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Computación y Sistemas

Print version ISSN 1405-5546

Comp. y Sist. vol.19 n.1 México Jan./Mar. 2015 



A Scatter Search Algorithm for Solving a Bilevel Optimization Model for Determining Highway Tolls


José Luis González Velarde1, José-Fernando Camacho-Vallejo2 and Gabriel Pinto Serrano1


1 Tecnológico de Monterrey, México.,

2 Universidad Autónoma de Nuevo León, Facultad de Ciencias Físico-Matemáticas, México.

Corresponding author is José Fernando Camacho Vallejo.


Article received on 27/11/2013.
Accepted on 08/11/2014.



The problem of determining optimal tolls established on a subset of arcs in a multicommodity capacitated transportation network is presented. The problem is formulated as a bilevel optimization problem where the upper level consists of an administrator who establishes tolls in some arcs of a network, while the lower level is represented by a group of users who travel along the shortest paths with respect to the travel cost. The objective is not only to increase the tolls, but also to maintain an optimal flow on the arcs of the network in order to maximize the leader's profit. If the leader sets very high toll values, the followers will be discouraged from using the tolled arcs, so the profit obtained from that decision is not going to be convenient for the leader. A methodology to solve this problem using optimization software at the lower level and the metaheuristic Scatter Search at the upper level is proposed.

Keywords: Bilevel programming, scatter search, toll optimization problem.





This research has been supported by Tecnológico de Monterrey-Research Group in Industrial Engineering and Numerical Methods 0822B01006, the Mexican National Council for Science and Technology (CONACyT) through grant SEP-CONACyT CB-2011-01-166397 and the Secretary of Public Education (SEP) within the Consolidation of the Academic Groups Program with the project PROMEP/103.5/12/4953 and the UANL within the PAICYT support for the project CE960-11. The authors also gratefully acknowledge the comments and suggestions of the reviewers.



1. Button, K. (2004). Road Pricing. Fairfax: Center for Transportation Policy Operations and Logistics.         [ Links ]

2. Mong-Sim, K. & Hong-Sun, W. (2003). Ant colony optimization for routing and load-balancing: survey and new directions. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, Vol. 33, No. 5, pp. 560-572.         [ Links ]

3. Patriksson, M. (1994). The Traffic Assignment Problem: Models and Methods. VSP International Science Publishers.         [ Links ]

4. Bergendor, P., Hearn, D., & Ramana, M. (1997). Congestion in Toll Pricing of Traffic Networks. Gainesville, FL, Center for Applied Optimization.         [ Links ]

5. Cropper, M. & Oates, W. (1992). Environmental economics: a survey. Journal of Economic Literature, Vol. 30, No. 2, pp. 675-740.         [ Links ]

6. Button, K. (2010). Transport Economics. MPG Books Group, UK.         [ Links ]

7. Labbé, M., Marcotte, P., & Savard, G. (1998). A bilevel model of taxation and its application to optimal highway pricing. Management Science, Vol. 44, pp. 1608-1622.         [ Links ]

8. Roch, S., Savard, G., & Marcotte, P. (2005). An approximation algorithm for Stackelberg network pricing. Networks, Vol. 46, No. 1, pp. 57-67.         [ Links ]

9. Didi-Biha, M., Marcotte, P., & Savard, G. (2006). Path-based formulations of a bilevel toll setting problem. Optimization with Multivalued Mappings Theory: Theory, Applications and Algorithms, S. Dempe, V. Kalashnikov (eds.), Springer, Vol. 2, pp. 29-50.         [ Links ]

10. Kalashnikov, V., Camacho, F., Askin R., & Kalashnikova, N. (2010). Comparing various algorithms performance: application to bilevel toll setting problem. International Journal of Innovating Computing, Information and Control (IJICIC), Vol. 6, No. 8, pp. 3529-3549.         [ Links ]

11. Brotcorne, L., Cirinei, F., Marcotte, P., & Savard, G. (2012). A tabu search algorithm for the network pricing problem. Computers & Operations Research, Vol. 39, pp. 2603-2611.         [ Links ]

12. Dimitriou, L., Tsekeris, T., & Stathopoulos, A. (2008). Genetic computation of road network design and pricing Stackelberg games with multi-class users. Applications of Evolutionary Computing, Springer, M. Giacobini et al. (Eds.), Vol. 4974, pp. 669-678.         [ Links ]

13. Laguna, M. & Martí, R. (2003). Scatter Search: Methodology and Implementations. Kluwer Academic Publishers.         [ Links ]

14. Zadeh, N. (1973). A bad network problem for the simplex method and other minimum cost flow algorithms. Mathematical Programming 5.         [ Links ]

15. Edmonds, K. (1972). Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM, Vol 19, No. 2, pp. 248-264.         [ Links ]

16. Anandalingam, G. & Friesz, T. (1992). Hierarchical optimization. Annals of Operations Research, Vol. 34, No. 1, pp. 1-11.         [ Links ]

17. Bard, J.F. (1998). Practical Bilevel Optimization. Algorithms and Applications. Kluwer Academic Publishers.         [ Links ]

18. Chandler, W. & Norton, R. (1977). Multi-Level Programming and development policy. Working Paper No. 258. World Bank.         [ Links ]

19. Dempe, S. (2003). Bilevel Programming: A survey. Preprint Technical University Bergakademie, Freiberg, Germany.         [ Links ]

20. Colson, B., Marcotte, P., & Savard, G. (2007). An overview of bilevel optimization. Annals of Operations Research, Vol. 153, pp. 235-256.         [ Links ]

21. Calvete, H.I. & Galé, C. (2010). A Multiobjective Bilevel Program for Production-Distribution Planning in a Supply Chain. Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems, Lecture Notes in Economics and Mathematical Systems, Vol. 634. Springer-Verlag, pp. 155-165.         [ Links ]

22. Candler, W. (1988). A linear bilevel programming algorithm: A comment. Computers & Operations Research, Vol.15, pp. 297-298.         [ Links ]

23. Clarke, P. & Westerberg, A. (1988). A note on the optimality conditions for the bilevel programming problem. Naval Research Logistics Quarterly, Vol.35, pp. 413-418.         [ Links ]

24. Haurie, A. Savard, G., & White, D. (1990). A note on: an efficient point algorithm for a linear two-stage optimization problem. Operations Research, Vol.38, pp. 553-555.         [ Links ]

25. Marcotte, P. & Savard, G. (1991). A note on the Pareto optimality of solutions to the linear bilevel programming problem. Computers & Operations Research, Vol.18, pp. 355-359.         [ Links ]

26. Glover, F. (1998). A template for scatter search and path relinking. Artificial Evolution: Third European Conference, Lecture Notes in Computer Science, Vol. 1363, pp. 13-54. Springer, Heidelberg, Germany.         [ Links ]

27. Martí, R. & Laguna, M. (2003). Scatter Search: Basic Design and Advanced Strategies. Inteligencia Artificial, Revista Iberoamericana de Inteligencia Artificial, Vol. 19, pp. 123-130.         [ Links ]

28. Moscato, P. (2000). Memetic Algorithms. Handbook of Applied Optimization, P. M. Pardalos and M. G. Resende (Eds.), Oxford University Press, USA.         [ Links ]

29. Nelder, J.A. & Mead, R. (1965). A simplex algorithm for function minimization. The Computer Journal, Vol. 7, No. 4, pp. 308-313.         [ Links ]

30. González-Velarde, J.L. & Martí, R. (2008). Adaptive memory programing for the robust capacitated international sourcing problem. Computers & Operations Research, Vol. 35, pp. 797-806.         [ Links ]

31. González-Velarde, J.L., Alvarez, A.M., & De Alba, K. (2005). Grasp Embedded Scatter Search for the Multicommodity Capacitated Network Design Problem. Journal of Heuristics, Vol. 11, No. 3, pp. 233-257.         [ Links ]

32. Camacho-Vallejo, J.F., Cordero, A.E., & González-Ramírez, R.G. (2014). Solving the Bilevel Facility Location Problem under Preferences by a Stackelberg Evolutionary algorithm. Mathematical Problems in Engineering, Vol. 2014, 14 p.         [ Links ]

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