simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Philippe LACOMME, Mohand LARABI Nikolay TCHERNEV LIMOS (UMR CNRS 6158), Clermont Ferrand, France IUP « Management et gestion des entreprises »

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution. Philippe LACOMME, Mohand LARABI Nikolay TCHERNEV LIMOS (UMR CNRS 6158), Clermont Ferrand, France IUP « Management et gestion des entreprises ». - PowerPoint PPT Presentation

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Page 1: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

Philippe LACOMME, Mohand LARABINikolay TCHERNEV

LIMOS (UMR CNRS 6158), Clermont Ferrand, FranceIUP « Management et gestion des entreprises »

Page 2: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 2

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolutionPlan

Plan

Introduction

Algorithm based framework

Computational evaluation

Conclusions and further works

Page 3: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 3

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Type of system under study: FMS based on AGV

FMS definition

Page 4: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 4

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Type of system under study: FMS based on AGV

AGV system

Guide path layout

Automated Guided Vehicles

Page 5: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 5

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Type of system under study: FMS based on AGV

Flexible machines

Page 6: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 6

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Type of system under study: FMS based on AGV

Flexible cells

Page 7: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 7

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Type of system under study: FMS based on AGV

Input/Output buffers

Page 8: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 8

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

AGV operating (1/2)

Loaded vehicle arriving at station k

Unloading at station k

Loading at station k

Loaded vehicle going to station ....

Loaded vehicle arriving at station i

Unloading at station i

Idle vehicle at station i

Page 9: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 9

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

AGV operating (2/2)

Loaded vehicle going to station ....

Loading at station j

Empty vehicle arriving at station j

There are two types of vehicle trips:

the first type of loaded vehicle trips ;

the second one is the empty vehicle trips.

Page 10: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 10

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Problem definition (1/5)

Problem definition

The scheduling problem under study can be defined in the following general form:

Given a particular FMS with several vehicles and a set of jobs, the objective is to determine the starting and completion times of operations for each job on each machine and the vehicle trips between machines according to makespan or mean completion time minimization.

Page 11: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 11

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Problem definition (2/5)

Problem definition : Example of solution

Empty trip

Page 12: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 12

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Problem definition (3/5)

Problem definition : Complexity

Combined problem of: (i) scheduling problem of the form

(n jobs, M machines, G general job shop, Cmax makespan), a well known NP-hard problem (Lenstra and Rinnooy Kan 1978);

(ii) a generic Vehicle Scheduling Problem (VSP) which is NP-hard problem (Orloff 1976).

max/// CGMn

Page 13: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 13

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Problem definition (4/5)

Problem definition : Assumptions in the literature All jobs are assumed to be available at the beginning of the

scheduling period. The routing of each job types is available before making scheduling

decisions. All jobs enter and leave the system through the load and unload

stations. It is assumed that there is sufficient input/output buffer space at

each machine and at the load/unload stations, i.e. the limited buffer capacity is not considered.

Vehicles move along predetermined shortest paths, with the assumption of no delay due to the congestion.

Machine failures are ignored. Limitations on the jobs simultaneously allowed in the shop are

ignored.

Page 14: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 14

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction

Problem definition (5/5)

Under these hypotheses the problem can be without doubt modelled as a job shop with several transport robots.

notation introduced by Knust 1999

J indicates a job shop, R indicates that we have a limited number of identical vehicles

(robots) and all jobs can be transported by any of the robots.

indicates that we have job-independent, but machine-dependant transportation times.

indicates that we have machine-dependant empty moving time.

The objective function to minimize is the makespan .

max', CttJR klkl

klt

'klt

Page 15: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 15

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

General template

General template

interface

Instance to solveand

frameworkparameters

Best solution

One solution evaluation

Non oriented disjunctive graphe

Oriented disjunctive graphe

Memetic algorithm

Longest path algorithm

MTS and OA

Page 16: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 16

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Disjunctive graph definition (1/3)

Non oriented disjunctive graph

consists of:

Vm : a set of vertices containing all machine operations;

Vt : a set of vertices containing all transport operations;

C : representing precedence constraints in the same job;

Dm : containing all machine disjunctions;

Dr : containing all transport disjunctions.

RmtM DDCVVG ,

Page 17: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 17

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Disjunctive graph definition (2/3)

*

M27

M35

M14

0

0

0

0

M3 M4 M15 4 1

M5 M1 M35 5 3

J1

J2

J3

Page 18: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 18

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Disjunctive graph definition (2/2)

0 *

M2 r1 M3 r2 M18 5

M3 r1 M4 M15 4

M5 M1 M35 5

40

0

0

1

3

Machine disjunctionproblem

Robot assignment problem

Robot disjunctionproblem

Page 19: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 19

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Disjunctive graph definition (3/3)

To obtain an oriented disjunctive graph we must : define a job sequence on machines ; define an assignment of robots to each transport

operation ; define a precedence (order) to transport operations

assigned to one robot.

Using two vectors:MTS which defines Machine and Transport SelectionsOA which defines Operation Assignments to each

robot

Page 20: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 20

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Disjunctive graph orientation (1/2)

0 *

M2 M3 M17 5

M3 M4 M15 4

M5 M1 M30 7

5 2 5 2

40

0

0

1

3

45 55

MTS 1 2 3 1 2 31

m2

2

m3

3

m5

1

m3

2

m4

3

m1

1

m1

2

m1

3

m3

Transport operations

Transport operations

Page 21: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 21

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Disjunctive graph orientation (2/2)

0 *

M2 r1 M3 r2 M17 2 5 3

M3 r1 M4 r2 M15 2 4 3

M5 r3 M1 r3 M30

5 2 5 2

40

0

0

1

3

7 3 45 55

MTS 1 2 1 2 33 1 2 1 2 33

tr11 tr21 tr31 tr12 tr22 tr32

1 2 3

OA r1

tr11

r1

tr21

r3

tr31

Machine operations

Machine operations

Machine operations

r2

tr12

r2

tr22

r3

tr32

Page 22: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 22

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Graph evaluation and Critical Path

0 *

M2 r1 M3 r2 M10 7 9 14 17

7 2 5 3

M3 r1 M4 r2 M10 14 16 20 23

5 2 4 3

M5 r3 M1 r3 M30 5 7 12 14

5 2 5 2

4

24

0

0

0

1

3

7 3 45 55

Makespan =24

Page 23: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 23

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Memetic algorithm

beginnpi := 0 ; // current iteration numberni := 0 ; // number of successive

unproductive iterationRepeatSelectSolution (P1,P2)C := Crossover(P1,P2)

LocalSearch(C) with probability pmInsertSolution(Pop,C)Sort(Pop)If (npi=np)

Restart(Pop,p)End If

Until (stopCriterion).End

interface

Instance to solveand

frameworkparameters

Best solution

One solution evaluation

Non oriented disjunctive graphe

Oriented disjunctive graphe

Memetic algorithm

Longest path algorithm

MTS and OA

Page 24: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 24

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Chromosome

1 2 3 1 2 31m2

2m3

3m5

1m3

2m4

3m1

1m1

2m1

3m3tr11tr21tr31 tr12tr22tr32

OA r1tr11

r1tr21

r3tr31

r2tr12

r2tr22

r3tr32

MTS

Chromosome is a representation of a solution

Makespan = 24

Page 25: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 25

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Local search (1/5)

For one iteration: Change one machine disjunction orientation (in the

critical path)OR

Change one robot disjunction orientation (in the critical path)

OR

Change one robot assignment.

Page 26: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 26

74

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Local search (2/5)

*

M2 r1 M3 r2 M10 7 9 14 17

7 2 5 3

r1 M4 r2 M114 16 20 23

2 4 3

r3 M1 r3 M35 7 12 14

2 5 2

4

24

0

0 M30

5

M50

5

0

0

1

3

3 45 55

MTS 1 2 3 1 2 31

m2

2

m3

3

m5

1

m3

2

m4

3

m1

1

m1

2

m1

3

m3

r1

tr11

r1

tr21

r3

tr31

r2

tr12

r2

tr22

r3

tr32

OA

tr11 tr21 tr31 tr12 tr22 tr32Change transport disjunction

Robot block

Page 27: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 27

0 *

M2 r1 M3 r2 M10 8 10 15 18

7 2 5 3

M3 r1 M4 r2 M10 5 7 18 22

5 2 4 3

M5 r3 M1 r3 M30 5 7 12 15

5 2 5 2

40

0

0

1

3

3 3 45 55

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Local search (3/5)

MTS 2 1 3 1 2 31

m2

2

m3

3

m5

1

m3

2

m4

3

m1

1

m1

2

m1

3

m3

r1

tr21

r1

tr11

r3

tr31

r2

tr12

r2

tr22

r3

tr32

OA

tr21 tr11 tr31 tr12 tr22 tr32

23

Makespan =23Machine block

Change machine disjunction

Page 28: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 28

r1

tr11

r3

0 *

M2 r1 M3 r2 M10 8 10 15 18

7 2 5 3

M3 r1 M4 r2 M10 5 7 18 22

5 2 4 3

M5 r3 M1 r3 M30 5 7 12 15

5 2 5 2

40

0

0

1

3

3 3 45 55

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Local search (4/5)

MTS 2 1 3 1 2 31

m2

2

m3

3

m5

1

m3

2

m4

3

m1

1

m1

2

m1

3

m3

r1

tr21

r3

tr31

r2

tr12

r2

tr22

r3

tr32

OA

tr21 tr11 tr31 tr12 tr22 tr32

23

Change robot assignement

r3

Page 29: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 29

4

r1

tr11

r3

0 *

M2 r1 M3 r2 M10 7 9 14 17

7 2 5 3

M3 r1 M4 r2 M10 5 7 17 21

5 2 4 3

M5 r3 M1 r3 M30 9 11 16 18

5 2 5 2

40

0

0

1

3

3 45 55

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework

Local search (5/5)

MTS 2 1 3 1 2 31

m2

2

m3

3

m5

1

m3

2

m4

3

m1

1

m1

2

m1

3

m3

r1

tr21

r3

tr31

r2

tr12

r2

tr22

r3

tr32

OA

tr21 tr11 tr31 tr12 tr22 tr32

22

Change robot assignement

r3

New transport disjunction is added

Page 30: Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

IESM 2009, MONTREAL – CANADA, May 13 TR 30

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation

Instances

Two types of experiments have been done using well known benchmarks in the literatures.

The first type of experiments concerns instances of:

Hurink J. and Knust S., "Tabu search algorithms for job-shop problems with a single transport robot", European Journal of Operational Research, Vol. 162 (1), pp. 99-111, 2005.

The second one with two identical robots from:

Bilge, U. and G. Ulusoy, 1995, A Time Window Approach to Simultaneous Scheduling of Machines and Material Handling System in an FMS, Operations Research, 43(6), 1058-1070.

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation

Experimental results (1/4)

Experiments on job-shop with one single robot on Hurink and Knust instances based on well-known 6x6 and 10x10 instances:J.F. Muth, G.L. Thompson, Industrial Scheduling, PrenticeHall, Englewood Cliffs, NJ, 1963.

Deviation in percentage from the best solution found byeach method to lower bound proposed by Hurink and Knust

Four methods proposed by Hurink and Knust Our method13,40 16,16 14,22 16,63 13,33

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation

Experimental results (2/4)

Experiments on Bilge & Ülusoy (1995) 40 instances

4 machines, 2 vehicles

10 jobsets,

5 - 8 jobs, 13 - 23 operations

4 different structures for FMS

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation

Experimental results (3/4)

Exemple of FMS structure

M1 M2 M3 M4

LU

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation

Experimental results (4/4)

Instances [7] BFS DEV% Instances [7] BFS DEV%Ex11 114 114 0 Ex61 129 129 0Ex12 90 90 0 Ex62 102 102 0Ex13 98 98 0 Ex63 105 105 0Ex14 140 140 0 Ex64 151 151 0Ex21 116 116 0 Ex71 134 134 0Ex22 82 82 0 Ex72 86 86 0Ex23 89 89 0 Ex73 93 93 0Ex24 134 134 0 Ex74 161 161 0Ex31 121 121 0 Ex81 167 167 0Ex32 89 89 0 Ex82 155 155 0Ex33 96 96 0 Ex83 155 155 0Ex34 148 148 0 Ex84 178 178 0Ex41 138 138 0 Ex91 129 127* -1,55Ex42 100 100 0 Ex92 106 106 0Ex43 102 102 0 Ex93 107 107 0Ex44 163 163 0 Ex94 149 149 0Ex51 110 110 0 Ex101 153 153 0Ex52 81 81 0 Ex102 139 139 0Ex53 89 89 0 Ex103 141 139* -1,42Ex54 134 134 0 Ex104 183 183 0

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Conclusion and further works

Conclusion

Step forwards the generalization of the disjunctive graph model including several robots;

Memetic algorithm based approach for a generalization of the job-shop problem;

Specific properties are derived from the longest path to generate neighbourhoods;

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Conclusion and further works

Further works

Additional constraints;

Axact methods;

Larger instances;