Abstract
Network routing problems are often modeled with the assumption that the network structure is deterministic, though they are often subject to uncertainty in many real-life scenarios. In this paper, we study the traveling salesman and the shortest path problems with uncertain topologies modeled by arc failures. We present the formulations that incorporate chance constraints to ensure reliability of the selected route considering all arc failure scenarios. Due to the computational complexity and large scales of these stochastic network optimization problems, we consider two cutting plane methods and a Benders decomposition algorithm to respectively solve them. We also consider to solve the reformulations of the problems obtained by taking the logarithm transformation of the chance constraints. Numerical experiments are performed to obtain results for comparisons among these proposed methods.
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References
Ahmed, S., & Papageorgiou, D. (2013). Probabilistic set covering with correlations. Operations Research, 61(2), 438–452.
Ahuja, R., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: Theory, algorithms and applications. Englewood Cliffs: Printice Hall.
Bard, J., & Miller, J. (1989). Probabilistic shortest path problems with budgetary constraints. Computers & Operations Research, 16(2), 145–159.
Beraldi, P., & Bruni, M. (2010). An exact approach for solving integer problems under probabilistic constraints with random technology matrix. Annals of Operations Research, 177, 127–137.
Beraldi, P., & Ruszczyński, A. (2002). The probabilistic set-covering problem. Operations Research, 50(6), 956–967.
Berman, O., & Simchi-Levi, D. (1989). The traveling salesman location problem on stochastic networks. Transportation Science, 23, 54–57.
Boginski, V., Commander, C., & Turko, T. (2009). Polynomial-time identication of robust network flows under uncertain arc failures. Optimization Letters, 3, 461–473.
Desrochers, M., & Laporte, G. (1991). Improvements and extensions to the miller-tucker-zemlin subtour elimination constraints. Operations Research Letters, 10(1), 27–36.
Fan, Y. Y., Kalaba, R. E., & Moore, J. E. (2005). Shortest paths in stochastic networks with correlated link costs. Computers and Mathematics with Applications, 49, 1549–1564.
Fischetti, M., & Monaci, M. (2012). Cutting plane versus compact formulations for uncertain (integer) linear programs. Mathematical Programming Computation, 4(3), 239–273.
Jaillet, P. (1988). A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Operations Research, 36(6), 929–936.
Laporte, G. (2010). A concise guide to the traveling salesman problem. Journal of the Operational Research Society, 61, 35–40.
Laporte, G., & Louveaux, F. V. (1993). The integer l-shaped method for stochastic integer programs with complete recourse. Operations Research Letters, 13(3), 133–142.
Laporte, G., Louveaux, F. V., & Mercure, H. (1994). A priori optimization of the probabilistic traveling salesman problem. Operations Research, 42(3), 543–549.
Mirchandani, P. B. (1976). Shortest distance and reliability of probabilistic networks. Computers & Operations Research, 3(4), 347–355.
Nie, Y., & Wu, X. (2009). Shortest path problem considering on-time arrival probability. Transportation Research Part B, 43, 597–613.
Polychronopoulos, G. H., & Tsitiklis, J. N. (1996). Stochastic shortest path problems with recourse. Networks, 27, 133–143.
Sigal, C. E., Pritsker, A. A. B., & Solberg, J. J. (1980). The stochastic shortest route problem. Operations Research, 28(5), 1122–1129.
Verweij, B., Ahmed, S., Kleywegt, A., Nemhauser, G., & Shapiro, A. (2003). The sample average approximation method applied to stochastic routing problems: A computational study. Computational Optimization and Applications, 24(2), 289–333.
Zheng, Q. P., Shen, S., & Shi, Y. (2015). Loss-constrained minimum cost flow under arc failure uncertainty with applications in risk-aware kidney exchange. IIE Transactions, 47(9), 961–977.
Acknowledgments
This work is in part supported by the AFRL Mathematical Modeling and Optimization Institute, and National Science Foundation through Grant CMMI-1355939. The authors would also like to thank the reviewers and Editors for their helpful suggestions and comments.
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Huang, Z., Zheng, Q.P., Pasiliao, E.L. et al. Exact algorithms on reliable routing problems under uncertain topology using aggregation techniques for exponentially many scenarios. Ann Oper Res 249, 141–162 (2017). https://doi.org/10.1007/s10479-016-2244-y
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DOI: https://doi.org/10.1007/s10479-016-2244-y