06: Flow Shop Scheduling
Phan Nguyen Ky Phuc
April 20, 2019
Contents
1 Flow Shop
1
1.1
Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
When number of machines in each work station is 1 . . . . . . . . . . . . . . . . . . . . . . .
1
1.3
When number of machines in each work station is more than 1 and process sequence on each
1.4
1
work station is not fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Sample Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Flow Shop
1.1
Assumption
• There are several work stations {1, 2, ...M } in the system
• Each job must be processed in work stations following the sequence of 1 → 2 → 3... → M
• Work station m has Nm identical machine, Nm ≥ 1
1.2
When number of machines in each work station is 1
It is assumed that in the flow shop the work station sequence on each job is 1 → 2 → 3... → M
Sets
M : set of work stations
J: set of jobs
Parameters
pij be processing time on workstation i of job j
Variables
Sij : starting time on work station i of job j
xjk : binary variable xjk = 1 if job j is processed before job k in the sequence; otherwise xjk = 0
Cmax : the make span
min Cmax
1
Ho Chi Minh City International University
Industrial Systems Engineering Department
Scheduling
Lecturer: Phan Nguyen Ky Phuc
subject to
Shj ≥ Sij + pij ,
∀j, h > i
Cmax ≥ Sij + pij ,
∀i, j
Sij ≥ Sik + pik − BigM × (1 − xkj ),
∀i, j, k
xjk + xkj = 1
1.3
When number of machines in each work station is more than 1 and process
sequence on each work station is not fixed
Sets
M : set of work stations
J: set of jobs
Parameters
pij be processing time on work stationi of job j
Ni number of machines on work station i
Variables
Smj : starting time on work station m of job j
Dmj : departure time on work station m of job j
Cmax : the make span
t
t
Umj
= 1 if t ≥ Smj otherwise Vmj
=0
t
t
Vmj
= 1 if t ≤ Dmj otherwise Umj
=0
t
t
Rmj
= 1 if Smj ≥ t ≥ Dmj otherwise Rmj
=0
min Cmax
subject to
Shj ≥ Smj + pmj ,
Cmax ≥ Dmj ,
∀j, h > m
∀m, j
Dmj ≥ Smj + pmj − 1,
∀m, j
Constraints for Ujt and Vjt
t
t ≥ Smj − 1 − BigM × (1 − Umj
), ∀j, t
t
t ≤ Smj − 1 + BigM × Umj
, ∀j, t
t
t ≤ Dmj + 1 + BigM × (1 − Vmj
), ∀j, t
t
t ≥ Dmj + 1 − BigM × Vmj
, ∀j, t
Constraints for Yjt
t
t
Rmj
≤ Umj
j, ∀j, t
t
t
Rmj
≤ Vmj
, ∀j, t
t
t
t
Rmj
≥ Vmj
+ Umj
− 1, ∀j, t
Workforce constraints
J
X
t
Rmj
≤ Nm ∀t, j
j=1
06: Flow Shop Scheduling
Page 2
Ho Chi Minh City International University
Industrial Systems Engineering Department
1.4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Scheduling
Lecturer: Phan Nguyen Ky Phuc
Sample Code
i n t numStation = . . . ;
i n t numJob = . . . ;
i n t numTime=30;
r a n g e r a n g e S t a t i o n = 1 . . numStation ;
r a n g e ra ngeJ ob = 1 . . numJob ;
r a n g e rangeTime = 0 . . numTime ;
i n t N[ r a n g e S t a t i o n ] = . . . ;
i n t p [ r a n g e S t a t i o n ] [ ra nge Job ] = . . . ;
i n t BigM=10000;
dvar i n t+ S [ r a n g e S t a t i o n ] [ ran geJ ob ] ;
dvar i n t+ D[ r a n g e S t a t i o n ] [ ra nge Job ] ;
dvar b o o l e a n U[ rangeTime ] [ r a n g e S t a t i o n ] [ ran geJ ob ] ;
dvar b o o l e a n V[ rangeTime ] [ r a n g e S t a t i o n ] [ ran geJ ob ] ;
dvar b o o l e a n R[ rangeTime ] [ r a n g e S t a t i o n ] [ ra nge Job ] ;
dvar f l o a t+ Cmax ;
e x e c u t e PRE{
c p l e x . t i l i m =60∗3;
}
m i n i m i z e Cmax ;
s u b j e c t to {
f o r a l l ( h i n r a n g e S t a t i o n , m i n r a n g e S t a t i o n , j i n ra nge Job : h>m) {
S [ h ] [ j ]>=S [m ] [ j ]+p [m ] [ j ] ;
23
24
}
25
26
27
28
29
f o r a l l ( m i n r a n g e S t a t i o n , j i n r ang eJob ) {
Cmax>=D[m ] [ j ] ;
D[m ] [ j ]>=S [m ] [ j ]+p [m ] [ j ] − 1 ;
}
30
31
32
33
34
35
36
forall ( j
t>=S [m ] [
t<=S [m ] [
t<=D[m ] [
t>=D[m ] [
}
i n rangeJob , t i n rangeTime , m i n r a n g e S t a t i o n ) {
j ]−1−BigM∗(1−U[ t ] [ m ] [ j ] ) ;
j ]−1+BigM∗U[ t ] [ m ] [ j ] ;
j ]+1+BigM∗(1−V[ t ] [ m ] [ j ] ) ;
j ]+1−BigM∗V[ t ] [ m ] [ j ] ;
forall ( j
R[ t ] [ m ] [
R[ t ] [ m ] [
R[ t ] [ m ] [
}
i n rangeJob
j ]<=U[ t ] [ m ] [
j ]<=V[ t ] [ m ] [
j ]>=V[ t ] [ m ] [
37
38
39
40
41
42
, t i n rangeTime , m i n r a n g e S t a t i o n ) {
j ];
j ];
j ]+U[ t ] [ m ] [ j ] − 1 ;
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
f o r a l l ( t i n rangeTime ,m i n r a n g e S t a t i o n ) {
sum ( j i n ran geJ ob )R[ t ] [ m ] [ j ]<=N[m ] ;
}
}
e x e c u t e WRITE{
v a r o f i l e =new I l o O p l O u t p u t F i l e ( " R e s u l t . t x t " ) ;
o f i l e . w r i t e l n ( "The o b j e c t i v e f u n c t i o n v a l u e : " , c p l e x . getObjValue ( ) ) ;
f o r ( var m in r a n g e S t a t i o n ) {
f o r ( v a r j i n ra nge Job ) {
o f i l e . w r i t e l n ( "S time on S t a t i o n [ " ,m, " ] o f j o b [ " , j , " ] i s : " , S [m ] [ j ] ) ;
o f i l e . w r i t e l n ( "D time on S t a t i o n [ " ,m, " ] o f j o b [ " , j , " ] i s : " , D[m ] [ j ] ) ;
}
}
o f i l e . w r i t e l n ( "−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−" ) ;
o f i l e . close () ;
06: Flow Shop Scheduling
Page 3
Ho Chi Minh City International University
Industrial Systems Engineering Department
59
Scheduling
Lecturer: Phan Nguyen Ky Phuc
}
Example 1
The Nm is given as Nm = [1, 2, 1]
\J
1
2
3
1
2
5
3
2
5
5
1
3
3
5
2
4
2
4
4
Example 2
Consider following job shop problem with revisit workstation is allowed
The Nm is given as Nm = [2, 1, 2, 2]
J
1
2
3
4
06: Flow Shop Scheduling
Sequence
1,2,3,1
2,3,1
4,1,3,2
1,4,2,4
Processing Time
p11 = 1, p21 = 2, p31 = 3, p11 = 2
p22 = 1, p32 = 2, p12 = 2
p43 = 4, p13 = 1, p33 = 2, p23 = 3
p14 = 2, p44 = 3, p24 = 2, p44 = 3
Page 4