Abstract
This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.
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References
Ağralı S., Geunes J.: A single-resource allocation problem with Poisson resource requirements. Optim. Lett. 3(4), 559–571 (2009)
Barnhart, C., Cohn, A.M.: The stochastic knapsack problem with random weights: a heuristic approach to robust transportation planning. In: Proceedings of Tristan III, Puerto Rico, pp. 17–23 (1998)
Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming: Theory and Algorithms, 2 edn. Wiley, New York (1993)
Carr S., Lovejoy W.: The inverse newsvendor problem: choosing an optimal demand portfolio for capacitated resources. Manage. Sci. 46(7), 912–927 (2000)
Carraway R.L., Schmidt R.L., Weatherford L.R.: An algorithm for maximizing target achievement in the stochastic knapsack problem with normal returns. Naval Res. Logist. 40, 161–173 (1993)
Dean, B.C., Goemans, M.X., Vondrak, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (2004)
Fortz, B., Labbé, M., Louveaux, F., Poss, M.: The knapsack problem with Gaussian weights. Technical report, Université Libre de Bruxelles, Brussels, Belgium (2008)
Garey M.R., Johnson D.S.: Computers and Intractability. W.H. Freeman and Company, New York (1979)
Goel, A., Indyk, P.: Stochastic load balancing and related problems. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pp. 579–586 (1999)
Henig M.: Risk criteria in a stochastic knapsack problem. Oper. Res. 38(5), 820–825 (1990)
Kleinberg J., Rabani Y., Tardos E.: Allocating bandwidth for bursty connections. SIAM J. Comput. 30(1), 191–217 (2000)
Kleywegt A., Papastavrou J.D.: The dynamic and stochastic knapsack problem with random sized items. Oper. Res. 49(1), 26–41 (2001)
Kleywegt A., Shapiro A., de Homem Mello T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2), 479–502 (2001)
Kosuch S., Lisser A.: Upper bounds for the 0-1 stochastic knapsack problem and a B &B algorithm. Ann. Oper. Res. 176, 77–93 (2010)
Martello S., Toth P.: Knapsack Problems, Algorithms and Computer Implementations. Wiley, New York (1990)
Merzifonluoğlu, Y.: Optimization models for integrated production, capacity and revenue management. PhD thesis, Department of Industrial and Systems Engineering, University of Florida (2006)
Morita H., Ishii H., Nishida T.: Stochastic linear knapsack programming problem and its applications to a portfolio selection problem. Eur. J. Oper. Res. 40(3), 329–336 (1989)
Papastavrou J.D., Rajagopalan S., Kleywegt A.: The dynamic and stochastic knapsack problem with deadlines. Manage. Sci. 42(12), 1706–1718 (1996)
Quesada I., Grossman I.E.: An lp/nlp based branch and bound algorithm for convex minlp optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)
Sniedovich M.: Preference order stochastic knapsack problems: methodological issues. J. Oper. Res. Soc. 31(11), 1025–1032 (1980)
Steinberg E., Parks M.S.: A preference order dynamic program for a knapsack problem with stochastic rewards. J. Oper. Res. Soc. 30(2), 141–147 (1979)
Taaffe K., Geunes J., Romeijn H.E.: Target market selection with demand uncertainty: the selective newsvendor problem. Eur. J. Oper. Res. 189(3), 987–1003 (2008)
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This work was supported by the National Science Foundation under grant no. DMI-0355533.
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Merzifonluoğlu, Y., Geunes, J. & Romeijn, H.E. The static stochastic knapsack problem with normally distributed item sizes. Math. Program. 134, 459–489 (2012). https://doi.org/10.1007/s10107-011-0443-5
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DOI: https://doi.org/10.1007/s10107-011-0443-5