Evaluation of the Economic Relationships on the Basis of Statistical Decision-Making in Complex Neutrosophic Environment

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. )is 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. )e real part of complex-valued degrees represents the amplitude term, while the imaginary part represents the phase term. )is 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.


Introduction
In mathematics, the word modeling refers to the process of representing real-world events in a mathematical form.
ere 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 (m) that attains real values from 0 to 1, like probability. Atanassov [2] developed the idea of intuitionistic fuzzy sets (IFSs) that also model fuzziness. e advancement in IFSs as compared to FSs is that IFSs discuss the degree of membership (m) as well the degree of nonmembership (n) 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 m � 0.8 and n � 0.5 because their sum exceeds 1. is 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 (m) and the degree of nonmembership (n) that are fuzzy numbers, provided that the sum m 2 + n 2 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 m � 0.9 and n � 0.7, then 0.9 2 + 0.7 2 � 0.81 + 0.49≰1. 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 m and n are fuzzy numbers and 0 ≤ m q + n q ≤ 1, where q is a positive integer. For q � 1 and q � 2, 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 (m), the degree of abstinence (a), and the degree of nonmembership (n). According to the NSs, the condition on the sum of the degrees is that 0 ≤ m + a + n ≤ 3.
is 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][17][18][19][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 decisionmaking, 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 (m C ). e degree of membership (m C ) acquires values from the unit circle in a complex plane. For an object n, the degree of membership is defined as m C (x) � τ C (x)e ρ C (x)2πi , where τ C and ρ C are fuzzy numbers and are known as the amplitude term and the phase term, respectively. e preeminence of CFSs over FSs is that CFSs are capable of modeling multidimensional problems. e 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 m C and nonmembership n C . 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. ese 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 (m C ), the degree of nonmembership (n C ), and the sum (m C ) q + (n C ) q 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 (m C ), degree of abstinence (a C ), and degree of nonmembership (n C ), such that each of these degrees is a fuzzy number, and their sum is restricted as 0 ≤ |m C | + |a C | + |n C | ≤ 3. Note that every complex-valued degree consists of two terms. Each of these terms is a fuzzy number representing two different entities. e 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 2 Complexity 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 prioritygenerating method of qROF preference relation. is 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. e 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. e 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.
is 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.
and ρ C (x) are known as the amplitude and phase terms, respectively.
e Cartesian product of two CFSs that is defined as Or equivalently
Note 1. For n � 1 and n � 2, the qROFS converts to an IFS and a Pythagorean fuzzy set (PFS), respectively.
Definition 8 (see [29]   ⟶ z n |z n ∈ C, |z n | ≤ 1 known as the degrees of membership and nonmembership, respectively, of Note 2. For n � 1 and n � 2, the CqROFS converts to n CIFS and a complex PFS (CPFS), respectively.

Main Results
is 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. 4 Complexity e Cartesian product of two CNSs ere is a condition that Complexity e Cartesian product of A And the CNR R is defined as Example 3. For a CNR R ...

Complexity
While on the other hand a CN irreflexive relation R ...

is said to be CN asymmetric relation if
If (x, y), m R              Complexity

⇒R
e CN composite relation is is a CN transitive relation, then for     ,

on (12) is
And a CN-order relation R  . (40) Now suppose that As R ...
is an equivalence relation and possesses the properties of CN symmetric relation and CN transitive relation, (41) and (42) imply that But according to CN composite relation, is also a CN order relation if the three properties for a CNorder relation hold: is also a CN reflexive relation: is also a CN antisymmetric relation. So is also a CN transitive relation. So Hence, R  is denoted and defined as Example 9. For a CN equivalence relation, e CN equivalence classes are Conversely, suppose en Complexity Hence, Similarly, suppose Since, is a CN symmetric relation. Now (62) Hence, (58) and (63) imply that R

Applications
is section presents application of the proposed notions. It shows the worth of CNRs in decision-making for a statistician. e application uses the concept of CNRs in economic statistics and talks about the economic relationships. Figure 1 depicts the economic factors whose relationships are discussed on the basis of statistical analysis in the following application.

Investigating the Economic Relationships Using Economic
Statistics. Economics conveys the way businesses, governments, societies, and individuals allocate their wealth and resources. It also provides knowledge for making everyday decisions. e international financial affairs are concerned with the economics. e policies made by the financial ministers are thoroughly worked out before being applied because they can make or break the progress and economy of a nation. ese economic policies are not just a shot in the dark but are properly calculated outcomes. ese calculations are carried out by statisticians, who collect and organize the data, present data in the form of tables, diagrams, and graphs and then finally analyze and interpret the data.
is process involves various statistical techniques at each stage, and Figure 2 summarizes each step.
An expert statistician carries out statistical analysis and makes some important decisions which are then implemented as the financial policies. e data at state levels are humongous and thus the probability of ambiguity, uncertainty, and imprecision is greater. So, CNSs and CNRs are worthwhile tools to cope with such kind of data and information. It will further improve the accuracy of the conclusions and thus help in making better decisions.
Let us consider a situation involving three economic factors, namely, investment (I), gross domestic product (GDP), and unemployment (U). A decision-making statistician assigns each of the factors a degree of membership, a degree of abstinence, and a degree of nonmembership after going through the statistical process. In current supposition, the results of the experiment are assumed to be applicable for twelve months or a year at most. Let

Complexity 13
Note that an ordered pair tells the impact of the first factor on the second one. Moreover, the degree of membership indicates the rate of influence. e degree of abstinence is taken to mean as the inability to decide that whether there is an influence of one factor on the other or not. And the degree of nonmembership indicates the rate of no influence. e amplitude term of each degree shows the strength of the impacts, and the phase terms refer to the time span in months. For example, the event ((I, U), 0.4e (0.75)12πi , 0.2e (0.25)12πi , 0.2e (0.25)12πi ) expresses the impacts of investment I on unemployment U. e numbers are translated in words as (i) e degree of membership indicates the rate of influence of I on U. In 0.4e (0.75)12πi , 0.4 is the amplitude terms, and (0.75)12 is the phase term. e translation is that influence of I on U is of degree 0.4 over the time span of 9 months, that is, very strong influence for a long duration.
(ii) e degree of abstinence 0.2e (0.25)12πi interprets the inability to determine the existence of any influence is 0.2 over the time span of 3 months. (iii) e degree of nonmembership 0.2e (0.25)12πi indicates that the degree of no influence is 0.2 over the period of 3 months.
Ignoring the CN reflexive relations and CN symmetric relations in (65), Table 1 shows the summary of all the impacts of one factor on the other. Now an illustration of types of CNRs for a decision maker is proposed. It can reduce the efforts of a statistician in decision-making. Consider some other two CNSs such that set B ...
Details of each of the relationships is given in Table 2.  is completely discussed through Table 3. Figure 3 portrays the impacts of all six factors on the investment I or viceversa. e vertical y-axis represents the reading of phase terms when translated to months. e horizontal x-axis is labeled with numbers 1 to 6, and each number represents a different factor that are mentioned just beneath the horizontal axis in the graph. Moreover, at each factor, there are three circles that represent the three degrees; green circles represent degree of membership, blue circles CIFS because the sum of phase terms of both the degrees of membership and nonmembership exceed 1, i.e., 1 + 0.16≰1 and 1 + 0.8≰1. Henceforth, NFSs and NFRs are superior tools to be applied in the above discussed scenario.

Conclusion
e main objective of this study was the introduction of the notion of CNRs which is a new concept.
e CNSs and CNRs are characterized by three complex-valued mappings whose values range in the unit circle in a complex plane. e three mapping are the degrees of membership, abstinence, and nonmembership. Moreover, each degree consists of two different terms that are described by the real and imaginary parts of the complex numbers. e real part is called the amplitude term which represents the degree or level of membership, abstinence, or nonmembership, while the imaginary part is called the phase term that represents the time lag, periodicity, or phase changes in the degrees of membership, abstinence, or nonmembership. Furthermore, the types of CNRs are also introduced with suitable examples. In addition, the properties of CNRs are derived along with some useful results. An economic policy comes through many statistical procedures, and ultimately a decision is made on the basis of statistical analysis and outcomes. Henceforth, an application has been presented which emphasizes the utility of the proposed method in decisionmaking processes. Finally, a comparative analysis has been carried out among the existing methods and the proposed method.

Data Availability
No data were used to support the study.   16 Complexity