A new initial basis for the simplex method combined with the nonfeasible basis method

Abstract

The simplex method is a popular method to solve a linear programming problem. It starts by choosing an initial basis associated with the origin point which is far from the optimal point in some problems. In this paper, we introduce a construction of an initial basis for starting the simplex method associated with an extreme point which is close to the optimal solution. This construction involves the angle between the coefficient vector of the objective function and the coefficient vector of each constraint. The small angles will be considered to construct a basis. If the proposed initial basis gives the infeasible solutions, then the nonfeasible basis method is performed to avoid the use of artificial variables. Not only the starting point is close to the optimal point, but also the artificial variable is needless. The results in the computational aspect show that the number of iterations and the running time of our method are less than other methods.

A Short Literature on Linear Programming Problem

“…Hence, it is observed that Nabli and Chahdoura [27] (2015) method helps us to reduce the number of iterations as well as time. Finally, Sangngern and Boonperm [28] (2020) proposed another different way to handle the simplex method.…”

“…Sangngern and Boonperm [28](2020) developed a new simplex method with the initial basis, that considers the angle between the objective function coefficient vectors and each constraint. This method uses restrictions with modest angles to generate an initial basis close to the ideal answer.…”

“…By using linear programming problems, computational experiments are randomly generated. Sangngern and Boonperm [28](2020) technique, the Two-phase method, Big-M method and non-feasible basis method solved the problems. Sangngern and Boonperm [28](2020) technique outperformed other issues by demonstrating its potential for tackling real-world optimization problems.…”

On the modifications of simplex method

“…Some traditional methods are introduced, such as Largest Increase Rules, Steeps Edge rules [2] and Random Edge rules [3], to reduce steps. Recently, many latest revised methods are proposed, including Greatest Improvement Rule [4,5], Quick Simplex Method [6,7] and Three-Phase Method [8]. Thus, in this study, a variety of methods will be presented and evaluated.…”