On stability in possibilistic linear equality systems with
Lipschitzian fuzzy numbers ∗
Robert Fullér
rfuller@abo.fi
Abstract
Linear equality systems with fuzzy parameters and crisp variables defined by the extension principle are called possibilistic linear equality systems. The study focuses on the problem of stability
(with respect to perturbations of fuzzy parameters) of the solution in these systems.
1
Introduction
Modelling real world problems mathematically we often have to find a solution to a linear system
ai1 x1 + · · · + ain xn = bi , i = 1, . . . , m,
(1)
where aij and bi are real numbers. It is known that the system of equations (1) generally belongs to the
class of ill-posed problems, so a small perturbation of the parameters ãij , b̃i may cause a large deviation
in the solution.
System (1) with fuzzy parameters ãij , b̃i and crisp variables xj is considered in a lot of paper [2,4,7].
Moreover, in [4] has been shown a stability property (with respect to changes of centres of fuzzy parameters) of the solution to the system (1) with symmetrical triangular fuzzy numbers. Our results are
connected with those presented ones in [4] and generalize and extend them. Namely, we shall prove that
the system (1) with Lipschitzian fuzzy numbers always belongs to the class of well-posed problems, so
a small perturbation of the fuzzy parameters may cause only a small deviation in the solution.
2
Preliminaries
A fuzzy number is a fuzzy set on the real axis, i.e. mapping ã : R → I = [0, 1] associating with each
real number t its grade of membership ã(t). To distinguish a fuzzy number from a crisp (non-fuzzy) one,
the former will always be denoted with a tilde ˜. By F we denote the set of all fuzzy numbers ã with
the membership function having the following properties: (i) ã is upper semicontinuous; (ii) ã(t) = 0,
outside of some interval [c, d]; (iii) there are real numbers a and b, c ≤ a ≤ b ≤ d such that ã is strictly
increasing on the interval [c, a], strictly decreasing on [b, d] and ã(t) = 1 for each x ∈ [a, b].
If ã, b̃ ∈ F and λ ∈ R then ã + b̃, ã − b̃, λã are defined by the Zadeh’s extension principle [9] in the
usual way. An α-level set of a fuzzy number ã is a non-fuzzy set denoted by [ã]α and is defined by
(
{t ∈ R|ã(t) ≥ α} if α > 0
α
[ã] =
cl(suppã)
if α = 0
∗
The final version of this paper appeared in R. Fullér, On stability in possibilistic linear equality systems with Lipschitzian
fuzzy numbers, Fuzzy Sets and Systems, 34(1990) 347-353. doi: 10.1016/0165-0114(90)90219-V
1
where cl(suppã) denotes the closure of the support of ã.
Let ã ∈ F be a fuzzy number. Then from (i)-(iii) it follows that the α-level set [ã]α is a nonempty
compact interval [a1 (α), a2 (α)] where a1 (α) = min[ã]α , a2 (α) = max[ã]α for each α ∈ I.
Let ã and b̃ ∈ F and [ã]α = [a1 (α), a2 (α)], [b̃]α = [b1 (α), b2 (α)]. It is well known that
[ã + b̃]α = [a1 (α) + b1 (α), a2 (α) + b2 (α)].
(2)
[ã − b̃]α = [a1 (α) − b2 (α), a2 (α) − b1 (α)].
(3)
The truth value of the assertion
”ã is equal to b̃”,
which we write ã = b̃ is Poss(ã = b̃) defined as
Poss(ã = b̃) = sup ã(t) ∧ b̃(t).
t∈R
It is easily verified that
Poss(ã = b̃) = (ã − b̃)(0)
(4)
We define a metric D [3] in F by the equation
D(ã, b̃) = sup max{|ai (α) − bi (α)|}
α∈I i=1,2
(5)
The following lemma can be proved directly by using the definition (5).
Lemma 2.1 Let ã, b̃, c̃ and d˜ ∈ F and λ ∈ R. Then
D(λã, λb̃) = |λ|D(ã, b̃),
˜ ≤ D(ã, b̃) + D(c̃, d),
˜
D(ã + c̃, b̃ + d)
˜ ≤ D(ã, b̃) + D(c̃, d).
˜
D(ã − c̃, b̃ − d)
Let L > 0 be a real number. By F(L) we denote the set of all fuzzy numbers ã ∈ F with membership
function satisfying the Lipschitz condition with constant L , i.e.
|ã(t) − ã(t0 )| ≤ L|t − t0 |, ∀t, t0 ∈ R
(6)
Let ã ∈ F(L) and [ã]α = [a1 (α), a2 (α)]. Then it is easily verified that
ã(a1 (α)) = α, ∀α ∈ I,
(7)
a1 (ã(t)) = t if a1 (0) ≤ t ≤ a1 (1).
(8)
In the following lemma we see that the linear combination of Lipschitzian fuzzy numbers is a Lipschitzian one, too.
Lemma 2.2 Let L > 0, λ 6= 0, µ 6= 0 be real numbers and let ã, b̃ ∈ F(L) be fuzzy numbers. Then
(i) λã ∈ F(L/|λ|),
(ii) λã + µb̃ ∈ F(L/(|λ| + |µ|).
2
Proof. (i) From the equation (λã)(u) = ã(u/λ) for each u ∈ R and (6) we have
|(λã)(t) − (λã)(t0 )| = |ã(t/λ) − ã(t0 /λ)| ≤
L
|t − t0 |
|λ|
for all t, t0 ∈ R. Thus (i) holds.
(ii) With the notations c̃ := λã, d˜ := µb̃ we need show that
˜
˜
|(c̃ + d)(u)
− (c̃ + d)(v)|
≤
L
|u − v|
|λ| + |µ|
for all u, v ∈ R.
Let u, v ∈ R be arbitrarily fixed. We assume without loss of generality that u < v. From the definition
of addition on fuzzy numbers (2) it follows that there are real numbers u1 , u2 , v1 , v2 with the properties
u = u1 + u2 , v = v1 + v2 , u1 ≤ v1 , u2 ≤ v2
(9)
˜ 2 ) = (c̃ + d)(u)
˜
c̃(u1 ) = d(u
(10)
˜ 2 ) = (c̃ + d)(v)
˜
c̃(v1 ) = d(v
(11)
Since from (8) and (i) we have
|c̃(u1 ) − c̃(v1 )| ≤
L
|u1 − v1 |,
|λ|
|c̃(u2 ) − c̃(v2 )| ≤
L
|u2 − v2 |,
|µ|
hence by (9)-(11) we get
˜
˜
|(c̃ + d)(v)
− (c̃ + d)(u)|
≤ min{
≤
L
L
|u1 − v1 |,
|u2 − v2 |}
|λ|
|µ|
L
L
(|v1 − u1 | + |v2 − u2 |) =
|v − u|
|λ| + |µ|
|λ| + |µ|
Which proves the lemma. The following lemma that will be used in the sequel shows one of the basic
properties of Lipschitzian fuzzy numbers.
Lemma 2.3 Let ã, b̃ ∈ F(L) and [ã]α = [a1 (α), a2 (α)], [b̃]α = [b1 (α), b2 (α)]. Suppose that D(ã, b̃) ≤
δ. Then
sup |ã(t) − b̃(t)| ≤ Lδ.
t∈R
Proof. Let t ∈ R be arbitrarily fixed. It will be sufficient to show that
|ã(t) − b̃(t)| ≤ Lδ
(12)
If t ∈
/ suppã ∪ suppb̃ then we obtain (12) trivially. Suppose that t ∈ suppã ∪ suppb̃. With no loss of
generality we will assume 0 ≤ b̃(t) < ã(t). Then either
1. t ∈ (b1 (0), b1 (1)),
2. t ≤ b1 (0),
3. t ∈ (b2 (1), b2 (0)),
3
4. t ≥ b2 (0),
must occur. In the case of 1) consider (6) with t and t0 = a1 (b̃(t)). Then from (7) and t = b1 (b̃(t)) and
(5) we have
|ã(t) − b̃(t)| = |ã(t) − ã(a1 (b̃(t)))| ≤ L|t − (a1 (b̃(t))| =
L|b1 (b̃(t)) − a1 (b̃(t))| ≤ LD(ã, b̃) ≤ Lδ.
In the case of 2) we have b̃(t) = 0; therefore from δ ≥ D(ã, b̃) ≥≥ |a1 (0) − b1 (0)| and t > a1 (0) we
get
|ã(t) − b̃(t)| = |ã(t) − 0| = |ã(t) − ã(a1 (0))| ≤
L|t − a1 (0)| = L(t − a1 (0)) ≤ L(b1 (0) − a1 (0)) ≤ Lδ.
In the cases of 3) and 4) the proofs are carried out in a similar manner.
3
Stability in possibilistic linear equality systems
Generalizing the system (1) consider the following possibilistic linear equality system
ãi1 x1 + · · · + ãin xn = b̃i , i = 1, . . . , m
(13)
where ãij , b̃i are fuzzy numbers. We denote by µi (x) the degree of satisfaction of the i-th equation at the
point x ∈ Rn in (13), i.e.
µi (x) = Poss(ãi1 x1 + · · · + ãin xn = b̃i ).
(14)
Following Bellman and Zadeh [1] the solution (or the fuzzy set of feasible solutions) of the system (13)
can be viewed as the intersection of the µi ’s such that
µ(x) = min µi (x)
(15)
i=1,m
A measure of consistency [2] for the system (13) can be
µ∗ = sup µ(x)
(16)
x∈R
In many important cases the fuzzy parameters ãij , b̃i of the system (13) are not known exactly [8] and
we have to work with their approximations ãδij , b̃δi such that
max(ãij , ãδij ) ≤ δ,
max D(b̃i , b̃δi ) ≤ δ,
i,j
i
(17)
where δ ≥ 0 is a real number. Then we get the following system with perturbed fuzzy parameters
ãδi1 x1 + · · · + ãδin xn = b̃δi , i = 1, . . . , m
(18)
In a similar manner we can define the solution and the measure of consistency of the perturbed system
(18)
µδ (x) = min µδi (x), µ∗ (δ) = sup µδ (x),
i=1,m
x∈Rn
where µδi (x) denotes the degree of satisfaction of the i-th equation at the point x ∈ Rn in (18). In the
following theorem we establish a stability property (with respect to perturbations (17)) of the solution to
the system (13).
4
Theorem 3.1 Let L > 0 and ãij , ãδij , b̃i , b̃δi ∈ F(L). If (17) holds, then
||µ − µδ ||C = sup |µ(x) − µδ (x)| ≤ Lδ,
(19)
x∈Rn
where µ(x) and µδ (x) are the solutions to the systems (13) and (18) respectively.
Proof. It is sufficient to show that
|µi (x) − µδi (x)| ≤ Lδ
for each x ∈ Rn and i = 1, . . . , m.
Let x ∈ Rn and i ∈ {1, . . . , m} be arbitrarily fixed. From (4) it follows that
X
n
µi (x) =
µδi (x)
ãij xj − b̃i (0),
=
j=1
X
n
ãδij xj
−
b̃δi
(0).
j=1
Applying Lemma 1. we have
D
X
n
ãij xj − b̃i ,
j=1
n
X
ãδij xj
−
b̃δi
≤
j=1
n
X
|xj |D(ãij , ãδij ) + D(b̃i , b̃δi ) ≤ δ(|x|1 + 1),
j=1
where |x|1 = |x1 | + · · · + |xn |. By Lemma 2. we have
n
X
ãij xj − b̃i ,
j=1
n
X
ãδij xj
j=1
−
b̃δi
L
∈F
|x|1 + 1
Finally, applying Lemma 3 we get
|µi (x) −
µδi (x)|
n
X
n
X
δ
δ
=
ãij xj − b̃i (0) −
ãij xj − b̃i (0) ≤
j=1
j=1
n
X
n
X
δ
δ
sup ãij xj − b̃i (t) −
ãij xj − b̃i (t) ≤
t∈R
j=1
j=1
L
δ(|x|1 + 1) = Lδ.
|x|1 + 1
Which proves the theorem.
Remark 3.1 From (19) it follows that
|µ∗ − µ∗ (δ)| ≤ Lδ,
where µ∗ , µ∗ (δ) are the measures of consistency for the systems (13) and (18) respectively.
Remark 3.2 It is easily checked that in the general case ãij , b̃i ∈ F the solution to the possibilistic
linear equality system (13) may be unstable (in metric C) under small variations in the membership
function of fuzzy parameters (in metric D).
5
Remark 3.3 Let X ∗ be the set of points x ∈ Rn for which µ(x) attains its maximum, if it exist. If
x∗ ∈ X ∗ , then x∗ is called a maximizing (or best) solution of the system (13). When the problem is to
find a maximizing solution to a possibilistic linear equality system (13), then according to Negoita [6],
we are led to solve the following optimization problem
max xn+1 ; xn+1 ≤ µi (x1 , . . . , xn ), i = 1, . . . , m, xj ∈ R, j = 1, . . . , n.
(20)
Finding the solutions of problem (20) generally requires use of nonlinear programming techniques, and
could be tricky [10]. However, if the fuzzy numbers in (13) are of trapezoidal form, then the problem (20)
turns into quadratically constrained programming problem [5].
4
Concluding remarks
In this paper we have shown that the solution and the measure of consistency of the system (13) have
a stability property with respect to changes of the fuzzy parameters. Nevertheless,the behavior of the
maximizing solution towards a small perturbations of the fuzzy parameters has not been described yet,
i.e. supposing that X ∗ 6= ∅, what can be said about the distance
ρ(x∗ (δ), X ∗ ) as δ ≥ 0,
where x∗ (δ) is a maximizing solution of perturbed problem (18) and ρ is a metric in Rn .
5
Aknowledgement
The author would like to express his gratitude to Dr S.A.Orlovsky for his helpful discussion.
References
[1] R.E.Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Management Sciences
17(1970), B141-B154.
[2] D.Dubois and H.Prade, System of linear fuzzy constraints, Fuzzy Sets and Systems, 3(1980), 3748.
[3] R.Goetschel and W.Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems,
10(1983), 87-99.
[4] M.Kovács, Fuzzification of ill-posed linear systems,in: D. Greenspan and P.Rózsa, Eds., Colloquia
mathematica Societitas János Bolyai 50, Numerical Methods, North-Holland, Amsterdam, 1988,
521-532.
[5] M.Kovács, F.P.Vasiljev and R.Fullér, On stability in fuzzified linear equality systems, Proceedings
of the Moscow State University, Ser. 15, 1(1989), 5-9 (in Russian), translation in Moscow Univ.
Comput. Math. Cybernet., 1(1989), 4-9.
[6] C.V.Negoita, Fuzzinies in management, ORSA/TIMS, Miami (Nov.1976).
[7] H.Tanaka, Fuzzy data analysis by possibilistic linear modells, Fuzzy Sets and Systems, 24(1987),
363-375.
6
[8] H.Tanaka,H.Ichihashi and K.Asai, A value of information in FLP problems via sensitivity analysis, Fuzzy Sets and Systems, 18(1986), 119-129.
[9] L.A.Zadeh, The concept of a linguistic variable and its application to approximate reasoning - 1,
Inform. Sci. 8(1975), 199-249.
[10] H.J.Zimmermann, Fuzzy set theory and mathematical programming, in: A.Jones et al. (eds.),
Fuzzy Sets Theory and Applications, 1986, D.Reidel Publishing Company, Dordrecht, 99-114.
6
Extensions
For more results on stability properties of possibilistic linear equality and inequality systems see:
• M.Kovács, F.P.Vasiljev and R.Fullér, On stability in fuzzified linear equality systems, Proceedings
of the Moscow State University, Ser. 15, 1(1989), 5-9 (in Russian), translation in Moscow Univ.
Comput. Math. Cybernet., 1(1989), 4-9 [MR 91c:94039] [Zbl.651-65028];
• R.Fullér, Well-posed fuzzy extensions of ill-posed linear equality systems, Fuzzy Systems and
Mathematics, 5(1991) 43-48.
7
Follow ups
The results of this paper have been mentioned later in the following works
in journals
A34-c11 Vroman A, Deschrijver G, Kerre EE, Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix, INTERNATIONAL JOURNAL OF COMPUTATIONAL INTELLIGENCE SYSTEMS, 1(2008), Number 3, pp. 248-261. 2008
http://dx.doi.org/10.2991/ijcis.2008.1.3.5
A34-c10 Vroman A, Deschrijver G, Kerre EE Solving systems of linear fuzzy equations by parametric
functions - An improved algorithm FUZZY SETS AND SYSTEMS, 158 (14): 1515-1534 JUL 16
2007
http://dx.doi.org/10.1016/j.fss.2006.12.017
Consequently, the exact solution does not exist and therefore the search for an alternative solution has a solid ground. There are already some alternative approaches
known in literature. Fuller [A34] considers a system of linear fuzzy equations with
Lipschitzian fuzzy numbers. He assigns a degree of satisfaction to each equation in the
system and then calculates a measure of consistency for the whole system. Abramovich
et al. [1] try to minimize the deviation of the left-hand side from the right-hand side
of the system with LR-type fuzzy numbers. Both methods try to approximate the exact solution, i.e. they try to minimize the error when one reenters the solution into the
system. (page 1516)
A34-c9 Rybkin VA, Yazenin AV On the problem of stability in possibilistic optimization, NTERNATIONAL JOURNAL OF GENERAL SYSTEMS, 30 (1): 3-22. 2001
7
A34-c8 E.B. Ammar and M.A.El-Hady Kassem, On stability analysis of multicriteria LP problems with
fuzzy parameters, FUZZY SETS AND SYSTEMS, 82(1996) 331-334. 1996
http://dx.doi.org/10.1016/0165-0114(95)00266-9
Fullér in [A34] introduced the stability of the FLP problems with fuzzy parameters. In
the present paper we investigate the stability of the solution in fuzzy . . . (page 331)
A34-c7 S.Jenei, Continuity in approximate reasoning, Annales Univ. Sci. Budapest, Sect. Comp.,
15(1995) 233-242. 1995
A34-c6 M.Kovács, Stable embeddings of linear equality and inequality systems into fuzzified systems,
FUZZY SETS AND SYSTEMS, 45(1992) 305-312. 1992
http://dx.doi.org/10.1016/0165-0114(92)90148-W
The obtained stability results are the generalizations of the stability properties of the
fuzzified linear systems proved in [A35, A34, 3,5]. (page 311)
in proceedings and in edited volumes
A34-c5 D.Dubois, E.Kerre, R.Mesiar and H.Prade, Fuzzy interval analysis, in: Didier Dubois and Henri
Prade eds., Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Volume 7, Kluwer
Academic Publishers, [ISBN 0-7923-7732-X], 2000, 483-581. 2000
Another type of fuzzy intervals is considered hy Fullér (l990): Lipschitzian fuzzy intervals M such that there is positive constant k such that |M (a) − M (a0 )| ≤ k|a − a0 |.
He proves that the class of Lipschitzian fuzzy intervals is closed under fuzzy addition
and scalar multiplication. (page 507)
A34-c4 V.A.Rybkin and A.V.Yazenin, Regularization and stability of possibilistic linear programming
problems, in: Proceedings of the Sixth European Congress on Intelligent Techniques and Soft
Computing (EUFIT’98), Aachen, September 7-10, 1998, Verlag Mainz, Aachen, Vol. I, 1998 3741. 1998
in books
A34-c3 R. Lowen, Fuzzy Set Theory, Kluwer Academic Publishers, [ISBN: 0-7923-4057-4], 1996.
A34-c2 Y.J.Lai and C.L.Hwang, Fuzzy Multiple Objective Decision Making, Lecture Notes in Economics and Mathematical Systems, No. 404, Springer Verlag, [ISBN: 978-3-540-57595-5], Berlin
1994.
A34-c1 Y.J.Lai and C.L.Hwang, Fuzzy Mathematical Programming, Methods and Applications, Lecture
Notes in Economics and Mathematical Systems, No. 394, Springer Verlag, [ISBN 3-540-56098X], Berlin 1992.
8