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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 3851520, 11 pages
Research Article

Feasible Initial Population with Genetic Diversity for a Population-Based Algorithm Applied to the Vehicle Routing Problem with Time Windows

Research Center in Engineering and Applied Sciences, Autonomous University of Morelos State, Avenida Universidad 1001, Colonia Chamilpa, 62209 Cuernavaca, MOR, Mexico

Received 23 August 2015; Accepted 17 January 2016

Academic Editor: Panos Liatsis

Copyright © 2016 Marco Antonio Cruz-Chávez and Alina Martínez-Oropeza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A stochastic algorithm for obtaining feasible initial populations to the Vehicle Routing Problem with Time Windows is presented. The theoretical formulation for the Vehicle Routing Problem with Time Windows is explained. The proposed method is primarily divided into a clustering algorithm and a two-phase algorithm. The first step is the application of a modified -means clustering algorithm which is proposed in this paper. The two-phase algorithm evaluates a partial solution to transform it into a feasible individual. The two-phase algorithm consists of a hybridization of four kinds of insertions which interact randomly to obtain feasible individuals. It has been proven that different kinds of insertions impact the diversity among individuals in initial populations, which is crucial for population-based algorithm behavior. A modification to the Hamming distance method is applied to the populations generated for the Vehicle Routing Problem with Time Windows to evaluate their diversity. Experimental tests were performed based on the Solomon benchmarking. Experimental results show that the proposed method facilitates generation of highly diverse populations, which vary according to the type and distribution of the instances.