In logistic problems, an Integral Multi-Commodity Network Design Problem on a time-space network is often used to model the problem of routing transportation means and assigning freight units to those means. In Physical Internet and Synchromodal networks an interactive planning approach is preferable, meaning that calculation times of a single planning step should be short. In this paper we provide ways to reduce the size of the problem formulation based on cutting planes, that are effective in reducing the computation time for Integer Linear Programming problem-based solution methods.

Central in this paper is a transportation network, in which containers are transported for multiple agents. This network is modelled by a Space Time Network, in which the travel time of modalities is fixed and independent of the occupancy of the network. To find the best allocation of containers to paths in this network, a flow problem can be solved. The System Optimal solution found then is the solution in which the total costs of the network are minimised. This paper introduces the idea of a fair User Equilibrium solution in such problem. The proposed approach changes the network, using a toll scheme, such that the fair User Equilibrium Solution in this changed network equals the System Optimal solution in the original network. This can be used to fairly redistribute the cost of the network among the users.