# Spatial Cluster-Based Model for Static Rebalancing Bike Sharing Problem

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Static Rebalancing Problem

#### 2.2. Dynamic Rebalancing Problem

## 3. Methodology

- Considering the users’ behavior in the network and determining the user-based rebalancing portion;
- Clustering the stations needed to be rebalanced;
- Minimizing the rebalancing tour length (cost);
- Implementing in a real scale case study.

#### 3.1. Notation

- Indices
- i: id of stations
- j: id of vehicles

- Problem parameters
- I: set of id of all stations in the network
- $IB{I}_{i}$: Number of entering bikes into the station i
- $OB{I}_{i}$: Number of exiting bikes from the station i
- ${B}_{i}$: Balance of the station i at the end of the day; Bi = IBIi − OBIi
- $Ca{p}_{j}$: Capacity of the vehicle

- Decision variables
- T: Tour for the vehicle; T = ${S}_{1}$, $\Delta {B}_{1}$, ${S}_{2}$, $\Delta {B}_{2}$, …, ${S}_{m}$, $\Delta {B}_{m}$
- $\Delta {B}_{m}$: Number of bikes delivered or picked up to/from m-th station
- ${S}_{m}$: id of the m-th station

- Objective function$$Min\sum _{i=1}^{m-1}D({S}_{i},{S}_{i+1})$$
- Constraints$$\Delta {B}_{m}\le Cap$$$$Ca{p}_{j}\ge 0$$$$abs(\Delta {B}_{m})\le abs\left({B}_{m}\right)$$$$\Delta {B}_{m}\times {B}_{m}<0$$$$\Delta {B}_{m}+{B}_{m}=0:i\in I$$

#### 3.2. Solution Approach

Algorithm 1: Genetic algorithm framework for the static rebalancing problem |

Algorithm 2: Initialize-Population function |

Algorithm 3: Cross-over function |

Algorithm 4: Mutation function |

#### 3.3. Data

## 4. Results

#### 4.1. Primary Network Analysis

#### 4.2. Temporal and Spatial Analysis

#### 4.3. Clustering

#### 4.4. Rebalancing

#### 4.4.1. Intra-Cluster

#### 4.4.2. Inter-Cluster

#### 4.5. Validation

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Dell’Amico, M.; Hadjicostantinou, E.; Iori, M.; Novellani, S. The bike sharing rebalancing problem: Mathematical formulations and benchmark instances. Omega
**2014**, 45, 7–19. [Google Scholar] [CrossRef] - Faroqi, H.; saadi Mesgari, M. Performance comparison between the multi-colony and multi-pheromone ACO algorithms for solving the multi-objective routing problem in a public transportation network. J. Navig.
**2016**, 69, 197–210. [Google Scholar] [CrossRef] - Faroqi, H.; Sadeghi-Niaraki, A. GIS-based ride-sharing and DRT in Tehran city. Public Transp.
**2016**, 8, 243–260. [Google Scholar] [CrossRef] - Li, Y.; Szeto, W.Y.; Long, J.; Shui, C.S. A multiple type bike repositioning problem. Transp. Res. Part B Methodol.
**2016**, 90, 263–278. [Google Scholar] [CrossRef] [Green Version] - Pal, A.; Zhang, Y. Free-floating bike sharing: Solving real-life large-scale static rebalancing problems. Transp. Res. Part C Emerg. Technol.
**2017**, 80, 92–116. [Google Scholar] [CrossRef] - Zhang, D.; Yu, C.; Desai, J.; Lau, H.Y.K.; Srivathsan, S. A time-space network flow approach to dynamic repositioning in bicycle sharing systems. Transp. Res. Part B Methodol.
**2017**, 103, 188–207. [Google Scholar] [CrossRef] - Wang, X.; Lindsey, G.; Schoner, J.E.; Harrison, A. Modeling bike share station activity: Effects of nearby businesses and jobs on trips to and from stations. J. Urban Plan. Dev.
**2015**, 142, 4015001. [Google Scholar] [CrossRef] - Caggiani, L.; Camporeale, R.; Ottomanelli, M.; Szeto, W.Y. A modeling framework for the dynamic management of free-floating bike-sharing systems. Transp. Res. Part C Emerg. Technol.
**2018**, 87, 159–182. [Google Scholar] [CrossRef] - Shui, C.S.; Szeto, W.Y. Dynamic green bike repositioning problem—A hybrid rolling horizon artificial bee colony algorithm approach. Transp. Res. Part D Transp. Environ.
**2018**, 60, 119–136. [Google Scholar] [CrossRef] - Forma, I.A.; Raviv, T.; Tzur, M. A 3-step math heuristic for the static repositioning problem in bike-sharing systems. Transp. Res. Part B Methodol.
**2015**, 71, 230–247. [Google Scholar] [CrossRef] - Cruz, F.; Subramanian, A.; Bruck, B.P.; Iori, M. A heuristic algorithm for a single vehicle static bike sharing rebalancing problem. Comput. Oper. Res.
**2017**, 79, 19–33. [Google Scholar] [CrossRef] - Alvarez-Valdes, R.; Belenguer, J.M.; Benavent, E.; Bermudez, J.D.; Muñoz, F.; Vercher, E.; Verdejo, F. Optimizing the level of service quality of a bike-sharing system. Omega
**2016**, 62, 163–175. [Google Scholar] [CrossRef] - Dondo, R.; Cerdá, J. A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows. Eur. J. Oper. Res.
**2007**, 176, 1478–1507. [Google Scholar] [CrossRef] - Chemla, D.; Meunier, F.; Calvo, R.W. Bike sharing systems: Solving the static rebalancing problem. Discr. Optim.
**2013**, 10, 120–146. [Google Scholar] [CrossRef] - Liu, J.; Sun, L.; Chen, W.; Xiong, H. Rebalancing bike sharing systems: A multi-source data smart optimization. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016; pp. 1005–1014. [Google Scholar]
- Schuijbroek, J.; Hampshire, R.; van Hoeve, W.J. Inventory rebalancing and vehicle routing in bike sharing systems. Eur. J. Oper. Res.
**2017**, 257, 992–1004. [Google Scholar] [CrossRef] [Green Version] - Erdoğan, G.; Battarra, M.; Calvo, R.W. An exact algorithm for the static rebalancing problem arising in bicycle sharing systems. Eur. J. Oper. Res.
**2015**, 245, 667–679. [Google Scholar] [CrossRef] [Green Version] - Erdoğan, G.; Laporte, G.; Calvo, R.W. The static bicycle relocation problem with demand intervals. Eur. J. Oper. Res.
**2014**, 238, 451–457. [Google Scholar] [CrossRef] [Green Version] - Regue, R.; Recker, W. Proactive vehicle routing with inferred demand to solve the bikesharing rebalancing problem. Transp. Res. Part E Logist. Transp. Rev.
**2014**, 72, 192–209. [Google Scholar] [CrossRef] - Waserhole, A.; Jost, V. Vehicle Sharing System Pricing Regulation: A Fluid Approximation. 2012. Available online: https://hal.archives-ouvertes.fr/hal-00727041v4/document (accessed on 8 June 2019).
- Contardo, C.; Morency, C.; Rousseau, L.M. Balancing a Dynamic Public Bike-Sharing System; Centre Interuniversitaire de Recherche sur les Réseaux d’Entreprise, la Logistique et le Transport (CIRRELT): Montreal, QC, Canada, 2012; Volume 4. [Google Scholar]
- Caggiani, L.; Camporeale, R.; Ottomanelli, M. A dynamic clustering method for relocation process in free-floating vehicle sharing systems. Transp. Res. Procedia
**2017**, 27, 278–285. [Google Scholar] [CrossRef] - Feng, Y.; Affonso, R.C.; Zolghadri, M. Analysis of bike sharing system by clustering: the Vélib’case. IFAC-PapersOnLine
**2017**, 50, 12422–12427. [Google Scholar] [CrossRef] - Ferreira, L.; Hitchcock, D.B. A comparison of hierarchical methods for clustering functional data. Commun. Stat.-Simul. Comput.
**2009**, 38, 1925–1949. [Google Scholar] [CrossRef] - Zaki, M.J.; Meira, W., Jr.; Meira, W. Data Mining and Analysis: Fundamental Concepts and Algorithms; Cambridge University Press: New York, NY, USA, 2014. [Google Scholar]
- Rousseeuw, P.J. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math.
**1987**, 20, 53–65. [Google Scholar] [CrossRef] [Green Version] - de Amorim, R.C. Feature relevance in ward’s hierarchical clustering using the L p norm. J. Classif.
**2015**, 32, 46–62. [Google Scholar] [CrossRef] - Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - Faroqi, H.; Mesbah, M.; Kim, J. Comparing Sequential with Combined Spatiotemporal Clustering of Passenger Trips in the Public Transit Network Using Smart Card Data. Math. Probl. Eng.
**2019**, 2019, 5070794. [Google Scholar] [CrossRef]

Reference | Type | Mathematical Approach | Objective |
---|---|---|---|

Dell’Amico et al. [1] | Static | mixed integer linear programming | Minimizing total traveling cost |

Cruz et al. [11] | Static | iterated local search (ILS) | Minimizing total traveling cost |

Chemla et al. [14] | Static | integer program | Minimizing total traveling cost |

Forma et al. [10] | Static | mixed integer linear programming | Minimize unserved users and the total traveling distance |

Liu et al. [15] | Static | mixed integer linear programming | Minimize the total traveling distance |

Schuijbroek et al. [16] | Static | Constraint Programming | Optimal vehicle routes |

Li et al. [4] | Static | mixed integer linear programming | Minimize the total cost |

Alvarez-Valdes et al. [12] | Static | a heuristic algorithm | Minimizing the overall cost of unsatisfied demands |

Pal and Zhang [5] | Static | mixed integer linear programming | Minimize the make-span of the fleet of rebalancing vehicles |

Erdogan et al. [17] | Static | Exact method Greedy | Minimize rebalancing costs |

Erdogan et al. [18] | Static | Exact method integer programming | Minimize travel and handling costs |

Shui and Szeto [9] | Dynamic | artificial bee colony algorithm | Minimizes the total unmet demand and the fuel and CO${}_{2}$ emission cost |

Zhang et al. [6] | Dynamic | mixed-integer problem | Minimizes the total unmet demand and route |

Caggiani et al. [8] | Dynamic | Travelling Salesman Problem | Minimize cost and maximization of user satisfaction |

Clusters | Visited Stations | Not-Visited Stations | Tour (km) |
---|---|---|---|

C1 | 19 | 11 | 33 |

C2 | 23 | 112 | 113 |

C3 | 23 | 3 | 24 |

C4 | 41 | 5 | 49 |

C5 | 37 | 4 | 47 |

C6 | 45 | 10 | 93 |

C7 | 45 | 8 | 82 |

C8 | 29 | 6 | 35 |

C9 | 52 | 6 | 82 |

C10 | 25 | 6 | 38 |

C11 | 25 | 7 | 46 |

C12 | 19 | 7 | 29 |

C13 | 9 | 5 | 8 |

Day | % Visited | Total Bikes Loaded/Unloaded | Tour (Km) |
---|---|---|---|

Monday | 67 | 659 | 676 |

Tuesday | 66 | 631 | 648 |

Wednesday | 64 | 684 | 619 |

Thursday | 70 | 724 | 720 |

Friday | 68 | 751 | 727 |

Saturday | 68 | 829 | 679 |

Sunday | 68 | 883 | 689 |

Day | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | C13 | Tour (Km) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Sunday | 0 | 0 | 0 | 2 | 13 | 0 | 0 | 0 | −21 | 7 | 0 | −1 | 0 | 28 |

Monday | 1 | 4 | −4 | 0 | 17 | −1 | 0 | 0 | −14 | −3 | 0 | 0 | 0 | 46 |

Tuesday | 0 | 0 | −2 | 3 | 10 | 1 | 0 | −1 | −11 | 0 | 0 | 0 | 0 | 14 |

Wednesday | 1 | 0 | −2 | 1 | 5 | −1 | 0 | 0 | −6 | 2 | 0 | 0 | 0 | 19 |

Thursday | 0 | 0 | 0 | 1 | 4 | 1 | 0 | 0 | −6 | −5 | 0 | 5 | 0 | 32 |

Friday | 0 | −1 | −4 | 5 | 25 | −1 | 0 | 0 | −25 | 4 | 0 | −3 | 0 | 40 |

Saturday | 0 | −1 | 0 | 1 | 37 | 3 | 0 | −2 | −41 | 5 | 0 | −2 | 0 | 46 |

Cluster | Scenario I | Scenario II |
---|---|---|

C1 | 31 | 21 |

C2 | 75 | 50 |

C3 | 26 | 23 |

C4 | 51 | 35 |

C5 | 50 | 35 |

C6 | 96 | 61 |

C7 | 88 | 68 |

C8 | 40 | 27 |

C9 | 80 | 57 |

C10 | 32 | 22 |

C11 | 53 | 32 |

C12 | 19 | 17 |

C13 | 8 | 8 |

Total (km) | 648 | 456 |

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**MDPI and ACS Style**

Lahoorpoor, B.; Faroqi, H.; Sadeghi-Niaraki, A.; Choi, S.-M.
Spatial Cluster-Based Model for Static Rebalancing Bike Sharing Problem. *Sustainability* **2019**, *11*, 3205.
https://doi.org/10.3390/su11113205

**AMA Style**

Lahoorpoor B, Faroqi H, Sadeghi-Niaraki A, Choi S-M.
Spatial Cluster-Based Model for Static Rebalancing Bike Sharing Problem. *Sustainability*. 2019; 11(11):3205.
https://doi.org/10.3390/su11113205

**Chicago/Turabian Style**

Lahoorpoor, Bahman, Hamed Faroqi, Abolghasem Sadeghi-Niaraki, and Soo-Mi Choi.
2019. "Spatial Cluster-Based Model for Static Rebalancing Bike Sharing Problem" *Sustainability* 11, no. 11: 3205.
https://doi.org/10.3390/su11113205