Minimal Energy Routing of a Leader and a Wingmate with Periodic Connectivity
We consider a route planning problem in which two unmanned vehicles are required to complete a set of tasks present at distinct locations, referred to as targets, with minimum energy consumption. The mission environment is hazardous, and to ensure a safe operation, the UVs are required to communicate with each other at every target they visit. The problem objective is to determine the allocation of the tasks to the UVs and plan tours for the UVs to visit the targets such that the weighted sum of the distances traveled by the UVs and the distances traveled by the communicating signals between them is minimized. We formulate this problem as an Integer program and show that naively solving the problem using commercially available off-the-shelf solvers is insufficient in determining scalable solutions efficiently. To address this computational challenge, we develop an approximation and a heuristic algorithm, and employ them to compute high-quality solutions to a special case of the problem where equal weights are assigned to the distances traveled by the vehicles and the communicating signals. For this special case, we show that the approximation algorithm has a fixed approximation ratio of 3.75. We also develop lower bounds to the optimal cost of the problem to evaluate the performance of these algorithms on large-scale instances. We demonstrate the performance of these algorithms on 500 randomly generated instances with the number of targets ranging from 6 to 100, and show that the algorithms provide high-quality solutions to the problem swiftly; the average computation time of the algorithmic solutions is within a fraction of a second for instances with at most 100 targets. Finally, we show that the approximation ratio has a variable ratio for the weighted case of the problem. Specifically, if ρ denotes the ratio of the weights assigned to the distances representing the communication and travel costs, the algorithm has an a posteriori ratio of $3 + \frac{3ρ}{4}$ when ρ ≥ 1, and $\frac{3}{ρ}$ + $\frac{3}{4}$ when ρ ≤ 1.