Solving the Asymmetric Traveling Salesman Problem with Periodic Constraints Giuseppe Paletta Dipartimento di Economia e Statistica, Universita ` della Calabria, 87036 Rende (CS), Italy Chefi Triki Dipartimento di Matematica, Universita ` di Lecce, via Arnesano, 73100 Lecce, Italy In this article we describe a heuristic algorithm to solve the asymmetrical traveling salesman problem with peri- odic constraints over a given m-day planning horizon. Each city i must be visited r i times within this time hori- zon, and these visit days are assigned to i by selecting one of the feasible combinations of r i 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 im- provement procedure that modifies the assigned com- bination 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. Compu- tational comparisons on special instances indicate en- couraging results. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(1), 31–37 2004 Keywords: asymmetric traveling salesman problem; periodic constraints; construction algorithm; improvement procedure 1. INTRODUCTION The necessity of formulating periodic constraints within routing problems arises in many real-life applications. Many distribution problems are, indeed, characterized by the fact that the cities should not be visited at each day of an m-day planning horizon, but rather a specified number of times. This is the case, for example, of some mail collection/ delivery problems, snow removal, refuse collection, grocery distribution, and fuel oil delivery. The periodicity aspect in the routing problems has at- tracted the interest of many researchers over the last 2 decades. A wide variety of well-known routing problems has been covered in this direction. Christofides and Beasley [3] have formulated the period vehicle routing problem (PVRP) and have solved the period traveling salesman problem (PTSP) by using heuristic algorithms. Other heu- ristics to solve the PTSP have been also proposed by Paletta [7], Chao et al. [1], and recently again by Paletta [8]. The PTSP problem has also been studied by Cordeau et al. [4], who have proposed a tabu search metaheuristic algorithm. They have also employed their tabu search approach to solve the PVRP and the multidepot vehicle routing problem. Gaudioso and Paletta [6] have solved a variant of the PVRP by using an algorithm based on a combination of a city- route assignment heuristic and a bin-packing algorithm for the route-vehicle assignment. The PVRP has also been studied by Chao et al. [2], who have proposed an efficient heuristic that first assigns the visit combinations by solving an integer linear programming model and solves a VRP for each day and then uses local improvement and reinitializa- tion techniques to improve the quality of the solution. This article represents a further step towards the com- pletion of the above-mentioned work. Indeed, we cover here another routing problem, namely the Periodic Asymmetric Traveling Salesman Problem (PATSP). In the next section, we will define the asymmetric trav- eling salesman problem, and describe the different ap- proaches for formulating periodicity constraints. Section 3 will be devoted to the development of a heuristic framework to solve the PATSP. The computational performance of the heuristic will be discussed in Section 4, and some conclud- ing remarks will be made in Section 5. 2. PROBLEM DESCRIPTION The PATSP represents a natural extension of the asym- metric traveling salesman problem to cover an m-day plan- ning horizon. Within this time horizon, each city i must be visited r i times, with at most one visit per day. These visits are assigned to i by selecting one of a given set of feasible combinations of r i visit days with the objective of minimiz- Received April 2002; accepted January 2004 Correspondence to: G. Paletta; e-mail: g.paletta@unical.it DOI 10.1002/net.20011 Published online in Wiley InterScience (www.interscience.wiley. com). © 2004 Wiley Periodicals, Inc. NETWORKS— 2004