# Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

# Simplex method (algorithm)

• Saul I. Gass
• Carl M. Harris
Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_953

A computational procedure for solving linear-programming problems of the form: Minimize (maximize) cx,v subject to Axv = b, xv > 0, where Av is an m × n matrix, cv is an n-dimensional row vector, bv is an m-dimensional column vector, and xv is an n-dimensional variable vector. The simplex method was developed by George B. Dantzig in the late 1940s. The method starts with a known basic feasible solution or an artificial basic solution, and, given that the problem is feasible, finds a sequence of basic feasible solutions (extreme-point solutions) such that the value of the objective function improves or does not degrade. Under a nondegeneracy assumption, the simplex algorithm will converge in a finite number of steps, as there are only a finite number of extreme points and extreme directions of the underlying convex set of solutions. At most, mvariables can be in the solution at a positive level. In each step (iteration) of the simplex method, a new basis is found and developed by...

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