Operation Research-Simplex Method Procedure And Solved Problems

Operation Research-Simplex Method Procedure And Solved Problems-73
This method consists of two phases and its general principle is the following: in the first phase, we start by searching an initial support with the Gauss-Jordan elimination method, then we proceed to the search of an initial feasible solution by solving an auxiliary problem having one artificial variable and an obvious feasible solution.

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In 1977, Gabasov and Kirillova [5] have generalized the simplex method and developed the primal support method which can start by any basis and any feasible solution and can move to the optimal solution by interior points or boundary points.

The latter is adapted by Radjef and Bibi to solve LPs which contain two types of variables: bounded and nonnegative variables [6]. developed the adaptive method to solve, particularly, linear optimal control problems [7].

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.

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).That is why many researchers have given a new interest for developing new initialization techniques.These techniques aim to find a good initial basis and a good initial point and use a minimum number of artificial variables to reduce memory space and CPU time.In 1984, Karmarkar presented for the first time an interior point algorithm competitive with the simplex method on large-scale problems [20].The efficiency of the simplex method and its generalizations depends enormously on the first initial point used for their initialization.LP is considered as the most important technique in operations research.Indeed, it is widely used in practice, and most of optimization techniques are based on LP ones.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].The results of the numerical comparison revealed that finding the initial support by the Gauss elimination method consumes much time, and transforming the equality constraints to inequality ones increases the dimension of the problem.Hence, the proposed approaches are competitive with the full artificial basis simplex method for solving small problems, but they are not efficient to solve large problems.


Comments Operation Research-Simplex Method Procedure And Solved Problems

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