Annals of Operations Research

, Volume 252, Issue 2, pp 383–406 | Cite as

Optimal duty rostering for toll enforcement inspectors

  • Ralf Borndörfer
  • Guillaume Sagnol
  • Thomas Schlechte
  • Elmar SwaratEmail author
S.I.: PATAT 2014


We present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, we developed a personalized crew rostering model, to schedule the crews of the tours. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The main focus of this paper extends our previous publications on how different requirements for the rostering can be modeled in detail. The second focus is on a bi-criteria analysis of the planning problem to find the balance between the control quality and the roster acceptance. Finally, computational results on real-world instances show the practicability of our method and how different input parameters influence the problem complexity.


Vehicle routing Crew rostering Integer Programming  Bi-criteria optimization 

Mathematics Subject Classification

90B20 90C06 



We thank Doris Ludwig-Schreiber and Christian Hoffmann from the German Federal office for Goods Transport (BAG) for initiating the project on optimal toll enforcement with the Zuse Institute Berlin. Furthermore, we thank the technical project managers Eduardo Pinto and Thomas Dankert for organising the project, and in addition, for giving a lot of technical support according to the installation and operation of our tool at the BAG. Our sincere thanks go also to Daniel Schneider, Ralf Haas and Uta Sperling from the planning department of the BAG, who provided us with many test instances, gave us excellent feedback, and helped us to understand the manifold requirements from real operations. Furthermore, we thank Hans-Stefan Madlung and his colleagues from IVU Traffic Technologies AG for the joint development of a small and flexible interface to exchange the data between TC-OPT and the commercial planning tool “IVU.Plan”. In addition, we want to thank two anonymous referees for improving this paper by their valuable comments.


  1. Archetti, C., Bianchessi, N., & Speranza, M. (2013). Optimal solutions for routing problems with profits. Discrete Applied Mathematics 161(4–5):547 – 557. In Seventh International Conference on Graphs and Optimization 2010. doi: 10.1016/j.dam.2011.12.021.
  2. Archetti, C., Speranza, M., & Vigo, D. (2014). Vehicle routing problems with profits. In P. Toth & D. Vigo (Eds.), Vehicle routing: Problems, methods, applications, MOS-SIAM series on optimization. Philadelphia: MOS and SIAM.Google Scholar
  3. Balakrishnan, N., & Wong, R. T. (1990). A network model for the rotating workforce scheduling problem. Networks, 20(1), 25–42. doi: 10.1002/net.3230200103.CrossRefGoogle Scholar
  4. Borndörfer , R., Omont, B., Sagnol, G., & Swarat, E. (2012a). A Stackelberg game to optimize the distribution of controls in transportation networks. In V. Krishnamurthy, Q. Zhao, M. Huang & Y. Wen (Eds.) Game theory for networks (Vol. 105, pp. 24–35). Berlin, Heidelberg: Springer, Lecture notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering. doi: 10.1007/978-3-642-35582-0_17.
  5. Borndörfer, R., Sagnol, G., & Swarat, E. (2012b). A case study on optimizing toll enforcements on motorways. In S. Ravizza & P. Holborn (Eds.) 3rd Student Conference on Operational Research, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, OpenAccess Series in Informatics (OASIcs), vol. 22, (pp. 1–10). doi: 10.4230/OASIcs.SCOR.2012.1,
  6. Borndörfer, R., Sagnol, G., & Swarat, E. (2012c). An IP approach to toll Enforcement optimization on German motorways. In D. Klatte & H. J. Lüthi, K. Schmedders (Eds.) Operations Research Proceedings 2011, Springer Berlin Heidelberg, Operations Research Proceedings, (pp. 317–322). doi: 10.1007/978-3-642-29210-1_51.
  7. Borndörfer, R., Buwaya, J., Sagnol, G., & Swarat, E. (2013). Optimizing toll enforcement in transportation networks: A game-theoretic approach. Electronic Notes in Discrete Mathematics, 41, 253–260. doi: 10.1016/j.endm.2013.05.100.CrossRefGoogle Scholar
  8. Burke, E. K., De Causmaecker, P., Berghe, G. V., & Van Landeghem, H. (2004). The state of the art of nurse rostering. Journal of Scheduling, 7(6), 441–499. doi: 10.1023/B:JOSH.0000046076.75950.0b.CrossRefGoogle Scholar
  9. Cappanera, P., & Gallo, G. (2004). A multicommodity flow approach to the crew rostering problem. Operations Research, 52(4), 583–596.CrossRefGoogle Scholar
  10. Castillo-Salazar, J. A., Landa-Silva, D., & Qu, R. (2014). Workforce scheduling and routing problems: literature survey and computational study. Annals of Operations Research. doi: 10.1007/s10479-014-1687-2.
  11. Ehrgott, M. (2005). Multicriteria optimization. Berlin: Springer.Google Scholar
  12. Ernst, A., Jiang, H., Krishnamoorthy, M., Nott, H., & Sier, D. (2001). Rail crew scheduling and rostering optimization algorithms. In S. Vo & J. Daduna (Eds.) Computer-aided scheduling of public transport, lecture notes in economics and mathematical systems (Vol. 505, pp. 53–71). Berlin: Springer. doi: 10.1007/978-3-642-56423-9_4
  13. Ernst, A., Jiang, H., Krishnamoorthy, M., Owens, B., & Sier, D. (2004a). An annotated bibliography of personnel scheduling and rostering. Annals of Operations Research, 127(1–4), 21–144. doi: 10.1023/B:ANOR.0000019087.46656.e2.CrossRefGoogle Scholar
  14. Ernst, A., Jiang, H., Krishnamoorthy, M., & Sier, D. (2004b). Staff scheduling and rostering: A review of applications, methods and models. European Journal of Operational Research, 153(1), 3–27. doi: 10.1016/S0377-2217(03)00095-X. (timetabling and Rostering).CrossRefGoogle Scholar
  15. Feillet, D., Dejax, P., & Gendreau, M. (2005). Traveling salesman problems with profits. Transportation Science 39(2):188–205. doi: 10.1287/trsc.1030.0079,
  16. Jiann-Sheng, W., & Tze-Chiang, L. (2010). Highway patrol officer scheduling using an optimization-based scheduling model. In Advanced Computer Theory and Engineering (ICACTE), 2010 3rd International Conference on (Vol. 2, pp. V2-552–V2-557). doi: 10.1109/ICACTE.2010.5579460
  17. Knauth, P., & Hornberger, S. (2003). Preventive and compensatory measures for shift workers. Occupational Medicine, 53(2), 109–116. doi: 10.1093/occmed/kqg049.CrossRefGoogle Scholar
  18. Kohl, N., & Karisch, S. (2004). Airline crew rostering: Problem types, modeling, and optimization. Annals of Operations Research, 127(1–4), 223–257.CrossRefGoogle Scholar
  19. Lau, H., Yuan, Z., & Gunawan, A. (2014). Patrol scheduling in urban rail network. Annals of Operations Research. doi: 10.1007/s10479-014-1648-9.
  20. Pita, J., Jain, M., Marecki, J., Ordóñez, F., Portway, C., Tambe, M., Western, C., Paruchuri, P., & Kraus, S. (2008). Deployed ARMOR protection: The application of a game theoretic model for security at the Los Angeles International Airport. In Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems: industrial track, International Foundation for Autonomous Agents and Multiagent Systems (pp. 125–132).Google Scholar
  21. Ralphs, T. K., Saltzman, M. J., & Wiecek, M. M. (2006). An improved algorithm for solving biobjective integer programs. Annals of Operations Research, 147(1), 43–70. doi: 10.1007/s10479-006-0058-z.CrossRefGoogle Scholar
  22. Santos, H., Toffolo, T., Gomes, R., & Ribas, S. (2014). Integer Programming Techniques for the Nurse Rostering Problem. Annals of Operations Research. doi: 10.1007/s10479-014-1594-6.
  23. Thorlacius, P., & Clausen, J. (2010). Scheduling of inspectors for ticket spot checking in urban rail transportation. Trafikdage på Aalborg UniversitetGoogle Scholar
  24. Toth, P., & Vigo, D. (2002). The vehicle routing problem. Philadelphia, USA: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
  25. Tsai, J., Kiekintveld, C., Ordonez, F., Tambe, M., & Rathi, S. (2009). IRIS-a tool for strategic security allocation in transportation networks. In Proceedings of the 8th international joint conference on Autonomous agents and multiagent systems: industrial track (pp. 37–44).Google Scholar
  26. Vansteenwegen, P., Souffriau, W., & Oudheusden, D. V. (2011). The orienteering problem: A survey. European Journal of Operational Research, 209(1), 1–10. doi: 10.1016/j.ejor.2010.03.045.CrossRefGoogle Scholar
  27. Weider, S. (2007). Integration of vehicle and duty scheduling in public transport. PhD thesis, Technische Universität Berlin, Berlin, DeutschlandGoogle Scholar
  28. Yin, Z., Jiang, A. X., Tambe, M., Kiekintveld, C., Leyton-Brown, K., Sandholm, T., et al. (2012). TRUSTS: Scheduling randomized patrols for fare inspection in transit systems using game theory. AI Magazine, 33(4), 59–72.Google Scholar
  29. Zhu, C., Hu, J., Wang, F., Xu, Y., & Cao, R. (2012). On the tour planning problem. Annals of Operations Research, 192(1), 67–86. doi: 10.1007/s10479-010-0763-5.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Ralf Borndörfer
    • 1
  • Guillaume Sagnol
    • 1
  • Thomas Schlechte
    • 1
  • Elmar Swarat
    • 1
    Email author
  1. 1.Zuse Institute BerlinBerlinGermany

Personalised recommendations