# TRAIN, a model of railroad passenger car allocation
#
# References:
# Robert Fourer, David M. Gay and Brian W. Kernighan, "A Modeling Language
# for Mathematical Programming." Management Science 36 (1990) 519-554.
### SCHEDULE SETS AND PARAMETERS ###
set cities;
set links within {c1 in cities, c2 in cities: c1 <> c2};
# Set of cities, and set of intercity links
param last > 0 integer; # Number of time intervals in a day
set times := 1..last; # Set of time intervals in a day
set schedule within
{c1 in cities, t1 in times,
c2 in cities, t2 in times: (c1,c2) in links};
# Member (c1,t1,c2,t2) of this set represents
# a train that leaves city c1 at time t1
# and arrives in city c2 at time t2
### DEMAND PARAMETERS ###
param section > 0 integer;
# Maximum number of cars in one section of a train
param demand {schedule} > 0;
# For each scheduled train:
# the smallest number of cars that
# can meet demand for the train
param low {(c1,t1,c2,t2) in schedule} := ceil(demand[c1,t1,c2,t2]);
# Minimum number of cars needed to meet demand
param high {(c1,t1,c2,t2) in schedule}
:= max (2, min (ceil(2*demand[c1,t1,c2,t2]),
section*ceil(demand[c1,t1,c2,t2]/section) ));
# Maximum number of cars allowed on a train:
# 2 if demand is for less than one car;
# otherwise, lesser of
# number of cars needed to hold twice the demand, and
# number of cars in minimum number of sections needed
### DISTANCE PARAMETERS ###
param dist_table {links} >= 0 default 0.0;
param distance {(c1,c2) in links} > 0
:= if dist_table[c1,c2] > 0 then dist_table[c1,c2] else dist_table[c2,c1];
# Inter-city distances: distance[c1,c2] is miles
# between city c1 and city c2
### VARIABLES ###
var U 'cars stored' {cities,times} >= 0;
# u[c,t] is the number of unused cars stored
# at city c in the interval beginning at time t
var X 'cars in train' {schedule} >= 0;
# x[c1,t1,c2,t2] is the number of cars assigned to
# the scheduled train that leaves c1 at t1 and
# arrives in c2 at t2
### OBJECTIVES ###
minimize cars:
sum {c in cities} U[c,last] +
sum {(c1,t1,c2,t2) in schedule: t2 < t1} X[c1,t1,c2,t2];
# Number of cars in the system:
# sum of unused cars and cars in trains during
# the last time interval of the day
minimize miles:
sum {(c1,t1,c2,t2) in schedule} distance[c1,c2] * X[c1,t1,c2,t2];
# Total car-miles run by all scheduled trains in a day
### CONSTRAINTS ###
account {c in cities, t in times}:
U[c,t] = U[c, if t > 1 then t-1 else last] +
sum {(c1,t1,c,t) in schedule} X[c1,t1,c,t] -
sum {(c,t,c2,t2) in schedule} X[c,t,c2,t2];
# For every city and time:
# unused cars in the present interval must equal
# unused cars in the previous interval,
# plus cars just arriving in trains,
# minus cars just leaving in trains
satisfy {(c1,t1,c2,t2) in schedule}:
low[c1,t1,c2,t2] <= X[c1,t1,c2,t2] <= high[c1,t1,c2,t2];
# For each scheduled train:
# number of cars must meet demand,
# but must not be so great that unnecessary
# sections are run
### DATA ###
data;
set cities := BO NY PH WA ;
set links := (BO,NY) (NY,PH) (PH,WA)
(NY,BO) (PH,NY) (WA,PH) ;
param dist_table := [*,*] BO NY 232
NY PH 90
PH WA 135 ;
param last := 48 ;
param section := 14 ;
set schedule :=
(WA,*,PH,*) 2 5 6 9 8 11 10 13
12 15 13 16 14 17 15 18
16 19 17 20 18 21 19 22
20 23 21 24 22 25 23 26
24 27 25 28 26 29 27 30
28 31 29 32 30 33 31 34
32 35 33 36 34 37 35 38
36 39 37 40 38 41 39 42
40 43 41 44 42 45 44 47
46 1
(PH,*,NY,*) 1 3 5 7 9 11 11 13
13 15 14 16 15 17 16 18
17 19 18 20 19 21 20 22
21 23 22 24 23 25 24 26
25 27 26 28 27 29 28 30
29 31 30 32 31 33 32 34
33 35 34 36 35 37 36 38
37 39 38 40 39 41 40 42
41 43 42 44 43 45 44 46
45 47 47 1
(NY,*,BO,*) 10 16 12 18 14 20 15 21
16 22 17 23 18 24 19 25
20 26 21 27 22 28 23 29
24 30 25 31 26 32 27 33
28 34 29 35 30 36 31 37
32 38 33 39 34 40 35 41
36 42 37 43 38 44 39 45
40 46 41 47 42 48 43 1
44 2 45 3 46 4 48 6
(BO,*,NY,*) 7 13 9 15 11 17 12 18
13 19 14 20 15 21 16 22
17 23 18 24 19 25 20 26
21 27 22 28 23 29 24 30
25 31 26 32 27 33 28 34
29 35 30 36 31 37 32 38
33 39 34 40 35 41 36 42
37 43 38 44 39 45 40 46
41 47 43 1 45 3 47 5
(NY,*,PH,*) 1 3 12 14 13 15 14 16
15 17 16 18 17 19 18 20
19 21 20 22 21 23 22 24
23 25 24 26 25 27 26 28
27 29 28 30 29 31 30 32
31 33 32 34 33 35 34 36
35 37 36 38 37 39 38 40
39 41 40 42 41 43 42 44
43 45 44 46 45 47 46 48
47 1
(PH,*,WA,*) 1 4 14 17 15 18 16 19
17 20 18 21 19 22 20 23
21 24 22 25 23 26 24 27
25 28 26 29 27 30 28 31
29 32 30 33 31 34 32 35
33 36 34 37 35 38 36 39
37 40 38 41 39 42 40 43
41 44 42 45 43 46 44 47
45 48 46 1 47 2 ;
param demand :=
[WA,*,PH,*] 2 5 .55 6 9 .01 8 11 .01
10 13 .13 12 15 1.59 13 16 1.69
14 17 5.19 15 18 3.55 16 19 6.29
17 20 4.00 18 21 5.80 19 22 3.40
20 23 4.88 21 24 2.92 22 25 4.37
23 26 2.80 24 27 4.23 25 28 2.88
26 29 4.33 27 30 3.11 28 31 4.64
29 32 3.44 30 33 4.95 31 34 3.73
32 35 5.27 33 36 3.77 34 37 4.80
35 38 3.31 36 39 3.89 37 40 2.65
38 41 3.01 39 42 2.04 40 43 2.31
41 44 1.52 42 45 1.75 44 47 1.88
46 1 1.05
[PH,*,NY,*] 1 3 1.05 5 7 .43 9 11 .20
11 13 .21 13 15 .40 14 16 6.49
15 17 16.40 16 18 9.48 17 19 17.15
18 20 9.31 19 21 15.20 20 22 8.21
21 23 13.32 22 24 7.35 23 25 11.83
24 26 6.61 25 27 10.61 26 28 6.05
27 29 9.65 28 30 5.61 29 31 9.25
30 32 5.40 31 33 8.24 32 34 4.84
33 35 7.44 34 36 4.44 35 37 6.80
36 38 4.11 37 39 6.25 38 40 3.69
39 41 5.55 40 42 3.29 41 43 4.77
42 44 2.91 43 45 4.19 44 46 2.53
45 47 4.00 47 1 1.65
[NY,*,BO,*] 10 16 1.23 12 18 3.84 14 20 4.08
15 21 1.47 16 22 2.96 17 23 1.60
18 24 2.95 19 25 1.71 20 26 2.81
21 27 1.77 22 28 2.87 23 29 1.84
24 30 2.95 25 31 1.91 26 32 3.12
27 33 1.93 28 34 3.31 29 35 2.00
30 36 3.40 31 37 2.08 32 38 3.41
33 39 2.69 34 40 4.45 35 41 2.32
36 42 3.40 37 43 1.80 38 44 2.63
39 45 1.52 40 46 2.23 41 47 1.25
42 48 1.79 43 1 .97 44 2 1.28
45 3 .48 46 4 .68 48 6 .08
[BO,*,NY,*] 7 13 .03 9 15 1.29 11 17 4.59
12 18 2.56 13 19 3.92 14 20 2.37
15 21 3.81 16 22 2.24 17 23 3.51
18 24 2.13 19 25 3.28 20 26 2.05
21 27 3.15 22 28 1.99 23 29 3.09
24 30 1.93 25 31 3.19 26 32 1.91
27 33 3.21 28 34 1.85 29 35 3.21
30 36 1.71 31 37 3.04 32 38 2.08
33 39 3.13 34 40 1.96 35 41 2.53
36 42 1.43 37 43 2.04 38 44 1.12
39 45 1.71 40 46 .91 41 47 1.32
43 1 1.80 45 3 1.13 47 5 .23
[NY,*,PH,*] 1 3 .04 12 14 4.68 13 15 5.61
14 16 3.56 15 17 5.81 16 18 3.81
17 19 6.31 18 20 4.07 19 21 7.33
20 22 4.55 21 23 7.37 22 24 4.73
23 25 7.61 24 26 4.92 25 27 7.91
26 28 5.19 27 29 8.40 28 30 5.53
29 31 9.32 30 32 5.51 31 33 10.33
32 34 9.21 33 35 18.95 34 36 11.23
35 37 16.85 36 38 7.29 37 39 10.89
38 40 5.41 39 41 8.21 40 42 4.52
41 43 6.99 42 44 3.92 43 45 6.21
44 46 3.44 45 47 5.17 46 48 2.55
47 1 1.24
[PH,*,WA,*] 1 4 .20 14 17 4.49 15 18 3.53
16 19 2.67 17 20 3.83 18 21 3.01
19 22 4.12 20 23 3.15 21 24 4.67
22 25 3.20 23 26 4.23 24 27 2.87
25 28 3.84 26 29 2.60 27 30 3.80
28 31 2.77 29 32 4.31 30 33 3.16
31 34 4.88 32 35 3.45 33 36 5.55
34 37 3.52 35 38 6.11 36 39 3.32
37 40 5.53 38 41 3.03 39 42 4.51
40 43 2.53 41 44 3.39 42 45 1.93
43 46 2.52 44 47 1.20 45 48 1.75
46 1 .88 47 2 .87 ;
end;
