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

1
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 

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

• Plan
• Introduction
• Algorithm based framework
• Computational evaluation
• Conclusions and further works

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
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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
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Introduction Type of system under study FMS
based on AGV
Flexible machines
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Introduction Type of system under study FMS
based on AGV
Flexible cells
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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
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Introduction AGV operating (1/2)
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Introduction AGV operating (2/2)

• There are two types of vehicle trips
• the first type of loaded vehicle trips
• the second one is the empty vehicle trips.

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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.

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Introduction Problem definition (2/5)

• Problem definition Example of solution

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Introduction Problem definition (3/5)

• Problem definition Complexity
• Combined problem of
• 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)
• a generic Vehicle Scheduling Problem (VSP) which
  is NP-hard problem (Orloff 1976).

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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.

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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 .

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

• General template

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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.

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Disjunctive graph
definition (2/3)
J1
J2

J3
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Disjunctive graph
definition (2/2)
5
8
M2
r1
M3
r2
M1
4
0
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M3
r1
M4
M1
0

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M5
M1
M3
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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
  Selections
• OA which defines Operation Assignments to each
  robot

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Disjunctive graph
orientation (1/2)
MTS
Transport operations
Transport operations
5
7
M2
M3
M1
4
0
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1
M3
M4
M1
0

0
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M5
M1
M3
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Disjunctive graph
orientation (2/2)
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MTS
1
2
1
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3
1
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Machine operations
Machine operations
Machine operations
2
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3
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M2
r1
M3
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M1
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0

0
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OA
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Graph evaluation
and Critical Path
Makespan 24
0
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Memetic algorithm

• begin
• npi  0  // current iteration number
• ni 0  // number of successive
• unproductive iteration
• Repeat
• SelectSolution (P1,P2)
• C Crossover(P1,P2)
• LocalSearch(C) with probability pm
• InsertSolution(Pop,C)
• Sort(Pop)
• If (npinp)
• Restart(Pop,p)
• End If
• Until (stopCriterion).
• End

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Chromosome
Chromosome is a representation of a solution
Makespan 24
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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.

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Local search (2/5)
tr11
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MTS
OA
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Local search (3/5)
tr21
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tr12
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tr32
MTS
OA
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Local search (4/5)
tr21
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tr12
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tr32
MTS
r3
OA
Change robot assignement
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M3
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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Algorithm based framework Local search (5/5)
tr21
tr11
tr31
tr12
tr22
tr32
MTS
r3
OA
Change robot assignement
0
7
9
14
17
2
5
3
7
M2
r1
M3
r2
M1
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0
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M3
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M4
r2
M1
r3

0
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M5
r3
M1
r3
M3
New transport disjunction is added
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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,
  Prentice
• Hall, Englewood Cliffs, NJ, 1963.

Deviation in percentage from the best solution found by each method to lower bound proposed by Hurink and Knust Deviation in percentage from the best solution found by each method to lower bound proposed by Hurink and Knust Deviation in percentage from the best solution found by each method to lower bound proposed by Hurink and Knust Deviation in percentage from the best solution found by each method to lower bound proposed by Hurink and Knust Deviation in percentage from the best solution found by each method to lower bound proposed by Hurink and Knust
Four methods proposed by Hurink and Knust Four methods proposed by Hurink and Knust Four methods proposed by Hurink and Knust Four methods proposed by Hurink and Knust Our method
13,40 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

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Simultaneous scheduling of machines and automated
guided vehicles graph modelling and resolution
Computational evaluation Experimental
results (4/4)
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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