## 3 Application and Results
### Case Study V: The MW9 Problem  
In this section, we apply the ALMO approach to solving the MW9 problem, which is part of the test suite introduced by [PERSON] and [PERSON] [28]. The MW9 problem is a constrained multi-objective optimization problem that presents a challenging landscape  
Figure 21: Case study IV: percentage of savings in constraint evaluations in different optimization intervals using ALMO in two different configurations. Figure 20: Case study IV: (**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 \(\beta\). due to its non-linear constraints and multiple objectives. The mathematical formulation of the MW9 problem can be described as follows:  
\[\text{Minimize }f_{1}(\mathbf{x}) =x_{1}g_{1}(\mathbf{x}), \tag{34}\] \[\text{Minimize }f_{2}(\mathbf{x}) =g_{1}(\mathbf{x})\Bigg{(}1-\left(\frac{f_{1}(\mathbf{x})}{g_{1}( \mathbf{x})}\right)^{0.6}\Bigg{)} \tag{35}\]  
where \(g_{1}(\mathbf{x})\) is defined in Section 3.4. The problem is subject to the following non-linear constraint:  
\[c_{1}(\mathbf{x}) =\min\{T_{1},T_{2}\cdot T_{3}\}\leq 0, \tag{36}\] \[T_{1}(\mathbf{x}) =\Big{(}1-0.64f_{1}^{2}(\mathbf{x})-f_{2}(\mathbf{x})\Big{)} \Big{(}1-0.36f_{1}^{2}(\mathbf{x})-f_{2}(\mathbf{x})\Big{)}\] (37) \[T_{2}(\mathbf{x}) =1.35^{2}-\left(f_{1}(\mathbf{x})+0.35\right))^{2}-f_{2}(\mathbf{ x})\] (38) \[T_{3}(\mathbf{x}) =1.15^{2}-\left(f_{1}(\mathbf{x})+0.15\right))^{2}-f_{2}(\mathbf{ x}). \tag{39}\]  
where the decision variables \(x_{1}\) to \(x_{15}\) are bounded as  
\[0\leq x_{1}\dots x_{15}\leq 1. \tag{40}\]  
The MW9 problem is known for its complicated Pareto front and complex constraint region, which together make it an ideal candidate to evaluate the efficiency and accuracy of the ALMO approach. Initially, the NSGA-II algorithm is used without any active learning to establish a baseline for comparing the benefits of the ALMO framework and to ensure the solutions matched those reported in the literature [22; 28], as shown in Figure 22.  
After validating the baseline solution with NSGA-II, the ALMO approach is applied to solve MW9 using different parameter settings for the active learner, as shown in Figure 20:0 The first experiment involves setting \(\phi=100\) and \(\beta=0.7\), resulting in an IGD of 0.013. This is followed by an experiment with \(\phi=200\) and \(\beta=0.9\), where the IGD improves to 0.003, demonstrating that increasing both \(\phi\) and \(\beta\) can lead to a better approximation of all feasible Pareto fronts. Figure 24 shows a Pareto-front solution where \(\beta\) is increased from 0.7 to 0.8 while having \(\phi=100\), and it improves the IGD to 0.004, demonstrating that this optimization problem needs a higher \(\beta\) value because of its complex constraint. To further investigate the behavior of the ALMO approach, different combinations of \(\phi\) and \(\beta\) are tested, as shown in Figure 25A. The results reveal that for \(\beta=0.8\), the percentage of infeasible solutions in the final Pareto front remains below 10%, whereas for \(\beta=0.9\), the infeasible solutions drop to around 5%. Compared to the MW1 problem discussed in Section 3.4, the percentage of infeasible solutions in MW9 is notably higher. This is likely due to the increased complexity of the MW9 problem, which makes it more difficult for the active learner to accurately predict feasible solutions. Figure 20:1 Case study V: Pareto-front solution found by NSGA-II without AL using _pymoo_ library. In terms of constraint evaluation savings, the results show that with \(\beta=0.8\), the ALMO approach achieves approximately 15 to 20% savings, while for \(\beta=0.9\), the savings are around 10%, as shown in Figure 25B. These savings are lower than those observed for the MW1 problem, again highlighting the increased difficulty of the MW9 problem for the active learner.