{i}) - 0.5 | + | I_{A^{C}} (x_{i}) - 0.5 |}{2}}$ (4) and

$\begin{matrix} E_{MT} (A) \\ = \sum_{i = 1}^{n} ω_{i} \frac{min {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}}{\max {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}} . \end{matrix}$ (5)

4 Similarity measures of SVNS based on entropy

In this section, we introduce a new way to construct the similarity measure of SVNS. In this way, we use the entropy of SVNS to determine the similarity measure of SVNS. In particular, the entropies given the Equation (2) - Equation (5) induce the similarity measures of SVNSs.

For two given SVNSs A, B ∈ SVNS (X). We define a new SVSN N (A, B) as following: $\begin{matrix} T_{N (A, B)} (x_{i}) = \frac{1 + | T_{A} (x_{i}) - T_{B} (x_{i}) |}{2}, \\ I_{N (A, B)} (x_{i}) = \frac{1 + | I_{A} (x_{i}) - I_{B} (x_{i}) |}{2}, \\ F_{N (A, B)} (x_{i}) = \frac{1 + | F_{A} (x_{i}) - F_{B} (x_{i}) |}{2} \end{matrix}$ (6) for all x_i ∈ X.

Then N (A, B) is a SVNS on X. In particular T_N(A,B) (x_i) ⩾0.5, I_N(A,B) (x_i) ⩾0.5 and F_N(A,B) (x_i) ⩾0.5 for all x_i ∈ X.

Theorem 4. If E is an entropy measure on SVNS (X) then S (A, B) = E (N (A, B)) will be a similarity measure of two SVNS sets A and B on X.

Proof.

(S1) Since 0 ⩽ E (C) ⩽1 for all C ∈ SVNS (X) then 0 ⩽ S (A, B) = E (N (A, B)) ⩽1 for all A, B ∈ SVNS (X).

(S2) It is obvious that S (A, B) = S (B, A) for all A, B ∈ SVNS (X).

(S3) Since T_N(A,A) (x_i) = I_N(A,A) (x_i) = F_N(A,A) (x_i) =0.5 for all x_i ∈ X then S (A, A) = E (N (A, A)) =1 for all A ∈ SVNS (X).

(S4) For all A, B, C ∈ SVNS (X), we have T_A (x_i) ⩽ T_B (x_i) ⩽ T_C (x_i), I_A (x_i) ⩾ I_B (x_i) ⩾ I_C (x_i) and F_A (x_i) ⩾ F_B (x_i) ⩾ F_C (x_i) for all x_i ∈ X. These imply that T_N(A,C) (x_i) ⩾ T_N(A,B) (x_i) ⩾0.5, I_N(A,C) (x_i) ⩾ I_N(A,B) (x_i) ⩾0.5 and F_N(A,C) (x_i) ⩾ F_N(A,B) (x_i) ⩾0.5 for all x_i ∈ X. So that S (A, C) = E (N (A, C)) ⩽ S (A, B) = E (N (A, B)). In the same way, we also have S (A, C) = E (N (A, C)) ⩽ S (B, C) = E (N (B, C)).□

From theorem 1 and theorem 2, using the entropy measures defined by Equation (2) and Equation (3) we obtain two similarity measures of two SVNSs A, B ∈ NS (X) as follow:

- with the entropy $E_{T} (A) = 1 - \frac{1}{n} \sum_{i = 1}^{n} \frac{| T_{A} (x_{i}) - 0.5 | + | F_{A} (x_{i}) - 0.5 | + | I_{A} (x_{i}) - 0.5 | + | I_{A^{C}} (x_{i}) - 0.5 |}{2}$

we have a similarity measure $\begin{matrix} S_{T} (A, B) = 1 - \frac{1}{4 n} \sum_{i = 1}^{n} {| T_{A} (x_{i}) - T_{B} (x_{i}) | \\ + | F_{A} (x_{i}) - F_{B} (x_{i}) | + | I_{A} (x_{i}) - I_{B} (x_{i}) | \\ + | I_{A^{C}} (x_{i}) - I_{B^{C}} (x_{i}) |} \\ = 1 - \frac{1}{4 n} \sum_{i = 1}^{n} {| T_{A} (x_{i}) - T_{B} (x_{i}) | \\ + | F_{A} (x_{i}) - F_{B} (x_{i}) | + 2 | I_{A} (x_{i}) - I_{B} (x_{i}) |} \end{matrix}$ (7)

- with the entropy $E_{MT} (A) = \frac{1}{n} \sum_{i = 1}^{n} \frac{min {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}}{\max {T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i}), I_{A^{C}} (x_{i})}}$

we have a similarity measure

$S_{MT} (A) = \frac{1}{n} \sum_{i = 1}^{n} \frac{min {| T_{A} (x_{i}) - T_{B} (x_{i}) | + 1, | F_{A} (x_{i}) - F_{B} (x_{i}) | + 1, | I_{A} (x_{i}) - I_{B} (x_{i}) | + 1, 1 - | I_{A} (x_{i}) - I_{B} (x_{i}) |}}{\max {| T_{A} (x_{i}) - T_{B} (x_{i}) | + 1, | F_{A} (x_{i}) - F_{B} (x_{i}) | + 1, | I_{A} (x_{i}) - I_{B} (x_{i}) | + 1, 1 - | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$ (8)

In general that, with ω = (ω₁, ω₂, . . . , ω_n) is the vector weight on X we have two similarity measures generated from Equation (7) and Equation (8) are

$S_{T}^{ω} (A, B) = \sum_{i = 1}^{n} ω_{i} \times {1 - \frac{| T_{A} (x_{i}) - T_{B} (x_{i}) | + | F_{A} (x_{i}) - F_{B} (x_{i}) | + 2 | I_{A} (x_{i}) - I_{B} (x_{i}) |}{4}}$ (9) and

$\begin{matrix} S_{MT} (A) = \sum_{i = 1}^{n} ω_{i} \frac{min {| T_{A} (x_{i}) - T_{B} (x_{i}) | + 1, | F_{A} (x_{i}) - F_{B} (x_{i}) | + 1, | I_{A} (x_{i}) - I_{B} (x_{i}) | + 1, 1 - | I_{A} (x_{i}) - I_{B} (x_{i}) |}}{\max {| T_{A} (x_{i}) - T_{B} (x_{i}) | + 1, | F_{A} (x_{i}) - F_{B} (x_{i}) | + 1, | I_{A} (x_{i}) - I_{B} (x_{i}) | + 1, 1 - | I_{A} (x_{i}) - I_{B} (x_{i}) |}} \end{matrix}$ (10)

4.1 Compare with some existing similarity measures

Let X ={ x₁, x₂, . . . , x_n } be a universal set. Given A, B ∈ SVNS (X). We recall some existing similarity measures of SVNSs. Ye [37–39 , 46] proposed some vector similarity measures of SVNSs.

+Cosine similarity measure [37] $S_{C} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \frac{T_{A} (x_{i}) T_{B} (x_{i}) + I_{A} (x_{i}) I_{B} (x_{i}) + F_{A} (x_{i}) F_{B} (x_{i})}{\sqrt{T_{A}^{2} (x_{i}) + I_{A}^{2} (x_{i}) + F_{A}^{2} (x_{i})} \sqrt{T_{B}^{2} (x_{i}) + I_{B}^{2} (x_{i}) + F_{B}^{2} (x_{i})}}$

+Dice similarity measure [38] $S_{D} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \frac{2 (T_{A} (x_{i}) T_{B} (x_{i}) + I_{A} (x_{i}) I_{B} (x_{i}) + F_{A} (x_{i}) F_{B} (x_{i}))}{T_{A}^{2} (x_{i}) + I_{A}^{2} (x_{i}) + F_{A}^{2} (x_{i}) + T_{B}^{2} (x_{i}) + I_{B}^{2} (x_{i}) + F_{B}^{2} (x_{i})}$

+Jaccard similarity measure [38] $S_{J} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \frac{T_{A} (x_{i}) T_{B} (x_{i}) + I_{A} (x_{i}) I_{B} (x_{i}) + F_{A} (x_{i}) F_{B} (x_{i})}{[\begin{matrix} T_{A}^{2} (x_{i}) + I_{A}^{2} (x_{i}) + F_{A}^{2} (x_{i}) + T_{B}^{2} (x_{i}) + I_{B}^{2} (x_{i}) + F_{B}^{2} (x_{i}) \\ - (T_{A} (x_{i}) T_{B} (x_{i}) + I_{A} (x_{i}) I_{B} (x_{i}) + F_{A} (x_{i}) F_{B} (x_{i})) \end{matrix}]}$

+Cosine function based similarity measures [41] $SC {OS}_{1} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} cos {\frac{π}{2} m ax {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$ $SC {OS}_{2} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} cos {\frac{π}{6} sum {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$

+Tangent similarity measure [43] ${ST}_{1} (A, B) = 1 - \frac{1}{n} \sum_{i = 1}^{n} tan {\frac{π}{4} m ax {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$ ${ST}_{2} (A, B) = 1 - \frac{1}{n} \sum_{i = 1}^{n} tan {\frac{π}{12} sum {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$

+Cotangent similarity measure [44] ${SCOT}_{1} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} cot {\frac{π}{4} + \frac{π}{4} m ax {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$ ${SCOT}_{2} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} cot {\frac{π}{4} + \frac{π}{12} sum {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$

+Similarity measure proposed by Ye [38 –40] ${SY}_{1} (A, B) = 1 - \frac{1}{3 n} \sum_{i = 1}^{n} {| T_{A} (x_{i}) - T_{B} (x_{i}) | + | F_{A} (x_{i}) - F_{B} (x_{i}) | + | I_{A} (x_{i}) - I_{B} (x_{i}) |}$ ${SY}_{2} (A, B) = 1 - \frac{1}{3 n} \sum_{i = 1}^{n} {{| T_{A} (x_{i}) - T_{B} (x_{i}) |}^{2} + {| F_{A} (x_{i}) - F_{B} (x_{i}) |}^{2} + {| I_{A} (x_{i}) - I_{B} (x_{i}) |}^{2}}^{\frac{1}{2}}$ ${SY}_{3} (A, B) = \frac{1 - \frac{1}{3 n} \sum_{i = 1}^{n} {| T_{A} (x_{i}) - T_{B} (x_{i}) | + | F_{A} (x_{i}) - F_{B} (x_{i}) | + | I_{A} (x_{i}) - I_{B} (x_{i}) |}}{1 + \frac{1}{3 n} \sum_{i = 1}^{n} {| T_{A} (x_{i}) - T_{B} (x_{i}) | + | F_{A} (x_{i}) - F_{B} (x_{i}) | + | I_{A} (x_{i}) - I_{B} (x_{i}) |}}$

+Similarity measure proposed by Ye and Du [46]. $SYD (A, B) = 1 - \frac{1}{n} \sum_{i = 1}^{n} max {| T_{A} (x_{i}) - T_{B} (x_{i}) |, | F_{A} (x_{i}) - F_{B} (x_{i}) |, | I_{A} (x_{i}) - I_{B} (x_{i}) |}$

To demonstrate the valid of our proposed measures, we consider the application of these measures in the pattern recognition problems as follow.

Given m patterns {A₁, A₂, . . . , A_m} in the form of SVNS on the universal set X ={ x₁, x₂, . . . , x_n }. Let C be a new sample in the form of SVNS on the universal set X ={ x₁, x₂, . . . , x_n }.

Question: What pattern C belongs to?

To deal with this problem, we implement in two steps:

Step 1: Compute the similarity measures S (A_i, C) for all i = 1, 2, . . . , m.

Step 2: Put the sample C belongs to pattern A_i*if S (A_i*, C) = max{ S (A_i, C) |i = 1, 2, . . . , m }

Example 4. Let X ={ x₁, x₂ } be a universal set. For two patterns SVNSs on X as follows: $A_{1} = {(x_{1}, 0.1, 0.2, 0.3), (x_{2}, 0.2, 0.6, 0.8)}$ $A_{2} = {(x_{1}, 0.15, 0.3, 0.45), (x_{2}, 0.8, 0.2, 0.6)}$

Let C be a new sample in the form of SVNS on X $C = {(x_{1}, 0.1, 0.2, 0.4), (x_{2}, 0.6, 0.6, 0.6)}$

The classification results based on similarity measures are shown in Table 2. In which, Null means that we non-determine to put C belongs to pattern A₁ or A₂.

Table 2
Comparison and decision in example 4

Similarity measures The results from C to Decision

A ₁ A ₂

S_C (A_i, C) 0.9486 0.9486 Null

S_D (A_i, C) 0.9385 0.9385 Null

S_J (A_i, C) 0.8860 0.8860 Null

SCOS₁ (A_i, C) 0.8984 0.8984 Null

SCOS₂ (A_i, C) 0.9728 0.9728 Null

ST₁ (A_i, C) 0.9077 0.8946 C belongs to pattern A ₁

ST₂ (A_i, C) 0.7982 0.7982 Null

SCOT₁ (A_i, C) 0.6818 0.6818 Null

SCOT₂ (A_i, C) 0.8377 0.8135 C belongs to pattern A ₁

SY₁ (A_i, C) 0.8833 0.8667 C belongs to pattern A ₁

SY₂ (A_i, C) 0.9088 0.9051 C belongs to pattern A ₁

SY₃ (A_i, C) 0.7910 0.7647 C belongs to pattern A ₁

SYD (A_i, C) 0.75 0.75 Null

S_T (A_i, C) (new measures) 0.9125 0.8375 C belongs to pattern A ₁

S_MT (A_i, C) (New measures) 0.8117 0.6234 C belongs to pattern A ₁

Similarity measures	The results from C to	Decision
S_C (A_i, C)	0.9486	0.9486	Null
S_D (A_i, C)	0.9385	0.9385	Null
S_J (A_i, C)	0.8860	0.8860	Null
SCOS₁ (A_i, C)	0.8984	0.8984	Null
SCOS₂ (A_i, C)	0.9728	0.9728	Null
ST₁ (A_i, C)	0.9077	0.8946	C belongs to pattern A ₁
ST₂ (A_i, C)	0.7982	0.7982	Null
SCOT₁ (A_i, C)	0.6818	0.6818	Null
SCOT₂ (A_i, C)	0.8377	0.8135	C belongs to pattern A ₁
SY₁ (A_i, C)	0.8833	0.8667	C belongs to pattern A ₁
SY₂ (A_i, C)	0.9088	0.9051	C belongs to pattern A ₁
SY₃ (A_i, C)	0.7910	0.7647	C belongs to pattern A ₁
SYD (A_i, C)	0.75	0.75	Null
S_T (A_i, C) (new measures)	0.9125	0.8375	C belongs to pattern A ₁
S_MT (A_i, C) (New measures)	0.8117	0.6234	C belongs to pattern A ₁

Example 5. Let X ={ x₁, x₂ } be a universal set. For two patterns SVNSs on X as follows: $A_{1} = {(x_{1}, 0.8, 0.8, 0.2), (x_{2}, 0.6, 0.8, 0.2)}$ $A_{2} = {(x_{1}, 1, 0.4, 0.4), (x_{2}, 1, 0.2, 0)}$

Let C be a new sample in the form of SVNS on X $C = {(x_{1}, 0.6, 0.6, 0.6), (x_{2}, 0.6, 0.2, 0.6)}$

The classification results based on similarity measures are shown in Table 3.

Table 3

Comparison and decision in example 5

Similarity measures	The results from C to		Decision
	A ₁	A ₂
S_C (A_i, C)	0.8122	0.8122	Null
S_D (A_i, C)	0.8056	0.8056	Null
S_J (A_i, C)	0.6850	0.6850	Null
SCOS₁ (A_i, C)	0.6984	0.6984	Null
SCOS₂ (A_i, C)	0.8898	0.8898	Null
ST₁ (A_i, C)	0.7597	0.7597	Null
ST₂ (A_i, C)	0.5828	0.5828	Null
SCOT₁ (A_i, C)	0.4172	0.4172	Null
SCOT₂ (A_i, C)	0.6134	0.6134	Null
SY₁ (A_i, C)	0.7	0.7	Null
SY₂ (A_i, C)	0.7982	0.7982	Null
SY₃ (A_i, C)	0.5385	0.5385	Null
SYD (A_i, C)	0.5	0.5	Null
S_T (A_i, C) (new measures)	0.675	0.75	C belongs to pattern A ₂
S_MT (A_i, C) (New measures)	0.4107	0.5982	C belongs to pattern A ₂

Two above examples on pattern recognition problems are demonstrated that our proposed measures are reasonable.

5 Applications of our proposed entropy and similarity measures to supplier selection problems

In this section, we applied our proposed similarity measures of SVNS in the multi criteria decision making. In MCDM problem, we have to find an optimal alternative from set of all feasible alternatives. Assume that, we have a set of m alternatives A ={ A₁, A₂, . . . , A_m } and a set of n criteria C ={ C₁, C₂, . . . , C_n }. In this problem, we consider each A_i is a SVNS on C, i.e. A_i ={ (C_j, D_ij) |C_j ∈ C } where D_ij = (T_ij, I_ij, F_ij) in which T_ij = T_ij (C_j) , I_ij = I_ij (C_j) , F_ij = F_ij (C_j) for all i = 1, 2, . . . , m, j = 1, 2, . . . , n. So that, we also consider D = [D_ij] _m×n is a SVNS matrix, and called a SVNS decision matrix.

The MCDM model use our proposed similarity measure of SVNSs

Input: a SVNS decision matrix.

Output: ranking of A ={ A₁, A₂, . . . , A_m }.

Procedure of proposed MCDM conclusion five steps as follow:

Step 1. Determine the weight ω_j of each criteria C_j. To do this we consider C_j is a set of SVNS of A_i for all j = 1, 2, . . . , n and i = 1, 2, . . . , m. It mean that C_j ={ (A_i, D_ij) |A_i ∈ A } for all j = 1, 2, . . . , n and i = 1, 2, . . . , m. Compute entropy measures e_j of this SVNS C_j for all j = 1, 2, . . . , n.

The weight ω_j of each criteria C_j is determined by $ω_{j} = \frac{1 - e_{j}}{\sum_{j = 1}^{n} (1 - e_{j})}$ (11)

for all j = 1, 2, . . . , n.

Step 2. Determine the SVNS best solution A^* and the SVNS worst solution A_* as follow $+ A^{*} = {(C_{j}, T_{j}^{*}, I_{j}^{*}, F_{j}^{*}) | C_{j} \in C}$

where $\begin{matrix} T_{j}^{*} = {\begin{matrix} \min_{i = 1, . . ., m} T_{ij} if C_{j} is a cost criteria \\ \max_{i = 1, . . ., m} T_{ij} if C_{j} is a benifite criteria \end{matrix}, \\ I_{j}^{*} = {\begin{matrix} \max_{i = 1, . . ., m} I_{ij} if C_{j} is a cost criteria \\ min_{i = 1, . . ., m} I_{ij} if C_{j} is a benifite criteria \end{matrix}, \\ F_{j}^{*} = {\begin{matrix} \max_{i = 1, . . ., m} F_{ij} if C_{j} is a cost criteria \\ min_{i = 1, . . ., m} F_{ij} if C_{j} is a benifite criteria \end{matrix} \end{matrix}$ (12) $+ A_{*} = {(C_{j}, T_{j}^{'}, I_{j}^{'}, F_{j}^{'}) | C_{j} \in C}$ where $\begin{matrix} T_{j}^{'} = {\begin{matrix} \max T_{ij} if C_{j} is a cost criteria \\ \min_{i = 1, . . ., m} T_{ij} if C_{j} is a benifite criteria \end{matrix}, \\ I_{j}^{'} = {\begin{matrix} \min_{i = 1, . . ., m} I_{ij} if C_{j} is a cost criteria \\ \max_{i = 1, . . ., m} I_{ij} if C_{j} is a benifite criteria \end{matrix}, \\ F_{j}^{'} = {\begin{matrix} \min_{i = 1, . . ., m} F_{ij} if C_{j} is a cost criteria \\ \max_{i = 1, . . ., m} F_{ij} if C_{j} is a benifite criteria \end{matrix} \end{matrix}$ (13)

Step 3. Calculate the similarity measures $S_{i}^{+}$ and $S_{i}^{-}$ from A_i to A^* and A_*

Step 4. Determine the relative closeness coefficient of A_i, (i = 1, 2, . . . , m) as ${CC}_{i} = \frac{S_{i}^{+}}{S_{i}^{+} + S_{i}^{-}}$ (14)

the similarity measures $S_{i}^{+}$ and $S_{i}^{-}$ from A_i to A^* and A_* for all i = 1, 2, . . . , m.

Step 5. Ranking A ={ A₁, A₂, . . . , A_m } with A_i ≻ A_k if CC_i ≻ CC_k for all i, k = 1, 2, . . . , m.

Example 6. We consider a supplier selection problem. A construction company wants to buy the material for his/her upcoming project. Have a set of six criteria C ={ C₁, C₂, C₃, C₄, C₅, C₆ } (where C₁: quality of material, C₂: price, C₃: services, C₄: delivery, C₅: technical support of required, C₆: behavior of material) for choosing the optimal alternative in a set of five alternatives A ={ A₁, A₂, A₃, A₄, A₅ }. In which, the criteria C₂ (price) is the cost criteria, and other criteria are benefit criteria. Here we convert the data sets from intuitionistic fuzzy set [12] to SVNS by adding the value of the third function as the smallest value of the first two values. The relationship between the choices and criteria is shown in the following Table 4.

Table 4

SVNS decision matrix

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ₁	(0.4,0.5,0.4)	(0.8,0.1,0.1)	(0.7,0.3,0.3)	(0.6,0.2,0.2)	(0.5,0.4,0.4)	(0.3,0.4,0.3)
A ₂	(0.7,0.2,0.2)	(0.5,0.3,0.3)	(0.3,0.4,0.3)	(0.8,0.1,0.1)	(0.2,0.6,0.2)	(0.4,0.5,0.4)
A ₃	(0.6,0.1,0.1)	(0.7,0.3,0.3)	(0.6,0.2,0.2)	(0.4,0.1,0.1)	(0.3,0.4,0.3)	(0.8,0.2,0.2)
A ₄	(0.5,0.4,0.4)	(0.3,0.4,0.3)	(0.8,0.1,0.1)	(0.7,0.2,0.2)	(0.6,0.1,0.1)	(0.7,0.1,0.1)
A ₅	(0.4,0.3,0.3)	(0.7,0.1,0.1)	(0.5,0.2,0.2)	(0.9,0.1,0.1)	(0.8,0,0)	(0.6,0.4,0.4)

Procedure of proposed MCDM conclusion five steps as follow:

• With entropy E_T and S_T

Step 1. Using the Equation (2) and Equation (11), we have the weight vector of C ={ C₁, C₂, C₃, C₄, C₅, C₆ } is ω = (0.1268, 0.1725, 0.169, 0.2289, 0.169, 0.1338).

Step 2. Using the Equation (12) and Equation (13) we get the SVNS best solution A^* and the SVNS worst solution A_* as follow Table 5.

Table 5

The SVNS best solution A^* and the SVNS worst solution A_*

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ^*	(0.7,0.1,0.1)	(0.3,0.4,0.3)	(0.8,0.1,0.1)	(0.9,0.1,0.1)	(0.8,0,0)	(0.8,0.1,0.1)
A _*	(0.4,0.5,0.4)	(0.8,0.1,0.1)	(0.3,0.4,0.3)	(0.4,0.2,0.2)	(0.2,0.6,0.4)	(0.3,0.5,0.4)

Step 3. We calculate the similarity measures $S_{i}^{+}$ and $S_{i}^{-}$ from A_i to A^* and A_* by using the Equation (9). These results are shown in Table 6.

Table 6

The similarity measures $S_{i}^{+}$ and $S_{i}^{-}$ from A_i to A^* by using the Equation (9)

	A ₁	A ₂	A ₃	A ₄	A ₅
S (A_i, A^*)	0.7288	0.7779	0.8436	0.9120	0.8576
S (A_i, A_*)	0.9236	0.8745	0.8088	0.7403	0.7948

Step 4. Determine the relative closeness coefficient of A_i, (i = 1, 2, . . . , 5) using the Equation (14) (see Table 7)

Table 7

The relative closeness coefficient of A_i, (i = 1, 2, . . . , 5) and their ranking

	A ₁	A ₂	A ₃	A ₄	A ₅
CC _i	0.4411	0.4708	0.5105	0.5520	0.5190
Ranking	5	4	3	1	2

Step 5. Ranking (in Table 7) A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁

Thus A₄ is the best alternative.

With the weight vector ω = (0.1268, 0.1725, 0.169, 0.2289, 0.169, 0.1338), we also compared our method of using similarity measures with the thirteen other similar measures shown in section 4 for this problem. Only Cosine similarity measure gives results ranking A₄ ≻ A₅ ≻ A₃ ≻ A₁ ≻ A₂, while our results and the remaining results all give the same ranking order of A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁. All results indicate that A4 has the highest rank, followed by A5, and then A3 (See Fig. 1).

Fig. 1

Ranking of alternative in example 6 using fifteen different similarity measures in section 4 with the weigh vector ω.

With entropy E _MT and S _MT

Step 1. Using the Equation (2) and Equation (11), we have the weight vector of C ={ C₁, C₂, C₃, C₄, C₅, C₆ } is ω′ = (0.1382, 0.1723, 0.1743, 0.2099, 0.1707, 0.1346).

Step 2. Using the Equation (12) and Equation (13) we get the SVNS best solution A^* and the SVNS worst solution A_* as follow Table 8.

Table 8

The SVNS best solution A^* and the SVNS worst solution A_*

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ^*	(0.7,0.1,0.1)	(0.3,0.4,0.3)	(0.8,0.1,0.1)	(0.9,0.1,0.1)	(0.8,0,0)	(0.8,0.1,0.1)
A _*	(0.4,0.5,0.4)	(0.8,0.1,0.1)	(0.3,0.4,0.3)	(0.4,0.2,0.2)	(0.2,0.6,0.4)	(0.3,0.5,0.4)

Step 3. We calculate the similarity measures $S_{i}^{+}$ and $S_{i}^{-}$ from A_i to A^* and A_* by using the Equation (10). These results are shown in Table 9.

Table 9

The similarity measures $S_{i}^{+}$ and $S_{i}^{-}$ from A_i to A^* by using the Equation (10)

	A ₁	A ₂	A ₃	A ₄	A ₅
S (A_i, A^*)	0.5371	0.6148	0.6855	0.8288	0.7449
S (A_i, A_*)	0. 8126	0.7543	0.6297	0.5509	0.6268

Step 4. Determine the relative closeness coefficient of A_i, (i = 1, 2, . . . , 5) using the Equation (14) (see Table 10)

Table 10

The relative closeness coefficient of A_i, (i = 1, 2, . . . , 5) and their ranking

	A ₁	A ₂	A ₃	A ₄	A ₅
CC _i	0.3979	0.4491	0.5212	0.6007	0.5431
Ranking	4	5	3	1	2

Step 5. Ranking (in Table 10) A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁

Thus A₄ is the best alternative.

In this case, we also compared our method of using similarity measures with the thirteen other similar measures shown in section 4 for this problem. Only Cosine similarity measure gives results ranking A₄ ≻ A₅ ≻ A₃ ≻ A₁ ≻ A₂, while our results and the remaining results all give the same ranking order of A₄ ≻ A₅ ≻ A₃ ≻ A₂ ≻ A₁. All results indicate that A4 has the highest rank, followed by A5, and then A3 (see Fig. 2).

Fig. 2

Ranking of alternative in example 6 using fifteen different similarity measures in section 4 with the weigh vector ω’.

6 Conclusion

In this paper, we introduce a new general concept of entropy measure of the SVNS, which overcomes the restriction of the previous entropy measures. We also show that some formulas satisfy the concept of our proposed entropy measure but do not satisfy the concept of entropy measure proposed by Wu et al. [34]. In contrast, the entropy formula by Wu et al. [34] is an entropy measure according to the concept of entropy measure of this paper. We also show an example to illustrate the robustness of our new entropy measure compared to some of the already knew entropy measures. At the same time, we also investigate several similarity measures which are induced from our new entropy measures and we apply these similarity measures to solve the multi-criteria decision making (MCDM) problem for selecting a supplier. In this paper, we also compared the obtained results with previous results. The new results have overcome the limitations of some previous results and have the same effect as other good results. In the future, we continue to study new entropy measures that satisfy this new concept and other measures that are generated from it. The researchers applied them to decision making [40,42, 40,42], classification, clustering in various fields such as: health, agriculture and service selection in smart cities. Other research directions can also be proposed and exploited on neutrosophic sets such as correlation coefficient [27, 35], cluster analysis [25, 27], difference and divergence/dissimilarity/similarity measures of neutrosophic sets (standard neutrosophic set) to deal with MCDM problems as in [1 , 42], combining with rough theory for processing knowledge on neutrosophic information systems as in [24, 25].

Footnotes

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 502.01 –2018.09.

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