An integer fixed-charge multicommodity flow (FCMF) model for train unit scheduling

An integer fixed-charge multicommodity flow (FCMF) model is used as the first part of a two-phase approach for train unit scheduling, and solved by an exact branch- and-price method. To strengthen knapsack constraints and deal with complicated scenarios arisen in the integer linear program (ILP) from the integer FCMF model, preprocessing is used by computing convex hulls of sets of points representing all possible train formations utilizing multiple unit types.