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Duality of the improved algebraic method (DIAM)

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  • Dalalah, Doraid
  • Lev, Benjamin

Abstract

In this note we present a variant of the improved algebraic method (IAM) using a duality analysis to solve linear programming (LP) problems where more insights to the method are presented. When the coordinates of all vertices are computed, any feasible point can be expressed as a linear combination of the vertices. The objective function is expressed as a weighted sum of its evaluation at the feasible vertices and the optimal point is associated with the highest/lowest coefficient of the weighted sum. In this work two adaptations of LP objective function are formulated in primal and dual domains. A simple LP bounds test is also presented which includes unbounded solution space in the IAM. The presented analysis can determine degeneracy and/or alternative optima from the dual parametric objective function. It also spots the optimal solution by intersecting the primal and dual parametric objective functions. The proposed approach is simple and enhances the understanding of the simplex method. We demonstrate several numerical examples to explain the proposed analysis.

Suggested Citation

  • Dalalah, Doraid & Lev, Benjamin, 2009. "Duality of the improved algebraic method (DIAM)," Omega, Elsevier, vol. 37(5), pages 1027-1035, October.
    • Handle: RePEc:eee:jomega:v:37:y:2009:i:5:p:1027-1035
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    1. Kuo, Ching-Chung & Nicholls, Gillian M., 2007. "A mathematical modeling approach to improving locomotive utilization at a freight railroad," Omega, Elsevier, vol. 35(5), pages 472-485, October.
    2. Arsham, Hossein & Adlakha, Veena & Lev, Benjamin, 2009. "A simplified algebraic method for system of linear inequalities with LP applications," Omega, Elsevier, vol. 37(4), pages 876-882, August.
    3. Shanmugam, Ramalingam & Johnson, Charles, 2007. "At a crossroad of data envelopment and principal component analyses," Omega, Elsevier, vol. 35(4), pages 351-364, August.
    4. Adlakha, Veena & Kowalski, Krzysztof & Vemuganti, R.R. & Lev, Benjamin, 2007. "More-for-less algorithm for fixed-charge transportation problems," Omega, Elsevier, vol. 35(1), pages 116-127, February.
    5. Wang, Guoqing & Cheng, T.C. Edwin, 2007. "Customer order scheduling to minimize total weighted completion time," Omega, Elsevier, vol. 35(5), pages 623-626, October.
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