PARTIE II

PARTIE IIPARTIE IIIntroduction à la Introduction à la

ModélisationModélisation & &

à l’ Optimisationà l’ Optimisation

Modèle GPIM Modèle GPIM vsvs GRP GRP Optimisation « Overhaul Policy B-H »Optimisation « Overhaul Policy B-H » Antinomie selon le REX pour la Antinomie selon le REX pour la

modélisationmodélisation MC/MP MC/MP GPIM_PLP GPIM_PLP vs vs GPIM_LLPGPIM_LLP

REF 1 : cf.Generalized proportional intensities models for repairable systems.

By D.F. PERCY & B.M. ALKALI. Journal of Management Mathematics(2006) 17,171-185.

REF 2 : cf. Discontinuous point processes for the analysis of REF 2 : cf. Discontinuous point processes for the analysis of repairable units .repairable units .

By R.CALABRIA & G. PULCINI. By R.CALABRIA & G. PULCINI. International Journal OF Reliability, Quality and Safety International Journal OF Reliability, Quality and Safety

Engineering (1999) Engineering (1999) Vol.6, N°.4, 361-382.Vol.6, N°.4, 361-382.

REF 3:P_PLP REF 3:P_PLP cf.Practical Methods for Modeling Repairable cf.Practical Methods for Modeling Repairable Systems with Time Trends and Repair Effects. Systems with Time Trends and Repair Effects.

by H. GUO, W. ZHAO & A. METTAS. IEEE(2006).by H. GUO, W. ZHAO & A. METTAS. IEEE(2006).L_LLP L_LLP cf.A New Stochastic Model for Systems Under cf.A New Stochastic Model for Systems Under

General Repair. General Repair. by H. GUO, W. ZHAO & A. METTAS. IEEE(2007).by H. GUO, W. ZHAO & A. METTAS. IEEE(2007).

BASELINE MODELSBASELINE MODELS

)1(** t

With the power-law intensity baseline functionWith the power-law intensity baseline functionLog Likelihood “ SIMPLE SYSTEM TYPE I ”Log Likelihood “ SIMPLE SYSTEM TYPE I ”

PARTIAL REPAIR (CM PARTIAL REPAIR (CM (PERCY) , (PERCY) , (GUO) , (GUO) , (CALABRIA)) (CALABRIA))

GRAPHIQUE par CALCUL ANALYTIQUE du modèle GRAPHIQUE par CALCUL ANALYTIQUE du modèle P_PLP/PIM_PLPP_PLP/PIM_PLP

P_PLP[P_PLP[_,t_] = (-1/_,t_] = (-1/)*Log[(1-)*Log[(1-**λλ*t^*t^)];)]; = 3, λ = 0.001. = 3, λ = 0.001.

= Exp[-]

With the log-linear intensity baseline functionWith the log-linear intensity baseline functionLog Likelihood “ SIMPLE SYSTEM TYPE I ”Log Likelihood “ SIMPLE SYSTEM TYPE I ”

PARTIAL REPAIR (CM PARTIAL REPAIR (CM (PERCY) , (PERCY) , (GUO) , (GUO) , (CALABRIA)) (CALABRIA))

Log Likelihood “ MULTI- SYSTEMS TYPE I ” Log Likelihood “ MULTI- SYSTEMS TYPE I ” (1/2)(1/2)

PARTIAL REPAIR (CM PARTIAL REPAIR (CM (GUO)) (GUO))

Log Likelihood “ MULTI- SYSTEMS TYPE I ” Log Likelihood “ MULTI- SYSTEMS TYPE I ” (2/2)(2/2)

PARTIAL REPAIR (CM PARTIAL REPAIR (CM (GUO)) (GUO))

ESTIMATIONSESTIMATIONS

GRAPHIQUES du C.I.F.GRAPHIQUES du C.I.F.

OPTIMISATION du OPTIMISATION du

REMPLACEMENTREMPLACEMENT

OPTIMISATION du OPTIMISATION du REMPLACEMENTREMPLACEMENT

OPTIMISATION du OPTIMISATION du REMPLACEMENTREMPLACEMENT

INFLUENCE SURINFLUENCE SUR L’ OPTIMISATIONL’ OPTIMISATION

VALIDATION d’un MODELE HPPVALIDATION d’un MODELE HPP

VERIFICATIONVERIFICATION

VALIDATION d’un MODELE VALIDATION d’un MODELE GRPGRP

C

?

?

RE-ANALYSERE-ANALYSE

CUMULATIF TIME ( hours )

MCF

4003002001000

50

40

30

20

10

0

Parameter, MLEMTBF

8,32612

Mean Cumulative Function " TUBER MACHINE "

LK_HPP = - 152.85

AIC_HPP = 307.7BIC_HPP = 309.52

GENERALIZED PROPORTIONAL INTENSINTIES GENERALIZED PROPORTIONAL INTENSINTIES MODELSMODELS

Whitout Covariates Whitout Covariates Log Likelihood “ SIMPLE SYSTEM ”Log Likelihood “ SIMPLE SYSTEM ”

GPIM GPIM ( CM ( CM + PM + PM ) )

EXAMPLE I : EXAMPLE I : SIMPLE SYSTEM (General SIMPLE SYSTEM (General Repair)Repair)

cf. Scheduling preventive maintenance for oil pumps using cf. Scheduling preventive maintenance for oil pumps using generalized proportional intensities models by D.F. PERCY & generalized proportional intensities models by D.F. PERCY &

B.M. ALKALI.B.M. ALKALI. International Transactions in Operational Research. 14 International Transactions in Operational Research. 14

(2007) 547-563.(2007) 547-563.

ESTIMATIONSESTIMATIONS

ESTIMATIONS « AUTEURS »ESTIMATIONS « AUTEURS »

EXAMPLE I I : EXAMPLE I I : SIMPLE SYSTEM (General SIMPLE SYSTEM (General Repair)Repair)

Cf. A pratical method of predicting the failure intensity of hydropower Cf. A pratical method of predicting the failure intensity of hydropower generating units.generating units.

By X. QIAN & Y. WUBy X. QIAN & Y. WUIEEE 2011IEEE 2011

ESTIMATIONSESTIMATIONS

ESTIMATIONS « AUTEURS »ESTIMATIONS « AUTEURS »

EXAMPLE III : EXAMPLE III : SIMPLE SYSTEM (General SIMPLE SYSTEM (General Repair)Repair)

Cf. Bayesian Prediction of the Overhaul Effect on a Repairable Cf. Bayesian Prediction of the Overhaul Effect on a Repairable SystemSystem

with Bounded Failure Intensity.with Bounded Failure Intensity.(International Journal of Quality, Statistics and Reliability (International Journal of Quality, Statistics and Reliability

2010).2010).

ESTIMATIONSESTIMATIONS

ESTIMATIONS « AUTEURS »ESTIMATIONS « AUTEURS »

OPTIMISATION de la M.P.OPTIMISATION de la M.P.

GPIM_PLP GPIM_PLP LK_V = LK_V = - 28.4311- 28.4311 = 1.13678e-14, = 1.13678e-14, = 7.68063, = 7.68063, c = 0.291656, c = 0.291656,

p = 0.432698p = 0.432698

0 50 100 150 200 250Tu.t 0.00

0.05

0.10

0.15

INTENSITY u.tINTENSITY FUNCTION GPIM_PLP vs ARAinfAGAN

GPIM_PLP GPIM_PLP LK_V = LK_V = - 28.4311- 28.4311 = 1.13678e-14, = 1.13678e-14, = 7.68063, = 7.68063, c = 0.291656, c = 0.291656,

p = 0.432698p = 0.432698

SIMULATION MONTE-CARLO SIMULATION MONTE-CARLO DU MODELE DU MODELE GPIM_PLPGPIM_PLP