14.10
Multiobjective Optimization
763
f
60
4
f1 = (x3)
40
f2 = (x6)
20
P
0
2
Q
x
1
2
4
3
5
6
7
8
Pareto optimal solutions.
Figure 14.9
In general, no solution vector X exists that minimizes all the k objective functions
simultaneously. Hence, a new concept, known as the Pareto optimum solution, is used
in multiobjective optimization problems. A feasible solution X is called Pareto optimal
if there exists no other feasible solution Y such that f i(Y) _ f i(X) for i = 1, 2, . . . , k
with f j(Y) < f i(X) for at least one j. In other words, a feasible vector X is called
Pareto optimal if there is no other feasible solution Y that 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 by f 1 = (x Š 3) 4 and f 2 = (x Š 6) 2,
their graphs are shown in Fig. 14.9. For this problem, all the values of x between 3
and 6 (points on the line segment PQ) 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 function U i(f i) is defined for each objective
depending on the importance of f i compared to the other objective functions. Then a
total or overall utility function U is defined, for example, as
U= ,
k
U i(f i)
(14.105)
i=1
The solution vector X_
is then found by maximizing the total utility U subjected to the
constraints g j(X) _ 0, j = 1, 2, . . . , m. A simple form of Eq. (14.105) is given by
U= ,
k
i=1
k
Ui = Š ,
i=1
w if
i(X)
(14.106)
764
Practical Aspects of Optimization
where w i is a scalar weighting factor associated with the ith objective function. This
method [Eq.(14.106)] is also known as the weighting 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 if U i(f i) denotes the utility function corresponding
to the ith objective function, the total undesirability is obtained as
UŠ
=
,
1
k
k
1
UiŠ 1 = ,
i=1
(14.107)
Ui
i=1
The solution of the problem is found by minimizing U
Š1
gj
14.10.3
subject to the constraints
(X) _ 0, j = 1, 2, . . . , m.
Global Criterion Method
In the global criterion method the optimum solution X_ 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
k
F (X) = ,
i=1
5
f (iX
i(X)
_)
Šf
f (iX
subject to
g (jX ) _
0,
i
_
i
j = 1, 2, . .
6p
)
(14.108)
.,m
where p is a constant (an usual value of p is 2) and Xi_
is the ideal solution for the
is obtained by minimizing f i(X) subject to the
ith objective function. The solution Xi_
constraints g j(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 function f i are specified as L(i)
and U (i) , respec_
tively, for i = 1, 2, . . . , k. Then the optimum solution X
is found by minimizing the
most important objective function, say, the rth one, as follows:
Minimize f
r(X)
subject to
g (jX ) _
0,
L(i) _ f i _ U (i),
j = 1, 2, . .
.,m
i = 1, 2, . . . , k, i = r
(14.109)
14.10
14.10.5
Multiobjective Optimization
765
Lexicographic Method
In the lexicographic method, the objectives are ranked in order of importance by the
designer. The optimum solutoin X _ is then found by minimizing the objective functions
starting with 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. Thus f 1(X) and f k(X) denote
the most and least important objective functions, respectively. The first problem is
formulated as
Minimize f
1(X)
subject to
and its solution X1_
formulated as
(14.110)
g (jX ) _ j = 1, 2, . . . , m
0,
and f1_ = f 1(X 1_) is obtained. Then the second problem is
Minimize f
2(X)
subject to
gj (X) _ 0,
.,m
j = 1, 2, . .
f (1X ) = f1_
(14.111)
_
The solution of this problem is obtained as X2_ and f2 = f (X _). This procedure
2
2
is repeated until all the k objectives have been considered. The ith problem is
given by
Minimize f i(X)
subject to
j = 1, 2, . .
gj (X) _ 0,
f (lX ) = f
l
_,
.,m
l = 1, 2, . . . , i Š 1
(14.112)
= f i(X i_). Finally, the solution obtained at
and its solution is found as Xi_ and
_) is taken as the desired solution
_
the
end
(i.e.,
X
fi
of the original multiobjective
k
X_
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 solution X_ 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
fi1/p
k
Minimize ff ,
j=1
(dj+
+d
Š p
ffi
j )
,
p ff 1
766
Practical Aspects of Optimization
subject to
f (jX
d j+
j = 1, 2, . .
.,m
j = 1, 2, . .
.,k
dj+ ff 0,
j = 1, 2, . .
.,k
djŠ ff 0,
j = 1, 2, . .
.,k
g (jX ) _
) + Š0,djŠ = b ,
j
+
(14.113)
Š
j = 1, 2, . . . , k
d j j = 0,
d
and
where bj is the goal set by the designer for the jth objective and
Š
+
d
j
respectively, the
underachievement and overachievement of the jth goal. The value of dj
p is based on the utility function chosen by the designer. Often the goal for the jth
objective, b j, is found by first solving the following problem:
are,
Minimizef j(X)
subject to
(14.114)
gj (X) _ 0,
j = 1, 2, . .
.,m
If the solution of the problem stated in Eq. (14.114) is denoted by X j_, then b
as b
14.10.7
j
j
is taken
= f j(X j_).
Goal Attainment Method
In the goal attainment method, goals are set as b i for the objective function f i(X), i =
1, 2, . . . , k. In addition, a weight w i > 0 is defined for the objective function f i (X) to
denote the importance of the ith objective function relative to other objective functions
in meeting the goal b i, i = 1, 2, . . . , k. Often the goal b i is found by first solving the
single objective optimization problem:
Minimizef i(X)
subject to
(14.115)
gj (X) _ 0; j = 1, 2, . . . , m
If the solution of the problem stated in Eq. (14.115) is denoted
then b i can
Xj_ as the optimum value of the objective f , f _
taken
= f (X i_). A scalar _ is
beintroduced
i i
as a design variable in addition to the n design variables x i, i = 1, 2, . . . , n. Then the
following problem is solved:
Find x
1,
x
2,
...,x
to minimize F (x
1,
x
n
and _
2,
. . . , x n, _ ) = _
subject to
(14.116)
g (jX ) _ 0; j = 1, 2, . . . , m
f (iX ) Š _ w
k
i
_ b i; i = 1, 2, . . . ,