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The Dynamic and Stochastic Knapsack Problem with Deadlines


  • Jason D. Papastavrou

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907-1287)

  • Srikanth Rajagopalan

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907-1287)

  • Anton J. Kleywegt

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907-1287)


In this paper a dynamic and stochastic model of the well-known knapsack problem is developed and analyzed. The problem is motivated by a wide variety of real-world applications. Objects of random weight and reward arrive according to a stochastic process in time. The weights and rewards associated with the objects are distributed according to a known probability distribution. Each object can either be accepted to be loaded into the knapsack, of known weight capacity, or be rejected. The objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward. The optimal decision rules are derived and are shown to exhibit surprising behavior in some cases. It is also shown that if the distribution of the weights is concave, then the decision rules behave according to intuition.

Suggested Citation

  • Jason D. Papastavrou & Srikanth Rajagopalan & Anton J. Kleywegt, 1996. "The Dynamic and Stochastic Knapsack Problem with Deadlines," Management Science, INFORMS, vol. 42(12), pages 1706-1718, December.
  • Handle: RePEc:inm:ormnsc:v:42:y:1996:i:12:p:1706-1718

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    Cited by:

    1. Grace Lin & Yingdong Lu & David Yao, 2007. "Stochastic Knapsack Problem Revisited: Switch-Over Policies and Dynamic Pricing," Papers 0708.1146,
    2. Pak, K. & Dekker, R., 2004. "Cargo Revenue Management: Bid-Prices for a 0-1 Multi Knapsack Problem," ERIM Report Series Research in Management ERS-2004-055-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    3. Moldovanu, Benny & Dizdar, Deniz & Gershkov, Alex, 2011. "Revenue maximization in the dynamic knapsack problem," Theoretical Economics, Econometric Society, vol. 6(2), May.
    4. Chiang, David Ming-Huang & Wu, Andy Wei-Di, 2011. "Discrete-order admission ATP model with joint effect of margin and order size in a MTO environment," International Journal of Production Economics, Elsevier, vol. 133(2), pages 761-775, October.
    5. Florian Zettelmeyer & Fiona Scott Morton & Jorge Silva-Risso, 2006. "Scarcity Rents in Car Retailing: Evidence from Inventory Fluctuations at Dealerships," NBER Working Papers 12177, National Bureau of Economic Research, Inc.
    6. Jacko, Peter & Niño-Mora, José, 2009. "An index for dynamic product promotion and the knapsack problem for perishable items," DES - Working Papers. Statistics and Econometrics. WS ws093111, Universidad Carlos III de Madrid. Departamento de Estadística.
    7. Clemens Thielen & Morten Tiedemann & Stephan Westphal, 2016. "The online knapsack problem with incremental capacity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(2), pages 207-242, April.
    8. Xiao, Baichun & Yang, Wei, 2010. "A revenue management model for products with two capacity dimensions," European Journal of Operational Research, Elsevier, vol. 205(2), pages 412-421, September.
    9. Kang, Seungmo & Ouyang, Yanfeng, 2011. "The traveling purchaser problem with stochastic prices: Exact and approximate algorithms," European Journal of Operational Research, Elsevier, vol. 209(3), pages 265-272, March.
    10. Wang, Xiubin & Regan, Amelia, 2006. "Dynamic yield management when aircraft assignments are subject to swap," Transportation Research Part B: Methodological, Elsevier, vol. 40(7), pages 563-576, August.
    11. J. Neil Bearden & Ryan O. Murphy & Amnon Rapoport, 2008. "Decision Biases in Revenue Management: Some Behavioral Evidence," Manufacturing & Service Operations Management, INFORMS, vol. 10(4), pages 625-636, June.
    12. 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.
    13. Kraft, Edwin R., 2002. "Scheduling railway freight delivery appointments using a bid price approach," Transportation Research Part A: Policy and Practice, Elsevier, vol. 36(2), pages 145-165, February.
    14. repec:spr:infosf:v:4:y:2002:i:1:d:10.1023_a:1015351025609 is not listed on IDEAS
    15. Zettelmeyer, Florian & Scott Morton, Fiona & Silva-Risso, Jorge, 2005. "Inventory Fluctuations and Price Discrimination: The Determinants of Price Variation in Car Retailing," Department of Economics, Working Paper Series qt6pf5s593, Department of Economics, Institute for Business and Economic Research, UC Berkeley.


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