Computation of lower bounds for a multiple depot, multiple vehicle routing problem with motion constraints

In this paper, the problem of planning paths for a collection of vehicles passing through a set of targets is considered. Each vehicle starts at a specified location (called a depot) and it is required that each target be on the path of at least one vehicle. Every vehicle has a motion constraint and the path of each vehicle must satisfy that constraint. In this article, we developed a method to compute lower bounds to this path planning problem by relaxing some of the constraints and posing it as a standard multiple traveling salesmen problem. For those problem instances where the distance between every pair of targets is at least 4 units, another method is developed to compute a lower bound using the convexity property of the length of such paths. The proposed bounds are numerically corroborated.