## 4 Computational Tests  
We tested the performance of the optimization model BLP-ESPLR and of the hybrid metaheuristic HQ-ACO using realistic data related to the city of Rome, Italy. The considered data instances have been defined not only taking into account the regulations of the city (in particular, the more recent updates regulating e-scooter sharing services imposing, e.g., to reduce the number of allowed companies to three, extending the overall service area to outer districts, see [27; 50; 51]) but also on the basis of discussions with sharing mobility professionals of companies that have operated in Rome. For defining the set of districts and subdistricts, we have considered the administrative division of the territory of Rome and, in each subdistrict, we have identified a set of parking potentially available for renting. The lots have been defined taking into account the relevance of the subdistricts, their position within Rome and their nature (business, residential, peripheral, presence of points of interest, etc.). The types of lots have been set to 6, taking into account their size and location with respect to landmarks and every lot may contain from 3 to 15 e-scooters. Larger lots are especially located in central areas and close to major public transportation hubs and are meant to host larger sets of e-scooters from all three active companies. The bounds on the total number of places, the bounds on the number of slot types and the upper bound on the number of car places that can be subtracted by reserved e-scooter lots have been defined by taking into account the features of the subdistricts (type of residents, public transportation presence, features of the urban fabric). The algorithm HQ-ACO has been coded in C/C++ and executed on a 2.70 GHz Windows machine equipped with 8 GB of RAM and using IBM ILOG CPLEX 12.5 as optimization software. Both CPLEX and HQ-ACO ran with a time limit of 3600 s. In the case of IQ-ACO, 3000 s are reserved for the solution construction phase and the remaining 600 s are reserved for the RINS-based exact neighborhood search. The ACO construction phase employs 10 ants with an evaporation factor \(\epsilon=0.1\), and the RINS search uses \(\rho=0.1\). For test purposes, we have considered 20 instances of increasing size from about 600 to about 4300 potential slots associated with up to about 33,000 potential single parking places. All ten instances consider a distribution of the parking lots that extends to all the districts of Rome, reflecting the enlargement of the overall service area imposed by the new e-scooter sharing plan agreed with the municipality of Rome. The distribution aims, in particular, at covering new outer zones that are located well beyond the railway ring surrounding more central districts (in Italian, the so-called \"Anello Ferroviario\") and that were excluded or poorly served in the previous regulation setting of the city of Rome. Lots located in peripheral areas are characterized by a (much) lower value than those in central historical districts. In the case of the first ten instances (ID from 1 to 10), given the lower total number of lots, the density of place distributions across the subdistricts is lower. In the case of instances with ID from 11 to 20, the number of potential lots sensibly increases, in particular, with the aim of potentially offering a higher density in central districts that are more afflicted by e-scooter wild parking. These lots in central areas tend to naturally have a substantially higher value than those located in suburbs. The upper bounds \(\text{UB}_{ds}^{\text{TOT}}\) and\(\text{UB}_{ds}^{\text{S}}\) have been set equal to 10% of the total number of potential places in each subdistrict and to 5% of the total number of car places in each subdistrict.