A genetic algorithm with a mixed region search for the asymmetric traveling salesman problem

This paper presents a genetic algorithm to solve the asymmetric traveling salesman problem. The genetic algorithm proposed in this study extends search space by purposefully generating and including infeasible solutions in the population. Instead of trying to maintain feasibility with crossover operations, it searches through both feasible and infeasible regions for good quality solutions. It is also shown in the article that the size of the infeasible region defined by solutions with subtours dominates that of a feasible region in the asymmetric traveling salesman problem. A comparative computational study using benchmark problems shows that the proposed genetic algorithm is a viable option for hard asymmetric traveling salesman problems. The asymmetric traveling salesman problem appears in various applications, such as vehicle routing problems, mixed Chinese postman problems, and scheduling problems with sequence dependent setups. Although there exist several heuristic procedures and branch and bound algorithms for it, the problem is still a difficult combinatorial optimization problem. The main purpose of the paper is to present a new genetic algorithm for the problem and to show its effectiveness. To give a justification for the algorithm, the paper also analyses the sizes of feasible and infeasible regions in the asymmetric traveling salesman problem. This analysis provides a basis for the choice of the solution representation (coding) scheme adopted in the genetic algorithm. The genetic operators that are well suited for this representation scheme are then identified for the problem.