New approaches for solving the Probabilistic Traveling Salesman Problem

The Probabilistic Traveling Salesman Problem (PTSP) is a TSP problem in which each customer requires a visit only with a given probability, independently of the others. The goal is to find an a priori tour of minimal expected length over all customers, with the strategy of visiting a random subset of customers in the same order as they appear in the a priori tour. The PTSP is NP-hard, and only very small instances may be solved by exact methods. Here, we concentrate on the design of algorithms (heuristics) for finding good suboptimal solutions to the problem. Given the analogies between the TSP and the PTSP, it is reasonable to expect that, like in the TSP, a good heuristic for the problem is composed by two components. The first component is a solution construction algorithm, which finds an initial solution. The second component is a local search algorithm, which tries to improve as much as possible the starting solution. A good heuristic algorithm usually integrates these two components, and repeats the sequence constructionimprovement of a solution several times, untill a good solution or some other termination criterion is not satisfied. In this thesis we propose new (for the PTSP) computational approaches for both solution construction and local search algorithms. As far as the solution construction algorithm is concerned, we investigate the potentialities of the ant colony optimization metaheuristic for the PTSP, which is a new approach for this type problem. This choice is motivated by the observation that, for the TSP, when adding to ant colony optimization algorithms local search procedures, the quality of the results obtained was close to that obtainable by other state-of-the-art methods. Also for the PTSP, it is reasonable to expect good performance of ant colony optimization algorithms with local search. As far as local search is concerned, we show that the current algorithms in the literature, 2-p-opt and 1-shift, are inadequate. This results pose the question whether a new fast local search operator for the PTSP can be derived. One possibility to design fast local search is to use an approximation of the objective function, which is faster to be computed than the exact one. Two types of approximations are proposed and analyzed in this thesis. Part of the research developed in this thesis has been published, or is about to be published, in the following papers: [6] L. Bianchi, L. M. Gambardella, and M. Dorigo. An ant colony optimization approach to the probabilistic traveling salesman problem. In Proceedings of PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, volume 2439 of Lecture Notes in Computer Science. Springer,