#### Abstract

Fuzzy sets and fuzzy logics are used to model events with imprecise, incomplete, and uncertain information. Researchers have developed numerous methods and techniques to cope with fuzziness or uncertainty. This research intends to introduce the novel concepts of complex neutrosophic relations (CNRs) and its types based on the idea of complex neutrosophic sets (CNSs). In addition, these concepts are supported by suitable examples. A CNR discusses the quality of a relationship using the degree of membership, the degree of abstinence, and the degree of nonmembership. Each of these degrees is a complex number from the unit circle in a complex plane. The real part of complex-valued degrees represents the amplitude term, while the imaginary part represents the phase term. This property empowers CNRs to model multidimensional variables. Moreover, some interesting properties and useful results have also been proved. Furthermore, the practicality of the proposed concepts is verified by an application, which discusses the use of the proposed concepts in statistical decision-making. Additionally, a comparative analysis between the novel concepts of CNRs and the existing methods is carried out.

#### 1. Introduction

In mathematics, the word modeling refers to the process of representing real-world events in a mathematical form. There are many ways to express the practical happenings in the mathematical form which depend on the nature of the problem. In real life, there are many occasions when one faces uncertainty, vagueness, and ambiguity. Fuzzy sets and logics introduced by Zadeh [1] are proved to be great tools at dealing with problems that involve doubts, vagueness, and imprecise information. Fuzzy sets (FSs) are characterized by a mapping called the degree of membership that attains real values from 0 to 1, like probability. Atanassov [2] developed the idea of intuitionistic fuzzy sets (IFSs) that also model fuzziness. The advancement in IFSs as compared to FSs is that IFSs discuss the degree of membership as well the degree of nonmembership of the events. Both degrees attain values from the unit interval provided that their sum is contained within the unit interval. Due to this constraint on the sum, a decision maker is bounded and limited in assigning the values to degrees of membership and nonmembership. For instance, a decision maker cannot assign and because their sum exceeds 1. This limitation affects the precision of the results. Henceforth, Yager [3] provided the notion of Pythagorean fuzzy sets (PFSs). A PFS is a generalization of IFS and FS that eases the constraints in IFS. Like IFSs, the PFSs also discuss the degree of membership and the degree of nonmembership that are fuzzy numbers, provided that the sum belongs to the unit interval. Although PFS provides a broader range of fuzzy numbers to be assigned as degrees as compared to an IFS, there are instances when someone needs to set both the degrees higher enough so that the sum disobeys the restrictions of PFSs. For example if and , then . Keeping this in mind, Yager [4] generalized IFSs and PFSs to devise the notion of q-rung orthopair fuzzy sets (qROFSs). A qROFS is one of the most powerful tools to tackle fuzziness when discussing the degree of membership and the degree of nonmembership. According to qROFSs, both the degrees and are fuzzy numbers and , where is a positive integer. For and , the qROFS transforms to an IFS and PFS, respectively. Garg [5] presented applications of PFSs in multiattribute decision-making process. Yang and Hussain [6, 7] introduced fuzzy entropy, distance, and similarity measures of PFSs with applications to multicriteria decision-making. Zhou et al. [8] also introduced divergence measure of PFSs and applied them in medical diagnosis. Yang et al. [9] gave the idea of belief and plausibility measures on IFSs with construction of belief-plausibility TOPSIS. Using the characteristic objects method, Faizi et al. [10] proposed IFSs in multicriteria group decision-making problems. Peng and Liu [11] devised information measures for qROFSs. Wei et al. [12] initiated the concept of qROF Heronian mean operators in multiple attribute decision-making, and Liu et al. [13] developed some cosine similarity measures and distance measures between q‐rung orthopair fuzzy sets.

Later, Smarandache [14] introduced neutrosophic sets (NSs) that are the generalization of FSs. In an NS, there are three independent fuzzy-valued mappings, i.e., the degree of membership , the degree of abstinence , and the degree of nonmembership . According to the NSs, the condition on the sum of the degrees is that . This theory permits the decision makers to freely assign any fuzzy value to an object as its degrees of membership, abstinence, and nonmembership. Wang et al. [15] devised single-valued NSs (SVNSs), Smarandache [16–20] scrupulously researched the NSs and provided several generalizations of NSs, Salama and Alblowi [21] worked on NS and neutrosophic topological spaces, Das et al. [22] applied the NS in decision-making, Khalil et al. [23] gave the combination of the SVNSs and their application in decision-making, and Sahin and Liu [24] presented the correlation coefficient of SVN hesitant FSs and applied them in decision-making. Hashim et al. [25] defined and applied the concept of neutrosophic bipolar fuzzy set in the preparation of medicines.

An idea of involving the complex numbers in the FS theory lead to the development of a new idea; complex FS (CFS) which was concocted by Ramot et al. [26]. A CFS is characterized by a complex-valued mapping, called the degree of membership . The degree of membership acquires values from the unit circle in a complex plane. For an object , the degree of membership is defined as , where and are fuzzy numbers and are known as the amplitude term and the phase term, respectively. The preeminence of CFSs over FSs is that CFSs are capable of modeling multidimensional problems. The phase term usually refers to time. Alkouri et al. [27] presented the concept of complex IFSs (CIFSs) that characterizes an object with a pair of complex-valued mappings, i.e., degrees of membership and nonmembership . Both the degrees belong to the unit circle in a complex plane and so does their sum. Equivalently, the amplitude and phase terms of both the degrees, the sum of amplitude terms, and the sum of phase terms are all fuzzy numbers. Moreover, Ullah et al. [28] introduced the concept of complex PFS (CPFS) that discusses the degree of membership and nonmembership. These degrees are complex numbers from a unit circle in complex plane provided that the sum of their squares is also a complex number in a unit circle. Furthermore, the CIFSs and CPFSs were generalized to complex qROFSs (CqROFSs) by Liu et al. [29] by updating the constraints on the sum of the degrees of membership and nonmembership. According to CqROFSs, the degree of membership , the degree of nonmembership , and the sum lie in a unit circle in a complex plane. Bi et al. [30] defined CF arithmetic aggregation operators, and Tamir et al. [31] presented an overview of theory and applications of CFSs and CF logic. Also, Tamir and Kandel [32] presented the axiomatic theory of CF logic and classes. Ma et al. [33] proposed the method of applying CFSs in multiple periodic factor prediction problems. Ngan et al. [34] generalized the CIFSs by space of quaternion numbers, Garg and Rani [35] offered the coefficient measure of CIFSs and their applications in decision-making, and Rani and Garg [36] introduced the CIF power aggregation operators and applied them in decision-making. Ali and Mahmood [37] gave the idea of Maclaurin symmetric mean operators for CqROFSs and presented their applications. Liu et al. [38] extended the prioritized weighted aggregation operators for decision-making under CqROFSs.

In addition, complex NSs (CNSs) were proposed by Ali and Smarandache [39]. A CNS is characterized by three complex-valued mapping, i.e., degree of membership , degree of abstinence , and degree of nonmembership , such that each of these degrees is a fuzzy number, and their sum is restricted as . Note that every complex-valued degree consists of two terms. Each of these terms is a fuzzy number representing two different entities. The advantage of CNSs over other CFSs and its generalizations is that CNSs discuss three independent degrees instead of two. Furthermore, it provides much more freedom to a decision maker because he/she can choose independently any value for each degree from [0, 1]. Broumi et al. [40] discussed the bipolar CNSs with applications. Furthermore, Gulistan et al. [41] introduced the CN subsemigroups and ideals. Ali and Mahmood [42], Al-Quran and Hassan [43], Manna et al. [44], and Dat et al. [45] applied the CNSs for decision-making, and Singh [46] used CNSs to analyze the air quality.

Klir and Folger [47] presented the concept of crisp relations (CRs) that are based on the crisp set theory. CRs describe the existence of a relationship between some events. Mendel [48] gave the concept of fuzzy relations (FRs), which are the extension of CRs. Like its predecessor, FRs also describe the existence of the relationship among the objects, but in addition, FRs also indicate the strength of the relationship by the degree of membership. If the value of degree of membership is nearer to 0, then it means the relationship is weak, and the value closer to 1 indicates the stronger relationship. For instance, a relationship with the degree of membership 0.5 is weaker than the relationship with the degree 0.6. Moreover, the notion of intuitionistic FRs (IFRs) was introduced by Burillo and Bustince [49]. IFRs describe the quality of relationship by degree of membership and degree of nonmembership, provided that their sum does not exceed 1. Ramot et al. [26] devised the notion of complex FR (CFR) which discusses the complex-valued degree of membership. Ejegwa [50] improved the composition relation for PFSs and applied the concept in medical diagnosis. Ramot et al. [51] worked on CF logic. Hu et al. [52] discovered the distances of CFSs and continuity of CF operations. Deschrijver and Kerre [53] worked on the composition of IFRs, Bustince and Burillo [54] studied the structures of the IFRs, Li et al. [55] proposed some preference relations based on qROFSs, and Zhang et al. [56] offered the concepts of additive consistency‐based priority‐generating method of qROF preference relation.

This paper aims to introduce the notion of complex NRs (CNRs) and its types such as inverse CNR, CN reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, composite, equivalence, order relations, and CN equivalence class. Besides these, some interesting properties and useful results have also been proved. Since CNRs carry three degrees, i.e., the degrees of membership, abstinence, and nonmembership, they define the quality of a relationship much efficiently. The complex degrees consist of two parts that are the amplitude and phase terms as discussed earlier, so CNRs are capable of describing the problems with time periods, phase changes or multidimensions. An application is also presented to illuminate the practicality of the proposed concepts. The application discusses the worth of the proposed work for a statistician who is supposed to make the decision for the economic policy. In the process of policy making, the data are collected, organized, and analyzed through statistical techniques such as percentages, averages, frequencies, and probabilities and then presented in the form of tables and graphs, and finally the interpretation of the information takes place. On large scales, the information is probably ambiguous, uncertain, or unclear that certainly affects the final decision. In order to cope with such issues, this study proposes a new method in the application.

This paper is organized such that Section 2 defines some fundamental concepts. Section 3 proposes the main objectives and results of the study. Application of CNSs and CNRs in investigating the economic relationships through statistical decision-making is presented in Section 4. Section 5 is named comparative analysis which compares the proposed work with the existing methods. Finally, the paper ends with a conclusion.

#### 2. Preliminaries

This section defines some fundamental concepts such as FSs, CFSs, Cartesian product of CFSs, CFRs, IFSs, CIFSs, PFSs, CPFSs, qROFSs, CqROFSs, NSs, and CNSs.

*Definition 1 (see [1]). *A fuzzy set (FS) of a referential set is characterized by a real-valued function known as the degree of membership of .

*Definition 2 (see [26]). *A complex FS (CFS) of a referential set is characterized by a complex-valued function known as the degree of membership of , where and . and are known as the amplitude and phase terms, respectively.

*Definition 3 (see [26]). *The Cartesian product of two CFSs and , , is given byThe function symbolizes the degree of membership of the Cartesian product that is defined asOr equivalently

*Definition 4 (see [26]). *A complex fuzzy relation (CFR) denoted by is any nonempty subset of , where and are CFSs.

*Example 1. *For given CFS,

The Cartesian product of to itself isand the CFR is

*Definition 5 (see [2]). *An intuitionistic FS (IFS) of a referential set is characterized by a pair of real-valued functions known as the degrees of membership and nonmembership, respectively, of as long as .

*Definition 6 (see [27]). *A complex IFS (CIFS) of a referential set is characterized by a pair of complex-valued functions and known as the degrees of membership and nonmembership, respectively, of where and . A CIFS has the condition that or equivalently, , ., are known as the amplitude terms, and , are known as the phase terms.

*Definition 7 (see [4]). *A q-rung orthopair FS (qROFS) of a referential set is characterized by a pair of real-valued functions known as the degrees of membership and nonmembership, respectively, of as long as , where is any natural number.

*Note 1. *For and , the qROFS converts to an IFS and a Pythagorean fuzzy set (PFS), respectively.

*Definition 8 (see [29]). *A complex qROFS (CqROFS) of a referential set is characterized by a pair of complex-valued functions and known as the degrees of membership and nonmembership, respectively, of where and . A CqROFS has the condition that or, equivalently, , , where is any natural number. , are known as the amplitude terms, and , are known as the phase terms.

*Note 2. *For and , the CqROFS converts to *n* CIFS and a complex PFS (CPFS), respectively.

*Definition 9 (see [14]). *A neutrosophic set (NS) of a referential set is characterized by three real-valued functions known as the degrees of membership, abstinence, and nonmembership, respectively, of as long as .

*Definition 10 (see [39]). *A complex NS (CNS) of a referential set is characterized by three complex-valued functions , , and known as the degrees of membership, abstinence, and nonmembership, respectively, of where , , and . A CNFS has the condition that or, equivalently, , . , , are known as the amplitude terms, and , , are known as the phase terms.

#### 3. Main Results

This section aims to define some new concepts in CNSs, like Cartesian product of CNSs and the CNRs. Moreover, types of CNRs are also introduced with examples. Furthermore, some interesting results and properties of these CNRs are obtained.

*Definition 11. *The Cartesian product of two CNSs and , , is given byThe functions , , and symbolize the degrees of membership, abstinence, andnonmembership of the Cartesian product . These functions are defined asThere is a condition thatOr equivalentlyThe conditions are and , where

*Definition 12. *A complex neutrosophic relation (CNR) denoted by is any nonempty subset of , where and are CNSs.

*Example 2. *For a given CNS,The Cartesian product of to itself isAnd the CNR is

*Definition 13. *The inverse CNR of a CNFRis defined as

*Example 3. *For a CNR from (12),The inverse CNR is given by

*Definition 14. *A CNR is said to be a CN reflexive relation ifWhile on the other hand a CN irreflexive relation implies

*Example 4. *Using (12), the CN reflexive relation and the CN irreflexive relation are

*Definition 15. *A CNR is said to be CN symmetric relation ifA CNR is said to be CN asymmetric relation ifIfthen is a CN antisymmetric relation.

*Example 5. *Using (12), the CN symmetric relation , the CN asymmetric relation , and the CN antisymmetric relation are

Theorem 1. *A CNR is a symmetric relation .*

*Proof. *Suppose that , thenBut .Hence, is a CN symmetric relation.

Conversely, suppose that is a CN symmetric relation, then forBut

Theorem 2. *For CN symmetric relations and , the intersection is also a CN symmetric relation.*

*Proof. *Suppose that and are two CN symmetric relations on a CNS . Then according to the definition of CNR, and is CNR on .

Now, suppose thatSince and are CN symmetric relations,

*Definition 16. *A CNR is said to be CN transitive relation if

*Example 6. *A CN transitive relation on (12) is

*Definition 17. *A CN composite relation combines the CNRs and such that for and ,

*Example 7. *For CNRs and ,The CN composite relation is

Theorem 3. *A CNR is a transitive relation .*

*Proof. *Suppose that is a CN transitive relation, then for and ,But . Hence .

Conversely, suppose that , then for and ,But

Thus, is a CN transitive relation.

*Definition 18. *If a CNR satisfies the conditions of CN reflexive relation, CN symmetric relation, and CN transitive relation, then is called a CN equivalence relation.

While a CNR satisfying the conditions of CN reflexive relation, CN antisymmetric relation and CN transitive relation are called a CN-order relation.

*Example 8. *A CN equivalence relation on (12) is