LP-Model for Job Shop Problems

LP-Model for Job Shop Problems
Günther Füllerer
June 23, 2005
LP-Model: Variables
Given Data:
n . . . number of jobs
m . . . number of machines
Di . . . due date of job i
pij . . . processing time of job i on machine j
seqij . . . position in sequence of job i on machine j
Decision Variables:
Cmax = max{Ci }
Ci . . . completion time of job i
Ti . . . tardiness of job i
(
1
if Ti > 0
δi =
0
otherwise
startij . . . start of job i on machine j
(
1
if job i follows job j on machine k
k
yij =
0
otherwise
LP-Model: Objective
n
n
n
i
n
i
n
Objective 1: min Cmax · M + ∑ Ti · 1/M + ∑ δi · 1/M + ∑ Ci · 1/M
i
n
Objective 2: min Cmax · 1/M + ∑ Ti · M + ∑ δi · 1/M + ∑ Ci · 1/M
i
n
i
n
i
n
Objective 3: min Cmax · 1/M + ∑ Ti · 1/M + ∑ δi · M + ∑ Ci · 1/M
i
n
i
n
i
i
i
n
Objective 4: min Cmax · 1/M + ∑ Ti · 1/M + ∑ δi · 1/M + ∑ Ci · M
Cmax ≥ starti,seqi,m + pi,seqi,m
∀i = 1, . . . , n
Ci ≥ starti,seqi,m + pi,seqi,m
∀i = 1, . . . , n
i
LP-Model: Constraints
Precedence constraints
starti,seqi,j +1 ≥ starti,seqi,j + pi,seqi,j
i = 1, . . . , n
j = 1, . . . , m − 1
Disjunction Constraints
startik + pik ≤ startjk + M · yijk
startjk + pjk ≤ startik + M
k = 1, . . . , m
· (1 − yijk )
i, j = 1, . . . , n ∧ i < j
k = 1, . . . , m
i, j = 1, . . . , n ∧ i < j
Bounds on Tardiness
Ti ≥ starti,seqi,m + pi,seqi,m − Di
i = 1, . . . , n
Ti ≤ δi · M
δi ≤ Ti · M
not needed
Non-negativity Constraints
Ti ≥ 0
i = 1, . . . , n
Ci ≥ 0
i = 1, . . . , n
Ti ≥ 0
i = 1, . . . , n
startij ≥ 0
i = 1, . . . , n
j = 1, . . . , m
M4
M3
M2
M1
0
4
J1
8
J2
12
J3
16
J4
20
J5
24
J6
28
32
36
40
44
48
52
56
44
48
52
56
Figure: Objective Cmax
M4
M3
M2
M1
0
4
J1
8
J2
12
J3
16
J4
20
J5
24
J6
28
32
36
40
Figure: Objective Total Tardiness
M4
M3
M2
M1
0
5
J1
10
J2
15
J3
20
J4
25
J5
30
J6
35
40
45
50
55
60
65
70
52
56
Figure: Objective Number of Tardy Jobs
M4
M3
M2
M1
0
4
J1
8
J2
12
J3
16
J4
20
J5
24
J6
28
32
36
40
44
Figure: Objective Total Flowtime
48