A three-phase simplex method for infeasible and unbounded linear programming problems

Abstract

The paper presents a modified artificial basis method MODART, which combine a big-M method with two-phase method. Unlike previous works, the sum of artificial variables in the objective function has not been used, but an additional constraint has been composed for this sum. In contrast with the classical implementation of the simplex method, in the Phase-0, we take into account the objective function of the initial problem and use the big-M method idea for the enlarged problem. If the solution found is infeasible for the initial problem, then Phase1 finds the feasible solution and Phase-2 the optimal solution. In this work, unbounded, infeasible and degenerate problems are mainly considered. Finally, in part five, suggested formulation and solution to the problems is given together with some computational experience. The main ideas are explained by simple examples. Over 60 years of age problem with M has been finally solved in present paper.