Cubic Vague Set and its Application in Decision Making.

From the hybrid nature of cubic sets, we develop a new generalized hybrid structure of cubic sets known as cubic vague sets (CVSs). We also define the concept of internal cubic vague sets (ICVSs) and external cubic vague sets (ECVSs) with examples and discuss their interesting properties, including ICVSs and ECVSs under both P and R-Order. Moreover, we prove that the R and R-intersection of ICVSs (or ECVSs) need not be an ICVS (or ECVS). We also derive the different conditions for P-union (P-intersection, R and R-intersection) operations of both ICVSs (ECVSs) to become an ICVS (ECVS). Finally, we introduce a decision-making based on the proposed similarity measure of the CVSs domain and a numerical example is given to elucidate that the proposed similarity measure of CVSs is an important concept for measuring entropy in the information/data. It will be shown that the cubic vague set has the novelty to accurately represent and model two-dimensional information for real-life phenomena that are periodic in nature.

1. Introduction

In order to transact with several complicated problems involving uncertainties in many fields such as engineering, economics, social and medical sciences, classical methods are found to be inadequate. In 1965, Zadeh [1] presented fuzzy sets which helped to handle uncertainty and imprecision. Fuzzy sets had since been applied in many directions especially in decision making such as multi fuzzy sets [2], complex multi fuzzy sets [3,4,5,6,7], vague soft set [8,9,10,11], multiparameterized soft set [12], multi Q-fuzzy soft matrix [13] and intuitionistic fuzzy sets [14].

In fuzzy set theory, the grade of membership of an object to a fuzzy set indicates the belongingness degree of the object to the fuzzy set, which is a point (single) value selected from the unit interval [0, 1]. In real life scenarios, a person may consider that an element belongs to a fuzzy set, but it is possible that person is not sure about it. Therefore, hesitation or uncertainty may exist in which the element can belong to the fuzzy set or not. The traditional fuzzy set is unable to capture this type of hesitation or uncertainty using only the single membership degrees. A possible solution is to use an intuitionistic fuzzy set [14] or a vague set [15] to handle this problem. The vague set [15] is an extension of fuzzy sets and regarded as a special case of context-dependent fuzzy set which has the ability to overcome the problems faced when using fuzzy sets by providing us with an interval-based membership which clearly separates the evidence for and against an element.

From the above existing literature, we can see that those studies mainly focus on the fuzzy set, interval fuzzy set, vague set and their entropies [16,17,18]. Later on, Jun et al. [19] gave the idea of cubic set and it was characterized by interval valued fuzzy set and fuzzy set, which is a more general tool to capture uncertainty and vagueness, since fuzzy set deals with single-value membership while interval valued fuzzy set ranges the membership in the form of intervals. They presented the ideas of internal and external cubic sets and their characteristics. The hybrid platform provided by a cubic set has the main advantage since it contains more information than a fuzzy set and an interval-valued fuzzy set. By using this concept, different problems arising in several areas can be solved by means of cubic sets as in the works of Rashid et al. [20], Ma et al. [21], Khan et al. [22], Jun et al. [23,24], Gulistan et al. [25], Khaleed et al. [26], Fu et al. [27] and Ashraf et al. [28].

As for the Pythagorean fuzzy set (PFS) and its generalizations, an entropy measure was defined by Yang and Hussein [29]. Thao and Smarandache [30] proposed a new entropy measure for Pythagorean fuzzy which discarded the use of natural logarithm, while Wang and Li [31] introduced Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making.

Vague sets have a more powerful ability than fuzzy sets to process fuzzy information to some degree. Human cognition is usually a gradual process. As a result, how to characterize a vague concept and further measure its uncertainty becomes an interesting issue worth studying. Nevertheless, the concept of simple vague set is insufficient to provide the information about the occurrence of ratings or grades with accuracy because information is limited, and it is also unable to describe the occurrence of uncertainty and vagueness well enough, when sensitive cases are involved in decision making problems. Hence, there is a pertinent need for us to introduce the novel concept of cubic vague set (CVS) by incorporating both the ideas of cubic set and vague set. The aim of this model to introduce the notion of cubic vague set by extending the range of the truth-membership function and the false-membership function from a subinterval of [0, 1] to the interval-based membership structure that allows users to record their hesitancy in assigning membership values. This feature and its ability to represent two-dimensional information makes it ideal to be used to handle uncertain and subjective information that are prevalent in most time-periodic phenomena in the real world. These reasons served as the motivation to choose the cubic vague set model and use it in decision making problem.

The contribution of the novel cubic vague set (CVS) in the decision making process is its ability to handle uncertainties, imprecise and vagueness information considering both the truth-membership and falsity-membership values, whereas cubic set can only process the uncertainties information without able to take into account the truth-membership and falsity-membership values. The core advantage of using CVS against CS will be illustrated by an example. Hence, this concept of cubic vague set (CVS) will further enrich the use of various fuzzy methods in decision making such as those current trends which include group decision making using complex q-rung orthopair fuzzy Bonferroni mean [32], air pollution model using neutrosophic cubic Einstein averaging operators [33] and medicine preparation using neutrosophic bipolar fuzzy set [34].

The flow of our research is as follows. Firstly, we examine the concept of cubic vague set (CVS), which is a hybrid of vague set and cubic set. Secondly, we define some concepts related to the notion of CVS as well as some basic operations namely internal cubic vague sets (ICVSs) and external cubic vague sets (ECVSs). The CVS will be used together with a generalized algorithm to determine the similarity measures between two CVSs for a pattern recognition problem. Finally, a numerical example is given to elucidate that the proposed similarity measure of CVS is an important concept for measuring the entropy of uncertain information.

The organization of the paper will be as follows. Fundamentals of vague set, cubic set and interval-valued vague set are presented in Section 2. In Section 3, the concept of a cubic vague set with P- and R-union and P- and R-intersection for CVSs, with various properties are introduced. In Section 4, the similarity measure between CVSs is shown, along with an illustrative example studied, followed by the conclusion in Section 5.

2. Preliminaries

In this section we now state certain useful definitions, properties and several existing results for vague sets and cubic sets that will be useful for our discussion in the next sections.

The notion of vague set theory was first introduced by Gau and Buehrer in 1993 [15] as an extension of fuzzy sets. It is an improvement to deal with the vagueness of problems involving complex data with a high level of uncertainty and imprecision. Some of the basic concepts are as follows:

(See [

A truth membership function and

a false membership function where

Thus, the grade of membership of u in the vague set A is bounded by a sub interval The vague set A is written as where the interval

(See [

(See [

(See [

Jun et al. [19] introduced the concept of a cubic set, as a novel hybrid structure of an interval-valued fuzzy set (IVFS) and a fuzzy set.

(See [ be a cubic set in X in which A is an IVFS and λ is a fuzzy set in X.

We will now introduce the concept of the interval-valued vague set to handle uncertainty of information, the grade of membership and the negation of x.

(See [ where

3. Cubic Vague Sets

In this section, we will define the concept of a cubic vague set (CVS) and internal/external cubic vague sets.

Let X be a universal set. A cubic vague set where

Let and a VS

Then the cubic vague set

Let X be a universal set and V be a non-empty vague set. A cubic vague set

Let X be a universal set and V be a non-empty vague set. A cubic vague set

Let X be a universal set and V be a non-empty set. A cubic vague set

Let

Let

Straightforward. □

Let where

Suppose that is an ICVS and ECVS. By using Definition 8 and Definition 9, we get and . Thus, or , and so . □

Let

(Equality)

(P-order)

(R-order)

The complement of

Let

Let

Since is also an ICVS (resp. ECVS) in X, we have (resp. ) for all . That means (resp. ). Thus, is an ICVS (resp. ECVS) in X. □

Let

Since is an ICVS in X, we have for . That means and

Hence and are ICVSs in X. □

The following example shows that the P-union and P-intersection of two ECVSs need not be an ECVS.

Let

Note that

Note that

The example below shows that the R-union and intersection of two ICVSs need not be an ICVS.

Let

Note that

Note that

The example below will show that the R-union and intersection of two ECVSs may not necessarily be an ECVS.

Let

Let

We give a condition of a R-union of two ICVSs to become an ICVS.

Let . Then the R-union ofandis an ICVS in X.

Let and be two ICVSs in X which satisfy the condition of Definition 7. Then and which means . Now apply the condition of Definition 7 that is so that is an ICVS in X. □

We give a condition of a R-intersection of two ICVSs to become an ICVS.

Let

Let and be ICVSs in X which satisfy the condition of Definition 1. Then and so . Now apply the condition of Definition 1 we get and therefore is an ICVS in X. □

Given two CVSs and . Suppose we exchange the for in the two CVSs and we denote the CVSs and , respectively. Then, for to ECVSs and in X, two cubic vague sets and may not be ICVSs in X as shown in the example below.

Let

Let

We give an example to show that the P-union of two ECVSs in X does not necessarily become an ICVS in X.

Let

We give a condition for P-union of two ECVSs to become an ICVS.

For two ECVSs

Let and be an ECVSs in X such that and are ICVSs in X. Then , , and for all . Now, for a given , we consider the cases:

and .

and .

and .

and .

We will illustrate the proof of the first case only because proofs of the remaining three cases are similar. Now, we get . Since and are ICVSs in X, we have and . It follows that

Hence is an ICVS in X. □

We give the condition of a P-intersection of two ECVSs to become an ICVS.

Let

The proof is similar to that of Theorem 7. □

For two ECVSs and in X, two CVSs and may not be ECVSs as shown in the following example.

Let

Let

For each , we get , , and which means . Then is an ECVS in X. □

We have given an example that shows the P-intersection for two ECVSs may not become an ECVS as in Example 3. Now we will add a condition for the P-itersection of two ECVSs to be an ECVS by using Definition 2.

Let

For each , substitute and

Then which is one of the and . We consider or only since the proof of the other cases are similar.

If , thus and so . Then thus .

If , thus and so . Suppose that, . Thus, then we get or of the case . This is the contradiction to and are ECVSs in X. For the case we get because .

Suppose that, . Thus, then we get or of the case . This is the contradiction to and are ECVSs in X. For the case we get because . Then P-intersection of and are ECVSs in X. □

We add a condition of a P-intersection of two CVSs to become both an ECVS and ICVS.

Let

For each , substitute and

Then which is one of the and . We take or only.

If , thus and so . This implies that . Thus, implies that . Thus, and .

If , thus, and so . Then and . Therefore, the P-intersection of and is an ECVS and ICVS in X. □

We provide the condition of a P-union of two ECVSs to become an ECVS.

Let

For each , substitute and

Then is one of the and . We consider or only.

If , thus, and so . This implies that hence,

If , thus and so . Suppose that We have, and or

That is a contradiction for that fact and are ECVSs in X in the first case. For the next case, we will show that because Suppose We have, which means or

It contradicts , for the fact and are ECVSs in X. In the case we get since . Thus, a P-union of and is an ECVS in X. □

Let

For each , substitute and

Then is one of the and . Consider the case of or .

If , thus, and . Then the first part of inequality and

If , then and . Suppose . Thus, which implies that or

For the case , it contradicts the fact that and are ECVSs in X. For the case we have since .

Suppose We have,

Hence, or

For the case , it is a contradiction since and are ECVSs in X. For the case we notice that since . Hence the R-union of and is an ECVS in X. □

For the R-intersection we provide the condition of two ECVSs to be an ECVS.

Let

The proof is similar to that of Theorem 13. □

For R-intersection we provide the condition of two CVSs to be both an ECVS and ICVS.

Let

The proof is similar to that of Theorem 11. □

For the R-union we provide the condition of two ICVSs to be an ECVS.

Let

Straightforward. □

For the R-intersection we provide the condition of two ICVSs to be an ECVS.

Let

Straightforward. □

For the R-union we provide the condition of two ECVSs to be an ICVS.

Let

Straightforward. □

4. Similarity Measure of Cubic Vague Sets

The most important mathematical tool for solving problems in pattern recognition and clustering analysis is similarity measure. Therefore, in this section we will propose the similarity measures between two CVSs, which will then be applied to a pattern recognition problem.

A real valued function

;

;

⟺

and , if , then and

Next, we give the similarity measurement between two CVSs.

Let

Application of the Similarity Measurement Method in a Pattern Recognition Problem

The measures of fuzzy sets and their hybrid methods help us to solve problems in many real-life areas, especially in the field of pattern recognition and image processing, among others. In this section, we will examine the similarity measures of two CVSs of a pattern recognition problem. We construct an algorithm, and suppose that is the ideal pattern.

Firstly, we construct an ideal VCS .

Then, we construct cubic vague sets , , on X for a sample patterns which are under consideration.

The similarity measures between the sample patterns , and ideal pattern are calculated using the formula given in Definition 14.

If then the pattern is to be recognized to belong to the ideal Pattern and if then the pattern is not to be recognized for an ideal Pattern .

Now, we provide a numerical application example for similarity measurement of two CVSs in a pattern recognition problem. Through the example, we will illustrate which one of the sample patterns belongs to the ideal pattern.

Consider a simple pattern recognition problem involving three sample patterns and an ideal pattern. The objective of the problem is to determine which one of the three sample patterns belongs to the ideal pattern. Let

[Step 1.] Construct an ideal CVS

[Step 2.] Construct CVSs

[Step 3.] Calculate the degree of similarity S between the three sample patterns

[Step 4.] Since

To show the advantage of our proposed method using CVS as compared to that of a cubic set [19], let us consider the decision making problem above. It can be seen that cubic set is unable to describe this problem, since it fails to capture the false membership portion of the data in assessing the alternative in the decision-making process.

Note that the CVS is a generalization of a cubic set by adding the concept of vague set to the definition of cubic set. Thus, as shown in the decision making problem above, the CVS has the ability to handle uncertainties, imprecise and vagueness information considering both the truth-membership and falsity-membership values, whereas cubic set can only handle the uncertainties information without taking into account the truth-membership and falsity-membership values. This indeed illustrates the core advantage of CVS against that of CS.

5. Conclusions

A new concept of a cubic set namely the cubic vague set is introduced by incorporating the features of a vague set and a cubic set. Several properties and theorems of cubic vague set are defined and proven involving ECVS or ICVS. We have derived different conditions for different operations of two ICVSs (ECVSs) to be an ICVS (ECVS). We have shown that the proposed set and corresponding algorithm can be applied to a decision making problem containing uncertainties. Our future research is finding ways to apply cubic vague set to groups, rings, numerical analysis [35,36,37] and more real life applications.

Table 1

Cubic vague set .

X	A_V	λ_V
a	[0.1,0.3],[0.3,0.7]	(0.5, 0.7)
b	[0.3,0.4],[0.5,0.6]	(0.1, 0.3)

Table 2

VCSs and .

X	A_V(x)	λ_V(x)	X	B_V(x)	ν_V(x)
l	〈[0.2,0.5],[0.5,0.5]〉	(0.1, 0.7)	l	〈[0.8,0.9],[0.8,0.8]〉	(0.9, 0.9)
m	〈[0.3,0.4],[0.3,0.3]〉	(0.7, 0.8)	m	〈[0.3,0.4],[0.5,0.6]〉	(0.8, 0.9)

Table 3

VCSs and .

X	A_V(x)	λ_V(x)	X	B_V(x)	ν_V(x)
k	〈[0.3,0.5],[0.2,0.2]〉	(0.5, 0.7)	k	〈[0.6,0.7],[0.4,0.7]〉	(0.35, 0.45)
l	〈[0.2,0.4],[0.1,0.1]〉	(0.1, 0.5)	l	〈[0,0.6],[0.7,0.8]〉	(0.3, 0.35)
m	〈[0.1,0.1],[0.1,0.3]〉	(0.4, 0.6)	m	〈[0.1,0.1],[0.7,0.8]〉	(0.2, 0.9)

Table 4

CVSs and .

X	A_V(x)	λ_V(x)	X	B_V(x)	ν_V(x)
l	〈[0.2,0.4],[0.4,0.5]〉	(0.6, 0.7)	l	〈[0.6,0.8],[0.7,0.8]〉	(0.2, 0.3)
m	〈[0.3,0.3],[0.4,0.5]〉	(0.2, 0.2)	m	〈[0.2,0.2],[0.1,0.3]〉	(0.4, 0.5)