# Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems
## 1 Introduction  
Bike sharing systems (BSSs) typically have a set of base stations that are strategically placed throughout a city and each station has a fixed number of docks. At the beginning of the day, each station is stocked with a predetermined number of bikes. Customers can pick and drop bikes from any station and are charged depending on the hiring duration. BSSs are capable of providing healthier living and greener environments while delivering fast movements for customers [1, 2, 3, 4, 5], e.g., Capital Bikeshare in Washington DC, Hubway in Boston, Mobike in Hangzhou, BIXI in Montreal, etc. Due to the individualistic and uncoordinated movements of customers, there is often starvation (empty base stations precluding bike pickup) or congestion (full base stations precluding bike return) of bikes at certain stations, which results in a significant loss of customer demand [6, 7]. To address this problem, a variety of systems [3, 8] employ the idea of repositioning idle bikes with the help of carrier vehicles [1, 9] during the day, by taking into account the movement of bikes by customers [10]. While above solutions of repositioning can help reduce imbalance, repositioning idle bikes using carrier vehicles incurs substantial routing and fuel costs while covering entire stations (The carrier vehicle is a truck to reposition idle bikes during the day using myopic and ad hoc methods so as to return to a predetermined configuration (e.g., each carrier vehicle can hold 30-40 bikes, its working distance is 5 km away)), as done by [11]. In addition, repositioning idle bikes using bike trailers just carries a few of bikes once and the range is limited (The bike trailer is an add-on to a bike that can carry a small number of bikes (e.g., each bike trailer can hold 4-6 bikes, its working distance is within 5 km) and is useful to relocate bikes to nearby stations). Meanwhile, they failed to take into account the available budget [8], working distance (Since nearby stations can be covered by bike trailers, we exploit thegeographical-proximity-based clustering method to obtain main stations to reduce the usage of carrier vehicles) and carrying capacity [12; 13], which may restrict the usage of bike trailers. In this paper, we propose an optimization model called (DRRPVT), which stands for **D**ynamically **R**epositioning and **R**outing **P**roblem with carrier **V**ehicles and bike **T**railers, to jointly consider the usage of carrier vehicles and bike trailers. We aim to better match the overall profit of hired bikes and consequently reduce the expected lost demand. Specifically, we build a profit objective function to calculate the value of carrier vehicle routing (i.e., fuel cost) and bike trailers (i.e., payment for the users of bike trailers), by considering a variety of constraints with respect to carrier vehicle routing and bike repositioning. Jointly considering both carrier vehicles and bike trailers is challenging in the sense so that we need to introduce new constraints to consider both carrier vehicles and bike trailers, and build a novel objective function to minimize the value of repositioning (and routing) and the loss of demand. To improve the efficiency of our approach with respect to large-scale problem with many stations (carrier vehicles and bike trailers), we design a clustering mechanism for computing main base stations to help reduce the computation time. In summary, our novel contributions can be described by three aspects:  
1. We propose an optimization model to improve upon the prior literature by bringing in simultaneous use of both carrier vehicles and bike trailers in the context of dynamic repositioning. While dynamic repositioning has been considered for bike trailers and carrier vehicles separately, previous work has not been considered jointly. 2. We propose a new profit objective function and additional constraints considering both carrier vehicles and bike trailers to reduce lost customer demand and increase overall profits.