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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 ...

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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 
                                 !;
          ) ,         ) 
                                           ,      *    ,      
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                                                  )                             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. Shimura, Fuzzy sets concept in rank ordering 
                                                     
                                                 )                              0.6
                                             ,      +
                    ⁄                        3 
               ) ) =                                                 =
                   +                                                !;0.6,0.4                    objects, J. Math. Anal.1 Appl., 43 (1973),  
                                 !;
 ) , ) 
                                           ,     +     ,      
                                      =1 3               9                                               [8]         S.  Elizabeth  and  L.  Sujatha  -  Application  Of 
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                                                                                                         Decision Making. 
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                                                  554
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...International journal of pure and applied mathematics volume no issn printed version on line url http www ijpam eu special issue fuzzy matrix with application in decision making c venkatesan p balaganesan j vimala associate professor department maher university faculty engineering technology chennai mathematic amit assistant srinivasan arts science college perambalur venkatmths gmail com balaganesanpp vimalathanshika abstract as is a most important definition comparison scientific social economic endeavour there exist several major approaches within the theories let be set here we have used ranking order to n variables defined universe x from deal vagueness imprecise determination relativity values where s for i are preferences an square called keywords function or rank different sets th smallest value row that introduction membership variable minimum problem decisions possible outcome new information way each rows conditions change time utility will lowest weights purpose thus determi...

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