###### Abstract  
Passenger demand for railway transportation rapidly increases in peak periods, and the transport capacity for existing trains is not sufficient. Railway companies usually adopt the strategy of adding additional trains in peak periods to meet the higher passenger demand. Designing a good operational plan for additional trains becomes a challenge for operators, though. A new optimization approach for designing an operational plan for additional trains is proposed in this paper. The number of trains, the operational plan, the stop plan, and the timetable for each train can be considered simultaneously in the new optimization approach, which will make it easier to design an operational plan for additional trains. A multi-objective nonlinear model with three objectives of minimizing total running distance, dwelling time, and unsatisfied passengers is proposed. Big-\(M\) is introduced to transform the nonlinear model into a linear model. The solver CPLEX is used to solve the transformed linear model and obtain the optimal operational plan. Small-scale numerical experiments are implemented to show the effectiveness of the optimization approach. The large-scale case of the Beijing-Shanghai railway corridor is studied to demonstrate that the optimization approach can be applied to real-word and large-scale situations.  
adding additional trains; operational plan; passenger demand; high-speed railway corridors; multi-objective nonlinear programming model +
Footnote †: journal: Applied Soft  
design an efficient operational plan for additional trains that maximizes the satisfaction of passenger demand and minimizes operating costs.  
In China, tickets are booked 30 days in advance. Passengers must board trains according to the information on the tickets. Tickets show the train number, price, departure time, and stations for getting on and off. Thus, the number of additional trains, operational zone, stop plan, and timetable for each additional train must be scheduled in advance. The number of additional trains is a complicated issue for a railway company. Too many additional trains will increase the operating costs, but too few additional trains cannot meet passenger demand. The operational zone specifies the origin and terminal stations for the additional train. The stop plan determines whether the train stops at the station in the operational zone. The timetable determines the departure and arrival times of each train for each station in its operational zone. Passenger distribution determines the number of passengers for each origin-destination (OD) pair on each train. The number of additional trains, operational zones, stop plans, timetables, and passenger distribution are the key elements of the operational plan for additional trains and are related to each other. Thus, the numbers of trains, operational zones, stop plans, timetables, and the passenger demand for additional trains are jointly optimized in this paper. Specifically, the purpose of this paper is to design an operational plan, including the number of trains, operational zones, stop plans, timetables, and passenger demand, for additional trains in peak periods. In this paper, the operational zones, timetables, and stop plans for existing trains are fixed, and the operation of additional trains cannot disturb the operation of existing trains. Passengers are accustomed to the timetables and stop plans of existing trains. Frequent changes in operational zones, timetables, and stop plans for existing trains cause inconvenience for passengers. The railway corridor is a part of the railway system, and a change in the operational plan for existing trains can disturb the whole railway system. However, passenger distribution for existing trains is reoptimized to make full use of the capacity of existing trains, which is not fully utilized in the off-peak period.