Shortest Path Problems with Resource Constraints

  • Stefan Irnich
  • Guy Desaulniers


In most vehicle routing and crew scheduling applications solved by column generation, the subproblem corresponds to a shortest path problem with resource constraints (SPPRC) or one of its variants.

This chapter proposes a classification and a generic formulation for the SPPRCs, briefly discusses complex modeling issues involving resources, and presents the most commonly used SPPRC solution methods. First and foremost, it provides a comprehensive survey on the subject.


Resource Constraint Sink Node Constraint Programming Short Path Problem Feasible Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aho, A. and Corasick, M. (1975). Efficient string matching: An aid to bibliographic search. Journal of the ACM, 18(6):333–340.MathSciNetGoogle Scholar
  2. Ahuja, R., Magnanti, T., and Orlin, J.(1993). Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  3. Arunapuram, S., Mathur, K., and Solow, D.(2003). Vehicle routing and scheduling with full truck loads. Transportation Science, 37(2):170–182.CrossRefGoogle Scholar
  4. Barnhart, C, Johnson, E., Nemhauser, G., Savelsbergh, M., and Vance, P. (1998). Branch-and-price: Column generation for solving huge integer programs. Operations Research, 46(3):316–329.MathSciNetGoogle Scholar
  5. Beasley, J. and Christofides, N. (1989). An algorithm for the resource constrained shortest path problem. Networks, 19:379–394.MathSciNetGoogle Scholar
  6. Bentley, J. (1980). Multidimensional divide-and-conquer. Communications of the ACM, 23(4):214–229.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Borndörfer, R., Grötschel, M., and Löbel, A. (2001). Scheduling duties by adaptive column generation. Technischer Bericht (ZIB-Report) 01-02, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Berlin.Google Scholar
  8. Chabrier, A. (2002). Vehicle routing problem with elementary shortest path based column generation. Technical Report, ILOG, Madrid.Google Scholar
  9. Desaulniers, G., Desrosiers, J., Dumas, Y., Marc, S., Rioux, B., Solomon, M. M., and Soumis, F. (1997). Crew pairing at Air France. European Journal of Operational Research, 97:245–259.CrossRefGoogle Scholar
  10. Desaulniers, G., Desrosiers, J., Erdmann, A., Solomon, M., and Soumis, F. (2002a). VRP with pickup and delivery. In: The Vehicle Routing Problem (P. Toth and D. Vigo, D., eds.), Chapter 9, pp. 225–242. Siam, Philadelphia.Google Scholar
  11. Desaulniers, G., Desrosiers, J., Ioachim, I., Solomon, M., Soumis, F., and Villeneuve, D. (1998). A unified framework for deterministic time constrained vehicle routing and crew scheduling problems. In: Fleet Management and Logistics (T. Crainic and G. Laporte, eds.), Chapter 3, pp. 57–93. Kluwer Academic Publisher, Boston, Dordrecht, London.Google Scholar
  12. Desaulniers, G., Desrosiers, J., Lasry, A., and Solomon, M. M. (1999). Crew pairing for a regional carrier. In: Computer-Aided Transit Scheduling (N. Wilson, ed.), Lecture Notes in Computer Science, Volume 471, pp. 19–41. Springer, Berlin.Google Scholar
  13. Desaulniers, G., Langevin, A., Riopel, D., and Villeneuve, B. (2002b). Dispatching and conflict-free routing of automated guided vehicles: An exact approach. Les Cahiers du GERAD G-2002-31, HEC, Montréal, Canada. Forthcoming in: International Journal of Flexible Manufacturing Systems.Google Scholar
  14. Desaulniers, G. and Villeneuve, D. (2000). The shortest path problem with time windows and linear waiting costs. Transportation Science, 34(3):312–319.CrossRefGoogle Scholar
  15. Desrochers, M., Desrosiers, J., and Solomon, M. (1992). A new optimization algorithm for the vehicle routing problem with time windows. Operations Research, 40(2):342–354.MathSciNetGoogle Scholar
  16. Desrochers, M. and Soumis, F. (1988). A generalized permanent labelling algorithm for the shortest path problem with time windows. Information Systems and Operations Research, 26(3):191–212.Google Scholar
  17. Desrochers, M. and Soumis, F. (1989). A column generation approach to the urban transit crew scheduling problem. Transportation Science, 23(1):1–13.Google Scholar
  18. Desrochers, M. (1986). La Fabrication d’horaires de travail pour les conducteurs d’autobus par une méthode de génération de colonnes. Ph.D Thesis, Centre de recherche sur les Transports, Publication #470, Université de Montréal, Canada.Google Scholar
  19. Desrosiers, J., Dumas, Y., Solomon, M., and Soumis, F. (1995). Time constrained routing and scheduling. In: Handbooks in Operations Research and Management Science (M. Ball, T. Magnanti, C. Monma, and G. Nemhauser, eds.), Volume 8, Network Routing, Chapter 2, pp. 35–139. Elsevier, Amsterdam.Google Scholar
  20. Desrosiers, J., Pelletier, P., and Soumis, F. (1983). Plus court chemin avec contraintes d’horaires. RAIRO, 17:357–377.Google Scholar
  21. Desrosiers, J., Soumis, F., Desrochers, M., and Sauve, M. (1986). Methods for routing with time windows. European Journal of Operational Research, 23:236–245.MathSciNetCrossRefGoogle Scholar
  22. Desrosiers, J., Soumis, F., and Desrochers, M. (1984). Routing with time windows by column generation. Networks, 14:545–565.Google Scholar
  23. de Silva, A. (2001). Combining constraint programming and linear programming on an example of bus driver scheduling. Annals of Operations Research, 108:277–291.zbMATHMathSciNetCrossRefGoogle Scholar
  24. Dror, M. (1994). Note on the complexity of the shortest path models for column generation in VRPTW. Operations Research, 42(5):977–978.zbMATHGoogle Scholar
  25. Dumas, Y., Desrosiers, J., and Soumis, F. (1991). The pick-up and delivery problem with time windows. European Journal of Operational Research, 54:7–22.CrossRefGoogle Scholar
  26. Dumas, Y., Soumis, F., and Desrosiers, J. (1990). Optimizing the schedule for a fixed vehicle path with convex inconvenience costs. Transportation Science, 24(2):145–152.MathSciNetGoogle Scholar
  27. Fahle, T., Junker, U., Karisch, S., Kohl, N., Sellmann, M., and Vaaben, B. (2002). Constraint programming based column generation for crew assignment. Journal of Heuristics, 8(1):59–81.CrossRefGoogle Scholar
  28. Feillet, D., Dejax, P., Gendreau, M., and Gueguen, C. (2004). An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems. Networks, 44(3):216–229.MathSciNetCrossRefGoogle Scholar
  29. Gamache, M., Soumis, F., and Marquis, G. (1999). A column generation approach for large-scale aircrew rostering problems. Operations Research, 47(2):247–263.Google Scholar
  30. Gamache, M., Soumis, F., Villeneuve, D., Desrosiers, J., and Gélinas, E. (1998). The preferential bidding system at Air Canada. Transportation Science, 32(3):246–255.Google Scholar
  31. Houck, D., Picard, J., Queyranne, M., and Vemuganti, R. (1980). The travelling salesman problem as a constrained shortest path problem: Theory and computational experience. Opsearch, 17:93–109.MathSciNetGoogle Scholar
  32. Ioachim, I., Gélinas, S., Desrosiers, J., and Soumis, F. (1998). A dynamic programming algorithm for the shortest path problem with time windows and linear node costs. Networks, 31:193–204.MathSciNetCrossRefGoogle Scholar
  33. Irnich, S. and Villeneuve, D. (2003). The shortest path problem with resource constraints and κ-cycle elimination for κ 3. Les Cahiers du GERAD G-2003-55, HEC, Montréal, Canada.Google Scholar
  34. Kohl, N. (1995). Exact methods for time constrained routing and related scheduling problems. Ph.D Thesis, Institute of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark.Google Scholar
  35. Kolen, A., Rinnooy-Kan, A., and Trienekens, H. (1987). Vehicle routing with time windows. Operations Research, 35(2):266–274.MathSciNetGoogle Scholar
  36. Kung, H., Luccio, F., and Preparata, F. (1975). On finding maxima of a set of vectors. Journal of the ACM, 22(4):469–476.MathSciNetCrossRefGoogle Scholar
  37. Larsen, J. (1999). Parallelization of the gehicle routing problem with time windows. Ph.D Thesis, Department of Mathematical Modelling, Technical Report, University of Denmark.Google Scholar
  38. Min, H. (1989). The multiple vehicle routing problem with simultaneous delivery and pick-up points. Transportation Research, 23:377–386.CrossRefGoogle Scholar
  39. Nemhauser, G. and Wolsey, L. (1988). Integer and Combinatorial Optimization. Wiley, New York.Google Scholar
  40. Powell, W. and Chen, Z. (1998). A generalized threshold algorithm for the shortest path problem with time windows. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science (P. Pardalos and D. Du, eds.), pp. 303–318. American Mathematical Society.Google Scholar
  41. Rousseau, L.-M., Focacci, F., Gendreau, M., and Pesant, G. (2003). Solving VRPTWs with constraint programming based column generation. Publication CRT-2003-10, Center for Research on Transportation, Université de Montréal, Canada.Google Scholar
  42. Ryan, D. and Foster, B. (1981). An integer programming approach to scheduling. In: Computer Scheduling of Public Transport Urban Passenger Vehicle and Crew Scheduling (A. Wren, ed.), pp. 269–280. North-Holland, Amsterdam.Google Scholar
  43. Savelsbergh, M. and Sol, M. (1998). Drive: Dynamic routing of independent vehicles. Operations Research, 46(4):474–490.Google Scholar
  44. Solomon, M. (1987). Algorithms for the vehicle routing and scheduling problem with time window constraints. Operations Research, 35(2):254–265.zbMATHMathSciNetCrossRefGoogle Scholar
  45. Vance, P., Barnhart, C, Johnson, E., and Nemhauser, G. (1997). Airline crew scheduling: A new formulation and decomposition algorithm. Operations Research, 45(2):188–200.Google Scholar
  46. Villeneuve, D. and Desaulniers, G. (2000). The shortest path problem with forbidden paths. Les Cahiers du GERAD G-2000-41, HEC, Montréal, Canada. Forthcoming in: European Journal of Operational Research.Google Scholar
  47. Xu, H., Chen, Z., Rajagopal, S., and Arunapuram, S. (2003). Solving a practical pickup and delivery problem. Transportation Science, 37(3):347–364.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Stefan Irnich
    • 1
  • Guy Desaulniers
    • 2
  1. 1.RWTH Aachen UniversityGermany
  2. 2.École Polytechnique and GERADCanada

Personalised recommendations