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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 nn fuzzy linear systems into 2n2n 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 AXBC (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 ,1im,1 jn b ij ij 1im, 1 j pare LR fuzzy numbers, is called a LR A(X, Xl, X r) A(X, Xl, X r) (AX, AXl, AX r) fuzzy matrix equations (LRFLME). (AX,AXl,AXr) 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 . AX AX B, AX l AX r Bl, (8) III. SOLVING FUZZY MATRIX EQUATION Theorem 3.1 The fuzzy linear system ~ ~ can be AX r AX 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 Rmnand Cbelong to Rmp. 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 mp fuzzy matrix X by X S C , (9) ~ l r (x , xl , xr ) . X (X, X , X ) ij ij ij np where S is the Moore-Penrose generalized inverse[20] of We also suppose A A A in which the elements matrix S . aij of matrix A and of aij 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 189 Advances in Intelligent Systems Research (AISR), volume 160 X AB, X l A ABl 1E FBl (10) X l Bl r A A r 2F E r A A . X B B r r X A A B l r 1EB FB O ~ l r l r 2FBl 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 AB, ( 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 XAB , 0.25 0.25 2 3 0.25 0.25 The product of two non negative matrices 190 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). REFERENCES [1] L.A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning”, Information Science, Vol 8, pp. 199-249, 1975. [2] D. Dubois, H. Prade, “Operations on fuzzy numbers”, Journal of Systems Science, Vol 9, pp. 613-626,1978. [3] S. Nahmias, Fuzzy variables, “Fuzzy Sets and Systems,” Vol 2, pp. 97-111, 1978. [4] M.L. Puri, D.A. Ralescu, “Differentials for fuzzy functions”, Journal of Mathematics Analyisis and Application, Vol 91 , pp. 552-558, 1983. [5] R. Goetschel, W. Voxman, “Elementary calculus”, Fuzzy Sets and Systems, Vol 18, pp. 31-43, 1986 . [6] C.X. Wu, M. Ma, “Embedding problem of fuzzy number space: Part I”, Fuzzy Sets and Systems, Vol 44, pp. 33-38, 1991. [7] C.X. Wu, M. 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