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A Linearization to the Sum of Linear Ratios Programming Problem

Author

Listed:
  • Mojtaba Borza

    (Department of Mathematical Sciences, Faculty of Science & Technology, UKM Bangi, Selangor 43600, Malaysia)

  • Azmin Sham Rambely

    (Department of Mathematical Sciences, Faculty of Science & Technology, UKM Bangi, Selangor 43600, Malaysia)

Abstract

Optimizing the sum of linear fractional functions over a set of linear inequalities (S-LFP) has been considered by many researchers due to the fact that there are a number of real-world problems which are modelled mathematically as S-LFP problems. Solving the S-LFP is not easy in practice since the problem may have several local optimal solutions which makes the structure complex. To our knowledge, existing methods dealing with S-LFP are iterative algorithms that are based on branch and bound algorithms. Using these methods requires high computational cost and time. In this paper, we present a non-iterative and straightforward method with less computational expenses to deal with S-LFP. In the method, a new S-LFP is constructed based on the membership functions of the objectives multiplied by suitable weights. This new problem is then changed into a linear programming problem (LPP) using variable transformations. It was proven that the optimal solution of the LPP becomes the global optimal solution for the S-LFP. Numerical examples are given to illustrate the method.

Suggested Citation

  • Mojtaba Borza & Azmin Sham Rambely, 2021. "A Linearization to the Sum of Linear Ratios Programming Problem," Mathematics, MDPI, vol. 9(9), pages 1-10, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:1004-:d:545738
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    References listed on IDEAS

    as
    1. Benson, Harold P., 2007. "A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem," European Journal of Operational Research, Elsevier, vol. 182(2), pages 597-611, October.
    2. Bartosz Sawik, 2012. "Downside Risk Approach for Multi-Objective Portfolio Optimization," Operations Research Proceedings, in: Diethard Klatte & Hans-Jakob Lüthi & Karl Schmedders (ed.), Operations Research Proceedings 2011, edition 127, pages 191-196, Springer.
    3. Y. Almogy & O. Levin, 1971. "A Class of Fractional Programming Problems," Operations Research, INFORMS, vol. 19(1), pages 57-67, February.
    4. H. P. Benson, 2002. "Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 1-29, January.
    5. A. Charnes & W. W. Cooper, 1962. "Programming with linear fractional functionals," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 9(3‐4), pages 181-186, September.
    6. Zvi Drezner & Siegfried Schaible & David Simchi‐Levi, 1990. "Queueing‐location problems on the plane," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(6), pages 929-935, December.
    7. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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