## 3 Results
### Results for Benchmark Instances  
In what follows, we present numerical results for the benchmark instances proposed in [13]. As suggested by the results of our calibration study, we used the 2 best values for the probability \(\rho=0.99,1.00\) to perform our experiments. For each benchmark instance and for each value of the probability \(\rho\), we ran 100 executions of our nearest-neighbor-based randomized algorithm (NNRA) with a time limit \(\tau=60\) s. Detailed experimental results are provided as Supplementary Materials.  
To compare our approach to other methods, we used IBM ILOG CPLEX 12.8 as the underlying solver of the formulation (1)-(12). A time limit of 3600 s of computation was set for the solver, and all other settings were preserved at their defaults. For each instance, the total number of vehicles found in the best solution of the NNRA, regardless of their types, was considered as the available quantity of each vehicle type in the formulation. In addition, we present the results obtained by the adaptive hybrid local search (HLS) of [PERSON] and [PERSON] [13]. To the best of our knowledge, HLS presents the best results to date. It is important to point out that HLS uses a time limit of 7200 s (used together with a convergence criterion), whereas NNRA uses only 60 s. Note, however, that the authors performed their computational study on a PC Pentium 4 3.00 GHz, and according to the website [[https://cpu.userbenchmark.com](https://cpu.userbenchmark.com)]([https://cpu.userbenchmark.com](https://cpu.userbenchmark.com)), the Intel Core i7-7500 2.7 GHz is 244% faster considering the peak overclocked single core mixed speed.  
In Table 6, along with the characterization of the instances (where \(n\) and \(t\) represent respectively the number of customers and the number of vehicle types), we present the results achieved by the alternative approaches. For CPLEX, we present the best feasible solution found (if any), the lower bound (if the solver was able to at least solve the root relaxation), and the solver computation time (if the solver was able to instantiate the model without an out-of-memory error). For HLS, we display the best and average solution values (over 10 executions) and the average computation time. For NNRA, we present the best and average solution values (over 100 executions), the standard deviation \(\sigma\), and the percentage difference \(\Delta(\%)\) between the results of the two heuristic algorithms, considering average solution values. A positive \(\Delta\) value means that HLS obtained a better result, while a negative \(\Delta\) value means that NNRA was able to find a better solution. We also present, for each instance, the average computation time to reach the best solution for each run (each execution, however, takes 60 s overall since we do not use any convergence criterion). The best known solutions are marked in bold font.  
Figure 2: Solution cost for a calibration instance with 100 customers for different values of \(\rho\).  
In Figure 3, we present the evolution of the computation time for each method as the size of the instances increases. Since the computational study for HLS was performed on a different machine, its computation time was rescaled by an appropriate factor of \(2.44^{-1}\). And in Figure 4, we present a comparative study of the quality of the solutions obtained by each method.