Approximate Reasoning for Solving Fuzzy
Linear Programming Problems ∗
Robert Fullér†
Department of Computer Science, Eötvös Loránd University,
P.O.Box 157, H-1502 Budapest 112, Hungary
Hans-Jürgen Zimmermann
Department of Operations Research, RWTH Aachen
Templergraben 64, W-5100 Aachen, Germany
Abstract
We interpret fuzzy linear programming (FLP) problems (where some or
all coefficients can be fuzzy sets and the inequality relations between fuzzy
sets can be given by a certain fuzzy relation) as multiple fuzzy reasoning
schemes (MFR), where the antecedents of the scheme correspond to the constraints of the FLP problem and the fact of the scheme is the objective of
the FLP problem. Then the solution process consists of two steps: first,
for every decision variable x ∈ Rn , we compute the maximizing fuzzy set,
M AX(x), via sup-min convolution of the antecedents/constraints and the
fact/objective, then an (optimal) solution to FLP problem is any point which
produces a maximal element of the set {M AX(x) | x ∈ Rn } (in the sense of
the given inequality relation). We show that our solution process for a classical (crisp) LP problem results in a solution in the classical sense, and (under
well-chosen inequality relations and objective function) coincides with those
suggested by [Buc88, Del87, Ram85, Ver82, Zim76]. Furthermore, we show
how to extend the proposed solution principle to non-linear programming
problems with fuzzy coefficients.
Keywords: Compositional rule of inference, multiple fuzzy reasoning, fuzzy
mathematical programming, possibilistic mathematical programming.
∗
Published in: Proceedings of 2nd International Workshop on Current Issues in Fuzzy Technologies, Trento, May 28-30, 1992, University of Trento, 1993 45-54.
†
Partially supported by the German Academic Exchange Service (DAAD) and Hungarian Research Foundation OTKA under contracts T 4281, I/3-2152 and T 7598.
1
1
Statement of LP problems with fuzzy coefficients
We consider LP problems, in which all of the coefficients are fuzzy quantities (i.e.
fuzzy sets of the real line R), of the form
maximize c̃1 x1 + · · · + c̃n xn
< b̃ , i = 1, . . . , m,
subject to ãi1 x1 + · · · + ãin xn ∼
i
(1)
where x ∈ Rn is the vector of decision variables, ãij , b̃i and c̃j are fuzzy quantities,
the operations addition and multiplication by a real number of fuzzy quantities are
< is given
defined by Zadeh’s extension principle [Zad75], the inequality relation ∼
by a certain fuzzy relation and the objective function is to be maximized in the
sense of a given crisp inequality relation ≤ between fuzzy quantities.
The FLP problem (1) can be stated as follows: Find x∗ ∈ Rn such that
c̃1 x1 + · · · + c̃n xn ≤ c̃1 x∗1 + · · · + c̃n x∗n
(2)
< b̃ , i = 1, . . . , m,
ãi1 x1 + · · · + ãin xn ∼
i
i.e. we search for an alternative, x∗ , which maximizes the objective function subject to constraints.
We recall now some definitions needed in the following.
A fuzzy number ã is a fuzzy quantity with a normalized (i.e. there exists a unique
a ∈ R, such that µã (a) = 1), continuous, fuzzy convex membership function of
bounded support. The real number a which maximizes the membership function
of the fuzzy number ã called the peak of ã, and we write peak(ã) = a.
A symmetric triangular fuzzy number ã denoted by (a, α) is defined as µã (t) =
1 − |a − t|/α if |a − t| ≤ α and µã (t) = 0 otherwise, where a ∈ R is the peak and
2α > 0 is the spread of ã.
We shall use the following crisp inequality relations between two fuzzy quantities
ã and b̃
ã ≤ b̃ iff max{ã,
˜
b̃} = b̃
(3)
ã ≤ b̃ iff ã ⊆ b̃
(4)
where max
˜ is defined by Zadeh’s extension principle.
The compositional rule of inference scheme with several relations (called Multiple
Fuzzy Reasoning Scheme) [Zad73] has the general form
Fact
Relation 1:
...
Relation m:
X has property P
X and Y are in relation W1
...
X and Y are in relation Wm
Consequence: Y has property Q
2
where X and Y are linguistic variables taking their values from fuzzy sets in classical sets U and V , respectively, P and Q are unary fuzzy predicates in U and V ,
respectively, Wi is a binary fuzzy relation in U × V , i=1,. . . ,m.
The consequence Q is determined by [Zad73]
Q=P ◦
m
Wi
i=1
or in detail,
µQ (y) = sup min{µP (x), µW1 (x, y), . . . , µW1 (x, y)}.
x∈U
2
Multiply fuzzy reasoning for solving FLP problems
We consider FLP problems as MFR schemes, where the antecedents of the scheme
correspond to the constraints of the FLP problem and the fact of the scheme is
interpreted as the objective function of the FLP problem. Then the solution process consists of two steps: first, for every decision variable x ∈ Rn , we compute the maximizing fuzzy set, M AX(x), via sup-min convolution of the antecedents/constraints and the fact/objective, then an (optimal) solution to the FLP
problem is any point which produces a maximal element of the set {M AX(x) |
x ∈ Rn } (in the sense of the given inequality relation).
We interpret the FLP problem
maximize c̃1 x1 + · · · + c̃n xn
< b̃ , i = 1, . . . , m,
subject to ãi1 x1 + · · · + ãin xn ∼
i
(5)
as Multiple Fuzzy Reasoning schemes of the form
Antecedent 1
...
Antecedent m
< b̃
Constraint1 (x) := ã11 x1 + · · · + ã1n xn ∼
1
...
< b̃
Constraintm (x) := ãm1 x1 + · · · + ãmn xn ∼
m
Fact
Goal(x) := c̃1 x1 + · · · + c̃n xn
Consequence
M AX(x)
where x ∈ Rn and the consequence (i.e. the maximizing fuzzy set) M AX(x) is
computed as follows
M AX(x) = Goal(x) ◦
m
i=1
3
Constrainti (x),
i.e.
µM AX(x) (v) = sup min{µGoal(x) (u), µConstraint1 (x) (u, v), . . . , µConstraintm (u, v)}.
u
We regard M AX(x) as the (fuzzy) value of the objective function at the point
x ∈ Rn . Then an optimal value of the objective function of problem (5), denoted
by M , is defined as
M := sup{M AX(x)|x ∈ Rn },
(6)
where sup is understood in the sense of the given inequality relation ≤. Finally, a
solution x∗ ∈ Rn to problem (5) is obtained by solving the equation
M AX(x) = M.
The set of solutions of problem (5) is non-empty iff the set of maximizing elements
of
(7)
{M AX(x)|x ∈ Rn }
is non-empty.
Remark 2.1 To determine the maximum of the set (7) with respect to the inequality relation ≤ is usually a very complicated process. However, this problem
can lead to a crisp LP problem [Zim78, Buc89], crisp multiple criteria parametric
linear programming problem (see e.g. [Del87, Del88, Ver82, Ver84]) or nonlinear
mathematical programming problem (see e.g. [Zim86]). If the inequality relation
for the objective function is not crisp, then we somehow have to find an element
from the set (7) which can be considered as a best choice in the sense of the given
fuzzy inequality relation [Ovc89, Orl80, Ram85, Rom89, Rou85, Tan84].
3
Extension to FMP problems with fuzzy coefficients
In this section we show how the proposed approach can be extended to non-linear
FMP problems with fuzzy coefficients. Generalizing the classical MP problem
maximize g(c, x)
subject to fi (ai , x) ≤ bi , i = 1, . . . , m,
where x ∈ Rn , c = (ci , . . . , ck ) and ai = (ai1 , . . . , ail ) are vectors of crisp
coefficients, we consider the following FMP problem
maximize g(c̃1 , . . . c̃k , x)
< b̃ , i = 1, . . . , m,
subject to fi (ãi1 , . . . ãil , x) ∼
i
4
where x ∈ Rn , c̃h , h = 1, . . . , k, ãis , s = 1, . . . , l, and b̃i are fuzzy quantities, the
functions g(c̃, x) and fi (ãi , x) are defined by Zadeh’s extension principle, and the
< is defined by a certain fuzzy relation. We interpret the above
inequality relation ∼
FMP problem as MFR schemes of the form
Antecedent 1:
...
Antecedent m:
< b̃
Constraint1 (x) := fi (ã11 , . . . ã1l , x) ∼
1
...
< b̃
Constraintm (x) := fm (ãm1 , . . . ãml , x) ∼
m
Fact:
Goal(x) := g(c̃1 , . . . , c̃k , x)
Consequence
M AX(x)
Then the solution process is carried out analogously to the linear case, i.e an optimal value of the objective function, M , is defined by (6), and a solution x∗ ∈ Rn
is obtained by solving the equation
M AX(x) = M.
4
Relation to classical LP problems
In this section we show that our solution process for classical LP problems results
in a solution in the classical sense. A classical LP problem can be stated as follows
max < c, x >
Ax≤b
(8)
Let X ∗ be the set of optimal solutions to (8), and if X ∗ is not empty, then let
v ∗ =< c, x∗ > denote the optimal value of its objective function.
In the following ā denotes the characteristic function of the singleton a ∈ R, i.e.
µā (t) = 1 if t = a and µā (t) = 0 otherwise.
Consider now the crisp problem (8) with fuzzy singletons and crisp inequality relations
max < c̄, x >
(9)
Āx≤b̄
where Ā = [āi,j ], b̄ = (b̄i ), c̄ = (c̄j ), and
āi1 x1 + · · · + āin xn ≤ b̄i iff ai1 x1 + · · · + ain xn ≤ bi ,


 1 if u = v and < ai , x > ≤ bi
i.e.
µāi1 x1 +···+āin xn ≤b̄i (u, v) =
0 if u = v and < ai , x > > bi

 0 if u = v
5
(10)
The following theorem can be proved directly by using the definition of the inequality relation (10).
Theorem 4.1 The sets of solutions of LP problem (8) and FLP problem (9) are
equal, and the optimal value of the FLP problem is the characteristic function of
the optimal value of the LP problem.
5
Crisp objective and fuzzy constraints
FLP problems with crisp inequality relations in fuzzy constraints and crisp objective function can be formulated as follows (see e.g. Negoita’s robust programming
[Neg81], Ramik and Rimanek’s approach [Ram85])
max < c, x >
s.t.
ãi1 x1 + · · · + ãin xn ≤ b̃i , i = 1, . . . , m.
(11)
It is easy to see that problem (11) is equivalent to the crisp MP:
max < c, x >
x∈X
(12)
where,
X = {x ∈ Rn | ãi1 x1 + · · · + ãin xn ≤ b̃i , i = 1, . . . , m}.
Now we show that our approach leads to the same crisp MP problem (12). Consider
problem (11) with fuzzy singletons in the objective function
max < c̄, x >
s.t.
ãi1 x1 + · · · + ãin xn ≤ b̃i , i = 1, . . . , m.
where the inequality relation ≤ is defined by


 1 if u = v and < ãi , x > ≤ b˜i
µãi1 x1 +···+ãin xn ≤b̃i (u, v) =


0 if u = v and < ãi , x > > b˜i
0 if u = v
Then we have
µM AX(x) (v) =
1 if v =< c, x > and x ∈ X
0 otherwise
Thus, to find a maximizing element of the set {M AX(x) | x ∈ Rn } in the sense
of the given inequality relation we have to solve the crisp problem (12).
6
6
Fuzzy objective and crisp constraints
Consider the FLP problem with fuzzy coefficients in the objective function and
fuzzy singletons in the constraints
maximize c̃1 x1 + · · · + c̃n xn
subject to āi1 x1 + · · · + āin xn ≤ b̄i , i = 1, . . . , m,
(13)
where the inequality relation for constraints is defined by (10) and the objective
function is to be maximized in the sense of the inequality relation (3), i.e.
< c̃, x > ≤ < c̃, x > iff max{<
˜
c̃, x >, < c̃, x >} =< c̃, x >
Then µM AX(x) (v), ∀v ∈ R, is the optimal value of the following crisp MP problem
maximize µc̃1 x1 +···+c̃n xn (v)
subject to Ax ≤ b,
and the problem of computing a solution to FLP problem (13) leads to the same
crisp multiple objective parametric linear programming problem obtained by Delgado et al. [Del87, Del88] and Verdegay [Ver82, Ver84].
7
Relation to Buckley’s possibilistic LP
We show that when the inequality relations in an FLP problem are defined in a
possibilistic sense then the optimal value of the objective function is equal to the
possibility distribution of the objective function defined by Buckley [Buc88].
Consider a possibilistic LP
maximize Z := c̃1 x1 + · · · + c̃n xn
subject to ãi1 x1 + · · · + ãin xn ≤ b̃i , i = 1, . . . , m.
(14)
The possibility distribution of the objective function Z, denoted by P oss[Z = z],
is defined by [Buc88]
P oss[Z = z] = sup min{P oss[Z = z | x], P oss[< ã1 , x >≤ b̃1 ], . . . , P oss[< ãm , x >≤ b̃m ]},
x
where P oss[Z = z | x], the conditional possibility that Z = z given x, is defined
by
P oss[Z = z | x] = µc̃1 x1 +···+c̃n xn (z).
The following theorem can be proved directly by using the definitions of P oss[Z =
z] and µM (v).
7
Theorem 7.1 For the FLP problem
maximize c̃1 x1 + · · · + c̃n xn
< b̃ , i = 1, . . . , m.
subject to ãi1 x1 + · · · + ãin xn ∼
i
(15)
< is defined by
where the inequality relation ∼
µã
< (u, v)
i1 x1 +···+ãin xn ∼ b̃i
=
P oss[< ãi , x >≤ b̃i ] if u = v
0
otherwise
and the objective function is to be maximized in the sense of the inequality relation
(4), the following equality holds
µM (v) = P oss[Z = v], ∀v ∈ Rn .
So, if the inequality relations for constraints are defined in a possibilistic sense and
the objective function is to be maximized in relation (4) then the optimal value of
the objective function of problem (15) is equal to the possibility distribution of the
objective function of possibilistic LP (14).
In a similar manner it can be swhon that the proposed solution concept is a generalization of Zimmermann’s method [Zim76] for solving LP problems with soft
constraints.
8
Example
We illustrate our approach by the following FLP problem
maximize c̃x
< b̃, x ≥ 0,
subject to ãx ∼
(16)
where ã = (1, 1), b̃ = (2, 1) and c̃ = (3, 1) are fuzzy numbers of symmetric
< is defined in a possibilistic sense, i.e.
triangular form, the inequality relation ∼
µãx < b̃ (u, v) =
∼
P oss[ãx ≤ b̃] if u = v,
0
otherwise
and the inequality relation for the values of the objective function is defined by
c̃x ≤ c̃x iff peak(c̃x) ≤ peak(c̃x ).
It is easy to compute that


 4 − v/2
if v/x ∈ [3, 4]
v/x − 2 if v/x ∈ [2, 3]
µM (v) = µM AX(2) (v) =

 0
otherwise
8
so, the unique solution to (16) is x∗ = 2. This result can be interpreted as: the
solution to problem (16) is equal to the solution to the following crisp LP problem
max 3x
s.t.
x ≤ 2, x ≥ 0,
where the coefficients are the peaks of the fuzzy coefficients of problem (16), and
the optimal value of problem (16), which can be called ”v is approximately equal
to 6”, can be considered as an approximation of the optimal value of the crisp
problem v ∗ = 6.
1
4
6
8
Figure 1. The membership function of the optimal value of problem (16).
Concluding remarks. We have interpreted FLP problems with fuzzy coefficients as MFR schemes, where the antecedents of the scheme correspond to the
constraints of the FLP problem and the fact of the scheme is the objective of the
FLP problem. Using the compositional rule of inference, we have shown a method
for finding an optimal value of the objective function and an optimal solution. In
the general case the computerized implementation of the proposed solution principle is not easy. To compute M AX(x) we have to solve a generally non-convex
and non-differentiable mathematical programming problem. However, the stability
property of the consequence in MFR schemes under small changes of the membership function of the antecedents [Ful91] guarantees that small rounding errors of
digital computation and small errors of measurement in membership functions of
the coefficients of the FLP problem can cause only a small deviation in the membership function of the consequence, M AX(x), i.e. every successive approximation
method can be applied to the computation of the linguistic approximation of the
exact M AX(x). As was pointed out in Section 2, to find an optimal value of the
objective function, M , from the equation (6) can be a very complicated process
(for related works see e.g. [Car82, Car87, Neg81, Orl80, Ver82, Ver84, Zim85,
Zim86) and very often we have to put up with a compromise solution [Rom89].
An efficient fuzzy-reasoning-based method is needed for the exact computation
of M . Our solution principle can be applied to multiple criteria mathematical pro9
gramming problems with fuzzy coefficients. This topic will be the subject of future
research.
Acknowledgements
The authors are indebted to Brigitte Werners of RWTH Aachen and the participants of the Workshop on Current Issues on Fuzzy Technologies 1992 for their
valuable comments and helpful suggestions.
References
[Buc88]
J.J.Buckley, Possibilistic linear programming with triangular fuzzy
numbers, Fuzzy sets and Systems, 26(1988) 135-138.
[Buc89]
J.J.Buckley, Solving possibilistic linear programming problems, Fuzzy
Sets and Systems, 31(1989) 329-341.
[Car82]
C.Carlsson, Tackling an MCDM-problem with the help of some results
from fuzzy sets theory, European Journal of Operational research,
3(1982) 270-281.
[Car87]
C.Carlsson, Approximate reasoning for solving fuzzy MCDM problems, Cybernetics and Systems: An International Journal, 18(1987)
35-48.
[Del87]
M.Delgado,J.L.Verdegay and M.A.Vila, Imprecise costs in mathematical programming problems, Control and Cybernetics, 16(1987) 114121.
[Del88]
M.Delgado,J.L.Verdegay and M.A.Vila, A procedure for ranking
fuzzy numbers using fuzzy relations, Fuzzy Sets and Systems,
26(1988) 49-62.
[Dub80]
D.Dubois and H.Prade, Systems of linear fuzzy constraints, Fuzzy Sets
and Systems 3(1980) 37-48.
[Ful91]
R.Fullér and H.-J.Zimmermann, On Zadeh’s compositional rule of
inference, in: R.Lowen and M.Roubens eds., Proc. of Fourth IFSA
Congress, Vol. Artifical Intelligence, Brussels, 1991 41-44.
[Neg81]
C.V.Negoita, The current interest in fuzzy optimization, Fuzzy Sets and
System, 6(1981) 261-269.
[Orl80]
S.A.Orlovski, On formalization of a general fuzzy mathematical problem, Fuzzy Sets and Systems, 3(1980) 311-321.
10
[Ovc89]
S.V.Ovchinnikov, Transitive fuzzy orderings of fuzzy numbers, Fuzzy
Sets and Systems, 30(1989) 283-295.
[Ram85]
J.Ramik and J.Rimanek, Inequality relation between fuzzy numbers
and its use in fuzzy optimization, Fuzzy Sets and Systems, 16(1985)
123-138.
[Rom89]
H.Rommelfanger,
Berlin, 1989.
[Tan84]
H.Tanaka and K.Asai, Fuzzy solution in fuzzy linear programming
problems, IEEE Transactions on Systems, Man, and Cybernetics, Vol.
SMC-14, 1984 325-328.
[Ver82]
J.L.Verdegay, Fuzzy mathematical programming, in: M.M.Gupta and
E.Sanchez eds., Fuzzy Information and Decision Processes, NorthHolland, 1982.
[Ver84]
J.L.Verdegay, Applications of fuzzy optimization in operational research, Control and Cybernetics, 13(1984) 230-239.
[Zad73]
L.A.Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. on Systems, Man and Cybernetics, Vol.SMC-3, No.1, 1973 28-44.
[Zad75]
L.A.Zadeh, The concept of linguistic variable and its applications to
approximate reasoning, Parts I,II,III, Information Sciences, 8(1975)
199-251; 8(1975) 301-357; 9(1975) 43-80.
[Zim78]
H.-J.Zimmermann, Fuzzy programming and linear programming with
several objective functions, Fuzzy Sets and Systems,1(1978) 45-55.
[Zim85]
H.-J.Zimmermann, Applications of fuzzy set theory to mathematical
programming, Information Sciences 36(1985) 29-58.
[Zim86]
H.-J.Zimmermann, Fuzzy Sets Theory and Mathematical Programming, in: A.Jones et al. (eds.), Fuzzy Sets Theory and Applications,
D.Reidel Publishing Company, 1986 99-114.
[Zim76]
H.-J.Zimmermann, Description and optimization of fuzzy system, International Journal of General Systems, 2(1976) 209-215.
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