By David M. Burton
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Extra resources for An introduction to abstract mathematical systems
100 NSGA-II SPEA2 % Max SP SP_6SC 95 SP_6TT SP_2SC_4TT SP_4SC_2TT SP_2SC_6TT SP_6SC_2TT 90 SP SP_6SC SP_6TT SP_2SC_4TT SP_4SC_2TT SP_2SC_6TT SP_6SC_2TT Assessed by Fig. 1. 00 Max Spread Max Spread 2D Morrison and De Jong Diversity Assessed By Fig. 2. The relative performance of multi-objective and single objective weighted sum genetic algorithms when assessed by diversity metrics An interesting observation is that sometimes the “best” result is not found by the GA that is running the same objective function weights as is being used to assess the runs, an example of this is shown in Figure 1 when assessing the runs with the SP_6SC_2TT objective, here the GA that was run with the SP_6SC_2TT objective is outperformed by both NSGA-II and SPEA2 albeit by a small amount.
Objective functions for weighted sum methods Formula for Calculation f = SP f = SP – (SC * 6) f = SP – (TT * 6) f = SP – (SC * 2) – (TT * 4) f = SP – (SC * 4) – (TT * 2) f = SP – (SC * 2) – (TT * 6) f = SP – (SC * 6) – (TT * 2) Name on Graphs SP SP_6SC SP_6TT SP_2SC_4TT SP_4SC_2TT SP_2SC_6TT SP_6SC_2TT The genetic algorithm was run ten times with each weighted sum objective function shown in Table 1, as well as ten times with both NSGA-II and SPEA2 multiobjective methods. Each of these runs were assessed by each of the objective functions in Table 1 as well as the “Max Spread”  and “Morrison and De Jong”  diversity measures.
3. Roughly speaking the ﬁtness of a GA individual is equal to the ﬁtness of the best solution generated by the PSO algorithm encoded into that GA chromosome. But, since the PSO algorithm uses pseudo-random numbers, it is very likely that successive runs of the same algorithm will generate completely different solutions. This problem can be handled in a standard manner: the PSO algorithm encoded by the GA individual is run multiple times (50 runs in fact) and the ﬁtness of the GA chromosome is averaged over all runs.
An introduction to abstract mathematical systems by David M. Burton