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Solving systems of linear fuzzy equations
Fuzzy Sets and Systems 43 (1991) 33-43 North-Holland
33
Solving systems of linear fuzzy equations J.J.
Buckley
Mathematics Department, University of Alabama at Birmingham, Birmingham, AL 35294, USA
Y. Qu Department of Mathematics and Mechanics, Taiyuan University of Technology, Taiyuan, Shanxi, People's Republic of China Received November 1989 Revised January 1990
Abstract: In this paper _we wish to construct solutions for $ to the fuzzy matrix equation AS =/~ when the elements in A and b are triangular fuzzy numbers. A is square and always non-singular. We argue that the previous method of solving for $, based on the extension principle and regular fuzzy arithmetic, should be abandoned since it too often fails to produce a solution. We present six new solutions of which we show that five are identical. We then adopt the common value X of these five new solutions as our solution to AS =/~. We show that Y? is a fuzzy vector which is the generalization to R n of real fuzzy numbers. We also show how these results pertain to, and extend, previous research on this problem within the area of interval analysis. Keywords: Algebra; fuzzy numbers.
1. Introduction This p a p e r c o n t i n u e s o u r r e s e a r c h on s o l v i n g fuzzy e q u a t i o n s . I n [1], using classical m e t h o d s b a s e d o n t h e e x t e n s i o n p r i n c i p l e , w e i n v e s t i g a t e d s o l u t i o n s to l i n e a r a n d q u a d r a t i c e q u a t i o n s w h e n t h e coefficients w e r e r e a l o r c o m p l e x fuzzy numbers. We concluded that too often these equations do not have a solution. This result p r o m p t e d us to i n v e s t i g a t e o t h e r s o l u t i o n s to fuzzy e q u a t i o n s . I n [2] we p r e s e n t e d o u r n e w s o l u t i o n c o n c e p t a n d s h o w e d t h a t t h e fuzzy q u a d r a t i c e q u a t i o n , w h e r e t h e coefficients a r e all r e a l fuzzy n u m b e r s , a l w a y s has a s o l u t i o n as a real o r c o m p l e x fuzzy n u m b e r . T h e s e results w e r e g e n e r a l i z e d to s y s t e m s o f n o n - l i n e a r e q u a t i o n s in [3]. I n [4] w e c o n t i n u e d to a p p l y o u r n e w s o l u t i o n i d e a to o b t a i n n e w s o l u t i o n s to t h e fuzzy first o r d e r initial v a l u e p r o b l e m . In this p a p e r we a r e c o n c e r n e d with solving t h e m a t r i x e q u a t i o n
Af =/~,
(1)
for f , w h e r e ,4 is a fuzzy m a t r i x a n d f a n d / ~ a r e fuzzy v e c t o r s . F o r t h e d i m e n s i o n s o f t h e p r o b l e m let A = [~i,3] b e a n x n s q u a r e m a t r i x o f real t r i a n g u l a r fuzzy n u m b e r s ~ j a n d let /~t= (bl . . . . . /~n) b e a n x 1 v e c t o r o f r e a l t r i a n g u l a r fuzzy n u m b e r s bi. W e will r e s t r i c t o u r d i s c u s s i o n to s q u a r e ,4 a n d l e a v e f o r f u t u r e r e s e a r c h t h e case w h e r e ,,i is n o t s q u a r e . 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. All rights reserved
J.J. Buckley, Y. Qu
34
We place a bar over a symbol if it represents a fuzzy matrix, vector, or a fuzzy subset of R m, m = 1, 2 . . . . . A real triangular fuzzy number N is defined by the three numbers (n~/n2/n3) where nl < n 2 < n3. Let y = t~(z ] 29) be the membership function for N. Then: (1) /~(z ] N) = 0 outside (n~,/73) and /~(z ] 29) = 1 at z = n2; and (2) the graph of y = (z ] 29) is a straight line from (nl, 0) to (n2, 1) (from (na, 1) to (n3, 0)) on [nl, n2] (on [n2, n 3 ] ) . All our fuzzy numbers aij and b i will be real triangular fuzzy numbers so let tii, = (aijl/aije/aij3) for all i , j and b, = (b~l/bia/bg3) for 1 ~
= (x I
I E)
¢},
(2)
for 0 < ¢ ~< 1. We separately define the 0-cut of/~, written/~(0), as the closure of the union of the /~(oc), 0 < o~~< 1. Now the o¢-cuts of the t/q and/~i are all closed intervals so let ~/iL(a')= [aijl(o¢), aij,(oO] and /~i(a') = [bit(or), bi,(a')]. Of course, flij(O) = [aijl, aij3] b,(O) = [bil, bi3], t~ij(l) = (aq2}, a n d /~i(1) = {bi2 }. If tiq (or bi) reduces to a real number, say a0-1 = aq2 = aij3 = a i j , then tiq(tr) = aq, 0 ~< te ~< 1. Let
a(o:) = FI aij(oO, i,j=l
(3)
b(cr) = ~/)i(o~),
(4)
i=1
for 0 ~< tr ~< 1. The sets a(cr) and b(tr) are the Cartesian product of the tr-cuts aij(oO a n d / ~ ( a 0 , respectively. As a first attempt at solving for £ in Eq. (1) let £t = (£1 . . . . . £n) be a vector of real (not necessarily triangular) fuzzy numbers £~. Let £i(c0 = [xi1(0¢), xi2(tr)], 0 ~< te ~< 1, a closed interval for all a~ and £g(0) = [Xil, xi3]. Substituting £ into Eq. (1) we obtain
/=1
aidj = b,,
(5)
for 1 <~i <~n, where regular fuzzy (based on the extension principle) multiplication and addition is used to evaluate the left-hand side of Eq. (5). Given the t~ij and/)i we now try to solve Eq. (5) for fuzzy numbers £i. Too often this procedure will produce no solution for £. The following example illustrates this phenomenon.
Exmnple 1. ,4 will be 2 x 2 with a l l = ( - 4 / - 2 / 0 ) , ti~2=-(121
= (1/2/3), (~22= 0 (real number zero) and /)1 =/~2 = ( - 1 / 0 / 1 ) . Taking m-cuts of Eq. (5), and then employing interval arithmetic [10] to evaluate the product and sums of intervals,
Solving systems of linear fuzzy equations
35
we obtain for cr = 0, min(0, -4x13) + min(x21, 3x20 = - 1 ,
(6)
max(0, - 4 x 1 0 + max(x23, 3x23) = 1,
(7)
min(-x13, --3X13) + 0 = -- 1,
(8)
m a x ( - x l l , --3X11) + 0 = 1.
(9)
Equations (8) and (9) imply that x ~ < 0 and x13>0. Then - x H = x ~ 3 = ½ . Equations (6) and (7) now imply that min(x21, 3 X 2 1 ) = 1 and max(x23, 3X23 ) = -~. It follows that x21 = 1 and x23 -~- -½ which is impossible. We conclude that using o r - c u t s and interval arithmetic to solve for the £i, Eq. (5) has no solution for the given air and g~. The only paper we know of that gives necessary and sufficient conditions on the tiij and t h e / ~ so that Eq. (5) has a solution for fuzzy numbers £i is [9]. However, these are very strong conditions since, as we have shown in Example 1 above, too often the system has no solution (using regular fuzzy arithmetic) for fuzzy numbers £~. Now let us introduce our first solution I to fuzzy A£ =/~. Set
g2(ol) = {x l A x = b, ai] E ai/(ol), b i E
/~i(l~')},
(10)
for 0 ~< 0¢~< 1. Define 2(1, a fuzzy subset of R n, by its membership function
181) =
sup{
Ix c
(11)
for x c g2(0), and define #(x I X1) = 0 when x does not belong to g2(0). There are two cases to consider: (1) A -1 exists for all aq c ~q(0); and (2) A is singular for some aq e ~q(0). We will only consider case (1) in this paper for the following two reasons. First, if A is singular for air c tiq(0), then the sets g2(cr) can be very complicated and difficult to describe completely. We will discuss this point more in the last section of this paper. So, case (2) becomes a topic for future research. Secondly, the research in interval analysis [6-8; 10, p. 59] describing the solution for x, for aij and b~ in intervals, only consider case (1). Also, the interval analysis research is concerned with only one interval for each aq and each b/. We are working with nested families of intervals for the aq and bi (the ti~j(o¢) and /~i(a~)). Therefore, in this initial paper on finding fuzzy set solutions to fi,£ =/~ we will follow the lead from interval analysis and assume that A is non-singular on a(0). Let I7" be a fuzzy subset of Rm, rn>~l, with membership function #(x I V) : Rm--~ [0, 1]. 17"is a fuzzy vector if and only if: (a) #(x [ I7") is upper semi-continuous (u.s.c.); (b) l?(cr) is compact, arcwise connected, and simply connected for 0 ~< cr ~< 1', and (c) I7"(1)4: 0. 1 A justification of why we called this a 'solution' to the fuzzy matrix equation is postponed until the end of Section 5.
J.J. Buckley, Y. Qu
36
Recall that g : R m ---->R is u.s.c, if and only if {x I g(x) >>-t} is closed for all real t. It is also known that g is u.s.c, if and only if xn---~x implies limsupg(x~)<~g(x). Clearly, I? is the generalization of a fuzzy number to R m, m i> 2. Since 17"will be a fuzzy subset of R m, m/> 2, we can not call it a fuzzy number but instead it will be called a fuzzy vector. In the next section we show that -~1 is a fuzzy vector and Xl(Cr) = f2(cr) all o:. In Section 3 we discuss three other solutions based on the form x = A - l b . We show that two of these are identical to -~'l. In Section 4 we introduce two more solutions based on fuzzy variables and possibility theory and prove that they also equal X1. We return to the classical solution discussed above (based on regular fuzzy arithmetic) in Section 5 and show that, when it exists, it is a subset o f - ~ l Our last section contains a brief summary and conclusions. Let us next introduce the last bit of notation needed in the rest of the paper. Let v = (all, al2 . . . . . ann) E R m, m = n 2. If v ~ a(0), then it produces a corresponding non-singular n × n matrix A = [a~j] whose elements come from v and ~(k) , • • • ) is a sequence in a(0) also v produces A - 1 . Suppose v , = ( a 1 1(k), t~12 converging to v = (all, alE . . . . ) in a(0). Let Vk produce Ak --- [a~k)] and A~-1 and let v produce A = [aij] and A -1. We see that Ak-->A and Akl-->A-k
2. First solution We assume that A -1 exists for all v ~ a(0). Theorem 1.
~r I
is a fuzzy vector.
ProoL (a) Let H:a(tr)×b(oO--->R n as H ( v , b ) = A - l b = x , for v~a(o:), b ~ b(c 0 , 0 ~< a~ ~< 1. Then f2(a 0 = H(a(o 0 × b(tr)), 0 ~< cr ~< 1. Therefore, S'2(tr) is compact, arcwise connected and simply connected because a(tr) x b(tr) has these properties and H is continuous. We next argue, in parts (b) and (c) below, that z'~rl(O~) = ~"~(~), 0 ~ (~" ~ 1.
(b) We first argue that if x ~ g2(0) and #(x IX1) = o~, then: (1) x ~ O(fl) for t r < f l ~ < l , when ~ r < l ; (2) x e K2(fl), 0~0; and (3) x e K2(cr). The first two assertions are obvious so let us prove the third assertion. We may assume that cr > 0. Choose 0 ~< ~n 1' tr and by (2) above we know that x e f2(0cn) all n. From (a) above we may choose (v,,, b,,)ea(oL,,)×b(o6,) so that H(v,, bn)=X all n. The sequence (vn, bn) belongs to compact a ( 0 ) × b ( 0 ) so there is a convergent subsequence (vnK, bnK)'-~(v, b ) e a(oc)× b(oc). Therefore, x,,K= H(v,,~, b,,,~)--->H(v, b_)=x and x e K2(a0. (c) We first show that Xl(O 0 = K2(o0, 0 < o~~< 1. We then argue that Xl(0) = Assume that 0 < o~~< 1. If x e f2(tr), then #(x L.(-1)/> tr so that x e ~'l(aO and C : ~rl(¢~"). Next let x ~ ~[rl(Ot') SO that #(x ] X1) = fl ~ Of. Since #(x I -~l) > 0 we must have x ~ g2(0). From part (3) of (b) we get x ~ g2(fl). But then, from part (2) of (b), we see that x ~ g2(c0 and X l ( o 0 c ~ ( a 0 . ~'-~(~')
Solving systems of linearfuzzy equations
37
We now show that I2(0)= X~(0). Recall that .~'1(0) = closure(o
(12)
Let us first argue that X I ( 0 ) c g2(0). We know that X l ( t r ) = g2(a:)c g2(0) for 0 < cr ~< 1. Hence, the union of the Xl(oc), 0 < tr <~ 1, is also a subset of f2(0). It follows, since g2(0) is compact, that the closure of the union of the X~(o<), 0 < tr ~< 1, which is X1(0) is a subset of g2(0). We next show that ~2(0) c X l ( 0 ) . Let x~g2(0), then ~ ( x l X l ) = t r > ~ 0 . If o<>0, then x e .,Y'~(o:) so that it is in XI(0). So now assume that tr = 0. We construct a sequence x~, each x~ belongs to the union of the X~(o<) and 0 < o<<~ 1, so that x~--->x. Then x e k l ( 0 ) and Q(0) ~,('1(0). There is a (v*, b*) e a(0) x b(0) so that H(v*, b*) = x. The point (v*, b*) must belong to the boundary of a(0) × b(0) since /z(x I X1) = 0. We may construct a sequence (v~, bn), all belonging to the interior of a(0) × b(0), so that (v,, b~)--> (v*, b*). We see that each (v~, b~) is in a(c~,)xb(tr~) for some cry>0. Let x~ = H ( v , , b~). Then x~ e g2(a:,) so that /z(x~ I -~1) ~> o:~ > 0 and x~ e .~'(cr~), for all n. So the sequence Xn belongs to the union of the Xl(o:), 0 < tr ~< 1. But x~ = H(v~, b,)--->H(v*, b*) = x and x belongs to )(l(0). That is, £2(0) c )(1(0). (d) We have shown that Xl(tr) = g2(a0, 0 <~ o: ~< 1. From (a) we conclude that .('1(o:) is compact, simply connected and arcwise connected, 0 ~< tr ~< 1. (e) Clearly, ~'x(1) 4= 0 since n ( a ( 1 ) × b(1)) e g2(1) = .~'1(1). (f) The last thing to show is that y =/~(x IX1) is u.s.c. For each 0 < ce~ < 1 we know that A'l(tr) = g2(tr) which is closed. It follows [11, Proposition 3.3] that ~(x IX1) is u.s.c. One may wonder if the tr-cuts of S l will be convex. The answer is no as shown by an example in [8]. The following simple example illustrates X1, a fuzzy vector solution to A£ =/~. Example 2. fi is 2 x 2 with i~11: t~22 ~---(1/2/3), ~i12= ti2t = 0 (real number zero), and bl = ( - 3 / - 2 / - 1 ) , /~2 = 1 (real number one). Then the determinant of A is never zero on a(0). Given o = ( a l l , 0 , 0, a22) E a(O), b t -- ( b l , 1) with bl ~/~l(0), the solution for x t -- (Xl, x2) is xl = bl/all and x2 = 1/a22. Then
= ((Xl, x2) l -3
1),
(13)
a rectangle. As tr grows from zero to one the rectangle f2(ce) gradually shrinks until g2(1) = { ( - 1 , 1)}.
3. Solutions based on
A-~b
Our next three solutions are based on A - t b = x. However, there are different methods of evaluating A - l b when the ~0 and/~i are fuzzy numbers. Let F( tr) = { Z - l b [ air E (lij(O[), bi
E
b/(o~')},
(14)
38
J.J. Buckley, Y. Qu
0 ~ tr ~< 1. F(tr) is called the united extension of the function F(v, b) = A - l b [10, p. 19]. Our second solution X2, a fuzzy subset of R n, is defined by its membership function ~(x 1~'2) = sup{re Ix • F(cr)}.
(15)
We may also evaluate A - l b , when A and /~ are fuzzy, using the extension principle. Let at(v) =
min
l~i,j<-n
{/~(aoltiij)} ,
(16)
at(b) = min {g(b~ I/~,)},
(17)
at(v, b ) = min{at(v), at(b)}.
(18)
l<.i<_n
and Define X3, a fuzzy subset of R", by its membership function ~t(x 1£'3) = sup(at(v, b) I a - l b = x } .
(19)
We have shown that X'2 =/t'3 and Xz(tr) =/~'3(tr) = F(o¢), 0 < tr ~< 1, in [5]. Still another method of evaluating A-~b, for fuzzy A and/~, is to substitute the intervals ~ij(a~) and/~(tr) for the aij and bi in A -1 and b and compute the result using interval arithmetic. For example, if A is 2 × 2 we would have F(v, b) = (xl, x2) t where x, = (az2b, - alzbz)/ A,
(20)
x2 = (-aeab, + a,lb2)/A,
(21)
for A = allaz2-a12a21. This is simply Cramer's Rule for finding x. We would substitute the intervals t~ij(ol) and/~i(tr) for the aij and b~ in Eq. (20) and (21) and compute intervals for x~ and x2 using interval arithmetic. We now need to generalize this simple example and review some results from interval analysis (see [101). Suppose F(v, b) = (fl(v, b) . . . . . f,,(v, b)) where X i =fii(l), b), 1 <~i ~ n. Each f,-(v, b) is x~ computed using Cramer's Rule. That is, f ( v , b) is the determinant of A, with its ith column replaced by b, divided by the determinant of A. Therefore, each f is a rational function of the aij and b~. The united extension of f is fii(ot) = { f ( v , b) I, (v, b) • a ( a ) x b(cr)},
(22)
for 0 ~ cry< 1, l<_i<~n. Next let f / b e the natural interval extension off,., 1 ~< i ~
Solving systems of linear fuzzy equations
39
An interval valued function G of interval variables /1 . . . . . I,, is inclusion monotonic if and only if li c Ji, 1 <<-i <- m, implies that G(I1 . . . . . Im) c G(J1. . . . . . J,,). Now each ~ is inclusion monotonic because it is the natural interval extension of rational function f~ ([10, p. 21]). Therefore, from Theorem 3.1 of [10] we conclude that2 f/(a0 c yi(tr), 0<~ o¢ << - 1, 1 <~i <-n. We would hope that the yi(ol), 0~< c~< 1, form the o~-cuts of some fuzzy number, 1 ~
(23)
for 1 ~
(24)
l~i~n
,~r 4 is a fourth possible solution to fits =/~ when .4 and/~ are fuzzy.
We now argue that X'2(tr)~X4(tr), 0 < c r < ~ l , X2(0¢) 4= d('4(tr). We show this result from proving
and it may happen that
n
l"( Ol) C [ I fii( OC) ~ ~-I Yi( 0¢) ~ i=1
i--1
~'~4(0(),
(25)
0 < t r ~ < l . We first show that F(ol)cIIT=~f(c~), 0 < t r < ~ l . Let x e F ( o : ) . Then xi=fii(v, b) for some v e a(tr) and b e b(ol), l<~i<-n. Hence, x belongs to FlP_-lf~(tr). We next show that I-[~'=lyi(tr) c)~4(a0. Let x e II~'=ayi(a0. Then xi • yi( o:), and lz(xi I Yi) >~ ol, 1 <<-i <~ n. Hence x e )(4(tr). We get X2(cr) = )(4(re), 0 < t r ~ 1 since F(rr)=/t'2(0¢) for 0 < a ~< 1 and we have already noted that f,.(cr) cy~(tr) for all cr and i. There are simple examples [3;5; 10, p. 22] of the united extension (here f~(tr)) being a proper subset of the interval extension (here yi(oc)). Also X2(tr) may not equal [l~'=lf/(cQ since I]~'=lf/(tr) will be a 'rectangle' in R ~ and X2(0¢) need not be 'rectangular'. Hence, we conclude that ,('2(a0 may not equal ,('4(tr), for 0 < tr < 1. We will not u s e .~r4 as a solution to the fuzzy matrix equation ,4S =/~. In practice, since the end points of the intervals y~(cr) may be computed in a finite number of arithmetic steps, y~(cQ is used to approximate f(tr), 1 ~
Proof. Clearly, f2(0¢) = F(tr), 0 <~ tr <~ 1, so X~ = X2. We have already noted that ~'Y2 ~ )~'3"
4. Solutions based on fuzzy variables
Our next two solutions are based on fuzzy variables and possibility theory. Consider real-valued fuzzy variables t/i~, 1 ~
40
J.J. Buckley, Y. Qu
values are restricted by possibility distributions tz(aijla~j), l < ~ i , j < ~ n , and Pt(bi I bi), 1 ~ i <- n, respectively. This means that Poss[a# - a#] = I*(aij ] ~#) for all i, j and Poss[/~i = bi] = Iz(bi ] bi) for all i. That is, the (li](bi) are now considered possibility distributions for the fuzzy variables t~ij(~) whose values are to be the a~j(b~) in the equations A x = b and x = A - l b . Assuming the ~ and ~i are all non-interactive [12], and joint possibility distribution is :r(v, b). Let S be a fuzzy variable whose values are possible solutions for x in the equation A x = b, when (v, b) • a(0) x b(0). Then define
Poss[,_¢ =x] = sup{at(v, b) lAx = b },
(26)
if A x = b for some (v, b) • a(0) × b(0), and set Poss[~{ = x] = 0 otherwise. Next let J" be a fuzzy variable whose possible values are the values of x given by A - l b , for (v, b) • a(0) x b(0). Then define
Poss[7" = x] = sup{g(v, b) I A - l b = x},
(27)
if A - l b = x for some (v, b) • a(0) x b(0), and set Poss[T = x] = 0 otherwise.
Theorem 3. (a) Poss[ { = x I = / , ( x I X1). (b) Poss[ = x] = I R3). Proof. (a) We argue that if Poss[• = x], or ~u(x [ 41) are positive (or zero), then the other is positive (or zero) and they are equal. Let/~(x [ 41) = tr > 0. We show that Poss[3 = x] I> a~. Now x • g2(tr) so there is a (v, b) • a(te) x b(tr) so that A x = b. Then ~r(v, b)/> a: and Poss[S = x]/> tr. Let Poss[S = x] = a~> 0. We show that #(x [ 41) I> tr. Let e > 0 be given. There is a fl, tr ~ fl i> tr - e, and a (v, b) • a(0) x b(0) so that ~r(v, b) = fl and A x = b. It follows that ( v , b ) • a ( f l ) x b ( f l ) and x • ~ ( f l ) . Hence ~u(x[41) l>/3 or /t(x ] 41) ~> tr - e. The result follows since e > 0 was arbitrary. (b) The proof is similar to (a) and is omitted.
We have shown that the two new solutions based on possibility theory agree with 41 = X2 = 43. Let 4 be the common value of all five solutions. 4 is then our new solution to the fuzzy matrix equation. Theorem 3 presents us with another justification of our new solution X. Consider solving A x = b for x when the values of the aij and bi are uncertain. We may model this uncertainty using random variables or fuzzy variables. In this paper we choose fuzzy variables tJij and/~,- to model this uncertainty. It is natural then to find all solutions for x using all the values of aq and bi at the same level of uncertainty. The values of the aq and b / a t the same level of uncertainty belong to the sets {aiy [ Poss[tJiy = aq] ~> 5} and {b i [ Poss[~i -- bi] >t ol}, respectively. However, these sets are identical to ~i~(ac) and /~(o¢), respectively. Therefore, we obtain Q(ct) = F(a 0 and our new solution X.
Solving systems of linear fuzzy equations
41
5. Classical solution This solution is based on Eq. (5). Let Z~, 1 ~
(28)
=
j=l
1 ~
[aqt( oO, aij,( oO][zjl( oO, zj2(a0] = [bil( O:), bi,(a0],
(29)
j=l
for 0 <~ o~~< 1, 1 <~i ~ n. Interval multiplication and addition is used to evaluate the left-hand side of Eq. (29). We now solve Eq. (29) for the zi~(ol), z~2(o0, 1 ~
(30)
Theorem 4. )(5 ~<) ( Proof. We show that )(s(a) c ~2(0c) = X(a0, 0 ~ o~~< 1. Let x • Xs(o 0. Then #(xi I Zi) ~> o~ for all i so that xi belongs to Zi(o 0, 1 <~ i <~n. That is, xi belongs to [zil(o0, zi2(oO], 1 ~ i <~n. Choose v in a(c 0. If v = (all . . . . . %,), then air belongs to [aql(a0, a0,(c0], 1 ~
j=l
aijxj • [bit( oO, bi,(o0],
l<~i<~n. That is, there is a v • a ( a : ) and a b • b ( o : ) so that A x = b .
(31) Hence
x • f2(o0.
Now Xs(cr)= II~'=1Zi(cr), 0<~ o~<~1, so o~-cuts of )(s(o0 are rectangles in R n. Since 2((o~) will Usually not be a rectangle in R" we would expect )(s(cr) to be a proper subset of ~2(_o0. Hence, we would usually expect A's to not equal X. We will not use Xs as a solution to the fuzzy matrix equation because: (1) too often it does not exist; and (2) when it does exist it may be a_proper subset of )(. There is still another problem associated with the Zi and Xs (see also [3, 5]). Suppose the solutions for the z~l(a) and z~2(o0 do define fuzzy numbers Zq. Now evaluate
aij2, = w/, j=l
(32)
42
J.J. Buckley, Y. Qu
1 ~ i ~ n, using the extension principle. It may happen that some I~,. does not equal/~i. -~s will not equal ,~ when the 2i are interactive. When the 2~ are noninteractive [12], then Eq. (30) gives the correct joint distribution of the Zi. However, when the Z, are interactive (which is usually the case) Eq. (30) does not produce the correct joint distribution. We believe that X gives the correct joint distribution for the solution to the fuzzy matrix equation. Let ,~7/~be the projection of ~" onto the x,-axis, 1 ~< i ~
6. Summary and conclusions This paper is concerned with finding solutions for ~ to the matrix equation .4~ -- 6 when the dij in ,4 and the/)i in/~ are triangular fuzzy numbers. We argued that the classical solution, based on the extension principle and regular fuzzy arithmetic, should be rejected since it too often fails to exist. Our new solution always exists. The discussion was restricted to square .4 which is always non-singular. We defined six new solutions and showed that five of them are identical and then adopted their common value ,~" as our solution to the fuzzy matrix equation. We showed that ,(" is a fuzzy vector, which is the generalization of a real fuzzy number to R n. If A turns out to be singular for some a~j in the s-cut of triangular fuzzy number d o then .(" may be far more complicated. The definition of X (from ~2(a0 in the Introduction) is still valid when A may be singular. Preliminary results show that s-cuts of .~" are comprised of components which may not be closed, can be unbounded, and could be overlapping (not disjoint). Future research will be concerned with describing .~" when A can be singular and when .4 is not square. The results of this paper are also important within the area of interval analysis. In interval analysis [6-8; 10, p. 59] the researchers are interested in solving A x -- b for x when the a~j in A and the b~ in b belong to intervals. They consider only one interval for each a# and only one interval for each bi. In this paper we did the same thing but for a nested family of intervals for each a~j and bg (the cr-cuts of the triangular fuzzy numbers) and showed how to put the solutions for x together to obtain fuzzy vector .('.
Acknowledgment The authors wish to thank an anonymous referee for suggestions to shorten the proof of T h e o r e m 1 and supplying reference [11].
Solving systems of linear fuzzy equations
43
References [11 J.J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38 (1990) 43-59. [2] J.J. Buckley and Y. Qu, Solving fuzzy equations: a new solution concept, Fuzzy Sets and Systems 39 (1991) 291-301. [3] J.J. Buckley, Solving fuzzy equations, J. Math. Anal. Appl., under review. [4] J.J. Buckley and Y. Qu, Fuzzy differential equations: new solutions, J. Math. Anal. Appl., under review. [5] J.J. Buckley and Y. Qu, On using or-cuts to evaluate fuzzy equations, Fuzzy Sets and Systems 38 (1990) 309-312. [6] E. Hansen, Interval arithmetic in matrix computations, Part I, SlAM J. Namer. Anal. 2 (19651 308-320. [7[ E. Hansen, Interval arithmetic in matrix computations, Part II, SlAM J. Namer. Anal. 4 (1967) 1-9. [8] E. Hansen, On the solution of linear algebraic equations with interval coefficients, Linear Algebra Appl. 2 (1969) 153-165. [9] H. Jiang, The approach to solving the simultaneous linear equations that coefficients are fuzzy numbers, J. Nat. Univ. Defence Technol. 3 (1986) 93-102 (in Chinese). [10] R.E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics (SIAM, Philadelphia, PA, 1979). [11] M.D. Weiss, Fixed points, separation, and induced topologies for fuzzy sets, J. Math. Anal. Appl. 50 (1975) 142-150. [12] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.
33
Solving systems of linear fuzzy equations J.J.
Buckley
Mathematics Department, University of Alabama at Birmingham, Birmingham, AL 35294, USA
Y. Qu Department of Mathematics and Mechanics, Taiyuan University of Technology, Taiyuan, Shanxi, People's Republic of China Received November 1989 Revised January 1990
Abstract: In this paper _we wish to construct solutions for $ to the fuzzy matrix equation AS =/~ when the elements in A and b are triangular fuzzy numbers. A is square and always non-singular. We argue that the previous method of solving for $, based on the extension principle and regular fuzzy arithmetic, should be abandoned since it too often fails to produce a solution. We present six new solutions of which we show that five are identical. We then adopt the common value X of these five new solutions as our solution to AS =/~. We show that Y? is a fuzzy vector which is the generalization to R n of real fuzzy numbers. We also show how these results pertain to, and extend, previous research on this problem within the area of interval analysis. Keywords: Algebra; fuzzy numbers.
1. Introduction This p a p e r c o n t i n u e s o u r r e s e a r c h on s o l v i n g fuzzy e q u a t i o n s . I n [1], using classical m e t h o d s b a s e d o n t h e e x t e n s i o n p r i n c i p l e , w e i n v e s t i g a t e d s o l u t i o n s to l i n e a r a n d q u a d r a t i c e q u a t i o n s w h e n t h e coefficients w e r e r e a l o r c o m p l e x fuzzy numbers. We concluded that too often these equations do not have a solution. This result p r o m p t e d us to i n v e s t i g a t e o t h e r s o l u t i o n s to fuzzy e q u a t i o n s . I n [2] we p r e s e n t e d o u r n e w s o l u t i o n c o n c e p t a n d s h o w e d t h a t t h e fuzzy q u a d r a t i c e q u a t i o n , w h e r e t h e coefficients a r e all r e a l fuzzy n u m b e r s , a l w a y s has a s o l u t i o n as a real o r c o m p l e x fuzzy n u m b e r . T h e s e results w e r e g e n e r a l i z e d to s y s t e m s o f n o n - l i n e a r e q u a t i o n s in [3]. I n [4] w e c o n t i n u e d to a p p l y o u r n e w s o l u t i o n i d e a to o b t a i n n e w s o l u t i o n s to t h e fuzzy first o r d e r initial v a l u e p r o b l e m . In this p a p e r we a r e c o n c e r n e d with solving t h e m a t r i x e q u a t i o n
Af =/~,
(1)
for f , w h e r e ,4 is a fuzzy m a t r i x a n d f a n d / ~ a r e fuzzy v e c t o r s . F o r t h e d i m e n s i o n s o f t h e p r o b l e m let A = [~i,3] b e a n x n s q u a r e m a t r i x o f real t r i a n g u l a r fuzzy n u m b e r s ~ j a n d let /~t= (bl . . . . . /~n) b e a n x 1 v e c t o r o f r e a l t r i a n g u l a r fuzzy n u m b e r s bi. W e will r e s t r i c t o u r d i s c u s s i o n to s q u a r e ,4 a n d l e a v e f o r f u t u r e r e s e a r c h t h e case w h e r e ,,i is n o t s q u a r e . 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. All rights reserved
J.J. Buckley, Y. Qu
34
We place a bar over a symbol if it represents a fuzzy matrix, vector, or a fuzzy subset of R m, m = 1, 2 . . . . . A real triangular fuzzy number N is defined by the three numbers (n~/n2/n3) where nl < n 2 < n3. Let y = t~(z ] 29) be the membership function for N. Then: (1) /~(z ] N) = 0 outside (n~,/73) and /~(z ] 29) = 1 at z = n2; and (2) the graph of y = (z ] 29) is a straight line from (nl, 0) to (n2, 1) (from (na, 1) to (n3, 0)) on [nl, n2] (on [n2, n 3 ] ) . All our fuzzy numbers aij and b i will be real triangular fuzzy numbers so let tii, = (aijl/aije/aij3) for all i , j and b, = (b~l/bia/bg3) for 1 ~
= (x I
I E)
¢},
(2)
for 0 < ¢ ~< 1. We separately define the 0-cut of/~, written/~(0), as the closure of the union of the /~(oc), 0 < o~~< 1. Now the o¢-cuts of the t/q and/~i are all closed intervals so let ~/iL(a')= [aijl(o¢), aij,(oO] and /~i(a') = [bit(or), bi,(a')]. Of course, flij(O) = [aijl, aij3] b,(O) = [bil, bi3], t~ij(l) = (aq2}, a n d /~i(1) = {bi2 }. If tiq (or bi) reduces to a real number, say a0-1 = aq2 = aij3 = a i j , then tiq(tr) = aq, 0 ~< te ~< 1. Let
a(o:) = FI aij(oO, i,j=l
(3)
b(cr) = ~/)i(o~),
(4)
i=1
for 0 ~< tr ~< 1. The sets a(cr) and b(tr) are the Cartesian product of the tr-cuts aij(oO a n d / ~ ( a 0 , respectively. As a first attempt at solving for £ in Eq. (1) let £t = (£1 . . . . . £n) be a vector of real (not necessarily triangular) fuzzy numbers £~. Let £i(c0 = [xi1(0¢), xi2(tr)], 0 ~< te ~< 1, a closed interval for all a~ and £g(0) = [Xil, xi3]. Substituting £ into Eq. (1) we obtain
/=1
aidj = b,,
(5)
for 1 <~i <~n, where regular fuzzy (based on the extension principle) multiplication and addition is used to evaluate the left-hand side of Eq. (5). Given the t~ij and/)i we now try to solve Eq. (5) for fuzzy numbers £i. Too often this procedure will produce no solution for £. The following example illustrates this phenomenon.
Exmnple 1. ,4 will be 2 x 2 with a l l = ( - 4 / - 2 / 0 ) , ti~2=-(121
= (1/2/3), (~22= 0 (real number zero) and /)1 =/~2 = ( - 1 / 0 / 1 ) . Taking m-cuts of Eq. (5), and then employing interval arithmetic [10] to evaluate the product and sums of intervals,
Solving systems of linear fuzzy equations
35
we obtain for cr = 0, min(0, -4x13) + min(x21, 3x20 = - 1 ,
(6)
max(0, - 4 x 1 0 + max(x23, 3x23) = 1,
(7)
min(-x13, --3X13) + 0 = -- 1,
(8)
m a x ( - x l l , --3X11) + 0 = 1.
(9)
Equations (8) and (9) imply that x ~ < 0 and x13>0. Then - x H = x ~ 3 = ½ . Equations (6) and (7) now imply that min(x21, 3 X 2 1 ) = 1 and max(x23, 3X23 ) = -~. It follows that x21 = 1 and x23 -~- -½ which is impossible. We conclude that using o r - c u t s and interval arithmetic to solve for the £i, Eq. (5) has no solution for the given air and g~. The only paper we know of that gives necessary and sufficient conditions on the tiij and t h e / ~ so that Eq. (5) has a solution for fuzzy numbers £i is [9]. However, these are very strong conditions since, as we have shown in Example 1 above, too often the system has no solution (using regular fuzzy arithmetic) for fuzzy numbers £~. Now let us introduce our first solution I to fuzzy A£ =/~. Set
g2(ol) = {x l A x = b, ai] E ai/(ol), b i E
/~i(l~')},
(10)
for 0 ~< 0¢~< 1. Define 2(1, a fuzzy subset of R n, by its membership function
181) =
sup{
Ix c
(11)
for x c g2(0), and define #(x I X1) = 0 when x does not belong to g2(0). There are two cases to consider: (1) A -1 exists for all aq c ~q(0); and (2) A is singular for some aq e ~q(0). We will only consider case (1) in this paper for the following two reasons. First, if A is singular for air c tiq(0), then the sets g2(cr) can be very complicated and difficult to describe completely. We will discuss this point more in the last section of this paper. So, case (2) becomes a topic for future research. Secondly, the research in interval analysis [6-8; 10, p. 59] describing the solution for x, for aij and b~ in intervals, only consider case (1). Also, the interval analysis research is concerned with only one interval for each aq and each b/. We are working with nested families of intervals for the aq and bi (the ti~j(o¢) and /~i(a~)). Therefore, in this initial paper on finding fuzzy set solutions to fi,£ =/~ we will follow the lead from interval analysis and assume that A is non-singular on a(0). Let I7" be a fuzzy subset of Rm, rn>~l, with membership function #(x I V) : Rm--~ [0, 1]. 17"is a fuzzy vector if and only if: (a) #(x [ I7") is upper semi-continuous (u.s.c.); (b) l?(cr) is compact, arcwise connected, and simply connected for 0 ~< cr ~< 1', and (c) I7"(1)4: 0. 1 A justification of why we called this a 'solution' to the fuzzy matrix equation is postponed until the end of Section 5.
J.J. Buckley, Y. Qu
36
Recall that g : R m ---->R is u.s.c, if and only if {x I g(x) >>-t} is closed for all real t. It is also known that g is u.s.c, if and only if xn---~x implies limsupg(x~)<~g(x). Clearly, I? is the generalization of a fuzzy number to R m, m i> 2. Since 17"will be a fuzzy subset of R m, m/> 2, we can not call it a fuzzy number but instead it will be called a fuzzy vector. In the next section we show that -~1 is a fuzzy vector and Xl(Cr) = f2(cr) all o:. In Section 3 we discuss three other solutions based on the form x = A - l b . We show that two of these are identical to -~'l. In Section 4 we introduce two more solutions based on fuzzy variables and possibility theory and prove that they also equal X1. We return to the classical solution discussed above (based on regular fuzzy arithmetic) in Section 5 and show that, when it exists, it is a subset o f - ~ l Our last section contains a brief summary and conclusions. Let us next introduce the last bit of notation needed in the rest of the paper. Let v = (all, al2 . . . . . ann) E R m, m = n 2. If v ~ a(0), then it produces a corresponding non-singular n × n matrix A = [a~j] whose elements come from v and ~(k) , • • • ) is a sequence in a(0) also v produces A - 1 . Suppose v , = ( a 1 1(k), t~12 converging to v = (all, alE . . . . ) in a(0). Let Vk produce Ak --- [a~k)] and A~-1 and let v produce A = [aij] and A -1. We see that Ak-->A and Akl-->A-k
2. First solution We assume that A -1 exists for all v ~ a(0). Theorem 1.
~r I
is a fuzzy vector.
ProoL (a) Let H:a(tr)×b(oO--->R n as H ( v , b ) = A - l b = x , for v~a(o:), b ~ b(c 0 , 0 ~< a~ ~< 1. Then f2(a 0 = H(a(o 0 × b(tr)), 0 ~< cr ~< 1. Therefore, S'2(tr) is compact, arcwise connected and simply connected because a(tr) x b(tr) has these properties and H is continuous. We next argue, in parts (b) and (c) below, that z'~rl(O~) = ~"~(~), 0 ~ (~" ~ 1.
(b) We first argue that if x ~ g2(0) and #(x IX1) = o~, then: (1) x ~ O(fl) for t r < f l ~ < l , when ~ r < l ; (2) x e K2(fl), 0~
Solving systems of linearfuzzy equations
37
We now show that I2(0)= X~(0). Recall that .~'1(0) = closure(o
(12)
Let us first argue that X I ( 0 ) c g2(0). We know that X l ( t r ) = g2(a:)c g2(0) for 0 < cr ~< 1. Hence, the union of the Xl(oc), 0 < tr <~ 1, is also a subset of f2(0). It follows, since g2(0) is compact, that the closure of the union of the X~(o<), 0 < tr ~< 1, which is X1(0) is a subset of g2(0). We next show that ~2(0) c X l ( 0 ) . Let x~g2(0), then ~ ( x l X l ) = t r > ~ 0 . If o<>0, then x e .,Y'~(o:) so that it is in XI(0). So now assume that tr = 0. We construct a sequence x~, each x~ belongs to the union of the X~(o<) and 0 < o<<~ 1, so that x~--->x. Then x e k l ( 0 ) and Q(0) ~,('1(0). There is a (v*, b*) e a(0) x b(0) so that H(v*, b*) = x. The point (v*, b*) must belong to the boundary of a(0) × b(0) since /z(x I X1) = 0. We may construct a sequence (v~, bn), all belonging to the interior of a(0) × b(0), so that (v,, b~)--> (v*, b*). We see that each (v~, b~) is in a(c~,)xb(tr~) for some cry>0. Let x~ = H ( v , , b~). Then x~ e g2(a:,) so that /z(x~ I -~1) ~> o:~ > 0 and x~ e .~'(cr~), for all n. So the sequence Xn belongs to the union of the Xl(o:), 0 < tr ~< 1. But x~ = H(v~, b,)--->H(v*, b*) = x and x belongs to )(l(0). That is, £2(0) c )(1(0). (d) We have shown that Xl(tr) = g2(a0, 0 <~ o: ~< 1. From (a) we conclude that .('1(o:) is compact, simply connected and arcwise connected, 0 ~< tr ~< 1. (e) Clearly, ~'x(1) 4= 0 since n ( a ( 1 ) × b(1)) e g2(1) = .~'1(1). (f) The last thing to show is that y =/~(x IX1) is u.s.c. For each 0 < ce~ < 1 we know that A'l(tr) = g2(tr) which is closed. It follows [11, Proposition 3.3] that ~(x IX1) is u.s.c. One may wonder if the tr-cuts of S l will be convex. The answer is no as shown by an example in [8]. The following simple example illustrates X1, a fuzzy vector solution to A£ =/~. Example 2. fi is 2 x 2 with i~11: t~22 ~---(1/2/3), ~i12= ti2t = 0 (real number zero), and bl = ( - 3 / - 2 / - 1 ) , /~2 = 1 (real number one). Then the determinant of A is never zero on a(0). Given o = ( a l l , 0 , 0, a22) E a(O), b t -- ( b l , 1) with bl ~/~l(0), the solution for x t -- (Xl, x2) is xl = bl/all and x2 = 1/a22. Then
= ((Xl, x2) l -3
1),
(13)
a rectangle. As tr grows from zero to one the rectangle f2(ce) gradually shrinks until g2(1) = { ( - 1 , 1)}.
3. Solutions based on
A-~b
Our next three solutions are based on A - t b = x. However, there are different methods of evaluating A - l b when the ~0 and/~i are fuzzy numbers. Let F( tr) = { Z - l b [ air E (lij(O[), bi
E
b/(o~')},
(14)
38
J.J. Buckley, Y. Qu
0 ~ tr ~< 1. F(tr) is called the united extension of the function F(v, b) = A - l b [10, p. 19]. Our second solution X2, a fuzzy subset of R n, is defined by its membership function ~(x 1~'2) = sup{re Ix • F(cr)}.
(15)
We may also evaluate A - l b , when A and /~ are fuzzy, using the extension principle. Let at(v) =
min
l~i,j<-n
{/~(aoltiij)} ,
(16)
at(b) = min {g(b~ I/~,)},
(17)
at(v, b ) = min{at(v), at(b)}.
(18)
l<.i<_n
and Define X3, a fuzzy subset of R", by its membership function ~t(x 1£'3) = sup(at(v, b) I a - l b = x } .
(19)
We have shown that X'2 =/t'3 and Xz(tr) =/~'3(tr) = F(o¢), 0 < tr ~< 1, in [5]. Still another method of evaluating A-~b, for fuzzy A and/~, is to substitute the intervals ~ij(a~) and/~(tr) for the aij and bi in A -1 and b and compute the result using interval arithmetic. For example, if A is 2 × 2 we would have F(v, b) = (xl, x2) t where x, = (az2b, - alzbz)/ A,
(20)
x2 = (-aeab, + a,lb2)/A,
(21)
for A = allaz2-a12a21. This is simply Cramer's Rule for finding x. We would substitute the intervals t~ij(ol) and/~i(tr) for the aij and b~ in Eq. (20) and (21) and compute intervals for x~ and x2 using interval arithmetic. We now need to generalize this simple example and review some results from interval analysis (see [101). Suppose F(v, b) = (fl(v, b) . . . . . f,,(v, b)) where X i =fii(l), b), 1 <~i ~ n. Each f,-(v, b) is x~ computed using Cramer's Rule. That is, f ( v , b) is the determinant of A, with its ith column replaced by b, divided by the determinant of A. Therefore, each f is a rational function of the aij and b~. The united extension of f is fii(ot) = { f ( v , b) I, (v, b) • a ( a ) x b(cr)},
(22)
for 0 ~ cry< 1, l<_i<~n. Next let f / b e the natural interval extension off,., 1 ~< i ~
Solving systems of linear fuzzy equations
39
An interval valued function G of interval variables /1 . . . . . I,, is inclusion monotonic if and only if li c Ji, 1 <<-i <- m, implies that G(I1 . . . . . Im) c G(J1. . . . . . J,,). Now each ~ is inclusion monotonic because it is the natural interval extension of rational function f~ ([10, p. 21]). Therefore, from Theorem 3.1 of [10] we conclude that2 f/(a0 c yi(tr), 0<~ o¢ << - 1, 1 <~i <-n. We would hope that the yi(ol), 0~< c~< 1, form the o~-cuts of some fuzzy number, 1 ~
(23)
for 1 ~
(24)
l~i~n
,~r 4 is a fourth possible solution to fits =/~ when .4 and/~ are fuzzy.
We now argue that X'2(tr)~X4(tr), 0 < c r < ~ l , X2(0¢) 4= d('4(tr). We show this result from proving
and it may happen that
n
l"( Ol) C [ I fii( OC) ~ ~-I Yi( 0¢) ~ i=1
i--1
~'~4(0(),
(25)
0 < t r ~ < l . We first show that F(ol)cIIT=~f(c~), 0 < t r < ~ l . Let x e F ( o : ) . Then xi=fii(v, b) for some v e a(tr) and b e b(ol), l<~i<-n. Hence, x belongs to FlP_-lf~(tr). We next show that I-[~'=lyi(tr) c)~4(a0. Let x e II~'=ayi(a0. Then xi • yi( o:), and lz(xi I Yi) >~ ol, 1 <<-i <~ n. Hence x e )(4(tr). We get X2(cr) = )(4(re), 0 < t r ~ 1 since F(rr)=/t'2(0¢) for 0 < a ~< 1 and we have already noted that f,.(cr) cy~(tr) for all cr and i. There are simple examples [3;5; 10, p. 22] of the united extension (here f~(tr)) being a proper subset of the interval extension (here yi(oc)). Also X2(tr) may not equal [l~'=lf/(cQ since I]~'=lf/(tr) will be a 'rectangle' in R ~ and X2(0¢) need not be 'rectangular'. Hence, we conclude that ,('2(a0 may not equal ,('4(tr), for 0 < tr < 1. We will not u s e .~r4 as a solution to the fuzzy matrix equation ,4S =/~. In practice, since the end points of the intervals y~(cr) may be computed in a finite number of arithmetic steps, y~(cQ is used to approximate f(tr), 1 ~
Proof. Clearly, f2(0¢) = F(tr), 0 <~ tr <~ 1, so X~ = X2. We have already noted that ~'Y2 ~ )~'3"
4. Solutions based on fuzzy variables
Our next two solutions are based on fuzzy variables and possibility theory. Consider real-valued fuzzy variables t/i~, 1 ~
40
J.J. Buckley, Y. Qu
values are restricted by possibility distributions tz(aijla~j), l < ~ i , j < ~ n , and Pt(bi I bi), 1 ~ i <- n, respectively. This means that Poss[a# - a#] = I*(aij ] ~#) for all i, j and Poss[/~i = bi] = Iz(bi ] bi) for all i. That is, the (li](bi) are now considered possibility distributions for the fuzzy variables t~ij(~) whose values are to be the a~j(b~) in the equations A x = b and x = A - l b . Assuming the ~ and ~i are all non-interactive [12], and joint possibility distribution is :r(v, b). Let S be a fuzzy variable whose values are possible solutions for x in the equation A x = b, when (v, b) • a(0) x b(0). Then define
Poss[,_¢ =x] = sup{at(v, b) lAx = b },
(26)
if A x = b for some (v, b) • a(0) × b(0), and set Poss[~{ = x] = 0 otherwise. Next let J" be a fuzzy variable whose possible values are the values of x given by A - l b , for (v, b) • a(0) x b(0). Then define
Poss[7" = x] = sup{g(v, b) I A - l b = x},
(27)
if A - l b = x for some (v, b) • a(0) x b(0), and set Poss[T = x] = 0 otherwise.
Theorem 3. (a) Poss[ { = x I = / , ( x I X1). (b) Poss[ = x] = I R3). Proof. (a) We argue that if Poss[• = x], or ~u(x [ 41) are positive (or zero), then the other is positive (or zero) and they are equal. Let/~(x [ 41) = tr > 0. We show that Poss[3 = x] I> a~. Now x • g2(tr) so there is a (v, b) • a(te) x b(tr) so that A x = b. Then ~r(v, b)/> a: and Poss[S = x]/> tr. Let Poss[S = x] = a~> 0. We show that #(x [ 41) I> tr. Let e > 0 be given. There is a fl, tr ~ fl i> tr - e, and a (v, b) • a(0) x b(0) so that ~r(v, b) = fl and A x = b. It follows that ( v , b ) • a ( f l ) x b ( f l ) and x • ~ ( f l ) . Hence ~u(x[41) l>/3 or /t(x ] 41) ~> tr - e. The result follows since e > 0 was arbitrary. (b) The proof is similar to (a) and is omitted.
We have shown that the two new solutions based on possibility theory agree with 41 = X2 = 43. Let 4 be the common value of all five solutions. 4 is then our new solution to the fuzzy matrix equation. Theorem 3 presents us with another justification of our new solution X. Consider solving A x = b for x when the values of the aij and bi are uncertain. We may model this uncertainty using random variables or fuzzy variables. In this paper we choose fuzzy variables tJij and/~,- to model this uncertainty. It is natural then to find all solutions for x using all the values of aq and bi at the same level of uncertainty. The values of the aq and b / a t the same level of uncertainty belong to the sets {aiy [ Poss[tJiy = aq] ~> 5} and {b i [ Poss[~i -- bi] >t ol}, respectively. However, these sets are identical to ~i~(ac) and /~(o¢), respectively. Therefore, we obtain Q(ct) = F(a 0 and our new solution X.
Solving systems of linear fuzzy equations
41
5. Classical solution This solution is based on Eq. (5). Let Z~, 1 ~
(28)
=
j=l
1 ~
[aqt( oO, aij,( oO][zjl( oO, zj2(a0] = [bil( O:), bi,(a0],
(29)
j=l
for 0 <~ o~~< 1, 1 <~i ~ n. Interval multiplication and addition is used to evaluate the left-hand side of Eq. (29). We now solve Eq. (29) for the zi~(ol), z~2(o0, 1 ~
(30)
Theorem 4. )(5 ~<) ( Proof. We show that )(s(a) c ~2(0c) = X(a0, 0 ~ o~~< 1. Let x • Xs(o 0. Then #(xi I Zi) ~> o~ for all i so that xi belongs to Zi(o 0, 1 <~ i <~n. That is, xi belongs to [zil(o0, zi2(oO], 1 ~ i <~n. Choose v in a(c 0. If v = (all . . . . . %,), then air belongs to [aql(a0, a0,(c0], 1 ~
j=l
aijxj • [bit( oO, bi,(o0],
l<~i<~n. That is, there is a v • a ( a : ) and a b • b ( o : ) so that A x = b .
(31) Hence
x • f2(o0.
Now Xs(cr)= II~'=1Zi(cr), 0<~ o~<~1, so o~-cuts of )(s(o0 are rectangles in R n. Since 2((o~) will Usually not be a rectangle in R" we would expect )(s(cr) to be a proper subset of ~2(_o0. Hence, we would usually expect A's to not equal X. We will not use Xs as a solution to the fuzzy matrix equation because: (1) too often it does not exist; and (2) when it does exist it may be a_proper subset of )(. There is still another problem associated with the Zi and Xs (see also [3, 5]). Suppose the solutions for the z~l(a) and z~2(o0 do define fuzzy numbers Zq. Now evaluate
aij2, = w/, j=l
(32)
42
J.J. Buckley, Y. Qu
1 ~ i ~ n, using the extension principle. It may happen that some I~,. does not equal/~i. -~s will not equal ,~ when the 2i are interactive. When the 2~ are noninteractive [12], then Eq. (30) gives the correct joint distribution of the Zi. However, when the Z, are interactive (which is usually the case) Eq. (30) does not produce the correct joint distribution. We believe that X gives the correct joint distribution for the solution to the fuzzy matrix equation. Let ,~7/~be the projection of ~" onto the x,-axis, 1 ~< i ~
6. Summary and conclusions This paper is concerned with finding solutions for ~ to the matrix equation .4~ -- 6 when the dij in ,4 and the/)i in/~ are triangular fuzzy numbers. We argued that the classical solution, based on the extension principle and regular fuzzy arithmetic, should be rejected since it too often fails to exist. Our new solution always exists. The discussion was restricted to square .4 which is always non-singular. We defined six new solutions and showed that five of them are identical and then adopted their common value ,~" as our solution to the fuzzy matrix equation. We showed that ,(" is a fuzzy vector, which is the generalization of a real fuzzy number to R n. If A turns out to be singular for some a~j in the s-cut of triangular fuzzy number d o then .(" may be far more complicated. The definition of X (from ~2(a0 in the Introduction) is still valid when A may be singular. Preliminary results show that s-cuts of .~" are comprised of components which may not be closed, can be unbounded, and could be overlapping (not disjoint). Future research will be concerned with describing .~" when A can be singular and when .4 is not square. The results of this paper are also important within the area of interval analysis. In interval analysis [6-8; 10, p. 59] the researchers are interested in solving A x -- b for x when the a~j in A and the b~ in b belong to intervals. They consider only one interval for each a# and only one interval for each bi. In this paper we did the same thing but for a nested family of intervals for each a~j and bg (the cr-cuts of the triangular fuzzy numbers) and showed how to put the solutions for x together to obtain fuzzy vector .('.
Acknowledgment The authors wish to thank an anonymous referee for suggestions to shorten the proof of T h e o r e m 1 and supplying reference [11].
Solving systems of linear fuzzy equations
43
References [11 J.J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38 (1990) 43-59. [2] J.J. Buckley and Y. Qu, Solving fuzzy equations: a new solution concept, Fuzzy Sets and Systems 39 (1991) 291-301. [3] J.J. Buckley, Solving fuzzy equations, J. Math. Anal. Appl., under review. [4] J.J. Buckley and Y. Qu, Fuzzy differential equations: new solutions, J. Math. Anal. Appl., under review. [5] J.J. Buckley and Y. Qu, On using or-cuts to evaluate fuzzy equations, Fuzzy Sets and Systems 38 (1990) 309-312. [6] E. Hansen, Interval arithmetic in matrix computations, Part I, SlAM J. Namer. Anal. 2 (19651 308-320. [7[ E. Hansen, Interval arithmetic in matrix computations, Part II, SlAM J. Namer. Anal. 4 (1967) 1-9. [8] E. Hansen, On the solution of linear algebraic equations with interval coefficients, Linear Algebra Appl. 2 (1969) 153-165. [9] H. Jiang, The approach to solving the simultaneous linear equations that coefficients are fuzzy numbers, J. Nat. Univ. Defence Technol. 3 (1986) 93-102 (in Chinese). [10] R.E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics (SIAM, Philadelphia, PA, 1979). [11] M.D. Weiss, Fixed points, separation, and induced topologies for fuzzy sets, J. Math. Anal. Appl. 50 (1975) 142-150. [12] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.