Topologically Consistent Fuzzy Clustering in a Neurodynamic Model of the Traveling Salesman Problem

Abstract

A new neurodynamic model for joint optimization of the traveling salesman problem route and fuzzy cluster partitioning of a set of vertices is proposed. The model’s energy function includes three components: a Hopfield-Tank term encoding route feasibility constraints; a fuzzy means term ensuring geometric compactness of clusters; and a topological matching term penalizing for discontinuities in cluster membership between adjacent vertices along the route. An iterative minimization scheme is developed. The following convergence properties are proven: monotonic energy decrease and convergence of the energy sequence to a limit value. Computational experiments on a test example showed that the proposed model finds solutions that coincide with the global optimum, while the resulting clusters differ significantly from the results of the classical fuzzy c-means algorithm, reflecting the topology of the optimal route. The results of the study can be used for problems of delivery territory zoning, production operation planning, and condensation of large factor models.