Progressively Interactive Evolutionary Multi-Objective Optimization ...

TABLE III
DISTANCE OF OBTAINED SOLUTION FROM THE MOST PREFERRED
SOLUTION, NUMBER OF FUNCTION EVALUATIONS, AND NUMBER OF DM
CALLS REQUIRED BY PI-NSGA-II-VF ON THE THREE-OBJECTIVE
DTLZ8 PROBLEM. ds = 0.1.
Minimum Median Maximum
Accuracy 0.0074 0.0230 0.0819
# of Function Eval. 3706 4625 4665
# of DM Calls 8 11 18
as n = 10M. So the five objective test problem which
we consider here has 50 number of decision variables. The
original problem once again is a minimization problem but
since we wish to maximize the objectives a negative sign
has been used before each of the objectives to turn the
problem into a maximization problem. The description of
the test problem for M number of objectives and n number
of variables is as follows:
⌊j n
M Maximize fj(x) = − ⌋
i=⌊(j−1) n
M
subject to gj(x) = f 2 M (x) + f 2 j
⌋
x0.1 i , j = 1, 2, . . . , M,
(x) − 1 ≥ 0,
j = 1, 2, . . . , M − 1,
0 ≤ xi ≤ 1, for i = 1, . . . , n,
(7)
For this test problem, the Pareto-optimal front is a curve
with f1 = f2 = . . . = fM−1. The Pareto-optimal curve
lies on the intersection of all M − 1 constraints. A two
dimensional plot of the Pareto-optimal front with fM and
any other objective represents a circular arc of radius 1.
The Pareto-optimal front for a five-objective DTLZ9 problem
is shown in Figure 8 with objectives f1 and f5. The other
objective values (f2, f3, f4) are equal to f1.
For this problem, we choose a non-linear DM-emulated
value function, as follows:
5
V (f) = 1/ (fi − ai) 2 , (8)
i=1
where a = (−0.175, −0.175, −0.175, −0.175, −0.4899) T .
This value function produces the most preferred point as
z ∗ = (−0.2, −0.2, −0.2, −0.2, −0.9798).
Table IV presents the obtained solutions by PI-NSGA-II-
VF with 50 population members. Table V shows the accuracy
TABLE IV
FINAL OBJECTIVE VALUES OBTAINED FROM PI-NSGA-II-VF FOR THE
FIVE-OBJECTIVE DTLZ9 PROBLEM.
z ∗ Best Median Worst
f1 -0.2000 -0.2012 -0.2023 -0.2610
f2 -0.2000 -0.2002 -0.2008 -0.2408
f3 -0.2000 -0.2008 -0.2024 -0.2111
f4 -0.2000 -0.2005 -0.2004 -0.2007
f5 -0.9798 -0.9798 -0.9797 -0.9797
measure, the number of overall function evaluations, and
0
−0.2
−0.4
f_5
−0.6
−0.8
−1
Pareto Optimal
Front
Most
Preferred Point
−1 −0.8 −0.6 −0.4 −0.2 0
f_1
Fig. 8. Final population members after termination of the algorithm for
five-objective DTLZ9 problem.
TABLE V
DISTANCE OF OBTAINED SOLUTION FROM THE MOST PREFERRED
SOLUTION, FUNCTION EVALUATIONS, AND THE NUMBER OF DM CALLS
REQUIRED BY PI-NSGA-II-VF FOR THE FIVE-OBJECTIVE DTLZ9
PROBLEM. ds = 0.01.
minimum median maximum
Accuracy 0.0015 0.0034 0.0742
# of Function Eval. 25452 29035 38043
# of DM Calls 59 63 89
the number of DM calls. Although the points close to the
most preferred point are obtained in each run, the higher
dimensionality of the problem requires more function evaluations
and DM calls compared to the three-objective test
problem. However, the above results are obtained for a strict
termination criterion with ds = 0.01. Smaller number of DM
calls and evaluations are expected if this termination criterion
is relaxed. In Table VI, the termination parameter ds has been
relaxed to 0.1. Once again we find that this leads to reduction
in function evaluations as well as reduction in the number of
DM-calls. The accuracy also becomes low.
TABLE VI
DISTANCE OF OBTAINED SOLUTION FROM THE MOST PREFERRED
SOLUTION, FUNCTION EVALUATIONS, AND THE NUMBER OF DM CALLS
REQUIRED BY PI-NSGA-II-VF FOR THE FIVE-OBJECTIVE DTLZ9
PROBLEM. ds = 0.1
minimum median maximum
Accuracy 0.0284 0.0673 1.3836
# of Function Eval. 8201 9273 12806
# of DM Calls 19 24 33
It is worth mentioning that the application of a usual EMO
(including the original NSGA-II) is reported to face difficulties
in converging to the entire five-dimensional Paretooptimal
front with an identical number of function evalua-
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