## 3 Application and Results
### Case Study IV: The MW1 Problem  
In this section, we demonstrate the application of the ALMO approach for solving the MW1 problem introduced by [PERSON] and [PERSON] in [28]. The problem has fifteen design variables, the following two objective functions, and one constraint. The mathematical formulation of the MW1 problem is as follows:  
Figure 16: Case study III: (**A**) percentage of infeasible solutions found in the optimal Pareto front; (**B**) overall savings in constraint evaluations in the optimization with different values of \(\phi\) and \(\beta\).  
Figure 17: Case study III: percentage of savings in constraint evaluations in different optimization intervals using ALMO in two different configurations.  
\[\text{Minimize }f_{1}(\mathbf{x}) =x_{1}, \tag{28}\] \[\text{Minimize }f_{2}(\mathbf{x}) =g_{1}(\mathbf{x})-0.85x_{1}, \tag{29}\]  
where the function \(g_{1}(\mathbf{x})\) is defined as  
\[g_{1}(\mathbf{x})=1+\sum_{i=2}^{15}\left(1-\exp\left(-10\left(x_{i}^{13}-0.5- \frac{i-1}{30}\right)^{2}\right)\right), \tag{30}\]  
subject to the following constraint:  
\[c(\mathbf{x})=f_{1}(\mathbf{x})+f_{2}(\mathbf{x})-1-0.5\sin(2 \pi l)^{8}\leq 0, \tag{31}\] \[l=\sqrt{2}f_{2}(\mathbf{x})-\sqrt{2}f_{1}(\mathbf{x}) \tag{32}\]  
where the decision variables \(x_{1}\) to \(x_{15}\) are bounded as  
\[0\leq x_{1}\dots x_{15}\leq 1. \tag{33}\]  
First, we apply the NSGA-II algorithm without any active learning to establish a baseline for comparing the benefits of the ALMO framework and to ensure that the solutions match the results presented in the literature [22; 28], as shown in Figure 18.  
We then apply the ALMO approach to the MW1 problem using two different configurations of the hyperparameters \(\phi\) and \(\beta\). The first configuration uses \(\phi=100\) and \(\beta=0.7\) and the second configuration uses \(\phi=200\) and \(\beta=0.9\). Both configurations achieve a similar IGD value of 0.001. Figure 19 shows all feasible Pareto fronts obtained from these two ALMO configurations. As demonstrated, both approaches yield near-optimal solutions, closely aligned with the true Pareto front.  
Figure 19: Case study IV: Pareto-front solution found using ALMO and two different configurations.  
Figure 18: Case study IV: Pareto-front solution found by NSGA-II without AL using _pymoo_ library.  
Next, we analyzed the effect of different values of \(\phi\) and \(\beta\) on the percentage of infeasible solutions in the final Pareto front. Figure 20A presents the results of this analysis, where \(\phi\) was varied along with \(\beta\). The findings show that at \(\beta=0.6\), approximately 7.5% of the solutions were infeasible. As \(\beta\) increased to 0.7, the infeasibility rate decreased to around 5%. Further increases in \(\beta\) led to a reduction of approximately 2.5% at \(\beta=0.8\), and, finally, the rate dropped to about 1% when \(\beta\) was increased to 0.9.  
Moreover, Figure 20B presents the overall savings in constraint evaluations across different combinations of \(\phi\) and \(\beta\). With \(\beta=0.7\), we achieved around a 65% saving in constraint evaluations, while with \(\beta=0.9\), the savings were approximately 40%. These savings indicate that the ALMO approach effectively reduces the computational burden by minimizing the number of constraint evaluations required to find feasible solutions.  
Finally, we investigate how the percentage savings in constraint evaluations increase over the optimization intervals. Figure 21 shows that as the optimization proceeds, the percentage savings increase. During the initial stages of the optimization, the savings are around 30% for \(\phi=100\) and \(\beta=0.7\). However, as the optimization progresses and the active learner becomes more confident, the savings increase significantly, reaching about 80% in the final stages. A similar trend can be seen for the ALMO configuration with \(\phi=200\) and \(\beta=0.9\), where the savings per interval increase from about 20% to 50%.