Computational Intelligence - August 2013 - 37
11
9
Lwall (m)
Location-Speed FC
Best-Location FC
7
3
1
Speed (m/s)
m
5
-1
-3
-5
-7
-9
-7
-5
-3
-1
m
1
3
5
7
1
0.8
0.6
0.4
0.2
0
50
0.6
100
150
0.4
Lwall (m)
Location-Speed FC
5
Speed (m/s)
m
7
3
-1
-6
Best-Location FC
-4
-2
0
300
350
400
0
50
100
150
200 250
Time Step
300
350
400
(b)
11
1
250
Location-Speed FC
Best-Location FC
0.2
(a)
9
200
2
4
6
7
1
0.8
0.6
0.4
0.2
0
50
0.6
100
150
200
0.4
Location-Speed FC
Best-Location FC
0.2
0
50
100
150
m
Time Step
(c)
(d)
200
Figure 11 (a) and (c) Robot wall-following trajectories. (b) and (d) The robot-wall distance value (Lwall), and the robot moving speed at each control
time step in the three test environments using the location-speed (-) and the best-location (- -) FCs in the three test environments in Example 2.
Table 2 The average diversity metric D of various algorithms in Example 1.
algoriThms
mo-aCor mo-aCor
mo-ega Nsgaii (q = 0.01) (q = 10)
mo-raCaCo(r) mo-raCaCo(C) mo-raCaCo
AvERAgE oF D 9.92
10.66
9.74
9.71
approach, and the same solution replacement strategy as in the
MO-RACACO. They differ from the MO-RACACO in the
way where a new solution is generated. In the MO-RACACO,
the population size N was set to 50. To determine the effect of
the adaptive elite-tournament solution number selection in (7),
the MO-RACACO with a constant number of L = N/2 was
applied to the same problems, and the method was denoted as
MO-RACACO(C). A modified MO-RACACO that implements the rule-based mutation operation in phase two by filling the original null vector in an inactive rule with a new solution vector, uniformly and randomly selected from the same
node column, was also applied to the same problems, and the
method was denoted as MO-RACACO(R). All MOPOAs
used the same population size and the same number of performance evaluations. Comparisons were made to show the
12.10
10.04
9.50
advantage of the new individual generation approach in the
MO-RACACO.
To evaluate the performance of the MOPOAs, the performance indicators employed were the coverage metric [40] and
the diversity metric [41]. The coverage metric C is applied to
obtain the dominance relationship between two optimization
algorithms. This metric is defined for two sets of obtained solutions A and B and is given by:
C (A, B) =
" b ! B 7a ! A: a ) b ,
,
B
(15)
where a ) b denotes a weakly dominates b. The value
C (A, B) = 1 means that all solutions in B are weakly
dominated by A. On the contrary, C (A, B) = 0 indicates that
August 2013 | IEEE ComputAtIonAl IntEllIgEnCE mAgAzInE
37
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