Engineering Optimization: Theory and Practice, Fourth Edition

14.10 Multiobjective Optimization 763

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f 1 = (x−3)^4
f 2 = (x−6)^2
P Q
x
f
Figure 14.9 Pareto optimal solutions.
In general, no solution vectorXexists that minimizes all thekobjective functions
simultaneously. Hence, a new concept, known as thePareto optimum solution, is used
in multiobjective optimization problems. A feasible solutionXis calledPareto optimal
if there exists no other feasible solutionYsuch thatfi(Y)≤fi( forX) i= 1 , 2 ,... , k
withfj( Y)< fi( for at least oneX) j. In other words, a feasible vectorXis called
Pareto optimal if there is no other feasible solutionYthat would reduce some objective
function without causing a simultaneous increase in at least one other objective function.
For example, if the objective functions are given byf 1 = (x− 3 )^4 andf 2 = (x− 6 )^2 ,
their graphs are shown in Fig. 14.9. For this problem, all the values ofxbetween 3
and 6 (points on the line segmentPQ) denote Pareto optimal solutions.
Several methods have been developed for solving a multiobjective optimization
problem. Some of these methods are briefly described in the following paragraphs.
Most of these methods basically generate a set of Pareto optimal solutions and use
some additional criterion or rule to select one particular Pareto optimal solution as the
solution of the multiobjective optimization problem.

14.10.1 Utility Function Method

In the utility function method, a utility functionUi(fi) s defined for each objectivei

depending on the importance officompared to the other objective functions. Then a

total or overall utility functionUis defined, for example, as

U=

∑k

i= 1

Ui(fi) (14.105)

The solution vectorX∗is then found by maximizing the total utilityUsubjected to the

constraintsgj( X)≤ 0 ,j= 1 , 2 ,... , m. A simple form of Eq. (14.105) is given by

U=

∑k

i= 1

Ui= −

∑k

i= 1

wifi(X) (14.106)