Recoverable robust knapsacks: the discrete scenario case
- 271 Downloads
The knapsack problem is one of the basic problems in combinatorial optimization. In real-world applications it is often part of a more complex problem. Examples are machine capacities in production planning or bandwidth restrictions in telecommunication network design. Due to unpredictable future settings or erroneous data, parameters of such a subproblem are subject to uncertainties. In high risk situations a robust approach should be chosen to deal with these uncertainties. Unfortunately, classical robust optimization outputs solutions with little profit by prohibiting any adaption of the solution when the actual realization of the uncertain parameters is known. This ignores the fact that in most settings minor changes to a previously determined solution are possible. To overcome these drawbacks we allow a limited recovery of a previously fixed item set as soon as the data are known by deleting at most k items and adding up to ℓ new items. We consider the complexity status of this recoverable robust knapsack problem and extend the classical concept of cover inequalities to obtain stronger polyhedral descriptions. Finally, we present two extensive computational studies to investigate the influence of parameters k and ℓ to the objective and evaluate the effectiveness of our new class of valid inequalities.
KeywordsRecoverable robustness Knapsack Extended cover inequalities
Unable to display preview. Download preview PDF.
- 2.Aissi, H., Bazgan, C., Vanderpooten, D.: Min-max and min-max regret versions of some combinatorial optimization problems: a survey. http://hal.archives-ouvertes.fr/docs/00/15/86/52/PDF/AN7LAMSADE_1-32.pdf (2007)
- 4.Beasley, J.E.: OR-Library: distributing test problems by electronic mail. J. of the OR Society 41: 1069–1072. http://people.brunel.ac.uk/~mastjjb/jeb/ (1990)
- 8.Büsing, C., Koster, A.M.C.A., Kutschka M.: Recoverable robust knapsacks: the discrete scenario case. Technical Report 018-2010, TU Berlin. http://www.math.tu-berlin.de/coga/publications/techreports/2010/ (2010)
- 10.Cicerone S., D’Angelo G.D., Di Stefano G., Frigioni D., Navarra A., Schachtebeck M., Schöbel A.: Recoverable robustness in shunting and timetabling Robust and Online Large-Scale Optimization, vol. 5868 of LNCS, 28–60. Springer, Berlin (2009)Google Scholar
- 14.Kalai, R., Vanderpooten, D.: Lexicographic α-robust knapsack problems: complexity results. IEEE International Conference on Services Systems and Service Managment 1103–1107 (2006)Google Scholar
- 16.Karp, R.: Reducibility among combinatorial problems. Complexity of Computer Computations. 85–103 (1972)Google Scholar
- 17.Kellerer H., Pferschy U., Pisinger D.: Knapsack Problems. Springer, (2004)Google Scholar
- 18.Klopfenstein, O., Nace, D.: Valid inequalities for a robust knapsack polyhedron—Application to the robust bandwidth packing problem. Networks Special Issue, Network Optimization (INOC 2009) (to appear)Google Scholar
- 19.Liebchen C., Lübbecke M.E., Möhring R.H., Stiller S.: The concept of recoverable robustness, linear programming recovery, and railway applications Robust and Online Large-Scale Optimization, vol 5868 of LNCS, 1–27. Springer, Berlin (2009)Google Scholar
- 20.Martello, S., Toth, P.: Knapsack problems: algorithms and computer implementations. Wiley (1990)Google Scholar