Róbert Fullér | Tibor Keresztfalvi | György Schuszter
Linguistic optimization under GoetschelVoxman defuzzification
TUCS Technical Report
No 285, May 1999
Linguistic optimization under GoetschelVoxman defuzzification
Róbert Fullér
Eötvös L. University, Department of Operations Research,
Múzeum krt. 6-8, H–1088 Budpaest, Hungary
Tibor Keresztfalvi
Eötvös L. University, Department of Applied Analysis,
Múzeum krt. 6-8, H–1088 Budpaest, Hungary
*Supported by the Bolyai scholarship of the
Hungarian Academy of Sciences
György Schuszter
Kandó K. College of Engineering,
Institute of Instrumentation and Automation,
Tavaszmező u. 15-17, H–1084 Budpaest, Hungary
TUCS Technical Report
No 285, May 1999
Abstract
We consider optimization problems in which the relation between the objective function and the decision variables is not known exactly but described
linguistically and represented in form of fuzzy if-then rules. To deal with linguistic variables represented by fuzzy sets with non-monotonic membership
functions, we suggest replacing Tsukamoto’s fuzzy reasoning method with another one based on Goetschel-Voxman defuzzification and give an alternative
procedure to obtain the crisp relationship between the objective function and
the decision variables. An optimal solution to the original fuzzy optimization
problem can be then obtained by solving the resulting crisp mathematical
programming program. We also give an example of the proposed procedure.
Keywords: Defuzzification, fuzzy optimization, linguistic variables, if-then
rules
1
Introduction
Let us consider the optimization problem of the form
max / min f (x);
subject to x ∈ X ⊂ Rn ,
where f (x) is not known exactly for any x ∈ X but the causal link between
x and f (x) is described linguistically using fuzzy if-then rules. According to
[2], we can write our problem in the following form:
max / min f (x);
subject to {R1 (x), . . . , Rm (x) | x ∈ X ⊂ Rn },
(1)
with
Ri (x): if x1 is Ai1 and . . . and xn is Ain then f (x) is Ci ,
where Aij and Ci (i = 1, . . . , m, j = 1, . . . , n) are fuzzy numbers. We want
to find crisp values x ∈ X that minimize/maximize f (x) in certain sense.
One can find a number of solution procedures for this kind of problems
proposed e.g. in [1]. Let us recall the one which is based on Tsukamoto’s
fuzzy reasoning method [5] and suggests determining the crisp value f (u)
corresponding to the crisp input u = (u1 , . . . , un ) ∈ X as follows:
1. Calculate the firing levels of individual rules by the product t-norm:
λi := Ai1 (u1 ) · . . . · Ain (un )
(i = 1, . . . , m).
2. Take the crisp rule output for each if-then rule:
zi := Ci−1 (λi )
(i = 1, . . . , m).
3. Aggregate the individual inputs by finding their weighted average (where
the associated weights are the calculated firing levels, respectively) and
get the overall system output:
z :=
λ1 z1 + . . . + λm zm
.
λ1 + . . . + λm
In this wise, our original optimization problem (1) turns into the following
crisp mathematical programming problem:
max / min f (u);
subject to u ∈ X ⊂ Rn .
We would like to point out that Tsukamoto’s fuzzy reasoning method
(specifically its second step) assumes that the conclusion part of each if-then
rule Ri is described by a fuzzy set Ci having strictly monotonic membership
1
function. On the other hand, the overall system output is calculated as the
weighted average of the individual rule outputs, which is actually the discrete
Center-of-Gravity of these outputs.
In this paper we suggest applying the Center-of-Gravity method also in
the second step of the procedure described above. This will allow our rule
base to contain also if-then rules with conclusion parts having not necessarily
monotonic membership functions.
2
Goetschel-Voxman defuzzification of
fuzzy intervals
We will represent the conclusion part of the if-then rules by fuzzy intervals
having trapezoidal membership functions:

a−x


1−
if x ∈ [a − α, a]


α

 1
if x ∈ [a, b]
C(x) := (a, b, α, β)(x) :=
(2)
x
−
b


1
−
if
x
∈
[b,
b
+
β]


β


0
otherwise
For us, it is more convenient now to describe a fuzzy interval with its level
sets:
[C]γ := {x ∈ R | C(x) ≥ γ}
(0 < γ ≤ 1)
and [C]0 := supp C. Obviously, the level sets of fuzzy intervals are always
crisp intervals. It is also easy to verify that by introducing the notations
a(γ) := min[C]γ ,
b(γ) := max[C]γ ,
we can describe the fuzzy interval (2) with its level sets as follows:
[C]γ = [a(γ), b(γ)] = [a − (1 − γ)α , b + (1 − γ)β]
(0 < γ ≤ 1).
The Goetschel-Voxman defuzzification method [3] defines the crisp value corresponding to a fuzzy interval as the weighted average of the centers of γ-level
sets. For the fuzzy interval C we have:
Z 1
a(γ) + b(γ)
γ·
dγ
2
0
GW (C) =
,
(3)
Z 1
γ dγ
0
i.e. the weight of the center of [C]γ is just γ. After some calculation, we can
easily obtain the following expression:
GW (C) = GW ((a, b, α, β)) =
2
a+b β−α
+
.
2
6
(4)
It is clear that by shearing a trapezoidal fuzzy interval at a certain (firing)
level λ ∈ [0, 1], the remaining part will also be of trapezoidal shape. The
sheared fuzzy interval Cλ is not normed to 1 anymore, see Figure 1. For its
level sets we have simply
(
[C]γ if 0 ≤ γ ≤ λ,
γ
[Cλ ] =
∅
if λ < γ.
According to this, the expression (3) must be slightly modified for Cλ as
follows:
Z λ
a(γ) + b(γ)
dγ
γ·
2
0
GW (Cλ) =
.
(5)
Z λ
γ dγ
0
This can easily be reduced to the form
GW (Cλ ) = GW ((a, b, α, β)λ) =
a+b β−α
+
(3 − 2λ),
2
6
(6)
which of course coincides with (4) if we substitute λ = 1.
1
λ
a−α a
Z
Z C
Z
Z
Z
Z
b
1
-
λ
b+β
Z Cλ
ZZ
Figure 1: Sheared fuzzy interval
3
Special cases
In most cases (in order to meet the requirements of Tsukamoto’s fuzzy reasoning method) rule bases consist of fuzzy if-then rules with conclusion parts
having skewed triangular shape, that is a zero-width peak and zero left or
right spreads, see Figure 2. Our expression (4) for these special cases reduces
to the following form:
α
,
6
β
GW ((b, b, 0, β)) = b + .
6
GW ((a, a, α, 0)) = a −
3
Shearing these skewed triangular fuzzy sets at level λ ∈ [0, 1] results in
trapezoidal fuzzy intervals with one of the spreads being equal to zero. It is
easy to verify that in these cases the Center-of-Gravity reads as follows:
3 − 2λ
α,
6
3 − 2λ
GW ((b, b, 0, β)λ ) = b +
β.
6
GW ((a, a, α, 0)λ) = a −
4
(7)
(8)
Example
Let us take a particular example from [2] with two decision variables x1 and
x2 and consider the following optimization problem:
min f (x1 , x2 );
subject to {x1 + x2 = 1/2,
(x1 , x2 ) ∈ [0, 1]2 ⊂ R2 }, (9)
where the correspondence (x1 , x2 ) 7→ f (x1 , x2 ) is given linguistically as
R1 (x1 , x2 ): if x1 is small and x2 is small then f (x1 , x2 ) is small,
R2 (x1 , x2 ): if x1 is small and x2 is big
then f (x1 , x2 ) is big,
and the linguistic values small and big are described with skewed triangular
fuzzy numbers:
small = (0, 0, 0, 1),
big = (1, 1, 1, 0),
see Figure 2.
1
HH
HH
H
small HHH
0
big
0
1
1
1
Figure 2: Skewed fuzzy intervals
Let us take an arbitrary crisp input (u1, u2 ) and follow the solution procedure described in the introduction but applying the Goetschel-Voxman
defuzzification method in the second step:
1. The firing levels are obviously
λ1 = (1 − u1 )(1 − u2 ),
4
λ2 = (1 − u1 )u2 .
(10)
2. We defined individual rule outputs as the Center-of-Gravity of sheared
fuzzy intervals and calculate them according to (7) and (8):
3 − 2λ1
6
3 − 2λ2
z2 = GW ((1, 1, 1, 0)λ2 ) = 1 −
6
z1 = GW ((0, 0, 0, 1)λ1 ) =
3. We obtain f (u1 , u2) as the weighted average of z1 and z2 with weights
λ1 and λ2 , respectively:
f (u1 , u2) =
3−2λ1
6
2
λ1 + (1 − 3−2λ
) λ2
6
,
λ1 + λ2
i.e. after substituting (10) in this expression we get:
f (u1 , u2 ) =
3 − 2(1 − u1 )(1 − u2 )
(1 − u2 ) +
6
3 − 2(1 − u1 )u2
u2 =
+ 1−
6
(11)
1 1
− (1 − u1 )(1 − 2u2).
2 3
Thus, we converted our initial fuzzy optimization problem into the following crisp mathematical programming problem:
=
f (u1 , u2) → min
subject to
u1 + u2 = 1/2,
(u1 , u2 ) ∈ [0, 1]2 ⊂ R2 .
Introducing the notation t := u1 and taking into account that u2 = 1/2 − t,
we have after some reduction the equivalent problem
f (t) =
1 1
+ (2t − 1)2 → min,
3 6
t ∈ [0, 1/2],
which has the optimal solution t∗ = 1/2, that is
u∗1 = 1/2,
u∗2 = 0
and the optimal value is
f (u∗ ) = f (u∗1, u∗2 ) = 1/3.
Firstly, we should note that this solution differs from that obtained in
[2] (u∗1 = u∗2 = 1/4, f (u∗) = 3/8). However, it reflects more fairly the fact
5
included in the rule base implicitly, namely that f (x) = f (x1 , x2 ) actually
depends just on (and correlates just with) x2 . We should also mention that
the optimal value f (u∗ ) is calculated else-ways in [2]. We can compare the
optimal values if we evaluate (11) for the input variables (u1 , u2) = (1/4, 1/4):
f (1/4, 1/4) = 3/8 = 0.375 > 1/3 ≈ 0.33.
On the other hand, this solution completely coincides with that obtained
in [4] where the Center-of-Gravity defuzzification was involved instead of the
Goetschel-Voxman method.
References
[1] C. Carlsson and R. Fullér, Optimization under fuzzy if-then rules, Fuzzy
Sets and Systems (1999) (to appear).
[2] R. Fullér, Fuzzy Reasoning and Fuzzy Optimization (TUCS General Publication, Turku, 1998).
[3] R. Goetschel and W. Voxman, Topological properties of fuzzy numbers,
Fuzzy Sets and Systems 9 (1983) 87–99.
[4] T. Keresztfalvi, Optimization with linguistic variables using the Centerof-Gravity defuzzification, in: C. Carlsson and R. Fullér eds., Frontiers
in Soft Decision Analysis, Studies in Fuzziness (Physica-Verlag, Heidelberg, 1999, to appear).
[5] Y. Tsukamoto, An approach to fuzzy reasoning method, in: M. M.
Gupta, R. K. Ragade and R. R. Yager eds., Advances in Fuzzy Set
Theory an Applications (North-Holland, New York, 1979).
6
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