The Efficiency of the Simplex Method: A Survey

• Published in: Management science Vol. 33; no. 3; pp. 301 - 334
• Main Author: Shamir, Ron
• Format: Journal Article
• Language: English
• Published: Linthicum, MD INFORMS 01.03.1987; Institute of Management Sciences; Institute for Operations Research and the Management Sciences
• Series: Management Science
• Subjects: Algorithms; Applied sciences; complexity; Efficiency; Ellipsoids; Exact sciences and technology; Linear programming; Management science; Mathematical analysis; Mathematical inequalities; Mathematical programming; Mathematische Optimierung; Monte Carlo simulation; Monte-Carlo methods; Operational research and scientific management; Operational research. Management science; Optimization; Polynomials; Polytopes; practical experience; probabilistic analysis; Probabilistic modeling; simplex algorithm; Simplex method; Vertices; Monte Carlo method; Linear programming; Simplex method; Mathematical programming
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.33.3.301

The Linear Programming Problem is by far the most widely used optimization model. Its impact on economic and government modeling is immense. The Simplex Method for solving the Linear Programming (LP) Problem, due to George Dantzig, has been an extremely efficient computational tool for almost four decades. The method has been the subject of intense investigations for many years, but some major aspects of its behavior are not fully understood yet. The purpose of this paper is to survey the body of knowledge on the efficiency of the Simplex Method, from both practical and theoretical points of view. Adopting the number of iterations (pivot steps) as the yardstick for efficiency, we survey four aspects of the issue: 1. Reports on practical experience of the method's performance on real-life LP problems. 2. Results on controlled (Monte-Carlo) experiments solving LP problems which were randomly generated according to some predetermined distributions. 3. Complexity results, including theoretical analyses on both upper and lower bounds for the performance of the Simplex as well as non-Simplex algorithms for LP. 4. Results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the Simplex Method. We discuss the consequences and limitations of the various studies. Special emphasis is given to open questions.