Depth-First Solutions for the Deliveryman Problem on Tree-Like Networks: An Evaluation Using a Permutation Model

The Deliveryman Problem, or DMP (also referred to as the Traveling Repairman Problem TRP), requires a single traveling service unit with a specified home location to serve a set of customers, each of which has a weight or priority. The objective is to find a tour which minimizes a weighted average of the times the customers wait for service. This paper examines the effectiveness of depth-first routes for the DMP on tree and cactus networks. Depth-first routes involve traversing the tree or cactus network in depth-first manner. The best depth-first solution for the DMP can be found very quickly. The evaluation of depth-first routes on general tree and cactus networks is based on a novel formulation for the DMP which employs permutation decision variables and includes as constraints necessary optimality conditions based on network structure. We are unaware of any previous explicit use of these permutation variables in the routing literature. This formulation leads to a Lagrangian relaxation which can be solved as a single-machine minimum weighted-completion time problem with forest precedence constraints. This subproblem is embedded in a subgradient optimization procedure and lower bounds on the costs of optimal DMP solutions are calculated. Computational tests were carried out to determine the quality of the depth-first solutions and the sharpness of the permutation lower bounds. Problem characteristics considered were problem size, variability in arc lengths, variability in customer weights, size of cycles and percentage of nodes in cycles.