Progressively Interactive Evolutionary Multi-Objective Optimization ...

F2
2
1.5
1
0.5
Most Preferred
Point
Lower Level Front
Upper Level Front
0
0 0.5 1
F1
1.5 2
Fig. 5. Pareto-optimal front for problem DS4. Final
parent population members have been shown
close to the most preferred point.
6.5 Problem DS5
F2
2
1.5
1
0.5
Lower Level Front
Most
Preferred
Point
Upper Level Front
0
0 0.5 1
F1
1.5 2
Fig. 6. Pareto-optimal front for problem DS5. Final
parent population members have been shown
close to the most preferred point.
Problem DS5 has been taken from [9]. A point on the Pareto-optimal front of the test problem
is chosen as the most-preferred point and then the PI-HBLEMO algorithm is executed to obtain
a solution close to the most preferred point. This problem is similar to problem DS4 except that
the upper level Pareto-optimal front is constructed from multiple points from a few lower level
Pareto-optimal fronts. There are K + L + 1 real-valued variables in this problem as well:
Minimize
F(x, y) =
(1 − x1)(1 + K j=2 x2j )y1
x1(1 + K j=2 x2j )y1
subject
to (x) ∈ argmin (x) f(x) =
(1 − x1)(1 +
,
K+L j=K+1 x2j )y1
x1(1 + K+L j=K+1 x2j )y1
,
G1(x) = (1 − x1)y1 + 1
2x1y1 − 2 + 1
5 [5(1 − x1)y1 + 0.2] ≥ 0,
[·] denotes greatest integer function,
−1 ≤ x1 ≤ 1, 1 ≤ y1 ≤ 2,
−(K + L) ≤ xi ≤ (K + L), i = 2, . . . , (K + L).
For the upper level Pareto-optimal front, xi = 0 for i = 2, . . . , (K+L), x1 ∈ [2(1−1/y1), 2(1−
0.9/y1)], y1 ∈ {1, 1.2, 1.4, 1.6, 1.8} (Figure 6). For this test problem we have chosen K = 5
and L = 4 (an overall 10-variable problem). This problem has similar difficulties as in DS4,
except that only a finite number of y1 qualifies at the upper level Pareto-optimal front and that
a consecutive set of lower level Pareto-optimal solutions now qualify to be on the upper level
Pareto-optimal front.
The Pareto-front, most-preferred point and the final population members from a particular
run are shown in Figure 6. Table 5 presents the function evaluations required by PI-HBLEMO
to produce the final solution and the function evaluations required by HBELMO to produce an
approximate Pareto-front.
7 Accuracy and DM calls
Table 6 represents the accuracy achieved and the number of decision maker calls required while
using the PI-HBLEMO procedure. In the above test problems the most preferred point which
126
(9)