A Novel grouping of Supremum and Infimum Function of a Fuzzy Set

A Novel grouping of Supremum and Infimum Function of a Fuzzy Set

K. Balasubramanian1

1. Department of Mathematics, Kongunadu College of Engineering and Technology,

Thottiam-621215.Trichy, TamilNadu

S. Subramanian2

2. Department of Mathematics, PRIST University,Tanjore.

TamilNadu

Abstract. In this study we suggest a new group decision-making method which is based on some basic fuzzy set operations. By using this method, we can get two types of results. First, we can identify which alternative is the best. The second result, a crucial point of this work, is the screening of decision- makers. The decision-makers should be serious and responsible in giving their opinions otherwise the process will eliminate them because of their inappropriate evaluations to the alternatives compared with that of the other decision-makers. We also discuss an application to demonstrate the process of the method.

Key Words and Phrases: Fuzzy sets; -level set; group decision-making.

INTRODUCTION

An individual or group is frequently faced with the problem o f choosing one alternative from a feasible a l t er nati ve set. For an individual people, t h e problem is the i dentification of the mo s t preferred alternative according to his/her preference structure. However, group decision making, except the above task, another important problem is how to aggregate the experts opinion to obtain an acceptable result for the group. In general, the preference relations take the form of multiplicative preference relations whose e l e m e n t s estimate the d o m i n a n c e of one alte rnati ve over another and take the form of exact numerical values. It is more suitable to provide their preferences by means of linguistic variables rather than numerical ones developed a practical method for group decision making with linguistic preference relations. To aggregate the preference information and to rank the given alternatives presented a method which is called the ordered weighted averaging operator.

SOME PRELIMINARIES

In this section, we describe some preliminary definitions of fuzzy set operations that will be used in this paper.

More details and historical background of fuzzy set theory can be found in [8, 17, 25].

A

In a universe U, a fuzzy set A is defined as

A

A x,

(x): x U ,

(x) [0,1

(1)

A

A A

Where the function (x) is called membership function. The value of the membership function (x) specifies the grade or degree to any element x in U Large value of (x) indicate higher degree of membership. We will identify any fuzzy set with its membership function and use these two concepts as interchanged.

Let A be a fuzzy set in the universe U as in (1). Then the support of A is defined as

A

A

inf A x : x U , (x) 0 & sup A x : x U , (x) 0 (2)

The cardinality of a crisp set A. denoted as A , is the number of elements of the set A and the cardinality of a fuzzy set A is

denoted as card A A (x)

xU

The mean relative cardinality of A is defined as mrcA

card A

(3)

(4)

And then the level set of A is defined as A

x : x U , (x)

sup p A

A

The proposed method

(5)

When using this method first we decision makers give their evaluations according their own opinions for all the considered alternatives in the form of a fuzzy set. Each alternative is evaluated with a value in the interval [0,1] from the point of view of each decision makers.

Let

A a1 ,a 2 …..an be an alternative set and let B b1 ,b2 …..bm

be a decision maker set in a finite universe set

A

U a & U b respect. Then this method can be described by the following steps in k- cycles. Step 1: let the evaluation of decision maker b j Bk

b j

b

j

A , A

( ) : A,

b j

( ) b j ( )

(6)

Step 2: we using the Arithmetic Mean for probability concept for the fuzzy set as

b

k

A ,

Abk

( ): A,

Abk

( ) 1

bk

i

k

b B

Abk

( )

(7)

Step 3: The distances between the sets

Ak (bi )

and

A

b

j

A for all bi Bk as

b

, A (b ) () : A, A (b ) () A (b ) () A (b ) ()

(8)

k

b i b i b i b k

CONCLUSION

The decision-making is a process of choosing the most desirable alternative among a set of alternatives and is, therefore, important in many disciplines including s o c i a l , physical, medical and engi neeri ng sciences. When choosing a preferable alternative, people m u s t set a priority fo r each available a l t e r n a t i v e . This process is not easy for an individual. It is even more difficult when there is more than one decision-maker involved in the process. How to aggregate the individual choices into a group preference has always been a hot topic.

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