A mixed-integer linear programming model to solve the Multidimensional Multi-Way Number Partitioning Problem

Abstract

This paper addresses a mixed-integer linear programming model for solving the Multidimensional Multi-Way Number Partitioning Problem (MDMWNPP), the most general version of the family of Number Partitioning Problems. First, a contextualization concerning the Two-Way Number Partitioning Problem (TWNPP), the Multi-Way Number Partitioning Problem (MWNPP), the Multidimensional Two-Way Number Partitioning Problem (MDTWNPP), and the approached problem are presented. MDMWNPP is a generalization of MDTWNPP for the number k of subsets greater than two. After the proposed model is introduced, some properties are shown, and the main constraints are proved. To validate the consistency of the proposed mathematical formulation, a comparison, using instances from the literature, between the proposed model for k=2 and a model from the literature for MDTWNPP, is performed. The proposed model showed competitive results against this specialized model and, besides, solved instances of the problem for values of k equals to 3 and 4.