Genetic Crossover Operators for the Capacitated Vehicle Routing Problem

Abstract

We study the capacitated vehicle routing problem (CVRP) which is a well-known NP-hard combinatorial optimization problem (COP). The aim of the problem is to serve different customers by a convoy of vehicles starting from a depot so that sum of the routing costs under their capacity constraints is minimized. Since the problem is very complicated, solving the problem using exact methods is almost impossible. So, one has to go for the heuristic/metaheuristic methods and genetic algorithm (GA) is broadly applied metaheuristic method to obtain near optimal solution to such COPs. So, this paper studies GAs to find solution to the problem. Generally, to solve a COP, GAs start with a chromosome set named initial population, and then mainly three operators-selection, crossover and mutation, are applied. Among these three operators, crossover is very crucial in designing and implementing GAs, and hence, numerous crossover operators were developed and applied to different COPs. There are two major kinds of crossover operators-blind crossovers and distance-based crossovers. We intend to compare the performance of four blind crossover and four distance-based crossover operators to test the suitability of the operators to solve the CVRP. These operators were originally proposed for the standard travelling salesman problem (TSP). First, these eight crossovers are illustrated using same parent chromosomes for building offspring(s). Then eight GAs using these eight crossover operators without any mutation operator and another eight GAs using these eight crossover operators with a mutation operator are developed. These GAs are experimented on some benchmark asymmetric and symmetric instances of numerous sizes and various number of vehicles. Our study revealed that the distance-based crossovers are much superior to the blind crossovers. Further, we observed that the sequential constructive crossover with and without mutation operator is the best one for the CVRP. This estimation is validated by Student's t-test at 95% confidence level. We further determined a comparative rank of the eight crossovers for the CVRP.