The phase 1 simplex algorithm on the objective hyperplane

This paper presents a new phase 1 simplex algorithm for solving linear programming problems. In the algorithm, first, the equation with the auxiliary objective function at the optimality as a new constraint is added to phase 1, and then a pivot is performed to generate one vertex on the associated objective hyperplane. This vertex may be feasible or not feasible. If it is feasible, the phase 1 simplex algorithm ends. Otherwise, proceed with the successive iterations fixed on the objective hyperplane. Next, three variants are presented to achieve a feasible vertex or the conclusion that the original problem is infeasible. Variant 1 is a dual simplex algorithm without the row ratio test. In variants 2 and 3, the ideas of the bounding hyperplane method and MBU dual simplex algorithm are applied, respectively. Finally, computational study is done to test the efficiencies of three variants comparing to the ordinary simplex algorithm on some standard test problems from NETLIB and MIPLIB, showing that variant 1 is simple in pivot rule, and variant 2 generally uses fewer iterations to solve those problems, and variant 3 generally spends less executive time at each iteration in the solution process.