Optimal hyperpaths with non-additive link costs

Non-additive fares are common in public transport as well as important for a range of other assignment problems. We discuss the problem of finding optimal hyperpaths under such conditions assuming a cost vector with a limited number of marginal decreasing costs depending on the number of links already traversed. We illustrate that these non-additive costs lead to violation of Bellman's optimality principle which in turn means that standard procedures to obtain optimal destination specific hyperpath trees are not feasible. To overcome the problem we introduce the concepts of a “travel history vector” and critical vs fixed nodes. The former records the expected number of traversed links until a node, and the latter distinguishes nodes for which the fare cost can be determined deterministically. With this we develop a 2-stage solution approach. In the first stage we test whether the optimal hyperpath can be obtained by backward search. If this is not the case, we propose a selective hyperpath generation to a (small) number of critical nodes and combine this with standard hyperpath search. We illustrate our approach by applying it to the Sioux Falls network showing that even for link cost functions (fare stages) with large step changes we are able to obtain all optimal hyperpaths in a reasonable computational time.