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 »
Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution
Plan
Plan

Introduction

Algorithm based framework

Computational evaluation

Conclusions and further works
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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
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)
Loaded vehicle
arriving at
station k
Unloading at
station k
Loaded vehicle
arriving at station i
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Loading at
station k
Unloading at
station i
Loaded vehicle
going to station ....
Idle vehicle at
station i
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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution
Introduction
AGV operating (2/2)
Empty vehicle
arriving at
station j
Loading at
station j
Loaded vehicle
going to station ....
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
Empty trip
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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:
(i) scheduling problem of the form n / M / G / Cmax
(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).
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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.
JR t kl , t kl' Cmax
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.

t kl indicates that we have job-independent, but machine-

dependant transportation times.

t kl' 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
interface
Instance to solve
and
framework
parameters
Best
solution
Memetic algorithm
MTS and OA
Non oriented
disjunctive graphe
One solution
evaluation
Longest path algorithm
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Oriented
disjunctive graphe
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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
G  VM Vt , C  Dm  DR 
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
M2
7
M3
5
M1
4
0
J2
J3
0
0
M3
0
M5
5
5
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M4
M1
4
5
M1
1
*
3
M3
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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution
Algorithm based framework
Disjunctive graph definition (2/2)
Machine
disjunction
problem
Robot
disjunction
problem
M2
8
r1
M3
5
r2
M1
4
0
0
0
M3
0
M5
5
r1
5
M4
M1
4
5
M1
1
*
3
M3
Robot assignment
problem
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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)
m2 m3 m5
MTS
1
2
3
m1 m1 m3
m3 m4 m1
1
2
3
1
2
3
1
Transport
operations
M2
0
0
M3
5
0
M5
5
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5
M3
5
0
1
3
2
3
Transport
operations
7
0
2
4
5
M4
2
M1
7
M1
5
4
5
4
M1
2
1
*
3
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)
tr11 tr21 tr31
MTS
1
2
3
1
2
3
Machine
operations
M2
7
0
7
0
0
0
r1
M3
5
M5
1
2
r1
r3
3
Machine
operations
2
2
2
1
5
M1
1
3
2
3
r2
3
M1
4
5
M4
2
Machine
operations
M3
5
0
5
tr12 tr22 tr32
3
4
5
r2
r3
5
3
2
4
M1
1
*
3
M3
OA r1 r1 r3 r2 r2 r3
tr11 tr21 tr31 tr12 tr22 tr32
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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
M2
7
7
0
r1
7
0
0
M3
5
0
M5
14
r1
5
5
r3
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2
2
5
M3
5
0
0
2
9
M4
7
M1
r2
17
3
M1
4
5
16
14
3
4
5
5
20
r2
12
r3
3
2
4
23
M1
14
24
1
*
3
M3
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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 (npi=np)
Restart(Pop,p)
End If
Until (stopCriterion).
End
IESM 2009, MONTREAL – CANADA, May 13
interface
Instance to solve
and
framework
parameters
Best
solution
Memetic algorithm
MTS and OA
Non oriented
disjunctive graphe
One solution
evaluation
Longest path algorithm
Oriented
disjunctive graphe
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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
m2 m3 m5 tr11 tr21tr31 m3 m4 m1 tr12tr22tr32 m1 m1 m3
MTS
1
2
3
1
2
3
OA
r1
r1
r3
r2
r2
r3
1
2
3
1
2
3
1
2
3
tr11 tr21 tr31tr12 tr22 tr32
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)
Change transport disjunction
m2 m3 m5 tr11 tr21 tr31 m3 m4 m1 tr12 tr22 tr32 m1 m1 m3
MTS
OA
1
2
3
1
2
3
1
2
3
r1 r1 r3 r2 r2 r3
1
2
1
3
2
3
Robot block
tr11 tr21 tr31 tr12 tr22 tr32
0
M2
7
7
0
r1
74
0
M3
5
0
M5
14
r1
5
5
r3
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2
2
5
M3
5
0
0
0
2
9
M4
7
M1
r2
17
3
M1
4
5
16
14
3
4
5
5
20
r2
12
r3
3
2
4
23
M1
14
24
1
*
3
M3
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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution
Algorithm based framework
Local search (3/5)
Change machine disjunction
m2 m3 m5 tr21 tr11 tr31 m3 m4 m1 tr12 tr22 tr32 m1 m1 m3
MTS
OA
1
2
3
2
1
3
1
2
3
1
2
1
3
2
Machine block
r1 r1 r3 r2 r2 r3
3
Makespan =23
tr21 tr11 tr31 tr12 tr22 tr32
0
M2
8
7
0
r1
3
0
0
M3
5
0
M5
5
r1
5
5
r3
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2
2
5
M3
5
0
0
2
10
M4
7
M1
r2
18
3
M1
4
5
7
15
3
4
5
5
18
r2
12
r3
3
2
4
22
M1
15
23
1
*
3
M3
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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution
Algorithm based framework
Local search (4/5)
m2 m3 m5 tr21 tr11 tr31 m3 m4 m1 tr12 tr22 tr32 m1 m1 m3
MTS
OA
1
2
3
2
1
3
1
2
3
r1 r3
r1 r3 r2 r2 r3
1
2
1
3
2
3
Change robot assignement
tr21 tr11 tr31 tr12 tr22 tr32
0
M2
8
7
0
r1
3
0
0
M3
5
0
M5
5
r3
r1
5
5
r3
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2
2
5
M3
5
0
0
2
10
M4
7
M1
r2
18
3
M1
4
5
7
15
3
4
5
5
18
r2
12
r3
3
2
4
22
M1
15
23
1
*
3
M3
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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution
Algorithm based framework
Local search (5/5)
m2 m3 m5 tr21 tr11 tr31 m3 m4 m1 tr12 tr22 tr32 m1 m1 m3
MTS
OA
1
2
3
2
1
3
1
2
3
r1 r3
r1 r3 r2 r2 r3
1
2
1
3
2
3
Change robot assignement
tr21 tr11 tr31 tr12 tr22 tr32
0
M2
7
7
2
r1
0
0
M3
5
5
r3
r1
2
5
M3
5
0
0
9
M4
r2
17
3
M1
4
5
7
14
3
4
5
17
r2
3
4
21
M1
22
1
*
4
0
0
M5
9
5
r3
2
11
M1
5
16
r3
2
18
3
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
Four methods proposed by Hurink and Knust
13,40
16,16
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14,22
16,63
Our method
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%
114 114
0
Ex11
90
90
0
Ex12
98
98
0
Ex13
140 140
0
Ex14
116 116
0
Ex21
82
82
0
Ex22
89
89
0
Ex23
134 134
0
Ex24
121 121
0
Ex31
89
89
0
Ex32
96
96
0
Ex33
148 148
0
Ex34
138 138
0
Ex41
100 100
0
Ex42
102 102
0
Ex43
163 163
0
Ex44
110 110
0
Ex51
81
81
0
Ex52
89
89
0
Ex53
134 134
0
Ex54
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Instances [7] BFS DEV%
129 129
0
Ex61
102 102
0
Ex62
105 105
0
Ex63
151 151
0
Ex64
134 134
0
Ex71
86
86
0
Ex72
93
93
0
Ex73
161 161
0
Ex74
167 167
0
Ex81
155 155
0
Ex82
155 155
0
Ex83
178 178
0
Ex84
129 127*
-1,55
Ex91
106 106
0
Ex92
107 107
0
Ex93
149 149
0
Ex94
153 153
0
Ex101
139 139
0
Ex102
141 139*
-1,42
Ex103
183 183
0
Ex104
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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;
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