Stochastic Maintenance Scheduling Problem

1. 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

2. Plan • Problem statement • Assumptions and objective • Genetic Algorithm template • Computational experiments • Future research

3. Problem statement 10 000 elementary tasks 8 majors operations for each coach 64 aggregated tasks for one TGV Objective: minimize the total duration

4. Physical description (1)

5. 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

6. Logical description (1) jobs sequence of treatment

7. Logical description (2)

8. 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

9. Random events modelization • : extra delay • pp : probability of random events occurrences

10. A template for stochastic problem (1)

11. 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

12. 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)

13. Genetic Algorithm template (2) One chromosome is: • Ordered set of jobs • Evaluation of the average cost • Evaluation of the standard deviation cost

14. 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

15. Stochastic Approach (1) • Replace statistical evaluation by mathematical evaluation • Based on shortest path computed in the disjunctive graph

16. Stochastic Approach (2) • Tasks duration  with    Y binomial law

17. Stochastic Approach (3) • So • Average : • Standard deviation : • Finally:

18. Results for the robust approach (Sevaux and Le Quéré, 2003)

19. Results for the robust approach Results with mathematical evaluation of criteria

20. Concluding remarks (1) • Stochastic maintenance problem • Two approaches: A robust approach A stochastic approach • Both approaches provides robust solutions

21. 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

22. Future Research • Improve mathematical analysis  Take into account all shortest paths • Improve modelization of the problem  modelize random variations