Grossone Methodology for Lexicographic Mixed-Integer Linear Programming Problems

In this work we have addressed lexicographic multi-objective linear programming problems where some of the variables are constrained to be integer. We have called this class of problems LMILP, which stands for Lexicographic Mixed Integer Linear Programming. Following one of the approach used to solve mixed integer linear programming problems, the branch and bound technique, we have extended it to work with infinitesimal/infinite numbers, exploiting the Grossone Methodology. The new algorithm, called GrossBB, is able to solve this new class of problems, by using internally the GrossSimplex algorithm (a recently introduced Grossone extension of the well-known simplex algorithm, to solve lexicographic LP problems without integer constraints). Finally we have illustrated the working principles of the GrossBB on a test problem.