IDEAS home Printed from https://ideas.repec.org/p/nbr/nberwo/0094.html
   My bibliography  Save this paper

Decision-Stage Method: Convergence Proof, Special Application, and Computation Experience

Author

Listed:
  • Vinay Dharmadhikari

Abstract

This paper presents a new method for obtaining exact optimal solutions for a class of discrete-variable non-linear resource-allocation problems. The new method is called the decision-state method because, unlike the conventional dynamic programming method which works only in the state space, the new method works in the state space and the decision space. It generates and retains only a fraction of the points in the state space at which the state functions are discontinuous; and thus overcomes to some extent the curse of dimensionality. It carries the cumulative decision-strongs associated with these points, and thus avoids the backtracking entailed by the conventional dynamic programming method for recovering the optimal decisions. A concise and complete statement of the method is given in Algorithm 2 and it is proved that the algorithm finds all exact optimal solutions. In addition the method is adapted for solving some problems with special structures such as block-angular or split-block-angular constraints and the resultant substantial advantages are demonstrated. The performance of Algorithm 2 on many resource-allocations problems is reported, along with investigations on many tactical decisions which have substantial impact on the performance. The performance of the computer implementation of Algorithm 2 is compared with that of the MMDP algorithm and it showed that for the class of problems at which the two are aimed, the decision-state Algorithm 2 performed better than MMDP algorithm both in terms of storage requirement and solution time. In fact, it achieved an order of magnitude saving in storage requirement.

Suggested Citation

  • Vinay Dharmadhikari, 1975. "Decision-Stage Method: Convergence Proof, Special Application, and Computation Experience," NBER Working Papers 0094, National Bureau of Economic Research, Inc.
    • Handle: RePEc:nbr:nberwo:0094
    as

    Download full text from publisher

    File URL: http://www.nber.org/papers/w0094.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. P. C. Gilmore & R. E. Gomory, 1966. "The Theory and Computation of Knapsack Functions," Operations Research, INFORMS, vol. 14(6), pages 1045-1074, December.
    2. A. M. Geoffrion & R. E. Marsten, 1972. "Integer Programming Algorithms: A Framework and State-of-the-Art Survey," Management Science, INFORMS, vol. 18(9), pages 465-491, May.
    3. Peter J. Wong, 1970. "A New Decomposition Procedure for Dynamic Programming," Operations Research, INFORMS, vol. 18(1), pages 119-131, February.
    4. Peter J. Wong & David G. Luenberger, 1968. "Reducing the Memory Requirements of Dynamic Programming," Operations Research, INFORMS, vol. 16(6), pages 1115-1125, December.
    5. G. L. Nemhauser & Z. Ullmann, 1969. "Discrete Dynamic Programming and Capital Allocation," Management Science, INFORMS, vol. 15(9), pages 494-505, May.
    Full references (including those not matched with items on IDEAS)

    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. Wilbaut, Christophe & Todosijevic, Raca & Hanafi, Saïd & Fréville, Arnaud, 2023. "Heuristic and exact reduction procedures to solve the discounted 0–1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 304(3), pages 901-911.
    2. Freville, Arnaud, 2004. "The multidimensional 0-1 knapsack problem: An overview," European Journal of Operational Research, Elsevier, vol. 155(1), pages 1-21, May.
    3. Setzer, Thomas & Blanc, Sebastian M., 2020. "Empirical orthogonal constraint generation for Multidimensional 0/1 Knapsack Problems," European Journal of Operational Research, Elsevier, vol. 282(1), pages 58-70.
    4. Throsby, C.D., 1973. "New Methodologies in Agricultural Production Economics: a Review," 1973 Conference, August 19-30, 1973, São Paulo, Brazil 181385, International Association of Agricultural Economists.
    5. Russo, Mauro & Sforza, Antonio & Sterle, Claudio, 2013. "An improvement of the knapsack function based algorithm of Gilmore and Gomory for the unconstrained two-dimensional guillotine cutting problem," International Journal of Production Economics, Elsevier, vol. 145(2), pages 451-462.
    6. Hanafi, Said & Freville, Arnaud, 1998. "An efficient tabu search approach for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 659-675, April.
    7. Parada Daza, Victor & Gomes de Alvarenga, Arlindo & de Diego, Jose, 1995. "Exact solutions for constrained two-dimensional cutting problems," European Journal of Operational Research, Elsevier, vol. 84(3), pages 633-644, August.
    8. Fujimoto, Masako & Yamada, Takeo, 2006. "An exact algorithm for the knapsack sharing problem with common items," European Journal of Operational Research, Elsevier, vol. 171(2), pages 693-707, June.
    9. M Hifi & M Michrafy, 2006. "A reactive local search-based algorithm for the disjunctively constrained knapsack problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(6), pages 718-726, June.
    10. Sbihi, Abdelkader, 2010. "A cooperative local search-based algorithm for the Multiple-Scenario Max-Min Knapsack Problem," European Journal of Operational Research, Elsevier, vol. 202(2), pages 339-346, April.
    11. Christian Meier & Dennis Kundisch & Jochen Willeke, 2017. "Is it Worth the Effort?," Business & Information Systems Engineering: The International Journal of WIRTSCHAFTSINFORMATIK, Springer;Gesellschaft für Informatik e.V. (GI), vol. 59(2), pages 81-95, April.
    12. Reinaldo Morabito & Vitória Pureza, 2010. "A heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem," Annals of Operations Research, Springer, vol. 179(1), pages 297-315, September.
    13. Jamain, Florian, 2014. "Représentations discrètes de l'ensemble des points non dominés pour des problèmes d'optimisation multi-objectifs," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/14002 edited by Bazgan, Cristina.
    14. Becker, Henrique & Buriol, Luciana S., 2019. "An empirical analysis of exact algorithms for the unbounded knapsack problem," European Journal of Operational Research, Elsevier, vol. 277(1), pages 84-99.
    15. Gomory, Ralph, 2016. "Origin and early evolution of corner polyhedra," European Journal of Operational Research, Elsevier, vol. 253(3), pages 543-556.
    16. Alexandre D. Jesus & Luís Paquete & Arnaud Liefooghe, 2021. "A model of anytime algorithm performance for bi-objective optimization," Journal of Global Optimization, Springer, vol. 79(2), pages 329-350, February.
    17. Jakob Puchinger & Günther R. Raidl & Ulrich Pferschy, 2010. "The Multidimensional Knapsack Problem: Structure and Algorithms," INFORMS Journal on Computing, INFORMS, vol. 22(2), pages 250-265, May.
    18. Simpson, Wendell P. & Patterson, James H., 1996. "A multiple-tree search procedure for the resource-constrained project scheduling problem," European Journal of Operational Research, Elsevier, vol. 89(3), pages 525-542, March.
    19. Song, X. & Chu, C.B. & Lewis, R. & Nie, Y.Y. & Thompson, J., 2010. "A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting," European Journal of Operational Research, Elsevier, vol. 202(2), pages 368-378, April.
    20. Bagloee, Saeed Asadi & Asadi, Mohsen, 2015. "Prioritizing road extension projects with interdependent benefits under time constraint," Transportation Research Part A: Policy and Practice, Elsevier, vol. 75(C), pages 196-216.

    More about this item

    Statistics

    Access and download statistics

    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:nbr:nberwo:0094. 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: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/nberrus.html .

    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.