For this type of problem [27], the optimisation algorithms have been applied in
a low-dimensional (N = 2) and a high-dimensional benchmark function (N = 30)
respectively. In MOLA, parameter wc is set to 1 for both 2-dimensional and 30-
dimensional cases.
MOGA and MOPSO find only 21 solutions in these cases, for the step length of the weights is set to 0.05. Thus,ψ1 andψ2can only take 21 different values. Conse- quently, only 21 solutions can be obtained. Lessening the step length will certainly lead to more solutions, however, the computation cost will increase accordingly and it will not ameliorate the distribution of the solutions.
When the dimensionality of Function I is two, the non-dominated solutions ob- tained by MOLA, MOGA and MOPSO are plotted in the objective space as shown in Fig. 3.5(a). In order to distinguish the notations of different algorithms, not all non-dominated solutions obtained by MOLA are plotted in the figures provided in this section, but only representative solutions in the objective space are selected to
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f1 f2 MOGA MOPSO MOLA 0.5 1 1.5 0 0.5 1 1.5 2 2.5 f1 f2 MOGA MOPSO MOLA (a) (b)
Figure 3.5: Pareto fronts obtained by MOGA, MOPSO and MOLA on Function I: (a) 2 dimensions; (b) 30 dimensions
show the intact shape of Pareto front obtained by MOLA. 3740 non-dominated so- lutions are found by MOLA with 10,000 FEs. Performing the process 30 times, the total number of solutions obtained by MOLA is not significantly different from each run. The fact that MOLA finds more non-dominated solutions suggests that it can provide more possible solutions that satisfy the optimisation targets. MOLA finds the smooth fronts (that can be considered as the Pareto front) which have the same shape and location in the objective space. For MOGA and MOPSO, 200 iterations (200×50 = 10,000FEs) are performed for each combination ofψ1andψ2. MOGA
can only find solutions at the ends of the Pareto front found by MOLA, due to the fact that the shape of the Pareto front is concave. The solutions of MOPSO gather around the two ends of the Pareto front found by MOLA, however, MOGA does not converge with 200 iterations. If the iterations are sufficient, the solutions obtained by MOGA will also flock to the two ends of the Pareto front obtained by MOLA.
In the case of 30 dimensions, for MOGA and MOPSO, 300 iterations (300 ×
50 = 15,000 FEs) are performed for each combination ofψ1 and ψ2. Similar to the case of 2 dimensions, the solutions of MOPSO flock to the ends of the front found by MOLA. MOGA cannot converge within 300 iterations, and its solutions begin to flock to the ends of the front when the number of iterations is increased to 1,000. With 15,000 FEs, MOLA finds 665 Pareto non-dominated solutions,
3.3 Compared with Weighted-sum Based Algorithms 82 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 f1 f2 MOGA MOPSO MOLA 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 f1 f2 MOGA MOPSO MOLA (a) (b)
Figure 3.6: Pareto fronts obtained by MOGA, MOPSO and MOLA on Function II: (a) 2 dimensions; (b) 30 dimensions
which are much more than those obtained by MOGA and MOPSO. The attraction of MOLA is greatly enhanced by the fact that the solutions found by MOLA can dominate those obtained by MOGA and MOPSO, as shown in Fig. 3.5(b). MOLA presents its superiority over the other two algorithms when the dimensionality of the multi-objective function is large.
Function II
The problem was proposed by Fonseca et al. [108]. Both a low dimensionality
(N = 2) and a high dimensionality (N = 30) of the benchmark function have been tested by the optimisation algorithms. In MOLA, parameterwcis set to 1 for both cases,i.e.2 dimensions and 30 dimensions.
This problem is a non-convex model which cannot be solved well by MOGA and MOPSO. In the case of 2 dimensions, the solutions obtained by MOGA and MOPSO locate at the two ends of the Pareto front in the objective space, as shown in Fig. 3.6(a). MOLA finds 746 non-dominated solutions with 10,000 FEs.
When the dimensionality of Function II is 30, MOLA performs much better than MOGA and MOPSO, as shown in Fig. 3.6(b). MOGA and MOPSO adopt 300
iterations for each combination of ψ1 and ψ2. MOGA can only find one solution
(1,1) in the objective space, while MOPSO finds three solutions, including (1,1) and
0 0.2 0.4 0.6 0.8 −0.5 0 0.5 1 1.5 2 f1 f2 MOGA MOPSO MOLA
Figure 3.7: Pareto fronts obtained by MOGA, MOPSO and MOLA on Function III
the other two solutions located on the two coordinates respectively. As for MOLA, it finds 401 non-dominated solutions with 15,000 FEs, and these solutions are evenly spread on the Pareto front.
Function III
The problem, which is a discontinuous Pareto front model, was proposed by Deb in 1998 [102]. The true Pareto font for this problem comprises four disconnected lines. In the case, the parameterwcis set to 0.001 for MOLA.
It can be seen from Fig. 3.7 that results obtained by MOGA and MOPSO with 400 iterations (400×50 = 20,000FEs) are far from satisfactory - only a paucity of solutions have converged to the front, and they are centralized at the ends of the first and fourth disconnected lines. On the second and third disconnected lines, no non-dominated solutions have been found by MOGA and MOPSO. MOLA finds 677 non-dominated solutions with 3,000 FEs, and they are distributed evenly over the four disconnected lines.
Function IV
This function is designed for disc break system, which is proposed by Osyczka et al. [109] in 1995. The objectives of the design are to minimise the mass of the brake and the stopping time. The search range of each variable is different, thus,
3.3 Compared with Weighted-sum Based Algorithms 84 0 1 2 3 4 0 5 10 15 20 f1 f2 MOGA MOPSO MOLA
Figure 3.8: Pareto fronts obtained by MOGA, MOPSO and MOLA on Function IV
parameter wc of MOLA is set to different values for different variables. In this
case, each variable is divided into ten grids, thus,wc,i = [xmax,i−xmin,i]/10, where
i= 1,· · · ,4.
MOPSO and MOGA obtain 21 points respectively with 2,000 iterations. The shape of the Pareto fronts obtained by algorithms MOGA and MOPSO is similar, as shown in the two upper lines in Fig. 3.8. The non-dominated solutions found by MOGA and MOPSO fall sparsely on the curve. This improvement is due to theF(X)is convex. However, in practical applications, usually it is not known in advance whether the target fitness function is convex or not, which is the problem confronted by algorithms MOGA and MOPSO. With 3,660 FEs, MOLA obtains 216 non-dominated solutions, which spread evenly on the Pareto front, i.e. the bottom curve in Fig. 3.8. It can be seen that the range of the Pareto front found by MOLA is wider than that obtained by MOGA and MOPSO, and MOLA can find more accurate results than other two algorithms.