Engineering Optimization: Theory and Practice, Fourth Edition

724 Modern Methods of Optimization

0

1

m m 1

m 2

y 2 yn

mn

(^0) y 1
(a) (b)
1
m
y 1 y 2 yn
0
(c)
1
m
y
Figure 13.5 Crisp and fuzzy sets:(a)crisp set;(b)discrete fuzzy set;(c)continuous fuzzy
set. [13.22], with permission of ASME.
B
A
B
A A
A

(a) (b) (c)

Figure 13.6 Basic set operations in crisp set theory:(a) AorBor both:A∪B;(b) Aand

B:A∩B;(c)notA:A. [13.22], with permission of ASME.

The result of this operation is shown in Fig. 13.7a. The intersection of the fuzzy sets

AandBis defined as

μA∩B(y)=μA(y)∧μB(y) =min[μA(y), μB(y)]

=

{

μA(y) if μA< μB

μB(y) if μA>μB

(13.50)

This operation is shown in Fig. 13.7b. The complement of a fuzzy setAis shown asA

in Fig. 13.7c, in which for everyμA(y) there is a corresponding, μA(y)= 1 −μA(y),

which defines the complement of the setA,A.