A Matheuristic Approach Based on Variable Neighborhood Search for the Static Repositioning Problem in Station-Based Bike-Sharing Systems
Abstract
:1. Introduction
1.1. Main Contributions of the Paper
- Introducing a novel s-BRP in SBBSs, this paper considers operational and damaged bikes and multiple visits to the depot and stations. It utilizes a heterogeneous fleet to implement a joint repositioning strategy, enabling the simultaneous movement (pick-up and delivery) of operational (usable) and damaged (unusable) bikes between stations and the depot.
- Designing a new arc flow formulation to model the novel static bike-sharing repositioning problem variant in the station-based bike-sharing systems proposed in this paper.
- Proposing two innovative approaches to optimize the number of bikes to be moved for each visit (loading instructions). The first is a heuristic approach, where we design two heuristic methods to compute the loading policy given the sequence of visits. Additionally, a second mathematical programming-based approach is proposed, in which we formulate the problem of finding the optimal loading instructions as an integer linear programming (ILP) model.
- Presenting an effective matheuristic based on a variable neighborhood search combined with several improving and perturbation algorithms, including an integer linear programming model to optimize loading policy.
1.2. The Outline of the Paper
2. Literature Review
2.1. Problem Characteristics
2.1.1. Fleet of Vehicles
2.1.2. Multiple Visits to the Stations
2.1.3. Subtypes of the s-BRP
2.1.4. Types of Bikes
2.2. Objectives
2.3. Optimization Methods
2.4. Summary
3. Problem Research
3.1. Characteristics for Stations and the Depot
- , the number of lockers available to store bikes (capacity).
- , the desired (or target) number of o-bikes at the station.
- , the initial number of o-bikes at the station.
- , the initial number of d-bikes at the station.
- , the station priority.
3.2. Characteristics for Vehicles and Routes
- The vehicles can transport both o-bikes and d-bikes.
- The total vehicle capacity is denoted by .
- Each vehicle route must start and end at the depot, and the route’s total time cannot exceed T, which represents the maximum time available to carry out the repositioning.
- It is not necessary to specify at which time the stations are visited by the vehicles.
- In each visit to a station, the vehicle can load/unload o-bikes and load d-bikes.
- The vehicle can also visit the depot several times, and on each visit, it can load/unload o-bikes and unload d-bikes.
- Any station can be visited several times by the same or different vehicles during the repositioning.
- The vehicles can only load o-bikes at a surplus station and unload o-bikes at a deficit station, so the stations cannot be used to store bikes temporarily. Similarly, the d-bikes can only be loaded at the stations and unloaded at the depot.
3.3. Function Objective Proposed
3.4. Notation for Feasible Solutions
- is the number of o-bikes loaded (if ) or unloaded (if ) at , and
- is the number of d-bikes loaded (if ) or unloaded (if ) at
3.5. Arc Flow Formulation
3.5.1. Elements to Configure Graph
- The nodes set is formed by:
- (i)
- A nodes set containing copies of each , each with demand 1 if or if .
- (ii)
- A nodes set containing copies of each , each with demand 1.
- (iii)
- A nodes set containing N copies of the depot, for potential intermediate visits of all vehicles.
- (iv)
- Two nodes, and , copies of the depot, acting as source and sink, respectively.
- The set of arcs is formed by:
- (i)
- A set of arcs formed by the arcs for all .
- (ii)
- A set of arcs formed by the arcs for all .
- (iii)
- A set of arcs formed by the arcs for all .
- (iv)
- A set of arcs formed by the arcs for all and or for all and .
- (v)
- An arc .
- Given , denotes the nodes of which it is a copy.
- Given , denotes the travel time if and 0 otherwise.
3.5.2. Assumptions
- The fleet of vehicles can make at most N intermediate visits to the depot. At each of these visits, o-bikes and/or d-bikes can be dropped off, but only o-bikes can be taken from the initial depot inventory.
- For the moment, that there are no conflicting stations.
3.5.3. Decision Variables
- : 1 if any vehicle uses the arc , 0 otherwise.
- : number of vehicles not used.
- : number of o-bikes in the vehicle that crosses the arc .
- : number of d-bikes in the vehicle that crosses the arc .
- : cumulative time of the route through the arc .
- : number of o-bikes picked up in the intermediate visit to the depot i.
- : number of o-bikes delivered in the intermediate visit to the depot i.
3.5.4. Mathematical Model
4. Construction of the Initial Solution
4.1. General Schema
Algorithm 1 RCH |
Input: Problem Instance I Input: Parameter Max. number of iterations
Return a feasible solution |
4.2. Updating States
4.2.1. States for Stations and the Depot
- the set of deficit station,
- the set of surplus station,
- the set of balanced station, and
- the set of stations with at least one d-bike.
4.2.2. States for the Vehicles
4.3. Selection of Successors
5. Approaches for Optimizing Loading Instructions
5.1. Heuristic Approach
5.1.1. Single-Vehicle Loading Heuristic: sv-LH
- Notations and General Definitions
- General Scheme
Algorithm 2 sv-LH |
Input: Solution
|
- Detailed Procedural Outline
5.1.2. Global Loading Heuristic: g-LH
5.2. Mathematical Programming-Based Approach
6. Framework of the Proposed Matheuristic
6.1. Variable Neighborhood Search
Algorithm 3 VND |
Input: Solution Input: Set of neighborhoods
|
6.2. General Variable Neighborhood Search
Algorithm 4 GVNS |
Input: Solution Input: Set of neighborhoods Input: Set of perturbation procedures
|
6.3. Improvement Moves
- Insert or change unsolved stations—An unsolved station is one that is still imbalanced or has damaged bikes in the current solution. This move tries to improve the current solution by inserting a new visit to an unsolved station (see Figure 4). For each unsolved station and any route, two possibilities are tried: to insert a visit to that station or to replace one visit (to another station) in the route with a visit to the unsolved station. Both situations are shown in Figure 4b and Figure 4c, respectively.
- Remove or relocate a visit—
- Relocate a segment intra-route—In this move, a segment of l consecutive visits, where , is removed from a route and re-inserted at another position in the same route, with the same or the reverse orientation. The segment may include intermediate visits to the depot. Both situations are shown in Figure 6b and Figure 6c, respectively.
- Relocate a segment inter-route—This move is similar to the one defining , but the segment is inserted in another route. Segments that include visits to conflicted stations are not considered. See Figure 7.
- Intra-route segment exchange—Two non-overlapping segments of a route, both with a number of visits between 1 and 4, change their positions in the route, may be with the reverse orientation in both cases. Segments with intermediate visits to the depot are also considered. See Figure 8.
- Inter-route segment exchange—This move is similar to the one defining , but the segments are taken from different routes. See Figure 9.
6.4. Perturbation Mechanisms
- (-Shift): The following operation is executed times. Randomly select two routes and (one of them may be empty), and then randomly select one visit i in and one visit j in . The visit i is removed from route and inserted just before visit j in route . Positions i and j are different from the first and last ones in the routes. See Figure 10.
- (-Swap): The following operation is executed times. Randomly select two routes and (one of them may be empty), and then randomly select one visit i in and one visit j in . The visit i of route is interchanged with the visit j in route . Positions i and j are different from the first and last ones in the routes. See Figure 11.
- (-Segment): The following operation is executed times. Chose a random integer L between 2 and 4 visits. Select randomly one route and one visit i in that route. Then, for each , a route , and a position j in it, are randomly chosen, and visit i is removed from and inserted in position j of . See Figure 12.
7. Computational Experiments
- Experiment Setup: Section 7.1 describes the instances based on real-world data used for testing and introduces the parameter information used in our solution method.
- First Experiment: Section 7.2 presents an exhaustive performance analysis for the randomized constructive heuristic (RCH) described in Section 4.
- Second Experiment: Section 7.3 provides an in-depth analysis of the general variable neighborhood search (GNVS) detailed in Section 6.2, evaluating its performance across various scenarios.
- Third Experiment: Section 7.4, we compare the results obtained by our method with the recommended parameter setting and the best solutions obtained over all the experiments made with any parameter setting while tuning the algorithm. Given that there are no previous studies on this model, it is difficult to assess the quality of the solutions obtained. Therefore, we analyze the results of the two phases of our matheuristic proposed in Section 6.
7.1. Instances and Parameter Setting
- Palma: This set contains 28 instances, based on the PBS of Palma de Mallorca, Spain, that were proposed in Alvarez-Valdes et al. [1]. Each instance contains 28 stations and one depot with an initial inventory of 10 usable bikes, i.e., . There are seven variants with respect to the imbalance and number of damaged bikes in the stations (corresponding to the seven days of a week). For each day, two fleet sizes (two and three vehicles with capacity ) and two variants of maximum time (2 and 4 h) were considered.
- Wien: These instances were taken from Rainer-Harbach et al. [13] and adapted to our problem by randomly replacing a few operating bikes in the stations with damaged bikes. These instances have 20, 30, 60, and 90 stations and one depot (with no initial inventory, i.e., ). Two values for the maximum time (4 and 8 h) and two fleet sizes were considered. For the instances with 20 or 30 stations, the fleet sizes considered are two and four, while for the instances with 60 or 90 stations, the fleet sizes are three and five vehicles. In all the cases, the capacity of the vehicles is 20. There are five instances in each group, making a total of 80 instances in this set.
7.2. Performance Analysis for RCH
7.3. Performance Analysis for GVNS
7.4. Results Comparison RCH and GVNS
8. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- 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]
- PBSC Urban Solutions. The Meddin Bike-sharing World Map Report. 2022. Available online: https://www.pbsc.com/ (accessed on 31 December 2023).
- DeMaio, P. Bike-sharing: History, impacts, models of provision, and future. J. Public Transp. 2009, 12, 41–56. [Google Scholar] [CrossRef]
- 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]
- Li, X.; Zhang, Y.; Sun, L.; Liu, Q. Free-Floating Bike Sharing in Jiangsu: Users’ Behaviors and Influencing Factors. Energies 2018, 11, 1664. [Google Scholar] [CrossRef]
- Haider, Z.; Nikolaev, A.; Kang, J.E.; Kwon, C. Inventory rebalancing through pricing in public bike sharing systems. Eur. J. Oper. Res. 2018, 270, 103–117. [Google Scholar] [CrossRef]
- Chemla, D.; Meunier, F.; Wolfler Calvo, R. Bike sharing systems: Solving the static rebalancing problem. Discret. Optim. 2013, 10, 120–146. [Google Scholar] [CrossRef]
- 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]
- Dell’Amico, M.; Iori, M.; Novellani, S.; Stützle, T. A destroy and repair algorithm for the bike sharing rebalancing problem. Comput. Oper. Res. 2016, 71, 149–162. [Google Scholar] [CrossRef]
- Ho, S.C.; Szeto, W.Y. A hybrid large neighborhood search for the static multi-vehicle bike-repositioning problem. Transp. Res. Part Methodol. 2017, 95, 340–363. [Google Scholar] [CrossRef]
- Kloimüllner, C.; Raidl, G.R. Full-load route planning for balancing bike sharing systems by logic-based Benders decomposition. Networks 2017, 69, 270–289. [Google Scholar] [CrossRef]
- Rainer-Harbach, M.; Papazek, P.; Hu, B.; Raidl, G.R. Balancing bicycle sharing systems: A variable neighborhood search approach. In Proceedings of the Evolutionary Computation in Combinatorial Optimization, EvoCOP 2013, Vienna, Austria, 3–5 April 2013; Lecture Notes in Computer Science. Middendorf, M., Blum, C., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 7832, pp. 121–132. [Google Scholar] [CrossRef]
- Rainer-Harbach, M.; Papazek, P.; Raidl, G.R.; Hu, B.; Kloimüllner, C. PILOT, GRASP, and VNS approaches for the static balancing of bicycle sharing systems. J. Glob. Optim. 2015, 63, 597–629. [Google Scholar] [CrossRef]
- Raviv, T.; Tzur, M.; Forma, I.A. Static repositioning in a bike-sharing system: Models and solution approaches. EURO J. Transp. Logist. 2013, 2, 187–229. [Google Scholar] [CrossRef]
- Wang, Y.; Szeto, W. Static green repositioning in bike sharing systems with broken bikes. Transp. Res. Part D Transp. Environ. 2018, 65, 438–457. [Google Scholar] [CrossRef]
- Casazza, M. Exactly Solving the Split Pickup and Split Delivery Vehicle Routing Problem on a Bike-Sharing System; Technical Report; [hal-01304433]; Università degli Studi di Milano: Milano, Italy; Université Paris: Paris, France, 2016; Volume 13. [Google Scholar]
- Di Gaspero, L.; Rendl, A.; Urli, T. A hybrid ACO+CP for balancing bicycle sharing systems. In Proceedings of the Hybrid Metaheuristics, HM 2013, Ischia, Italy, 23–25 May 2013; Lecture Notes in Computer Science. Blesa, M.J., Blum, C., Festa, P., Roli, A., Sampels, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 7919, pp. 198–212. [Google Scholar] [CrossRef]
- Di Gaspero, L.; Rendl, A.; Urli, T. Constraint-based approaches for balancing bike sharing systems. In Proceedings of the Principles and Practice of Constraint Programming, CP 2013, Uppsala, Sweden, 16–20 September 2013; Lecture Notes in Computer Science. Schulte, C., Ed.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 8124, pp. 758–773. [Google Scholar] [CrossRef]
- Di Gaspero, L.; Rendl, A.; Urli, T. Balancing bike sharing systems with constraint programming. Constraints 2016, 21, 318–348. [Google Scholar] [CrossRef]
- Espegren, H.M.; Kristianslund, J.; Andersson, H.; Fagerholt, K. The static bicycle repositioning problem-literature survey and new formulation. In Proceedings of the Computational Logistics, ICCL 2016, Lisbon, Portugal, 7–9 September 2013; Lecture Notes in Computer Science. Paias, A., Ruthmair, M., Voß, S., Eds.; Springer: Cham, Switzerland, 2016; Volume 9855, pp. 337–351. [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]
- Papazek, P.; Raidl, G.R.; Rainer-Harbach, M.; Hu, B. A PILOT/VND/GRASP hybrid for the static balancing of public bicycle sharing systems. In Proceedings of the Computer Aided Systems Theory—EUROCAST 2013, Canaria, Spain, 10–15 February 2013; Lecture Notes in Computer Science. Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 8111, pp. 372–379. [Google Scholar] [CrossRef]
- Papazek, P.; Kloimüllner, C.; Hu, B.; Raidl, G.R. Balancing bicycle sharing systems: An analysis of path relinking and recombination within a GRASP hybrid. In Proceedings of the Parallel Problem Solving from Nature—PPSN XIII, PPSN 2014, Ljubljana, Slovenia, 13–17 September 2014; Lecture Notes in Computer Science. Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J., Eds.; Springer: Cham, Switzerland, 2014; Volume 8672, pp. 792–801. [Google Scholar] [CrossRef]
- Raidl, G.R.; Hu, B.; Rainer-Harbach, M.; Papazek, P. Balancing bicycle sharing systems: Improving a VNS by efficiently determining optimal loading operations. In Proceedings of the Hybrid Metaheuristics, HM 2013, Ischia, Italy, 23–25 May 2013; Lecture Notes in Computer Science. Blesa, M.J., Blum, C., Festa, P., Roli, A., Sampels, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 7919, pp. 130–143. [Google Scholar] [CrossRef]
- Schuijbroek, J.; Hampshire, R.C.; van Hoeve, W.J. Inventory rebalancing and vehicle routing in bike sharing systems. Eur. J. Oper. Res. 2017, 257, 992–1004. [Google Scholar] [CrossRef]
- Sorensen, K.; Vergeylen, N. The bike request scheduling problem. In Proceedings of the Computer Aided Systems Theory—EUROCAST 2015, Canaria, Spain, 8–13 February 2015; Lecture Notes in Computer Science. Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A., Eds.; Springer: Cham, Switzerland, 2015; Volume 9520, pp. 294–301. [Google Scholar] [CrossRef]
- Lahoorpoor, B.; Faroqi, H.; Sadeghi-Niaraki, A.; Choi, S.M. Spatial Cluster-Based Model for Static Rebalancing Bike Sharing Problem. Sustainability 2019, 11, 3205. [Google Scholar] [CrossRef]
- Zhang, S.; Xiang, G.; Huang, Z. Bike-Sharing Static Rebalancing by Considering the Collection of Bicycles in Need of Repair. J. Adv. Transp. 2018, 2018, 1–18. [Google Scholar] [CrossRef]
- Kinable, J. A reservoir balancing constraint with applications to bike-sharing. In Proceedings of the Integration of AI and OR Techniques in Constraint Programming, CPAIOR 2016, Banff, AB, Canada, 29 May–1 June 2016; Lecture Notes in Computer Science. Quimper, C.G., Ed.; Springer: Cham, Switzerland, 2016; Volume 9676, pp. 216–228. [Google Scholar] [CrossRef]
- Kloimüllner, C.; Papazek, P.; Hu, B.; Raidl, G.R. A cluster-first route-second approach for balancing bicycle sharing systems. In Proceedings of the Computer Aided Systems Theory—EUROCAST 2015, Las Palmas de Gran Canaria, Spain, 8–13 February 2015; Lecture Notes in Computer Science. Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A., Eds.; Springer: Cham, Switzerland, 2015; Volume 9520, pp. 439–446. [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]
- Tang, Q.; Fu, Z.; Zhang, D.; Qiu, M.; Li, M. An Improved Iterated Local Search Algorithm for the Static Partial Repositioning Problem in Bike-Sharing System. J. Adv. Transp. 2020, 2020, 1–15. [Google Scholar] [CrossRef]
- Lv, C.; Zhang, C.; Lian, K.; Ren, Y.; Meng, L. A hybrid algorithm for the static bike-sharing re-positioning problem based on an effective clustering strategy. Transp. Res. Part Methodol. 2020, 140, 1–21. [Google Scholar] [CrossRef]
- Du, M.; Cheng, L.; Li, X.; Tang, F. Static rebalancing optimization with considering the collection of malfunctioning bikes in free-floating bike sharing system. Transp. Res. Part E Logist. Transp. Rev. 2020, 141, 102012. [Google Scholar] [CrossRef]
- Kaspi, M.; Raviv, T.; Tzur, M. Detection of unusable bicycles in bike-sharing systems. Omega 2016, 65, 10–16. [Google Scholar] [CrossRef]
- Kaspi, M.; Raviv, T.; Tzur, M. Bike sharing systems: User dissatisfaction in the presence of unusable bicycles. IIE Trans. 2017, 49, 144–458. [Google Scholar] [CrossRef]
- Castro Fernandez, A. The contribution of bike-sharing to sustainable mobility in Europe. Ph.D. Thesis, Technischen Universität Wien, Vienna, Austria, 2011. [Google Scholar]
- DeMaio, P.; Meddin, R. The Bike-Sharing World—First Week of June 2014. 2014. Available online: http://bike-sharing.blogspot.com.es/2014/06/the-bike-sharing-world-first-week-of.html (accessed on 31 December 2023).
- Benchimol, M.; Benchimol, P.; Chappert, B.; De la Taille, A.; Laroche, F.; Meunier, F.; Robinet, L. Balancing the stations of a self service bike hire system. RAIRO Oper. Res. 2011, 45, 37–61. [Google Scholar] [CrossRef]
- Erdoğan, G.; Battarra, M.; Wolfler Calvo, R. An exact algorithm for the static rebalancing problem arising in bicycle sharing systems. Eur. J. Oper. Res. 2015, 245, 667–679. [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]
- Erdoğan, G.; Laporte, G.; Wolfler Calvo, R. The static bicycle relocation problem with demand intervals. Eur. J. Oper. Res. 2014, 238, 451–457. [Google Scholar] [CrossRef]
- Kadri, A.A.; Kacem, I.; Labadi, K. A branch-and-bound algorithm for solving the static rebalancing problem in bicycle-sharing systems. Comput. Ind. Eng. 2016, 95, 41–52. [Google Scholar] [CrossRef]
- Mladenović, N.; Hansen, P. A variable neighborhood algorithm: A new metaheuristic for combinatorial optimization. In Proceedings of the Abstract for Papers presented at Optimization Days, Montreal, QC, Canada, 8–10 May 1995; p. 112. [Google Scholar]
- Mladenović, N.; Hansen, P. Variable neighborhood search. Comput. Oper. Res. 1997, 24, 1097–1100. [Google Scholar] [CrossRef]
- Daza-Escorcia, J.M.; Alvarez-Martinez, D. A Matheuristic Multi-Start Algorithm for a Novel Static Repositioning Problem in Public Bike-Sharing Systems. In Proceedings of the Metaheuristics, MIC 2014, Lorient, France, 4–7 June 2024; Lecture Notes in Computer Science. Sevaux, M., Ed.; Springer Nature: Cham, Switzerland, 2014; Volume 14754, pp. 188–203. [Google Scholar] [CrossRef]
- CPLEX. Optimization Studio ILOG IBM CPLEX. 2017. Available online: https://www.ibm.com/us-en/marketplace/ibm-ilog-cplex (accessed on 31 December 2023).
- Hansen, P.; Mladenović, N. Variable neighborhood search. In Handbook of Metaheuristics; Glover, F., Gary, K., Eds.; Kluwer Academic Publishers: Boston, MA, USA, 2003; Chapter 6; pp. 145–184. [Google Scholar]
Instance | 1000 | 2000 | 3000 | 4000 | 5000 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Type | T (h) | obj | (s) | obj | (s) | obj | (s) | obj | (s) | obj | (s) | |||||||
Palma | 28 | 2 | 2 | 1.67 | 3.98 | 1.7 | 1.26 | 4.61 | 1.7 | 0.83 | 2.64 | 1.7 | 0.69 | 0.77 | 1.7 | 0.35 | 0.29 | 1.0 |
Palma | 28 | 2 | 4 | 0.27 | 3.65 | 1.0 | 0.27 | 4.38 | 1.0 | 0.26 | 2.66 | 1.0 | 0.25 | 0.84 | 1.0 | 0.17 | 0.31 | 1.0 |
Palma | 28 | 3 | 2 | 1.08 | 1.32 | 1.7 | 0.66 | 3.53 | 1.7 | 0.65 | 4.29 | 1.7 | 0.51 | 0.77 | 1.7 | 0.23 | 0.29 | 1.0 |
Palma | 28 | 3 | 4 | 0.18 | 2.71 | 1.0 | 0.18 | 3.78 | 1.0 | 0.17 | 3.02 | 1.0 | 0.17 | 1.01 | 1.0 | 0.12 | 0.30 | 1.0 |
Wien | 20 | 2 | 4 | 32.33 | 6.45 | 2.0 | 32.53 | 3.29 | 2.0 | 29.94 | 0.71 | 2.0 | 29.94 | 3.33 | 2.0 | 9.35 | 0.49 | 2.0 |
Wien | 20 | 2 | 8 | 2.50 | 5.91 | 2.0 | 3.28 | 3.94 | 2.0 | 2.69 | 0.84 | 2.0 | 2.69 | 4.13 | 2.0 | 0.56 | 0.63 | 1.6 |
Wien | 20 | 3 | 4 | 6.50 | 3.93 | 3.0 | 6.32 | 3.11 | 3.0 | 7.30 | 0.75 | 3.0 | 7.30 | 0.55 | 3.0 | 1.20 | 0.56 | 2.8 |
Wien | 20 | 3 | 8 | 2.27 | 1.95 | 2.0 | 2.06 | 3.10 | 2.0 | 1.67 | 0.87 | 2.0 | 1.67 | 0.56 | 2.0 | 0.38 | 0.62 | 1.6 |
Wien | 30 | 2 | 4 | 104.13 | 4.88 | 2.0 | 98.74 | 1.49 | 2.0 | 95.14 | 1.31 | 2.0 | 95.14 | 0.85 | 2.0 | 54.77 | 0.80 | 2.0 |
Wien | 30 | 2 | 8 | 15.56 | 0.72 | 2.0 | 15.35 | 0.94 | 2.0 | 12.36 | 2.00 | 2.0 | 12.17 | 1.07 | 2.0 | 0.91 | 1.23 | 2.0 |
Wien | 30 | 3 | 4 | 60.95 | 2.97 | 3.0 | 53.36 | 0.76 | 3.0 | 52.36 | 1.51 | 3.0 | 52.36 | 0.92 | 3.0 | 19.17 | 1.11 | 3.0 |
Wien | 30 | 3 | 8 | 7.13 | 2.23 | 2.6 | 5.93 | 0.95 | 2.6 | 8.11 | 2.02 | 2.6 | 7.12 | 1.47 | 2.8 | 0.61 | 1.44 | 2.0 |
Wien | 60 | 3 | 4 | 216.75 | 2.12 | 3.0 | 213.15 | 2.46 | 3.0 | 214.94 | 6.35 | 3.0 | 214.94 | 2.25 | 3.0 | 146.58 | 2.70 | 3.0 |
Wien | 60 | 3 | 8 | 76.17 | 3.17 | 3.0 | 68.97 | 3.28 | 3.0 | 64.98 | 0.38 | 3.0 | 63.98 | 3.14 | 3.0 | 20.98 | 4.71 | 3.0 |
Wien | 60 | 5 | 4 | 134.36 | 3.25 | 5.0 | 126.76 | 2.85 | 5.0 | 124.96 | 8.96 | 5.0 | 124.96 | 2.70 | 5.0 | 61.56 | 3.61 | 5.0 |
Wien | 60 | 5 | 8 | 16.68 | 6.94 | 5.0 | 13.87 | 4.14 | 5.0 | 15.65 | 0.44 | 5.0 | 13.86 | 5.24 | 5.0 | 0.81 | 4.92 | 4.6 |
Wien | 90 | 3 | 4 | 441.55 | 6.67 | 3.0 | 438.54 | 7.02 | 3.0 | 431.35 | 0.55 | 3.0 | 431.35 | 6.20 | 3.0 | 350.17 | 4.20 | 3.0 |
Wien | 90 | 3 | 8 | 267.18 | 14.04 | 3.0 | 256.58 | 12.38 | 3.0 | 251.57 | 1.17 | 3.0 | 251.57 | 11.93 | 3.0 | 146.78 | 8.15 | 3.0 |
Wien | 90 | 5 | 4 | 347.55 | 11.41 | 5.0 | 336.76 | 10.65 | 5.0 | 331.16 | 0.88 | 5.0 | 331.16 | 6.95 | 5.0 | 231.57 | 6.22 | 5.0 |
Wien | 90 | 5 | 8 | 108.57 | 15.42 | 5.0 | 94.18 | 23.81 | 5.0 | 90.38 | 1.67 | 5.0 | 90.38 | 24.57 | 5.0 | 16.38 | 9.75 | 5.0 |
Instance | 1000 | 2000 | 3000 | 4000 | 5000 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Type | T (h) | obj | (s) | obj | (s) | obj | (s) | obj | (s) | obj | (s) | ||
Palma | 28 | 2 | 2 | 0.32 | <1 | 0.30 | <1 | 0.30 | <1 | 0.30 | 1.59 | 0.30 | 5.49 |
Palma | 28 | 2 | 4 | 0.16 | <1 | 0.15 | <1 | 0.15 | <1 | 0.15 | 1.64 | 0.15 | 6.56 |
Palma | 28 | 3 | 2 | 0.21 | <1 | 0.20 | <1 | 0.20 | <1 | 0.20 | 1.56 | 0.20 | 5.65 |
Palma | 28 | 3 | 4 | 0.11 | <1 | 0.10 | 1.07 | 0.10 | <1 | 0.10 | 1.67 | 0.10 | 5.11 |
Wien | 20 | 2 | 4 | 7.35 | <1 | 7.15 | 1.14 | 7.15 | <1 | 5.94 | 2.99 | 5.95 | 8.91 |
Wien | 20 | 2 | 8 | 0.52 | <1 | 0.52 | 1.26 | 0.52 | <1 | 0.51 | 4.47 | 0.51 | 11.03 |
Wien | 20 | 3 | 4 | 0.73 | <1 | 0.73 | 1.61 | 0.73 | <1 | 0.72 | 4.00 | 0.72 | 12.10 |
Wien | 20 | 3 | 8 | 0.35 | <1 | 0.35 | 1.31 | 0.35 | <1 | 0.34 | 4.65 | 0.34 | 11.73 |
Wien | 30 | 2 | 4 | 52.58 | <1 | 51.18 | 1.37 | 51.18 | <1 | 49.38 | 3.44 | 47.58 | 10.37 |
Wien | 30 | 2 | 8 | 0.82 | <1 | 0.83 | 3.29 | 0.82 | <1 | 0.82 | 13.28 | 0.81 | 30.08 |
Wien | 30 | 3 | 4 | 16.55 | <1 | 14.76 | 2.36 | 14.76 | <1 | 13.37 | 8.68 | 13.18 | 22.84 |
Wien | 30 | 3 | 8 | 0.58 | <1 | 0.55 | 3.27 | 0.55 | <1 | 0.54 | 14.71 | 0.54 | 32.63 |
Wien | 60 | 3 | 4 | 141.38 | <1 | 136.18 | 3.48 | 136.18 | <1 | 134.78 | 10.50 | 135.19 | 34.14 |
Wien | 60 | 3 | 8 | 13.57 | <1 | 12.38 | 10.90 | 12.58 | <1 | 11.37 | 63.54 | 11.18 | 174.19 |
Wien | 60 | 5 | 4 | 51.78 | <1 | 50.97 | 6.09 | 50.97 | <1 | 48.18 | 31.67 | 47.77 | 90.94 |
Wien | 60 | 5 | 8 | 0.69 | <1 | 0.67 | 17.65 | 0.67 | <1 | 0.67 | 95.77 | 0.72 | 225.27 |
Wien | 90 | 3 | 4 | 344.77 | <1 | 341.18 | 4.63 | 341.18 | <1 | 336.99 | 18.47 | 338.38 | 47.41 |
Wien | 90 | 3 | 8 | 136.79 | <1 | 133.99 | 13.30 | 130.79 | <1 | 131.39 | 73.93 | 128.99 | 200.97 |
Wien | 90 | 5 | 4 | 218.78 | <1 | 215.98 | 8.90 | 215.98 | <1 | 211.18 | 45.52 | 207.59 | 124.11 |
Wien | 90 | 5 | 8 | 10.37 | <1 | 6.97 | 35.98 | 6.37 | <1 | 4.78 | 882.47 | 5.37 | 786.36 |
Instance | Phase I | Phase II | PBS’s States | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Type | T (h) | BKS | obj | (s) | obj | (s) | dif (%) | Before | After | Imp (%) | ||||
Palma | 28 | 2 | 2 | 0.30 | 0.35 | <1 | 1.0 | 0.30 | 5.5 | 1.0 | 11.5 | 32.1 | 0.0 | 100 |
Palma | 28 | 2 | 4 | 0.15 | 0.17 | <1 | 1.0 | 0.15 | 6.6 | 1.0 | 11.8 | 32.1 | 0.0 | 100 |
Palma | 28 | 3 | 2 | 0.20 | 0.23 | <1 | 1.0 | 0.20 | 5.6 | 1.0 | 11.5 | 32.1 | 0.0 | 100 |
Palma | 28 | 3 | 4 | 0.10 | 0.12 | <1 | 1.0 | 0.10 | 5.1 | 1.0 | 11.8 | 32.1 | 0.0 | 100 |
Wien | 20 | 2 | 4 | 5.55 | 9.35 | <1 | 2.0 | 5.95 | 8.9 | 2.0 | 38.6 | 136.0 | 5.0 | 96.5 |
Wien | 20 | 2 | 8 | 0.51 | 0.56 | <1 | 1.6 | 0.51 | 11.0 | 1.6 | 10.2 | 136.0 | 0.0 | 100 |
Wien | 20 | 3 | 4 | 0.72 | 1.20 | <1 | 2.8 | 0.72 | 12.1 | 2.6 | 22.5 | 136.0 | 0.0 | 100 |
Wien | 20 | 3 | 8 | 0.34 | 0.38 | <1 | 1.6 | 0.34 | 11.7 | 1.6 | 10.0 | 136.0 | 0.0 | 100 |
Wien | 30 | 2 | 4 | 46.98 | 54.77 | <1 | 2.0 | 47.58 | 10.4 | 2.0 | 13.5 | 207.8 | 46.6 | 77.7 |
Wien | 30 | 2 | 8 | 0.81 | 0.91 | 1.2 | 2.0 | 0.81 | 30.1 | 2.0 | 10.8 | 207.8 | 0.0 | 100 |
Wien | 30 | 3 | 4 | 11.98 | 19.17 | 1.1 | 3.0 | 13.18 | 22.8 | 3.0 | 34.2 | 207.8 | 12.2 | 94.2 |
Wien | 30 | 3 | 8 | 0.54 | 0.61 | 1.4 | 2.0 | 0.54 | 32.6 | 2.0 | 10.6 | 207.8 | 0.0 | 100 |
Wien | 60 | 3 | 4 | 133.58 | 146.58 | 2.7 | 3.0 | 135.19 | 34.1 | 3.0 | 7.8 | 381.2 | 134.2 | 64.8 |
Wien | 60 | 3 | 8 | 10.18 | 20.98 | 4.7 | 3.0 | 11.18 | 174.2 | 3.0 | 46.8 | 381.2 | 10.2 | 97.3 |
Wien | 60 | 5 | 4 | 46.78 | 61.56 | 3.6 | 5.0 | 47.77 | 90.9 | 5.0 | 22.7 | 381.2 | 46.8 | 87.8 |
Wien | 60 | 5 | 8 | 0.66 | 0.81 | 4.9 | 4.6 | 0.72 | 225.3 | 4.0 | 10.5 | 381.2 | 0.0 | 100 |
Wien | 90 | 3 | 4 | 334.79 | 350.17 | 4.2 | 3.0 | 338.38 | 47.4 | 3.0 | 3.4 | 617.4 | 337.4 | 45.3 |
Wien | 90 | 3 | 8 | 126.59 | 146.78 | 8.1 | 3.0 | 128.99 | 200.9 | 3.0 | 12.0 | 617.4 | 128.0 | 79.2 |
Wien | 90 | 5 | 4 | 204.59 | 231.57 | 6.2 | 5.0 | 207.59 | 124.1 | 5.0 | 10.4 | 617.4 | 206.6 | 66.5 |
Wien | 90 | 5 | 8 | 4.37 | 16.38 | 9.7 | 5.0 | 5.37 | 786.4 | 5.0 | 68.1 | 617.4 | 4.4 | 99.3 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Daza-Escorcia, J.M.; Álvarez-Martínez, D. A Matheuristic Approach Based on Variable Neighborhood Search for the Static Repositioning Problem in Station-Based Bike-Sharing Systems. Mathematics 2024, 12, 3573. https://doi.org/10.3390/math12223573
Daza-Escorcia JM, Álvarez-Martínez D. A Matheuristic Approach Based on Variable Neighborhood Search for the Static Repositioning Problem in Station-Based Bike-Sharing Systems. Mathematics. 2024; 12(22):3573. https://doi.org/10.3390/math12223573
Chicago/Turabian StyleDaza-Escorcia, Julio Mario, and David Álvarez-Martínez. 2024. "A Matheuristic Approach Based on Variable Neighborhood Search for the Static Repositioning Problem in Station-Based Bike-Sharing Systems" Mathematics 12, no. 22: 3573. https://doi.org/10.3390/math12223573
APA StyleDaza-Escorcia, J. M., & Álvarez-Martínez, D. (2024). A Matheuristic Approach Based on Variable Neighborhood Search for the Static Repositioning Problem in Station-Based Bike-Sharing Systems. Mathematics, 12(22), 3573. https://doi.org/10.3390/math12223573