# STOCHASTIC SHORTEST PATH PROBLEM WITH UNCERTAIN DELAYS

### Jianqiang Cheng, Stefanie Kosuch, Abdel Lisser

#### Abstract

This paper considers a stochastic version of the shortest path problem, the Stochastic Shortest Path Problem with Delay Excess Penalty on directed, acyclic graphs. In this model, the arc costs are deterministic, while each arc has a random delay, assumed normally distributed. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. In order to solve the model, a Stochastic Projected Gradient method within a branch-and-bound framework is proposed and numerical examples are given to illustrate its effectiveness. We also show that, within given assumptions, the Stochastic Shortest Path Problem with Delay Excess Penalty can be reduced to the classic shortest path problem.

#### References

- Ahuja, R. K., Magnanti, T. L., and Orlin, J. B. (1993). Network Flows: Theory, Algorithms and Applications. Prentice Hall, New Jersey.
- Beasley, J. E. (2010). Or-library. http://people.brunel.ac.uk/ ~mastjjb/jeb/info.html.
- Bellman, R. E. (1958). On a routing problem. Quarterly of Applied Mathematic, 16(87-90).
- Dijkstra, E. W. (1959). A note on two problems in connection with graphs. Numerische Mathematik, 1(269- 271).
- Ford, L. R. and Fulkerson, D. R. (1962). Flows in Networks. Princeton University Press, Princeton.
- Hutsona, K. R. and Shierb, D. R. (2009). Extended dominance and a stochastic shortest path problem. Computers and Operations Research, 36(584-596).
- Ji, X. (2005). Models and algorithm for stochastic shortest path problem. Applied Mathematics and Computation, 170(503-514).
- Kosuch, S. and Lisser, A. (2010). Upper bounds for the 0- 1 stochastic knapsack problem and a b&b algorithm. Annals of Operations Research, 176(77-93).
- Luenberger, D. G. and Ye, Y. (2008). Linear and Nonlinear Programming. Springer, New York, 3rd edition.
- Mirchandani, P. B. and Soroush, H. (1985). Optimal paths in probabilistic networks: a case with temporary preferences. Computers and Operations Research, 12(365-381).
- Murthy, I. and Sarkar, S. (1996). A relaxation-based pruning technique for a class of stochastic shortest path problems. Transportation Science, 30(220-236).
- Nikolova, E., Kelner, J., Brand, M., and Mitzenmacher, M. (2006). Stochastic shortest paths via quasi-convex maximization. In Proceedings of European Symposium of Algorithms, pages 552-563. Springer.
- Ohtsubo, Y. (2003). Minimization risk models in stochastic shortest path problems. Mathematical Methods of Operations Research, 57(79-88).
- Ohtsubo, Y. (2008). Stochastic shortest path problems with associative accumulative criteria. Applied Mathematics and Computation, 198(1)(198-208).
- Polychronopoulos, G. H. and Tsitsiklis, J. N. (1996). Stochastic shortest path problems with recourse. Networks, 27(133-143).
- Provan, J. S. (2003). A polynomial-time algorithm to find shortest paths with recourse. Networks, 41(115-125).
- Sahinidis, N. V. (2004). Optimization under uncertainty: state-of-the-art and opportunities. Computers and Chemical Engineering, 28(971-983).
- Verweij, B., Ahmed, S., Kleywegt, A. J., Nemhauser, G., and Shapiro, A. (2003). The sample average approximation method applied to stochastic routing problems: A computational study. Computational Optimization and Applications, 24(2-3)(289-333).

#### Paper Citation

#### in Harvard Style

Cheng J., Kosuch S. and Lisser A. (2012). **STOCHASTIC SHORTEST PATH PROBLEM WITH UNCERTAIN DELAYS** . In *Proceedings of the 1st International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-8425-97-3, pages 256-264. DOI: 10.5220/0003725102560264

#### in Bibtex Style

@conference{icores12,

author={Jianqiang Cheng and Stefanie Kosuch and Abdel Lisser},

title={STOCHASTIC SHORTEST PATH PROBLEM WITH UNCERTAIN DELAYS},

booktitle={Proceedings of the 1st International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2012},

pages={256-264},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003725102560264},

isbn={978-989-8425-97-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 1st International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - STOCHASTIC SHORTEST PATH PROBLEM WITH UNCERTAIN DELAYS

SN - 978-989-8425-97-3

AU - Cheng J.

AU - Kosuch S.

AU - Lisser A.

PY - 2012

SP - 256

EP - 264

DO - 10.5220/0003725102560264