Enhanced Algorithms for Tricriteria Shortest Path Problems with Two Bottleneck Objective Functions

This paper deals with a tricriteria path problem involving two bottleneck objective functions and a cost. It presents two methods for computing shortest paths in subnetworks, obtained by restricting the set of arcs according to the bottleneck values in order to find the minimal complete set of Pareto-optimal solutions. These procedures are enhanced by using the objective values of the determined shortest paths to reduce the number of considered subnetworks, and thus the number of solved shortest path problems. Two algorithms are introduced, evaluated and compared with the previous literature. Results for random instances with 5 000 nodes, an average degree of 100 and 200 distinct bottleneck values show that one of them solves 15 times fewer shortest path problems than before, and the other 43 times fewer, while the CPU times are improved 11 and 29 times, respectively. On average the new algorithms computed the minimal complete set in 7 000 node networks with 200 bottleneck values in less than four minutes. Moreover two variants of these methods are introduced. Their aim is to choose the solutions with the best bottleneck values when the cost is the same. For random problems this leads to a maximum improvement of 30% and 9% for costs in [1,10] and [1,100], respectively.