Let G=(V,E) be a graph with time-dependent edges where the cost of a path p through the graph is determined by a vector functions F(p)=[f_1(p),f_2(p), \dots, f_n(p)], where f_1,f_2,...,f_n are independent objective functions. Where n>1 there is no clear idea of what a ``best'' solution is, instead we turn to the idea of Pareto-optimality to define the efficiency of a path. Given the set of paths P through the network, a path p' is Pareto-optimal if for every p in P for all the objective functions (f_i(p) >= f_i(p')).
The problem of planning itineraries on a transportation system involves computing the set of optimal paths through a time-dependent network where the cost of a path is determined by more than one, possibly non-linear and non-additive, cost function. This thesis introduces an algorithmic toolkit for finding the set of Pareto-optimal paths in time-dependent networks in the presence of multiple objective functions.
Multi-criteria path optimization problems are known to be NP-Hard, however, by exploiting geometric and periodic properties of the dynamic graphs that model transit networks we show that it is possible to compute the Pareto-optimal solutions sets rapidly without using heuristics. We show that we can solve the itinerary problem in the presence of response time constraints for a large scale graph.