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International Journal of Pure and Applied Mathematics
Volume 116 No. 23 2017, 551-554
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
Special Issue ijpam.eu
FUZZY MATRIX WITH APPLICATION IN DECISION MAKING
1 2 3
C.Venkatesan, P.Balaganesan, J.Vimala
1 Associate Professor, Department of Mathematics, MAHER University, Faculty of
Engineering and Technology, Chennai
2Associate Professor, Department of Mathematic, AMIT University,Chennai,
3J.Vimala, Assistant Professor, Department of Mathematics, Srinivasan Arts and
Science college, Perambalur,
1 2 3
venkatmths@gmail.com, balaganesanpp@gmail.com, vimalathanshika@gmail.com
Abstract: As fuzzy decision making is a most important Definition 2.2 Comparison Matrix
scientific, social and economic endeavour, there exist
several major approaches within the theories of fuzzy Let = , ……. , , ……… be the set
decision making. Here we have used the ranking order to
of n variables defined on universe X.From a matrix of
deal with the vagueness in imprecise determination of relativity values ⁄ where ’s for i=1 to n, are n
preferences.
variables defined on an universe X.The matrix = a
⁄
square matrix of order n with is called the
Keywords: Decision making, Relativity function,
comparison matrix (or) C- matrix.
Comparison matrix and Ranking. The C-matrix is used to rank different fuzzy sets.
th
The smallest value in the i row of the C- matrix, that is
1. Introduction ′ ⁄
= !"# $ ," = 1 &' # is the membership value
th ′
of the i variable. The minimum of / " = 1 &' # , that
The problem in making decisions is that the possible
outcome, the value of new information, the way the is the smallest value in each of the rows of the C – matrix
conditions change with time, the utility of each outcome- will have the lowest weights for ranking purpose. Thus
ranking, the variables , ……. are determined by
action pair and our preferences for each action is ′ ′ ′
ordering the membership values , ……. .
typically vague, ambiguous and fuzzy.
2. Pre-Requisites 3. Illustrative Example
Definition 2.1 Relativity function A piece of property is evaluated so that it best suits a
client’s needs. Different available pieces of properties
Let x and y be variables defined on a set X. the relativity may have different benefits when compared to each other
⁄ and to the needs of the client. Assume that four pieces of
function denoted as is defined as
the property are available and the client compares from
⁄
= (1)
, criteria ) ,) ,) and ) with each other and to his
needs. * +
Where be the membership function of x with
The pair wise function as follows:
respect to y and be the membership function of y
with respect to x. Then the relativity function is a ) =1 , ) =0.5, ) =0.3 and
, , , *
- - -
measurement of the membership value of preferring (or) 1 =0.2
, +
⁄ -
choosing x over y. The relativity function can be
) =0.8 , ) =1, ) =0.4 and
, , , *
regarded as the membership of preferring variable x over 3 3 3
) =0.6
, +
the variable y. Equation (1) can be extended for many 3
) =0.5, ) =0.9, ) =1 and
, , , *
variables. 7 7 7
) =0.95
, +
7
) =0.7, ) =0.4, ) =0.2 and
, , , *
9 9 9
) =1
, +
9
551
International Journal of Pure and Applied Mathematics Special Issue
Develop a comparison matrix based on this information ) 0.95
, +
⁄ 7
) ) = =
and determine the overall ranking. + * !;0.95,0.2
!;
) , )
, + , *
=1 7 9
Solution The comparison matrix = = < ⁄ ? is given
= >
The relativity function by
⁄
= ) ) ) ) ′ FG
* + @ =min' &ℎE " H'I
!; ,
) 1 1 1 1 1
To find the comparison matrix and ranking:
)
) ⁄) = 1 ; ) ⁄) = 1; ) ⁄) = 1 ; 0.625 1 1 0.667 0.625
* * = ) J K
) ⁄) = 1 * 0.6 0.444 1 0.211 0.211
+ + )
+ 0.286 1 1 1 0.286
) 0.8
,
⁄ 3 The extra column to the right of the comparison
) ) = =
!;0.8 ,0.5
!;
) , )
, , matrix C is the minimum value for each of the rows.
=1 3 - The ranking is ) ,) ,) ;#L ) . The best suits a
* +
) 0.5 client is ) .
,
⁄ 7
) ) = =
* !;0.5,0.3
!;
) , )
, , *
=1 7 - 4. Conclusion
) 0.7
,
⁄ 9
) ) = = The fuzzy decision model in which overall ranking (or)
+ !;0.7,0.2
!;
) , )
, , +
=1 9 - ordering of different fuzzy sets are determined by using
comparison matrix. When we compare objects that are
) 0.5
, fuzzy or vague, we may have a situation where there is a
⁄ -
) ) = =
!;0.5,0.8 contradiction of transitivity in the ranking. This form of
!;
) , )
, ,
- 3 non transitive ranking can be accommodated by means of
=0.625 relativity function which is defined as a measurement of
)
,
⁄ 7
) ) = the membership value of choosing one variable over the
*
!;
) , )
, , *
7 0.9 3 other. Hence in this paper, Fuzzy Matrix with
=!;0.9,0.4 = 1 Applications in Decision Making mainly deals with fuzzy
matrix.
) 0.4
,
⁄ 9
) ) = =
+ !;0.4,0.6 References
!;
) , )
, , +
9 3
=0.667 [1] Meenakshi .A.R (2008), “Fuzzy Matrix” Theory
) 0.3
, *
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) , )
, * ,
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⁄ 3
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, * ,
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552
International Journal of Pure and Applied Mathematics Special Issue
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