Double-Valued Neutrosophic Sets, their Minimum Spanning Trees, and Clustering Algorithm

1 Introduction

Clustering plays a vital role in data mining, pattern recognition, and machine learning in several scientific and engineering fields. Clustering categorises the entities into groups (clusters); that is, it gathers similar samples into the same class and gathers the dissimilar ones into different classes. Clustering analysis has traditionally been kind of hard, which strictly allocates an object to a specific or particular class. It cannot be directly ascertained which class they should belong to, as a great deal of objects have no rigid restrictions. Therefore, it is required to divide them softly.

The fuzzy set theory introduced by Zadeh [29] provides a constructive analytic tool for such “soft” divisions of sets. The concept of fuzzy division was put forth by Ruspini [16], which marked the start of fuzzy clustering analysis. Zadeh’s fuzzy set theory [29] was extended to the intuitionistic fuzzy set (A-IFS), in which each element is assigned a membership degree and a non-membership degree by Atanassov [1]. A-IFS seems to be more suitable in dealing with data that has fuzziness and uncertainty than the fuzzy set. A-IFS was further generalised into the notion of interval-valued intuitionistic fuzzy set (IVIFS) by Atanassov and Gargov [2].

Because of the fuzziness and uncertainty of the real world, the attributes of the samples are often represented with A-IFSs; there is a need to cluster intuitionistic fuzzy data. Zahn [30] proposed the clustering algorithm using the minimum spanning tree (MST) of the graph. He had defined several criteria of edge inconsistency for detecting clusters of various shapes, and proposed a clustering algorithm using MST. Then, Chen et al. [4] put forward a maximal tree clustering method of the fuzzy graph by constructing the fuzzy similarity relation matrix and used the threshold of fuzzy similarity relation matrix to cut the maximum spanning tree, and obtained the classification.

Dong et al. [6] had given a hierarchical clustering algorithm based on fuzzy graph connectedness. Three MST algorithms were introduced and applied to clustering gene expression data by Xu et al. [24]. Two intuitionistic fuzzy MST (IF-MST) clustering algorithms to deal with intuitionistic fuzzy information was proposed and then extended to clustering interval-valued intuitionistic fuzzy information by Zhao et al. [33]. Furthermore, Zhang and Xu [31] introduced an MST algorithm-based clustering method under a hesitant fuzzy environment. The graph theory-based clustering algorithm [3], [4], [6], [30], [31] is an active research area.

To represent uncertain, imprecise, incomplete, inconsistent, and indeterminate information that are present in the real world, the concept of a neutrosophic set from the philosophical point of view was proposed by Smarandache [19]. The neutrosophic set is a prevailing framework that generalises the concept of the classic set, fuzzy set, intuitionistic fuzzy set (IFS), interval-valued fuzzy set, IVIFS, paraconsistent set, paradoxist set, and tautological set. Truth membership, indeterminacy membership, and falsity membership are represented independently in the neutrosophic set. However, the neutrosophic set generalises the above-mentioned sets from the philosophical point of view, and its functions T_A(x), I_A(x), and F_A(x) are real standard or non-standard subsets of ]⁻0, 1⁺[; that is, T_A(x): X→ ]⁻0, 1⁺[, I_A(x):X→ ]⁻0, 1⁺[, and F_A(x):X→ ]⁻0, 1⁺[, with the condition ⁻0≤supT_A(x)+supI_A(x)+supF_A(x)≤3⁺. It is difficult to apply the neutrosophic set in this form in real scientific and engineering areas.

To overcome this difficulty, Wang et al. [22] introduced a Single-Valued Neutrosophic Set (SVNS), which is an instance of a neutrosophic set. SVNS can deal with indeterminate and inconsistent information, which fuzzy sets and IFSs are incapable of. Ye [25], [26], [27] presented the correlation coefficient of SVNS and its cross-entropy measure, and applied them to single-valued neutrosophic decision-making problems. Recently, Ye [28] proposed a Single-Valued Neutrosophic Minimum Spanning Tree (SVN-MST) clustering algorithm to deal with the data represented by SVNSs. Owing to the fuzziness, uncertainty, and indeterminate nature of many practical problems in the real world, neutrosophy has found application in many fields including social network analysis [17], image processing [5], [18], [32], and socio-economic problems [20], [21]. Liu et al. have applied neutrosophy to group decision problems and multiple attribute decision-making problems [8], [9], [10], [11], [12], [13], [14], etc.

To provide more accuracy and precision to indeterminacy in this paper, the indeterminacy value present in the neutrosophic set has been classified into two based on membership: one as indeterminacy leaning towards truth membership and the other as indeterminacy leaning towards false membership. When the indeterminacy I can be identified as indeterminacy that is more of the truth value than the false value but cannot be classified as truth, it is considered to be indeterminacy leaning towards truth (I_T). When the indeterminacy can be identified to be indeterminacy that is more of the false value than the truth value but cannot be classified as false, it is considered to be indeterminacy leaning towards false (I_F). The single value of indeterminacy has vagueness, for one is not certain of whether the indeterminacy is favouring truth or false membership. When the indeterminacy is favouring or leaning towards truth and when it is favouring or leaning towards false is captured, the results or outcome will certainly be better than when a single value is used. Indeterminacy leaning towards truth and indeterminacy leaning towards falsity makes the indeterminacy involved in the scenario to be more accurate and precise. This modified neutrosophic set is defined as a Double-Valued Neutrosophic Set (DVNS).

Consider the scenario where the expert’s opinion is requested about a particular statement; he/she may state that the possibility in which the statement is true is 0.6 and in which the statement is false is 0.5, the degree in which he/she is not sure but thinks it is true is 0.2, and the degree in which he/she is not sure but thinks it is false is 0.1. Using a dual-valued neutrosophic notation, it can be expressed as x (0.6, 0.2, 0.1, 0.5). Assume another example: suppose there are 10 voters during a voting process; two people vote yes, two people vote no, three people are for yes but still undecided, and two people are favouring towards a no but still undecided. Using a dual-valued neutrosophic notation, it can be expressed as x (0.2, 0.3, 0.3, 0.2). However, these expressions are beyond the scope of representation using the existing SVNS. Therefore, the notion of a dual-valued neutrosophic set is more general and it overcomes the aforementioned issues.

This paper is organised into eight sections. Section 1 is introductory in nature. The basic concepts related to this paper are recalled in Section 2. Section 3 introduces DVNSs, their set operators, and discusses their properties. The distance measure of DVNS is defined, and its properties are discussed in Section 4. Section 5 proposes the Double-Valued Neutrosophic Minimum Spanning Tree (DVN-MST) clustering algorithm. Illustrative examples of the DVN-MST clustering algorithm and comparison of the DVN-MST clustering algorithm with other clustering algorithms such as SVN-MST, IF-MST, and FMST is carried out in Section 7. Conclusions and future work are provided in Section 8.

2 Preliminaries/Basic Concepts

3 DVNSs and their Properties

Indeterminacy deals with uncertainty that is faced in every sphere of life by everyone. It makes research/science more realistic and sensitive by introducing the indeterminate aspect of life as a concept. There are times in the real world where the indeterminacy I can be identified to be indeterminacy that has more of the truth value than the false value but cannot be classified as truth. Similarly, in some cases, the indeterminacy can be identified to be indeterminacy that has more of the false value than the truth value but cannot be classified as false. To provide more sensitivity to indeterminacy, this kind of indeterminacy is classified into two. When the indeterminacy I can be identified as indeterminacy that is more of the truth value than the false value but cannot be classified as truth, it is considered to be indeterminacy leaning towards truth (I_T). Whereas in case the indeterminacy can be identified to be indeterminacy that is more of the false value than the truth value but cannot be classified as false, it is considered to be indeterminacy leaning towards false (I_F). Indeterminacy leaning towards truth and indeterminacy leaning towards falsity makes the indeterminacy involved in the scenario to be more accurate and precise. It provides a better and detailed view of the existing indeterminacy.

The definition of DVNS is as follows:

Definition 3: Let X be a space of points (objects) with generic elements in X denoted by x. A DVNS A in X is characterised by truth membership function T_A(x), indeterminacy leaning towards truth membership function I_TA(x), indeterminacy leaning towards falsity membership function I_FA(x), and falsity membership function F_A(x). For each generic element x∈X, there are

TA(x), ITA(x), IFA(x), FA(x)∈[0, 1],

and

0≤TA(x)+ITA(x)+IFA(x)+FA(x)≤4.

Therefore, a DVNS A can be represented by

A={〈x, TA(x), ITA(x), IFA(x), FA(x)〉|x∈X}.

A DVNS A is represented as

when X is continuous. It is represented as

when X is discrete.

To illustrate the applications of DVNS in the real world, consider parameters that are commonly used to define the quality of service of semantic web services, like capability, trustworthiness, and price, for illustrative purposes. The evaluation of the quality of service of semantic web services [23] is used to illustrate set-theoretic operation on DVNSs.

Example 1: Let X=[x₁, x₂, x₃] where x₁ is capability, x₂ is trustworthiness, and x₃ is price. The values of x₁, x₂, and x₃ are in [0, 1]. They are obtained from the questionnaire of some domain experts; their option could be a degree of “good service,” indeterminacy leaning towards a degree of “good service,” indeterminacy leaning towards a degree of “poor service,” and a degree of “poor service.” A is a DVNS of X defined by

A=〈0.3, 0.4, 0.2, 0.5〉/x1+〈0.5, 0.1, 0.3, 0.3〉/x2+〈0.7, 0.2, 0.1, 0.2〉/x3.

B is a DVNS of X defined by

B=〈0.6, 0.1, 0.3, 0.2〉/x1+〈0.2, 0.1, 0.2, 0.4〉/x2+〈0.4, 0.1, 0.1, 0.3〉/x3.

Definition 4: The complement of a DVNS A denoted by c(A) is defined as

1. T_c(A)(x) =F_A(x),

2. I_Tc(A)(x)=1–I_TA(x),

3. I_Fc(A)(x)=1–I_FA(A)(x),

4. F_c(A)(x)=T_A(x),

for all x in X.

Example 2: Consider the DVNS A defined in Example 1. Then, the complement of A is

c(A)=〈0.5, 0.6, 0.8, 0.3〉/x1+〈0.3, 0.9, 0.7, 0.5〉/x2+〈0.2, 0.8, 0.9, 0.7〉/x3.

Definition 5: A DVNS A is contained in DVNS B, A⊆B, if and only if

1. T_A(x) ≤T_B(x),

2. I_TA(x)≤I_TB(x),

3. I_FA(x)≤I_FB(x),

4. F_A(x)≥F_B(x),

for all x in X.

Note that by the definition of containment relation, X is a partially ordered set and not a totally ordered set.

For example, let A and B be the DVNSs as defined in Example 1, then A is not contained in B and B is not contained in A.

Definition 6: Two DVNSs A and B are equal, denoted as A=B, if and only if A⊆B and B⊆A.

Theorem 1:A⊆B if and only if c(B)⊆c(A).

Proof.

A⊆B⇔TA≤TB, ITA≤ITB, IFA≤IFB, FB≥FA⇔FA≤FB, 1−ITB≤1−ITA, 1−IFB≤1−IFA, TB≥TA⇔c(B)⊆c(A).□

Definition 7: The union of two DVNSs A and B is a DVNS C, denoted as C=A∪B, whose truth membership, indeterminacy leaning towards truth membership, indeterminacy leaning towards falsity membership, and falsity membership functions are related to those of A and B by the following:

1. T_C(x) =max(T_A(x), T_B(x)),

2. I_TC(x)=max(I_TA(x), I_TB(x)),

3. I_FC(x)=max(I_FA(x), I_FB(x)),

4. F_C(x)=min(F_A(x), F_B(x)),

for all x in X.

Example 3: Consider the DVNSs A and B defined in Example 1. Then,

A∪B=〈0.6, 0.4, 0.3, 0.2〉/x1+〈0.5, 0.1, 0.3, 0.3〉/x2+〈0.7, 0.2, 0.1, 0.2〉/x3.

Theorem 2.A∪

B is the smallest DVNS containing both A and B.

Proof. It is direct from the definition of the union operator.□

Definition 8: The intersection of two DVNSs A and B is a DVNS C, denoted as C=A∩B, whose truth membership, indeterminacy leaning towards truth membership, indeterminacy leaning towards falsity membership, and falsity membership functions are related to those of A and B by the following:

1. T_C(x) =min(T_A(x), T_B(x)),

2. I_TC(x)=min(I_TA(x), I_TB(x)),

3. I_FC(x)=min(I_FA(x), I_FB(x)),

4. F_C(x)=max(F_A(x), F_B(x)),

for all x∈X.

Example 4: Consider the DVNSs A and B as defined in Example 1. Then,

A∩B=〈0.3, 0.1, 0.2, 0.5〉/x1+〈0.2, 0.1, 0.2, 0.4〉/x2+〈0.4, 0.1, 0.1, 0.3〉/x3.

Theorem 3:A∩

B is the largest DVNS contained in both A and B.

Proof. It is straightforward from the definition of intersection operator.□

Definition 9: The difference of two DVNSs D, written as D=A\B, whose truth membership, indeterminacy leaning towards truth membership, indeterminacy leaning towards falsity membership, and falsity membership functions are related to those of A and B by

1. T_D(x) =min(T_A(x), F_B(x)),

2. I_TD(x)=min(I_TA(x), 1−I_TB(x)),

3. I_FD(x)=min(I_FA(x), 1−I_FB(x)),

4. F_D(x)=min(F_A(x), T_B(x)),

for all x in X.

Example 5: Consider A and B the DVNSs defined in Example 1. Then,

A\B=〈0.2, 0.4, 0.2, 0.5〉/x1+〈0.4, 0.1, 0.3, 0.2〉/x2+〈0.3, 0.2, 0.1, 0.2〉/x3.

Three operators, called truth favourite (Δ), falsity favourite (∇), and indeterminacy neutral (∇), are defined over DVNSs. Two operators, truth favourite (Δ) and falsity favourite (∇), are defined to remove the indeterminacy in the DVNSs and transform it into IFSs or paraconsistent sets. Similarly, a DVNS can be transformed into an SVNS by applying the indeterminacy neutral (∇) operator that combines the indeterminacy values of the DVNS. These three operators are unique on DVNSs.

Definition 10: The truth favourite of a DVNS A is written as B=ΔA, whose truth membership and falsity membership functions are related to those of A by

1. T_B(x) =min(T_A(x)+I_TA(x), 1),

2. I_TB(x)=0,

3. I_FB(x)=0,

4. F_B(x)=F_A(x),

for all x in X.

Example 6: The truth favourite of the DVNS A given in Example 1 is B=ΔA, where

B=〈0.7, 0, 0, 0.5〉/x1+〈0.6, 0, 0, 0.2〉/x2+〈0.9, 0, 0, 0.2〉/x3.

Definition 11: The falsity favourite of a DVNS A, written as B=∇A, whose truth membership and falsity membership functions are related to those of A by

1. T_B(x) =T_A(x),

2. I_TB(x)=0,

3. I_FB(x)=0,

4. F_B(x)=min(F_A(x)+I_FA(x), 1),

for all x in X.

Example 7: Let A be the DVNS defined in Example 1. Then, B is the falsity favourite of the DVNS B=∇A:

B=〈0.3, 0, 0, 0.7〉/x1+〈0.5, 0, 0, 0.6〉/x2+〈0.7, 0, 0, 0.3〉/x3.

Definition 12: The indeterminacy neutral of a DVNS A, written as B=∇A, whose truth membership, indeterminate membership, and falsity membership functions are related to those of A by

1. T_B(x) =T_A(x),

2. I_TB(x)=min(I_TA(x)+I_TB(x), 1),

3. I_FB(x)=0,

4. F_B(x)=F_A(x),

for all x in X.

Example 8: Consider the DVNS A defined in Example 1. Then, the indeterminacy neutral of the DVNS B=∇A, is

B=〈0.3, 0.6, 0, 0.5〉/x1+〈0.5, 0.4, 0, 0.3〉/x2+〈0.7, 0.3, 0, 0.2〉/x3.

Proposition 1: The following set theoretic operators are defined over DVNSs A, B, and C.

1. (Property 1) (commutativity):

   A∪B=B∪A,   A∩B=B∩A,   A×B=B×A.

2. (Property 2) (associativity):

   A∪(B∪C)=(A∪B)∪C,A∩(B∩C)=(A∩B)∪C,A×(B×C)=(A×B)×C.

3. (Property 3) (distributivity):

   A∪(B∩C)=(A∪B)∩(A∪C),A∩(B∪C)=(A∩B)∪(A∩C).

4. (Property 4) (idempotency):

   A∪A=A,   A∩A=A,ΔΔA=ΔA,   ∇∇A=∇A.

5. (Property 5)

   A∩ϕ=ϕ, A∩X=X,

   where Tϕ=Iϕ=0, Fϕ=1 and T_X=I_TX=I_FX=1, F_X=0.

6. (Property 6)

   A∪ϕ=A, A∩X=A,

   where Tϕ=I_Iϕ=I_Fϕ=0, Fϕ=1 and T_X=I_TX=I_FX=1, F_X=0.

7. (Property 7) (absorption):

   A∪(A∩B)=A, A∩(A∪B)=A.

8. (Property 8) (De Morgan’s laws):

   c(A∪B)=c(A)∩c(B), c(A∩B)=c(A)∪c(B).

9. (Property 9) (involution):

   c(c(A))=A.

The definition of complement, union, and intersection of DVNSs and the DVNS itself satisfy most properties of the classical set, fuzzy set, IFS, and SNVS. Similar to the fuzzy set, IFS, and SNVS, it does not satisfy the principle of middle exclude.

4 Distance Measures of DVNS

The distance measures over DVNSs is defined in the following and the related algorithm for determining the distance is given.

Consider two DVNSs A and B in a universe of discourse, X=x₁, x₂, …, x_n, which are denoted by

A={〈xi, TA(xi), ITA(xi), IFA(xi), FA(xi)〉|xi∈X}, andB={〈xi, TB(xi), ITB(xi), IFB(xi), FA(xi)〉|xi∈X},

where T_A(x_i), I_TA(x_i), I_FA(x_i), F_A(x_i), T_B(x_i), I_TB(x_i), I_FB(x_i), F_B(x_i) ∈[0, 1] for every x_i∈X. Let w_i(i=1, 2, …, n) be the weight of an element x_i(i=1, 2, …, n), with w_i≥0(i=1, 2, …, n) and ∑i = 1nwi=1.

Then, the generalised double-valued neutrosophic weighted distance is defined as follows:

where λ>0.

Equation (3) reduces to the double-valued neutrosophic weighted Hamming distance and the double-valued neutrosophic weighted Euclidean distance, when λ=1, 2, respectively. The double-valued neutrosophic weighted Hamming distance is given as

where λ=1 in Eq. (3).

The double-valued neutrosophic weighted Euclidean distance is given as

where λ=2 in Eq. (3).

The algorithm to obtain the generalised double-valued neutrosophic weighted distance d_λ(A, B) is given in Algorithm 1.

The following proposition is given for the distance measure:

Proposition 2: The generalised double-valued neutrosophic weighted distance d_λ(A, B) for λ>0 satisfies the following properties:

1. (Property 1) d_λ(A, B)≥0

2. (Property 2) d_λ(A, B)=0 if and only if A=B

3. (Property 3) d_λ(A, B)=d_λ(B, A)

4. (Property 4) If A⊆B⊆C, C is a DVNS in X, then d_λ(A, C)≥d_λ(A, B) and d_λ(A, C)≥d_λ(B, C).

It can be easily seen that d_λ(A, B) satisfies (Property 1) to (Property 3); hence, only (Property 4) is proved. Let A⊆B⊆C, then 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 every x_i∈X. Then, the following relations are obtained:

|TA(xi)−TB(xi)|λ ≤ |TA(xi)−TC(xi)|λ,|TB(xi)−TC(xi)|λ ≤ |TA(xi)−TC(xi)|λ,|ITA(xi)−ITB(xi)|λ ≤ |ITA(xi)−ITC(xi)|λ,|ITB(xi)−ITC(xi)|λ ≤ |ITA(xi)−ITC(xi)|λ,|IFA(xi)−IFB(xi)|λ ≤ |IFA(xi)−IFC(xi)|λ,|IFB(xi)−IFC(xi)|λ ≤ |IFA(xi)−IFC(xi)|λ,|FA(xi)−FB(xi)|λ ≤ |FA(xi)−FC(xi)|λ, and|FB(xi)−FC(xi)|λ ≤ |FA(xi)−FC(xi)|λ.

Hence,

|TA(xi)−TB(xi)|λ+|ITA(xi)−ITB(xi)|λ+|IFA(xi)−IFB(xi)|λ+|FA(xi)−FB(xi)|λ ≤ |TA(xi)−TC(xi)|λ+ ≤ |ITA(xi)−ITC(xi)|λ+≤ |IFA(xi)−IFC(xi)|λ+|FA(xi)−FC(xi)|λ.|TB(xi)−TC(xi)|λ+|ITB(xi)−ITC(xi)|λ+|IFB(xi)−IFC(xi)|λ+|FB(xi)−FC(xi)|λ ≤ |TA(xi)−TC(xi)|λ+|ITA(xi)−ITC(xi)|λ+|IFA(xi)−IFC(xi)|λ+|FA(xi)−FC(xi)|λ.

Combining the above inequalities with the distance formula given in Eq. (3), the following is obtained:

dλ(A, B)≥dλ(A, C) and dλ(B, C)≥dλ(A, C).

for λ>0. Thus, (Property 4) is obtained.

The double-valued neutrosophic distance matrix D is defined in the following.

Definition 13: Let A_j(j=1, 2, …, m) be a collection of m DVNSs; then, D=(d_ij)_m×m is called a double-valued neutrosophic distance matrix, where d_ij=d_λ(A_i, A_j) is the generalised double-distance-valued neutrosophic between A_i and A_j, and its properties are as follows:

1. 0 ≤d_ij≤1 for all i, j=1, 2, …, m;

2. d_ij=0 if and only if A_i=A_j;

3. d_ij=d_ji for all i, j=1, 2, …, m.

The algorithm to calculate the double-valued neutrosophic weighted distance matrix D is given in Algorithm 2.

The following section provides the DVN-MST clustering algorithm.

5 DVN-MST Clustering Algorithms

In this section, a DVN-MST clustering algorithm is proposed as a generalisation of the IF-MST and SVN-MST clustering algorithms.

Let X={x₁, x₂, …, x_n} be an attribution space and the weight vector of an element x_i(i=1, 2, …, n) be w={w₁, w₂, …, w_n}, with w_i≥0(i=1, 2, …, n) and ∑i = 1nwi=1. Consider that A_j(j=1, 2, …, m) is a collection of m DVNSs, which has m samples that need to be clustered. Then, they are represented in the following form: Aj={〈xi, TAj​(xj), ITAj​(xj), IFAj​(xj), FAj​(xj)〉|xj∈X}. Algorithm 3 provides the DVN-MST clustering algorithm. The description of the algorithm is as follows:

Step 1: Calculate the distance matrix D=d_ij=d_λ(A_i, A_j) by Algorithm 2 (take λ=2). The double-valued neutrosophic distance matrix D=(d_ij)_m×m obtained is

D=[0d12…d1m⋮⋮⋮dm1dm2…0].

Step 2: The double-valued neutrosophic graph G(V, E), where every edge between A_i and A_j(i, j=1, 2, …, m) is assigned the double-valued neutrosophic weighted distance d_ij, is an element of the double-valued neutrosophic distance matrix D=(d_ij)_m×m, which represents the dissimilarity degree between the samples A_i and A_j. The double-valued neutrosophic graph G(V, E) is represented as a graph.

Step 3: Construct the MST of the double-valued neutrosophic graph G(V, E).

1. The sorted list of distances of edges of G(V, E) in increasing order by weights is constructed.

2. Keep an empty subgraph S of G(V, E) and select the edge e with the smallest weight to add in S, where the end points of e are disconnected.

3. The smallest edge e is added to S and deleted from the sorted list.

4. The next smallest edge is selected and if no cycle is formed in S, it is added to S and deleted from the list.

5. Repeat process 4 until the subgraph S has (m−1) edges.

Thus, the MST of the double-valued neutrosophic graph G(V, E) is obtained and illustrated as a graph.

Step 4: Select a threshold r and disconnect all the edges of the MST with weights greater than r to obtain a certain number of clusters; list it as a table. The clustering results induced by the subtrees do not depend on some particular MST [30], [31].

6 Illustrative Examples

Two descriptive examples are presented and utilised to demonstrate the real-world applications and the effectiveness of the proposed DVN-MST clustering algorithm. This example is adopted from Ref. [28].

Example 9: A car market is going to classify eight different cars of A_j(j=1, 2, …, 8). For each car, the six evaluation factors (attributes) are as follows: x₁, fuel consumption; x₂, coefficient of friction; x₃, price; x₄, comfortable degree; x₅, design; x₆, security coefficient. The characteristics of each car under the six attributes are represented by the form of DVNSs, and then the double-valued neutrosophic data are as follows:

A1={〈x1, 0.3, 0.2, 0.1, 0.5〉, 〈x2, 0.6, 0.3, 0.4, 0.1〉, 〈x3, 0.4, 0.3, 0.2, 0.3〉,〈x4, 0.8, 0.1, 0.2, 0.1〉, 〈x5, 0.1, 0.3, 0.4, 0.6〉, 〈x6, 0.5, 0.2, 0.1, 0.4〉},A2={〈x1, 0.6, 0.3, 0.2, 0.3〉, 〈x2, 0.5, 0.4, 0.1, 0.2〉, 〈x3, 0.6, 0.2, 0.2, 0.1〉,〈x4, 0.7, 0.2, 0.3, 0.1〉, 〈x5, 0.3, 0.1, 0.4, 0.6〉, 〈x6, 0.4, 0.3, 0.2, 0.3〉},A3={〈x1, 0.4, 0.2, 0.3, 0.4〉, 〈x2, 0.8, 0.2, 0.3, 0.1〉, 〈x3, 0.5, 0.3, 0.2, 0.1〉,〈x4, 0.6, 0.1, 0.2, 0.2〉, 〈x5, 0.4, 0.1, 0.2, 0.5〉, 〈x6, 0.3, 0.2, 0.1, 0.2〉},A4={〈x1, 0.2, 0.4, 0.3, 0.4〉, 〈x2, 0.4, 0.5, 0.2, 0.1〉, 〈x3, 0.9, 0.2, 0.3, 0.0〉,〈x4, 0.8, 0.2, 0.2, 0.1〉, 〈x5, 0.2, 0.3, 0.4, 0.5〉, 〈x6, 0.7, 0.3, 0.2, 0.1〉},A5={〈x1, 0.2, 0.3, 0.2, 0.3〉, 〈x2, 0.3, 0.2, 0.1, 0.6〉, 〈x3, 0.5, 0.1, 0.2, 0.4〉,〈x4, 0.7, 0.1, 0.3, 0.1〉, 〈x5, 0.4, 0.2, 0.2, 0.4〉, 〈x6, 0.3, 0.2, 0.1, 0.6〉},A6={〈x1, 0.3, 0.2, 0.1, 0.4〉, 〈x2, 0.2, 0.1, 0.4, 0.7〉, 〈x3, 0.4, 0.2, 0.1, 0.5〉,〈x4, 0.8, 0.0, 0.1, 0.1〉, 〈x5, 0.4, 0.3, 0.2, 0.5〉, 〈x6, 0.2, 0.1, 0.2, 0.7〉},A7={〈x1, 0.4, 0.4, 0.1, 0.3〉, 〈x2, 0.5, 0.3, 0.2, 0.1〉, 〈x3, 0.6, 0.1, 0.2, 0.2〉,〈x4, 0.2, 0.3, 0.1, 0.7〉, 〈x5, 0.3, 0.1, 0.2, 0.5〉, 〈x6, 0.7, 0.2, 0.2, 0.1〉}A8={〈x1, 0.4, 0.1, 0.1, 0.2〉, 〈x2, 0.6, 0.1, 0.1, 0.1〉, 〈x3, 0.8, 0.2, 0.2, 0.1〉,〈x4, 0.7, 0.2, 0.3, 0.1〉, 〈x5, 0.1, 0.1, 0.2, 0.8〉, 〈x6, 0.2, 0.1, 0.1, 0.8〉}.

Let the weight vector of the attribute x_i(i=1, 2, …, 6) be w=(0.16, 0.12, 0.25, 0.2, 0.15, 0.12)^T; then, the DVN-MST clustering algorithm given in Algorithm 3 is used to group the eight different cars of A_j(j=1, 2, …, 8).

Step 1: Calculate the distance matrix D=d_ij=d_λ(A_i, A_j) by Algorithm 2 (take λ=2). The double-valued neutrosophic distance matrix D=(d_ij)_m×m is obtained as follows:

D=[00.144650.137750.189340.172920.182210.241450.187680.1446500.122070.151080.16830.22130.208870.156520.137750.1220700.189930.184860.222430.198560.180830.189340.151080.1899300.226720.283460.234840.219150.172920.16830.184860.2267200.112470.25110.196020.182210.22130.222430.283460.1124700.297740.233830.241450.208870.198560.234840.25110.2977400.2650.187680.156520.180830.219150.196020.233830.2650].

Step 2: The double-valued neutrosophic graph G(V, E), where every edge between A_i and A_j(i, j=1, 2, …, 8) is assigned the double-valued neutrosophic weighted distance d_ij, is an element of the double-valued neutrosophic distance matrix D=(d_ij)_m×m, which represents the dissimilarity degree between the samples A_i and A_j. The double-valued neutrosophic graph G(V, E) is shown in Figure 1.

Figure 1:

Double-Valued Neutrosophic Graph G.

Step 3: Construct the MST of the double-valued neutrosophic graph G(V, E).

1. The sorted list of distances of edges of G in increasing order by weights is d₅₆≤d₃₂≤d₃₁≤d₁₂≤d₄₂≤d₈₂≤ d₅₂≤d₅₁≤d₃₈≤d₁₆≤d₃₅≤d₈₁≤d₄₁≤d₃₄≤d₈₅≤d₇₃≤d₇₂≤d₈₄≤d₂₆.

2. Keep an empty subgraph S of G and add the edge e with the smallest weight to S, where the end points of e are disconnected.

3. The edge between A₅ and A₆, d₅₆=0.11247, is the smallest; it is added to S and deleted from the sorted list.

4. The next smallest edge is selected from G and if no cycle is formed in S, it is added to S and deleted from the list.

5. Repeat process 4 until the subgraph S has (7−1) edges or spans eight nodes.

Thus, the MST of the double-valued neutrosophic graph G(V, E) is obtained, as illustrated in Figure 2.

Figure 2:

DVN-MST S of Graph G.

Step 4: Select a threshold r and disconnect all the edges of the MST with weights greater than r to obtain a certain number of subtrees (clusters), as listed in Table 1.

To compare the DVN-MST clustering algorithm with the SVN-MST clustering algorithm, IF-MST clustering algorithm, and fuzzy MST clustering algorithm, the following example discussed in Refs. [28], [30], [31] is introduced for comparative convenience.

Example 10: For the completion of an operational mission, the six sets of operational plans are made (adapted from Refs. [28], [30], [31]). To group these operational plans with respect to their comprehensive function, a military committee has been set up to provide assessment information on them. The attributes that are considered here in assessment of the six operational plans, A_j(j=1, 2, …, 6), are based on the effectiveness of (i) operational organisation (x₁) and (ii) operational command (x₂). The weight vector of the attributes x_i(i=1, 2) is w=(0.45, 0.55)^T. The military committee evaluates the performance of the six operational plans A_j(j=1, 2, …, 6) with respect to the attributes x_i(i=1, 2) and provides the DVNSs as follows:

A1={〈x1, 0.7, 0.2, 0.05, 0.15〉, 〈x2, 0.6, 0.3, 0.4, 0.2〉},A2={〈x1, 0.4, 0.3, 0.04, 0.35, 〉, 〈x2, 0.8, 0.1, 0.3, 0.1〉},A3={〈x1, 0.55, 0.2, 0.02,0.25, 〉, 〈x2, 0.7, 0.1, 0.05, 0.15〉},A4={〈x1, 0.44, 0.2, 0.3, 0.35, 〉, 〈x2, 0.6, 0.2, 0.1, 0.2〉},A5={〈x1, 0.5, 0.15, 0.2, 0.35,〉, 〈x2, 0.75, 0.1, 0.1, 0.2〉},A6={〈x1, 0.55, 0.2, 0.3, 0.25, 〉, 〈x2, 0.57, 0.2, 0.3, 0.15〉}.

Then, the DVN-MST clustering algorithm is utilised to group these operational plans A_j(j=1, 2, …, 6).

Step 1: Calculate the distance matrix D=d_ij=d_λ(A_i, A_j) by Algorithm 3 (take λ=2). The double-valued neutrosophic distance matrix D=(d_ij)_m×m is obtained as follows:

D=[00.171790.166790.181350.181060.117920.1717900.123040.150530.117280.146210.166790.1230400.121420.0797730.14530.181350.150530.1214200.0792460.0919440.181060.117280.0797730.07924600.120360.117920.146210.14530.0919440.120360].

Step 2: The double-valued neutrosophic graph G(V, E) where every edge between A_i and A_j(i, j=1, 2, …, 8) is assigned the double-valued neutrosophic weighted distance d_ij coming from an element of the double-valued neutrosophic distance matrix D=(d_ij)_m×m, which represents the dissimilarity degree between the samples A_i and A_j. Then, the double-valued neutrosophic graph G(V, E) is shown in Figure 3.

Figure 3:

Double-Valued Neutrosophic Graph G for Operational Plans.

Step 3: Construct the MST of the double-valued neutrosophic graph G(V, E).

1. The sorted list of distances of edges of G in increasing order by weights is d₅₄≤d₃₅≤d₄₆≤d₅₂≤d₆₁≤d₅₆≤ d₄₃≤d₃₂≤d₆₃≤d₂₆≤d₄₂≤d₁₃≤d₁₂≤d₅₁≤d₄₁

2. Keep an empty subgraph S of G and add the edge e with the smallest weight to S, where the end points of e are disconnected.

3. The edge between A₅ and A₄, d₅₄=0.079246, is the smallest; it is added to S and deleted from the sorted list.

4. The next smallest edge is selected from G and if no cycle formed in S, it is added to S and deleted from the list.

5. Repeat process 4 until the subgraph S spans six nodes or has (5−1) edges.

The MST S of the double-valued neutrosophic graph G(V, E) is obtained, as shown in Figure 4.

Figure 4:

DVN-MST S for Operational Plans.

Step 4: Select a threshold r and disconnect all the edges of the MST with weights greater than r to obtain clusters, as listed in Table 2.

7 Comparison of DVN-MST Clustering Algorithm with Other Clustering Algorithms

For comparative purposes, the DVN-MST clustering algorithm, SVN-MST clustering algorithm, IF-MST clustering algorithm, and fuzzy clustering algorithm are applied to the problem given in Example 10. In Example 10, the DVN-MST clustering algorithm is applied to the military committee evaluation of the performance of the six operational plans; its related clustering results are given in Table 2.