# Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems
## 6 Experiments
### Data Set and Criteria  
We employ data sets of two leading bike sharing systems in Capital Bikeshare ([[http://www.capitalbikeshare.com/system-data](http://www.capitalbikeshare.com/system-data)]([http://www.capitalbikeshare.com/system-data](http://www.capitalbikeshare.com/system-data)), accessed on 1 st quarter of 2017) and Hubway ([[http://hubwaydatachallenge.org/trip-history-data/](http://hubwaydatachallenge.org/trip-history-data/)]([http://hubwaydatachallenge.org/trip-history-data/](http://hubwaydatachallenge.org/trip-history-data/)), accessed on 1 st quarter of 2017), and a synthetic data set which is derived from multiple real data sets. We generate the synthetic data set by: (1) taking a subset of the stations from real-world data sets, (2) taking customer demand, station capacity, geographical location of stations, initial distribution, bid values and, value model drawn from real-world data sets.  
The Hubway system consists of 95 base stations, 3 vehicles, and 10 trailers; the Capital Bikeshare system consists of 305 base stations, 10 vehicles, and 35 trailers. A synthetic data set system consists of 60 base stations, 2 vehicles, and 7 trailers. We employ \(k\)-means clustering to generate 12 main stations (5 base stations grouped into 1 main stations), which are typically within 5 km of each other. We take 7 h of planning horizon in the morning peak (5 a.m.-12 p.m.) and take 19 h of planning horizon in the whole day (5 a.m.-12 a.m.). The duration of each decision epoch is considered as 30 min. The demand scenarios were generated from three months of historical trip data. Once the distribution of bikes and vehicles across the stations at time step \(t+1\) is obtained, the information is utilized to compute the repositioning strategy for trailers at time step \(t+1\). This iterative process continues until we reach the last decision epoch, as shown in the second paragraph of Section Problem Formulation (i.e., assumptions 1-3).  
Since our goal is to avoid people from going back to using private vehicles due to unavailability of bikes, the key comparison metric is the total amount of lost demand. To ensure that the amount of lost demand is reduced at no extra fuel cost to the operators, we also consider the total profit as a metric. Let \(U\) denote the profit and \(\mathcal{E}\) denote the lost demand reduction. Firstly, let \(v\) denote the DRRPV approach which stands for dynamically repositioning and routing problem with carrier vehicles, let \(t\) denote the DRRPT approach which stands for dynamically repositioning and routing problem with bike trailers, and \(vt\) denote the DRRPVT approach. Furthermore, let \(G_{v}\) denote the the gain of profit with our approach DRRPVT in comparison to the benchmark approaches DRRPV, which are computed by:  
\[G_{v}=\frac{\mathcal{U}_{vt}-\mathcal{U}_{v}}{\mathcal{U}_{v}}.\]  
Let \(G_{t}\) denote the the gain of profit with our approach DRRPVT in comparison to the benchmark approaches DRRPT, which is computed by:  
\[L_{v}=\frac{\mathcal{E}_{vt}-\mathcal{E}_{v}}{\mathcal{E}_{v}}.\]  
Let \(L_{t}\) denote the reduction in lost demand with our approaches DRRPVT in comparison to the benchmark approaches DRRPT, which is computed by:\[L_{t}=\frac{\mathcal{E}_{vt}-\mathcal{E}_{t}}{\mathcal{E}_{t}}.\]  
In terms of scalability, LDD improves over MILP and the use of main stations on top of LDD further improves performance. The runtime is primarily employed to measure scalability and whether we are able to get a high-quality solution within a reasonable amount of time. The duality gap is significantly employed to measure scalability and whether we are able to get the convergence of LDD to near optimal solutions.