Advances in Intelligent Systems Research (AISR), volume 160
                                   3rd International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2018)
              
                  Approximate Solution of LR Fuzzy Matrix System 
                                                                       1                  1                     2,*
                                                     Dequan Shang , Xingmin Wei  and Xiaobin Guo  
                            1College of Public Health, Gansu University of Traditional Chinese Medicine, Lanzhou 730000, China 
                                2College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China 
                                                                        *
                                                                         Corresponding author 
              
              
                  Abstract—In this paper we proposed a general model for                  In this paper, we consider a class of LR fuzzy matrix 
             solving the fuzzy matrix equation AX=B in which A is a crisp              equation    ~     ~   by a complete matrix method. 
             matrix and B is an arbitrary LR fuzzy numbers matrix. The                           AX  B
             original fuzzy equation is converted to a system of linear matrix                                  II.  PRELIMINARIES 
             equations by the embedding approach. The fuzzy solution is                   Definition 2.1 A fuzzy number ~ is said to be a LR fuzzy 
             derived from solving model and a sufficient condition for the                                                    M
             existence of strong LR fuzzy solution is analyzed. Two illustrating       number if   
             examples are given to show the effectiveness of the proposed 
             method.                                                                                               m  x
                                                                                                               L(        ), x  m ,   0,  
                 Keywords—LR fuzzy numbers; matrix analysis; fuzzy matrix                                      
                                                                                                      ~ ( x)  
             equations; fuzzy approximate solutions                                                   M             x  m
                                                                                                               R(  ), x  m,   0,
                                       I.  INTRODUCTION                                                        
                 To this day, there has a great enormous investigation in the          wherem,  and  are called the mean value, left and right 
             study of fuzzy mathematics. From 1976 to 1991, some                       spreads of ~ , respectively. The function               , which is 
             scholars studied fuzzy mathematics theory and obtained a                               M                                     L(.)
             series of results. Zadeh [1], Dubois et al.[2] and Nahmias [3]            called left shape function satisfies: 
             firstly introduced the concept of fuzzy numbers and arithmetic               (1) L(x)  L(x),  
             operations. Meanwhile, Puri and Ralescu [4], Goetschell et 
             al.[5] and Wu Congxin et al.[6-7] investigated the structure of              (2) L(0)  1 and  L(1)  0,  
             fuzzy number spaces. The uncertainty of the parameters is 
             involved in the process of actual mathematical modeling,                     (3) L(x)   is a non increasing on  [0,        ) . 
             which is often represented by fuzzy numbers. Now the theory 
             and computing method of fuzzy linear systems are still playing               The definition of a right shape function R(.)   is similar to 
             an important role to the fuzzy mathematics and its  that of L(.) . 
             applications.   
                 In 1998, Friedman et al.[8] proposed a method that they                  Definition 2.2 For arbitrary LR fuzzy numbers 
                                                                                         ~                   and ~                   , we have 
             can transform  nn  fuzzy linear systems into  2n2n                       M (m,,)LR              N(n,,)LR
             crisp function linear equation  SX(r)Y(r) by using fuzzy                    (1) Addition 
             set decomposition theorem and embedding method. Later, S.                                ~     ~
             Abbasbandy et al.[9], T. Allahviranloo et al.[10-11], B. Zheng                                                
             et al.[12] studied some specific fuzzy linear systems such as                           M  N  m,,  LR  n, , LR       (1) 
             dual fuzzy linear systems, general fuzzy linear systems, dual                                     (m  n,   ,    )LR .
             full fuzzy linear systems and general dual fuzzy linear systems. 
             In recent years, T. Allahviranloo, N. Babbar et al.[13-14]                        Subtraction 
             proposed many new approaches and theories of fuzzy linear                    (2)
             systems. In 2009, Allahviranloo et al. [15] firstly discussed the                          ~     ~               
             fuzzy linear matrix equations (FLME) with form                                            M  N  m,,  LR  n, , LR     (2) 
             of   ~      ~ by the Kronecker product of the matrices. In 
                AXBC                                                                                             (m  n,   ,    )        .
             2011, Gong and Guo[16] considered a class of fuzzy matrix                                                                         LR
             equations     ~     ~ and obtained its fuzzy least squares 
                         AX  B                                                           (3) Scalar multiplication 
             solutions by the embedding approach. In 2012, M. Otadi et                                     ~
             al.[17]   considered a kind of fully fuzzy matrix                                          M  (m,, )
             equation ~      ~    ~ . Recently, Guo et al.[18-19] proposed a                                                   LR
                       X  A  B                                                                                                                       (3) 
             computing method for complex fuzzy linear                                                           (m, , ),  0,
             systems    ~    ~ and investigated fuzzy matrix equation with                                       
                      Cz  w                                                                                   
             the form ~       ~ based on angular fuzzy numbers.                                                  (m, , )RL,  0.
                       XA  B                                                                                    
              
                                                     Copyright © 2018, the Authors.  Published by Atlantis Press.                               188
                             This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
                                                                                                     Advances in Intelligent Systems Research (AISR), volume 160
                                                                                                                                                                  
                                                                  A matrix ~                        ~ is called a LR fuzzy                                            if                                           else a               0,1 i  m,1 j  n . 
                               Definition 2.3                                           A  (aij )                                                                         a ij  0, a ij  aij                                      ij
                        matrix, if each element ~ of ~ is a LR fuzzy number.                                                                                                                                                            ~            ~
                                                                          aij          A                                                                                     For fuzzy matrix equation XA  B , we can express it as 
                               Definition 2.4 The matrix system 
                                                                                                                                                                                               (A  A)(X, X l, X r)  (B, Bl, Br) .    (7) 
                                                                                              ~          ~                    ~
                                             a            a          a x                             x          x 
                                              11            12                    1n   11               12                   1p                                          Since 
                                                                                              ~          ~                   ~
                                             a            a           a x                            x          x 
                                              21             22                   2n   21               22                   2p  
                                                                                                                                                                                                 (kx ,kxl ,kxr ),                                               k 0,
                                                                                                                                                                                         ~                   ij          ij          ij
                                             a             a           a ~                           ~                   ~                                                           kx                                                                                       ,  
                                              m1             n2                   mn x                 x          x                                                                         ij                               r                l
                                                                                             n1           n2                   np                                                                    (kx ,kx ij, kx ij),                                          k 0
                                                                           ~         ~                    ~                                                                                                      ij
                                                                        b           b         b 
                                                                         11           12                  1p                                                        we have 
                                                                        ~           ~          ~ 
                                                                           b         b                   b
                                                                     21              22                  2p              (4) 
                                                                                                                                                                                                                             l              r
                                                                                                                                                                                    ~           (AX , AX , AX ),                                                           A  0,
                                                                                                                                                                                AX                                                                                                          
                                                                        ~            ~                    ~                                                                                                                      r                  l
                                                                           b          b           b                                                                                               (AX ,  AX ,  AX ),                                                        A  0
                                                                         m1            m2                   mp                                                                                  
                        where                                                               are crisp numbers and ~ ,                                                        So the Eq.(7) can be rewritten as 
                                        a ,1im,1 jn                                                                                             b
                                           ij                                                                                                          ij
                         1im,  1 j pare LR fuzzy numbers, is called a LR                                                                                            A(X, Xl, X r) A(X, Xl, X r)  (AX, AXl, AX r)
                        fuzzy matrix equations (LRFLME). 
                                                                                                                                                                       (AX,AXl,AXr)
                               Using matrix notation, we have                                                                                                                                                  l                r              r                l                    l       r
                                                                                             ~         ~                                                               (A X A X, A X A X , A X A X )(B,B ,B ). 
                                                                                         AX B,                (5) 
                                                                                                                                                                             Thus we get 
                               A fuzzy numbers matrix ~ is called a solution of the fuzzy 
                                                                          ~         X                ~        ~
                        matrix equation (5) if  X satisfies AX  B .                                                                                                                                                AX  AX  B,
                                                                                                                                                                                                                    AX l  AX r  Bl,           (8) 
                                                   III.  SOLVING FUZZY MATRIX EQUATION                                                                                                                             
                               Theorem 3.1 The fuzzy linear system                                                            ~         ~ can be                                                                    AX r  AX l  Br.
                                                                                                                          AX  B                                                                                   
                        extended into the following system of linear matrix equations 
                                                                                                                                                                             or 
                                                       (A  A)X  B,
                                                                                                                                                                                                ( A   A ) X  B,
                                                        A                       A  X l                           Bl        (6)                                                                                                                                       . 
                                                                                                                                                                                          A                       A  X l                          B l 
                                                              A                   A  X r                          Br                                                                                                                                        
                                                                                                                                                                                                 A                   A  X r                         B r 
                                                                                                                                                                                                                                                                      
                        where ~                                l       r    .  
                                       X  (X , X , X )                                                                                                                      Theorem 3.2 Let S belong to Rmnand Cbelong to Rmp. 
                               Proof.  We denote the right fuzzy matrix ~ with                                                                                                                                                     
                           ~                                                                                                                B                         Then the minimal                                         X           solution of the matrix 
                          B  (B, Bl, Br)  (b , blij, brij )                                                   and the unknown  equation SX Cis expressed by 
                                                      ~                    ij                        mp
                        fuzzy matrix X by                                                                                                                                                                                                     
                                                                                                                                                                                                                               X S C ,                (9) 
                                             ~                         l         r          (x , xl , xr )                              . 
                                            X  (X, X , X )                                      ij          ij          ij    np                                   where S is the Moore-Penrose generalized inverse[20] of 
                               We also suppose A  A  A in which the elements                                                                                      matrix S . 
                          aij of matrix A and of aij matrix A are determind by the                                                                                       Proof. The proof is straight forward.   
                        following way:                                                                                                                                       In order to solve the fuzzy matrix equation (5), we need to 
                        if a           0, a               a else a                      0,1 i  m,1 j  n ;                                                   consider the systems of linear equations (6). It seems that we 
                                   ij                   ij          ij                 ij                                                                             have obtained the minimal solution of the function linear 
                                                                                                                                                                      system (10) as 
                         
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                                                      Advances in Intelligent Systems Research (AISR), volume 160
                                                                                        
                               X  AB,                                                          X l     A       ABl         1E FBl 
                                                                            (10)                                                              
                                X l                    Bl                                 r     A       A   r        2F E r
                                       A         A          .                            X                       B                  B  
                                   r                      r 
                                X        A       A  B                                              l       r
                                                                                                  1EB FB 
                                                                                                                    O
                                          ~            l   r               l    r                  2FBl EBr
                 Definition 3.1 Let X  (X , X , X ).  If (X , X , X ) is                                          
             the minimal solution of Eqs.(10), such that X l  0, X r  0, we 
             call ~            l    r  is a strong LR fuzzy minimal solution of           is non negative in nature. i.e. X l  0 and X r  0 . 
                  X  (X , X , X )
             fuzzy equation (5). Otherwise, the said to a weak LR fuzzy                                        IV. 
             minimal solution of fuzzy matrix equation (5) given                                                    NUMERICAL EXAMPLES 
             by         ~    ~ ,                                                              Example 4.1. Consider the following fuzzy matrix system 
                       X  xij
             where                                                                                  ~     ~
                                                                                                   x      x    1      1      (3,1,1)     (2,1, 2)
                                                                                                   11      12                                       .  
                                                                                                   ~     ~                                          
                                                                                                   x21   x22   1     1      (2,1,1)      (1, 0,1) 
                          (x , xlij , xrij ), xlij  0, xrij  0,
                              ij                                                             Let 
                          (x ,0,max{xlij,xrij}), xlij  0, xrij  0,
                    ~         ij                                                                 ~
                    x                                                         (11)              X  (X,Xl,Xr)
                      ij  (x ,max{ xl ,xr },0), xl  0,xr  0,
                               ij           ij     ij        ij         ij                                                                                 
                                                                                                       x     x  xl11       xl12   xr11    xr12 
                                                                                                      11      12  ,             ,              ,
                          (x ,xlij,xrij), xlij  0, xrij  0.                                       x      x   l          l    r         r  
                              ij                                                                       21      22   x 21   x 22   x 21    x 22 
                           i, j  1,, m.                                                                                 1    0  0       1
                                                                                                         A  A  A                        .  
                 Theorem 3.3 If                                                                                           0    1 1        0 
                                                                                                                                             
                  (A  A) (A  A))  0, (A  A) (A  A))  0,
             the fuzzy matrix equation (6) has a strong LR fuzzy minimal                  and 
             solution as follows: 
                                                                                                   ~          l    r    3    2 1 1 1 2  
                                        ~                                                          B  (B, B , B )             ,       ,       .
                                       X  (X , X l, X r )                                                              2    3 1 0 1 1
                                                                                                                                               
             where                                                                            By the Theorem 3.1., the original fuzzy matrix equation is 
                                                                                          equivalent to the following linear system 
                                  X  AB,
                                                                                                        ( A  A)X  B,
                                 Xl                     Bl       (12)                         
                                          A         A         .                                  A          A  X l         Bl   
                                 Xr A              A  Br                                                                         
                                                                                                                        r        r 
                                                                                                         A           A X  B 
                 Proof.  Since Bl and Br are the left and right spreads 
             fuzzy matrix ~ ,       l     and     r      . It means      l    r  is a       The matrix S is obviously not invertible one, its M-P 
                             B B 0             B 0                 (B , B )             inverse is 
             non negative matrix. 
                 Let                                                                                                   1   0    0   1
                                                                                                                    1 0    1    1   0        
                        E   F                                                                               S                       0 .
                 S                                                                                              4 0    1    1   0
                        F    E                                                                                                      
                                                                                                                     1   0    0   1
                    1 (A  A)  (A  A)        (A  A) (A  A)                                                         
                                                                                 
                                                                     .
                    2(A  A ) (A  A )              (A  A ) (A  A ) 
                                                                                            From Eqs. (10), the minimal solution of model is as 
                                                                                          follows: 
                 We know the condition that S  0is equivalent to E  0  
             and         .                                                                                    0.25 0.25 3 2            0.25 0.25
                                                                                                                             
                  F  0                                                                                                                           
                                                                                                                             
                                                                                                  XAB                                           ,
                                                                                                                             
                                                                                                                                                  
                                                                                                              0.25     0.25 2 3         0.25     0.25
                                                                                                                             
                 The product of two non negative matrices                                                                                           
              
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                                                                   Advances in Intelligent Systems Research (AISR), volume 160
                                                                                                             
                                                                                2       2                  [13]  N. Babbar, A. Kumar, A. Bansal, “Solving full fuzzy linear systems with 
                                                                                                                arbitrary triangular fuzzy numbers”, Soft Computing, Vol 17, pp. 
                          X l        A           A  Bl              1 2        3                        691-702, 2013. 
                                                                                                   [14]  R. Ghanbari, “Solutions of fuzzy LR algeraic linear systems using linear 
                          X r        A           A  Br              4 2 3                                 programs”, Applied Mathematical and Modelling, Vol 39, pp. 
                                                                                     
                                                                                2       2                        5164-5173, 2015. 
                                                                                                           [15]  T. Allahviranloo, N. Mikaeilvand, M. Barkhordary, “Fuzzy linear matrix 
                                                                                                                     equations”, Fuzzy Optimization and Decision Making,Vol 37, pp. 
                i.e.,                                                                                                165-177, 2009. 
                                                                                                               [16]  Z.T. Gong, X.B. Guo, “Inconsistent fuzzy matrix equations and its fuzzy 
                      ~       ~                                                                                      least squares solutions,” Applied Mathematical Modelling, Vol.35, pp. 
                      x       x                                         
                     11        12     (0.25,0.50,0.50)             ( 0.25,0.50,0.75)                             1456-1469, 2011. 
                                                                                              .
                    ~        ~                                                                             [17]  M. Otadi, M.Mosleh, “Solving fuzzy matrix equations”,  Applied 
                      x       x                                         
                     21        22     (0.25,0.50,0.50)             ( 0.25,0.75,0.50)                             Mathematical Modelling, Vol. 36, pp. 6114-6121, 2012.   
                               ~                                                                               [18]  X.B. Guo, K. Zhang, “Minimal solution of complex fuzzy linear 
                     Since X is an appropriate LR fuzzy numbers matrix, we                                           systems”, Advances in Fuzzy Systems, pp. 1-9, 2016.   
                obtained the solution of the fuzzy matrix system is a strong LR                                [19]  X.B. Guo, K. Zhang,  “Solving fuzzy matrix equation of the form 
                fuzzy solution by Definition 8.                                                                      XA=B”, Journal of Intelligent and Fuzzy Systems, Vol. 32, pp. 
                                                                                                                     2771-2778, 2017.   
                                                  V.  CONCLUSION                                               [20]  Ben-Israel, T.N.E. Greville, “Generalized inverses: Theory and 
                                                                                                                     applications, second edition”, Springer-Verlag, 2003. 
                     In this work, we discussed a class of LR fuzzy matrix 
                equations         ~       ~ by a full matrix method. The model we 
                               AX  B
                constructed for solving original fuzzy matrix system was made 
                of two crisp systems of linear matrix equations. The existence 
                condition of strong LR fuzzy solution was also studied. Our 
                result will enrich fuzzy linear system theory and can be 
                applied to solve all kinds of fuzzy matrix equations.   
                                                CKNOWLEDGEMENT 
                                             A
                     This research work was supported by the National Natural 
                Science Foundation of China (no.11761062) and the Scientific 
                Research Project of Gansu Province Colleges and Universities 
                                        
                (no.2017A-058).
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