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Solving knapsack problems with S-curve return functions


  • AgralI, Semra
  • Geunes, Joseph


We consider the allocation of a limited budget to a set of activities or investments in order to maximize return from investment. In a number of practical contexts (e.g., advertising), the return from investment in an activity is effectively modeled using an S-curve, where increasing returns to scale exist at small investment levels, and decreasing returns to scale occur at high investment levels. We demonstrate that the resulting knapsack problem with S-curve return functions is NP-hard, provide a pseudo-polynomial time algorithm for the integer variable version of the problem, and develop efficient solution methods for special cases of the problem. We also discuss a fully-polynomial-time approximation algorithm for the integer variable version of the problem.

Suggested Citation

  • AgralI, Semra & Geunes, Joseph, 2009. "Solving knapsack problems with S-curve return functions," European Journal of Operational Research, Elsevier, vol. 193(2), pages 605-615, March.
  • Handle: RePEc:eee:ejores:v:193:y:2009:i:2:p:605-615

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    References listed on IDEAS

    1. Paul H. Zipkin, 1980. "Simple Ranking Methods for Allocation of One Resource," Management Science, INFORMS, vol. 26(1), pages 34-43, January.
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    3. Ginsberg, William, 1974. "The multiplant firm with increasing returns to scale," Journal of Economic Theory, Elsevier, vol. 9(3), pages 283-292, November.
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    5. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    6. Gabriel R. Bitran & Arnoldo C. Hax, 1981. "Disaggregation and Resource Allocation Using Convex Knapsack Problems with Bounded Variables," Management Science, INFORMS, vol. 27(4), pages 431-441, April.
    7. Leonard M. Lodish, 1971. "Callplan: An Interactive Salesman's Call Planning System," Management Science, INFORMS, vol. 18(4-Part-II), pages 25-40, December.
    8. Thomas L. Morin & Roy E. Marsten, 1976. "An Algorithm for Nonlinear Knapsack Problems," Management Science, INFORMS, vol. 22(10), pages 1147-1158, June.
    9. Andris A. Zoltners & Prabhakant Sinha, 1980. "Integer Programming Models for Sales Resource Allocation," Management Science, INFORMS, vol. 26(3), pages 242-260, March.
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    Cited by:

    1. Duijzer, L.E. & van Jaarsveld, W.L. & Wallinga, J. & Dekker, R., 2015. "Dose-optimal vaccine allocation over multiple populations," Econometric Institute Research Papers EI2015-29, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Zschocke, Mark S. & Mantin, Benny & Jewkes, Elizabeth M., 2013. "Mature or emerging markets: Competitive duopoly investment decisions," European Journal of Operational Research, Elsevier, vol. 228(3), pages 612-622.
    3. Srivastava, Vaibhav & Bullo, Francesco, 2014. "Knapsack problems with sigmoid utilities: Approximation algorithms via hybrid optimization," European Journal of Operational Research, Elsevier, vol. 236(2), pages 488-498.
    4. Vahideh Sadat Abedi, 2017. "Allocation of advertising budget between multiple channels to support sales in multiple markets," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(2), pages 134-146, February.


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