# Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems
## 3 Problem Formulation  
Our bike sharing problem is formally defined by the following tuple:  
\[\langle\mathcal{S},\mathcal{V},\mathcal{F},\mathcal{C}^{\#},\mathcal{C}^{*},d ^{\#},d^{*},\sigma,R,P,\hat{P},D,B\rangle,\]  
where  
* \(\mathcal{S}\) denotes the set of base stations. * \(\mathcal{V}\) denotes the set of vehicles used for repositioning, which is restricted to carrier vehicles only. * \(\mathcal{F}\) denotes samples of customer requests for the future time steps with \(F^{t}_{s,s^{\prime}}\) indicating the number of customer requests between stations \(s\) and \(s^{\prime}\), which start at decision epoch \(t\) and end at decision epoch \(t+1\). * \(\mathcal{C}^{\#}\) denotes the capacity of stations with \(C^{\#}_{s}\) indicating capacity of station \(s\). * \(\mathcal{C}^{*}\) denotes the capacity of carrier vehicles or bike trailers with \(C^{*}_{v}\) indicating capacity of vehicle \(v\) or bike trailer \(v\). * \(d^{\#}\) denotes the distribution of bikes at stations with \(d^{\#,t}_{s}\) indicating the number of bikes at station \(s\) at decision epoch \(t\). * \(d^{*}\) denotes the distribution of bikes in vehicles with \(d^{*,t}_{v}\) indicating the number of bikes in vehicle \(v\) at decision epoch \(t\). * \(\sigma\) denotes the distribution of carrier vehicles at stations, with \(\sigma^{t}_{v,s}\) set to be \(1\) if vehicle \(v\) is present at station \(s\) at decision epoch \(t\) and \(0\) otherwise. * \(R\) denotes the revenue of bikes being hired, with \(R^{t}_{s,s^{\prime}}\) indicating the revenue from station \(s\) to \(s^{\prime}\) which starts at decision epoch \(t\) and ends at decision epoch \(t+1\). * \(D\) denotes the actual distance with \(D_{s,s^{\prime}}\) indicating the distance between stations \(s\) and \(s^{\prime}\). * \(B\) denotes the total budget for all trailers to bid. In other words, the total amount of value spent on trailers should not be larger than \(B\). * \(\hat{P}\) denotes the value for executing the task of bike trailer with \(\hat{P}_{s,s^{\prime}}\) indicating the value for executing the task of bike trailer picking up idle bikes at station \(s\) and dropping off them at station \(s^{\prime}\). * \(P\) denotes the routing value (e.g., fuel cost) for vehicles traveling with \(P_{s,s^{\prime}}\) indicating the routing value for vehicles traveling from station \(s\) to \(s^{\prime}\), which depends on the distance between the two stations. We made the following assumptions for the ease of explanation and representation:  
1. We assumed that users who carry bikes and trailers at decision epoch \(t\) always return their bikes at the beginning of the decision epoch \(t+1\). The duration of each decision epoch was 30 min. We evaluated different duration impacts on runtime performance. We chose 30 min as the default setting for the duration of time step. 2. We sampled the empirical distribution of the real historical data of customer requests to simulate customer requests for the future time steps. We produced three types of demand scenarios: (1) We took the real demand data for 60 weekdays. We used 20 days of demand scenarios for training purpose and other 40 days of demand for testing. (2) We generated 100 demand scenarios, where the arrival demand at each station is generated using Poisson distribution with the mean computed from historical data. Similar to [PERSON] et al. [6], we assume that customers reach their destination station with a fixed probability. (3) We generated 100 demand scenarios, where demand for each origin destination [OD] pair at each time step is computed using Poisson distribution. For the demand scenario types 2 and 3, we used 30 demand scenarios for training and 70 demand scenarios for testing [9]. We assumed that the numbers of extra bikes is the lost demand at the time of return. To ensure that the capacity constraints were considered for the stations, we transferred extra bikes to the nearest available station if the number of bikes exceeds the station capacity.