1 1.2. Main concepts 7 This plot shows the feasible objective space, representing trade-o!s between market cost and performance for the set of solutions. By setting objectives to minimise market cost and maximise performance, we identify non-dominated solutions, forming a Pareto front. Solutions on a Pareto front are optimal as they cannot improve in one objective without sacrificing the other (Deb, 2005). MOEAs employ natural selection, where the best individuals are more likely to reproduce and pass on to the next generations (Ojha, Singh, Chakraborty, et al., 2019). The following example illustrates how they work. For functions f1(x)=(x→2) 2 and f2(x)=→(x+2) 2, the minimum and maximum values are at x =2 and x =→2, respectively. However, when optimising both objectives simultaneously, we will obtain a set of non-dominated solutions instead. For this, we use an MOEA called the NSGA-II (Deb, 2005), and the results are shown in Figure 1.6(a)-(b). We observe that the first Pareto front ranges from→2 to +2 for both functions. This means that all solutions in this set are optimal for minimising f1(x) while maximising f2(x). Figure 1.6(c) displays convergence of the algorithm over generations, and Figure 1.6(d) shows x values in the first Pareto front across generations, showing convergence of x between →2 and +2. 0 60 120 180 240 300 Generations 25 0 25 50 75 100 Mean func. value f1(x) f2(x) 0 60 120 180 240 300 Generations 7.5 5.0 2.5 0.0 2.5 5.0 x in 1st Pareto front Variable x (a) (c) (d) (b) f1(x) f1(x) = (x 2)2 Variable x f2(x) f2(x)= (x+2)2 4 2 0 2 4 0 10 20 30 40 50 4 2 0 2 4 50 40 30 20 10 0 Pareto front Figure 1.6: Solving f1(x) = (x→2) 2 and f2(x) =→(x+2) 2 using MOEAs (example). In (a) Pareto front for f1(x); (b) Pareto front for f2(x); (c) convergence; and (d) convergence of solutions in the 1st Pareto front.
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