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April 2018, 14(2): 447-472. doi: 10.3934/jimo.2017055

## The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty

 1 Av. Pedro de Alba, San Nicolás de los Garza, NL 66450, México, PhD Student in Program for Economy and Enterprise at the University of Málaga 2 Universidad Autónoma de Nuevo León, Av. Pedro de Alba, San Nicolás de los Garza, NL 66450, México 3 Universidad de Málaga, Campus El Ejido S/N, Málaga, 29071, España

* Corresponding author: Paulina Avila-Torres

Received  January 2016 Revised  November 2016 Published  June 2017

Fund Project: CONACyT, AUIP, DoA, Spanish MINECO and Andalucia Goverment

The urban transport planning process has four main activities: Network design, Timetable construction, Vehicle scheduling and Crew scheduling; each activity has subactivities. In this paper the authors work with the subactivities of timetable construction: minimal frequency calculation and departure time scheduling. The authors propose to solve both subactivities in an integrated way. The developed mathematical model allows multi-period planning and it can also be used for multimodal urban transportation systems. The authors consider demand uncertainty and the authors employ fuzzy programming to solve the problem. The authors formulate the urban transportation timetabling construction problem as a bi-objective problem: to minimize the total operational cost and to maximize the number of multi-period synchronizations. Finally, the authors implemented the SAUGMECON method to solve the problem.

Citation: Paulina Ávila-Torres, Fernando López-Irarragorri, Rafael Caballero, Yasmín Ríos-Solís. The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty. Journal of Industrial & Management Optimization, 2018, 14 (2) : 447-472. doi: 10.3934/jimo.2017055
##### References:
 [1] P. Avila and F. López, Two multiobjective metaheuristics for solving the integrated problem of frequencies calculation and departures planning in an urban transport system, Annals of Management Science, 3 (2014), 29-42. [2] R. Baskaran and K. Krishnaiah, Simulation model to determine frequency of a single bus route with single and multiple headways, Int. J. Business Performance and Supply Chain Modelling, 4 (2012), 40-59. [3] L. Cadarso and A. Marín, Integration of timetable planning and rolling stock in rapid transit networks, Annals of Operations Research, 199 (2012), 113-135. doi: 10.1007/s10479-011-0978-0. [4] L. Campos and J. L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems, 32 (1989), 1-11. doi: 10.1016/0165-0114(89)90084-5. [5] A. Ceder, Public Transit Planning and Operation: Theory, Modeling and Practice, 1 edition, Elsevier, USA, 2007. [6] P. Chakroborty, Genetic algorithms for optimal urban transit network design, Computer-Aided Civil and Infrastructure Engineering, 18 (2003), 184-200. doi: 10.1111/1467-8667.00309. [7] H. Chen, Stochastic optimization in computing multiple headways for a single bus line, Proceedings of the 35th Annual Simulation Symposium, (2002), 316-323. [8] C. Daraio, D. Marco, F. Di Costa, C. Leporelli, G. Matteucci and A. Nastasi, Efficiency and effectiveness in the urban public transport sector: A critical review with directions for future research, European Journal of Operational Research, 248 (2016), 1-20. [9] G. Desaulniers and M. D. Hickman, Public transit, in Handbook in OR & MS (eds C. Barnhart and G. Laporte), Elsevier, (2007), 69-127. [10] A. Eranki, A model to create bus timetables to attain maximum synchronization considering waiting times at transfer stops, Thesis University of South Florida, 2004. [11] H. Fazlollahtabar and M. Saidi-Mehrabad, Optimizing multi-objective decision making having qualitative evaluation, Journal of Industrial and Management Optimization, 11 (2016), 747-762. doi: 10.3934/jimo.2015.11.747. [12] Y. Hadas and M. Shnaiderman, Public-transit frequency setting using minimum-cost approach with stochastic demand and travel time, Transportation Research Part B: Methodological, 46 (2012), 1068-1084. doi: 10.1016/j.trb.2012.02.010. [13] O. J. Ibarra-Rojas and Y. A. Rios-Solis, Synchronization of bus timetabling, Transportation Research Part B: Methodological, 46 (2012), 599-614. doi: 10.1016/j.trb.2012.01.006. [14] J. Jensen, O. Nielsen and C. Prato, Public transport optimisation emphasising passengers' travel behaviour, Thesis Technical University of DenmarkDanmarks Tekniske Universitet, 2015. [15] L. Linzhong, Y. Juhua, M. Haibo, L. Xiaojing and W. Fang, Exact algorithms for multi-criteria multi-modal shortest path with transfer delaying and arriving time-window in urban transit network, Applied Mathematical Modeling, 38 (2014), 2613-2629. doi: 10.1016/j.apm.2013.10.059. [16] S. H. Nasseri and E. Behmanesh, Linear programming with triangular fuzzy numbers--A case study in a finance and credit institute, Fuzzy Information and Engineering, 5 (2013), 295-315. doi: 10.1007/s12543-013-0151-3. [17] F. Perez, T. Gomez and R. Caballero, Un modelo difuso para la selección de carteras de proyectos con incertidumbre en los costes, Revista Electrónica de Comunicaciones y Trabajos de ASEPUMA, 13 (2012), 129-143. [18] F. Perez and T. Gomez, Multiobjective project portfolio selection with fuzzy constraints, Annals of Operation Research, 245 (2016), 7-29. doi: 10.1007/s10479-014-1556-z. [19] T. Rasmussen, M. Anderson, O. Nielsen and C. Prato, Timetable-based simulation method for choice set generation in large-scale public transport networks, EJTIR, 16 (2016), 467-489. [20] V. Sahinidis Nikolaos, Optimization under uncertainty: State-of-the-art and opportunities, Computers and Chemical Engineering, 28 (2004), 971-983. [21] Y. Shangyao, C. Chin-Jen and T. Ching-Hui, Inter-city bus routing and timetable setting under stochastic demands, Transportation research part A, 40 (2006), 572-586. [22] L. Sun, Z. Gao and Y. Wang, A Stackelberg game management model of the urban public transport, Journal of Industrial and Management Optimization, 8 (2012), 507-520. doi: 10.3934/jimo.2012.8.507. [23] W. Y. Szeto and W. Yongzhong, A simultaneous bus route design and frequency setting problem for Tin Shui Wai, Hong Kong, European Journal of Operational Research, 209 (2011), 141-155. doi: 10.1016/j.ejor.2010.08.020. [24] S. L. Tilahun and H. C. Ong, Bus timetabling as a fuzzy multiobjective optimization problem using preference based genetic algorithm, Promet -Traffic & Transportation, 24 (2012), 183-191. doi: 10.7307/ptt.v24i3.311. [25] I. Verbas, C. Frei, H. Mahmassani and R. Chan, Stretching resources: Sensitivity of optimal bus frequency allocation to stop-level demand elasticities, Public Transport, 7 (2015), 1-20. [26] I. Verbas and H. Mahmassani, Exploring trade-offs in frequency allocation in a transit network using bus route patterns: Methodology and application to large-scale urban systems, Transportation Research Part B: Methodological, 81 (2015), 577-595. [27] Y. Wang, X. Zhu and L. B. Wu, Integrated multimodal metropolitan transportation model, Procedia Social and Behavioral Sciences, 96 (2013), 2138-2146. doi: 10.1016/j.sbspro.2013.08.241. [28] J. Zhang, T. Arentze and H. Timmermans, A multimodal transport network model for advanced traveler information system, Journal of Ubiquitous System and Pervasive Networks, 4 (2012), 21-27. [29] W. Zhang and M. Reimann, A simple augmented e-constraint method for multi-objective mathematical integer programming problems, European Journal of Operations Research, 234 (2014), 15-24. doi: 10.1016/j.ejor.2013.09.001. [30] F. Zhao and Z. Xiaogang, Optimization of transit route network, vehicle headways and timetables for large-scale transit networks, European Journal of Operational Research, 186 (2008), 841-855. doi: 10.1016/j.ejor.2007.02.005. [31] Y. Zhu, B. Mao, L. Liu and M. Li, Timetable design for urban rail line with capacity constraints Discrete Dynamics in Nature and Society 2015 (2015), Art. ID 429219, 11 pp. doi: 10.1155/2015/429219.

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##### References:
 [1] P. Avila and F. López, Two multiobjective metaheuristics for solving the integrated problem of frequencies calculation and departures planning in an urban transport system, Annals of Management Science, 3 (2014), 29-42. [2] R. Baskaran and K. Krishnaiah, Simulation model to determine frequency of a single bus route with single and multiple headways, Int. J. Business Performance and Supply Chain Modelling, 4 (2012), 40-59. [3] L. Cadarso and A. Marín, Integration of timetable planning and rolling stock in rapid transit networks, Annals of Operations Research, 199 (2012), 113-135. doi: 10.1007/s10479-011-0978-0. [4] L. Campos and J. L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems, 32 (1989), 1-11. doi: 10.1016/0165-0114(89)90084-5. [5] A. Ceder, Public Transit Planning and Operation: Theory, Modeling and Practice, 1 edition, Elsevier, USA, 2007. [6] P. Chakroborty, Genetic algorithms for optimal urban transit network design, Computer-Aided Civil and Infrastructure Engineering, 18 (2003), 184-200. doi: 10.1111/1467-8667.00309. [7] H. Chen, Stochastic optimization in computing multiple headways for a single bus line, Proceedings of the 35th Annual Simulation Symposium, (2002), 316-323. [8] C. Daraio, D. Marco, F. Di Costa, C. Leporelli, G. Matteucci and A. Nastasi, Efficiency and effectiveness in the urban public transport sector: A critical review with directions for future research, European Journal of Operational Research, 248 (2016), 1-20. [9] G. Desaulniers and M. D. Hickman, Public transit, in Handbook in OR & MS (eds C. Barnhart and G. Laporte), Elsevier, (2007), 69-127. [10] A. Eranki, A model to create bus timetables to attain maximum synchronization considering waiting times at transfer stops, Thesis University of South Florida, 2004. [11] H. Fazlollahtabar and M. Saidi-Mehrabad, Optimizing multi-objective decision making having qualitative evaluation, Journal of Industrial and Management Optimization, 11 (2016), 747-762. doi: 10.3934/jimo.2015.11.747. [12] Y. Hadas and M. Shnaiderman, Public-transit frequency setting using minimum-cost approach with stochastic demand and travel time, Transportation Research Part B: Methodological, 46 (2012), 1068-1084. doi: 10.1016/j.trb.2012.02.010. [13] O. J. Ibarra-Rojas and Y. A. Rios-Solis, Synchronization of bus timetabling, Transportation Research Part B: Methodological, 46 (2012), 599-614. doi: 10.1016/j.trb.2012.01.006. [14] J. Jensen, O. Nielsen and C. Prato, Public transport optimisation emphasising passengers' travel behaviour, Thesis Technical University of DenmarkDanmarks Tekniske Universitet, 2015. [15] L. Linzhong, Y. Juhua, M. Haibo, L. Xiaojing and W. Fang, Exact algorithms for multi-criteria multi-modal shortest path with transfer delaying and arriving time-window in urban transit network, Applied Mathematical Modeling, 38 (2014), 2613-2629. doi: 10.1016/j.apm.2013.10.059. [16] S. H. Nasseri and E. Behmanesh, Linear programming with triangular fuzzy numbers--A case study in a finance and credit institute, Fuzzy Information and Engineering, 5 (2013), 295-315. doi: 10.1007/s12543-013-0151-3. [17] F. Perez, T. Gomez and R. Caballero, Un modelo difuso para la selección de carteras de proyectos con incertidumbre en los costes, Revista Electrónica de Comunicaciones y Trabajos de ASEPUMA, 13 (2012), 129-143. [18] F. Perez and T. Gomez, Multiobjective project portfolio selection with fuzzy constraints, Annals of Operation Research, 245 (2016), 7-29. doi: 10.1007/s10479-014-1556-z. [19] T. Rasmussen, M. Anderson, O. Nielsen and C. Prato, Timetable-based simulation method for choice set generation in large-scale public transport networks, EJTIR, 16 (2016), 467-489. [20] V. Sahinidis Nikolaos, Optimization under uncertainty: State-of-the-art and opportunities, Computers and Chemical Engineering, 28 (2004), 971-983. [21] Y. Shangyao, C. Chin-Jen and T. Ching-Hui, Inter-city bus routing and timetable setting under stochastic demands, Transportation research part A, 40 (2006), 572-586. [22] L. Sun, Z. Gao and Y. Wang, A Stackelberg game management model of the urban public transport, Journal of Industrial and Management Optimization, 8 (2012), 507-520. doi: 10.3934/jimo.2012.8.507. [23] W. Y. Szeto and W. Yongzhong, A simultaneous bus route design and frequency setting problem for Tin Shui Wai, Hong Kong, European Journal of Operational Research, 209 (2011), 141-155. doi: 10.1016/j.ejor.2010.08.020. [24] S. L. Tilahun and H. C. Ong, Bus timetabling as a fuzzy multiobjective optimization problem using preference based genetic algorithm, Promet -Traffic & Transportation, 24 (2012), 183-191. doi: 10.7307/ptt.v24i3.311. [25] I. Verbas, C. Frei, H. Mahmassani and R. Chan, Stretching resources: Sensitivity of optimal bus frequency allocation to stop-level demand elasticities, Public Transport, 7 (2015), 1-20. [26] I. Verbas and H. Mahmassani, Exploring trade-offs in frequency allocation in a transit network using bus route patterns: Methodology and application to large-scale urban systems, Transportation Research Part B: Methodological, 81 (2015), 577-595. [27] Y. Wang, X. Zhu and L. B. Wu, Integrated multimodal metropolitan transportation model, Procedia Social and Behavioral Sciences, 96 (2013), 2138-2146. doi: 10.1016/j.sbspro.2013.08.241. [28] J. Zhang, T. Arentze and H. Timmermans, A multimodal transport network model for advanced traveler information system, Journal of Ubiquitous System and Pervasive Networks, 4 (2012), 21-27. [29] W. Zhang and M. Reimann, A simple augmented e-constraint method for multi-objective mathematical integer programming problems, European Journal of Operations Research, 234 (2014), 15-24. doi: 10.1016/j.ejor.2013.09.001. [30] F. Zhao and Z. Xiaogang, Optimization of transit route network, vehicle headways and timetables for large-scale transit networks, European Journal of Operational Research, 186 (2008), 841-855. doi: 10.1016/j.ejor.2007.02.005. [31] Y. Zhu, B. Mao, L. Liu and M. Li, Timetable design for urban rail line with capacity constraints Discrete Dynamics in Nature and Society 2015 (2015), Art. ID 429219, 11 pp. doi: 10.1155/2015/429219.
Transport planning process [5]
Departure times[5]
Types of synchronization nodes [13]
Multiperiod Scheduling Urban Transportation Problem. $S_{h}$ is the scheduling horizon, $T^v$ are the time periods. In each $T^v$ demand is considered almost constant
Differences of how to represent departures
First departure
Consecutive departure
Last departure
Flowchart to determine the frequency [5]
Window time for synchronization
Correlation
Execution time effect
Cost vs. Synchronization (Instance 20)
Cost vs. Synchronization (Instance 24)
Cost behaviour in relation to instance parameters
Synchronizations behaviour in relation to instance parameters
Literature review
 Authors Frequency Transfer nodes Bunching nodes Cost Multimodal Uncertain Demand Multiperiod Integration Chen et al. x x x Chakroborty x x x Zhao & Zeng x x x Szeto & Wu x x Hadas & Shnaiderman x x x Baskaran & Krishnaiah x x x Tilahun & Ong x x Liu et al. x x Zhang et al. x x Wang et al. x x Eranki x Ibarra-Rojas et al. x x x Avila et al. x x x x x x x x
 Authors Frequency Transfer nodes Bunching nodes Cost Multimodal Uncertain Demand Multiperiod Integration Chen et al. x x x Chakroborty x x x Zhao & Zeng x x x Szeto & Wu x x Hadas & Shnaiderman x x x Baskaran & Krishnaiah x x x Tilahun & Ong x x Liu et al. x x Zhang et al. x x Wang et al. x x Eranki x Ibarra-Rojas et al. x x x Avila et al. x x x x x x x x
Characteristics of instances
 Parameter Low level High level Routes 8 20 Periods 2 12 Segments 10 150 Sync. nodes 2 12 Headways 5-10 5-20
 Parameter Low level High level Routes 8 20 Periods 2 12 Segments 10 150 Sync. nodes 2 12 Headways 5-10 5-20
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