Engineering Optimization: Theory and Practice, Fourth Edition

764 Practical Aspects of Optimization

wherewiis a scalar weighting factor associated with theith objective function. This

method [Eq.(14.106)] is also known as theweighting function method.

14.10.2 Inverted Utility Function Method

In the inverted utility function method, we invert each utility and try to minimize or

reduce the total undesirability. Thus ifUi(fi) enotes the utility function correspondingd

to theith objective function, the total undesirability is obtained as

U−^1 =

∑k

i= 1

Ui−^1 =

∑k

i= 1

1

Ui

(14.107)

The solution of the problem is found by minimizingU−^1 subject to the constraints

gj( X)≤ 0 ,j= 1 , 2 ,... , m.

14.10.3 Global Criterion Method

In the global criterion method the optimum solutionX∗is found by minimizing a

preselected global criterion,F (X), such as the sum of the squares of the relative

deviations of the individual objective functions from the feasible ideal solutions. Thus

X∗is found by minimizing

F(X)=

∑k

i= 1

{

fi(X∗i)−fi(X)

fi(X∗i)

}p

subject to (14.108)

gj( X)≤ 0 , j= 1 , 2 ,... , m

wherepis a constant (an usual value ofpis 2) andX∗i is the ideal solution for the

ith objective function. The solutionX∗iis obtained by minimizingfi( subject to theX)

constraintsgj( X)≤ 0 ,j= 1 , 2 ,... , m.

14.10.4 Bounded Objective Function Method

In the bounded objective function method, the minimum and the maximum acceptable

achievement levels for each objective functionfiare specified asL(i)andU(i), respec-

tively, fori= 1 , 2 ,... , k. Then the optimum solutionX∗is found by minimizing the

most important objective function, say, therth one, as follows:

Minimizefr(X)

subjectto

gj(X)≤ 0 , j= 1 , 2 ,... , m

L(i)≤fi≤U(i), i= 1 , 2 ,... , k, i

=r (14.109)