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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 , * ) ⁄) = - = and Application, MJP publishers. * !;0.3,0.5 !; ) , ) , * , - 7 [2] Bellman.R. and Zadeh.L.A. – “Decision making =0.6 in a fuzzy environment.” Management Science. ) 0.4 , * [3] George J.Klir/Bo Yuan (2005) - “Fuzzy sets and ⁄ 3 ) ) = = * !;0.4,0.9 Fuzzy Logic” Theory and Applications, prentice hall of !; ) , ) , * , 3 7 India Private limited. =0.444 [4] Kim, K.H and Roush, F.W - “Generalized fuzzy ) 0.2 , * ⁄ 9 ) ) = = matrices.” Fuzzy Sets Sys. * + !;0.2,0.95 !; ) , ) , * , + 9 7 [5] Meenakshmi.A.R. and Sriram,S (2003)– “Some =0.211 Remakrs on Regular Fuzzy matrices” Annamalai ) 0.2 , + ⁄ - University, Science Journal. ) ) = = + !;0.2,0.7 !; ) , ) [6] Mizumoto – Fuzzy Theory and its Applications. , + , - 9 Science Publications. =0.286 [7] M. 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