Solving the Traveling Salesman Problem through Extended Changing Crossover Operators

In order to efficiently obtain an approximate solution of the traveling salesman problem (TSP), a new method of extended changing crossover operators (ECXO) which can change any crossover operator of both genetic algorithms (GAs) and ant colony optimization (ACO) for another crossover operator at any time is proposed. In ECXO, all the cyclic paths which are produced by some operator are not only able to be delivered to another operator and are but also able to be updated by it. Through ECXO, we can succeed to search for the global optimum solution while maintaining diversity of the cyclic paths. In this study, we investigate an ECXO which mainly controlls edge assembly crossover (EAX), since recently EAX is found to be powerful to find out the global optimum solution by preserving good edges of previous tours and maintaining diversity of cyclic paths in the population. EAX represents any tour as a set of edges that connect two cities. With EAX a parent can exchange his edges with another parent’s ones reciprocally to create sub-cyclic paths, before restructuring a cyclic path by combining them with making distances minimum. EAX is an extension of edge exchange crossover (EXX) with which a parent can change his edge for another parent’s edge and he can make arrangement of his edges reverse or foward. Our C experiments show that EAX is superior to both edge recombination crossover (EX) and ACO from points of view of ability to combine parents’ sub-paths together to create a global optimum solution if parents’ sub-paths are locally optimal, that with EAX alone any child spends much time on creating locally optimal sub-paths because he can not take into consideration of edges’ lengths when he generates sub-cyclic paths, that since with EX or ACO any child or any ant selects the next city he visits by edges’ lengths or by tours’ lengths deposited on edges as pheromone, he can only generate local optimum paths and can not select an optimum one by considering global links of edges, he is apt to fall into the local optimum solution. Hence we study ECXO (ACO(or EX)(→EXX)→EAX) in which we generate cyclic-paths which are locally optimum and which have variety of arranging edges with EX, ACO and EXX in early generations, and we generate cyclic-paths with EAX to create global optimum solution efficiently after generations, where EAX succeeds cyclic-paths EX, ACO and EXX generate. The efficiency of ECXO is verified by C experiments using the data of lin318 which is a midium sized TSP in TSPLIB. Table 1 demonstrates its verification. Table 1 shows best length, average length, the number of optimum trials, relative error which is defined as ((average length)/(optimal length) 1), computational time to find the best length, and average computational time to find it. The average length is the average value of fifteen best lengths. They are obtained from 15 independent runs by using different seed-ids which range from one to fifteen, where seed-id generates different initial random tours for lin318, where EAX has Ncross of 30. From Table 1, it has been shown that four ECXOs as well as EAX can find best lengths and their relative errors are low. Four ECXOs can find the best solution earlier than EAX and its improvement-ratio ranges from 10% to 30%. Among them ECXO (EX→EAX) has least amount of mean time to find the optimum one. In Fig. 1 we compare four ECXOs and EAX on the convergence speed to the optimal length of 42,029. The data is obtained from a trial of seed_id of one with Ncross of 100. It shows that firstly ECXO (EX → EXX → EAX) find the best solution at 2,224 sec and that secondly ECXO (EX → EAX), thirdly EAX, fourthly ECXO (ACO→EAX), and finally ECXO(ACO → EXX→EAX) find it at 3,055 sec. In this paper it has been also shown that we can select the optimal generation to change crossover operators based on diversity and convergence of chromosomes. The efficiency of ECXO is also verified by bench-mark test using d198 and pcb442 in TSPLIB. Table 1. Comparison of ECXOs with EAX, EX, EXX and ACO on best length, average length, the number of optimal trials, relative error, and computational time to find the best length. Results are obtained from fifteen independent trials. Ncross=30 in EAX. The optimum length of lin318 is 42,029 in TSPLIB