# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 3 Methodology
### Rebalancing Strategy for Model Validation  
In order to provide an example of practical application that can be supported by the proposed model, the rebalancing method from [7] was chosen. There, the rebalancing problem was decomposed into two smaller problems (bike station prioritization, station-to-rebalancer assignment) that were solved individually as follows: the bike station prioritization was provided by a fuzzy logic-controlled genetic algorithm; the assignment between the stations and rebalancing agents (trucks) was obtained by applying an inference mechanism. Real data from Citi Bike New York BSSs were used for the validation of the solution. An example of results for the bike station prioritization and assignment to trucks can be seen in Table 2:  
The presented results have the corresponding graphic representation in Figure 9: \begin{table}
\begin{tabular}{l l} \hline \hline
**Truck (Index) Assigned** & **Bike Stations (Index, Prioritized)** \\ \hline ALL & \(4\longrightarrow 5\longrightarrow 2\longrightarrow 8\longrightarrow 3 \longrightarrow 1\longrightarrow 7\longrightarrow 9\longrightarrow 6\) \\ \hline
1 & \(4\longrightarrow 2\longrightarrow 9\longrightarrow 6\) \\
2 & \(5\longrightarrow 3\longrightarrow 1\) \\
3 & \(8\longrightarrow 7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example result for the rebalancing strategy [7]. Figure 8: Probabilistic function fit. Figure 7: Visual examples of allowed behavior outside CTI. In Figure 9, the location labeled \"0\" is the depot of the trucks, and the correspondence (color \(\leftrightarrow\) truck index) is as follows: blue \(\leftrightarrow\) 1, green \(\leftrightarrow\) 2, red \(\leftrightarrow\) 3. The locations labeled \"1\",\"2\",...,\"9\" correspond to bike stations, as mentioned in Table 2.  
Reasoning for the Attached Model  
Although Ref. [7] features a comparative performance analysis by using different numbers of trucks (2,3,...,10) in order to prove effectiveness, the method is still reliant on another source of decision for the final number of trucks allocated for each subsystem. Moreover, when rebalancing is considered, it must be taken into account that even though all stations from the subsystem are highly dynamical, most of the time not all of them will be simultaneously critical, but a percentage instead (<100%), i.e., it is not the number of bike stations (\(m\) from Equation (5)) but the overall criticality (defined in Equation (7)) of a subsystem that matters when choosing the number of rebalancing trucks. Based on the considerations above, the degree of subsystem demand (for rebalancing) is defined in the equation below:  
\[D=\overline{x}\cdot m \tag{14}\]  
The relationship between optimal fleet size and different subsystem demands can also be visualized in Figure 10.  
For exemplification purposes, inside Figure 10, based on the available results from [7], the actual rebalancing cost is drawn, in the form of the continuous black line for which the subsystem demand was maximum (\(D=m=50\)--marked on top of line); dashed lines are used for other different demands (\(D=10,15,20,\ldots,45\)--marked on top of lines). The red point corresponds to the optimal number of trucks, as a trade-off between the initial fleet acquisition cost and the rebalancing cost; the blue line is the scaled truck acquisition cost, which grew linearly, based on the number of trucks involved, considering that each additional truck represented a fixed cost. The scaling of the truck acquisition cost is formulated below:  
\[\text{FK}(\text{Tr})=\alpha\cdot K_{t}\cdot\text{Tr}, \tag{15}\]  
where \(\text{FK}:\mathbb{N}\rightarrow\mathbb{R}^{+}\) is the scaled fleet cost; \(\text{Tr}\in\mathbb{N}\) is the number of trucks allocated for the subsystem; \(\alpha\in\mathbb{R}^{+}\) is a constant scaling factor; \(K_{t}\in\mathbb{R}^{+}\) is the fixed cost of the acquisition of one truck. Figure 9: Rebalancing routes.