A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem

Pablo Adasme, Rafael Andrade, Janny Leung, Abdel Lisser

2016

Abstract

In the context of combinatorial optimization, recently some efforts have been made by extending classical optimization problems under the two-stage stochastic programming framework. In this paper, we introduce the two-stage stochastic traveling salesman problem (STSP). Let G = (V,ED ∪ES) be a non directed complete graph with set of nodes V and set of weighted edges ED ∪ES where ED ∩ES = 0/. The edges in ED and ES have deterministic and uncertain weights, respectively. Let K = {1,2,··· ,|K|} be a given set of scenarios referred to the uncertain weights of the edges in ES. The STSP consists in determining Hamiltonian cycles of G, one for each scenario s ∈ K, sharing the same deterministic edges while minimizing the sum of the deterministic weights plus the expected weight over all scenarios associated with the uncertain edges. We propose two compact models and a formulation with an exponential number of constraints which are adapted from the classic TSP. One of the compact models allows to solve instances with up to 40 nodes and 5 scenarios to optimality. Finally, we propose an iterative procedure that allows to compute optimal solutions and tight lower bounds within very small CPU time.

References

  1. Adasme, P., Andrade, R., Letournel, M., and Lisser, A. (2013). A polynomial formulation for the stochastic maximum weight forest problem. ENDM, 41:29-36.
  2. Adasme, P., Andrade, R., Letournel, M., and Lisser, A. (2015). Stochastic maximum weight forest problem. Networks, 65(4):289-305.
  3. Bertazzi, L. and Maggioni, F. (2014). Solution approaches for the stochastic capacitated traveling salesmen location problem with recourse. J Optim Theory Appl, 166(1):321-342.
  4. Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C. (2009). Introduction to algorithms. MIT Press and McGraw-Hill, Cumberland, RI, USA.
  5. Escoffier, B., Gourves, L., Monnot, J., and Spanjaard, O. (2010). Two-stage stochastic matching and spanning tree problems: Polynomial instances and approximation. Eur J Oper Res, 205:19-30.
  6. Flaxman, A. D., Frieze, A., and Krivelevich, M. (2006). On the random 2-stage minimum spanning tree. Random Struct Algor, 28:24-36.
  7. Gaivoronski, A., Lisser, A., Lopez, R., and Xu, H. (2011). Knapsack problem with probability constraints. J Global Optim, 49:397-413.
  8. Gavish, B. and Graves, S. C. (1978). The travelling salesman problem and related problems. Operations Research Centre, Massachusetts Institute of Technology.
  9. Letchford, A. N., Nasiri, S. D., and Theis, D. O. (2013). Compact formulations of the steiner traveling salesman problem and related problems. European Journal of Operational Research, 228:83-92.
  10. Maggioni, F., Perboli, G., and Tadei, R. (2014). The multipath traveling salesman problem with stochastic travel costs: a city logistics computational study. Transportation Research Procedia, 1(3):528-536.
  11. Miller, C. E., Tucker, A. W., and Zemlin, R. A. (1960). Integer programming formulations and travelling salesman problems. J. Assoc. Comput. Mach., 7:326-329.
  12. Shapiro, A., Dentcheva, D., and Ruszczynski, A. (2009). Lectures on stochastic programming: Modeling and theory. MOS-SIAM Series on Optimization, Philadelphia.
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Paper Citation


in Harvard Style

Adasme P., Andrade R., Leung J. and Lisser A. (2016). A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem . In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES, ISBN 978-989-758-171-7, pages 163-169. DOI: 10.5220/0005738801630169


in Bibtex Style

@conference{icores16,
author={Pablo Adasme and Rafael Andrade and Janny Leung and Abdel Lisser},
title={A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem},
booktitle={Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},
year={2016},
pages={163-169},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005738801630169},
isbn={978-989-758-171-7},
}


in EndNote Style

TY - CONF
JO - Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,
TI - A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem
SN - 978-989-758-171-7
AU - Adasme P.
AU - Andrade R.
AU - Leung J.
AU - Lisser A.
PY - 2016
SP - 163
EP - 169
DO - 10.5220/0005738801630169