Medical Diagnosis and Life Span of Sufferer Using Interval Valued Complex Fuzzy Relations

Fuzzy set theory resolved the crux of modeling uncertainty, vagueness, and imprecision. Many researchers have contributed to the development of the theory. This paper intends to define the innovative concept of the interval valued complex fuzzy relations (IVCFRs) using the proposed Cartesian product of two interval valued complex fuzzy sets (IVCFSs). Moreover, the types of IVCFRs are devised with some exciting results and properties. Furthermore, a couple of prodigious applications have been established as an illustration of the modeling capabilities of the proposed structures. The concept of interval valued complex fuzzy (IVCF) composite relations is used in the medical diagnosis of patients on the basis of symptoms. The inclusion of phase term in the grade of membership of IVCFRs facilitated modeling the periodic diseases. Additionally, another application of the Cartesian products and the IVCF equivalence relations is proposed that studies the life expectancies or mortality rates of patients with certain diseases. In addition, the effects of multiple illnesses on the life expectancy of a patient are also deliberated through IVCFRs. The proposed framework is also compared with the existing structures in the field of fuzzy set theory.


I. INTRODUCTION
Uncertainty is inevitable in different areas of life. Human opinions are mostly unclear. In sciences, the errors and miscalculations during experiments also lead to inaccuracy and uncertain results. Sometimes the data and information are ambiguous and confusing. Mathematicians tend to solve many practical problems through mathematics after writing them in the mathematical form. The process of converting reality into a mathematical structure is known as modeling. Modeling uncertainty, vagueness, and imprecision has been a crux in mathematics. In 1965, Zadeh [1] resolved this crux by introducing the theory of fuzzy sets (FSs). A fuzzy set allows assigning each element of the set, a function whose range is the unit interval [0, 1]. This function is called the grade of membership of an element. From the concept of relations in classical set theory initiated by Klir and Folger [2], The associate editor coordinating the review of this manuscript and approving it for publication was Juan Wang .
Mendel [3] conceived the idea of fuzzy relations (FRs) and introduced the concept of relations in FS theory. Classical relations only indicate the presence or absence of a relationship between any two objects. At the same time, the FRs also add the description of the quality of the relationship through the grade of membership. The values of grades of memberships closer to one represent a higher quality and good relationships, and the lower grades of membership specify the poor quality relationships. The concept of FRs is broader than that of classical relations.
In 1975, Zadeh [4] introduced the concept interval valued FSs (IVFSs). The IVFSs are the generalization of Zadeh's FSs. These sets describe the grade of membership in the form of an interval that is a subset of the unit interval. Since it is difficult for an expert to express his/her certainty by an exact real number, so it is suitable to choose an interval that expresses the certainty level. Therefore, the IVFSs help in modeling the uncertainty along with the consequences of any ignorance, mistakes, and confusion of experts.
Bustince and Burillo [5] concocted the concepts of interval valued fuzzy relations (IVFRs), which generalize classical relations and FRs. Deschrijver and Kerre [6] studied the relationships between some extensions of fuzzy set theory. Yao [7] compared the fuzzy sets and rough sets, Chiang and Lin [8] worked on the correlation of fuzzy sets. Ashtiani et al. [9] extended the method of fuzzy TOPSIS based on IVFSs, Zeng et al. [10], [11] discussed the entropy of IVFSs and their relationships with different measures. Guijun and Xiaoping [12] presented the applications of IVF numbers, while Bustince et al. [13] constructed the IVFSs from matrices and applied them to detect the edge. Gorzalczany [14] proposed a mothed of inference in approximate reasoning based on IVFSs. De Baets and Kerre [15] presented the applications of FRs. Yeh and Bang [16] applied FRs to clustering analysis. Braae and Rutherford [17] used FRs in the control setting. Delgado et al. [18] proposed the method of ranking fuzzy numbers via FRs. Barrenechea et al. [19] used IVFRs in the generation of fuzzy edge images. Bustince and Burillo [5] applied IVFRs to approximate reasoning. Elkano et al. [20] used aggregation functions for the composition of IVFRs. Bentkowska et al. [21] proposed the decision-making methods with IVFRs.
Ramot et al. [22] developed the conception of complex FSs (CFSs) by the involvement of complex numbers in FS theory. In a CFS, the grade of membership is considered to be a complex number in a unit circle of the complex plane instead of a real number in the unit interval. Since the complex numbers comprise of two different numbers, i.e., the real number and an imaginary number. So the grade of membership C ( ) of an object in a CFS is of the form C ( ) = C (x) e 2π℘ C (x)i , where C (x) and ℘ C (x) are real numbers in the unit interval. C (x) is said to be the amplitude term which often denotes the level of membership, while the ℘ C (x) is said to be the phase term. Usually, the phase term represents the variable with phase changes or periodic nature. The CFSs facilitate the model of multidimensional problems, especially the time-dependent ones. Moreover, Ramot et al. [22], [23] also introduced the complex fuzzy relations (CFRs) and set operators in the environment of CFSs. Zhang et al. [24] discussed the operation properties and δ-equalities of CFSs. Li et al. [25]- [27] comprehensively worked on the applications of CFSs. Zhang et al. [28] established the delta-equalities of CFRs. Imtiaz et al. [29], [30] worked on the image development in ξ -CF morphisms and the structural properties of ξ -CFSs and their applications. Plenty of research work is carried out on CFSs and CFRs in [31]- [35].
Greenfield [36] initiated the notion of interval valued complex FSs (IVCFSs) in 2016. The grades of memberships are complex valued intervals in the unit circle of the complex plane. Similar to CFSs, each membership function of IVCFSs consists of two intervals that are subsets of the unit interval. Likewise, there are amplitude and phase terms but in the form of intervals that enable to take any ignorance, confusion, or mistakes of experts into account. Dai et al. [35], [37] derived the distance measure between the IVCFSs. Selvachandran et al. [38], Bustince and Burillo [5] came up with the applications of IVCFSs, Singh et al. [39] studied the IVCF concept lattice and its granular decomposition, and Shuaib et al. [40] worked on -interval valued fuzzification of Lagrange's theorem of -IVF subgroups.
This paper introduces the novel concept of interval valued complex fuzzy relations (IVCFRs) by defining the Cartesian product of two IVCFSs. Also, the score function for an IVCFS is defined. Moreover, different types of IVCFRs are also defined, such as inverse IVCFR, IVCF reflexive relation, IVCF irreflexive relation, IVCF symmetric relation, IVCF asymmetric relation, IVCF anti-symmetric relation, IVCF transitive relation, IVCF composite relation, IVCF equivalence relation, IVCF order relation, and IVCF equivalence class. In addition, suitable examples have also been provided. Furthermore, some interesting results and properties of IVCFRs are discussed. Additionally, a couple of applications are presented. One of the applications uses the concepts of IVCF composite relations to diagnose the patients based on symptoms they are experiencing.
In contrast, the other application uses the idea of IVCFRs to study the life expectancies of patients with different medical conditions. Besides this, the increased risks of mortality due to multiple illnesses are also deliberated. Lastly, the comparative study has also been carried out among the IVCFRs and other structures in theory.
The benefit of an interval valued grade of membership over a crisp valued grade is that an interval value covers the mistakes, confusions, and errors made by the decisionmaker. It facilitates the professionals to assign a value with some plus-minus errors. Further, the complex valued grades have the advantage of modeling problems with periodic nature and phase changes. Henceforth, this paper chooses to introduce the interval valued complex values for modeling and solving some medical problems. In the future, these concepts can be stretched to some other generalizations of fuzzy sets such as complex Pythagorean fuzzy and t-spherical fuzzy environments, which can be utilized in different types of decision-making processes through aggregation operators and other techniques. These new structures could be useful in various fields of sciences, engineering, economics, statistics, physics, and information technology.
In the following section, the definitions of some fundamental conceptions and their examples are presented. Section III defines the novel concepts of IVCFRs and their types, including an example of each concept. Section IV incorporates the applications of proposed concepts in medical diagnosis and the life expectancies of ill persons. In section V, the reasons for using the IVCFRs for the proposed application problems are explained by comparing them with other structures. Finally, the research work is concluded in the last section.

II. PRELIMINARIES
In this section, some fundamental concepts are reviewed. FSs, CFSs, IVFSs, IVCFSs, and IVFRs are defined with appropriate examples.
Definition 1 [1]: For a non-empty set , a fuzzy set (FS) on is characterized by a real valued function : → [0, 1], where ( ) is the grade of membership of ∈ . An FS is expressed as:

=
, ( ) : ∈ Definition 2 [22]: For a non-empty set , a complex FS (CFS) on is characterized by a complex valued function C : → { : ∈ C, | | ≤ 1}, where C is the grade of membership of ∈ . A CFS is expressed as: Since C is a complex number, so it is of the from: and ℘ C (x) ∈ [0, 1] are called amplitude term and phase term, respectively.
Remark: CFS is a generalization of FS. By setting phase terms equal to zero in a CFS, we end up with an FS.
Definition 3 [4]: For a non-empty set , an interval-valued FS (IVFS) on is characterized by an interval as a grade of membership ( ) and expressed as: Henceforth, an IVCFS is of the following form: : ∈ Definition 5 [5]: If 1 = j , 1 j : j ∈ and 2 = k , 2 ( k ) : k ∈ for jk ∈ N are two IVFSs on a non-empty set , Then the Cartesian product of 1 and 2 is 1] is the grade of membership of 1 × 2 and defined as: Or equivalently ( 1 × 2) j , k are the left and right endpoints of the membership interval respectively such that Definition 6 [5]: Any subset of the Cartesian product of two IVFSs is known as an IVF relation (IVFR) .

III. MAIN RESULTS
In this section, the novel concepts of IVCFRs, the Cartesian product of two IVCFSs, and the types of IVCFRs are defined such as inverse IVCFR, IVCF reflexive relation, IVCF irreflexive relation, IVCF symmetric relation, IVCF asymmetric relation, IVCF anti-symmetric relation, IVCF transitive relation, IVCF composite relation, IVCF equivalence relation, IVCF order relation, and IVCF equivalence class. Moreover, some results and properties of these concepts are proposed. Besides, the score function of an IVCFS is also defined.
for j, k ∈ N are two IVCFSs on a non-empty set , then the Cartesian product of 1 and 2 is Similarly , symbolize the left and right phase terms of the interval, respectively.
Definition 8: Any subset of the Cartesian product of two IVCFSs is known as an IVCF relation (IVCFR) .
Example 2: Consider two IVCFSs Then the Cartesian product from 1 to 2 is given as The subset of 1 × 2 is an IVCFR given as: the score function ϕ of is defined as: On the other hand, 2 is called an IVCF irreflexive relation if Then, the Cartesian product is Then, the IVCF reflexive relation 1 is And the IVCF irreflexive relation 2 is Example 5: Using Eq. (2), the IVCF symmetric relation 1 , IVCF asymmetric relation 2 and IVCF anti-symmetric relation 3 are Hence, is an IVCF symmetric relation.
On the other hand, let is an IVCF symmetric relation, then Hence, = −1 . Theorem 2: If 1 and 2 are IVCF symmetric relations, then 1 ∩ 2 is an IVCF symmetric relation too.
Proof: Let 1 and 2 are IVCF symmetric relations on an IVCFS . Then by definition of IVCFR, 1 and 2 are subsets of the Cartesian product × .
Hence, 1 ∩ 2 is an IVCF symmetric relation. Definition 13: An IVCFR is called an IVCF transitive relation if Definition 14: Let 1 and 2 be two IVCFRs then an IVCF composition relation 1 • 2 is defined as If Hence, is an IVCF transitive relation.
On the other hand, let Since is an IVCF equivalence relation, so by transitive property Therefore, (4) and (5) ⇒ = • . Theorem 5: If is an IVCF order relation, then the inverse IVCFR −1 is also an IVCF order relation. Proof: To prove that the inverse IVCFR −1 of an IVCF order relation is also an IVCF order relation. We need to check all three conditions of an IVCF order relation for −1 .
1. Using the fact that is an IVCF order relation, so by reflexive property Hence, −1 is an IVCF reflexive relation.
2. Since is an IVCF order relation and thus holds antisymmetry, so Since is an IVCF order relation and thus holds transitivity, so Hence, −1 is an IVCF transitive relation. Thus, −1 holds all the three conditions for an IVCF order relation, so the proof completes.
Definition 16: For an IVCF equivalence relation and ∈ , the equivalence class of modulo is symbolized and defined as Example 9: Consider an IVCF equivalence relation given in (3), Proof: Given that is an IVCF equivalence relation, suppose that The symmetric property of implies that, The transitive property of implies that, The transitive property of implies that, The symmetric property of implies that, Thus, Eq. (6) and Eq. (7) ⇒ x = .

IV. APPLICATIONS
This section presents a couple of applications of the proposed conceptions. The first application uses the concepts of IVCF composite relations to diagnose the patients based on their symptoms. The second application employs IVCFRs to study the life expectancies of patients with different medical conditions. And the effects of multiple illnesses on the life expectancy of a patient are also worked out.

A. MEDICAL DIAGNOSIS
In this subsection, the concept of IVCFR is used for medical diagnosis. Consider the set of patients under inspection, the set of symptoms and the set of diagnosis . The methodology being followed defines a relation 1 between the set of patients and the set of symptoms . 1 takes on grades of membership based on the severity of symptoms the patients are facing. The higher grades of membership indicate a more substantial relation and vice versa. Similarly, another relation 2 is defined between the sets of symptoms and diagnosis .
The grades of membership of 2 indicate the degrees of relationships between the diseases and the symptoms that is determined by the doctor. Finally, to diagnose the patients, the composite relation between 1 and 2 is found. Figure 1 describes the process of modeling and calculations.
In this application, the IVCF composite relations are used to diagnose patients with periodic fever syndrome. Periodic fever syndrome is a set of conditions in which a child has recurrent episodes of fever over time that are generally accompanied by the same symptoms. Every episode of the fever usually lasts roughly the same length of time. These disorders are genetic conditions that are very rare. The most common types of this syndrome in children are periodic fever, aphthous stomatitis, pharyngitis, and familial Mediterranean fever (FMF). Figures 2 and 3 show the areas that these diseases affect.
Aphthous stomatitis or recurrent aphthous ulcers is an oral mucosal lesion that may cause significant morbidity. One or more discrete, shallow and painful ulcers appear on the unattached oral mucous membranes. Each of the ulcers  typically lasts about a week and heals without scarring. Some larger ulcers can scar when recovering and may last several weeks or months. Pharyngitis is an infection of the pharynx or tonsils caused by the pharynx's swelling between the tonsils and larynx. It is triggered by viral infections such as common colds and bacterial infections such as Streptococcus of group A. Viral pharyngitis often clears up on its own within a week or so.
The mesentery is a tissue that connects the intestines to the abdominal wall. Mesenteric adenitis causes inflammation and swelling in the lymph nodes within the mesentery. Lymph nodes are small bean-shaped structures that are part of the lymphatic system, which contribute in immunity. Figure 3 depicts the lymph nodes.
FMF is a genetic illness that causes recurrent episodes of fever, typically accompanied by pain in the abdomen, chest, or joints.   Suppose that five children are sick and under treatment in a hospital. The set of children is = {Ben, Bilal, David, Sara, Sophia}. These kids have different symptoms that reoccur over time. The set of symptoms is = fever, abdominal pain, sore throat, oral cavity ulceration, chest pain .
An IVCFR 1 is defined that relates every patient to each of the symptoms and assigns the grade of membership. fever, abdominal pain, sore throat, oral cavity ulceration, chest pain .
An IVCFR 1 is defined that relates every patient to each of the symptoms and assigns the grades of membership. The amplitude term of the grade of membership indicates the strength and severity of the symptoms, while the phase term represents the period of the symptoms. The values closer to 1 describe very severe symptoms, and the values closer to zero describe mild symptoms. The period is measured in weeks. For phase term equal to 1, the period will be maximum. In this application, the maximum possible period is two weeks. For example, Bilal experiences the abdominal pain of [0.7, 0.9] e 2π[0.3,0.5]i . Here, the amplitude term interval [0.7, 0.9] tells that he experiences very severe abdominal pain. The pain lasts from 3 days to a week as the phase term interval 2π[0.3, 0.5]i translates. These membership values depend on the condition of the patient. Table 1 summarizes the IVCFR 1 between patients and the symptoms. After defining the IVCFRs 1 and 2 , the process of the diagnosis is carried out by calculating the IVCF composite relation 3 = 1 • 2 . The diagnosis is based on the symptoms each child is experiencing. Numerous results appear during the composition process. So, the score function ϕ is used to decide on the choice of suitable grades of membership for each relation.

B. ILLNESS AND LIFE EXPECTANCY
In this subsection, the IVCFRs are used to study the risks and fatality rates of different illnesses. Moreover, the effects of multiple diseases on an individual's life expectancy are explored via IVCFRs. Figure 4 shows the stepwise process that is being followed in this application.
This application considers four diseases; cardiovascular disease (CVD), diabetes, cancer, and chronic obstructive pulmonary disease (COPD). The CIVFS • A contains each of these diseases, and their grades of memberships indicate immortality or non-fatality with respect to the duration of illness. The amplitude term of the grade of membership represents the life expectancy of a patient with a certain illness. On the other hand, the phase term represents the duration of being ill.  Figure 5 portrays the life expectancy of patients with each ill condition. The red bars embody the lower grades of memberships, while the yellow bars embody the upper grades of memberships. But a patient having diabetic conditions accompanied by cardiovascular disease has a higher risk of mortality. The IVCFR helps in finding the rate of mortality. As (CVD, Diabetes) Figure 6 displays the comparison of life expectancies of three different people; a healthy person compared to a diabetes patient and a patient having diabetes VOLUME 9, 2021 accompanied by cardiovascular disease.
An IVCF transitive relation is helpful in situations where an indirect relation is required. Suppose that in the above IVCFR , the effects of CVD and COPD are unknown, then it can be found via IVCF transitive relation between the elements (CVD, Diabetes)  Furthermore, the life expectancy of patients with more than two illnesses is calculated by repeating the previous process, i.e., find the Cartesian product  without repetition to avoid lengthy calculations. Because all the relations are equivalence and thus symmetric, so they are ignored. Also, the IVCFR 1 contains an element of the Cartesian product

V. COMPARATIVE ANALYSIS
In this section, the concept of IVCFRs is compared with some predefined competing structures in the theory of fuzzy sets such as FSs, CFSs, and IVFSs. An FS is a set that is characterized by a fuzzy number called a grade of membership. The associated relations are known as FRs. The grades of membership of an FR represent the effects of the first object over the second in an ordered pair. In the proposed application, the interval values are preferred over the crisp membership grades because the interval provides comfort to a decision-maker while assigning values. It covers the mistakes, errors, and confusion of the decision-makers. In the case of crisp values, a mistake or confusion of a person can lead to erroneous and flawed results.
Another difference is the complex-valued grade of membership. A complex-valued grade of membership has two parts; real and imaginary, called the amplitude term and the phase term, respectively. This composition of complex grades enables them to model problems with dual dimensions. Two parts of a number can represent two different quantities. Usually, the phase term is used to represent the time. The issues in the applications used complex valued grades of memberships to model the issues. The amplitude term is used to tell the strength of symptoms and the level of disease a person is suffering from. The phase term is used to determine the time period of certain conditions. An FR and an IVFR do not have the ability to model such problems that include time. Let us solve the problem with crisp values in the environment of CFSs and CFRs. By using Eq.   Now again, taking the Cartesian product and defining the IVCFR 1 ⊆ B× ∪ B×B× . Table 6 shows the results of 1 . There is always uncertainty in such types of predictions. The exact number for the life expectancy is very difficult to find, which will always lead to some errors in the final calculations. Henceforth to cope with this issue, this paper preferred the interval values.

VI. CONCLUSION
In this paper, the novel concepts of interval valued complex fuzzy relations (IVFRs)) and their types have been devised, including the IVCF equivalence relations, IVCF order relations, IVCF composite relations, and IVCF equivalence classes. Additionally, the Cartesian product of two IVCFSs is also defined. The properties of IVCFRs are argued, and some of their exciting results have been derived. Moreover, these innovative concepts of IVCFRs are utilized to model a couple of in medical situations. A set of sick children were diagnosed with different periodic diseases diagnosis on the basis of the severity and the duration of certain patients' symptoms. Furthermore, another application is proposed that discusses the life expectancies of the ill persons with respect to the duration of being sick. Also, the Cartesian product and IVCFRs are used to examine the life expectancies expectancy of the patients when several diseases affect them. Finally, the advantages of the proposed concept of IVCFRs over the existing frameworks are discussed. In the future, these concepts can be extended to the other generalizations of FSs, which will develop some really interesting structures that could be used in various decision-making processes such as group decision-making, multiattribute and multicriteria decision-making.