Abstract
This paper proposes a streamlined form of simplex method which provides some great benefits over traditional simplex method. For instance, it does not need any kind of artificial variables or artificial constraints; it could start with any feasible or infeasible basis of an LP. This method follows the same pivoting sequence as of simplex phase 1 without showing any explicit description of artificial variables which also makes it space efficient. Later in this paper, a dual version of the new method has also been presented which provides a way to easily implement the phase 1 of traditional dual simplex method. For a problem having an initial basis which is both primal and dual infeasible, our methods provide full freedom to the user, that whether to start with primal artificial free version or dual artificial free version without making any reformulation to the LP structure. Last but not the least, it provides a teaching aid for the teachers who want to teach feasibility achievement as a separate topic before teaching optimality achievement.
Highlights
Linear programming has been an indispensable area in the progress of the computational sciences [1]
Since simplex method is so far still practically the best known pivot algorithm for solving LPs [10], this has arose the need of developing more general linear program solving methods in which one may start from an arbitrary initial basic solution
In this paper we have presented a dual version of our method which is an artificial constraint free version of the dual simplex method
Summary
Linear programming has been an indispensable area in the progress of the computational sciences [1]. Since simplex method is so far still practically the best known pivot algorithm for solving LPs [10], this has arose the need of developing more general linear program solving methods in which one may start from an arbitrary initial basic solution (not necessarily a feasible one). The term auxiliary form is commonly used in the context of simplex method as a special purpose linear program constructed by incorporating a sufficient number of artificial variables into the system in order to develop a pseudo feasible basis, and appending an objective function of minimizing sum of all artificial variables to reach a feasible basis to the original system. The value of xÀB is showing infeasibility of the current basis B (Note: xþB and xÀB work just like slack and artificial variables in usual simplex method). By using the paradigm (described above ), we may construct the following augmented matrix showing basic variable corresponding to each row
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