A Lagrangian Algorithm for Multiple Depot Traveling Salesman Problem With Revisit Period Constraints

This work presents a Multiple Depot Traveling Salesman Problem with revisit period constraints. The revisit period constraints are relevant to persistent routing applications, where these constraints represent maximum time between successive visits to a target. This problem is first posed as a Mixed Integer Linear Program. The coupling constraints in the primal problem are then relaxed via Lagrangian relaxation. Minimizing the resulting Lagrangian over the primal variables can be separated into an individual subproblem for each salesman. An algorithm to solve the subproblems to near-optimality that scales well for larger instances is presented. The dual function is then maximized, where three different dual update methods are studied: the subgradient method, the ellipsoid algorithm, and a bundle algorithm. Further, a primal reconstruction algorithm is presented to reconstruct feasible solutions from the solution of the dual algorithm. The quality of these solutions are compared to the optimal solutions obtained from the CPLEX MILP optimizer. The results of extensive numerical testing show that the dual algorithm presented was computationally efficient, capable of finding high quality solutions, and scale well compared to CPLEX. Further, it was seen that the bundle method used showed better convergence than the other update methods within the dual algorithm. Note to Practitioners— This work was motivated by the problem of routing UAVs in the presence of timing constraints, specifically motivated by military applications. This problem has constraints on the frequency at which targets are visited, such that high priority targets must be visited more frequently. Similar prior work exists in the literature for a variety of routing problems for different timing or resource constraints. Techniques which are successful on one set of constraints may be less useful when applied to different constraints. We present a Lagrangian based approach to the multiple depot traveling salesman variant. The results show the utility of a Lagrangian based technique to this problem, as both the solution quality and computation time scale well with problem size. The Lagrangian based technique presented here is seen to provide, on average, high quality solutions with improved computational time over a branch-and-bound algorithm, which solves the problem exactly. While the solutions returned from the algorithm are on average of good quality, they do exhibit some variance. Alternative algorithms may give more consistent results. The technique presented in this paper is general and may be applied to other routing problems where the revisit period constraints are applied. Future research includes application of this method to dynamic environments, where the targets’ states are unknown to the UAVs except when being monitored. Further, alternative techniques can be developed for this problem and compared to both the branch-and-bound and the Lagrange algorithm presented here.