3 rd International Conference of Iranian Operations Research Society May 5-6, 2010 Faculty of Mathematics and Computer Science, Amirkabir University of Technology 1 A solution procedure for the Generalized Covering Salesman Problem Zahra Naii-Azimi, Majid Salari, Paolo Toth University of Bologna – DEIS – {zahra.najiazimi2, majid.salari2, paolo.toth}@unibo.it Abstract: Given n nodes, the covering salesman problem is to identify the minimum length tour “covering” all the nodes, i.e. the minimum length tour visiting a subset of the n nodes and such that each node not on the tour is within a predetermined distance from the nodes on the tour. In the Generalized Covering Salesman Problem (GCSP) each node i needs to be covered at least i k times and there is a visiting cost associated with each node. This problem has three variants; in the first case, each node can be visited by the tour at most once, in the second version visiting a node more than once is possible but it is not allowed to stay overnight, and finally, in the third variant the tour can visit each node more than once consecutively. We propose an improvement procedure based on Integer Linear Programming (ILP) techniques. Computational results on benchmark instances from the literature show the effectiveness of the proposed approach. Keywords: Covering Salesman Problem, Generalized Covering Salesman Problem, Heuristic Procedures, Integer Linear Programming. 1. INTRODUCTION The Traveling Salesman Problem (TSP) is one of the best known problems in Operations Research. Given n nodes, the goal is to find the minimum length tour of the nodes, such that the salesman, starting from a node, visits each node exactly once and returns to the starting node [1]. Defined by Current [2], the Covering Salesman Problem (CSP) is to find the minimum length tour of a subset of n given nodes, such that each node i not on the tour is within a predefined covering distance i d from a node that is on the tour. Current and Schilling [3] referred to some real world examples, such as routing of rural healthcare delivery teams where the assumption of visiting each city is not valid since it is sufficient for all cities to be near to some stops on the tour (the inhabitants of those cities which are not in the tour are expected to go to their nearest stop). They suggested a heuristic for the CSP in which in the first step the Set Covering Problem (SCP) over the given nodes is solved. Then the algorithm finds the optimal TSP tour of the nodes generated by solving the SCP. Arkin and Hassin [4] introduced a geometric version of the Covering Salesman Problem. In this problem each of the n nodes specifies a compact set in the plan, its neighborhood, within which the salesman should meet the stop [4]. The goal is computing the shortest length tour that intersects all of the neighborhoods and returns to the initial node. They presented simple heuristics for constructing tours for a variety of neighborhood types [4]. Since sometimes in the real world applications some cities need to be covered more than once, and there is a cost for staying in a city for one night, such as the cost of hotel, parking or other fees that commonly occur in practice, Salari et al. [5] introduced the Generalized Covering Salesman Problem (GCSP). They divided this problem into three variants: Binary GCSP, Integer GCSP without overnight and Integer GCSP with overnight and developed two local search heuristics, LS1 and LS2, for these variants [5]. 2. PROBLEM DEFINITION Given a directed graph ( ) A N G , = with { } n N ,..., 2 , 1 = and } , : , { N j i j i A ∈ > < = as the node and the arc sets, respectively, each node i can cover a subset of nodes i D and has a predetermined coverage demand i k . i F is the fixed cost associated with node i and the solution is feasible if each node i is covered at least i k times by the nodes in the tour. The goal is minimizing the total cost including the tour length and the cost associated with