Operation Research-Simplex Method Procedure And Solved Problems

Linear programming is a mathematical discipline which deals with solving the problem of optimizing a linear function over a domain delimited by a set of linear equations or inequations.

The first formulation of an economical problem as a linear programming problem is done by Kantorovich (1939, [1]), and the general formulation is given later by Dantzig in his work [2].

This method is extended to solve general linear and convex quadratic problems [8–18].

In 1979, Khachian developed the first polynomial algorithm which is an interior point one to solve LP problems [19], but it’s not efficient in practice.

These test problems are practical linear programs modelling various real-life problems arising from several fields such as oil refinery, audit staff scheduling, airline scheduling, industrial production and allocation, image restoration, multisector economic planning, and data fitting.

It has been shown that our approach is competitive with our implementation of the primal simplex method and the primal simplex algorithm implemented in the known open-source LP solver LP_SOLVE.In [25–31], crash procedures are developed to find a good initial basis.In [32], a two-phase support method with one artificial variable for solving linear programming problems was developed.Copyright © 2012 Mohand Bentobache and Mohand Ouamer Bibi.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In this work, we will first extend the full artificial basis technique presented in [7], to solve problems in general form, then we will combine a crash procedure with a single artificial variable technique in order to find an initial support feasible solution for the initialization of the support method.This technique is efficient for solving practical problems.In his experimental study, Millham [23] shows that when the initial basis is available in advance, the single artificial variable technique can be competitive with the full artificial basis one.Wolfe [24] has suggested a technique which consists of solving a new linear programming problem with a piecewise linear objective function (minimization of the sum of infeasibilities).We develop a single artificial variable technique to initialize the primal support method for solving linear programs with bounded variables.We first recall the full artificial basis technique, then we will present the proposed algorithm.


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