Heuristic Approaches for the Probabilistic Traveling Salesman Problem

The probabilistic traveling salesman problem (PTSP) was introduced by Jaillet [6] as an extension of the classical traveling salesman problem (TSP). In the PTSP, the salesman does not necessarily have to visit every node, but a probability of requiring a visit is given for each node. The goal is to find an a-priori tour that includes every node and minimizes the expected length of an a-posteriori tour which contains the nodes with the given probabilities and skips the other ones. A real world application of the PTSP would be e.g. to plan an every-day tour for the postman. After each delivery he always proceeds to the next address on the pre-planned tour for which a delivery is required. Formally we are given a complete graph G = 〈V,E〉 with V containing n nodes and E containing m edges. Each edge (i, j) ∈ E is assigned a cost dij > 0 and each node v ∈ V has a probability pv of being visited. If the probabilities pv are equal for every node, the problem is called homogeneous and otherwise it is heterogeneous [5]. A solution is an a-priori tour T = 〈v1, . . . , vn〉 over all nodes with vn connected to v1 and its expected costs are defined as