The challenge of maximizing the diversity of a collection of points arises in a variety of settings, and the growing interest of dealing with diversity resulted in an effort to study these problems in the last few years. Generally speaking, maximizing diversity consists in selecting a subset of points from a given set in such a way that a measure of their diversity is maximized. Different objective functions have been proposed to capture the notion of diversity, being the sum and the minimum of the distances between the selected points the most widely used. However, in all these models, the number of points to be selected is established beforehand, which in some settings can be unrealistic. In this paper, we target a variant recently introduced in which the model includes capacity values, which reflects the real situation in many location problems. We propose a mathematical model and a heuristic based on the Scatter Search methodology to maximize the diversity while satisfying the capacity constraint. Scatter search is a memetic algorithm hybridizing evolutionary global search with a problem-specific local search. Our empirical analysis with previously reported instances shows that the mathematical model implemented in Gurobi solves to optimality many more instances than the previous published model, and the heuristic outperforms a very recent development based on GRASP. We present a statistical analysis that permits us to draw significant conclusions.

最大化设置点的多样性的挑战出现在各种环境中，并且处理多样性的兴趣日益浓厚，导致人们在过去几年中努力研究这些问题。一般而言，最大化多样性包括以给定集合选择点的子集的方式，以使它们的多样性的度量最大化。已经提出了不同的目标函数来捕获多样性的概念，即最广泛使用的选定点之间距离的总和和最小值。但是，在所有这些模型中，要预先确定要选择的点数，在某些设置中这可能是不现实的。在本文中，我们针对最近引入的一个变体，其中模型包含容量值，该值反映了许多位置问题中的实际情况。我们提出了一种基于分散搜索方法的数学模型和启发式算法，以在满足容量约束的同时最大程度地提高多样性。散布搜索是一种模因算法，将进化全局搜索与特定于问题的本地搜索混合在一起。我们对以前报告的实例进行的经验分析表明，与以前发布的模型相比，在Gurobi中实现的数学模型可以将最优实例求解得更多，并且启发式算法的性能优于最近基于GRASP的开发。我们提出了一项统计分析，使我们可以得出重要的结论。散布搜索是一种模因算法，将进化全局搜索与特定于问题的本地搜索混合在一起。我们对以前报告的实例进行的经验分析表明，与以前发布的模型相比，在Gurobi中实现的数学模型可以将最优实例求解得更多，并且启发式算法的性能优于最近基于GRASP的开发。我们提出了一项统计分析，使我们可以得出重要的结论。散布搜索是一种模因算法，将进化全局搜索与特定于问题的本地搜索混合在一起。我们对以前报告的实例进行的经验分析表明，与以前发布的模型相比，在Gurobi中实现的数学模型可以将最优实例求解得更多，并且启发式算法的性能优于最近基于GRASP的开发。我们提出了一项统计分析，使我们可以得出重要的结论。