# Area of Operation Planning for Free-Floating Car Sharing Systems
## 4 Results and Discussion
### Solution Stability  
To investigate the dependence of an optimal solution's structure on the specific set of trips included from a given trip data set (like \"OEU\" or \"trip\"), we also performed computational experiments on the ten bootstrapped instances that we created for each _original_ instance (see Section 3.3).  
As in Section 4.1, we focus our analysis on the relative agreement (i.e., overlap) between solutions. However, in contrast to our previous experiments, we now compare those ten bootstrapped instances derived from the same trip data set, using the same revenue metric and the same budget factor \(\beta\).  
The first four lines in Figure 9 show the overlap between the optimal solutions for all bootstrapped instances pertaining to a certain configuration (trip data set and revenue metric). These results indicate strong agreement between their corresponding solutions on the composition of the optimal area of operation within a given configuration. Both configurations based on the taxi trip data set, as well as the \"OEU\" configuration with revenue coefficients \(N_{ij}\), consistently show solution overlaps in excess of 90%. While the \"OEU\" configuration with duration-based trip revenues does lag notably behind the other three in terms of inter-solution agreement for instances with budget factors \(\beta\leq 1.75\), it does (mostly) catch up to the others for higher values of \(\beta\). These consistently strong to very strong agreements between solutions for a configuration are especially noteworthy when considering the fact that we are now looking at the intersection of ten individual areas of operation, where even small differences between the individual pairs of solutions can lead to large disagreement when all are considered at once.  
The situation does, however, change when looking at the agreement between solutions across instance configurations for the same budget factor \(\beta\): The overlap between the  
Figure 8: Solutions to instances of both data sets (highlighted in teal), each using revenue matrix \(\mathbf{N}\) (number of trips), for three exemplary budget factor values \(\beta\in\{1,1.5,2\}\). The semi-transparent area shaded in red denotes ELOOP’s current AoO.  
solutions hovers around 60%, unsurprisingly even worse than the agreement between only the original instances with a given \(\beta\).  
When considering the ten solutions of each configuration and budget value \(\beta\) simultaneously, familiar patterns emerge. Figure 10 shows that solutions for instances based on the taxi data set tend to remain more compact and closer to the city center, whereas ones based on \"OEU\" trips branch out more towards the outer sub-districts. We also observe the aforementioned strong agreement between solutions within a configuration. A large majority of sub-districts is colored solidly blue, indicating that they were selected by all candidate solutions. Moreover, the ones with more reddish hues, which indicates that they were selected less frequently, tend to be selected more often as the available budget grows, again hinting at a monotonic growth pattern.  
Figure 10: _Cont._  
Figure 9: Solution overlap for different configurations of the _bootstrapped_ instances.