## 4 Computational Tests
In Table 1, we visualize the results of the computational tests and we denote by \"\(ID\)\" the index number of the data instance, by \"_Lots_\" the total number of potential lots available, by \"_Places_\" the total number of parking places (we recall that each e-scooter parking lot may contain a different number of places). Then, in the set of columns that follow: (1) \"_Places (act)_\" is the number of places that are offered for rental in the best solution found; (2) \"\(\Delta v\) (_best)_\" and \"\(\Delta v\)% (_best_)\" are the difference expressed in EUR and the corresponding percentage difference between the value of the best solution found by CPLEX within the time limit and the value of the best solution found by HQ-ACO within the limit (we note that a positive percentage indicates a better performance of HQ-ACO with respect to CPLEX); (3) \"\(\Delta v\) (_avg)_\" and \"\(\Delta v\)% (_avg)_\" are the difference expressed in EUR and the corresponding percentage difference between the value of the best solution found by CPLEX within the time limit and the average value of feasible solutions found by HQ-ACO within the limit (also in this case, a positive percentage indicates a better performance of HQ-ACO with respect to CPLEX). Looking at the results of Table 1, it is possible to make the following major observations. * For the first three instances of smaller size 1-3, CPLEX and HQ-ACO find an optimal solution to the problem (in this case, the percentage difference of their value \(\Delta v\) % is then equal to zero). For these instances, since CPLEX finds an optimal solution, we also note that the average value of feasible solutions found by HQ-ACO is naturally smaller than the best solution found by CPLEX. * For all the remaining instances, CPLEX is not able to find an optimal solution within the time limit and, for those of larger size from 13 to 20, it even encounters difficulties in identifying good quality solutions. This should not be a surprise: even though it has a neat mathematical structure, the problem BLP-ESPLR is NP-Hard and its difficulty rapidly increases as the size of the instances grows and the distribution of lots and parking becomes more articulated and introduces more complex correlation through the three distinct set of bound constraints. In contrast, HQ-ACO is able to find solutions of higher quality that, on average guarantee an increase in value of 13% and may exceed 20% in the case of the larger instances, obtaining more than a 25% increase for the instance with ID = 19. Looking more closely at the structure of the best solutions found, it is interesting to note that they tend to activate as many lots as possible including, in particular, those with a higher number of spaces, since they are typically associated with higher lot rental value per single space. Thanks to multiple bound constraints involving subdistricts and lot types, the distribution of the lots results fair, without neglecting outer districts that are often less considered when designing public transportation plans. * For all the instances from ID 4 to ID 20, the average value of the feasible solutions identified by HQ-ACO is always higher than that of the best solution found by CPLEX. Naturally, this situation does not hold in the case of the three instances with ID 1-3 for which CPLEX finds an optimal solution. In the case of larger instances, the average value can be more than 10% higher with respect to CPLEX, granting a substantial amount of additional money from rental. * Concerning computational time, looking at the final set of columns of the table, it can be noted that CPLEX needs a substantial amount of time to identify its best solution, taking more than 3000 s for most of the instances, with an average increase of about 28% with respect to the time that HQ-ACO requires to find its best solution. In Figure 2, we visually compare the value of the best solution found by CPLEX and HQ-ACO and the average value of solutions found by HQ-ACO within the time limit.