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In China, the transportation capacity of high-speed railways is gradually sufficient to provide services for high-value express freight besides meeting passenger demands. However, similar as the passengers, express freight demands fluctuate and show clear peak and trough periods daily. Therefore, optimizing running numbers of express freight trains on high-speed railway by periods is quite necessary to guarantee the revenues of railway industry and to meet the various requirements of all consignees simultaneously. First, a space-period-pattern three-dimensional network with virtual arcs is built to describe the departure period selection and the stopping or skip-stopping operations. Second, by constructing the arcs' impedance function, the user equilibrium principle is introduced to optimize the express freight flow distribution for each pattern in each period. To elaborate the comprehensive goal of balancing the relationship between profits and the flow distribution, a bi-level programming model is established. The upper model addresses the railway industry's maximum profits, and the lower model addresses the minimum and similar impedance values of the final express freight flow distribution. Finally, through the use of a hybrid algorithm that combines the heuristic genetic algorithm with the Frank-Wolfe algorithm, an experimental case of 10 stations on the Beijing-Xi'an high-speed railway corridor is used to validate the study. The results show that the express freight volume is reasonably distributed to each kind of virtual arc, the impedance value of each pattern is minimized and almost equalized, and the profits of railway industry are maximized by optimizing the number and departure time of trains by each pattern in each period on the basis of meeting all the express freight demands.

Optimizing the Number of Express Freight Trains on a High-speed Railway Corridor by the Departure Period