Algorithms for a stochastic selective travelling salesperson problem

In this paper, the selective travelling salesperson problem with stochastic service times, travel times, and travel costs (SSTSP) is addressed. In the SSTSP, service times, travel times and travel costs are known a priori only probabilistically. A non-negative value of reward for providing service is associated with each customer and there is a pre-specified limit on the duration of the solution tour. It is assumed that not all potential customers can be visited within this tour duration limit, even under the best circumstances. And, thus, a subset of customers must be selected. The objective of the SSTSP is to design an a priori tour that visits each chosen customer once such that the total profit (total reward collected by servicing customers minus travel costs) is maximized and the probability that the total actual tour duration exceeds a given threshold is no larger than a chosen probability value. We formulate the SSTSP as a chance-constrained stochastic program and propose both exact and heuristic approaches for solving it. Computational experiments indicate that the exact algorithm is able to solve small- and moderate-size problems to optimality and the heuristic can provide near-optimal solutions in significantly reduced computing time.