Mathematical models and solving methods for diversity and equity optimization

Discrete diversity optimization basically consists of selecting a subset of elements of a given set in such a way that the sum of their pairwise distances is maximized. Equity, on the other hand, refers to minimizing the difference between the maximum and the minimum distances in the subset of selected elements to balance their diversity. Both problems have been studied in the combinatorial optimization literature, but recently major drawbacks in their classic mathematical formulations have been identified. We propose new mathematical models to overcome these limitations, including multi-objective optimization, and heuristics to solve large-size instances of them. Specifically, we propose a matheuristic based on the CMSA framework for diversity and a GRASP heuristic for equity. Our extensive experimentation compares the original models with the new proposals by analyzing the solutions of our heuristics and those of the previous approaches, both from a single objective and a bi-objective paradigm. We also evaluate their quality with respect to the optimal solutions obtained with CPLEX, size permitting. Statistical analysis allows us to draw significant conclusions.