Abstract
We examine a routing problem in which network arcs fail according to independent failure probabilities. The reliable h-path routing problem seeks to find a minimum-cost set of h ≥ 2 arc-independent paths from a common origin to a common destination, such that the probability that at least one path remains operational is sufficiently large. For the formulation in which variables are used to represent the amount of flow on each arc, the reliability constraint induces a nonconvex feasible region, even when the integer variable restrictions are relaxed. Prior arc-based models and algorithms tailored for the case in which h = 2 do not extend well to the general h-path problem. Thus, we propose two alternative integer programming formulations for the h-path problem in which the variables correspond to origin-destination paths. Accordingly, we develop two branch-and-price-and-cut algorithms for solving these new formulations, and provide computational results to demonstrate the efficiency of these algorithms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andreas, A.K.: Mathematical programming algorithms for robust routing and evacuation problems. PhD thesis. Department of Systems and Industrial Engineering, The University of Arizona, Tucson, Arizona (2006)
Andreas, A.K., Smith, J.C.: Mathematical programming algorithms for two-path routing problems with reliability considerations. Working Paper, Department of Systems and Industrial Engineering, The University of Arizona, Tucson, Arizona (2006)
Bard J.F. and Miller J.L. (1989). Probabilistic shortest path problems with budgetary constraints. Comput. Oper. Res. 16(2): 145–159
Barnhart C., Hane C.A., Johnson E.L. and Sigismondi G. (1995). A column generation and partitioning approach for multi-commodity flow problems. Telecommun. Syst. 3: 239–258
Barnhart C., Hane C.A. and Vance P.H. (2000). Using branch-and-price-and-cut to solve origin-destination integer multicommodity flow problems. Oper. Res. 48(2): 318–326
Chen Z.L. and Powell W.B. (1998). A generalized threshold algorithm for the shortest path problem with time windows. In: Pardalos, P.M. and Du, D.-Z. (eds) Network Design: Connectivity and Facilities Location, Discrete Mathematics and Theoretical Computer Science, pp 303–318. American Mathematical Society, Providence, RI
Desrochers M. and Soumis F. (1988). A generalized permanent labelling algorithm for the shortest path problem with time windows. INFOR 26(3): 191–212
Desrosiers J., Dumas Y., Solomon M.M. and Soumis F. (1995). Time constrained routing and scheduling. In: Ball, M.O., Magnanti, T.L., Monma, C.L., and Nemhauser, G.L. (eds) Network Routing, volume 8 of Handbooks in Operations Research and Management Science, pp 35–139. Elsevier, Amsterdam
Dijkstra E.W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik 1: 269–271
Dumitrescu I. and Boland N. (2001). Algorithms for the weight constrained shortest path problem. Int. Trans. Oper. Res. 8: 15–29
Dumitrescu I. and Boland N. (2003). Improved preprocessing, labeling and scaling algorithms for the weight-constrained shortest path problem. Networks 42(3): 135–153
Elimam A.A. and Kohler D. (1997). Case study: Two engineering applications of a constrained shortest-path model. Eur. J. Oper. Res. 103: 426–438
Falk J.E. and Soland R.M. (1969). An algorithm for separable nonconvex programming problems. Manage. Sci. 15: 550–569
Fortune S., Hopcroft J. and Wyllie J. (1980). The directed subgraph homeomorphism problem. Theor. Comp. Sci. 10: 111–121
Glover F., Glover R. and Klingman D. (1984). The threshold shortest path problem. Networks 14: 25–36
Lübbecke M.E. and Desrosiers J. (2005). Selected topics in column generation. Oper. Res. 53(6): 1007–1023
Sherali H.D. and Smith J.C. (2001). Improving discrete model representations via symmetry considerations. Manage. Sci. 47(10): 1396–1407
Sherali H.D. and Tuncbilek C.H. (1992). A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique. J. Global Optim. 2: 101–112
Suurballe J.W. (1974). Disjoint paths in a network. Networks 4: 125–145
Vanderbeck, F.: Decomposition and column generation for integer programs. PhD thesis, Université Catholique de Louvain, Belgium (1994)
Vanderbeck, F.: Branching in branch-and-price: a generic scheme. Working Paper, Applied Mathematics, University Bordeaux 1, F-33405 Talence Cedex, France (2006)
Vanderbeck F. and Wolsey L.A. (1996). An exact algorithm for IP column generation. Oper. Res. Lett. 19: 151–159
Wilhelm W.E. (2001). A technical review of column generation in integer programming. Optim. Eng. 2: 159–200
Zabarankin M., Uryasev S. and Pardalos P.M. (2002). Optimal risk path algorithms. In: Murphey, R. and Pardalos, P.M. (eds) Cooperative Control and Optimization, pp 273–303. Kluwer Academic Publishers, Boston
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Andreas, A.K., Smith, J.C. & Küçükyavuz, S. Branch-and-price-and-cut algorithms for solving the reliable h-paths problem. J Glob Optim 42, 443–466 (2008). https://doi.org/10.1007/s10898-007-9254-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-007-9254-x