Engineering Optimization: Theory and Practice, Fourth Edition

726 Modern Methods of Optimization

Constraint

0

x (in.)

1

m

Objective function

Design (decision)

Figure 13.8 Concept of fuzzy decision. [13.22], with permission of ASME.

subject to

gj(X)∈Gj, j= 1 , 2 ,... , m (13.55)

whereGjdenotes the fuzzy interval to which the functiongj( should belong. ThusX)

the fuzzy feasible region,S, which denotes the intersection of allGj, is defined by the

membership function

μS( X)= min

j = 1 , 2 ,...,m

{μGj[gj(X)]} (13.56)

Since a design vectorXis considered feasible whenμS( X)> 0 , the optimum design is

characterized by the maximum value of the intersection of the objective function and

the feasible domain:

μD(X∗) =maxμD( X), X∈D (13.57)

where

μD( X)=min

{

μf( X), min

j = 1 , 2 ,...,m

μGj[gj(X)]

}

(13.58)

13.6.3 Computational Procedure

The solution of a fuzzy optimization problem can be determined once the membership

functions offandgjare known. In practical situations, the constructions of the mem-

bership functions is accomplished with the cooperation and assistance of experienced

engineers in specific cases. In the absence of other information, linear membership

functions are commonly used, based on the expected variations of the objective and

constraint functions. Once the membership functions are known, the problem can be

posed as a crisp optimization problem as

FindXandλwhich maximizeλ