AbstractLinear programming is a very well known and deeply applied field of optimization theory. One of its most famous and used algorithms is the so called Simplex algorithm, independently proposed by Kantorovič and Dantzig, between the end of the 30s and the end of the 40s. Even if extremely powerful, the Simplex algorithm suffers of one initialization issue: its starting point must be a feasible basic solution of the problem to solve. To overcome it, two approaches may be used: the two-phases method and the Big-M method, both presenting positive and negative aspects. In this work we aim to propose a non-Archimedean and non-parametric variant of the Big-M method, able to overcome the drawbacks of its classical counterpart (mainly, the difficulty in setting the right value for the constant M). We realized such extension by means of the novel computational methodology proposed by Sergeyev, known as Grossone Methodology. We have validated the new algorithm by testing it on three linear programming problems.