1

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

7

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:

9

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

10

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

11

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)

12

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

13

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

14

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

16

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

17

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

18

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

27

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