**6. Results**

The results of each instance (LU1, LU2 and MU1) are represented as a scatter plot where the x-axis represents the maximum walking distance (lower is better) and the yaxis represents the number of covered users (higher is better). Execution in this work is performed on a single core of an Intel Xeon L5640 (2.26 GHz) with courtesy of University of Luxembourg HPC.

#### *6.1. Result of LU1 Instance*

Figure 7 presents the obtained Pareto fronts from eight different algorithms and Table 6 provides the numerical results obtained for all evaluated algorithms. To simplify the table, only two extreme points on both Pareto fronts are shown. The highest achieved coverage is 391 users, which is yielded by the iterative heuristic algorithm, the exact method, and NSGA-II. On the other hand, the lowest distance of 93.5 m is achieved using PolySCIP.

**Table 6.** Numerical results in LU1 instance. Only extreme solutions from the two Pareto fronts are mentioned.


**Figure 7.** Results in the validation phase. The higher the value on the x axis (moving toward the right), the better the distance objective is. The higher the value on y axis, the better the user coverage objective is.

Overall, results from NSGA-II are debatably superior when both objectives are considered. The results are close to the optimum (106.4 from NSGA-II and distance-focused iterative heuristic and 93.546 m from PolySCIP). In fact, the distance result from NSGA-II is even better than its iterative counterpart as it covers more users. Figure 8 shows the stations and their respective coverage in LU1 instance.

**Figure 8.** A map showing a solution from NSGA-II that yields the highest user coverage and the highest maximum walking distance. Covered buildings are depicted as nodes inside polygons.

For the IGD indicator (see Table 7), NSGA-II yields 3.02. This value is in accordance to Figure 7 between a true Pareto and an approximated fronts. It can be seen that NSGA-II achieved some of the solutions on the true Pareto front—especially the highest user coverage solution.

**Table 7.** Comparing Pareto fronts for PolySCIP and NSGA-II.


As for the spread indicator, the true Pareto front yields 0.488, while the NSGA-II Pareto front yields 0.525. This means the diversity in the exact method Pareto front is better than NSGA-II's. This was due to the fact that the coverage objective overwhelmed the distance objective leading to a cluster of solutions in the upper right region in Figure 7. The hypervolume of the true Pareto front is 0.449 and the NSGA-II Pareto front yields 0.351. The difference occurs because some solutions of NSGA-II are dominated by PolySCIP's.

It is essential to note that PolySCIP was applicable for the LU1 instance due to its small size. However due to the FPP complexity, for larger instances like LU2 and MU1, PolySCIP became an enviable approach. This is elaborated in Sections 6.2 and 6.3.

#### *6.2. Result of LU2 Instance*

There is only one Pareto front from NSGA-II in Figure 9 since PolySCIP cannot deliver the solutions even after 18 days and has a memory usage of 84 GB. The plot shows that NSGA-II yields a higher coverage than the iterative heuristic coverage algorithm when it takes the iterative methods' solutions as seed solutions in the initial populations. The highest achieved user coverage is 8421 users with a maximum walking distance of 399.8 m. On the other hand, the lowest maximum walking distance achieved is 135.7 m with only 47 covered users. In Table 8, the best results from each category (i.e., simple heuristic, iterative heuristic, and NSGA-II), are compared. It reveals that NSGA-II achieves the highest user coverage and lowest walking distance among all algorithms. It can be observed in Figure 9 that even though some residential buildings are located close to carsharing stations, they are not covered. This is because the entrances of those buildings (determined during the snapping process) are mapped on the opposite streets, which are not covered by the stations. However, the number of such buildings is marginal and can be neglected.

**Figure 9.** NSGA-II's Pareto front and heuristic algorithms' solutions for LU2. The higher the value on the x axis (moving towards the right), the better the distance objective is. The higher the value on the y axis, the better the user coverage objective is.



#### *6.3. Result of MU1 Instance*

Due to the larger input size of the MU1 instance and FPP being NP-hard, PolySCIP cannot be employed. It takes 17 h to come up with one solution for iterative heuristic algorithms, while it takes 26 min for NSGA-II to come up with an estimated front (read Table 9). Their respective results are presented in Figure 10. The obtained results are consistent with LU1 and LU2 instances. Table 9 presents the execution time of all algorithms. Although simple heuristic algorithms take only 7 min to find a solution, the results are not comparable to the others, which are more complex. From the results, NSGA-II also achieves higher user coverage and shorter walking distance than the heuristics.

**Table 9.** Execution time for NSGA-II and heuristic algorithms on MU1. The measured time depicts the execution time each algorithm takes to locate 100 stations.


**Figure 10.** NSGA-II Pareto front and solutions of heuristic algorithms for the MU1 instance.

Figure 11 illustrates the NSGA-II fleet placement solution, which maximizes the number of covered users. Red pins mark locations of the carsharing stations, while green polygons show designated parking areas in inner Munich. All heuristic algorithms and NSGA-II are also compared using 72 stations in MU1 instance to compare with the manual allocation. Figure 12 shows that the manual allocation is of a lesser quality than some of the (meta-)heuristic algorithms, the iterative coverage and bi-objective version in particular, and NSGA-II. The comparison between the best results from each category of algorithm and manual allocation is also shown in Table 10. The difference in results in terms of user coverage is up to 50% (manual allocation being on the lower end), while the walking distance is similar. The results also further stress the benefit of Pareto front in decision making since it offers more options to choose from compared to the heuristic algorithms.

**Figure 11.** NSGA-II fleet placement solution which maximises user coverage in the city of Munich. Red pins are locations of the carsharing station. Green zones indicate the inner area of Munich.

 **Figure 12.** NSGA-II approximated Pareto front and solutions of heuristic algorithms in MU1 instance compared to the manual allocation (72 stations).

From the MU1 instance results, iterative heuristic approaches may still be possible but usually at the expense of extremely high computation time. Moreover, discovering suitable heuristics is problematic in its own. This is where the use of metaheuristic algorithms (e.g., NSGA-II) is proven to be more effective in term of solution qualities and computation time.


**Table 10.** Comparing the best results from each algorithm categories and manual allocation in MU1 instance.
