# Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems

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## Abstract

**:**

## 1. Introduction

`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.

- 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.
- 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. While profit objective function has been considered for reduce lost customer demand and increase overall profits separately, previous work has not been considered jointly.
- We design a clustering mechanism for computing main base stations to help improve the efficiency of optimization model with respect to large-scale problem with many stations.

## 2. Related Work

#### 2.1. Static Repositioning Using Carrier Vehicles

#### 2.2. Dynamic Repositioning Using Carrier Vehicles

#### 2.3. Dynamic Repositioning Using Bike Trailers

`DRRPVT`approach aims to leverage the advantage of using both carrier vehicles, which are able to take a large number of bikes and move to longer distance, and bike trailers, which are able to move to short distance with limited cost and allow self-sustaining, by considering the expected profit and the loss demands reduction of repositioning and routing solution [22,23].

## 3. Problem Formulation

- $\mathcal{S}$ 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 ${F}_{s,{s}^{\prime}}^{t}$ 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 ${d}_{s}^{\#,t}$ indicating the number of bikes at station s at decision epoch t.
- ${d}^{*}$ denotes the distribution of bikes in vehicles with ${d}_{v}^{*,t}$ indicating the number of bikes in vehicle v at decision epoch t.
- $\sigma $ denotes the distribution of carrier vehicles at stations, with ${\sigma}_{v,s}^{t}$ 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 ${R}_{s,{s}^{\prime}}^{t}$ 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.
- $\widehat{P}$ denotes the value for executing the task of bike trailer with ${\widehat{P}}_{s,{s}^{\prime}}$ 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 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.
- 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 Shu 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. Once the distribution of bikes across the stations for time step $t+1$ was obtained, we utilized this information to compute the repositioning strategy for trailers and vehicles for time step $t+1$. This iterative process continued until we reached the last decision epoch.
- Customers can rent a bike for 30 min or more, and they have to know in advance at which station they will return the bike. On the other hand, they return their bikes to the nearest available station if the destination station is full, and they leave the system if they encounter an empty station.

`DRRPVT`approach is to maximize the expected profit over the entire time horizon. Let U denote the sum of revenue of hired bikes and the fuel cost of vehicles and the value of bike trailers. We provide an optimization model for a given

`DRRPVT`. Specifically, we provide a mixed integer linear programming (MILP for short) that computes a profit-maximizing repositioning and routing solution. The objective is shown in Equation (1):

**Objective:**To represent the trade-off between lost demand (or alternatively the revenue from customer trips) and the value $\widehat{P}$ of bike trailers and the value P of using carrier vehicles, we employ the dollar value of both quantities and combine them into the overall profit at any decision epoch in Expression (1). The

**notation**used in the formulation are shown:

- ${y}_{s,v}^{+,t}$ denotes the number of bikes picked up from station s by vehicle v at decision epoch t.
- ${y}_{s,v}^{-,t}$ denotes the number of bikes dropped at station s by vehicle v at decision epoch t.
- ${z}_{s,{s}^{\prime},v}^{t}$ denotes whether vehicle v picks up bikes from station s at decision epoch t and drops off at station ${s}^{\prime}$ at decision epoch $t+1$.
- ${a}_{s,v}^{+,t}$ denotes the number of bikes picked up from station s by bike trailer v at decision epoch t.
- ${a}_{s,v}^{-,t}$ denotes the number of bikes dropped off at station s by bike trailer v at decision epoch t.
- ${b}_{s,{s}^{\prime}v}^{t}$ denotes a binary decision variable that is set to be 1 if bike trailer v picks up bikes from station s in at decision epoch t and returns bikes to station ${s}^{\prime}$ in at decision epoch $t+1$ else 0 otherwise.
- ${x}_{s,{s}^{\prime}}^{t}$ denotes the number of hired bikes moving from station s at decision epoch t to station ${s}^{\prime}$ at decision epoch $t+1$.

## 4. Constraints

**C1–C15**) that we exploit in our bike sharing system, where constraints (

**C1–C4**) are newly created in this paper, while constraints (

**C5–C8**) have presented by [3,8] and constraints (

**C9–C15**) have presented by [10].

#### 4.1. C1: Preservation of Bike Flows in and Out of Station

#### 4.2. C2: Preservation of Bikes Flows between Any Two Stations Follow the Transition Dynamics Observed in the Data

#### 4.3. C3: Value of Task for Bike Trailer

#### 4.4. C4: Ensuring the Budget Feasibility

#### 4.5. C5: Preservation of Bikes Flows in and Out of Vehicles

#### 4.6. C6: Preservation of Vehicles Flows in and Out of Stations

#### 4.7. C7: A Maximum of One Vehicle Can Be Present in One Station at Any Time Step

#### 4.8. C8: Vehicles Can Only Pick Bikes Up or Drop Bikes Off at a Station If They Are Present at That Station

#### 4.9. C9: Trailer Capacity Is Not Exceeded While Picking Up Bikes

#### 4.10. C10: Total Number of Bikes Picked Up from a Station Is Less Than the Number of Available Bikes

#### 4.11. C11: Station Capacity Is Not Exceeded While Dropping Off Bikes

#### 4.12. C12: Total Traveling Distance for a Trailer Is Bounded by a Threshold Value

#### 4.13. C13: A Trailer Can Only Pick Bikes Up or Drop Bikes Off at Exactly One Station

#### 4.14. C14: A Trailer Should Return the Exact Number of Bikes Picked Up

#### 4.15. C15: Station and Vehicle Capacities Are Not Exceeded When Repositioning Bikes

**C1**–

**C15**, our task is to calculate which vehicles reposition bikes from state s to ${s}^{\prime}$, i.e., z, and which trailers reposition bikes from s to ${s}^{\prime}$, i.e., b, by optimizing Equation (1).

## 5. Our `DRRPVT` Approach

**Identifying the right constraints to be dualized:**This step is crucial to ensure that the resulting subproblems are easy to solve and the resulting bound derived from the dual solution is tight during the LDD process. If the right constraints are not dualized, then the underlying Lagrangian-based optimization may not be decomposable or it may take significantly more time than the original MILP to find the desired solution.**Extraction of a primal solution from an infeasible dual solution:**The primal extraction process is important to derive a valid bound (heuristic solution) during the LDD process. In many cases, the solution obtained by solving the decomposed dual slaves can be infeasible with respect to the original formulation and, hence, the overall approach can potentially lead to slower convergence and poor solutions.**Decompose the original problem into a master problem and two slaves:**As highlighted in Equation (1), only constraints (8) contain a dependency between routing and repositioning variables. We dualize constraints (8) using the dual variables, ${\alpha}_{s,t,v}$ and obtain the Lagrangian function as Equation (2).

`DRRPVT`is shown in Algorithm 1. We will present the main steps of Algorithm 1 in the following subsections.

Algorithm 1 An overview of our DRRPVT approach |

Input:$\langle \mathcal{S},\mathcal{V},\mathcal{F},{\mathcal{C}}^{\#},{\mathcal{C}}^{*},{d}^{\#},{d}^{*},\sigma ,P,\widehat{P},R,D,B\rangle $Output: $y,z,a,b$1: $\tilde{S}=CalculateMainStations$($\mathcal{S},D$) 2: $\alpha =0$, $iter=0$ 3: while$[p-(\rho 1+\rho 2)]\le \delta $do4: $\rho 1,y,a,b\leftarrow SolveReposition({\alpha}^{iter})$ 5: $\rho 2,z\leftarrow SolveRouting({\alpha}^{iter})$ 6: ${\alpha}^{iter+1}\leftarrow {[{\alpha}^{iter}+\gamma \times ({y}^{+}+{y}^{-}-{C}^{*}\times {\sum}_{\tilde{S}}{z}_{\xb7,\tilde{s},\xb7})]}_{+}$ 7: $p,{y}_{p},{a}_{p},{b}_{p}\leftarrow ExtractPrimal(z)$ 8: $iter\leftarrow iter+1$ 9: end while10: $SolvingIncentivizeTrailerTask$(${d}^{\#},F,a$) |

#### 5.1. Calculating Main Stations

#### 5.2. Repositioning Bikes and Routing for Vehicles

- $y,a,b$ capture the solution to the repositioning problem.
- z captures the solution to the routing problem.

`DRRPVT`model, the routing variables are completely independent with repositioning variables.

`DRRPVT`. It is crucial to ensure that the resulting subproblems are easy to solve and the resulting bound derived from the dual solution is tight during the LDD process. We first decompose the original problem into a master problem (i.e., Equation (2)) and two slaves SolveReposition and SolveRouting. As highlighted, only constraint (8) contains dependencies between routing and repositioning variables, i.e., ${\alpha}_{v,s,t}$. Thus, we dualize constraint (8) using the dual variables and obtain the Lagrangian function in Equation (2). The first three terms in Equation (2) corresponding to the repositioning problem are given in Equation (3) and the last term corresponding to the routing problem is given in Equation (4), respectively, i.e.,

**master**in order to reduce violations of the dualized constraints: ${max}_{\alpha \ge 0}L\left(\alpha \right)$. This

**master**optimization problem is solved iteratively using a subgradient descent method applied on the dual variables $\alpha $, i.e., Step 6 of Algorithm 1, where $\gamma $ is a step-size parameter. The algorithm terminates when the difference between the primal objective (defined as p in Algorithm 1) and the dual objective (the sum of the slave’s objectives ${\rho}_{1},{\rho}_{2}$) is less than a predetermined threshold value $\delta $. In order to compute the best primal solution in conjunction with the dual solution, it is important to obtain a primal solution after each iteration from the solutions of the slaves.

#### 5.3. Incentivize Trailer Tasks

## 6. Experiments

`DRRPVT`approach with novel mechanism (LDD + Main station) outperforms two baselines which deal with dynamically repositioning and routing problem using vehicles [3] or trailers [10]. (2) Our

`DRRPVT`approach remains robust with respect to variation of the number of stations, vehicles and trailers.

#### 6.1. Data Set and Criteria

`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:

`DRRPVT`in comparison to the benchmark approaches

`DRRPT`, which is computed by:

`DRRPVT`in comparison to the benchmark approaches

`DRRPV`, which is computed by:

`DRRPVT`in comparison to the benchmark approaches

`DRRPT`, which is computed by:

#### 6.2. Comparison against Baselines

`DRRPVT`with respect to different time periods, i.e., the peak period and the whole day.

`DRRPVT`) in comparison to the benchmark approaches (

`DRRPV`and

`DRRPT`) on the two real-world data sets. Based on the aggregate results, our approach

`DRRPVT`is always able to outperform both

`DRRPV`and

`DRRPT`with respect to both of the profit gain and lost demand reduction. From Table 1, our approach performs much better in Hubway than Capital Bikeshare comparing to baselines. This is because the number of users hiring bikes in Hubway is much larger than Capital Bikeshare. When more bikes are hired, the most of the lost demand will occur. The percentage gain in profit in the peak hours is much higher because

`DRRPVT`is able to better match the supply of bikes with the demand for bikes. As expected, the more users hire bikes, the better our approach performs. Similar results can be found in Table 2.

#### 6.3. Utility of DRRPVT

`DRRPVT`that LDD and main stations can both improve the MILP, we provide three sets of results. Based on the same budgets and resources, LDD can solve more large-scale stations problems than MILP in the same times.

**Runtime performance:**We compare the runtime performance of the

`DRRPVT`with LDD to DRRPVT with MILP as shown Figure 2a. The x-axis denotes the scale of problem where we vary the number of stations from 5 to 60. The y-axis denotes the total time taken to solve problem in seconds. Except on small instances (e.g., 10 stations),

`DRRPVT`outperforms the MILP with respect to runtime. The MILP is unable to finish within a cutoff time of 3 h for any problem with more than 20 stations, whereas

`DRRPVT`is able to obtain near optimal solutions on problems with 60 stations with less than 3 h.

`DRRPVT`becomes relatively stable after reaching 35 stations (red curve of Figure 2a). It could be easily speeded up by running our approach in a server of higher performance in real-world applications. Meanwhile, we have observed reductions trend in runtime when using main station clustering on problems with 100–200 stations and it scaled in similar trend with respect to using vs. not using main stations.

**Duality gap:**We demonstrate the convergence of LDD to near optimal solutions. LDD achieves an optimal solution if the duality gap, i.e., the gap between primal and dual solutions, becomes zero. Figure 2b show that the duality gap for the instances with 30 stations (grouped into 6 main stations). For these larger problems we are able to obtain a solution with the duality gap of less than 1%.

#### 6.4. Theorem

`DRRPVT`with the ratio of base stations to main stations in Figure 3a. The x-axis denotes the scale of the ratio from 0 to 60. The y-axis denotes the data of total profit from

`DRRPVT`. The total profit is able to obtain maximum on the ratio with 5. Secondly, we compare the profit of the

`DRRPVT`with the ratio of base stations to vehicles in Figure 3b. The x-axis denotes the scale of the ratio from 0 to 60. The y-axis denotes the data of total profit from

`DRRPVT`. The total profit is able to obtain maximum on the ratio with 20. Finally, we compare the profit of the

`DRRPVT`with the ratio of base stations to trailers in Figure 3c. The x-axis denotes the scale of the ratio from 0 to 60. The y-axis denotes the data of total profit from

`DRRPVT`. The total profit is able to obtain maximum on the ratio with 3.

## 7. Conclusions

`DRRPVT`.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Runtime comparison between the global MILP and LDD, (

**b**) Duality gap in the synthetic data set with 30 stations.

**Figure 3.**The maximal profit of the

`DRRPVT`with the ratio of base stations to: (

**a**) main stations, (

**b**) vehicles, and (

**c**) trailers.

Data Sets | ${\U0001d4d6}_{\mathit{v}}$ | ${\U0001d4db}_{\mathit{v}}$ | ${\U0001d4d6}_{\mathit{t}}$ | ${\U0001d4db}_{\mathit{t}}$ |
---|---|---|---|---|

Hubway | 2.42% | 23.57% | 2.18% | 26.91% |

Capital Bikeshare | 1.97% | 14.42% | 1.25% | 17.38% |

Data Sets | ${\U0001d4d6}_{\mathit{v}}$ | ${\U0001d4db}_{\mathit{v}}$ | ${\U0001d4d6}_{\mathit{t}}$ | ${\U0001d4db}_{\mathit{t}}$ |
---|---|---|---|---|

Hubway | 4.63% | 29.71% | 4.26% | 31.12% |

Capital Bikeshare | 4.25% | 19.39% | 4.11% | 24.45% |

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## Share and Cite

**MDPI and ACS Style**

Zheng, X.; Tang, M.; Liu, Y.; Xian, Z.; Zhuo, H.H. Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems. *Appl. Sci.* **2021**, *11*, 7227.
https://doi.org/10.3390/app11167227

**AMA Style**

Zheng X, Tang M, Liu Y, Xian Z, Zhuo HH. Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems. *Applied Sciences*. 2021; 11(16):7227.
https://doi.org/10.3390/app11167227

**Chicago/Turabian Style**

Zheng, Xinghua, Ming Tang, Yuechang Liu, Zhengzheng Xian, and Hankz Hankui Zhuo. 2021. "Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems" *Applied Sciences* 11, no. 16: 7227.
https://doi.org/10.3390/app11167227