A General Variable Neighborhood Search with Mixed VND for the multi-Vehicle multi-Covering Tour Problem

The well-known Vehicle Routing-Allocation Problem (VRAP) receives recently more attention than the classical routing problems. This article deals with a special case of the VRAP named the multi-vehicle multi-Covering Tour Problem (mm-CTP-p). More precisely, the mm-CTP-p is a generalized variant of the multi-vehicle Covering Tour Problem (m-CTP-p). In both problems, the objective is to find a minimum length set of vehicle routes while satisfying the total demands by visiting vertices by the route or covering vertices which does not included in any route. But, in the m-CTP-p, the demand of a vertex can be satisfied with only one coverage whereas in the mm-CTP-p, a vertex must be covered several times to be completely served. Indeed, a vertex is covered if it lies within a specified distance of at least one vertex of a route. We develop a General Variable Neighborhood Search algorithm (GVNS) with a mixed Variable Neighborhood Descent (mixed-VND) method to solve the problem. Experiments were conducted using benchmark instances from the literature. Extensive computational results on mm-CTP-p problems show the performance of our method.