Bi-Objective Search for the Traveling Salesman Problem with Time Windows and Vacant Penalties

This paper investigates a Traveling Salesman Problem with Time Windows and Vacant Penalties (TSP-TW-VP), which plans a path to service a set of machines at different locations within their respective time windows while minimizing two objective functions: the finish time and penalty for machine vacancy. There is often no single solution that optimizes both objectives simultaneously, and the problem thus seeks the Pareto-optimal solutions. TSP-TW-VP generalizes TSP-TW and is therefore NP-hard. To solve the problem, this paper develops an algorithm called Search with Look-Ahead Pruning (S-LAP) that is guaranteed to find all Pareto-optimal solutions for TSP-TW-VP. S-LAP gains computational efficiency by introducing a novel look-ahead pruning rule, and a fast dominance checking method based on both the objective functions and path history. Experimental results show that the proposed look-ahead pruning and fast dominance can speed up the search for 2-8 times over 4 different datasets.