1 Introduction
2 Mathematical model
2.1 Assumptions and parameters
Parameters | Explanations |
---|---|
j
| Station number of line l, j = 1, 2, …, N |
\(V_{\hbox{max} }\)
| Maximum speed of the trains |
\(L_{j}\)
| Distance between stations j −1 and j |
\(r_{j}\)
| Travel time between stations j −1 and j, j = 2, 3,…, N, \(r_{j} = 60\;L_{j} /V_{\hbox{max} }\) |
\(D_{i,j}\)
| Departure time of train i from stop j |
\(A_{i,j}\)
| Arrival time of train i at stop j |
\(\tau_{i,j}\)
| Dwell time of train i at stop j; uniform dwell time is τ |
\(H_{i,j}\)
| Headway between trains i and i −1 at station j, assumed as two constants; i.e., H
ij
= h |
\(W_{i,jk}\)
| Number of passengers on train i, boarding at stop j and about to alight at station k, 1 ≤ j < k ≤ N |
\(S_{i,jk}\)
| Number of passengers skipped by train i at station j, who intend to alight at station k, 1 ≤ j < k ≤ N |
\(S_{i,j}\)
| Total number of passengers skipped by train i at station j |
\(B_{i,j}\)
| Number of passengers boarding train i at station j |
\(V_{i,j}\)
| Number of passengers alighting from train i at station j |
\(\partial\)
| Constant parameter for the acceleration time of trains |
\(\beta\)
| Constant parameter for the deceleration time of trains |
\(\xi ,\eta\)
| Preference variables depending on the situation in which the destination station is skipped |
\(\lambda_{j,k}\)
| Arrival rate of passengers heading to station k from station j |
\(\lambda_{j}\)
| Total arrival rate of passengers at station j |
\(y_{i,j}\)
| A binary variable for the stop-skipping decision of train i at stop j; i.e., y
i,j
= 0 if stop j is skipped and y
i,j
= 1 otherwise |
2.2 Mathematical formulation
2.3 Objective function
3 Algorithm
4 Numerical simulations
4.1 Data settings
O | D | Number of boarding passengers | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | ||
1 | 0 | 150 | 180 | 100 | 220 | 190 | 210 | 220 | 250 | 260 | 240 | 230 | 215 | 200 | 190 | 170 | 130 | 150 | 110 | 70 | 3,485 |
2 | 0 | 0 | 160 | 170 | 220 | 200 | 190 | 180 | 160 | 230 | 210 | 220 | 270 | 260 | 210 | 180 | 140 | 160 | 120 | 60 | 3,340 |
3 | 0 | 0 | 0 | 110 | 130 | 170 | 180 | 240 | 280 | 300 | 210 | 280 | 220 | 190 | 140 | 110 | 80 | 70 | 70 | 50 | 2,830 |
4 | 0 | 0 | 0 | 0 | 120 | 180 | 160 | 150 | 100 | 190 | 220 | 260 | 250 | 230 | 190 | 140 | 130 | 100 | 60 | 80 | 2,560 |
5 | 0 | 0 | 0 | 0 | 0 | 80 | 120 | 160 | 150 | 180 | 200 | 210 | 240 | 260 | 250 | 230 | 200 | 150 | 100 | 60 | 2,590 |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 60 | 100 | 120 | 150 | 180 | 160 | 220 | 180 | 200 | 140 | 130 | 90 | 60 | 20 | 1,810 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 80 | 120 | 160 | 170 | 200 | 230 | 250 | 200 | 150 | 160 | 120 | 80 | 50 | 1,970 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 50 | 80 | 130 | 160 | 200 | 230 | 190 | 140 | 110 | 90 | 70 | 60 | 1,510 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 150 | 180 | 100 | 220 | 190 | 210 | 170 | 110 | 90 | 50 | 80 | 1,550 |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 160 | 170 | 130 | 200 | 150 | 180 | 130 | 100 | 70 | 90 | 1,380 |
11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 | 140 | 160 | 140 | 120 | 100 | 120 | 90 | 80 | 1,050 |
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 110 | 130 | 160 | 150 | 130 | 110 | 100 | 60 | 950 |
13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 160 | 130 | 120 | 80 | 130 | 140 | 150 | 910 |
14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 90 | 110 | 80 | 130 | 80 | 70 | 560 |
15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 80 | 115 | 135 | 90 | 110 | 530 |
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 90 | 110 | 80 | 90 | 370 |
17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 120 | 70 | 100 | 290 |
18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 70 | 60 | 130 |
19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 70 | 70 |
20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Alighting passengers | 0 | 150 | 340 | 380 | 690 | 820 | 920 | 1,130 | 1,230 | 1,700 | 1,900 | 2,090 | 2,445 | 2,640 | 2,450 | 2,190 | 1,915 | 1,975 | 1,510 | 1,410 | 27,885 |
4.2 Simulation results
4.2.1 Optimal SOS
Station stop | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Y
ij
| 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
Station stop | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|
Y
ij
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
4.2.2 SOS model with different constraints
Station stop | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Y
ij
| 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
Station stop | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|
Y
ij
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 |
Station stop | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Y
ij
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
Station stop | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|
Y
ij
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |