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| Model parameters |
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| Set of urban rail lines, indexed by l, l′. The upstream and downstream directions of a line are considered as two separated lines. |
| Set of two terminal stations and transfer stations on line l, Sl =, where nl + 1 is the number of stations on line l |
| Transfer at station s from line l to l′ |
| Departure time of the last train from the starting station on line l |
| , | Lower and upper bound for the running time of the last train from station s-1 to station s on line l |
| , | Lower and upper bound for the dwell time of the last train at station s on line l |
| Upper bound for the total travel time of the last train on line l |
| Departure time of the penultimate train from station s on line l |
| Minimal safety headway between successive departure of trains on line l |
| Minimal safety headway between the departure of the penultimate train and the arrival of the last train on line l |
| Transfer walking time of passengers at transfer direction i |
| Probability distribution function of transfer walking time at transfer direction i |
| Transfer demand at transfer direction i |
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| Decision variables |
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| Running time of the last train from station s-1 to station s on line l |
| Dwell time of the last train at station s on line l |
| Arrival time of the last train at station s on line l |
| Departure time of the last train from station s on line l |
| Violation of upper bound for the dwell time at station s on line l |
| Transfer redundant time for the last-train passengers in transfer direction i |
| Binary variable indicating transfer accessibility of transfer direction i with the fixed transfer walking time |
| Continuous variable indicating transfer accessibility of transfer direction i with the heterogeneous transfer walking time |
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