We introduce a class of incremental network design problems focused on investigating the optimal choice and timing of network expansions. We concentrate on an incremental network design problem with shortest paths. We investigate structural properties of optimal solutions, we show that the simplest variant is NP-hard, we analyze the worst-case performance of natural greedy heuristics, and we derive a 4-approximation algorithm.

This paper describes a method for scheduling trains on the Hunter Valley Coal Chain rail network. Coal for a particular ship is railed from different mines to stockpiles at one of the Port's terminals. The coal producers decide which mines will supply each order in what proportion, so there is no flexibility in the allocation of mines to cargoes. We are presented with a list of tonnes of coal which need to be transported from specified load points at mines to specified stockpiles at the port. The operators of the rail network provide a number of paths, with specified arrival and departure times, that can be used for coal movement. The requirement to assign coal trains to these existing paths makes this rail scheduling problem different to most of those discussed in the literature. In this paper we describe the problem in detail, demonstrate that it is very large making it difficult to solve with commercial MILP solvers, and show that our Lagrangian heuristic is able to produce high quality solutions in a reasonable amount of time.