, Volume 39, Issue 2, pp 449–464 | Cite as

Optimizing bus-size and headway in transit networks

  • Luigi dell’OlioEmail author
  • Angel Ibeas
  • Francisco Ruisánchez


Optimization models for calculating the best size for passenger carrying vehicles in urban areas were popular during the 1980s. These studies were abandoned in the ‘90s concluding that it was more efficient to use smaller buses at higher frequencies. This article returns to this controversial question, starting from the point of view that any calculation of bus size can only be made after considering the demand for each of the routes on the system. Therefore, an optimization model for sizing the buses and setting frequencies on each route in the system is proposed in accordance with the premises detailed below. The proposed model is a bi-level optimization model with constraints on bus capacity. The model allows buses of different sizes to be assigned to public transport routes optimizing the headways on each route in accordance with observed levels of demand. At the upper level the model considers the optimization of the system’s social and operating costs, these are understood to be the sum of the user’s and operator’s costs. At the lower level there is an assignment model for public transport with constraints on vehicle capacity which balances the flows for bus sizes and headways at each iteration. By graphically representing the results of the model applied to a real case, a series of useful conclusions are reached for the management and planning of a fleet of public transport vehicles.


Bi-level optimization model Bus size Headway Vehicle capacity 



The authors would like to thank the Ministry of Science and Education, and the Ministry of Development of the Government of Spain, as this work is fruit of work done in the following research projects: TRA2006-14663, PT-2006-027-06IAPP, INTERCOR.


  1. Abdulaal, M., y LeBlanc, L.J.: Continuous equilibrium network design models. Transp. Res. 13B, 19–32 (1979)CrossRefGoogle Scholar
  2. Alonso, B., Moura, J.L., dell’Olio, L., Ibeas, A.: Bus stop location under different levels of network congestion and elastic demand. TRANSPORT. (2011, In Press)Google Scholar
  3. Chiriqui, C.: Reseaux de transport en commun: Les prolemesdecheminementetd’acces, p. 11. Center of Transport Research, University of Montreal, Publication, Montreal (1974)Google Scholar
  4. Chriqui, C., y Robilland, P.: Common bus lines. Transpor. Sci. 9, 115–121 (1975)CrossRefGoogle Scholar
  5. De Cea, J., y Fernandez, J.E.: Transit assignment for congested public transport systems: an equilibrium model. Transp. Sci. 27, 133–147 (1993)CrossRefGoogle Scholar
  6. dell’Olio, L., Ibeas, A., y Moura, J.L.: A bi-level mathematical programming model to locate bus stops and optimize frequencies. Transportation Research Record, Journal of Transportation Research Board, 1971, pp. 23–31, (2006)Google Scholar
  7. dell’Olio,L., Ibeas, A., y Moura, J.L.: Headway and vehicle size optimization in urban bus transit networks. Third international symposium of transport simulation (ISTS08), Monash University, Monash, 2008Google Scholar
  8. Florian, M.: A traffic equilibrium model of travel by car and public transit modes. Transp. Sci. 2, 166–179 (1977)CrossRefGoogle Scholar
  9. Glaister, S.: Bus deregulation, competition and vehicle size. J. Transp. Econ. Policy 20 2, 217–244 (1986)Google Scholar
  10. Hooke, R., y Jeeves, T.A.: Direct search solution of numerical and statistical problems. J. Assoc. Comput. Mach. 1961, 212–229 (1961)CrossRefGoogle Scholar
  11. Ibeas, A., Moura, J.L., dell’Olio, L., de yOrtuzar, J.D.: Costing school transport in Spain. Transp. Plan. Technol. 29(6), 483–501 (2006)CrossRefGoogle Scholar
  12. Ibeas, A., dell’Olio, L., Alonso, B., Sáinz, O.: Optimizing bus stop spacing in urban areas. Transp. Res. Part E 46(3), 446–458 (2010)CrossRefGoogle Scholar
  13. Jansson, J.O.: A simple bus line model for optimisation of service frequency and bus size. J. Transp. Econ. Policy14 1, 53–80 (1980)Google Scholar
  14. Jara-Dίaz, S.R., Gschwender, A.: Towards a general microeconomic model for the operation of public transport. Transp. Rev. 23(4), 453–469 (2003a)CrossRefGoogle Scholar
  15. Jara-Dίaz, S.R., Gschwender, A.: From the single line model to the spatial structure of transit services: Corridors or direct? J. Transp. Econ. Policy 37(2), 261–277 (2003b)Google Scholar
  16. Jara-Dίaz, S.R., Gschwender, A.: The effect of financial constraints on the optimal design of public transport services. Transp. 36(1), 65–75 (2009)CrossRefGoogle Scholar
  17. Mohring, H.: Optimization and scale economies in urban bus transportation. Am. Econ. Rev. 62, 591–604 (1972)Google Scholar
  18. Mohring, H.: Transportation economics. Ballinger Publisher Comp., Cambridge (1976)Google Scholar
  19. Nakamura, S.: Numerical analysis and graphic visualization whit matlab. 1st edn. Prentice Hall PTR, Upper Saddle River, New Jersey (1997)Google Scholar
  20. Nguyen, S., y Dupuis, C.: An efficient method for computing traffic equilibrium in networks with asymmetric transportation costs. Transp. Sci. 18, 185–202 (1984)CrossRefGoogle Scholar
  21. Oldfield, R.H., y Bly, P.H.: An analytic investigation of optimal bus size. Transp. Res. Part B Methodol. 22(5), 319–337 (1988)CrossRefGoogle Scholar
  22. SECTU: Strategic urban transport study for Santiago: final report. Chile. (1989)Google Scholar
  23. Vickrey, W.: Some implications of marginal cost pricing for public utilities. Am. Econ. Rev. 45, 605–620 (1955)Google Scholar
  24. Vijayakumar, S.: Optimal vehicle size for road-based urban public transport in developing countries. Transp. Rev. 6 2, 193–212 (1986)Google Scholar
  25. Walters, A.A.: Externalties in urban buses. J. Urban Econ. 11, 60–72 (1982)CrossRefGoogle Scholar
  26. Webster F.V. and Oldfield, R.H.: A theoretical study of bus and car travel in Central London. In: Transport and Road Research Laboratory Report LR451, TRRL, Crowthorne, United Kingdom (1972)Google Scholar
  27. Webster, F.V.: A theoretical estimate of the effect of London car commuters transferring to bus travel. In: Road Research Laboratory Report LR165, TRRL, Crowthorne, United Kingdom (1968)Google Scholar
  28. Wong, S.C., Yang, H.: Reserve capacity of a signal-controlled road network. Transp. Res. B n 30l, 397–402 (1997)CrossRefGoogle Scholar
  29. Yang, H.: Sensitivity analysis for the elastic-demand network equilibrium problem with application. Transp. Res. B n. 31, 55–70 (1997)CrossRefGoogle Scholar
  30. Yang, H., Bell, M.: Traffic restraint, road pricing and network equilibrium. Transp. Res. B n. 31, 303–314 (1997)CrossRefGoogle Scholar
  31. Yang, H., Michael, G.H.B.: Models and algorithm for the road network design: a review and some new development. Transp. Rev. n 18, 257–278 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Luigi dell’Olio
    • 1
    Email author
  • Angel Ibeas
    • 1
  • Francisco Ruisánchez
    • 1
  1. 1.Department of TransportationUniversity of CantabriaCantabriaSpain

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