Linear programming with infinite, finite, and infinitesimal values in the right-hand side

Abstract

The goal of this work is to propose a new type of constraint for linear programs: inequalities having infinite, finite, and infinitesimal values in the right-hand side. Because of the nature of such constraints, the feasible region polyhedron becomes more complex, since its vertices can be represented by non-purely finite coordinates, and so is the optimum of the problem. The introduction of such constraints enlarges the class of linear programs, where those described by finite values only become a special case. To tackle optimization problems over such polyhedra, there is a need for an ad-hoc solving routine: this work proposes a generalization of the Simplex algorithm, which is able to solve common linear programs as corner cases. Finally, the study presents three relevant applications that can benefit from the use of these novel constraints, making the use of the extended Simplex algorithm essential. For each application, an exemplifying benchmark is solved, showing the effectiveness of the proposed routine.