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# Application of Circulant Triangular Fuzzy Number Matrix in Swine Flu: Analytical Essay

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## Abstract

The major and challenging process of Swine Flu conﬁrmation has promoted attempts to model it with the use of circulant triangular fuzzy numbers. In this paper, calculate the Six different indications using occurrence relationship (Ro) and conformability relationship (Rc) based on expert medical reports and analysis of related patients with Swine Flu and present some operations on circulant triangular fuzzy numbers matrices (TFNMs). The first row of the circulant(TFNMs) play important role in this study.

## Key Words:

Triangular fuzzy numbers matrices (TFNMs), Circulant triangular fuzzy numbers matrices (CTFNMs), Occurrence relation (Ro), Conformability relation(Rc), and Swine Flu.

## I. Introduction

In 1965, computer scientist L.A.Zadeh used the theory of a fuzzy concept for the first time in a scientific sense. Certain models for understanding and teaching the medical diagnosis process using fuzzy set theory vary in the degree to which they attempt to deal with different aspects of complications, such as

1. Relative important of symptoms,
2. Symptom patterns of disease stage,
3. Relation between diseases themselves,
4. Stages of hypothesis formation,
5. Preliminary and ﬁnal diagnosis within-diagnosis process.

Such models also form the basis for a computerized medical expert program, which is useful for physicians in the diagnosis of certain identified categories of disease, which encourages us to create a model related to SWINE FLU disease, which is commonly observed in India. Implemented the concept TFM. In this paper, we give few basic dentition recalls TFM and CFM and its operations. In section, II discussed the deﬁnition of CTFM and its operations. In section III, Swine Flu diagnosis problems using CTFM relations are discussed. Finally, section IV presents the conclusion of this work.

## II. Preliminaries

### Deﬁnition 2.1.

A TFM represented as, where are all real value denotes the membership height and the left-hand and right-hand spreads of the mean value respectively, and membership height is defined as follows:

A TFN is said to be normalized if = 1 and it can be represented as.

If then is called standardized FN. Throughout this paper, we used normalized TFN.

### Deﬁnition 2.2

TFNM of order m×n is defined as =(aij)m×n where aij = (aijL,aijM,aijU) is the ijth element of.aijL,aijUare the left and right spreads of aij respectively andaijM is the middling value.

### Deﬁnition 2.3

Let = (aij)n×n and = (bij)n×n be two CTFM of same order. then

• (+) = (aij + bij)n×n where aij + bij = (aijL + bijL,aijM + bijM,aijU + bijU) is the ijth element of (+)
• (ii) Subtraction Operation
• (-) = (aij −bij)n×n where aij −bij = (aijL −bijU,aijM −bijM,aijU −bijL) is the ijth element of (-) The same condition holds for CTF membership number.

### Deﬁnition 2.4

Let = (aij)m×p and = (bij)p×n be two CTFNM. Then the Multiplication Operation:

• (.) = (cij)m×n, where (cij) = p X k=1 aik ·bkj for i = 1,2,...,m and j = 1,2,....,n.

### Deﬁnition 2.5

Let and be two fuzzy relations on (p,q) and (q,r) respectively then the max-avg composition is denoted as is defined as

• (P,R) = {[(p,q),max r

### Deﬁnition 2.6

Let = (aij)n×n where aij = (aijL,aijM,aijU) and = (bij)n×n where bij = (bijL,bijM,bijU) be two TFNM of same order. Then the maximum operation on it is given by Lmax = max(,) = (sup{aij,bij}) where sup{aij,bij} = (sup(aijL,bijL),sup(aijM,bijM),sup(aijU,bijU)) is the ijth element of max(,).

### Deﬁnition 2.7

Let = (a1,a2,a3) be a TFN then AM() = . The same condition holds for TM membership number.

### Deﬁnition 2.8

An n×n circulant matrix has the form

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Thus a circulant matrix is determined by its first row.

### Deﬁnition 2.9

A Fuzzy matrix = [aijA᷉ ] is said to be CFM if all the elements of the can are determined completely by its first row. Suppose the first row of is [a1,a2,………..an]Then any element of can be determined [ throughout the element of the first row] as aij= a1(n –i+j+1) with a1(n+ k) = a1k

A CFM is a form of

With entries in [ 0 , 1 ].

### Deﬁnition 2.10

Let A᷉= [aijA᷉ ] ϵ circulant FMm×n, according to the definition in the representation of the complement of the FM which is denoted by ° and then ° is called CF complement matrix if ° = [(1-aijA᷉)]m×n for all aijA᷉ [0,1]. Then the matrix obtained from the so-called membership value would be the following ° = aijA᷉ = [(1- aijA᷉ ] for all i and j.

### Deﬁnition 2.11

A TFNM is said to be a circulant TFNM if all the elements of the can are determined completely by its first row. Suppose the first row of is then any element aij of can be determined (throughout the element of the first row) as aij= a1(n –i+j+1) witha1(n+ k) = a1k

A CTFM in the form of

## III. Medical diagnosis fuzzy matrix

Let be the set of symptoms of certain diseases, is a set of diseases and is a set of patients. The elements of CTFNMare deﬁned as = (aij)m×l where aij = (aijL,aijM,aijU) is the ijth element of , 0 ≤ aijL ≤ aijM ≤ aijU ≤ 10. Here aijL is the lower bound, aijM is the middling value and aijU is the upper bound.

### Procedure 3.1

1. Step 1: Construct a CTFNM(F,) over, where F is a mapping given by F: → F(), and F() is a set of all CTF sets. This matrix is denoted by which is the fuzzy occurrence matrix or symptom-disease CTFM.
2. Step 2: Construct a CTFNM(F,) over, where F is a mapping given by F: → F(), and F() is a set of all CTF sets. This matrix is denoted by which is the fuzzy conformation matrix or symptom-disease CTF number matrix.
3. Step 3: Construct another CTFNM(F1,) over, where F1 is a mapping given by F1: → F(). This matrix is denoted by which is the patient-symptom CTFNM.
4. Step 4: Convert the elements of CTFNM into its membership function as follows: Membership function of aij = (aijL,aijM,aijU) is deﬁned Asif 0 ≤ aijL≤ aijM≤ aijU≤1, where 0 ≤≤ 1. Now the matrix, and are converted into Circulant triangular fuzzy membership matrices namely, and.
5. Step 5: Compute the following relation matrices. = ()mem()()mem it is calculated using Deﬁnition 2.5. = ()mem()()mem and = ()mem() (J(−)()mem), where J is the triangular fuzzy membership matrix in which all entries are (1, 1, 1). (J(−)()mem) is the complement of ()mem and it is called a non-symptom-disease triangular fuzzy membership matrix. and are calculated using subtraction operation and = max{,}. The elements of ,,,is of the form yij = (yijL,yijM,yijU) where 0 ≤ yijL ≤ yijM ≤ yijU ≤ 1. = (−). It is calculated using a subtraction operation. The elements of is of the form zij = (zijL,zijM,zijU) ∈ [−1,1] where zijL ≤ zijM ≤ zijU.
6. Step 6: Calculate = AM(zij) and Row’i = Maximum of ith row which helps the decision maker to strongly conﬁrm the disease for the patient.

### Case Study:

Suppose there are three patients, and in a hospital with symptoms of fever, sore throat, nausea, and vomiting problem. Let the possible diseases relating to the above symptoms be Normal viral fever and early stage of Swine Flu and Final stage of Swine Flu.

Step 1: We consider the set = {,}as universal set where, and represent the symptoms fever, sore throat, nausea and vomiting problem respectively and the set = {,} where, and represent the parameters Normal viral fever and Early stage of Swine Flu and Final stage of Swine Flu. respectively. Suppose that

• F() = [< e1,(1,2,3) >,< e2,(2,3,4) >,< e3,,(1,3,4) >]
• F() = [< e1,(1,3,4) >,< e2,(1,2,3) >,< e3,(2,3,4) >]
• F() = [< e1,(2,3,4) >,< e2,(1,3,4) >,< e3,(1,2,3) >]

The CTFNM(F,) is a parameterized family (F(), F(), F()) of all TFC number matrix over the set S and are determined from expert medical documentation. Thus the CTFNM(F,) represents a relation matrix and it gives an approximate description of the CTFNM medical knowledge of the two diseases and their symptoms given by =

1. Step 2:

Again we take the set = {,}as a universal set where, and represent the symptoms of fever, sore throat, nausea, and vomiting problem respectively and the set = {,} were, and represent the parameters of Normal viral fever and Early stage of Swine Flu and Final stage of Swine Flu respectively. Suppose that

• F() = [,,]
• F() = [,,]
• F() = [,,]

The CTFNM(F,) is a parameterized family (F(), F(), F()) of all triangular fuzzy Circulant number matrix over the set S and are determined from expert medical documentation. Thus the CTFNM(F,) represents a relation matrix and it gives an approximate description of the CTFNM medical knowledge of the two diseases and their symptoms given by =

1. Step 3:

Consider = {,,}as the universal set where ,and represent patients respectively and = {,,} as the set of parameters suppose that, F1() = [,,] F1() = [,< ,(1,3,4) >, ] F1() = [,,] The CTFNM(F1,) is another parameterized family of triangular fuzzy number matrix and gives a collection of approximate description of the patient-symptoms in the hospital. Thus the triangular fuzzy number matrix (F1,) represents a relation matrix called the patient-symptom matrix given by =

1. Step 4:
2. Step 5: The following relation matrices are computed
3. Step 6: From the above analysis it is obvious that patients P1 and P3 suffer from Swine Flu whereas patient P3 faces Normal Viral fever.

## IV. Conclusion

Medicine is one of the fields where fuzzy set theory was recognized early on. The doctor generally gathers patient knowledge from the results of the past history, laboratory test results, and other investigative procedures like x-rays and ultra-sonic rays, etc. The information provided by each of these sources brings varying degrees of ambiguity with it. Thus the best and most accurate definitions of disease entities often use ambiguous linguistic words. Thus in this paper, a Fuzzy set structure has been used to model the medical diagnostic process and decision-making process in several different approaches.

## References

1. L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353.
2. H.K.Baruah, Towards forming a field of fuzzy sets, International Journal of Energy, Information and Communications, 2(1) (2011) 16 - 20.
3. H.K.Baruah, The theory of fuzzy sets: beliefs and realities, International Journal of Energy, Information and Communication, 2(2) (2011) 1 - 22.
4. R.Bellman and L.A.Zadeh, Decision Making in a fuzzy environment, Management Science, 17 (1970) B144-B164.
5. A.O.Esogbue and R.C.Elder, Fuzzy diagnosis decision models, Fuzzy Sets and Systems, 3 (1980) 1 - 9.
6. A.R.Meenakshi, Fuzzy Matrix theory, and applications, MJP Publishers, 2008.
7. E.Sanchez, Resolution of composite fuzzy relation equations, Information and Control, 30 (1976) 38 – 48
8. A.K.Shyamal and M.Pal, Triangular fuzzy matrices, Iranian Journal of Fuzzy Systems, 4(1) (2007) 75-87
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