Extensions to K-Medoids with Balance Restrictions over the Cardinality of the Partitions

Abstract

The zones design occurs when small areas or basic geographic units (BGU) must be grouped into acceptable zonesunder the requirements imposed by the case study. These requirements can be the generation of intra-connectedand/or compact zones or with the same amount of habitants, clients, communication means, public services, etc. Inthis second point to design a territory, the selection and adaptation of a clustering method capable of generatingcompact groups while keeping balance in the number of objects that form each group is required.

The classic partitioning stands out (also known as classification by partition among the clustering or classificationmethods [1]). Its properties are very useful to create compact groups.

An interesting property of the classification by partitions resides in its capability to group different kinds of data.When working with geographical data, such as the BGU, the partitioning around medoids algorithms have givensatisfactory results when the instances are small and only the objective of distances minimization is optimized. Inthe presence of additional restrictions, the K-medoids algorithms, present weaknesses in regard to the optimalityand feasibility of the solutions.

In this work we expose 2 variants of partitioning around medoids for geographical data with balance restrictions overthe number of objects within each group keeping the optimality and feasibility of the solution. The first algorithmconsiders the ideas of k-meoids and extends it with a recursive constructive function to find balanced solutions. Thesecond algorithm searches for solutions taking into account a balance between compactness and the cardinality of thegroups (multiobjective). Different tests are presented for different numbers of groups and they are compared withsome results obtained with Lagrange Relaxation. This kind of grouping is needed to solve aggregation for TerritorialDesign problems.