Fuzzy Modeling of Mathematical Programming Problems
and Interactive Processing
A Way for Realistic Decision Making and for Reducing Information Costs
Heinrich J. Rommelfanger
Institute of Statistics and Mathematics, J.W. Goethe-University of Frankfurt
Mertonstrasse 12-23, D-60054 Frankfurt am Main, Germany
Abstract.
This paper will demonstrate through the analysis of
(multiobjective) linear programming problems, that fuzzy
modeling in combination with an interactive solution
process presents an adequate answer to the information
dilemma of real problems. Instead of an extensive
gathering of information ex ante, the acquisition of
additional information will be oriented at set aims and
carried out under consideration of cost-benefit-relations.
Moreover, some interactive solution algorithms, e.g.
FUPAL, offer the possibility to solve mixed integer linear
programming problems.
1. Introduction
Empirical surveys reveal that Linear Programming is
one of the most frequently applied OR techniques in realworld problems, (see e.g. Kivijärvi, Korhonen and
Wallenius [1]; Lilien [2]; Tingley [8]; Meyer zu Selhausen
[3]). However, given the power of LP one could have
expected even more applications; a lot of conceivable LPapplications offered in literature are not used in practice.
This might be due to the fact that LP requires much of
well-defined and precise data which involve high
information costs. In real-world applications certainty,
reliability and precision of data is often illusory. The
stochastic approach has not proved to be very efficient for
this.
Yet, since the seminal paper on "Fuzzy Sets" by Lofti A.
Zadeh in 1965, there exists a convenient and powerful way
of modeling vague data without having recourse to
stochastic concepts. Fuzzy set literature usually justifies the
application of fuzzy numbers by the fact, that they allow an
adequate mathematical shaping of inaccuracy, which is not
of stochastic nature. Instead of using "average numbers"
for only vaguely known data fuzzy numbers and fuzzy
intervals make it possible to model the subjective
imaginations of a decision maker as precisely as he can
express them. If real problems are modeled by means of
fuzzy systems the chances for getting a wrong picture of
reality and by that selecting a solution which does not
correspond to the original problem will immensely be
reduced. This circumstance signifies a deciding factor and
illustrates one considerable advantage fuzzy models can
offer.
However, another and in my opinion even more important advantage of fuzzy models is frequently disregarded in literature. Deterministic and stochastic models
require a vast amount of information to ensure the identification of at least acceptable "average numbers", so that
the probability of incorrect modeling is kept as low as
possible. Furthermore, the optimal solution of a LP only
depends on a limited number of constraints and, thus, much
of the information collected have little impact on the
solution. Therefore, it would be sufficient to work with
vague data in these sectors.
This paper will demonstrate that fuzzy modeling in
combination with an interactive solution process presents
an adequate answer to the information dilemma of real
problems. Instead of an extensive gathering of information
ex ante, the acquisition of additional information will be
oriented at set aims and carried out under consideration of
cost-benefit-relations. In the first step of the interactive
solution process, the fuzzy system is modeled by using only
the information which the decision maker (DM) can
provide without any expensive acquisition. Knowing a first
“compromise solution“ the DM can perceive which further
information is required and is able to justify the decision by
comparing additional advantages and arising costs. In
doing so, the compromise solutions are improved step by
step. This procedure obviously offers the possibility to
limit the acquisition and processing of information to the
relevant components and therefore information costs will
be considerably reduced.
Moreover, some interactive solution algorithms, e.g.
FUPAL, offer the possibility to solve mixed integer linear
programming problems.
2. FULPAL
FULPAL (FUzzy Linear Programming based on Aspiration Levels) is an interactive solution algorithm for
solving fuzzy linear programming problems (FLP problem)
of the general type
~
~
~
C1x1  C 2 x 2  C n x n  m~
ax
subject to
~
~
~
~~
A i1x1  A i2 x 2  A in x n  Bi ,
i = 1,..., m,
~
 is a
fuzzy inequality relation that will be defined later.
As each real number a can be modeled as a fuzzy
number
~
A  {( x, f A ( x )) x R}
1 if x = a
with fA(x) = 
,
0 otherwise
the general system (1) includes the following special cases:
(i) the objective function is crisp, i.e.
z(x) = c1x1 + c2x2 + ... + cnxn  max;
(2)
(ii) some or all constraints are crisp, i. e.
gi(x) = ai1x1 + ai2x2 + ... + ainxn  bi;
(3)
(iii) some or all constraints have the soft form
~~
gi(x) = ai1x1 + ai2x2 + ... + ainxn  Bi .
(4)
These special cases may be combined.
The basic characteristics of FULPAL 3.0 are:
 The right-hand sides of the constraints are modeled as
~
Bi  ( b i ; b i , b i ; b i  A , b i ) A ,  , where  A is the
membership degree of the crisp aspiration level
A
A
 b i  i
~
 ) ,
C kj  ( c kj; c kj; c kj; c kj
(1)
x1 , x 2 , ..., x n  0 ,
~ ~
~
where C j, A ij and Bi are (flat) fuzzy numbers and
bi
~
 ; a ; a ; a  ) or
A ij  ( a ij
ij ij ij
a ij  a ij   ij
a ij  a ij   ij
a ij
a ij
~
 ; a ; a ; a  )
Figure 2: A ij = (a ij
ij ij ij
~
FULPAL is based on the inequality relation "  R ",
proposed by ROMMELFANGER [1988]:
~
~ ~
A i ( x )  R Bi
 n


( 5)
  ( a ij   ij )x j  b i   i
  j1
 ( x )   ( a ( x ))  Max.
( 6)
Bi i
 i
It is composed of the "pessimistic index" (5) and the fuzzy
goal (6).
The membership function
1
if a i ( x )  b i

 B   B ( a i ( x )) if b i  a i ( x )  b i   i
i
i

if b i   i  a i ( x )
0
may be interpreted as the subjective evaluation of the
n
needed quantity
a i (x) =  a ij x j
j 1
with regard to the
~
right-hand side B i .
,
 Bi ( a i ( x ))
1
A
bi

bi

b i A b i  b i   i
Figure 1: Membership function of
~
~
B i = Bi  ( b i ; b i , b i ; b i  A , b i ) A , 
 The coefficients are modeled as
ai (x )
bi  i
~
Figure 3: Inequality relation "  R "
~
The inequality relation "  R " has the advantage, that a
possible surplus a i (x) - bi directly influences the decision
~
process. Moreover "  R " coincides with the usual
interpretation of the inequality relation in soft constraints;
in the special case of deterministic inequalities it
~
corresponds to the classical "  "-relation. Thus "  R " is a
~
 )  or
The coefficients C kj  ( c kj; c kj; c kj; c kj
~
 ; a ; a ; a  )
A ij  ( a ij
ij ij ij
general definition for inequality relations in optimization
models.
are crisp real numbers, fuzzy
~
numbers or fuzzy intervals. The right hand sides B i are
 FULPAL uses a satisfying approach,
~ ~ ~
N k  R Z k ( x ) , k = 1, 2, ..., K,
crisp real numbers or fuzzy numbers where the left spreads
~
are zero. In FULPAL the B i are symbolized as ( b i , b i ) .
where the fuzzy aspiration levels
~

N k  ( n k ; n k A , n k ; n k ; n k )  A , 
for the objective
functions are changed step by step by improving the
crisp aspiration levels


n k A  n k  k A ,
K
1
 For getting a more realistic extended addition of the
left-hand sides of fuzzy constraints, a more flexible
extended addition, based on Yager's parametrized tnorm Tp may be used. Obviously, the inequality re~
lation  R is a special case (p  +) of the inequality
~
relation  KR :
~
~
~
A i ( x ) KR Bi


a i ( x )   i ( x , p )  b i   i
 
 i ( x )   Bi ( a i ( x ))  Max

( 7)
( 6)
For simplification, only 5 selected values for p are
proposed in FULPAL.
3. Example
The advantages of fuzzy modeling in combination with
the interactive solution algorithm FULPAL will be demonstrated by means of an numerical example.
We look at a fuzzy multicriteria linear programming
system with 8 variables, 4 objective functions and 10
constraints:
~
~
~
C11x1  C12 x 2    C18 x8
~
~
~
C21x1  C22 x 2    C28 x8
~
 Max
~
~
~
C31x1  C32 x 2    C38 x8
~
~
~
C41x1  C42 x 2    C48 x8
subject to
~
~
~
~~
A i1x1  A i2 x 2  A i8 x8  Bi ,
n k A , k = 1, 2, ..., K.
In FULPAL, the highest level for the objective Z k , k =
1, ..., K, is defined as
z k = c k 'x k  Z k ( x k ) = Max Z k ( x )
 B ( a1( x )), ...,  B ( a m ( x ))) ,
1
m
1
x1, x 2 , , x 8  0 .
FULPAL assists the decision maker by proposing
intervals [z k , z k ] for specifying the crisp aspiration level

 The total satisfaction of the DM with a solution x is
expressed by the compromise objective function
( x )  Min(  Z ( c1( x )), ...,  Z ( c K ( x )),
1
Unfortunately, it is not enough space for presenting the
extensive tables containing the coefficients and the right
hand sides or the results in this paper.
(8)
i = 1, 2, ..., 8;
x X R
z k = Max c k 'x = Max ( c k1x1   c kn x n )
x X R
(9)
xX R
with
 x  b  i  1,..., m 

a i1x1  a in
n
1 
i



X R  x  R a i1x1  a in x n  b i i  m1,..., m ,


x1, ..., x n  0




i.e. z k stands for the greatest value, the single objective
function Z k ( x ) can take on X R .
The lower boundaries z k , k = 1, ..., K, are defined as
 K

K


z k = Min  Min c k 'x *h , Min c k 'x **
(10)
h 
h 1
 h 1

 hk

hk
where x **
k is an optimal solution of the crisp LP-system
c k 'x  c k1x1  c kn x n  Max
subject to
x  X R a i1x1    a in x n  b i , i  1, ..., m .
(11)
If there exist more than one solution in (9) or (11) then
the solutions should be used which lead to the greatest
value of z k .
Obviously we have z k  n k and n k  z k . In FULPAL
we set z k  n k and n k  z k .
Therefore, in the first iteration step and after each
alteration of the coefficients and the right-hand side the
algorithm calculate these values z k and z k ,
k = 1, 2, 3, 4.
In order to get easily a first solution, there exist the
possibility that the aspiration levels were automatically
fixed as
b i  3  i
4
for the constraints and as
3 n   1 n for the constraints.
4 k
4 k
If the result of the first iteration step have a total
satisfaction value    A  0.5 the solution satisfies all
aspiration levels. Therefore, the DM can accept this
solution or he can decide to increase the aspiration levels.
On the other hand, if    A  0.5 there exist no solution
which satisfies all aspiration levels. In this case the DM has
to decrease at least one level. In FULPAL this can be done
comfortable by can shifting the buttons on the special
window. In the Figure 4 the dotted small boxes mark the
old aspiration level, the smooth small boxes the new
aspiration levels of the objectives and the green small
boxes the new aspiration levels of the constraints.
In the oral presentation the interactive solution process
will be demonstrated step by step with the aid of this
example.
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