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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 »
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
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
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
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
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
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
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
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.
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.
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
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
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.
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
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
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 ,
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
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
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 Selections
OA which defines Operation Assignments to each robot
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 M3
0 75 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
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 M3
05 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
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 M1
0 7 9 14 177 2 5 3
M3 r1 M4 r2 M1
0 14 16 20 235 2 4 3
M5 r3 M1 r3 M3
0 5 7 12 145 2 5 2
4
24
0
0
0
1
3
7 3 45 55
Makespan =24
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
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 (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
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 31
m2
2
m3
3
m5
1
m3
2
m4
3
m1
1
m1
2
m1
3
m3tr11tr21tr31 tr12tr22tr32
OA r1
tr11
r1
tr21
r3
tr31
r2
tr12
r2
tr22
r3
tr32
MTS
Chromosome is a representation of a solution
Makespan = 24
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.
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 M1
0 7 9 14 177 2 5 3
r1 M4 r2 M1
14 16 20 232 4 3
r3 M1 r3 M3
5 7 12 142 5 2
4
24
0
0 M3
05
M5
05
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 tr32
Change transport disjunction
Robot block
IESM 2009, MONTREAL – CANADA, May 13 TR 27
0 *
M2 r1 M3 r2 M1
0 8 10 15 187 2 5 3
M3 r1 M4 r2 M1
0 5 7 18 225 2 4 3
M5 r3 M1 r3 M3
0 5 7 12 155 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
IESM 2009, MONTREAL – CANADA, May 13 TR 28
r1
tr11
r3
0 *
M2 r1 M3 r2 M1
0 8 10 15 187 2 5 3
M3 r1 M4 r2 M1
0 5 7 18 225 2 4 3
M5 r3 M1 r3 M3
0 5 7 12 155 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
IESM 2009, MONTREAL – CANADA, May 13 TR 29
4
r1
tr11
r3
0 *
M2 r1 M3 r2 M1
0 7 9 14 177 2 5 3
M3 r1 M4 r2 M1
0 5 7 17 215 2 4 3
M5 r3 M1 r3 M3
0 9 11 16 185 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
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.
IESM 2009, MONTREAL – CANADA, May 13 TR 31
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 byeach method to lower bound proposed by Hurink and Knust
Four methods proposed by Hurink and Knust Our method
13,40 16,16 14,22 16,63 13,33
IESM 2009, MONTREAL – CANADA, May 13 TR 32
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
IESM 2009, MONTREAL – CANADA, May 13 TR 33
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
IESM 2009, MONTREAL – CANADA, May 13 TR 34
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
IESM 2009, MONTREAL – CANADA, May 13 TR 35
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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