13.6 Optimization of Fuzzy Systems 725
mAUB(y)
mB(y)
mABU(y) m(y)
mA(y) mA(y)
(a)
A
y
(b)
y
(c)
y
1 1 1
mB(y)
mA(y) A
mA–(y)
Figure 13.7 Basic set operations in fuzzy set theory:(a)union;(b)intersection;(c)comple-
ment. [13.22], with permission of ASME.
13.6.2 Optimization of Fuzzy Systems
The conventional optimization methods deal with selection of the design variables that
optimizes an objective function subject to the satisfaction of the stated constraints.
For a fuzzy system, this notion of optimization has to be revised. Since the objective
and constraint functions are characterized by the membership functions in a fuzzy
system, a design (decision) can be viewed as the intersection of the fuzzy objective
and constraint functions. For illustration, consider the objective function: “The depth of
the crane girder(x)should be substantially greater than 80 in.” This can be represented
by a membership function, such as
μf(x)=
{
0 if x <80 in.
[1+(x− 80 )−^2 ]−^1 if x≥ 8 0 in.
(13.51)
Let the constraint be “The depth of the crane girder(x)should be in the vicinity of
83 in.” This can be described by a membership function of the type
μg(x) =[ 1 +(x− 83 )^4 ]−^1 (13.52)
Then the design (decision) is described by the membership function,μD(x) as,
μD(x)=μf(x)∧μg(x)
=
0 x<80 in.
min{[1+(x− 80 )−^2 ]−^1 , 1 [ +(x− 83 )^4 ]−^1 }
if x≥80 in.
(13.53)
This relationship is shown in Fig. 13.8.
The conventional optimization problem is usually stated as follows:
FindXwhich minimizesf (X)
subject to
g
(l)
j ≤gj(X)≤g
(u)
j , j=^1 ,^2 ,... , m (13.54)
where the superscriptslandudenote the lower and upper bound values, respectively.
The optimization problem of a fuzzy system is stated as follows:
FindXwhich minimizesf (X)
