1 mixed integer linear model for fms scheduling based on agvs: job-shop with a single transport...
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Mixed Integer Linear Model for FMS scheduling based on AGVs: Job-Shop with a Single Transport
Robot
Mathieu BECART, Philippe LACOMME, Aziz MOUKRIM, Nikolay TCHERNEV
LIMOS CNRS UMR 6158 HeuDiaSyC CNRS UMR 6599
2
Summary
1. FMS presentation
2. Objectives and assumptions
3. MILP formulation
4. Benchmarks
5. Concluding remarks
3
Flexible Manufacturing System
• M stations
• One or more vehicles
• One load/unload station
• Each job follows a given sequence of operations
Station 1
Load/Unload Station
Station 2 Station 3 Station 4
Vehicle
4
FMS: station
• Limited input buffer capacity
• Limited output buffer capacity
Vehicle
output buffer
input buffer
machine
station
job
5
Deadlock phenomenon
Vehicle
input buffer
output buffer
1 2
5 4 3
input buffer
output buffer
10 9
6 7 8
Station 1 Station 2
input buffer
output buffer
input buffer
output buffer
Station 4 Station 3
Load/Unload station
6
FMS: Job Type
Job 1: L/U (M1,10) (M3,20) (M4,6) L/U
Job 2: L/U (M2,5) (M1,14) (M3,12) L/U
FMS: Management rules
• Management policy of the vehicle: FIFO, STT, MOQS, …
• Management rule for buffers: FIFO
• Maximal number of jobs simultaneously allowed to avoid deadlock
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Scope and purposeObjectives
• Exact resolution for small and medium scale instances
• Mixed Integer Linear Programing formulation
Constraints of interest
• Only one vehicle
• Nonpreemptive operations
• Deadheading transport times
• Limited input/output buffers capacity
• Management rule of buffers (FIFO)
8
Problem formulation
Global optimization between job processing and job transportation
Notations:
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Set of constraints
J ob Station Vehicle
Buffers Machine
- Precedence of job sequence operations - Maximal number of jobs in the system
- Management rule - I nput/ ouput storage
- Sequencing of treatments
- Transport of jobs
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Set of constraints
• Precedence constraints
• Sequencing constraints
• Transport constraints
• Storage constraints for input buffers
• Storage constraints for output buffers
• Maximal number of jobs simultaneously allowed
• Buffers managements rule constraints
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Precedence constraints
Processing order of operations according to each job sequence of
treatement
illgildklcikil OutpIntTsTsi
station l
station k
d cIn ilp g il Out il
tkl
timeTs ik + c Ts il
c
(d)
(d)
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Sequencing constraints
No more than one job processed on the same station at the same time
)1(2111122
2122211
liilglililili
liilglililili
bHpOutTsOutTs
HbpOutTsOutTs
i
i
station l
time
job i2
job i1
Ts i2,l-Outi2,l-pgi2,l
Process
Process
Ts i1,l-Outi1,l
Conflict: 2 jobs in process
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Transport constraints
Only one loaded/deadheading move of the vehicle at the same time
)1(212111122222
212122211111
kkiikikldlkcki
kkiikikldlkcki
bHTsvtTs
HbTsvtTs
tk2l2
time
c
station l1
vl2k1
Ts i2k2
Ts i1l1
station k1
station k2
station l2Ts i2l2
Ts i1k1
tk1l1
Conflict: 2 jobs in move
Ts i1k1+c+tk1l1+d
Ts i2k2+c+tk2l2+d
c
vl1k2
loadedtransport
deadheadingtransport
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Storage constraints for input buffers
miimii
mimimiimiidlmcli
lmlimiimiimimi
bZ
OutTsHZHHbtTs
tTsHHZHbOutTs dc
1212
1112122
2121211
0
)1(
1
2
21iBZ m
imii
station m
time
job i2
job i1
station l
Ts i2l
job i2
Ts i2l + c+t lm + d
In
Ts i1m -Out i1m
job i2 is waitingfor i1 performs its
processZ i2i1m =1
In
15
Storage constraints for output buffers
miimii
mimiimiimimi
mimimiimiimi
bY
TsHYHHbOutTs
OutTsHHYHbTs
1212
1121222
2212121
0
)1(
112
21iSY m
imii
station m
time
job i 2
job i 1
Ts i2m -Out i2m
Ts i1m
job i1 is waiting fori2 leaves the ouput
bufferY i1i2m =1
Out
Out
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Limited number of jobs simultaneously allowed
1
)/(1)/(
)/(1)/(
1212
112122
212121
0
)1(
iiii
ULiiiiiULi
ULiiiiiULi
bW
TsHWHHbTs
TsHHWHbTs
2
12 11i
ii iNW
M3
M2
M1
L/U
2 jobs in thesystem
time
loaded transportdeadheading transport
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Buffers management rule constraints
station l
time
job i2
job i1
FIFO for both input and output buffers
station l
time
job i2
job i1
station l
time
job i 2
job i 1
station l
time
job i 2
job i 1
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Evaluation of the model(Bilge and Ulusoy, 1995) (Liu and MacCarthy, 1997)
Managing constraints Bilge and Ulusoy’s
formulation
Liu and MacCarthy’s formulation
Formulation proposed
Precedence constraints YES YES YES
Sequencing constraints YES YES YES
Transport constraints YES YES YES
Loading/ unloading time for vehicle moves
YES NO YES
Set up times between operations
NO YES NO
Number of vehicles 1 1 1
Input buffers storage constraints
NO YES YES
Output buffers storage constraints
NO NO YES
Limited number of jobs in the system
NO NO YES
I nput/ output station buffer managing rule
NO NO YES (FIFO)
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BenchmarksBilge and Ulusoy instances
4 Layouts, 10 JobsetsLAYOUT 2
Station 2
Station 1
Station 3
Station 4
Load/Unload
station
0
90
150
0 30 60
75
135
J ob 1 (L/ U) (0) M1 (10) M4 (18) (L/ U) (0)
J ob 2 (L/ U) (0) M2 (10) M4 (18) (L/ U) (0)
J ob 3 (L/ U) (0) M1 (10) M3 (20) (L/ U) (0)
J ob 4 (L/ U) (0) M2 (10) M3 (15) M4 (12) (L/ U) (0)
J ob 5 (L/ U) (0) M1 (10) M2 (15) M4 (12) (L/ U) (0)
J ob 6 (L/ U) (0) M1 (10) M2 (15) M3 (12) (L/ U) (0)
(L/ U) M1 M2 M3 M4 (L/ U)
(L/ U) 0 4 6 8 6 0
M1 6 0 2 4 2 6
M2 8 12 0 2 4 8
M3 6 10 12 0 2 6
M4 4 8 10 12 0 4
(L/ U) 0 4 6 8 6 0
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Computational experiments
Sun Entreprise 450 with 4 Ultra Sparc II processors 450 MHz under Sun Solaris 7 OS with
2 Go of central memory
< 1min
30min
Number of jobs N Optimal makespan
Computational time (sec.)
N=1 92 0.07 2 jobs N=2 72 0.06 N=1 142 0.24 N=2 100 0.54
3 jobs
N=3 92 0.49 N=1 201 0.84 N=2 129 16.02 N=3 116 8.06
4 jobs
N=4 116 6.25 N=1 260 5.53 N=2 162 755.58 N=3 144 1093.78 N=4 138 1162.24
5 jobs
N=5 138 1765.40
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Concluding remarks
• MILP for FMS with one vehicle
• Great number of management constraints taken into account: limited input/output buffers capacity, managing rule of buffers (FIFO), maximal number of jobs in the system
• Instances of Bilge and Ulusoy, 1995
• Optimal resolution for small and medium scale instances
22
Future research
• Cutting plane approach
• Extend the model for more than one vehicle
• Extend the model to stochastic transportation times (robustness)
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Transport constraints
Deadheading vehicle move from L/U station to stations in the system taken into account
liililULULi
liiULili
liiliULi
HdTsvTs
dHTsTs
HdTsTs
2112
2121
2112
)/()/(
)/(
)/(
)1(
tk(L/U)
time
c
c
station l
v(L/U)l
Ts i2k
Ts i1l
station L/U
station k
Ts i2(L/U)
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Buffers management rule constraints (FIFO)
station l
time
job i2
job i1
(a) (b) (c)
Ts i2k2+tk2l
Ts i1k1+tk1l
Ts i2l-Outi2l-p i2l
Ts i1l-Outi1l-p i1l
Ts i2l
Ts i1l
25
miimii
mimimiimiidlmcli
lmlimiimiimimi
bZ
OutTsHZHHbtTs
tTsHHZHbOutTsdc
1212
1112122
2121211
0
)1(
Storage constraints for input buffers
)2.(1
1)1.(112
12
12A
OutTstTs
bZA
mimiclmdli
mii
mii
Theorem:
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ProofProof
10
)1(
1
12
11122
21211
112
12
mii
mimimiidlmcli
lmlimiimimi
mimiclmdli
mii
Z
OutTsZHtTs
tTsHZOutTs
OutTstTs
b
dc
1
1
12
112
12
mii
mimiclmdli
mii
Zthatproveand
OutTstTs
bAssume
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mimiclmdli
clmdlimimi
mii OutTsHtTs
tTsOutTsthenZIf
112
211
120
This is impossible since
mimiclmdli OutTstTs112
This requires1
12miiZ
28
mimiclmdli
mii
mii
OutTstTs
b
thatproveandZAssume
112
12
12
1
1
mii
mimiclmdli
clmdlimiimimi
mii
b
HOutTsHtTs
tTsHHHbOutTs
thenZIf
12
112
21211
12
1
1
112
miibSo
mimidlmcli
lmlimimi
OutTstTs
tTsHOutTsanddc
112
211