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Solving a Freight Railcar Flow Problem Arising in Russia

We consider a variant of the freight railcar flow problem. In this 
problem, we need 1) to choose a set of transportation demands between 
stations in a railroad network, and 2) to fulfill these demands by 
appropriately routing the set of available railcars, while 
maximizing the total profit. We formulate this problem as a 
multi-commodity flow problem in a large space-time graph. Three 
approaches are proposed to solve the Linear Programming relaxation 
of this formulation: direct solution by an LP solver, a column 
generation approach based on the path reformulation, and a ``column 
generation for extended formulations'' approach. In the latter, the 
multi-commodity flow formulation is solved iteratively by dynamic 
generation of arc flow variables. Three approaches have been tested 
on a set of real-life instances provided by one of the largest 
freight rail transportation companies in Russia. Instances with up 
to 10 millions of arc flow variables were solved within minutes of 
computational time.