The artificial-free technique along the objective direction for the simplex algorithm

Abstract

The simplex algorithm is a popular algorithm for solving linear programming problems. If the origin point satisfies all constraints then the simplex can be started. Otherwise, artificial variables will be introduced to start the simplex algorithm. If we can start the simplex algorithm without using artificial variables then the simplex iterate will require less time. In this paper, we present the artificial-free technique for the simplex algorithm by mapping the problem into the objective plane and splitting constraints into three groups. In the objective plane, one of variables which has a nonzero coefficient of the objective function is fixed in terms of another variable. Then it can split constraints into three groups: the positive coefficient group, the negative coefficient group and the zero coefficient group. Along the objective direction, some constraints from the positive coefficient group will form the optimal solution. If the positive coefficient group is nonempty, the algorithm starts with relaxing constraints from the negative coefficient group and the zero coefficient group. We guarantee the feasible region obtained from the positive coefficient group to be nonempty. The transformed problem is solved using the simplex algorithm. Additional constraints from the negative coefficient group and the zero coefficient group will be added to the solved problem and use the dual simplex method to determine the new optimal solution. An example shows the effectiveness of our algorithm.