Solving the asymmetric traveling salesman problem with periodic constraints

In this article we describe a heuristic algorithm to solve the asymmetrical traveling salesman problem with periodic constraints over a given m-day planning horizon. Each city i must be visited ri times within this time horizon, and these visit days are assigned to i by selecting one of the feasible combinations of ri visit days with the objective of minimizing the total distance traveled by the salesman. The proposed algorithm is a heuristic that starts by designing feasible tours, one for each day of the m-day planning horizon, and then employs an improvement procedure that modifies the assigned combination to each of the cities, to improve the objective function. Our heuristic has been tested on a set of test problems purposely generated by slightly modifying known test problems taken from the literature. Computational comparisons on special instances indicate encouraging results.