Engineering Optimization: Theory and Practice, Fourth Edition

766 Practical Aspects of Optimization

subject to

gj( X)≤ 0 , j= 1 , 2 ,... , m

fj(X)+dj+−dj−=bj, j= 1 , 2 ,... , k

dj+≥ 0 , j= 1 , 2 ,... , k (14.113)

dj−≥ 0 , j= 1 , 2 ,... , k

d+jdj−= 0 , j= 1 , 2 ,... , k

wherebj is the goal set by the designer for thejth objective anddj+anddj−are,

respectively, the underachievement and overachievement of thejth goal. The value of

pis based on the utility function chosen by the designer. Often the goal for thejth

objective,bj, is found by first solving the following problem:

Minimizefj(X)

subjectto (14.114)

gj( X)≤ 0 , j= 1 , 2 ,... , m

If the solution of the problem stated in Eq. (14.114) is denoted byX∗j, thenbjis taken

asbj=fj(X∗j).

14.10.7 Goal Attainment Method

In the goal attainment method, goals are set asbifor the objective functionfi( X), i=

1 , 2 ,... , k. In addition, a weightwi> is defined for the objective function 0 fi( toX)

denote the importance of theith objective function relative to other objective functions

in meeting the goalbi, i= 1 , 2 ,... , k. Often the goalbiis found by first solving the

single objective optimization problem:

Minimizefi(X)

subject to (14.115)

gj(X)≤ 0 ;j= 1 , 2 ,... , m

If the solution of the problem stated in Eq. (14.115) is denotedX∗j thenbican be

taken as the optimum value of the objectivefi, fi∗= f(X∗i) A scalar. γis introduced

as a design variable in addition to thendesign variablesxi, i= 1 , 2 ,... , n. Then the

following problem is solved:

Findx 1 , x 2 ,... , xnandγ

to minimizeF (x 1 , x 2 ,... , xn, γ )=γ

subject to (14.116)

gj(X)≤ 0 ;j= 1 , 2 ,... , m

fi (X)−γwi≤bi; i= 1 , 2 ,... , k