Fuzzy Sets and Systems 109 (2000) 55–58www.elsevier.com/locate/fss

Duality in fuzzy linear systemsMing Ma, M. Friedman, A. Kandel∗

Department of Computer Science and Engineering, University of South Florida, 4202 E. Fowler Avenue,ENB 118, Tampa, FL 33620, USA

Received May 1997; received in revised form March 1998

Abstract

According to fuzzy arithmetic, dual fuzzy linear system can not be replaced by a fuzzy linear system. In this paper weinvestigate the existence of a solution of duality fuzzy linear equation systems. Two necessary and su�cient conditions forthe solution existence are given. c© 2000 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy linear equation system; Duality; Embedding method; Nonnegative matrix

1. Introduction

The topic of fuzzy linear systems which attractedincreasing interest for some time, in particular inrelation to fuzzy neural network, has been rapidlygrowing in recent years [1,2,5,6].In a previous paper [5] we investigated a general

model for solving an n× n fuzzy linear system whosecoe�cient matrix is crisp and the right-hand side col-umn is an arbitrary fuzzy vector. We use the paramet-ric form of fuzzy numbers [7] and replace the originalsystem by a (2n)× (2n) representation. This enablesus to treat this problem using the theory of positivematrix [8]. In this paper, we will apply the same ap-proach to solve the dual fuzzy linear systems.In Section 2 we present necessary preliminaries re-

lated to fuzzy numbers. The dual fuzzy linear systemis de�ned and discussed in the Section 3, followed bya concluding remark in Section 4.

∗ Corresponding author. Tel.: 813 974 4232; fax: 813 974 5456;E-mail address: kandel@babbage.csee.usf.edu.

2. Preliminaries

In this section we recall the basic notations of fuzzynumber arithmetic and fuzzy linear system.We start by de�ning the fuzzy number.

De�nition 1. A fuzzy number is a fuzzy setu :R1→ I = [0; 1] which satis�es(i) u is upper semicontinuous.(ii) u(x)= 0 outside some interval [c; d].(iii) There are real numbers a; b: c6a6b6d for

which1. u(x) is monotonic increasing on [c, a].2. u(x) is monotonic decreasing on [b, d].3. u(x)= 1; a6x6b.

The set of all the fuzzy numbers is denoted by E1.An equivalent parametric de�nition is given in

[7] as:

De�nition 2. A fuzzy number u is a pair (u; u) offunctions u(r); u(r); 06r61 which satisfy the

0165-0114/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(98)00102 -X

56 M. Ma et al. / Fuzzy Sets and Systems 109 (2000) 55–58

following requirements:(i) u(r) is a bounded monotonic increasing left-

continuous function.(ii) u(r) is a bounded monotonic decreasing left-

continuous function.(iii) u(r)6u(r); 06r61.

Using the extension principle [11], the addition andthe scalar multiplication of fuzzy numbers are de�nedby

(u+ v)(x)= sups+t=x

min((u(s); v(t)); (1)

(ku)(x)= u(x=k); k 6= 0; (2)

(0u)= 0 (3)

for u; v∈E1; k ∈R1. Equivalently, for arbitrary u=(u; u); v=(v; v) and k ∈R1 we may de�ne addition(u+ v) and multiplication by k as

(u+ v)(r)= u(r) + v(r);

(u+ v)(r)= u(r) + v(r);(4)

(ku)(r)= ku(r); (ku)(r)= ku(r); k¿0; (5)

(ku)(r)= ku(r); (ku)(r)= ku(r); k60: (6)

In a previous paper [5] we have investigated a fuzzysystem of linear equations systems as

a11x1 + a12x2 + · · ·+ a1nxn=y1;a21x1 + a22x2 + · · ·+ a2nxn=y2;...

an1x1 + an2x2 + · · ·+ annxn=yn;

(7)

where the coe�cient matrix A=(aij); 16i; j6n is acrisp n× n matrix and yi ∈ e1; 16i6n. This systemis called a fuzzy linear system (FLS).By the previous de�nition of the addition and scalar

multiplication between fuzzy numbers, Eq. (7) can bereplaced by the following parametric system:

n∑j=1

aijxj =n∑j=1

aijxj =yi;

n∑j=1

aijxj =n∑j=1

aijxj = �yi:

(8)

If for a particular i: aij¿0; 16j6n, we simplyget

n∑j=1

aijxj =y i;n∑j=1

aij �xj = �yi: (9)

In general, however an arbitrary equation for eitheryior �yi may include a linear combination of xj’s and

�xj’s. Consequently, in order to solve the system givenby Eq. (8) one must solve a (2n) × (2n) crisp linearsystem where the right-hand side column is the vector(y1; y2; : : : ; y

n; �y1; �y2; : : : �yn)

T.Let us now rearrange the linear system of Eq. (8) so

that the unknowns are xi; (− �xi); 16i6n and the right-hand side column is Y =(y

1; y2; : : : ; y

n;− �y1;− �y2; : : : ;

− �yn)T. We get the (2n)× (2n) linear systems11x1 + s12x2 + · · · s1nxn + s1; n+1(− �x1)

+s1; n+2(− �x2) + · · ·+ s1;2n(− �xn)=y1;...

sn1x1 + sn2x2 + · · · snnxn + sn; n+1(− �x1)

+sn; n+2(− �x2) + · · ·+ sn;2n(− �xn)=yn;

sn+1;1x1 + sn+1;2x2 + · · · sn+1; nxn+sn+1; n+1(− �x1) + sn+1; n+2(− �x2)

+ · · ·+ sn+1;2n(− �xn)=− �y1;...

s2n;1x1 + s2n;2x2 + · · · s2n; nxn + s2n; n+1(− �x1)

+s2n; n+2(− �x2) + · · ·+ s2n;2n(− �xn)=− �yn;

(10)

where sij are determined as follows:

aij¿0⇒ sij = aij; si+n; j+n= aij;

aij¡0⇒ si; j+n=−aij; si+n; j =−aij (11)

and any sij which is not determined by Eq. (11) iszero. Using matrix notation we obtain

SX =Y; (12)

M. Ma et al. / Fuzzy Sets and Systems 109 (2000) 55–58 57

where S =(sij); 16i; j62n and

X =

x1...

xn− �x1...

− �xn

; Y =

y1

...

yn

− �y1...

− �yn

: (13)

The structure of S implies that sij¿0; 16i; j6nand that

S =

(B C

C B

); (14)

where B contains the positive entries of A, C the abso-lute values of the negative entries of A and A=B−C.The linear system of Eq. (7) is now a (2n) × (2n)

crisp function linear system and can be uniquelysolved for X , if and only if the matrix S is nonsingu-lar. We recall the following lemmas in [5].

Lemma 1. The matrix S is singular if and only if thematrices A=B− C and B+ C are both singular.

Lemma 2. The unique solution X of Eq. (12) is afuzzy vector for arbitrary Y if and only if S−1 isnonnegative; i.e.

(S−1)ij¿0; 16i; j62n: (15)

Based on these fundamental results, an algorithmfor calculating fuzzy solution of the Eq. (8) was de-signed in [5].

3. A dual fuzzy linear system

Usually, there is no inverse element for an arbi-trary fuzzy number u∈E1, i.e. there exists no elementv∈E1 such thatu+ v=0: (16)

Actually, for all non-crisp fuzzy number u∈E1 wehave

u+ (−u) 6= 0: (17)

Therefore, the fuzzy linear equation system

AX =BX + Y (18)

cannot be equivalently replaced by the fuzzy linearequation system

(A− B)X =Y (19)

which had been investigated. In the sequel, we willcall the fuzzy linear system

AX =BX + Y; (20)

where A=(aij); B=(bij); 16i; j6n are crisp coe�-cient matrix and Y a fuzzy number vector, a dual fuzzylinear system.

Theorem 1. Let A=(aij); B=(bij); 16i; j6n; benonnegative matrices. The dual fuzzy linear sys-tem (18) has a unique fuzzy solution if and onlyif the inverse matrix of A − B exists and has onlynonnegative entries.

Proof. By Eqs. (4)–(6), the dual fuzzy linear system

n∑i=1

aijxi=n∑i=1

bijxi + yj (21)

is equivalent to (since ai; j¿0 and bi; j¿0 for all i; j)

n∑i=1

aijxi=n∑i=1

bijxi + yj;

n∑i=1

aij �xi=n∑i=1

bij �xi + �yj:

(22)

It follows thatn∑i=1

(aij − bij)xi=yj; (23)

n∑i=1

(aij − bij)�xi= �yj: (24)

If (A−B)−1 exists, Eqs. (23) and (24) have uniquesolutions (xi)n1; ( �xi)

n1 and clearly if (A − B)−1i; j ¿0 for

all i; j (xi; �xi)n1 is a fuzzy number vector.

The following theorem guarantees the existenceof a fuzzy solution for a general case. Consider the

58 M. Ma et al. / Fuzzy Sets and Systems 109 (2000) 55–58

dual fuzzy linear system (18), and transform itsn× n coe�cient matrix A and B into (2n)× (2n)matrices as in the Eqs. (10)–(11). De�ne matricesS =(si; j); T =(ti; j); 16i; j6n by

aij¿0⇒ sij = aij; si+n; j+n= aij;

aij¡0⇒ si; j+n=−aij; si+n; j =−aij;bij¿0⇒ tij = bij; ti+n; j+n= bij;

bij¡0⇒ ti; j+n=−bij; ti+n; j =−bij

(25)

while all the remaining sij; ti; j are taken zero.

Theorem 2. The dual fuzzy linear equation system(18) has a unique fuzzy solution if and only if theinverse matrix of S − T exists and nonnegative.

Proof. Using the form of Eq. (10), we obtain thatthe system (18) is equivalent to the function equationsystem

SX =TX + Y; (26)

where X; Y are given by Eq. (13). Consequently,

(S − T )X =Y (27)

and a solution exists if and only if S−T is nonsingular.If in addition (S − T )−1ij ¿0 for all i; j, then by thevirtue of Lemma 2, the solution X provides a fuzzysolution.

4. Concluding remarks

In a previous paper [5], we have shown severalnecessary and su�cient conditions for the existence

of a solution to the fuzzy linear system Ax=Y . Dueto nonnegative matrix theory, a real fuzzy solution forfuzzy linear system Ax=Y is very rare even when thecoe�cient matrix is crisp. Therefore, a weak solutionwas proposed in [5]. In this paper, we have investi-gated a dual fuzzy linear system by means of nonneg-ative matrix theory. It has been shown that it is morelikely to have a fuzzy solution for the dual fuzzy linearsystem.

References

[1] J.J. Buckley, Fuzzy eigenvalues and input–output analysis,J. Fuzzy Sets and Systems 34 (1990) 187–195.

[2] J.J. Buckley, Y. Qu, Solving linear and quadratic fuzzyequations, J. Fuzzy Sets and Systems 38 (1990) 43–49.

[3] D. Dubois, H. Parde, Operations on fuzzy numbers, J. SystemsSci. 9 (1978) 613–626.

[4] D. Dubois, H. Parde, Fuzzy Sets and Systems: Theory andApplications, Academic Press, New York, 1980.

[5] M. Friedman, Ma Ming, A. Kandel, Fuzzy linear systems,J. Fuzzy Sets and Systems, to appear.

[6] R. Goetschell, W. Voxman, Eigen fuzzy number sets, J. FuzzySets and Systems 16 (1985) 75–85.

[7] R. Goetschell, W. Voxman, Elementary calculus, J. FuzzySets and Systems 18 (1986) 31–43.

[8] H. Minc, Nonnegative Matrices, Wiley, New York, 1988.[9] M. Mizumoto, K, Tanaka, The four operations of arithmetic

on fuzzy numbers, Systems Comput. Controls 7 (5) (1976)73–81.

[10] C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets toSystems Analysis, Wiley, New York, 1975.

[11] R.R. Yager, On the lack of inverses on fuzzy arithmetic,J. Fuzzy Sets and Systems 4 (1980) 73–82.

[12] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965)338–353.