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A Linearization Procedure for Quadratic and Cubic Mixed-Integer Problems

Author

Listed:
  • Muhittin Oral

    (Université Laval, Quebec, Canada)

  • Ossama Kettani

    (Université Laval, Quebec, Canada)

Abstract

Several techniques of linearization have appeared in the literature. The technique of F. Glover, which seems to be the most efficient, linearizes a binary quadratic integer problem of n variables by introducing n new continuous variables and 4n auxiliary linear constraints. The new technique proposed in this paper is not only useful in linearizing binary quadratic and cubic integer problems, but also applicable to the case of quadratic and to a certain class of cubic “mixed-integer” problems. It is shown that the new technique further reduces the number of auxiliary linear constraints from 4n to n , while keeping the number of new continuous variables at n for the binary quadratic integer problem of n variables. And, it requires, in the case of a certain class of cubic mixed-integer problems having 2n of 0–1 variables, only 3n auxiliary linear constraints and the same number of new continuous variables. The analytical superiority of the new linearization technique has also been observed, in terms of the number of iterations and execution times, through a computational experiment conducted on a set of randomly generated binary quadratic integer problems.

Suggested Citation

  • Muhittin Oral & Ossama Kettani, 1992. "A Linearization Procedure for Quadratic and Cubic Mixed-Integer Problems," Operations Research, INFORMS, vol. 40(1-supplem), pages 109-116, February.
    • Handle: RePEc:inm:oropre:v:40:y:1992:i:1-supplement-1:p:s109-s116
    DOI: 10.1287/opre.40.1.S109
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    Citations

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    Cited by:

    1. Oral, Muhittin & Kettani, Ossama & Cinar, Unver, 2001. "Project evaluation and selection in a network of collaboration: A consensual disaggregation multi-criterion approach," European Journal of Operational Research, Elsevier, vol. 130(2), pages 332-346, April.
    2. Oral, Muhittin, 2010. "E-DEA: Enhanced data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 207(2), pages 916-926, December.
    3. Stefan Helber & Daniel Böhme & Farid Oucherif & Svenja Lagershausen & Steffen Kasper, 2016. "A hierarchical facility layout planning approach for large and complex hospitals," Flexible Services and Manufacturing Journal, Springer, vol. 28(1), pages 5-29, June.
    4. Fishburn, Peter C. & LaValle, Irving H., 1996. "Binary interactions and subset choice," European Journal of Operational Research, Elsevier, vol. 92(1), pages 182-192, July.
    5. Crama, Y. & Pascual J., R. & Torres, A., 2004. "Optimal procurement decisions in the presence of total quantity discounts and alternative product recipes," European Journal of Operational Research, Elsevier, vol. 159(2), pages 364-378, December.
    6. Sina Abbasi & Ilias Vlachos & Shabnam Rekabi & Mohammad Talooni, 2023. "Designing the Distribution Network of Essential Items in the Critical Conditions of Earthquakes and COVID-19 Simultaneously," Sustainability, MDPI, vol. 15(22), pages 1-23, November.
    7. Cherif, Mohamed Sadok & Chabchoub, Habib & Aouni, Belaid, 2008. "Quality control system design through the goal programming model and the satisfaction functions," European Journal of Operational Research, Elsevier, vol. 186(3), pages 1084-1098, May.
    8. Richard J. Forrester & Warren P. Adams & Paul T. Hadavas, 2010. "Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(1), pages 1-12, February.
    9. Shahzad Bhatti & Michael Lim & Ho-Yin Mak, 2015. "Alternative fuel station location model with demand learning," Annals of Operations Research, Springer, vol. 230(1), pages 105-127, July.
    10. Aouni, Belaid & Ben Abdelaziz, Foued & Martel, Jean-Marc, 2005. "Decision-maker's preferences modeling in the stochastic goal programming," European Journal of Operational Research, Elsevier, vol. 162(3), pages 610-618, May.
    11. Kharrat, Aida & Chabchoub, Habib & Aouni, Belaid & Smaoui, Soulef, 2007. "Serial correlation estimation through the imprecise Goal Programming model," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1839-1851, March.
    12. Scott Kolodziej & Pedro Castro & Ignacio Grossmann, 2013. "Global optimization of bilinear programs with a multiparametric disaggregation technique," Journal of Global Optimization, Springer, vol. 57(4), pages 1039-1063, December.
    13. Allouche, Mohamed Anis & Aouni, Belaïd & Martel, Jean-Marc & Loukil, Taïcir & Rebaï, Abdelwaheb, 2009. "Solving multi-criteria scheduling flow shop problem through compromise programming and satisfaction functions," European Journal of Operational Research, Elsevier, vol. 192(2), pages 460-467, January.
    14. Lakhal, Salem & Martel, Alain & Kettani, Ossama & Oral, Muhittin, 2001. "On the optimization of supply chain networking decisions," European Journal of Operational Research, Elsevier, vol. 129(2), pages 259-270, March.

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