Table 1 Definition of variables

Train index

t

Segment index

s

Segment sequence number in train route

j

Station index

i

Set of trains

T

Set of segments |S| = m

S

Set of stations |J| = m + 1

J

Direction indicator for train t, p(t) = 0 for an inbound train and p(t) = 1 for an outbound train

p(t)

Segment index of the jth traveling segment in a route for train i, σ (t,j) = j for outbound trains, σ (t,j) = m + 1-j for inbound trains

σ(t, j)

Downstream station number of the jth traveling segment in a given route for train t, b(t,j) = j foroutbound trains, b(t,j) = m -j for inbound trains

b(t, j)

Planned departure time for train t at its first station

k_t

Free running time for train t at segment s

f _(t,s)

Minimum required station dwell time before train t entering segment s

d _(t,s)

Maximum allowed station dwell time before train i entering segment j

\( {\overline{d}}_{\left(t,s\right)} \)

Minimum headway between arrival and departure times of two consecutive trains at segment j

h _s

Minimum headway between arrival times of two consecutive trains at station u

g _i

Entering time for train t at segment s, i.e., start time for job t on machine s

o _t,s

Leaving time for train t at segment s, i.e., end time for job t on machine s

c _t,s

Binary Variable,1 if train t is scheduled before train t’ on segment s, 0 otherwise

\( {A}_{t,{t}^{\prime },s} \)

Sufficiently large constant

M