Progressively Interactive Evolutionary Multi-Objective Optimization ...

is at most 10% of the maximum difference in value functions
between ≻-class of points.
A little thought will reveal that the above optimization
problem attempts to find a value function for which the
minimum difference in the value function values between
the ordered pairs of points is maximum. Considering all the
expressions, we have the following optimization problem:
Maximize ǫ,
subject to Sm(Pi) ≥ 0, i = 1, 2, . . . , η, and m = 1, 2,
S2(Pi) + k2S1(Pi) ≥ 0, i = 1, 2, . . . , η,
k1S2(Pi) + S1(Pi) ≥ 0, i = 1, 2, . . . , η,
V (Pi) − V (Pj) ≥ ǫ, for all (i, j) pairs
satisfying Pi ≻ Pj,
|V (Pi) − V (Pj)| ≤ δV , for all (i, j) pairs
satisfying Pi ≡ Pj.
(4)
Figure 2 considers five (η = 5) hypothetical points (P1 =
(3.5, 3.7), P2 = (2.6, 4.0), P3 = (5.9, 2.2), P4 = (0.0, 6.0),
and P5 = (15.0, 0.5)) and a complete ranking of the points (P1
being best and P5 being worst). Due to a complete ranking, we
do not have the fourth constraint set. The solution to the above
optimization problem results in a value function, the contours
(iso-utility curves) of which are drawn in the figure. The value
function obtained after the optimization is as follows:
V (f1, f2) = (f1 + 4.3229)(f2 + 0.9401).
The asymptotes of this value function are parallel to f1 and
f2 axes. The optimized value of ǫ is 2.0991. It is interesting to
note the preference order and other restrictions are maintained
by the obtained value function.
f 2
11
10
9
8
7
6
5
4
3
2
1
0
P4
P2
P1
P3
0 2 4 6 8 10 12 14 16
f
1
Fig. 2. Value function found by optimization.
Interestingly, if the DM provides a different preference
information: P1 is preferred over P2, P2 is preferred over P3
and no preference exists among P3, P4 and P5, a different
value function will be obtained. We re-optimize the resulting
problem with the above preference information on the same
set of five points used in Figure 2 and obtain the following
value function:
V (f1, f2) = (f1 + 5.9355)(f2 + 1.6613).
P5
34
Figure 3 shows the corresponding value function contours.
The contour makes a clear distinction between solutions in
pairs P1-P2 and P2-P3 (within an optimized value ǫ), however,
there is no distinction among P3, P4 and P5 (with 0.1ǫ), to
establish the given preference structure. Since a value function
maintaining a difference (ǫ) between points in pairs P1-P2 and
P2-P3 is needed and a maximum gap of 10% of ǫ is needed,
a somewhat greater ǫ value to that found in the previous case
is obtained here. The optimized ǫ value is found to be 2.2645
in this case.
Fig. 3. Revised value function with a different preference information.
2) Polynomial Value Function for M Objectives: The above
suggested methodology can be applied to any number of
objectives. For a general M objective problem the value
function can be written as follows:
V (f) = (f1 + k11f2 + k12f3 + . . . + k 1(M−1)fM + l1)×
(f2 + k21f3 + k22f4 + . . . + k 2(M−1)f1 + l2)×
. . .
(fM + kM1f1 + kM2f4 + . . . + k M(M−1)fM−1 + lM )
(5)
The above value function can be expressed more elegantly as
follows:
⎛
M M
V (f) = ⎝
i=1
j=1
Kijfj + K i(M+1)
⎞
⎠ . (6)
Since each term in the value function can be normalized, we
can introduce an additional constraint M
j=1 Kij = 1 for each
term denoted by i. As discussed below, Kij ≥ 0 for j ≤ M
and for each i, however K i(M+1) can take any sign. In the
remainder of the paper, we follow the value function definition
given in equation 5.
In the formulation it should be noted that the subscripts
of the objective functions change in a cyclic manner as we
move from one product term to the next. The number of
parameters in the value function is M 2 . The optimization
problem formulation for the value function suggested above
contains M 2 + 1 variables (kij and li). The variable ǫ is to be
maximized. The second set of constraints (strictly increasing
property of V ) will introduce non-linearity. To avoid this,
we simplify the above constraints by restricting the strictly
increasing property of each term Sk, instead of V itself. The