14.10 Multiobjective Optimization 765
14.10.5 Lexicographic Method
In the lexicographic method, the objectives are ranked in order of importance by the
designer. The optimum solutoinX∗is then found by minimizing the objective functions
startingwith the most important and proceeding according to the order of importance
of the objectives. Let the subscripts of the objectives indicate not only the objective
function number, but also the priorities of the objectives. Thusf 1 ( andX) fk( denoteX)
the most and least important objective functions, respectively. The first problem is
formulated as
Minimizef 1 (X)
subject to (14.110)
gj( X)≤ 0 , j= 1 , 2 ,... , m
and its solution X∗ 1 and f 1 ∗=f 1 (X∗ 1 ) is obtained. Then the second problem is
formulated as
Minimizef 2 (X)
subjectto
gj( X)≤ 0 , j= 1 , 2 ,... , m
f 1 (X)=f 1 ∗ (14.111)
The solution of this problem is obtained asX∗ 2 andf 2 ∗=f 2 (X∗ 2 ) This procedure.
is repeated until all the k objectives have been considered. The ith problem is
given by
Minimizefi(X)
subjectto
gj(X)≤ 0 , j= 1 , 2 ,... , m
fl(X)=fl∗, l= 1 , 2 ,... , i− 1 (14.112)
and its solution is found asX∗i andfi∗=fi(X∗i) Finally, the solution obtained at.
the end (i.e.,X∗k) s taken as the desired solutioni X∗of the original multiobjective
optimization problem.
14.10.6 Goal Programming Method
In the simplest version of goal programming, the designer sets goals for each objective
that he or she wishes to attain. The optimum solutionX∗is then defined as the one that
minimizes the deviations from the set goals. Thus the goal programming formulation
of the multiobjective optimization problem leads to
Minimize
∑k
j= 1
(d+j +dj−)p
1 p/
, p≥ 1
