Abstract
Simplex-type algorithms perform successive pivoting operations (or iterations) in order to reach the optimal solution. The choice of the pivot element at each iteration is one of the most critical steps in simplex-type algorithms. The flexibility of the entering and leaving variable selection allows to develop various pivoting rules. This chapter presents six pivoting rules used in each iteration of the simplex algorithm to determine the entering variable: (i) Bland’s rule, (ii) Dantzig’s rule, (iii) Greatest Increment Method, (iv) Least Recently Considered Method, (v) Partial Pricing rule, and (vi) Steepest Edge rule. Each technique is presented with: (i) its mathematical formulation, (ii) a thorough numerical example, and (iii) its implementation in MATLAB. Finally, a computational study is performed. The aim of the computational study is twofold: (i) compare the execution time of the presented pivoting rules, and (ii) highlight the impact of the choice of the pivoting rule in the number of iterations and the execution time of the revised simplex algorithm.
This is a preview of subscription content, log in via an institution to check access.
References
Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (1990). Linear programming and network flows. New York: John Wiley & Sons, Inc.
Benichou, M., Gautier, J., Hentges, G., & Ribiere, G. (1977). The efficient solution of large–scale linear programming problems. Mathematical Programming, 13, 280–322.
Bland, R. G. (1977). New finite pivoting rules for the simplex method. Mathematics of Operations Research, 2(2), 103–107.
Clausen, J. (1987). A note on Edmonds-Fukuda’s pivoting rule for the simplex method. European Journal of Operations Research, 29, 378–383.
Dantzig, G. B. (1963). Linear programming and extensions. Princeton, NJ: Princeton University Press.
Forrest, J. J., & Goldfarb, D. (1992). Steepest-edge simplex algorithms for linear programming. Mathematical Programming, 57(1–3), 341–374.
Fukuda, K. (1982). Oriented matroid programming. Ph.D. Thesis, Waterloo University, Waterloo, Ontario, Canada.
Fukuda, K., & Terlaky, T. (1999). On the existence of a short admissible pivot sequence for feasibility and linear optimization problems. Pure Mathematics and Applications, 10(4), 431–447
Gärtner, B. (1995). Randomized optimization by simplex-type methods. Ph.D. thesis, Freien Universität, Berlin.
Goldfarb, D., & Reid, J. K. (1977). A practicable steepest–edge simplex algorithm. Mathematical Programming, 12(3), 361–371.
Harris, P. M. J. (1973). Pivot selection methods for the Devex LP code. Mathematical Programming, 5, 1–28.
Klee, V., & Minty, G. J. (1972). How good is the simplex algorithm. In O. Shisha (ed.) Inequalities – III. New York and London: Academic Press Inc.
Maros, I., & Khaliq, M. H. (1999). Advances in design and implementation of optimization software. European Journal of Operational Research, 140(2), 322–337.
Murty, K. G. (1974). A note on a Bard type scheme for solving the complementarity problem. Opsearch, 11, 123–130.
Papadimitriou, C. H., & Steiglitz, K. (1982). Combinatorial optimization: algorithms and complexity. Englewood Cliffs, NJ: Prentice-Hall, Inc.
Ploskas, N. (2014). Hybrid optimization algorithms: implementation on GPU. Ph.D. thesis, Department of Applied Informatics, University of Macedonia.
Ploskas, N., & Samaras, N. (2014). Pivoting rules for the revised simplex algorithm. Yugoslav Journal of Operations Research, 24(3), 321–332.
Ploskas, N., & Samaras, N. (2014). GPU accelerated pivoting rules for the simplex algorithm. Journal of Systems and Software, 96, 1–9.
Świetanowski, A. (1998). A new steepest edge approximation for the simplex method for linear programming. Computational Optimization and and Applications, 10(3), 271–281.
Terlaky, T., & Zhang, S. (1993). Pivot rules for linear programming: a survey on recent theoretical developments. Annals of Operations Research, 46(1), 203–233.
Thomadakis, M. E. (1994). Implementation and evaluation of primal and dual simplex methods with different pivot-selection techniques in the LPBench environment. A research report. Texas A&M University, Department of Computer Science.
Van Vuuren, J. H., & Grundlingh, W. R. (2001). An active decision support system for optimality in open air reservoir release strategies. International Transactions in Operational Research, 8(4), 439–464.
Wang, Z. (1989). A modified version of the Edmonds-Fukuda algorithm for LP problems in the general form. Asia-Pacific Journal of Operational Research, 8(1), 55–61.
Zadeh, N. (1980). What is the worst case behavior of the simplex algorithm. Technical report, Department of Operations Research, Stanford University.
Zhang, S. (1991). On anti-cycling pivoting rules for the simplex method. Operations Research Letters, 10, 189–192.
Ziegler, G. M. (1990). Linear programming in oriented matroids. Technical Report No. 195, Institut für Mathematik, Universität Augsburg, Germany.
Author information
Authors and Affiliations
6.1 Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Ploskas, N., Samaras, N. (2017). Pivoting Rules. In: Linear Programming Using MATLAB® . Springer Optimization and Its Applications, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-319-65919-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-65919-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65917-6
Online ISBN: 978-3-319-65919-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)