Train Platforming Problem - Geoinformatics for Intelligent Transportation - page 149

1 if train i arrives before train j arrives

for i ; j 2 U ; i \ j : y aa

ij ¼

0 otherwise

1 if train i arrives before train j departs

ð i ; j Þ2 V ad

: y ad

for

ij ¼

0 otherwise

1 if train i departs before train j arrives

ð i ; j Þ2 V da : y da

for

ij ¼

0 otherwise

1 if train i departs before train j arrives

ð i ; j Þ2 V dd

: y dd

for

ij ¼

0 otherwise

After these preliminaries, the mathematical model can be written as follows:
minimise X
i 2 U

c i ð u i þ v i Þ ð 1 Þ

maximise X
i 2 U

X

p ik x ik ð 2 Þ

k 2 K ð i Þ

subject to

v i þ t P i u i þ t P i þ t min 8 i 2 U ð 3 Þ

v i þ t P i u j þ t P j þ t Cn

8 j 2 U ; i 2 W ð j Þ ð 4 Þ

i 0 u i þ t P i þ h ð i ; k ; i 0 ; k 0 Þ

aa

u i 0 þ t Pa y aa

M ð 1

ii 0 Þ M ð

1 x ik Þ M ð 1 x i 0 k 0 Þ

8ð i ; i 0 Þ2 V aa

; k 2 K ð i Þ ; k 0 2 K ð i 0 Þ : a ð I i ; k ; I i 0 ; k 0 Þ

ð 5 Þ

u i þ t P i u i 0 þ t Pa þ h ð i 0 ; k 0 ; i ; k Þ

aa

My aa

ii 0 M ð

1 x ik Þ M ð 1 x i 0 k 0 Þ

i 0

ð 6 Þ

8ð i ; i 0 Þ2 V aa

; k 2 K ð i Þ ; k 0 2 K ð i 0 Þ : a ð I i ; k ; I i 0 ; k 0 Þ

Constraints (7) (12) are speci ed for the other combinations of arrival —

-

departure and have similar meaning as ( 5 ) and ( 6 ).

da

u i þ t P i v i 0 þ t Pd þ h ð i 0 ; k ; i ; k Þ My aa

ii 0 M ð

1 x ik Þ M ð 1 x i 0 k Þ

ð 7 Þ

i 0

8ð i ; j Þ2 U ; i \ j ; k 2 K ð i Þ\ K ð i 0 Þ

da

u i 0 þ t Pa

i 0 v i þ t P i þ h ð i ; k ; i 0 ; k Þ

M ð 1 y aa

ii 0 Þ M ð

1 x ik Þ M ð 1 x i 0 k Þ

ð 8 Þ

8ð i ; j Þ2 U ; i \ j ; k 2 K ð i Þ\ K ð i 0 Þ

y aa

8 i ; j 2 U ; i 6¼ j ; I i ¼ I j ; t P i t Pa

ij ¼

ð 9 Þ

1

j