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Average-Case Performance of Rollout Algorithms for Knapsack Problems

Author

Listed:
  • Andrew Mastin

    (Massachusetts Institute of Technology)

  • Patrick Jaillet

    (Massachusetts Institute of Technology)

Abstract

Rollout algorithms have demonstrated excellent performance on a variety of dynamic and discrete optimization problems. Interpreted as an approximate dynamic programming algorithm, a rollout algorithm estimates the value-to-go at each decision stage by simulating future events while following a heuristic policy, referred to as the base policy. While in many cases rollout algorithms are guaranteed to perform as well as their base policies, there have been few theoretical results showing additional improvement in performance. In this paper, we perform a probabilistic analysis of the subset sum problem and 0–1 knapsack problem, giving theoretical evidence that rollout algorithms perform strictly better than their base policies. Using a stochastic model from the existing literature, we analyze two rollout methods that we refer to as the exhaustive rollout and consecutive rollout, both of which employ a simple greedy base policy. We prove that both methods yield a significant improvement in expected performance after a single iteration of the rollout algorithm, relative to the base policy.

Suggested Citation

  • Andrew Mastin & Patrick Jaillet, 2015. "Average-Case Performance of Rollout Algorithms for Knapsack Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 964-984, June.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:3:d:10.1007_s10957-014-0603-x
    DOI: 10.1007/s10957-014-0603-x
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    References listed on IDEAS

    as
    1. Nicola Secomandi, 2001. "A Rollout Policy for the Vehicle Routing Problem with Stochastic Demands," Operations Research, INFORMS, vol. 49(5), pages 796-802, October.
    2. Gennady Diubin & Alexander Korbut, 2003. "The average behaviour of greedy algorithms for the knapsack problem: General distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(3), pages 449-479, August.
    3. Luca Bertazzi, 2012. "Minimum and Worst-Case Performance Ratios of Rollout Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 378-393, February.
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    Cited by:

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    2. Goodson, Justin C. & Thomas, Barrett W. & Ohlmann, Jeffrey W., 2017. "A rollout algorithm framework for heuristic solutions to finite-horizon stochastic dynamic programs," European Journal of Operational Research, Elsevier, vol. 258(1), pages 216-229.

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