Progressively Interactive Evolutionary Multi-Objective Optimization ...

Table 1. Total function evaluations for the upper and lower level (21 runs) for DS1.
6.2 Problem DS2
Algo. 1, Algo. 2, Best Median Worst
Savings Total LL Total UL Total LL Total UL Total LL Total UL
FE FE FE FE FE FE
HBLEMO 2,819,770 87,582 3,423,544 91,852 3,829,812 107,659
PI-HBLEMO 329,412 12,509 383,720 12,791 430,273 10,907
HBLEMO
P I−HBLEMO 8.56 7.00 8.92 7.18 8.90 9.87
Problem DS2 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 uses discrete values of y1 to determine
the upper level Pareto-optimal front. The overall problem is given as follows:
⎧
⎨ cos(0.2π)y1 + sin(0.2π)
u1(y1) =
⎩
|0.02 sin(5πy1)|,
for 0 ≤ y1 ≤ 1,
y1 − (1 − cos(0.2π)), y1 ⎧
> 1
⎨ − sin(0.2π)y1 + cos(0.2π)
u2(y1) =
⎩
(5)
|0.02 sin(5πy1)|,
for 0 ≤ y1 ≤ 1,
0.1(y1 − 1) − sin(0.2π), for y1 > 1.
Minimize F(x, y) =
⎛
u1(y1) +
⎜
⎝
K
2
j=2 yj + 10(1 − cos( π
K yi))
+τ K i=2 (xi − yi) 2
− r cos γ π
x1
2 y1
u2(y1) + K
2
j=2 yj + 10(1 − cos( π
K yi))
+τ K i=2 (xi − yi) 2
− r sin γ π
⎞
⎟
⎠
x1
2 y1
,
subject to (x) ∈
f(x) =
x
argmin (x)
2 1 + K i=2 (xi − yi) 2
K i=1 i(xi − yi) 2
,
−K ≤ xi ≤ K, i = 1, . . . , K,
0.001 ≤ y1 ≤ K, −K ≤ yj ≤ K, j = 2, . . . , K,
(6)
Due to the use of periodic terms in u1 and u2 functions, the upper level Pareto-optimal front
corresponds to only six discrete values of y1 (=0.001, 0.2, 0.4, 0.6, 0.8 and 1). r = 0.25 has
been used.
In this test problem the upper level problem has multi-modalities, thereby causing an algorithm
difficulty in finding the upper level Pareto-optimal front. A value of τ = −1 has been
used, which introduces a conflict between upper and lower level problems. The results have been
produced for 20 variables test problem.
The Pareto-optimal front for this test problem is shown in Figure 3. The most-preferred
solution is marked on the Pareto-front along with the final population members obtained from a
particular run of the PI-HBLEMO algorithm. Table 2 presents the function evaluations required
to arrive at the best solution using PI-HBLEMO. The function evaluations required to achieve
an approximated Pareto-frontier using the HBELMO algorithm is also reported.
6.3 Problem DS3
Problem DS3 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. In this test problem, the variable y1 is considered
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