## 5 Computational Results
### Algorithm Results  
For testing our algorithm, we set a 100 s time limit on every iteration and a global time limit of 1800 s and compared the results to the best found solution by CPLEX 12.6 in a time limit of 1800 s. Concerning the algorithm parameters, we analyzed its performance for three relaxation strategies, i.e., \(\alpha_{1},\alpha_{2}\) and \(\alpha_{3}\) (see Section 4.2), respectively, and tested the algorithm with regards to its  
to the CPLEX results. In addition, we can see that the average objective function values are much higher than for the CPLEX results for all data classes. The effect of getting stuck in local optima is compensated for higher values of \(l\) resulting in much better objective function values but with also a small increase in solution times. Furthermore, for the class A instances, CPLEX is not able to find an optimal solution for any instance in the given time limit of 1800 s. With \(l=1\), the algorithm was able to find better solutions for six instances with a reduction in the average solution time of 92.2%. While the average objective values for \(l=1\) for class A are higher than those from CPLEX, this changes for \(l=10\). For \(l=10\), the algorithm is able to find 37 better solutions and a reduction in the average objective values of 7.2%. Also, the number of iterations rise resulting in larger computation times; these times are still much lower (78.8%) than for CPLEX. The best results for all the data classes are for the list length of \(l=50\). Analyzing the performance of the algorithm for each class of instances, we can observe that for the class A instances, the algorithm is able to find 48 better solutions in half of the computation time of CPLEX. Regarding the class B instances, 49 optimal solutions could be found by CPLEX in an average time of 212.23 s. For \(l=50\), the algorithm is able to find 48 of these solutions resulting in an average objective value that is only 0.1% higher than the CPLEX results while the computation time reduces by 20.8% compared to CPLEX. For the class C instances, it can be observed that 20 instances could be solved to optimality by CPLEX and the overall average CPLEX time is very high with 1354.50 s. While the algorithm is not able to find a better solution for any of the 50 instances for \(l=1\) this number increases to 6 instances for \(l=10\) and 12 instances for \(l=20\). For \(l=50\) our algorithm is able to find 40 better or same-quality solutions for the instances that could be solved by CPLEX. We can observe an average increase of 0.2% in the objective values due to the 10 instances where the algorithm was not able to find a better solution. The small increases in the objective function values are superseded by the significant reduction of the computational time, which is a decrease of 86.6% in this case in comparison to solving the optimization problem at once with a general solver. Concluding this section, we can see several effects for the single parameter of LAHCM \(l\). Reducing our approach to a hill-climbing matheuristic without late-acceptance criterion, i.e., setting \(l=1\) leads to solutions that get stuck in a local optimum soon for all data classes. Setting \(l=10\) is most effective for the solution quality of class A in terms of better solutions found. That is, for 37 instances better solutions can be found in comparison to the CPLEX results. However, this list size is not sufficient to overcome local optima for a large number of the instances for classes B and C. While for \(l=20\) the solution quality of the method increases, the best results can be seen for \(l=50\). As we can observe, a strict solution quality improvement for longer list sizes we can also observe that choosing a good value for \(l\) is critical regarding the computational time of the method. In Table 13, we test LAHCM with \(l=100\) for the instances where we could not find better or optimal results for \(l=50\) compared to the CPLEX results.