IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v193y2009i2p605-615.html
   My bibliography  Save this article

Solving knapsack problems with S-curve return functions

Author

Listed:
  • AgralI, Semra
  • Geunes, Joseph

Abstract

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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377-2217(07)01124-1
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Paul H. Zipkin, 1980. "Simple Ranking Methods for Allocation of One Resource," Management Science, INFORMS, vol. 26(1), pages 34-43, January.
    2. Duncan M. Holthausen, Jr. & Gert Assmus, 1982. "Advertising Budget Allocation under Uncertainty," Management Science, INFORMS, vol. 28(5), pages 487-499, May.
    3. Muralidharan S. Kodialam & Hanan Luss, 1998. "Algorithms for Separable Nonlinear Resource Allocation Problems," Operations Research, INFORMS, vol. 46(2), pages 272-284, April.
    4. 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.
    5. Anantaram Balakrishnan & Joseph Geunes, 2003. "Production Planning with Flexible Product Specifications: An Application to Specialty Steel Manufacturing," Operations Research, INFORMS, vol. 51(1), pages 94-112, February.
    6. C. P. M. van Hoesel & A. P. M. Wagelmans, 2001. "Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 339-357, May.
    7. Ginsberg, William, 1974. "The multiplant firm with increasing returns to scale," Journal of Economic Theory, Elsevier, vol. 9(3), pages 283-292, November.
    8. Kurt M. Bretthauer & Bala Shetty, 1995. "The Nonlinear Resource Allocation Problem," Operations Research, INFORMS, vol. 43(4), pages 670-683, August.
    9. Andris A. Zoltners & Prabhakant Sinha & Philip S. C. Chong, 1979. "An Optimal Algorithm for Sales Representative Time Management," Management Science, INFORMS, vol. 25(12), pages 1197-1207, December.
    10. 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.
    11. Leonard M. Lodish, 1971. "Callplan: An Interactive Salesman's Call Planning System," Management Science, INFORMS, vol. 18(4-Part-II), pages 25-40, December.
    12. Thomas L. Morin & Roy E. Marsten, 1976. "An Algorithm for Nonlinear Knapsack Problems," Management Science, INFORMS, vol. 22(10), pages 1147-1158, June.
    13. Andris A. Zoltners & Prabhakant Sinha, 1980. "Integer Programming Models for Sales Resource Allocation," Management Science, INFORMS, vol. 26(3), pages 242-260, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Westerink-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. Westerink-Duijzer, L.E. & Schlicher, L.P.J. & Musegaas, M., 2019. "Fair allocations for cooperation problems in vaccination," Econometric Institute Research Papers EI2019-06, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. Lotty E. Westerink‐Duijzer & Loe P. J. Schlicher & Marieke Musegaas, 2020. "Core Allocations for Cooperation Problems in Vaccination," Production and Operations Management, Production and Operations Management Society, vol. 29(7), pages 1720-1737, July.
    6. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Patriksson, Michael, 2008. "A survey on the continuous nonlinear resource allocation problem," European Journal of Operational Research, Elsevier, vol. 185(1), pages 1-46, February.
    2. Kameshwaran, S. & Narahari, Y., 2009. "Nonconvex piecewise linear knapsack problems," European Journal of Operational Research, Elsevier, vol. 192(1), pages 56-68, January.
    3. Zhang, Bin & Hua, Zhongsheng, 2008. "A unified method for a class of convex separable nonlinear knapsack problems," European Journal of Operational Research, Elsevier, vol. 191(1), pages 1-6, November.
    4. Zhang, Jianzhong & Xu, Chengxian, 2010. "Inverse optimization for linearly constrained convex separable programming problems," European Journal of Operational Research, Elsevier, vol. 200(3), pages 671-679, February.
    5. DePaolo, Concetta A. & Rader, David Jr., 2007. "A heuristic algorithm for a chance constrained stochastic program," European Journal of Operational Research, Elsevier, vol. 176(1), pages 27-45, January.
    6. De Waegenaere, A.M.B. & Wielhouwer, J.L., 2001. "A Partial Ranking Algorithm for Resource Allocation Problems," Other publications TiSEM 8b2e0185-36f9-43df-8a3d-d, Tilburg University, School of Economics and Management.
    7. Patriksson, Michael & Strömberg, Christoffer, 2015. "Algorithms for the continuous nonlinear resource allocation problem—New implementations and numerical studies," European Journal of Operational Research, Elsevier, vol. 243(3), pages 703-722.
    8. 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.
    9. De Waegenaere, A.M.B. & Wielhouwer, J.L., 2001. "A Partial Ranking Algorithm for Resource Allocation Problems," Discussion Paper 2001-40, Tilburg University, Center for Economic Research.
    10. Kurt M. Bretthauer & Bala Shetty & Siddhartha Syam, 2003. "A specially structured nonlinear integer resource allocation problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 770-792, October.
    11. Mohammadivojdan, Roshanak & Geunes, Joseph, 2018. "The newsvendor problem with capacitated suppliers and quantity discounts," European Journal of Operational Research, Elsevier, vol. 271(1), pages 109-119.
    12. Lee, Zu-Hsu & Deng, Shiming & Lin, Beixin & Yang, James G.S., 2010. "Decision model and analysis for investment interest expense deduction and allocation," European Journal of Operational Research, Elsevier, vol. 200(1), pages 268-280, January.
    13. Jamie Fravel & Robert Hildebrand & Laurel Travis, 2024. "Continuous Equality Knapsack with Probit-Style Objectives," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1060-1076, September.
    14. Hanan Luss, 1999. "On Equitable Resource Allocation Problems: A Lexicographic Minimax Approach," Operations Research, INFORMS, vol. 47(3), pages 361-378, June.
    15. Drexl, Andreas & Haase, Knut, 1996. "Fast approximation methods for sales force deployment," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 411, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    16. Johannes Habel & Sascha Alavi & Nicolas Heinitz, 2023. "A theory of predictive sales analytics adoption," AMS Review, Springer;Academy of Marketing Science, vol. 13(1), pages 34-54, June.
    17. SYAFITRI, Utami & SARTONO, Bagus & GOOS, Peter, 2015. "D- and I-optimal design of mixture experiments in the presence of ingredient availability constraints," Working Papers 2015003, University of Antwerp, Faculty of Business and Economics.
    18. Chase Rainwater & Joseph Geunes & H. Edwin Romeijn, 2014. "Resource-Constrained Assignment Problems with Shared Resource Consumption and Flexible Demand," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 290-302, May.
    19. 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.
    20. Bretthauer, Kurt M. & Ross, Anthony & Shetty, Bala, 1999. "Nonlinear integer programming for optimal allocation in stratified sampling," European Journal of Operational Research, Elsevier, vol. 116(3), pages 667-680, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:193:y:2009:i:2:p:605-615. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.