Mathematical Programming

, Volume 157, Issue 1, pp 277–296 | Cite as

Robust optimization approach for a chance-constrained binary knapsack problem

Full Length Paper Series B


We consider a certain class of chance-constrained binary knapsack problem where each item has a normally distributed random weight that is independent of the other items. For this problem we propose an efficient pseudo-polynomial time algorithm based on the robust optimization approach for finding a solution with a theoretical bound on the probability of satisfying the knapsack constraint. Our algorithm is tested on a wide range of random instances, and the results demonstrate that it provides qualified solutions quickly. In contrast, a state-of-the-art MIP solver is only applicable for instances of the problem with a restricted number of items.


Knapsack problem Combinatorial optimization Chance-constrained programming Robust optimization 

Mathematics Subject Classification

90C15 90C27 90C59 



This research was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant 2011-0027301).


  1. 1.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bhalgat, A., Goel, A., Khanna, S.: Improved approximation results for stochastic knapsack problems. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, pp. 1647–1665 (2011)Google Scholar
  4. 4.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  5. 5.
    Calafiore, G., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory App. 130(1), 1–22 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cohn, A., Barnhart, C.: The stochastic knapsack problem with random weights: a heuristic approach to robust transportation planning. In: Proceedings of the Triennial Symposium on Transportation Analysis (TRISTAN III) (1998)Google Scholar
  7. 7.
    Fortz, B., Labbé, M., Louveaux, F., Poss, M.: The knapsack problem with gaussian weights. Technical report. Université Libre de Bruxelles, Brussels, Belgium (2008)Google Scholar
  8. 8.
    Goel, A., Indyk, P.: Stochastic load balancing and related problems. In: 40th Annual Symposium on Foundations of Computer Science, pp. 579–586 (1999)Google Scholar
  9. 9.
    Goerigk, M.: A note on upper bounds to the robust knapsack problem with discrete scenarios. Ann. Oper. Res. 223(1), 461–469 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Goyal, V., Ravi, R.: A PTAS for the chance-constrained knapsack problem with random item sizes. Oper. Res. Lett. 38(3), 161–164 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Han, J., Lee, K., Lee, C., Park, S.: Exact algorithms for a bandwidth packing problem with queueing delay guarantees. INFORMS J. Comput. 25(3), 585–596 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Iida, H.: A note on the max–min 0–1 knapsack problem. J. Comb. Optim. 3(1), 89–94 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Kleinberg, J., Rabani, Y., Tardos, É.: Allocating bandwidth for bursty connections. SIAM J. Comput. 30(1), 191–217 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kleywegt, A., Papastavrou, J.: The dynamic and stochastic knapsack problem. Oper. Res. 46(1), 17–35 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kleywegt, A., Papastavrou, J.: The dynamic and stochastic knapsack problem with random sized items. Oper. Res. 49(1), 26–41 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Klopfenstein, O., Nace, D.: A robust approach to the chance-constrained knapsack problem. Oper. Res. Lett. 36, 628–632 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Klopfenstein, O., Nace, D.: Cover inequalities for robust knapsack sets—application to the robust bandwidth packing problem. Networks 59(1), 59–72 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kosuch, S., Lisser, A.: Upper bounds for the 0–1 stochastic knapsack problem and a B and B algorithm. Ann. Oper. Res. 176, 77–93 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lee, C., Lee, K., Park, K., Park, S.: Technical note—branch-and-price-and-cut approach to the robust network design problem without flow bifurcations. Oper. Res. 60(3), 35–53 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Martello, S., Toth, P.: Knapsack Problems. Wiley, New York (1990)MATHGoogle Scholar
  22. 22.
    Merzifonluoğlu, Y., Geunes, J., Romeijn, H.E.: The static stochastic knapsack problem with normally distributed item sizes. Math. Program. 134(2), 459–489 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Monaci, M., Pferschy, U.: On the robust knapsack problem. SIAM J. Optim. 23(4), 1956–1982 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Monaci, M., Pferschy, U., Serafini, P.: Exact solution of the robust knapsack problem. Comput. Oper. Res. 40, 2625–2631 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pisinger, D.: A minimal algorithm for the 0–1 knapsack problem. Oper. Res. 45(5), 758–767 (1997)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Savelsbergh, M.: A branch-and-price algorithm for the generalized assignment problem. Oper. Res. 45, 831–841 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sbihi, A.: A cooperative local search-based algorithm for the multiple-scenario max–min knapsack problem. Eur. J. Oper. Res. 202(2), 339–346 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Talla Nobibon, F., Leus, R.: Complexity results and exact algorithms for robust knapsack problems. J. Optim. Theory Appl. 161(2), 533–552 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Taniguchi, F., Yamada, T., Kataoka, S.: Heuristic and exact algorithms for the max–min optimization of the multi-scenario knapsack problem. Comput. Oper. Res. 35(6), 2034–2048 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Yu, G.: On the max–min 0–1 knapsack problem with robust optimization applications. Oper. Res. 44(2), 407–415 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.CORE, Université catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Department of Industrial and Systems EngineeringKorea Advanced Institute of Science and TechnologyDaejeonRepublic of Korea
  3. 3.Department of Industrial Engineering & Institute for Industrial Systems InnovationSeoul National UniversitySeoulRepublic of Korea
  4. 4.Department of Industrial and Management EngineeringHankuk University of Foreign StudiesGyeonggi-doRepublic of Korea

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