An Integer Programming Formulation for the Maximally Diverse Grouping Problem

TL;DR

This paper introduces a new integer programming formulation for the Maximally Diverse Grouping Problem that overcomes previous restrictions, enabling exact solutions for more general and practical scenarios.

Contribution

It proposes a novel integer linear programming model that handles unequal group sizes and diverse distance measures, broadening the problem's solvability and benchmarking capabilities.

Findings

Formulation is applicable to unequal group sizes.

Supports various distance measures beyond Manhattan.

Provides a basis for benchmarking heuristic methods.

Abstract

The Maximally Diverse Grouping Problem (MDGP) is the problem of assigning a set of elements to mutually disjoint groups in order to maximise the overall diversity between the elements. Because the MDGP is NP-complete, most studies have focused on heuristic solution approaches, as compared to exact solution approaches, to the problem. On the one hand, heuristic solution approaches, although common in practice, do not guarantee a global optimal solution. On the other hand, studies that have reformulated the problem as an integer linear programme, which can be solved using exact solution approaches, are either restricted to groups of equal size or restricted to the use of the Manhattan distance. The present paper presents a new integer linear programming formulation that is not subjected to either of these restrictions, and can therefore be used to establish useful benchmarks for the performance of heuristics in a broader range of applications moving forward.