# Fuzzy Matrix Solution (FMS)

Categories: The Matrix

## ABSTRACT

In the paper we analysis the salary of three companies using Fuzzy Matrix Solution (FMS), With the help of product of fuzzy matrix under usual matrix multiplication this define compatible operation Analog to product that the product again happens to be a fuzzy matrix by introducing max-min Operation and min-max Operation. Finally, we conclude the salary of three companies as high or low using fuzzy matrix.

Keywords: FMS, visual basic, indication relation, max-min operation.

## INTRODUCTION

Fuzzy set theory was introduced by Lotfi A.

Zadeh in the year 1956. Fuzzy set in any set which allows having a member in the interval [0,1] and it is known membership function fuzzy environment has the potential to solve uncertainty. In the 20th century, the rise of the service economy made salaried employment even more common in developed countries, where the relative share of industrial production jobs declined, and the share of executive, administrative, computer, marketing salary is typically determined by comparing market pay rater for people performing similar work in similar industries in the same region.

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A salary is a form of payment from an employer to an employee, which may be specified in an employment contract In this paper, we are analyzing the results of company using fuzzy matrix with the help of a product of fuzzy matrices by introducing max-min operation & min-max operation. Finally, we conclude that the Result of the company is high or low.

## DEFINITIONS

### Fuzzy matrix:

A of order mxn is defined as A=[]mxn where aijt is the membership value of the elements aij in A, where A = [aijt]

### Fuzzy set:

A Fuzzy set is a pair (u,m) where u is a set and M:u [0,1] a membership function.

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The reference set u is called universe of discourse and for each XEU, the value m(x)is called grade of m.

### Multiplication of Fuzzy Matrix:

The Product of two fuzzy matrices under usual matrix multiplication is not a fuzzy matrix. So that we need to define compatible operation analogs to product that the product again happens to be a fuzzy matrix. However even for this new operation if the product XY is to be defined we need the number of columns of X is equal to the number of rows of Y. The types of operations which can have are max-min operation and min-max operation.

### CRISP SET:

A set means any well-defined collection of objects. The words aggregate, class or collection are also used in place of the word set of set. But the use of the word set is common. In general capital letters like A, B, C, D.etc are used to denote the sets and lower letters a, b, c, d, to denote the objects or elements belonging to these sets. We express the relation between an object and a set to which it belongs by writing, A. There are three basic methods by which sets can be represented.

List method – A = {a1, a2, a3 an}

Set builder form

A characteristic function. But here we use the first method.

## CALCULATION

Entering data in Matrix format for occurrence relation, confirmative relation, and fuzzy relations denoted by RO, RC and RS respectively.

Matrix for performance relation is RP = CxS, which indicates the frequency of performance of company C which gives the Result salary S.

### COMPANY 1

Calculation of RP

 RP S1 S2 S3 C1 0.6 0.2 0 C2 0.2 0.6 0.2 C3 0.6 0 0.2

Matrix for Confirmative relation RC = W ? S corresponds to the degree to which the Workers’ ability confirms the Salary.

Calculation of RC

 RS S1 S2 S3 W1 1.0 0 0 W2 0 1.0 0 W3 0 0 1.0

Now assume a fuzzy relation RS = C ? W, Specified Marks C1, C2 and C3 for three Students W1, W2 and W3 as follows:

Calculation of RS

 RC C1 C2 C3 W1 0.6 0.2 0.6 W2 0.2 0.6 0 W3 0 0.2 0.2

We can now calculate the result using four indications in four different stages:

• The performance indication relation of result
• The conformability Indication relation of result
• The non-conformability Indication relation of result
• The non-specification Indication relation of result

The performance indication relation of result is calculated by IR1 =RS*RP

 IR1 S1 S2 S3 W1 0.6 0.2 0.2 W2 0.2 0.6 0.2 W3 0.2 0.2 0.2

The conformability Indication relation of result is calculated by IR2=RS*RC

 IR2 S1 S2 S3 W1 0.6 0.2 0.6 W2 0.2 0.6 0 W3 0 0.2 0.2

The non-conformability Indication relation of result is calculated by IR3=RS*(1-RP)

 IR3 S1 S2 S3 W1 0.4 0.6 0.6 W2 0.6 0.4 0.6 W3 0.2 0.2 0.2

The non-specification Indication relation of result is calculated by IR4=(1-RS)*RP

 IR4 S1 S2 S3 W1 0.4 0.6 0.2 W2 0.6 0.4 0.2 W3 0.6 0.6 0.2

### COMPANY II

Entering data in Matrix format for occurrence relation, confirmative relation and fuzzy relations denoted by RO, RC and RS respectively.

Matrix for performance relation is RP = CxS, indicates the frequency of performance of company C which gives the Result salary S.

Calculation of RP

 RP S1 S2 S3 C1 0 0.6 0.4 C2 0.8 0 0.2 C3 0.2 0.4 0.4

Matrix for Confirmative relation RC = WxS corresponds to the degree to which the Workers ability confirms the Salary.

Calculation of RC

 RC S1 S2 S3 W1 1.0 0 0 W2 0 1.0 0 W3 0 0 1.0

Now assume a fuzzy relation RS = C ? W , Specified Marks C1, C2 and C3 for three Students W1, W2 and W3 as follows:

Calculation of RS

 RS C1 C2 C3 W1 0 0.8 0.2 W2 0.6 0 0.4 W3 0.4 0.2 0.4

The performance indication relation of result is calculated by IR1 =RS*RP

 IR1 S1 S2 S3 W1 0.8 0.2 0.2 W2 0.2 0.6 0.4 W3 0.2 0.4 0.4

The conformability Indication relation of result is calculated by IR2=RS*RC

 IR2 S1 S2 S3 W1 0 0.8 0.2 W2 0.6 0 0.4 W3 0.4 0.2 0.4

The non-conformability Indication relation of result is calculated by IR3=RS*(1-RP)

 IR3 S1 S2 S3 W1 0.2 0.8 0.8 W2 0.6 0.4 0.6 W3 0.4 0.4 0.4

The non-specification Indication relation of result is calculated by IR4=(1-RS)*RP

 IR4 S1 S2 S3 W1 0.2 0.6 0.4 W2 0.8 0.4 0.4 W3 0.8 0.6 0.4

### COMPANY III

Matrix for performance relation is RP = CxS, which indicates the frequency of performance of company C which gives the Result salary S.

Calculation of RP

 RP S1 S2 S3 C1 0.8 0 0.2 C2 0.2 0.4 0.4 C3 0.2 0.4 0.4

Matrix for Confirmative relation RC = W ? S corresponds to the degree to which the Workers’ ability confirms the Salary.

Calculation of RC

 RC S1 S2 S3 W1 1.0 0 0 W2 0 1.0 0 W3 0 0 1.0

Now assume a fuzzy relation RS = C ? W , Specified Marks C1, C2 and C3 for three Students W1, W2 and W3 as follows:

Calculation of RS

 RS C1 C2 C3 W1 0.8 0.2 0.2 W2 0 0.4 0.2 W3 0.2 0.4 0.4

The performance indication relation of result is calculated by IR1 =RS*RP

 IR1 S1 S2 S3 W1 0.8 0.2 0.2 W2 0.2 0.4 0.4 W3 0.2 0.4 0.4

The conformability Indication relation of result is calculated by IR2=RS*RC

 IR2 S1 S2 S3 W1 0.8 0.2 0.2 W2 0 0.4 0.2 W3 0.2 0.4 0.4

The non-conformability Indication relation of result is calculated by IR3=RS*(1-RP)

 IR3 S1 S2 S3 W1 0.2 0.8 0.8 W2 0.4 0.4 0.4 W3 0.4 0.4 0.4

The non-specification Indication relation of result is calculated by IR4=(1-RS)*R

 IR4 S1 S2 S3 W1 0.2 0.4 0.4 W2 0.8 0.4 0.4 W3 0.8 0.4 0.4

## REFERENCES

1. M.Geethalakshmi, N.Jose Praveena, A.Rajkumar., result analysis of Students using fuzzy matrices international journal of scientific research publications, volume 2, issue 4, April 2012.
2. Adalassing, K.P., Fuzzy sets theory in Medical diagnosis, IEEE Trans. System, Man, Cybernetics, 16, 1986, 260-265.
3. George J. Klir, Fuzzy sets and Fuzzy logic theory and applications PHI New Delhi.
4. E. Sanchez, Eigen Fuzzy sets and fuzzy relation, J. Math. Anal. Appl. 81, 1981, 399-421