Strong Solutions of the Fuzzy Linear Systems

S trong S olutions of the Fuzz y Linear System s Şahin Emrah A mrahov 1 and I m an N. As kerzade 1 Abstract We consider a fuzzy linear system with crisp coefficient matrix and with an arbitrary fuzzy nu m ber in par am etric f orm on the right -hand side. It is known that the well-know n existence and uniqueness theorem of a strong fuzzy sol ution is equivalent t o the following: The coefficient matr ix is the product of a permutation matrix and a diagonal matrix. This means tha t this theorem can be applicable only for a special form of linear systems, namely, only when the system consists of equations, each of which has exactly one variable. We prove an existence and uniq ueness theorem, which can be use on more g eneral system s. The necessary and suffic ient conditions of the theo rem are dependent on both the coefficient matrix and t he right - hand side. This theorem is a generalization of the well-known existence and uniqueness theorem for the strong solution. Keywords : Fuz zy linear sy stem, Fuzzy number, Strong solution, Nonneg ative matrix 1 Introduction Fuzzy linear system s ar ise in many branches of science and t echno logy such as economics, socia l sciences, telecomm unications, image process ing etc. [Hu et al (2000), Zhang et al (2003), Wu et al (2004), Trivedi et al (2005), Zhou et al (2006), Chen et al (2006), Li at al (2007, 2008)]. I n this paper, we consider the fuzzy linear systems whose coefficients matrix i s crisp and right-hand side colum n is an arbitrary fuzzy number vector. These system s have studied by m any authors such as, Peeva (1992), Friedman et al (1998, 200 0), Asady et al (2005), Abbasbandy et al (2007), Ez zati (2008, 2011), Gasilov et al (2009). T here are two approaches for solving the system. Most of the authors try to find th e solution in the form of v ector of fuzzy numbers. But in this approach, there may be examples having solution s which are not vecto rs of fuzzy numbers, regardless t o which method i s used known up t o this point. The broades t m odel 1 Ankara University, Computer Engineering Departme nt in this approach i s given by Friedman et al (1998). They transformed the system into n n 2 2  crisp system by using embeddi ng method of Cong-Xin and Ming (1991). After applying the method, if the found solution is a vector of fuzzy numbers, then the solution is called strong solution, if it is not, then it is called weak sol ution. Friedman et al (2000) also studied duality fuzzy linear systems. They proved necessary and sufficien t conditions i ndepe ndently from the right-hand side of the system for the existence of a strong fuzzy solution. R ecently, Ez zati ( 2011) proposed a new method for solving fuzzy linear systems whose coefficients matrix is crisp and right -hand side column is an arbitrary fuzzy number vector. He uses embedding method of Cong - Xin and Ming t oo. He transform s t he Fredman et al’s (1 998) n n 2 2  system into two n n  systems. Gasilov et al (2009 ) suggested a second approach for linear system s, which is a geom etric approach based on the propertie s of linear transformations. Unlike the first approach, in this approach, the solution is considered to be a fuzzy set of vectors. Therefore, the suggested approach can be applied to all linear square sy stems. Gasilov et al (2009) apply this approach t o non- square linear system s and differential equation systems too (2011) . Comparison analy sis between th ese two approache s is done by Wierzchon (2010) . In this paper we analy ze F riedman et al ’s existence and uniqueness t heorem for strong solution. It is known that these brought up conditions are equival ent to the following : The coefficient matrix can be obt ained by multiplying a permutation matrix by a diagonal matrix. This shows that t hese conditions are applicable only in narrow cases. That is, conditions hold for on ly when the sy stem consists of equations, each o f which has exactly one variable. Geom etric proof of the theorem of Friedm an et al is handled by Gasilov et al (2009). For obtaining an existence theorem which can be used on more general systems; we searched for conditions dependent on both the coefficient matrix and the right-hand side. The existe nce and uniquene ss theorem of the strong solution prov ed in this paper is in fact more general and coincides with the theorem of Friedman et al (1998) in special cases. This paper is organized in 6 sections. In Secti on 2, we state some basic definitions of fuzzy sets and num bers. In Section 3, we define fuzz y linear system s (FLS) and tr ansform FLS to linear system s of crisp equation s as done in Friedm an et al (1998, 200 0). In t his section we also define strong and weak solutions. In Section 4, we prove the necessary and sufficient conditions for existence of unique strong solution of t he auxiliary cri sp linear system correspondin g t o FLS. In Section 5, we give analogous conditions for FLS . In Section 6 we conclude the pape r with some rem arks. 2 Preliminaries As Dubois et al (1978) we define a fuzzy number u in parametric form . Definition 1 . A f uzz y number u in parametric form is a pair   , uu of functions ( ), ( ), 0 1 u r u r r  , which satisfy the following requirements: 1. () ur is a bounded m onotonically increa sing left continu ous function ov er [0,1] 2. () ur is a bounded m onotonically decreas ing left continuou s function ov er [0,1] 3. () ur  () ur , 01 r  The set of all these fuzzy numbers is denoted by 1 E . A popular type of fuzzy number is triangular numbers u = ( , , ) a c b with the membership function , () , xa a x c ca x xb c x b cb                where , c a c b  . For triangular numbers we have r a c a r u ) ( ) (    and ( ) ( ) u r b c b r    . We can represen t a crisp num ber a by () ur = () ur = a , 01 r  . Definition 2 . For two arbitrary fuzzy numbers u and v the equality u = v means that ( ) ( ) u r v r  and ( ) ( ) u r v r  for all   1 , 0  r . Definition 3 . For two arbitra ry fuzzy num bers u and v addition is d efined as ( ( )( ) , ( )( )) ( ( ) ( ), ( ) ( )) u v u v r u v r u r v r u r v r        Definition 4 . For any fuzzy nu m ber u and real number 0 k  multiplication by positive real number is de fined as ( ( ) , ( ) ) k u k u r k u r  Definition 5 . For any fuzzy number u and real number 0 k  multiplication by negative real number is de fined as ( ( ) , ( )) ku k u r k u r  3 Fuzzy linear syste ms Definition 6. Let a crisp n x n coefficient m atrix ( ), 1 , ij A a i j n    and fuzzy numbers 1 ,1 i b E i n    be given. Then the n x n system of equation s 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 ... , ... , ... nn nn n n nn n n a x a x a x b a x a x a x b a x a x a x b          (1) is called a fuzz y linear system (FLS). Definition 7 . A fuzzy number vector 12 ( , , ... ) T n x x x is called a solution of t he fuzzy system if the fuzzy num bers 12 , ,... n x x x satisfy the system ( 1) in the sense of defin itions 2-5. Following Friedm an et al (1998) we introduce the notations below : 12 12 ( , , .. . , , , ... ) T n n x x x x x x x     12 12 ( , , .. . , , , ... ) T n n b b b b b b b     S= ( ),1 , 2 ij s i j n  , where ij s are determined as follows: ij j n i ij n j i ij ij n j n i ij ij ij a s a s a a s a s a               , , , , 0 , , 0 (2) and any ij s which is not det ermined by (2) is zero. Using matrix notation w e have Sx b  (3) The structure o f S implies that 0 ij s  and that BC S CB     (4) where B contains the posi tive elem ents of A , C contains the abso lute value o f th e negative elements of A and A = BC  . An example in the work of Friedm an et al (1998) shows that the matrix S may be singula r even if A is nonsingula r. Theorem1 (Friedm an et al (1998 )) The m atrix S is nonsingular i f and only if the m atrices A = BC  and BC  are both nonsing ular. The following example show s that the solu tion of t he crisp linear system (3) does not define a fuzz y solution of the sy stem (1) even if S is nonsingular. Example . Conside r the following fuzzy system 1 2 1 1 2 2 2 x x b x x b   (5) where 1 11 ( ( ), ( )) b b r b r  and 2 22 ( ( ), ( )) b b r b r  . We will write b riefly 1 11 ( , ) b b b  , 2 22 ( , ) b b b  . The extended m atrix is 1 0 0 1 1 2 0 0 0 1 1 0 0 0 1 2 S        (6) Since 11 12 A      and 11 12 BC     are nonsingular matrices by the Theorem 1, S is nonsing ular matrix. Therefore the crisp system (3) has a unique solution. The solution of the system (3) for the m atrix (6) is     1 1 2 1 21 21 33 x b b b b b          21 1 1 2 1 12 33 x b b b b b          2 2 1 21 12 33 x b b b b          21 2 2 1 21 33 x b b b b     Note that 1 11 ( , ) x x x  and 2 22 ( , ) x x x  are not necessarily fuzzy numbers. From the condition 1 1 xx  we have 21 21 2( ) b b b b    (7) From the condition 2 2 xx  we get 12 12 b b b b    (8) Therefore 1 11 ( , ) x x x  and 2 22 ( , ) x x x  are fuzzy numbers if and only if the inequalities (7) and (8) hold. As in Friedman et al (1998) if t he solution of the crisp system (3) defines fuzzy sol ution to the fuzzy system (1) this solution is calle d strong solution. Definition 8 . If 12 12 ( , , ... , , , ... ) T n n x x x x x x x     is a solution of (3) and for each 1 in  the inequalities i i xx  hold, then the solution 12 12 ( , , ... , , , ... ) T n n x x x x x x x     is called a strong solution of the system (3). Definition 9 . If 12 12 ( , , ... , , , ... ) T n n x x x x x x x     is a solution of (3) and for som e   1, in  the inequality i i xx  hold, th en t he solution 12 12 ( , , ... , , , ... ) T n n x x x x x x x     is ca lled a weak solution of the system (3). 4 Necessary and su fficient conditions for the existence o f a strong solution Let us define 12 ( , , ... ) n b b b b  (9) and 12 ( , , ... ) n b b b b  (10) Theorem 2 . Let BC S CB     be a nonsingula r matrix. The system (3) has a strong solution if and only if   1 ( ) 0 B C b b     (11) Proof . Sim ilar to (9) and (1 0), we can define: 12 ( , , ... ) n x x x x  12 ( , ,... ) n x x x x  From the system (3) we obtain: B C x b C B x b                     Hence B x Cx b  (12) C x B x b    (13) From (12) and (13) we h ave ( ) ( ) B C x B C x b b      ( )( ) B C x x b b     By the Theorem 1, the matrix BC  is nonsingular. Therefore 1 ( ) ( ) x x B C b b      (14) If the system ( 3) has a strong solution t hen, by the Definition 8, we have 0 xx  . Hence the inequality (11) holds. Conversely, if the inequality (11) holds, by (14), we have 0 xx  5 The necessary and sufficient conditions for th e existence of a unique strong solution of FLS By the Theorem s 1 and 2, we hav e the following result: Theorem 3. The FLS (1) has a un ique strong solution if and only if the fol lowing condition s hold: 1) The matrices A = BC  and BC  are both nonsing ular. 2)   1 ( ) 0 B C b b     Example. Conside r the following fuzzy sy stem again. 1 2 1 1 2 2 2 x x b x x b   Here 1 11 ( ( ), ( )) b b r b r  and 2 22 ( ( ), ( )) b b r b r  . For this exam ple, we have the m atrices 11 12 A      10 12 B     01 00 C     11 12 BC     1 21 () 11 BC        The matrices 11 12 A      and 11 12 BC     are nonsingular matrices. Applying the Theorem 3, we note that for existence of a unique strong fuzzy solution, the necessary and sufficient cond ition is 1 1 2 2 21 0 11 bb bb            Hence 1 1 bb  21 21 2( ) b b b b    . Corollary. The system (1) has a unique fuzzy soluti on for an ar bi trary 12 ( , , ... ) T n b b b b  if and only if   1 BC   is nonnegativ e, in other wo rds   1 0 ,1 , ij B C i j n      This result is equivalent to the Lemm a 2 of Fri edm an et al (2000 ) which states that the system (1) has a unique f uz zy sol ution for arbitra ry 12 ( , , ... ) T n b b b b  if and only if 1 S  is nonneg ative. But it turns out that this condition restricts the coefficient matrix A to a very specific case. The entries of the matrices S and   BC  are nonneg ative too. It is a well-know n fact from li near al geb ra [Anton et al (2003)] t hat if a nonsing ular matrix and its inverse are both nonnegative matrice s (i.e. matrices with nonnegative entrie s), then the matrix is a generalized per m utation matrix (monomial matrix). Furthermore, a nonsingular matrix is a generalized pe rmutation m atrix if and only if it can be written as a product of a nonsingular diag onal m atrix and a perm utation m atrix. Henc e we can apply this corollary or Lemma 2 of Friedm an et al (2000) on ly on a small cla ss of FLS. 6 Conclusion Friedman and e t al (1998, 2010) proved severa l necessary and sufficient conditions independent of right- hand side for the existence of a strong solution to the FLS. In this paper, we point out that t he se conditions are applicable only in certain narrow cases. W e have suggested a generalized version of these conditions which ar e additionally dependent on the r ight-hand side of the system. 7 References Abbasbandy, S.; A llahviranloo, T.; Ezz ati, R . ( 2007): A method for solving fuzzy linear genera l systems. The Journal o f Fuzzy Mathemat ics , vol. 15, no. 4, pp. 8 81-889. Anton, H.; Busby, R. C . (2003): Contempora ry Linear Alg ebra . John Wiley . 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