## 4 Computational Results
### Parameters Setting  
In the case of the MA-RK v1, the values for the parameters corresponding to the size of the population \(N\), the threshold value \(\beta\), and the maximum number of generations \(D\) were set as 1000, 0.1, and 100, respectively. In the case of MA-RK v2, we set a threshold of 0.4. These values were obtained after performing a preliminary analysis over a subset of instances randomly selected. In addition, several tests with different number of iterations were conducted, finding that, for all the analyzed instances, the MA-RK stops improving when it reaches 80% of the maximum number of iterations, depending on the instance. Lastly, to evaluate consistency, each instance was executed 10 times, and the best front obtained is reported. The next sections show the numerical results computed over the test instances. Experimental Results  
The first set of experiments aims at evaluating the performance of the formulation concerning optimality (optimally solved instances), and the effectiveness of the MA-RK comparing the results with those obtained by the resolution of the model. We first present the results for Instances FNO1 and FNO2 (up to 15 nodes). The following metrics were used to compare the performance of the exact and approximation procedures:  
* Number of points on the front (the larger, the better)
* CPU time (in seconds) (the shorter, the better)
* \(k\)-distance [47] (the smaller, the better)* Hypervolume [48] (the larger, the better)
* The coverage of the fronts [48] computed of one front over another, denoted by c(X\({}^{\prime}\), X\({}^{\prime\prime}\)) (the higher, the better)  
These metrics have shown their successful implementation in biobjective VRPs [19; 39]. The number of non-dominated points measures the ability of each method to find efficient fronts. Table 1 summarizes these results for Instances FNO1 and FNO2. Figures 7 and 8 show the Pareto front for each instance and method. A point to highlight in Figures 7 and 8 and Table 1 is that, for both instances, the Pareto front obtained by Gurobi is densely crowded. Additionally, notice that both versions of MA-RK performed differently over the solved instances. In particular, for Instance FNO1, the second version of the algorithm produced a front that is closer to the optimal front obtained by CPLEX. On the contrary,  
Figure 8: Pareto front for Instance FNO2. Figure 7: Pareto front for Instance FNO1. for Instance FNO2, both algorithms produced fronts near to the optimal front and, in particular, the MA-RK v1 produced a more dense front than the MA-RK v2. Regarding the computational time, the summarized results are presented in Table 2. In this case, the exact method required around 1 h for solving the FNO1 instance (12 nodes), whereas the time required to solve the FNO2 (15 nodes) instance was almost four times as long. In particular, both versions of the metaheuristic required less than 1 s to obtain the solutions. This fact, in conjunction with the metric of the quantity of non-dominated points, supports the evidence that, in general, the MA-RK algorithm performed very well. Regarding the density of the fronts, Table 3 shows the average \(k\)-distance value of all points on the efficient frontiers for each instance, with \(k=3\). Specifically, the MA-RK produced fronts with more density than AUGMECON2. In particular, for the FNO2 instance, it can be seen that the NSGA-v1 obtained the minimum values for the maximum and average distances, while the MA-RK v2 obtained a denser front for Instance FNO1. To verify the efficiency of the MA-RK (in both versions), we used the hypervolume metric. Table 4 displays the obtained results. Again, the MA-RK v2 provided a better value of hypervolume for Instance FNO1, while the MA-RK v1 performed better on Instance FNO2. Finally, we used the coverage measure (considering only the strict domination). Tables 5 and 6 exhibit the results.