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Using the parametric approach to solve the continuous-time linear fractional max–min problems

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  • Ching-Feng Wen
  • Hsien-Chung Wu

Abstract

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm. Copyright Springer Science+Business Media, LLC. 2012

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  • Ching-Feng Wen & Hsien-Chung Wu, 2012. "Using the parametric approach to solve the continuous-time linear fractional max–min problems," Journal of Global Optimization, Springer, vol. 54(1), pages 129-153, September.
    • Handle: RePEc:spr:jglopt:v:54:y:2012:i:1:p:129-153
    DOI: 10.1007/s10898-011-9751-9
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    References listed on IDEAS

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    1. Ching-Feng Wen & Yung-Yih Lur & Yan-Kuen Wu, 2010. "A recurrence method for a special class of continuous time linear programming problems," Journal of Global Optimization, Springer, vol. 47(1), pages 83-106, May.
    2. Ching-Feng Wen & Hsien-Chung Wu, 2011. "Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems," Journal of Global Optimization, Springer, vol. 49(2), pages 237-263, February.
    3. Schaible, Siegfried & Ibaraki, Toshidide, 1983. "Fractional programming," European Journal of Operational Research, Elsevier, vol. 12(4), pages 325-338, April.
    4. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
    5. Lisa Fleischer & Jay Sethuraman, 2005. "Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and Extensions," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 916-938, November.
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    Cited by:

    1. Hsien-Chung Wu, 2019. "Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems," Mathematics, MDPI, vol. 7(5), pages 1-50, May.
    2. Hsien-Chung Wu, 2021. "Robust Solutions for Uncertain Continuous-Time Linear Programming Problems with Time-Dependent Matrices," Mathematics, MDPI, vol. 9(8), pages 1-52, April.
    3. Ching-Feng Wen, 2013. "Continuous-Time Generalized Fractional Programming Problems. Part I: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 365-399, May.
    4. Ching-Feng Wen, 2013. "Continuous-Time Generalized Fractional Programming Problems, Part II: An Interval-Type Computational Procedure," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 819-843, March.

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