Abstract
We consider an integer stochastic knapsack problem (SKP) where the weight of each item is deterministic, but the vector of returns for the items is random with known distribution. The objective is to maximize the probability that a total return threshold is met or exceeded. We study several solution approaches. Exact procedures, based on dynamic programming (DP) and integer programming (IP), are developed for returns that are independent normal random variables with integral means and variances. Computation indicates that the DP is significantly faster than the most efficient algorithm to date. The IP is less efficient, but is applicable to more general stochastic IPs with independent normal returns. We also develop a Monte Carlo approximation procedure to solve SKPs with general distributions on the random returns. This method utilizes upper- and lower-bound estimators on the true optimal solution value in order to construct a confidence interval on the optimality gap of a candidate solution.
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References
Brooke, A., Kendrick, D., and Meeraus, A., GAMS: A User’s Guide, The Scientific Press, San Francisco (1992).
Carraway, R.L., Morin, T.L., and Moskowitz, H., Generalized Dynamic Programming for Stochastic Combinatorial Optimization, Operations Research, 37, 819- STOCHASTIC KNAPSACK PROBLEM 167 829 (1989).
CPLEX Manual, Using the CPLEXTM Callable Library and CPLEX T M Mixed Integer Library, CPLEX Optimization, Inc., Incline Village, Nevada, 1993.
Carraway, R.L., Schmidt, R.L., and Weatherford, L.R., An Algorithm for Maximizing Target Achievement in the Stochastic Knapsack Problem with Normal Returns, Naval Research Logistics, 40, 161–173 (1993).
Chiu, S.Y., Lu, L., and Cox, L.A., Optimal Access Control for Broadband Services: Stochastic Knapsack with Advance Information, European Journal of Operations Research, 89, 127–134 (1996).
Dantzig, G.B., Discrete-Variable Extremum Problems, Operations Research, 5, 266–277 (1957).
Dreyfus, S.E. and Law, M.L., The Art and Theory of Dynamic Programming, Academic Press, New York (1977).
Dupacovâ, J. and Wets, R.J.-B., Asymptotic Behavior of Statistical Estimators and of Optimal Solutions of Stochastic Optimization Problems, The Annals of Statistics, 16, 1517–1549 (1988).
Gavious, A. and Rosberg, Z., A Restricted Complete Sharing Policy for a Stochastic Knapsack Problem in B-ISDN, IEEE Transactions on Communications, 42, 23752379 (1994).
Geoffrion, A.M., Solving Bicriterion Mathematical Programs, Operations Research, 15, 39–54 (1967).
Greenberg, H.J., Dynamic Programming with Linear Uncertainty, Operations Research, 16, 675–678 (1968).
Henig, M.I., Risk Criteria in the Stochastic Knapsack Problem, Operations Research, 38, 820–825 (1990).
Ishii, H. and Nishida, T., Stochastic Linear Knapsack Problem: Probability Maximization Model, Mathematica Japonica, 29, 273–281 (1984).
King, A.J. and Wets, R.J.-B., Epi-Consistency of Convex Stochastic Programs, Stochastics, 34, 83–91 (1991).
Law, A.M. and Kelton, W.D., Simulation Modeling and Analysis, McGraw-Hill, New York (1991).
Mak, W.K., Morton, D.P., and Wood, R.K., Monte Carlo Bounding Techniques for Determining Solution Quality in Stochastic Programs, Technical Report, The University of Texas at Austin (1997).
Marchetti-Spaccamela, A. and Vercellis, C., Stochastic On-Line Knapsack Problems, Mathematical Programming, 68, 73–104 (1995).
Morita, H., Ishii, H., and Nishida, T., Stochastic Linear Knapsack Programming Problem and Its Application to a Portfolio Selection Problems, European Journal of Operations Research, 40, 329–336 (1989).
Norkin, V.I., Pflug, G.Ch., and Ruszczyfiski, A., A Branch and Bound Method for Stochastic Global Optimization, Working Paper, IIASA (1996).
Nemhauser, G.L., and Ullman, Z., Discrete Dynamic Programming and Capital Allocation, Management Science, 15, 494–505 (1969).
Papastavrou, J.D.; Rajagopalan, S., Kleywegt, A.J., Discrete Dynamic Programming and Capital Allocation, Management Science, 42, 1706–1718 (1996).
Prékopa, A., Stochastic Programming, Kluwer Academic Publishers, Dordrecht (1995).
Ross, K.W., Multiservice Loss Models for Broadband Telecommunication Networks, Springer-Verlag, London (1995).
Ross, K.W. and Tsang, D.H.K., The Stochastic Knapsack Problem, IEEE Transactions on Communications, 37, 740–747 (1989).
Sniedovich, M. Preference Order Stochastic Knapsack Problems: Methodological Issues, Journal of the Operational Research Society, 31, 1025–1032 (1980).
Steinberg, E., and Parks, M.S., A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards, Journal of the Operational Research Society, 30, 141–147 (1979).
Wets, R.J.-B. (1989): Stochastic Programming, in G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd (eds.) Handbooks in Operations Research and Management Science, Elsevier Science Publishers, Amsterdam.
Weingartner, M.H., and Ness, D.N., Methods for the Solution of Multidimensional 0/1 Knapsack Problems, Operations Research, 15, 83–103 (1967).
XA, Professional Linear Programming System, Version 2.2, Sunset Software Technology, San Marino, California (1993).
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Morton, D.P., Wood, R.K. (1998). On a Stochastic Knapsack Problem and Generalizations. In: Woodruff, D.L. (eds) Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search. Operations Research/Computer Science Interfaces Series, vol 9. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2807-1_5
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DOI: https://doi.org/10.1007/978-1-4757-2807-1_5
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