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Robust optimization approach for a chance-constrained binary knapsack problem

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Abstract

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.

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References

  1. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  4. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  5. Calafiore, G., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory App. 130(1), 1–22 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  7. Fortz, B., Labbé, M., Louveaux, F., Poss, M.: The knapsack problem with gaussian weights. Technical report. Université Libre de Bruxelles, Brussels, Belgium (2008)

  8. Goel, A., Indyk, P.: Stochastic load balancing and related problems. In: 40th Annual Symposium on Foundations of Computer Science, pp. 579–586 (1999)

  9. Goerigk, M.: A note on upper bounds to the robust knapsack problem with discrete scenarios. Ann. Oper. Res. 223(1), 461–469 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  12. Iida, H.: A note on the max–min 0–1 knapsack problem. J. Comb. Optim. 3(1), 89–94 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  14. Kleinberg, J., Rabani, Y., Tardos, É.: Allocating bandwidth for bursty connections. SIAM J. Comput. 30(1), 191–217 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kleywegt, A., Papastavrou, J.: The dynamic and stochastic knapsack problem. Oper. Res. 46(1), 17–35 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kleywegt, A., Papastavrou, J.: The dynamic and stochastic knapsack problem with random sized items. Oper. Res. 49(1), 26–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Klopfenstein, O., Nace, D.: A robust approach to the chance-constrained knapsack problem. Oper. Res. Lett. 36, 628–632 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Klopfenstein, O., Nace, D.: Cover inequalities for robust knapsack sets—application to the robust bandwidth packing problem. Networks 59(1), 59–72 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martello, S., Toth, P.: Knapsack Problems. Wiley, New York (1990)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Monaci, M., Pferschy, U.: On the robust knapsack problem. SIAM J. Optim. 23(4), 1956–1982 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Monaci, M., Pferschy, U., Serafini, P.: Exact solution of the robust knapsack problem. Comput. Oper. Res. 40, 2625–2631 (2013)

    Article  MathSciNet  Google Scholar 

  25. Pisinger, D.: A minimal algorithm for the 0–1 knapsack problem. Oper. Res. 45(5), 758–767 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Savelsbergh, M.: A branch-and-price algorithm for the generalized assignment problem. Oper. Res. 45, 831–841 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Talla Nobibon, F., Leus, R.: Complexity results and exact algorithms for robust knapsack problems. J. Optim. Theory Appl. 161(2), 533–552 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yu, G.: On the max–min 0–1 knapsack problem with robust optimization applications. Oper. Res. 44(2), 407–415 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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).

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Authors and Affiliations

  1. CORE, Université catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    Jinil Han

  2. Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Jinil Han & Sungsoo Park

  3. Department of Industrial Engineering & Institute for Industrial Systems Innovation, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 151-742, Republic of Korea

    Kyungsik Lee

  4. Department of Industrial and Management Engineering, Hankuk University of Foreign Studies, San 89, Wangsan-ri, Mohyu-myun, Yongin, Gyeonggi-do, 449-791, Republic of Korea

    Chungmok Lee & Ki-Seok Choi

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  1. Jinil Han
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  2. Kyungsik Lee
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  3. Chungmok Lee
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  4. Ki-Seok Choi
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  5. Sungsoo Park
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Correspondence to Kyungsik Lee or Sungsoo Park.

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Han, J., Lee, K., Lee, C. et al. Robust optimization approach for a chance-constrained binary knapsack problem. Math. Program. 157, 277–296 (2016). https://doi.org/10.1007/s10107-015-0931-0

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