Stochastic Maintenance
Scheduling Problem
G. Fleury, P. Lacomme,
M. Sevaux
Laboratoire de
Mathématiques
Clermont-Ferrand
UMR 6620
Laboratoire
d’Informatique
Clermont-Ferrand
UMR 6158
LAMIH
Valenciennes
UMR 8530
1
Plan
 Problem
statement
 Assumptions and objective
 Genetic Algorithm template
 Computational experiments
 Future research
2
Problem statement
Maintenance
10 000 elementary tasks
8 majors operations for each coach
64 aggregated tasks for one TGV
Objective: minimize the total
duration
3
Physical description (1)
coaches 1, 2
CTA1
coaches 3, 4
TSC1
CTA2
IP
coaches 5, 6
CTA3
TSC2
coaches 7, 8
CTA4
4
Physical description (2)

CTAx
(caisses TGV et Automoteur):
Dis-assembling tasks
Re-assembling tasks
Works insided coaches

IP (industries privées):
Sand blasting by external companies

TSCx
(tôlerie, stucture de caisse):
Handling the tollery
Renovation of external parts of coaches
5
Logical description (1)
jobs sequence of treatment
6
Logical description (2)
7
A stochastic problem (1)
 Processing
time of jobs are
submitted to variations
 Robust
solutions are required to
avoid periodic computation of new
schedule

Minimization of the makespan is also
required
8
Random events modelization


 : extra delay
pp : probability of random events
occurrences
9
A template for stochastic
problem (1)
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A template for stochastic
problem (2)

Optimization phase:


Searching process based of statistic
performances of solutions
Robustness evaluation of solutions



Replications
Average cost of solution
Standard deviation of solutions
11
Genetic Algorithm template (1)
Construct a random initial set of solutions
Repeat
Select P1 and P2 based on the inverse function of
the fitness rank distribution
Apply XOver operator
Evaluate C
With probability P then
Mutate C (swap two random points p and q)
Until (a maximal number of iterations is reached).
 See (Sevaux and Le Quéré, 2003)
12
Genetic Algorithm template (2)
One chromosome is:
– Ordered set of jobs
– Evaluation of the average cost
– Evaluation of the standard deviation cost
13
Robust Approach

Principles
Compute
which is a evaluation of the average cost over n
replications

Compute
which is evaluation of the standard deviation over n
replications


Problems
Very costly for a computational point of view
14
Stochastic Approach (1)


Replace statistical evaluation by
mathematical evaluation
Based on shortest path computed in
the disjunctive graph
15
Stochastic Approach (2)

Tasks duration

with


 Y binomial law
16
Stochastic Approach (3)

So

Average :

Standard deviation :

Finally:
17
Results for the robust approach
(Sevaux and Le Quéré, 2003)
18
Results for the robust approach
Results with mathematical evaluation of criteria
19
Concluding remarks (1)
 Stochastic
 Two
maintenance problem
approaches:
A robust approach
A stochastic approach
 Both
approaches provides robust
solutions
20
Concluding remarks (2)
 Robust
Approach
High quality solutions
Post analysis provide results very closed to
the evaluations
Time consuming
 Stochastic
Approach
Satisfactory evaluation of soluitons
Very short computational time
21
Future Research
 Improve
mathematical analysis
 Take into account all shortest paths
 Improve
modelization of the problem
 modelize random variations
22