Abstract
The purpose of this paper is to introduce a new pivot rule of the simplex algorithm. The simplex algorithm first presented by George B. Dantzig, is a widely used method for solving a linear programming problem (LP). One of the important steps of the simplex algorithm is applying an appropriate pivot rule to select the basis-entering variable corresponding to the maximum reduced cost. Unfortunately, this pivot rule not only can lead to a critical cycling (solved by Bland’s rules), but does not improve efficiently the objective function. Our new pivot rule 1) solves the cycling problem in the original Dantzig’s simplex pivot rule, and 2) leads to an optimal improvement of the objective function at each iteration. The new pivot rule can lead to the optimal solution of LP with a lower number of iterations. In a maximization problem, Dantzig’s pivot rule selects a basis-entering variable corresponding to the most positive reduced cost; in some problems, it is well-known that Dantzig’s pivot rule, before reaching the optimal solution, may visit a large number of extreme points. Our goal is to improve the simplex algorithm so that the number of extreme points to visit is reduced; we propose an optimal improvement in the objective value per unit step of the basis-entering variable. In this paper, we propose a pivot rule that can reduce the number of such iterations over the Dantzig’s pivot rule and prevent cycling in the simplex algorithm. The idea is to have the maximum improvement in the objective value function: from the set of basis-entering variables with positive reduced cost, the efficient basis-entering variable corresponds to an optimal improvement of the objective function. Using computational complexity arguments and some examples, we prove that our optimal pivot rule is very effective and solves the cycling problem in LP. We test and compare the efficiency of this new pivot rule with Dantzig’s original pivot rule and the simplex algorithm in MATLAB environment.
Highlights
Our goal is to improve the simplex algorithm so that the number of extreme points to visit is reduced; we propose an optimal improvement in the objective value per unit step of the basis-entering variable
Using computational complexity arguments and some examples, we prove that our optimal pivot rule is very effective and solves the cycling problem in Linear programming (LP)
The tableau format of the simplex method follows: Table 1 format reports the value of the objective function z0 = cBT B−1b, the basis variables xB = B−1b, the reduced cost row, which consist of c j = c j − z j = c j − cBT B−1 Aj for non basic variables. ∀j, c j > 0, the LP is at optimal solution
Summary
Linear programming (LP) has been one of the most dynamic areas of applied mathe-. J. Dantzig’s original pivot rule cannot prevent cycling in linear programming and takes a lot of iterations in some cases [3]. To prevent this weakness, many research studies tried to improve the simplex algorithm, via the pivot rule by reducing the number of iterations and the solution time [4]-[6]. Some computational results are reported, comparing the number of iterations from this new rule to Dantzig’s original pivot rule. We report the computational results by testing and comparing the number of iterations from this new rule to Dantzig’s original pivot rule in MATLAB environment. Etoa comparing the speed and the number of iterations from this new pivot rule to classical simplex rule and conclusions drawn
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