New Subtour-Based Crossover Operator for the TSP

Genetic algorithm (GA) is a very useful method for the global search of large search space and has been applied to various problems. It has two kinds of important search mechanisms, crossover and mutation. Because the performance of GA depends on these operators, a large number of operators have been developed for improving the performance of GA. Especially many researchers have more interested in crossover operator than mutation operator because crossover operator has charge of the responsibility of local search. We only deal with crossover operator. In this paper we propose subtour preservation crossover (SPX), which uses a similar subtour enumeration method to other subtour-based crossovers but has an amount of difference in method that generates a valid tour. SPX generates offsprings as many as we wish by using genetic information of parents propagated over a lot of generations. And the most severe drawback of subtour-based crossovers is they cannot generate a different offspring when two identical parents are selected for crossover. At our experiments, in case of over 200 generations, the average number of times which two identical parents are selected is over 20. That is, it shows the identical parents are selected about over 30% for total crossovers (Pc = 0.6). So if we do not consider a supplementary method for avoiding this problem, the improvement of solutions will be more and more difficult because crossover operator cannot generate new offspring. So our method for generating new offspring is as follows. If identical parents are selected in SPX, it first generates two solutions randomly and then applies a local search method to each solution for competing with good solution existing already in population. Figure 1 indicates the procedure of SPX. The first step of SPX is to enumerate common subtours and then reconnect each subtour using reconnection rule. Reconnection rule is as follow.