Solving the multi-vehicle multi-covering tour problem

The well-known multi-vehicle covering tour problem (m-CTP) involves finding a minimum-length set of vehicle routes passing through a subset of vertices, subject to constraints on the length of each route and the number of vertices that it contains, such that each vertex not included in any route is covered. Here, a vertex is considered as covered if it lies within a given distance of at least a vertex of a route. This article introduces a generalized variant of the m-CTP that we called the multi-vehicle multi-covering Tour Problem (mm-CTP). In the mm-CTP, a vertex must be covered at least not only once but several times. Three variants of the problem are considered. The binary mm-CTP where a vertex is visited at most once, the mm-CTP without overnight where revisiting a vertex is allowed only after passing through another vertex and the mm-CTP with overnight where revisiting a vertex is permitted without any restrictions. We first propose graph transformations to convert the last two variants into the binary one and focus mostly on solving this variant. A special case of the problem is then formulated as an integer linear program and a branch-and-cut algorithm is developed. We also develop a Genetic Algorithm (GA) that provides high-quality solutions for the problem. Extensive computational results on the new problem mm-CTP as well as its other special cases show the performance of our methods. In particular, our GA outperforms the current best metaheuristics proposed for a wide class of CTP problems.