# Types of Complex Fuzzy Relations with Applications in Future Commission Market

Authors: Madad Khan; Muhammad Zeeshan; Seok-Zun Song; Sohail Iqbal
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce types of relations on complex fuzzy sets such as the complex fuzzy (CF) inverse relation, complex fuzzy reflexive relation, complex fuzzy symmetric relation, complex fuzzy antisymmetric relation, complex fuzzy transitive relation, complex fuzzy irreflexive relation, complex fuzzy asymmetric relation, complex fuzzy equivalence relation, and complex fuzzy-order relation. We study some basic results and particular examples of these relations. Moreover, we discuss the applications of complex fuzzy relations in Future Commission Market (FCM). We show that the introduction of CF relations to applications of FCMs can give a significant method for describing the temporal dependence between parameters of a Future Commission Market.


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## Body

## 1. Introduction

Models reflecting the phenomena of real life with just choices of truth and falsehood are not enough to reflect the true reality of the problems. The explanation for this is that the models have many complications, which is why a framework needs to be built to deal with the models’ ill-defined situations. There are now two ways to deal with these kinds of situations, one is to find the problems’ numerical solutions and the other is to create a numerical model. We get numerical solutions to the problems in both cases. The second is about the fuzzy set theory, which includes the theory of probability, the theory of fuzzy soft sets, the theory of intuitionist fuzzy sets, and most specifically, the theory of neutrosophic sets. The later theory for dealing with problems involving complexities is more generalized. One of the acceptable examples of these theories is the fuzzy differential equations theory, which is more generalized than the differential equations to solve problems of everyday life with greater precision.Zadeh [1] gave the description of a fuzzy set (FS) in 1965, which is similar to a probability function. For models of real-world problems in different branches of science, a fuzzy set plays a vital role. Fuzzy set theory has many applications in operational research, decision making, medicine, engineering design, psychology, quantum physics, image processing, mathematical chemistry, biological classification, thermodynamics, economics, and nonequilibrium. Dubois et al. in [2] discussed the applications of fuzzy sets in approximate reasoning and information systems, pattern recognition and image processing, decision analysis, operation research and statistics, and modeling and control of systems. Ngan et al. in [3] provided two numerical examples of applying the complex t-norm and t-conorm to multicriteria decision making in the context of medicine-related problems using medical datasets. Nisren et al. introduced the concept of complex multifuzzy soft expert set (CMFSES) and discussed the application of a complex multifuzzy expert soft set in decision-making problems [4]. Poodeh studied and evaluated a randomized-learning approach to train this neurofuzzy system and proposed a machine-learning algorithm, which is designed for fast training of a compact, accurate forecasting model [5]. Singh in [6] introduced a method to provide an effective way to analyze the uncertainty and vagueness in a complex (or dynamic) dataset using a complex vague concept lattice. Xindong et al. in [7] discussed the relationship between the distance measure, the similarity measure, the entropy, and the inclusion measure for Pythagorean fuzzy sets. They showed the efficiency of the proposed similarity measure in pattern recognition, clustering analysis, and medical diagnosis. Moreover, Xindong et al. in [8] studied deeper insights into the decision-making problem based on the interval-valued fuzzy soft set. Xindong et al. provided two novel algorithms in decision-making problems under a Pythagorean fuzzy environment [9]. Naz et al. developed a new decision-making approach based on graph theory to deal with the multiattribute decision-making problems. They utilized the numerical examples concerning the energy project selection and software evaluation to show the detailed implementation procedure and reliability of our method in solving multiattribute decision-making problems under hesitant fuzzy, interval-valued hesitant fuzzy, and a hesitant triangular fuzzy environment [10].Ramot et al. in [11] first gave the concept of a complex fuzzy set (CFS). The generalization of a real number set introduced by Gauss in 1795 is the complex number set. Accordingly, a CFS is the extension of a fuzzy set, the range of which extends from a closed interval [0, 1] to a disc of radius one in a complex plane. The membership function of CFS C is denoted as λCu and defined on the universal U as for any u∈U a complex value in the disc of radius one in a complex plane. Thus, all values of λCu exist inside a circle of radius one in a complex plane and λCu=aCueipCu, where i=−1. The term pCu is said to be phase term, aCu is said to be an amplitude term, and both of these are real valued with aCu∈0,1. The CFS C is represented as u,λCu|u∈U.Imprecise, inconsistent, and incomplete knowledge of the periodic nature cannot be treated by fuzzy sets and intuitionistic fuzzy sets. These theories refer to various fields of research, but in both sets, there is one significant weakness, that is, a lack of capacity to discuss two-dimensional phenomena. Ramot presented a complex fuzzy set to address this challenge. The phase term of the CFS plays a crucial role in defining the functionality of the complex fuzzy set model. This term differentiates a model of the CFS from all other models available in the literature. The ability of a complex fuzzy set to depict two-dimensional phenomena makes it superior to the handling of vague and intuitive details prevalent in time-periodic phenomena. Complex fuzzy sets, their classes, and logic play an important role in applications including periodic event prediction and advanced control systems. A complex fuzzy set is somewhat similar to a Fourier transform; in reality, it is the particular form of the Fourier transformation by limiting the range of the Fourier transformation to a complex disc unit. Fourier transform has a lot of applications in various fields such as in signals and systems, communication, astronomy, geology, and optics. A complex fuzzy set can also be used in models such as the Fourier transform. Several other real-life phenomena are vague and cannot be modeled using one-dimensional variables. For example, objects can be represented as a collection of measurements in pattern recognition and are seen as vectors in a multidimensional space. These multidimensional variables cannot be expressed through a simple combination of variables, particularly the consideration of fuzzy sets. These types of sets can be expressed via complex classes. For periodic phenomena, a complex fuzzy set is very useful. Ramot et al. proposed that the intermittent problems or repeated-problem phenomenon be more precisely modeled using the phase term of the complex fuzzy set membership, such as describing the effect of two countries’ financial measures on each other over time. He suggested that signal processing is yet another attractive area of operation for a complex fuzzy set. Xueling et al. in [12] proposed the model for identifying the reference signal out of largely interested signals by using complex fuzzy sets. In addition, it is used to convey solar activity (solar maximum and solar minimum) by means of the average sunspot number [11]. Dick suggested that one of the beneficial applications of complex fuzzy sets is to use it to represent relatively periodic behavioral phenomena [13]. Traffic congestions in a big city are aperiodic phenomena that never repeat themselves. Complex fuzzy logic can also be used to solve those forms of issues more easily and reliably than fuzzy logic. Akram et al. [14] introduced the concept of competition graphs under a complex fuzzy environment. They described an application in the ecosystem. Moreover, Akram et al. [15] discussed the complex Pythagorean Dombi fuzzy graph (CPDFG). They utilized CPDFAA and CPDFGA operators in solving a decision-making numerical example. Akram et al. [16] gave the notion of the complex Pythagorean fuzzy planar graph (CPFPG), and an extension of a Pythagorean fuzzy planar graph is presented to study the planarity. Moreover, Akram et al. proposed a new graph, a complex Pythagorean fuzzy competition graph by combining the complex Pythagorean fuzzy information with a competition graph. They also investigate the two extensions of complex Pythagorean fuzzy competition graphs, namely, complex Pythagorean fuzzy k-competition and complex Pythagorean fuzzy p-competition graphs [17].The idea of relations is one of the most important notions in pure and applied science. Science has been defined as the discovery of the relation between events, objects, and states. Relations are associations that remain at the very core of the majority of science and engineering methodological approaches. Fuzzy relations in fuzzy theory are important concepts and have been commonly used in many fields, such as fuzzy control, fuzzy clustering, and uncertainty reasoning. In fuzzy diagnosis and fuzzy modeling, they also play a significant role. How to estimate and compare them is a significant issue when fuzzy relations are used in practice. Some researchers have carried out ambiguous measurements of fuzzy relations.Klir studied the crisp relations in [18]. A crisp relation shows the existence or absence of association, interconnectedness, or interaction between the parameters. The relation between two sets is denoted by RM,N, and its membership function is represented by λRm,n, where m∈M and n∈N. The membership function λRm,n has two values either 1 or 0. The generalization of a crisp relation is the fuzzy relation. The fuzzy relation was discussed by Mendel in [19]. Fuzzy relations show a degree of the presence or absence of association, interaction, or interconnectedness between the elements of two or more fuzzy sets. A fuzzy relation between two fuzzy sets M and N is denoted by RM,N, and its membership function is represented by λRm,n, where m∈M and n∈N. All the values of λRm,n∈0,1. Fuzzy relations play a vital role in a fuzzy logic system. Triapathi et al. in [20] used the complex fuzzy relations to obtain diagnostic conclusions about diabetes by restricting grade values to symptoms of a disease from 0 to 1. Majid in [21] discussed some important compositions of fuzzy relations for predicting scores in cricket. Moreover, it studied the restoration and the identification of the causes (diagnosis) through the observed effects (symptoms) on the basis of fuzzy relations.In crisp relations and fuzzy relations, there is one significant weakness, which is a lack of capacity to examine two-dimensional phenomena. Ramot discussed the complex fuzzy relations in [11], which is the generalization of crisp relations and fuzzy relations. Complex fuzzy relations represent both the degree of the presence or absence of association, interaction, or interconnectedness and the phase of association, interaction, or interconnectedness between the elements of two or more crisp sets. For any two crisp sets M and N, the complex fuzzy relation is denoted by RM,N. The relation RM,N may be represented as the set of ordered pairs RM,N=m,n,λRm,n/m,n∈M×N. The membership function of a complex fuzzy relation is denoted by λRm,n, and all its values lie within the unit circle in the complex plane. Ramot et al. in [11] discussed the applications of complex fuzzy relations in Future Commission Merchant.The purpose of this article is two-fold. The first half aims to present the theoretical foundations of the types of complex fuzzy relations. In any field of mathematics, we have needed such relations to solve problems. We can solve lots of problems with the help of these relations easily. The second half aims to present these theoretical foundations and key techniques in Future Commission Market, decision making, and the principle of the types of complex fuzzy relations in a coherent manner. The purpose of these innovative concepts is to provide a new approach with useful mathematical tools to address the fundamental problem of decision making (FCM problem). The generality of these new concepts is given special importance, illustrating how many interesting uncertainty problems can be formulated easily. These applied contexts provide solid evidence of the wide applications of the complex fuzzy relation approach to the model. This article will stimulate the interest in types of complex fuzzy relations and their application in various fields of science.Now in this paper, we define some types of complex fuzzy relations such as the complex fuzzy inverse relation, complex fuzzy reflexive relation, complex fuzzy symmetric relation, complex fuzzy antisymmetric relation, complex fuzzy transitive relation, complex fuzzy irreflexive relation, complex fuzzy asymmetric relation, complex fuzzy equivalence relation, complex fuzzy-order relation, and complex fuzzy equivalence class and discuss the particular examples of these relations. We also study some basic results. Moreover, we discuss the applications of complex fuzzy relations in Future Commission Merchant.


## 2. Preliminaries

We will discuss here the types of complex fuzzy relations and also discuss particular examples of these relations.Definition 1.
(See [11]). A CFSS, defined on a universal setU, is represented by a grade valueλCuwhose codomain is the disc of the radius on in the complex plane. Mathematically, the grade value of CFSCcan be represented byλCu=aCueipCuwhereaCuandpCuare known as an amplitude term and phase term, respectively. These two terms are real valued andaCu∈0,1.
Mathematically, CFS can be expressed as a set of ordered pairs given by(1)C=u;λCu:u∈U.Definition 2.
(See [22]). LetXm,m=1,2,3,…,MbeMCFS defined onUandλCmu=aCmueipCmutheir membership functions. The complex fuzzy Cartesian product ofCmdenoted byC1×C2×C3×⋯×Cmis specified by a function(2)λC1×C2×C3×⋯×Cmu=aC1×C2×C3×⋯×CmueipC1×C2×C3×⋯×Cmu,=minaC1u1,aC2u2,…,aCmumeiminpC1u1,pC2u2,…,pCmum.Definition 3.
(See [11]). For any two crisps setsXandY, the fuzzy relationRX,Yis a fuzzy subset of the product spaceX×Y. The grade value of the fuzzy relation is represented byλX×Yx,y, wherex∈Xandy∈Y. All the values of grade value lie in the closed interval0,1.
The fuzzy relation may be represented as the set of ordered pairs(3)RX,Y=x,y,λX×Yx,yx,y∈X×Y.Definition 4.
(See [11]). For any two crisp setsXandY, the complex fuzzy relationRX,Yis a complex fuzzy subset of the product spaceX×Y. The grade value of the complex fuzzy relation is represented byλX×Yx,y, wherex∈Xandy∈Y. All the values of the grade value of the complex fuzzy relation lie in unit disc in a complex plane.
The complex fuzzy relation may be represented as the set of ordered pairs:(4)RX,Y=G=x,y,λX×Yx,yx,y∈X×Y,where(5)λX×Yx,y=minaXx,aXyeiminpXx,pYyx,y.Example 1.
For any complex fuzzy setC=0.8eiπ/1+1ei2π/2, the product spaceX×Xis(6)C×C=0.8eiπ1,1+0.8eiπ1,2+0.8eiπ2,1+1ei2π2,2.
So, the complex fuzzy relation onC is subset of C×C is given by(7)RC,C=G=0.8eiπ1,1+0.8eiπ1,2+1ei2π2,2.


## 3. Types of Complex Fuzzy Relations

Definition 5.
IfGis a complex fuzzy relation on a complex fuzzy setC, then the inverse relationG−1inCis defined by(8)G−1=λC×Cv,uλC×Cu,v∈G.Definition 6.
IfGis a complex fuzzy relation on a complex fuzzy setC,Gis said to be complex fuzzy reflexive relation if for allu∈UandλCu∈C⟹λC×Cu,u∈G.Example 2.
For any complex fuzzy setC=0.4ei2π/1+1eiπ/2/2+0.5eiπ/3defined on any universal setU=1,2,3, the product spaceC×Cis(9)C×C=0.4ei2π1,1+0.4eiπ/21,2+0.4eiπ1,3+0.4eiπ/22,1+1eiπ/22,2+0.5eiπ/22,3+0.4eiπ3,1+0.5eiπ/23,2+0.5eiπ3,3.
So, the complex fuzzy reflexive relation onC is subset of C×C is given by(10)G=0.4ei2π1,1+1eiπ/22,2+0.5eiπ3,3.Definition 7.
IfGis a complex fuzzy relation on a complex fuzzy setC,Gis said to be a complex fuzzy symmetric relation ifλC×Cu,v∈G⟹λC×Cv,u∈G.Example 3.
For any complex fuzzy setC=0.4ei2π/a+1eiπ/2/b+0.5eiπ/cdefined on any universal setU=a,b,c, the product spaceC×Cis(11)C×C=0.4ei2πa,a+0.4eiπ/2a,b+0.4eiπa,c+0.4eiπ/2b,a+1eiπ/2b,b+0.5eiπ/2b,c+0.4eiπc,a+0.5eiπ/2c,b+0.5eiπc,c.
So, the complex fuzzy symmetric relation onC is given by(12)G=0.4ei2πa,a+1eiπ/2b,b+0.5eiπc,c+0.4eiπ/2a,b+0.4eiπ/2b,a.Definition 8.
IfGis a complex fuzzy relation on a complex fuzzy setC,Gis said to be a complex fuzzy antisymmetric relation ifλC×Cu,v∈GandλC×Cv,u∈G⟹λC×Cu,v=λC×Cv,u.Example 4.
For any complex fuzzy setC=0.5ei2π/x+1eiπ/2/y+0eiπ/zdefined on any universal setU=a,b,c, the product spaceC×Cis(13)C×C=0.5ei2πx,x+0.5eiπ/2x,y+0eiπx,z+0.5eiπ/2y,x+1eiπ/2y,y+0eiπ/2y,z+0eiπz,x+0eiπ/2z,y+0eiπz,z.
So, the complex fuzzy antisymmetric relation onC is given by(14)G=0.5ei2πx,x+1eiπ/2y,y+0eiπz,z.Definition 9.
IfGis a complex fuzzy relation on a complex fuzzy setC,Gis said to be complex fuzzy transitive relation ifλC×Cu,v∈GandλC×Cv,w∈G⟹λC×Cu,w∈G.Example 5.
For any complex fuzzy setC=0.5ei2π/x+1eiπ/2/y+0eiπ/zdefined on any universal setU=a,b,c, the product spaceC×Cis(15)C×C=0.5ei2πx,x+0.5eiπ/2x,y+0eiπx,z+0.5eiπ/2y,x+1eiπ/2y,y+0eiπ/2y,z+0eiπz,x+0eiπ/2z,y+0eiπz,z.
So, the complex fuzzy transitive relation onC is given by(16)G=0.5ei2πx,x+1eiπ/2y,y+0eiπz,z+0.5eiπ/2x,y+0eiπx,z.Definition 10.
IfGis a complex fuzzy relation on a complex fuzzy setC,Gis said to be a complex fuzzy irreflexive relation if for allu∈UandλCu∈C⟹λC×Cu,u∉G.Example 6.
For any complex fuzzy setC=0.4ei2π/1+1eiπ/2/2+0.5eiπ/3defined on any universal setU=1,2,3, the product spaceC×Cis(17)C×C=0.4ei2π1,1+0.4eiπ/21,2+0.4eiπ1,3+0.4eiπ/22,1+1eiπ/22,2+0.5eiπ/22,3+0.4eiπ3,1+0.5eiπ/23,2+0.5eiπ3,3.
So, the complex fuzzy irreflexive relation onC is subset of C×C is given by(18)G=0.4ei2π1,1+1eiπ/22,2+0.4eiπ1,3+0.4eiπ/22,1.Definition 11.
IfGis a complex fuzzy relation on a complex fuzzy setC,Gis said to be a complex fuzzy asymmetric relation ifλC×Cu,v∈G⟹λC×Cv,u∉G.Example 7.
For any complex fuzzy setC=0.4ei2π/a+1eiπ/2/b+0.5eiπ/cdefined on any universal setU=a,b,c, the product spaceC×Cis(19)C×C=0.4ei2πa,a+0.4eiπ/2a,b+0.4eiπa,c+0.4eiπ/2b,a+1eiπ/2b,b+0.5eiπ/2b,c+0.4eiπc,a+0.5eiπ/2c,b+0.5eiπc,c.
So, the complex fuzzy asymmetric relation onC is given by(20)G=0.4ei2πa,a+1eiπ/2b,b+0.5eiπc,c+0.4eiπ/2a,b.Definition 12.
A relationGis said to be a complex fuzzy equivalence relation if it satisfies the following conditions:(i)
G is reflexive(ii)
G is symmetric(iii)
G is transitiveExample 8.
For any complex fuzzy setC=0.8ei2π/u+0.5eiπ/2/v+0.2eiπ/vdefined on any universal setU=1,2,3, the product spaceC×Cis given by(21)C×C=0.8ei2πu,u+0.5eiπ/2u,v+0.2eiπu,w+0.5eiπ/2v,u+0.5eiπ/2v,v+0.2eiπ/2v,w+0.2eiπw,u+0.2eiπ/2w,v+0.2eiπw,w.
So, the complex fuzzy equivalence relation onC is given by(22)G=0.8ei2πu,u+0.5eiπ/2v,v+0.2eiπw,w+0.2eiπ/2v,w+0.2eiπ/2w,v.Definition 13.
A relationGis said to be a complex fuzzy-order relation if it satisfies the following conditions:(i)
G is reflexive(ii)
G is antisymmetric(iii)
G is transitiveExample 9.
For any complex fuzzy setC=0.8ei2π/u+0.5eiπ/2/v+0.2eiπ/vdefined on any universal setU=1,2,3, the product spaceC×Cis given by(23)C×C=0.8ei2πu,u+0.5eiπ/2u,v+0.2eiπu,w+0.5eiπ/2v,u+0.5eiπ/2v,v+0.2eiπ/2v,w+0.2eiπw,u+0.2eiπ/2w,v+0.2eiπw,w.
So, the complex fuzzy-order relation onC is given by(24)G=0.8ei2πu,u+0.5eiπ/2v,v+0.2eiπw,w+0.5eiπ/2u,v+0.2eiπ/2v,w.Definition 14.
LetCbe a complex fuzzy set andGbe a complex fuzzy equivalence relation inC. IfλCu∈C, then the complex fuzzy equivalence class ofλCumoduloGis the setGλCudefined by(25)GλCu=λCvλCv,u∈G.Example 10.
For any complex fuzzy setC=1ei1.2π/1+0.5eiπ/2+0eiπ/2/3defined on any universal setU=1,2,3, the product spaceC×Cis given by(26)C×C=1ei1.2π1,1+0.5eiπ1,2+0eiπ/21,3+0.5eiπ2,1+0.5eiπ2,2+0eiπ/22,3+0eiπ/23,1+0eiπ/23,2+0eiπ/23,3.
Also, let(27)G=1ei1.2π1,1+0.5eiπ2,2+0.5eiπ2,1+0eiπ/23,2+0eiπ/23,3,
be the complex fuzzy relation inC. Then, the complex fuzzy equivalence class of λC1 is given by(28)GλC1=λC1,λC2.
The complex fuzzy equivalence class ofλC2 is given by(29)GλC2=λC2,λC3.
Also, the complex fuzzy equivalence class ofλC3 is given by(30)GλC3=λC3.Definition 15.
LetCbe a complex fuzzy set andGbe a complex fuzzy relation inC. For anyλC×Cu,v∈GandλC×Cv,w∈G,λC×Cu,w∈G°Gfor allu,v,w∈U(universal set).G°Gis called the complex fuzzy composition relations.


## 4. Main Results

Proposition 1.
IfGandHare symmetric relations in a complex fuzzy setC, thenG∩His also a complex fuzzy symmetric relation inC.Proof.
Suppose thatG and H are complex fuzzy symmetric relations in a complex fuzzy set C. Since G⊆C×C and H⊆C×C, G∩H⊆C×C. Therefore, G∩H is a complex fuzzy relation in C.
LetλC×Cu,v∈G∩H; then, λC×Cu,v∈G and λC×Cu,v∈H. But, G and H are symmetric. Therefore λC×Cv,u∈G and λC×Cv,u∈H, so that λC×Cv,u∈G∩H.Proposition 2.
LetGbe a complex fuzzy relation in a complex fuzzy setC; then,Gis symmetric ifG=G−1.Proof.
SupposeG is a complex fuzzy symmetric relation; then,(31)λC×Cu,v∈G⟺λC×Cv,u∈G⟺λC×Cu,v∈G−1.
Thus,(32)G=G−1.
Conversely, letG=G−1; then,(33)λC×Cu,v∈G⟺λC×Cu,v∈G−1⟺λC×Cv,u∈G.
Thus,G is a complex fuzzy symmetric relation.Proposition 3.
LetGbe a complex fuzzy relation in a complex fuzzy setC; then,Gis transitive ifG°G⊆G.Proof.
SupposeG is transitive. Assume that λC×Cu,w∈G°G. Then, there exists v∈U such that λC×Cu,v∈G and λC×Cv,w∈G. Since G is complex fuzzy transitive relation and, hence, by the transitive property, λC×Cu,w∈G.
Thus,(34)G°G⊆G.
Conversely, suppose thatG°G⊆G. Then, λC×Cu,v∈G and λC×Cv,w∈G⟺λC×Cu,w∈G°G⊆G. Thus, λC×Cu,w∈G, and hence, G is transitive.Proposition 4.
IfGis a complex fuzzy equivalence relation in a complex fuzzy setC, thenG°G=G.Proof.
LetλC×Cu,v∈G°G; then, there exists w∈U such that λC×Cu,w∈G and λC×Cw,v∈G, but G is a complex fuzzy equivalence relation in C, so G is a complex fuzzy transitive relation. Therefore, by the transitive property, λC×Cu,v∈G. Thus,(35)G°G⊆G.
Conversely, assume thatλC×Cu,v∈G. Since G is a complex fuzzy reflexive relation, for v∈U⟹λC×Cv,v∈G. Now, λC×Cu,v∈G and λC×Cv,v∈G. Since G is a complex fuzzy transitive relation, so by the transitive property, λC×Cu,v∈G°G. Thus,(36)G⊆G°G.
From (35) and (36), we have(37)G=G°G.Proposition 5.
The inverse of a complex fuzzy-order relation in a complex fuzzy setCis also a complex fuzzy-order relation inC.Proof.
LetG be a complex fuzzy-order relation in a complex fuzzy set C. To show that G−1 is also a complex fuzzy-order relation in C, we have to satisfy the three conditions of a complex fuzzy-order relation.
SinceG is a complex fuzzy-order relation, for any u∈U⟹λC×Cu,u∈G⟹λC×Cu,u∈G−1. Thus, G−1 is a complex fuzzy reflexive relation.
LetλC×Cu,v∈G−1 and λC×Cv,u∈G−1. Then, λC×Cv,u∈G and λC×Cu,v∈G, but G is a complex fuzzy-order relation in C. Therefore, λC×Cu,v=λC×Cv,u, and hence, G−1 is antisymmetric.
Assume thatλC×Cu,v∈G−1 and λC×Cv,w∈G−1. Then, λC×Cv,u∈G and λC×Cw,v∈G. Since G is a complex fuzzy transitive relation. By the transitive property, λC×Cw,u∈G. Thus, λC×Cu,w∈G−1, and hence, G−1 is a complex fuzzy transitive relation in C. Since G−1 satisfies all the properties of a complex fuzzy-order relation, G−1 is a complex fuzzy-order relation.Theorem 1.
LetGbe any complex fuzzy equivalence relation in a complex fuzzy setC. Then,λC×Cv,u∈Gif and only ifGλCu=GλCv.Proof.
Suppose thatλC×Cu,v∈G. Let λCw∈GλCu. Then, λC×Cw,u∈G. Now, λC×Cw,u∈G and λC×Cu,v∈G. But, since G is a complex fuzzy equivalence relation in a complex fuzzy set C, by the transitive property, λC×Cw,v∈G, so that λCw∈GλCv. Thus,(38)GλCu⊆GλCv.
LetλCw∈GλCv; then, λC×Cw,v∈G and also λC×Cu,v∈G. Since G is a complex fuzzy symmetric relation, so by the symmetric property, λC×Cv,u∈G. Also, G is a complex fuzzy equivalence relation. Therefore, by the transitive property, λC×Cw,u∈G. Thus, λCw∈GλCu, and hence,(39)GλCv⊆GλCu.
From (38) and (39), we have(40)GλCv=GλCu.
Conversely, suppose thatGλCv=GλCu. Since G is a complex fuzzy equivalence relation, so by the reflexive property, λC×Cu,u∈G. By definition of a complex fuzzy equivalence class of modulo G,(41)λCu∈GλCu=GλCvλCu∈GλCv.
Hence,λC×Cu,v∈G.


## 5. Applications

In this section, we will discuss the application of complex fuzzy relations in Future Commission Market.We are going to discuss a real-life application of newly defined types/properties of complex fuzzy relations. Ramot et al. in [11] discussed the application of the complex fuzzy relation in Future Commission Merchant. The physical meaning of the Ramot and proposed model is the same, but here, we will show that how the types of complex fuzzy relations play a significant role in real-life applications.


### 5.1. Decision-Making Method

LetU be a collection of financial indicators or indexes of any country. Possible components of this collection are import, export, agriculture, unemployment, and development rate, that is, U=import,export,agriculture,unemployment,devlopment rate. Let(42)C=λCimport,λCexport,λCagriculture,λCunemployment,λCdevelopment raterepresent the complex fuzzy sets and λC×Cx,y represent the membership function of the complex fuzzy relations.The membership functionλC×Cx,y is complex-valued, with a phase term and amplitude term. The amplitude term shows the grade of influence of one parameter on another parameter. The amplitude term with a value close to zero shows more influence, while a value close to zero shows a small influence. The phase term refers to the “phase” of control or time lag that characterizes one parameter’s effect on another parameter.Consider, for example, we find the relation setG for the complex fuzzy set C, that is, the cross product of C.(43)G=λC×Cimport,import,λC×Cimport,export,λC×Cimport,agriculture,λC×Cimport,unemployment,λC×Cimport,development rate,λC×Cexport,import,λC×Cexport,export,λC×Cexport,agriculture,λC×Cexport,unemployment,λC×Cexport,development rate,λC×Cagriculture,import,λC×Cagriculture,export,λC×Cagriculture,agriculture,λC×Cagriculture,unemployment,λC×Cagriculture,development rate,λC×Cunemployment,import,λC×Cunemployment,export,λC×Cunemployment,agriculture,λC×Cunemployment,unemployment,λC×Cunemployment,development rate,λC×Cdevelopment rate,import,λC×Cdevelopment rate,export,λC×Cdevelopment rate,agriculture,λC×Cdevelopment rate,unemployment,λC×Cdevelopment rate,development rate.The relationG is an equivalence relation, that is, G is a complex fuzzy reflexive, complex fuzzy symmetric, and complex fuzzy transitive relation. From this relation, we can easily determine the influence of one parameter on another parameter. For example, we take λC×Cimport,agriculture=minaCimport,aCagricultureeiminpCimport,pCagriculture/import,agriculture.TheminaC(import), aC(agriculture) shows the degree of influence of import on agriculture or agriculture on import of a country with respect to time.(44)minpCimport,pCagriculture.This means that the degree of influence depends on themin value of the parameter. Moreover, if we know the influence of import on agriculture and agriculture on development rate, then by the complex fuzzy transitive relation, we can determine the influence of import on the development rate. Similarly, we have to find the influence of each parameter on the other. The more significant relation is the complex fuzzy transitive relation because if we know the degree of influence of the first parameter on the second parameter and second parameter on the third parameter, then by the complex fuzzy transitive relation, we can find the degree of influence of the first parameter on the third parameter. Moreover, the complex fuzzy symmetric relation shows that the degree of influence of the first parameter on the second parameter and the second parameter on the first parameter is the same.Example 11.
Consider a complex fuzzy set(45)C=0.4eiπdevelopment rate+0.8eiπ/2agriculture+0.2ei2πexport.
Then, the relationG on C×C is(46)G=C×C=min0.4,0.4eiminπ,πdevelopment rate,development rate+min0.4,0.8eiminπ,π/2development rate,dgriculture+min0.4,0.2eiminπ,2πdevelopment rate,export+min0.8,0.4eiminπ/2,πagriculture,development rate+min0.8,0.8eiminπ/2,π/2agriculture,agriculture+min0.8,0.2eiminπ/2,2πagriculture,export+min0.2,0.4eimin2π,πexport,development rate+min0.2,0.8eimin2π,π/2export,agriculture+min0.2,0.2eiminπ/2,πexport,export,G=C×C=0.4eiπdevelopment rate,development rate+0.4,eiπ/2development rate,dgriculture+0.2eiπdevelopment rate,export+0.4eiπ/2agriculture,development rate+0.8eiπ/2agriculture,agriculture+0.2eiπ/2agriculture,export+0.2eiπexport,development rate+0.2eiπ/2export,agriculture,+0.2eiπ/2export,export.
From the abovementioned relationG (complex fuzzy equivalence relation), we can determine the influence of the development rate, agriculture, and export on each other. For example,(47)λC×Cagriculture,development rate=0.4eiπ/2∈G.
Here, 0.4 shows the degree of influence of agriculture on the development rate with respect to half a month. Moreover, the membership functionλC×C(agriculture, export), that is,(48)λC×Cagriculture,export=0.2eiπ/2∈G,shows that the degree of influence of agriculture on export is 0.2 with respect to the half a month.
If we compare the influence of agriculture on the development rate and export, we find that the degree of influence of agriculture on export is less than the degree of influence on the development rate.
From the abovementioned relationG, we have the same degree of influence of agriculture on the development rate and the development rate on agriculture.
Moreover, if we have the degree of influence of agriculture on the development rate and the degree of influence of the development rate on export, then by complex fuzzy transitive relations, we can obtain the degree of influence of agriculture on export.
In the second example, we will discuss the degree of influence of American financial indexes on China’s financial indexes and China’s financial indexes on Saudi Arabia’s financial indexes.Example 12.
LetC1,C2, andC3represent the set of America, China, and Saudi Arabia’s financial indexes. A possible collection of these sets are import, export, growth rate, interest rate, and unemployment rate. They are all complex-valued functions. Let the complex fuzzy relationλC1×C2x,yrepresent the relation of influence of American financial indexes on China financial indexes and the complex fuzzy relationλC2×C3x,yrepresent the relation of influence of China financial indexes on Saudi Arabia financial indexes, wherex,yrepresent any two parameters.
Consider(49)C1=λC1import,λC1export,λC1growth rate,λC1interest rateλC1unemployment rate,C2=λC2import,λC2export,λC2growth rate,λC2interest rateλC2unemployment rate,C3=λC3import,λC3export,λC3growth rate,λC3interest rateλC3unemployment rate.
Now, the complex fuzzy relationG1 between C1 and C2 is given by(50)G1=C1×C2=λC1×C2import,import,λC1×C2import,export,λC1×C2import,growth rate,λC1×C2import,interest rate,λC1×C2import,unemployment rate,λC1×C2export,import,λC1×C2export,export,λC1×C2export,growth rate,λC1×C2export,interest rate,λC1×C2export,unemployment rate,λC1×C2growth rate,import,λC1×C2growth rate,export,λC1×C2growth rate,growth rate,λC1×C2growth rate,interest rate,λC1×C2growth rate,unemployment rate,λC1×C2interest rate,import,λC1×C2interest rate,export,λC1×C2interest rate,growth rate,λC1×C2interest rate,interest rate,λC1×C2interest rate,unemployment rate,λC1×C2unemployment rate,import,λC1×C2unemployment rate,export,λC1×C2unemployment rate,growth rate,λC1×C2unemployment rate,interest rate,λC1×C2unemployment rate,unemployment rate.
The relationG1 shows the relation of influence of American financial indexes on China financial indexes.
The complex fuzzy relation betweenC2 and C3 is given by(51)G2=C2×C3=λC2×C3import,import,λC2×C3import,export,λC2×C3import,growth rate,λC2×C3import,interest rate,λC2×C3import,unemployment rate,λC2×C3export,import,λC2×C3export,export,λC2×C3export,growth rate,λC2×C3export,interest rate,λC2×C3export,unemployment rate,λC2×C3growth rate,import,λC2×C3growth rate,export,λC2×C3growth rate,growth rate,λC2×C3growth rate,interest rate,λC2×C3growthrate,unemploymentrate,λC2×C3interest rate,import,λC2×C3interest rate,export,λC2×C3interest rate,growth rate,λC2×C3interest rate,interest rate,λC2×C3interest rate,unemployment rate,λC2×C3unemployment rate,import,λC2×C3unemployment rate,export,λC2×C3unemployment rate,growth rate,λC2×C3unemployment rate,interest rate,λC2×C3unemployment rate,unemployment rate.
The relationG2 represents the relation of influence of China’s financial indexes on Saudi Arabia’s financial indexes.
From relationG1 and G2, we have the relation of influence of American financial indexes on China’s financial indexes and China financial indexes on Saudi Arabia financial indexes. By the complex fuzzy composition relation, we can find the relation of the influence of American financial indexes on Saudi Arabia’s financial indexes.
The relationG represents the relation of influence of American financial indexes on Saudi Arabia financial indices, that is,(52)G1=C1×C3=λC1×C3import,import,λC1×C3import,export,λC1×C3import,growth rate,λC1×C3import,interest rate,λC1×C3import,unemployment rate,λC1×C3export,import,λC1×C3export,export,λC1×C3export,growth rate,λC1×C3export,interest rate,λC1×C3export,unemployment rate,λC1×C3growth rate,import,λC1×C3growth rate,export,λC1×C3growth rate,growth rate,λC1×C3growth rate,interest rate,λC1×C3growth rate,unemployment rate,λC1×C3interest rate,import,λC1×C3interest rate,export,λC1×C3interest rate,growth rate,λC1×C3interest rate,interest rate,λC1×C3interest rate,unemployment rate,λC1×C3unemployment rate,import,λC1×C3unemployment rate,export,λC1×C3unemployment rate,growth rate,λC1×C3unemployment rate,interest rate,λC1×C3unemployment rate,unemployment rate.
For example, the membership functionλC1×C2 (import, export) ∈ G1 shows the degree of influence of American import on a China export and the membership function λC2×C3(export, growth rate) ∈ G2 shows the degree of influence of China export on a Saudi Arabia growth rate. By the complex fuzzy composition relation of these two relations, we have the membership function λC1×C3(import, growth rate)∈G, which is more significant and gives the degree of influence of American import on the Saudi Arabia growth rate.
Moreover, the types of complex fuzzy relations play a vital role in applications. If we have known the degree of influence of American financial indexes on China financial indexes, then by the inverse complex fuzzy relations, we can find the degree of China financial indexes on American financial indexes.
Similarly, the complex fuzzy transitive relation is the most important type of complex fuzzy relations and plays a major role in applications. For example, if we have the degree of the influence of import on the interest rate and the interest rate on the unemployment rate, that is,λC1×C3(import, interest rate) and λC1×C3(interest rate, unemployment rate), then by complex fuzzy transitive relations, we can easily find the degree of influence of import on interest rate λC1×C3(import, interest rate).
The abovementioned process may be applied to discuss the degree of financial indexes of more than three countries.Example 13.
LetC1=0.8eiπ/2//import,0.5eiπ/export,1ei2π/interestrate,0.7ei3π/2/unemploymnt rate,C2=0.5ei3π/2/import, 0.9ei2π/export, 0.3eiπ/interest rate, 0.4eiπ/2/unemployment rate, , andC3=0.6ei5π/2/import, 1eiπ/2/export0.7eiπ/3/interest rate, 0.5eiπ/unemployment rate, represent the sets of American, China, and Saudi Arabia’s financial indexes. Then, the relationG1onC1×C2is(53)G1=C1×C2=0.5eiπ/2import,import,0.8eiπ/2import,export,0.3eiπ/2import,interest rate,0.4eiπ/2import,unemployment rate,0.5ei3π/2export,import,0.5eiπexport,export,0.3eiπexport,interest rate,0.4eiπ/2export,unemployment rate,0.5ei3π/2interest rate,import,0.9ei2πinterest rate,export,0.3eiπinterest rate,interest rate,0.4eiπ/2interest rate,unemployment rate,0.5ei3π/2unemployment rate,import,0.7ei3π/2unemployment rate,export,0.3eiπunemployment rate,interest rate,0.4eiπ/2unemployment rate,unemployment rate.
The relationG1 shows the relation of influence of American financial indexes on China financial indexes.
The complex fuzzy relation betweenC2 and C3 is given by(54)G2=C2×C3=0.5ei3π/2import,import,0.5eiπ/2import,export,0.5eiπ/3import,interest rate,0.5ei3π/2import,unemployment rate,0.6ei5π/2export,import,0.9eiπ/2export,export,0.7eiπ/3export,interest rate,0.5eiπexport,unemployment rate,0.3ei5π/2interest rate,import,0.3eiπ/2interest rate,export,0.3eiπ/3interest rate,interest rate,0.3eiπinterest rate,unemployment rate,0.4eiπ/2unemployment rate, import,0.4eiπ/2unemployment rate,export,0.4eiπ/3unemployment rate,interest rate,0.4eiπ/2unemployment rate,unemployment rate.
The relationG2 represents the relation of influence of China’s financial indexes on Saudi Arabia’s financial indexes. By the complex fuzzy composition relation, we can find the relation of the influence of American financial indexes on Saudi Arabia’s financial indexes. The relation G3 represents the relation of influence of American financial indexes on Saudi Arabia financial indices, that is,(55)G3=C1×C3=0.6eiπ/2import,import,0.8eiπ/2import,export,0.7eiπ/2import,interest rate,0.5eiπ/2import,unemployment rate,0.5eiπexport,import,0.5eiπ/2export,export,0.5eiπ/3export,interest rate,0.5eiπexport,unemployment rate,0.6ei2πinterest rate,import,1eiπ/2interest rate,export,0.7eiπ/3interest rate,interest rate,0.5eiπinterest rate,unemployment rate,0.6ei3π/2unemployment rate,import,0.7eiπ/2unemployment rate, export,0.7eiπ/3unemployment rate,interest rate,0.5eiπunemployment rate,unemployment rate.
The abovementioned three relationsG1, G2, and G3 show the degree of influence of American financial indexes on China financial indexes, China financial indexes on Saudi Arabia financial indexes, and American financial indexes on Saudi Arabia financial indexes, respectively. Moreover, by the inverse complex fuzzy relations, we can obtain the degree of influence of China financial indexes on American financial indexes, Saudi Arabia financial indexes on China financial indexes, and Saudi Arabia financial indexes on American financial indexes.
Moreover, if we have the degree of the influence of import on the interest rate and the interest rate on the unemployment rate, then by complex fuzzy transitive relations, we can easily find the degree of influence of import on the interest rate. For example,
0.5eiπ/3/import,interest rate, ∈G and 0.3eiπ/(interest rate, unemployment rate)∈G; then, by complex fuzzy transitive relations, we have 0.5ei3π/2/(import, unemployment rate).


## 5.1. Decision-Making Method

LetU be a collection of financial indicators or indexes of any country. Possible components of this collection are import, export, agriculture, unemployment, and development rate, that is, U=import,export,agriculture,unemployment,devlopment rate. Let(42)C=λCimport,λCexport,λCagriculture,λCunemployment,λCdevelopment raterepresent the complex fuzzy sets and λC×Cx,y represent the membership function of the complex fuzzy relations.The membership functionλC×Cx,y is complex-valued, with a phase term and amplitude term. The amplitude term shows the grade of influence of one parameter on another parameter. The amplitude term with a value close to zero shows more influence, while a value close to zero shows a small influence. The phase term refers to the “phase” of control or time lag that characterizes one parameter’s effect on another parameter.Consider, for example, we find the relation setG for the complex fuzzy set C, that is, the cross product of C.(43)G=λC×Cimport,import,λC×Cimport,export,λC×Cimport,agriculture,λC×Cimport,unemployment,λC×Cimport,development rate,λC×Cexport,import,λC×Cexport,export,λC×Cexport,agriculture,λC×Cexport,unemployment,λC×Cexport,development rate,λC×Cagriculture,import,λC×Cagriculture,export,λC×Cagriculture,agriculture,λC×Cagriculture,unemployment,λC×Cagriculture,development rate,λC×Cunemployment,import,λC×Cunemployment,export,λC×Cunemployment,agriculture,λC×Cunemployment,unemployment,λC×Cunemployment,development rate,λC×Cdevelopment rate,import,λC×Cdevelopment rate,export,λC×Cdevelopment rate,agriculture,λC×Cdevelopment rate,unemployment,λC×Cdevelopment rate,development rate.The relationG is an equivalence relation, that is, G is a complex fuzzy reflexive, complex fuzzy symmetric, and complex fuzzy transitive relation. From this relation, we can easily determine the influence of one parameter on another parameter. For example, we take λC×Cimport,agriculture=minaCimport,aCagricultureeiminpCimport,pCagriculture/import,agriculture.TheminaC(import), aC(agriculture) shows the degree of influence of import on agriculture or agriculture on import of a country with respect to time.(44)minpCimport,pCagriculture.This means that the degree of influence depends on themin value of the parameter. Moreover, if we know the influence of import on agriculture and agriculture on development rate, then by the complex fuzzy transitive relation, we can determine the influence of import on the development rate. Similarly, we have to find the influence of each parameter on the other. The more significant relation is the complex fuzzy transitive relation because if we know the degree of influence of the first parameter on the second parameter and second parameter on the third parameter, then by the complex fuzzy transitive relation, we can find the degree of influence of the first parameter on the third parameter. Moreover, the complex fuzzy symmetric relation shows that the degree of influence of the first parameter on the second parameter and the second parameter on the first parameter is the same.Example 11.
Consider a complex fuzzy set(45)C=0.4eiπdevelopment rate+0.8eiπ/2agriculture+0.2ei2πexport.
Then, the relationG on C×C is(46)G=C×C=min0.4,0.4eiminπ,πdevelopment rate,development rate+min0.4,0.8eiminπ,π/2development rate,dgriculture+min0.4,0.2eiminπ,2πdevelopment rate,export+min0.8,0.4eiminπ/2,πagriculture,development rate+min0.8,0.8eiminπ/2,π/2agriculture,agriculture+min0.8,0.2eiminπ/2,2πagriculture,export+min0.2,0.4eimin2π,πexport,development rate+min0.2,0.8eimin2π,π/2export,agriculture+min0.2,0.2eiminπ/2,πexport,export,G=C×C=0.4eiπdevelopment rate,development rate+0.4,eiπ/2development rate,dgriculture+0.2eiπdevelopment rate,export+0.4eiπ/2agriculture,development rate+0.8eiπ/2agriculture,agriculture+0.2eiπ/2agriculture,export+0.2eiπexport,development rate+0.2eiπ/2export,agriculture,+0.2eiπ/2export,export.
From the abovementioned relationG (complex fuzzy equivalence relation), we can determine the influence of the development rate, agriculture, and export on each other. For example,(47)λC×Cagriculture,development rate=0.4eiπ/2∈G.
Here, 0.4 shows the degree of influence of agriculture on the development rate with respect to half a month. Moreover, the membership functionλC×C(agriculture, export), that is,(48)λC×Cagriculture,export=0.2eiπ/2∈G,shows that the degree of influence of agriculture on export is 0.2 with respect to the half a month.
If we compare the influence of agriculture on the development rate and export, we find that the degree of influence of agriculture on export is less than the degree of influence on the development rate.
From the abovementioned relationG, we have the same degree of influence of agriculture on the development rate and the development rate on agriculture.
Moreover, if we have the degree of influence of agriculture on the development rate and the degree of influence of the development rate on export, then by complex fuzzy transitive relations, we can obtain the degree of influence of agriculture on export.
In the second example, we will discuss the degree of influence of American financial indexes on China’s financial indexes and China’s financial indexes on Saudi Arabia’s financial indexes.Example 12.
LetC1,C2, andC3represent the set of America, China, and Saudi Arabia’s financial indexes. A possible collection of these sets are import, export, growth rate, interest rate, and unemployment rate. They are all complex-valued functions. Let the complex fuzzy relationλC1×C2x,yrepresent the relation of influence of American financial indexes on China financial indexes and the complex fuzzy relationλC2×C3x,yrepresent the relation of influence of China financial indexes on Saudi Arabia financial indexes, wherex,yrepresent any two parameters.
Consider(49)C1=λC1import,λC1export,λC1growth rate,λC1interest rateλC1unemployment rate,C2=λC2import,λC2export,λC2growth rate,λC2interest rateλC2unemployment rate,C3=λC3import,λC3export,λC3growth rate,λC3interest rateλC3unemployment rate.
Now, the complex fuzzy relationG1 between C1 and C2 is given by(50)G1=C1×C2=λC1×C2import,import,λC1×C2import,export,λC1×C2import,growth rate,λC1×C2import,interest rate,λC1×C2import,unemployment rate,λC1×C2export,import,λC1×C2export,export,λC1×C2export,growth rate,λC1×C2export,interest rate,λC1×C2export,unemployment rate,λC1×C2growth rate,import,λC1×C2growth rate,export,λC1×C2growth rate,growth rate,λC1×C2growth rate,interest rate,λC1×C2growth rate,unemployment rate,λC1×C2interest rate,import,λC1×C2interest rate,export,λC1×C2interest rate,growth rate,λC1×C2interest rate,interest rate,λC1×C2interest rate,unemployment rate,λC1×C2unemployment rate,import,λC1×C2unemployment rate,export,λC1×C2unemployment rate,growth rate,λC1×C2unemployment rate,interest rate,λC1×C2unemployment rate,unemployment rate.
The relationG1 shows the relation of influence of American financial indexes on China financial indexes.
The complex fuzzy relation betweenC2 and C3 is given by(51)G2=C2×C3=λC2×C3import,import,λC2×C3import,export,λC2×C3import,growth rate,λC2×C3import,interest rate,λC2×C3import,unemployment rate,λC2×C3export,import,λC2×C3export,export,λC2×C3export,growth rate,λC2×C3export,interest rate,λC2×C3export,unemployment rate,λC2×C3growth rate,import,λC2×C3growth rate,export,λC2×C3growth rate,growth rate,λC2×C3growth rate,interest rate,λC2×C3growthrate,unemploymentrate,λC2×C3interest rate,import,λC2×C3interest rate,export,λC2×C3interest rate,growth rate,λC2×C3interest rate,interest rate,λC2×C3interest rate,unemployment rate,λC2×C3unemployment rate,import,λC2×C3unemployment rate,export,λC2×C3unemployment rate,growth rate,λC2×C3unemployment rate,interest rate,λC2×C3unemployment rate,unemployment rate.
The relationG2 represents the relation of influence of China’s financial indexes on Saudi Arabia’s financial indexes.
From relationG1 and G2, we have the relation of influence of American financial indexes on China’s financial indexes and China financial indexes on Saudi Arabia financial indexes. By the complex fuzzy composition relation, we can find the relation of the influence of American financial indexes on Saudi Arabia’s financial indexes.
The relationG represents the relation of influence of American financial indexes on Saudi Arabia financial indices, that is,(52)G1=C1×C3=λC1×C3import,import,λC1×C3import,export,λC1×C3import,growth rate,λC1×C3import,interest rate,λC1×C3import,unemployment rate,λC1×C3export,import,λC1×C3export,export,λC1×C3export,growth rate,λC1×C3export,interest rate,λC1×C3export,unemployment rate,λC1×C3growth rate,import,λC1×C3growth rate,export,λC1×C3growth rate,growth rate,λC1×C3growth rate,interest rate,λC1×C3growth rate,unemployment rate,λC1×C3interest rate,import,λC1×C3interest rate,export,λC1×C3interest rate,growth rate,λC1×C3interest rate,interest rate,λC1×C3interest rate,unemployment rate,λC1×C3unemployment rate,import,λC1×C3unemployment rate,export,λC1×C3unemployment rate,growth rate,λC1×C3unemployment rate,interest rate,λC1×C3unemployment rate,unemployment rate.
For example, the membership functionλC1×C2 (import, export) ∈ G1 shows the degree of influence of American import on a China export and the membership function λC2×C3(export, growth rate) ∈ G2 shows the degree of influence of China export on a Saudi Arabia growth rate. By the complex fuzzy composition relation of these two relations, we have the membership function λC1×C3(import, growth rate)∈G, which is more significant and gives the degree of influence of American import on the Saudi Arabia growth rate.
Moreover, the types of complex fuzzy relations play a vital role in applications. If we have known the degree of influence of American financial indexes on China financial indexes, then by the inverse complex fuzzy relations, we can find the degree of China financial indexes on American financial indexes.
Similarly, the complex fuzzy transitive relation is the most important type of complex fuzzy relations and plays a major role in applications. For example, if we have the degree of the influence of import on the interest rate and the interest rate on the unemployment rate, that is,λC1×C3(import, interest rate) and λC1×C3(interest rate, unemployment rate), then by complex fuzzy transitive relations, we can easily find the degree of influence of import on interest rate λC1×C3(import, interest rate).
The abovementioned process may be applied to discuss the degree of financial indexes of more than three countries.Example 13.
LetC1=0.8eiπ/2//import,0.5eiπ/export,1ei2π/interestrate,0.7ei3π/2/unemploymnt rate,C2=0.5ei3π/2/import, 0.9ei2π/export, 0.3eiπ/interest rate, 0.4eiπ/2/unemployment rate, , andC3=0.6ei5π/2/import, 1eiπ/2/export0.7eiπ/3/interest rate, 0.5eiπ/unemployment rate, represent the sets of American, China, and Saudi Arabia’s financial indexes. Then, the relationG1onC1×C2is(53)G1=C1×C2=0.5eiπ/2import,import,0.8eiπ/2import,export,0.3eiπ/2import,interest rate,0.4eiπ/2import,unemployment rate,0.5ei3π/2export,import,0.5eiπexport,export,0.3eiπexport,interest rate,0.4eiπ/2export,unemployment rate,0.5ei3π/2interest rate,import,0.9ei2πinterest rate,export,0.3eiπinterest rate,interest rate,0.4eiπ/2interest rate,unemployment rate,0.5ei3π/2unemployment rate,import,0.7ei3π/2unemployment rate,export,0.3eiπunemployment rate,interest rate,0.4eiπ/2unemployment rate,unemployment rate.
The relationG1 shows the relation of influence of American financial indexes on China financial indexes.
The complex fuzzy relation betweenC2 and C3 is given by(54)G2=C2×C3=0.5ei3π/2import,import,0.5eiπ/2import,export,0.5eiπ/3import,interest rate,0.5ei3π/2import,unemployment rate,0.6ei5π/2export,import,0.9eiπ/2export,export,0.7eiπ/3export,interest rate,0.5eiπexport,unemployment rate,0.3ei5π/2interest rate,import,0.3eiπ/2interest rate,export,0.3eiπ/3interest rate,interest rate,0.3eiπinterest rate,unemployment rate,0.4eiπ/2unemployment rate, import,0.4eiπ/2unemployment rate,export,0.4eiπ/3unemployment rate,interest rate,0.4eiπ/2unemployment rate,unemployment rate.
The relationG2 represents the relation of influence of China’s financial indexes on Saudi Arabia’s financial indexes. By the complex fuzzy composition relation, we can find the relation of the influence of American financial indexes on Saudi Arabia’s financial indexes. The relation G3 represents the relation of influence of American financial indexes on Saudi Arabia financial indices, that is,(55)G3=C1×C3=0.6eiπ/2import,import,0.8eiπ/2import,export,0.7eiπ/2import,interest rate,0.5eiπ/2import,unemployment rate,0.5eiπexport,import,0.5eiπ/2export,export,0.5eiπ/3export,interest rate,0.5eiπexport,unemployment rate,0.6ei2πinterest rate,import,1eiπ/2interest rate,export,0.7eiπ/3interest rate,interest rate,0.5eiπinterest rate,unemployment rate,0.6ei3π/2unemployment rate,import,0.7eiπ/2unemployment rate, export,0.7eiπ/3unemployment rate,interest rate,0.5eiπunemployment rate,unemployment rate.
The abovementioned three relationsG1, G2, and G3 show the degree of influence of American financial indexes on China financial indexes, China financial indexes on Saudi Arabia financial indexes, and American financial indexes on Saudi Arabia financial indexes, respectively. Moreover, by the inverse complex fuzzy relations, we can obtain the degree of influence of China financial indexes on American financial indexes, Saudi Arabia financial indexes on China financial indexes, and Saudi Arabia financial indexes on American financial indexes.
Moreover, if we have the degree of the influence of import on the interest rate and the interest rate on the unemployment rate, then by complex fuzzy transitive relations, we can easily find the degree of influence of import on the interest rate. For example,
0.5eiπ/3/import,interest rate, ∈G and 0.3eiπ/(interest rate, unemployment rate)∈G; then, by complex fuzzy transitive relations, we have 0.5ei3π/2/(import, unemployment rate).


## 6. Comparison

There are many applications of crisp relations and fuzzy relations, particularly in fuzzy logic systems, diagnostic of symptoms, and decision making. But, there is one significant weakness, which is a lack of capacity to examine two-dimensional phenomena. They cannot deal with two-dimensional parameters. Ramot et al. in [11] introduced complex fuzzy relations which is the generalization of a fuzzy relation. Complex fuzzy relations show a degree of the presence or absence of association, interaction, or interconnectedness between two-dimensional parameters. Ramot et al. in [11] discussed the application of the complex fuzzy relation in Future Commission Merchant. He studied the degree of influence of parameters with respect to complex fuzzy relations. We explore this concept in detail and used the types of complex fuzzy relations such as the complex fuzzy transitive relation and complex fuzzy equivalence relation. The method we proposed here gives the degree of influence of the financial indexes of the three countries. This method can be used to find the degree of influence of financial indexes of more than three countries by using complex fuzzy transitive relations or composition of complex fuzzy relations. Moreover, this approach provides the degree of influence of the two countries financial indexes that do not have a direct relation by using the types of complex fuzzy relations. Similarly, by the inverse complex fuzzy relations, we can obtain the degree of influence of financial indexes on each other. However, our designed model is not complete, but it is stuck with a shortage of theoretical support. For applications, the types of complex fuzzy relations can be useful. Therefore, it will be significant for future work.


## 7. Conclusions

In this paper, we have discussed some new types of complex fuzzy relations such as the complex fuzzy inverse relation, complex fuzzy reflexive relation, complex fuzzy symmetric relation, complex fuzzy antisymmetric relation, complex fuzzy transitive relation, complex fuzzy irreflexive relation, complex fuzzy asymmetric relation, complex fuzzy equivalence relation, complex fuzzy-order relation, and complex fuzzy equivalence class. We have presented some basic results and examples of these relations. Moreover, we have discussed the application of complex fuzzy relations in Future Commission Market. The complex fuzzy relation may be used in geology, signals and systems, and engineering fields for the identification of reference signals [23].



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*Source: 6685977-2021-03-28.xml*