# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 3 Methodology
### Practical Evaluation for the Proposed Model
During the evaluation, the average (Equation (25)), the worst case (Equation (26)), and the best case (Equation (27)) of all the unworking times are used as performance indicators:  
\[\overline{\omega}=\frac{\sum_{B_{i}\in B}\sum_{\Gamma\in T_{i}\cap\Lambda_{ REF}}\xi_{t}(B_{i})}{m\cdot|T_{i}|}, \tag{25}\]  
\[\omega_{max}=\max_{B_{i}\in B}\frac{\sum_{\Gamma\in T_{i}\cap\Lambda_{REF}} \xi_{t}(B_{i})}{|T_{i}|}, \tag{26}\]  
\[\omega_{min}=\min_{B_{i}\in B}\frac{\sum_{\Gamma\in T_{i}\cap\Lambda_{REF}} \xi_{t}(B_{i})}{|T_{i}|}, \tag{27}\]  
where \(\xi_{t}(B_{i})\) is equivalent to the function \(\xi(B_{i})\) defined in Equation (2), at moment \(t\), which returns the value 1 when the bike station \(B_{i}\) has critical load, and 0 otherwise; \(\overline{\omega}\), \(\omega_{max}\), \(\omega_{min}\) are the average, worst case, and best case unworking times, respectively, with lower values corresponding to better performance. The average is computed using the data of all the bike stations (\(B_{i}\in B\)) inside the analyzed subsystems, including all the days that figure in the model identification (\(|T_{i}|\) denoting the number of days covered by the time interval \(T_{I}\)) for which traffic data are collected during the reference time interval (\(\Delta t_{REF}\) from Equation (13)), obtaining a daily unworking time that describes the usual bike station. The worst and best cases are calculated in a similar manner, focusing on the bike station with the highest and lowest unworking time, respectively. Apart from the operational goals, the practical evaluation is also performed with respect to the end-user (bike rider) expectation, by determining the lost customer rate [58] of the rebalanced subsystem. The lost customer rate is defined as the ratio of customers who have arrived at a bike station and are not able to fulfill their expectation of either renting or returning a bike, because of critical load. It is therefore defined in Equation (28) as follows:  
\[\phi=\frac{k_{lost}}{k_{lost}+k_{uill}}, \tag{28}\]  
where \(\phi\) is the lost customer rate, \(k_{lost}\) represents the number of bikes that cannot be rented/returned, due to critical load at the bike station, and \(k_{uill}\) represents the number of bikes that are rented/returned successfully. During the evaluation, the average (\(\overline{\phi}\)), the worst case (\(\phi_{max}\)), and the best case (\(\phi_{min}\)), computed in a similar manner as Equations (25), (26), and (27), respectively, are used as performance indicators, with the same interpretation that lower values mean better performance. In the context of unworking time and lost customer rate, in order to assess the model effectiveness, a distinction is made between assisted (by the model) rebalancing and unassisted. Specifically, \"assisted\" means that before applying the rebalancing strategy [7] the model is used to determine the cost-effective number of trucks, while \"unassisted\" means that an arbitrary number of trucks are used, as defined in Equation (29):  
\[\widetilde{\text{Tr}}=\widehat{\text{Tr}}+e_{\text{Tr}}, \tag{29}\]  
where \(\widetilde{\text{Tr}}\) is the arbitrary number of trucks; \(\widehat{\text{Tr}}\) is the cost-effective number of trucks (defined in Equation (16)); and \(e_{\text{Tr}}\) is the deviation from the cost-effective, and has a non-zero integer value. When \(e_{\text{Tr}}>0\), the overall unworking time and lost customer rate are, of course, expected to increase or at least remain the same, but this would imply an excessive acquisition of trucks, which, in turn, would lead to low profitability of the system. Alternatively, to maintain profitability, the extra cost could be passed to the end user by increasing the ride prices, which, in turn, would make BSSs less attractive. Instead, the validation analysis is more pertinent when \(e_{\text{Tr}}<0\).