Engineering Optimization: Theory and Practice, Fourth Edition

13.6 Optimization of Fuzzy Systems 723

The set [0, 1] is called avaluation set.A setAis called afuzzy setif the valuation set
is allowed to be the whole interval [0, 1]. The fuzzy setAis characterized by the set
of all pairs of points denoted as

A= {x, μA(x) }, x∈X (13.45)

whereμA(x) is called themembership functionofxinA. The closer the value of
μA(x) s to 1, the morei xbelongs toA. For example, letX= {62 64 66 68 70
72 74 76 78 80}be possible temperature settings of the thermostat (◦F) in an
air-conditioned building. Then the fuzzy setAof “comfortable temperatures for human
activity” may be defined as

A= {( 62 , 0. 2 ) ( 64 , 0. 5 ) ( 66 , 0. 8 ) ( 68 , 0. 95 ) ( 70 , 0. 85 ) ( 72 , 0. 75 )

( 74 , 0. 6 ) ( 76 , 0. 4 ) ( 78 , 0. 2 ) ( 80 , 1. 0 )} (13.46)

where a grade of membership of 1 implies complete comfort and 0 implies complete
discomfort. In general, ifXis a finite set,{x 1 , x 2 ,... , xn} he fuzzy set ont Xcan be
expressed as

A=μA(x 1 )|x 1 +μA(x 2 )|x 2 +· · · +μA(xn)|xn=

∑n

i= 1

μA(xi)|xi (13.47)

or in the limit, we can expressAas

A=

∫

x

μA(x)|x (13.48)

Crisp set theoryis concerned with membership of precisely defined sets and is
suitable for describing objective matters with countable events. Crisp set theory is
developed using binary statements and is illustrated in Fig. 13.5a, which shows the
support fory 1 with no ambiguity. Since fuzzy set theory is concerned with linguistic
statements of support for membership in imprecise sets, a discrete fuzzy set is denoted
as in Fig. 13.5b, where the degree of support is shown by the membership values,μ 1 ,
μ 2 ,... , μn, corresponding toy 1 , y 2 ,... , yn, respectively. The discrete fuzzy set can
be generalized to a continuous form as shown in Fig. 13.5c.
The basic crisp set operations of union, intersection, and complement can be rep-
resented on Venn diagrams as shown in Fig. 13.6. Similar operations can be defined
for fuzzy sets, noting that the setsAandBdo not have clear boundaries in this case.
The graphs ofμAandμBcan be used to define the set-theoretic operations of fuzzy
sets. The union of the fuzzy setsAandBis defined as

μA∪B(y)=μA(y)∨μB(y) =max[μA(y), μB(y)]

=

{

μA(y) if μA>μB

μB(y) if μA< μB

(13.49)