sim.vam: Virtual age simulation model
In rcqls/VAM: Virtual Age Models
sim.vam R Documentation
Virtual age simulation model
Description
sim.vam is used to define a virtual age model for Corrective Maintenance (CM) and planned Preventive Maintenance (PM). The object define with sim.vam can be used to simulate realizations of the CM-PM process thanks to simulate method. The last simulated data set, produced with the simulate method, is also memorized inside the sim.vam object, in such way that the characteristics of the model can be plotted thanks to plot method. The object define with sim.vam can also be used to define a PM policy.
Usage
sim.vam(formula)
Arguments
formula
a symbolic description of the virtual age model and observations, or a mle.vam class object for which the estimation method has been launched at least one time. When formula is mle.vam object, the model considered corresponds to the plug in estimator, that is to say the output of the formula.mle.vam function. Otherwise, the formula specifications are described below.
For simulation, with simulate method, formula follows one of following form:
- Only CM:
Response~(CMeff|InitDist)
- CM and one type of PM effect:
Response~(CMeff|InitDist)&(PMeff|PMpolicy)
- CM and several types of PM effect:
Response~(CMeff|InitDist)&(PMeff_1+...+PMeff_n|PMpol)
Response~(CMeff|InitDist)&(PMeff_1+...+PMeff_n|PMpol_1*...*PMpol_k)
Response is description of the data set. For simulation, there is no previously defined data set since it will be generated by the simulation method. Consequently, response has not necessary to be defined in sim.vam object formula. Its default value is System&Time&Type in the case where several systems are simultaneously considered and Time&Type otherwise. Time and Type respectively correspond to the data frame column names of the successive events times and the corresponding events types. When the data set is of data frame class type (and not a list) and when several systems are simultaneously considered, System represents the column name specifying for each event to which system it corresponds.
In others cases than simulation (where the PM policy is useless), formula can possibly be simplify to:
-
Response~(CMeff|InitDist)&(PMeff)
-
Response~(CMeff|InitDist)&(PMeff_1+...+PMeff_n)
-
Response~(CMeff|InitDist)&(PMeff_1+...+PMeff_n)
More details of formula specifications are given under ‘Details’.
Details
From a mathematical point of view the CM-PM process can be specified by its failure intensity. The general failure intensity of a virtual age model verifies for t >= 0
\lambda(t) = V'(t) h(V(t))
where h is a deterministic function characterizing the time to failure hazard rate of the new unmaintained system (specified by InitDist in formula). V(.) is an adapted process, called virtual age, that takes into account the effects of the different maintenances, and V'(.) denotes its derivative. Let us denote C[k] the successive maintenance times and V(t | k) the virtual age at time t given that k maintenance times have already been observed. V(t | 0) denotes the virtual age of a new never maintained system, it is assumed to be equal to t. Consequently, the cumulative intensity or compensator of the CM-PM process, \Lambda verifies for C[k] <= t <= C[k+1],
\Lambda(t)-\Lambda(C[k]) = H(V(t | k)) - H(V(C[k] | k))
with H(t) = \int_{0}^{t} h(s) ds.
In order to be able to simulate the CM-PM process, one has also to specify how PM are planned. This is done thanks to the PM policy denoted PMpol or PMpol_1, ...., PMpol_k in formula.
Maintenance effects
CMeff, PMeff, PMeff_1, ....,, PMeff_n specified respectively the effect of CM, of PM (where only one type of PM effect is considered) and of the different types of PM effect. Notice that only one type of CM effect can be considered. The available maintenance effects are :
AGAN()for As Good As New maintenance. An AGAN maintenance renews the system. That is to say,
after an AGAN maintenance done at time C[k], the virtual age is equal to V(t | k) = V(t-C[k] | 0) = t-C[k] up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]).
ABAO()for As Bad As Old maintenance. An ABAO maintenance has no effect on the propensity of the system to break done, it continues to evolve similarly to what it was before maintenance. That is to say,
after an ABAO maintenance done at time C[k], the virtual age is equal to V(t | k) = V(t | k-1) up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]).
AGAP()for As Good As Previous maintenance. An AGAP maintenance puts the system exactly in the same state as it was just after the last previous maintenance. That is to say, after an AGAP maintenance done at time C[k], the virtual age is equal to V(t | k) = V(t-(C[k]-C[k-1]) | k-1) up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]).
QAGAN()for Quasi As Good As New maintenance. A QAGAN maintenance puts back the virtual age to zero without necessary renewing the system. That is to say, after a QAGAN maintenance done at time C[k], the virtual age is equal to V(t | k) = V(t | k-1)-V(C[k] | k-1) up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]).
ARAInf(rho)for Arithmetic Reduction of Age model with infinite memory. This is also equivalent to a Kijima type II model with deterministic constant effect. The effect of such maintenance is to reduce the virtual age from a quantity proportional to its value just before maintenance. That is to say, after an ARAInf maintenance done at time C[k], the virtual age is equal to
V(t | k) = V(t | k-1) - \rho V(C[k] | k-1) up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]). rho is the corresponding maintenance effect parameter vector, it is of size 1 and represents \rho.
ARA1(rho)for Arithmetic Reduction of Age model with memory one. This also equivalent to a Kijima type I model with deterministic constant effect. The effect of such maintenance is to reduce the virtual age from a quantity proportional to the supplement of age accumulated since the last maintenance. That is to say, after an ARA1 maintenance done at time C[k], the virtual age is equal to
V(t | k) = V(t | k-1) - \rho ( V(C[k] | k-1) - V(C[k-1] | k-1) ) up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]). rho is the corresponding maintenance effect parameter vector, it is of size 1 and represents \rho.
ARAm(rho) or ARAm(rho | m)for Arithmetic Reduction of Age model with memory m. In the ARAInf model, maintenance is supposed to reduce the whole virtual age, whereas in the ARA1 model, it is supposed to reduce only the supplement of age accumulated since the last maintenance. The idea of the ARAm model is to proposed an intermediate effect on the m last times between maintenances. Consequently, an ARAm model with m=1 corresponds to an ARA1 model, and an ARAm model with m bigger than the number of maintenance time observations corresponds to an ARAInf model. But, the bigger m is, in the ARAm model, the less efficient (from an algorithmic point of view) are the different algorithms of the VAM package.
The default value for m is 1. rho is the corresponding maintenance effect parameter vector, it is of size 1 and has the same meaning as for ARAInf and ARA1 models.
QR(rho)for Quasi Renewal maintenance model. This also equivalent to the geometric model. Let Y_1, Y_2, Y_3, ... be iid random variables. The classical geometric process assumes that the successive times between failures follows the same distribution as Y_1, a Y_2, a^2 Y_3, .... Consequently, after a QR maintenance done at time C[k], the virtual age is equal to
V(t | k) = \rho ( V(t | k-1) - V(C[k] | k-1) ) up to the next maintenance time C[k+1] (for C[k] < t <= C[k+1]). A QR maintenance modifies the virtual age time speed. rho is the corresponding maintenance effect parameter vector, it is of size 1 and represents \rho = 1/a.
GQR(rho) or GQR(rho | fun)for Generalized Quasi Renewal maintenance model. This model is similar to the extended geometric process. The geometric evolution of the times between maintenances in the QR model often seems to much explosive to be realistic from a practical point of view. The idea of the GQR model is simply to replace a^n by a^(f(n)) where f() stands for a non decreasing function that can be specified by the fun argument. The possible values for fun are
identity : (default value) f(x)=x, in this case the GQR model corresponds to a QR model,
sqrt : f(x)=x^(0.5),
log : f(x)=log(x+1).
A GQR maintenance modifies the virtual age time speed.
rho is the corresponding maintenance effect parameter vector, it is of size 1.
GQR_ARAInf(rho) or GQR_ARAInf(rho | fun)-
A GQR maintenance puts back the virtual age to zero and modifies the virtual age time speed.
An ARAInf maintenance does not modify the virtual age time speed but restores the virtual age into an intermediate condition between its value before maintenance and zero. The GQR_ARAInf model mixes both effects.
rho is the corresponding maintenance effect parameter vector, it is of size 2 : rho[1] characterizes the GQR effect and rho[2] characterizes the ARAInf effect.
GQR_ARA1(rho) or GQR_ARA1(rho | fun)-
A GQR maintenance puts back the virtual age to zero and modifies the virtual age time speed.
An ARA1 maintenance does not modify the virtual age time speed but restores the virtual age into an intermediate condition between its value before maintenance and zero. The GQR_ARA1 model mixes both effects.
rho is the corresponding maintenance effect parameter vector, it is of size 2 : rho[1] characterizes the GQR effect and rho[2] characterizes the ARA1 effect.
GQR_ARAm(rho) or GQR_ARAm(rho | fun) or GQR_ARAm(rho | m) or GQR_ARAm(rho | m, fun) or GQR_ARAm(rho | fun, m)-
The GQR_ARAm model mixes simultaneously GQR effect and ARAm effect.
rho is the corresponding maintenance effect parameter vector, it is of size 2 : rho[1] characterizes the GQR effect and rho[2] characterizes the ARAm effect.
Time to failure distribution of the new unmaintained system
InitDistr characterizes the hazard rate of the new unmaintained system h(.). The available distributions are:
Weibull(alpha,beta)for a Weibull initial hazard rate with parametrization h(t) = \alpha \beta t^(\beta-1) with \alpha > 0 and \beta > 0.
LogLinear(alpha,beta)for a log-linear hazard rate with parametrization h(t) = \alpha exp(\beta t) with \alpha > 0.
Weibull3(alpha,beta,c)for a three parameter Weibull initial hazard rate with parametrization h(t) = \alpha \beta (t+c)^(\beta-1) with \alpha > 0, \beta > 0 and c > 0.
PM policies
PMpol or PMpol_1, ...., PMpol_k characterize the PM policy. Some of the PM policies available can manage simultaneously several PM effect types. That is why the number of PM effect arguments (PMeff_1, ...., PMeff_n) in formula is not necessary the same as the number of PM policy arguments (PMpol_1, ...., PMpol_k). Whatever the first PM policy argument refers to the corresponding first PM effect arguments and so on. The different PM policies available are:
Periodic(by,from=0,prob=1)a PM is done every by units of times from time from. This PM policy can manage k>1 different PM effect types if prob is a vector of length k. In this case, each PM time as a probability prob[j] to be of j th type (for the different PM effect considered) for every j between 1 and k.
AtTimes(times,cycle=TRUE)-
a PM is done every at every successive times of vector times. If cycle==TRUE, possible PM times are extended after the last times of vector times by recycling, that is to say by reusing the times between events of vector times.
AtIntensity(level,model=NULL)a PM is done as soon as the failure intensity reaches level. By default level refers to the failure intensity computed with the model defined by formula. But, it can also refer to another model denoted model. model is a sim.vam object depending of another formula denoted formula_bis. The number of different PM effects has to be the same in formula and formula_bis. The grammar used to write formula_bis is the same as this of formula. But in formula_bis the PM policy is useless, so it has not to be necessarily defined. For example in the case where only one type of PM effect is considered in formula, then formula_bis can equivalently follows the form : ~(CMeff_bis|InitDist_bis)&(PMeff_bis|PMpolicy_bis) or ~(CMeff_bis|InitDist_bis)&(PMeff_bis).
AtVirtualAge(level,model=NULL)a PM is done as soon as the virtual age reaches level. By default level refers to the virtual age computed with the model defined by formula. But, it can also refers to another model similarly to AtIntensity PM policy.
AtFailureProbability(level,model=NULL)a PM is done as soon as the conditional probability of failure reaches level. By default level refers to the conditional failure probability computed with the model defined by formula. But, it can also refers to another model similarly to AtIntensity PM policy.
Value
The function produces an object of class sim.vam which contains the virtual age model considered.
Author(s)
L. Doyen and R. Drouilhet
References
Classes of imperfect repair models based on reduction of failure intensity or virtual age:
L. Doyen, O. Gaudoin,
Reliability Engineering and System Safety, Elsevier, 2004, 84 (1), pp.45-56.
Modelling and assessment of aging and efficiency of corrective and planned preventive maintenance:
L. Doyen, O. Gaudoin, IEEE Transactions on Reliability, 2011, 60 (4), pp.759-769.
See Also
simulate.sim.vam for simulation.
plot.sim.vam for plotting characteristics of the model.
Examples
##########
# Simulation model:
# ARA Infinite CM with \rho=0.4
# Weibull initial intensity h(t)=0.001*2.5*t^(1.5)
simCM<-sim.vam( T & U ~ (ARAInf(.4) | Weibull(.001,2.5)))
(simulate(simCM,30))
plot(simCM,'i')
##########
# Simulation model:
# Weibull h(t)=0.001*2.5*t^(1.5)
# CM ARA1 (rhoMC=0.6)
# PM ARA Infinite (rhoMP=0.4)
# PM at fixed failure intensity level
simCMPM<-sim.vam( ~ (ARA1(.9) | Weibull(.001,2.5)) & (ARAInf(.4) | AtIntensity(0.2)))
(simulate(simCMPM,50))
plot(simCMPM,'i')
###########
# Simulation model:
# Weibull h(t)=0.001*2.5*t^(1.5)
# CM ARA infinite (rhoMC=0.3)
# periodic PM randomly chosen upon two types
# PM ARA Infinite (rhoMP=0.6) with probability 0.6
# PM ARA Infinite (rhoMP=-0.2) with probability 0.4
simCMPM<-sim.vam( ~ (ARAInf(.3) | Weibull(.001,2.5)) & (ARAInf(.6)+ARAInf(-.2) | Periodic(12,prob=c(0.6,0.4))))
(simulate(simCMPM,20,nb.system=3))
plot(simCMPM,'i',system.index=3)
##########
# Combined PM policies:
# Weibull h(t)=0.001*2.6*t^(1.6)
# CM ABAO
# periodic PM randomly chosen upon two types
# PM ARA Infinite (rhoMP=0.6) with probability 0.6
# PM ARA Infinite (rhoMP=-0.2) with probability 0.4
# and AGAN PM renew the system as soon as the failure intensity reaches the level 0.3
simCMPM<-sim.vam( ~ (ABAO() | Weibull(.001,2.6)) & (ARAInf(.6)+ARAInf(-.2)+AGAN() | Periodic(12,prob=c(0.6,0.4))*AtIntensity(0.3)))
(simulate(simCMPM,15))
plot(simCMPM,'i')
###########
# PM policy not using the same model as for simulation:
# The model used for PM planning
modMPplan <- model.vam( ~ (ARA1(.87) | Weibull(.0015,2.7)) & (ARAInf(.44)))
# The model used for failure time simulation
# a PM is done as soon as the failure intensity of the previous model reaches level 0.3
simCMPM<-sim.vam( ~ (ARA1(.9) | Weibull(.001,2.5)) & (ARAInf(.35) | AtIntensity(0.3,modMPplan)) )
simData<-simulate(simCMPM,20)
modMPplan_plot<-model.vam(Time & Type ~ (ARA1(.87) | Weibull(.0015,2.7)) & (ARAInf(.44)),data=simData)
# The failure intensity of the model used for PM planning
# a PM is done when this failure intensity reaches the level 0.3
plot(modMPplan_plot,'i',col='blue')
# The failure intensity of the model used for failure simulation
plot(simCMPM,'i-cm-pm',col='darkblue',add=TRUE)
###########
# Simulation under the estimated model
simARAInf<-sim.vam( ~ (ARAInf(.4) | Weibull(.001,2.5)))
simData<-simulate(simARAInf,30)
mleARAInf <- mle.vam(Time & Type ~ (ARAInf(.5) | Weibull(1,3)),data=simData)
coef(mleARAInf)
# The following instruction creates a simulator using the plug in estimated model
sim_mle<-sim.vam(mleARAInf)
(simulate(sim_mle,30))
rcqls/VAM documentation built on Jan. 14, 2024, 9:07 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
| sim.vam | R Documentation |
Virtual age simulation model
Description
sim.vam is used to define a virtual age model for Corrective Maintenance (CM) and planned Preventive Maintenance (PM). The object define with sim.vam can be used to simulate realizations of the CM-PM process thanks to simulate method. The last simulated data set, produced with the simulate method, is also memorized inside the sim.vam object, in such way that the characteristics of the model can be plotted thanks to plot method. The object define with sim.vam can also be used to define a PM policy.
Usage
sim.vam(formula)
Arguments
formula |
a symbolic description of the virtual age model and observations, or a
More details of formula specifications are given under ‘Details’. |
Details
From a mathematical point of view the CM-PM process can be specified by its failure intensity. The general failure intensity of a virtual age model verifies for t >= 0
\lambda(t) = V'(t) h(V(t))
where h is a deterministic function characterizing the time to failure hazard rate of the new unmaintained system (specified by InitDist in formula). V(.) is an adapted process, called virtual age, that takes into account the effects of the different maintenances, and V'(.) denotes its derivative. Let us denote C[k] the successive maintenance times and V(t | k) the virtual age at time t given that k maintenance times have already been observed. V(t | 0) denotes the virtual age of a new never maintained system, it is assumed to be equal to t. Consequently, the cumulative intensity or compensator of the CM-PM process, \Lambda verifies for C[k] <= t <= C[k+1],
\Lambda(t)-\Lambda(C[k]) = H(V(t | k)) - H(V(C[k] | k))
with H(t) = \int_{0}^{t} h(s) ds.
In order to be able to simulate the CM-PM process, one has also to specify how PM are planned. This is done thanks to the PM policy denoted PMpol or PMpol_1, ...., PMpol_k in formula.
Maintenance effects
CMeff, PMeff, PMeff_1, ....,, PMeff_n specified respectively the effect of CM, of PM (where only one type of PM effect is considered) and of the different types of PM effect. Notice that only one type of CM effect can be considered. The available maintenance effects are :
AGAN()for As Good As New maintenance. An AGAN maintenance renews the system. That is to say, after an AGAN maintenance done at time
C[k], the virtual age is equal toV(t | k) = V(t-C[k] | 0) = t-C[k]up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]).ABAO()for As Bad As Old maintenance. An ABAO maintenance has no effect on the propensity of the system to break done, it continues to evolve similarly to what it was before maintenance. That is to say, after an ABAO maintenance done at time
C[k], the virtual age is equal toV(t | k) = V(t | k-1)up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]).AGAP()for As Good As Previous maintenance. An AGAP maintenance puts the system exactly in the same state as it was just after the last previous maintenance. That is to say, after an AGAP maintenance done at time
C[k], the virtual age is equal toV(t | k) = V(t-(C[k]-C[k-1]) | k-1)up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]).QAGAN()for Quasi As Good As New maintenance. A QAGAN maintenance puts back the virtual age to zero without necessary renewing the system. That is to say, after a QAGAN maintenance done at time
C[k], the virtual age is equal toV(t | k) = V(t | k-1)-V(C[k] | k-1)up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]).ARAInf(rho)for Arithmetic Reduction of Age model with infinite memory. This is also equivalent to a Kijima type II model with deterministic constant effect. The effect of such maintenance is to reduce the virtual age from a quantity proportional to its value just before maintenance. That is to say, after an ARAInf maintenance done at time
C[k], the virtual age is equal toV(t | k) = V(t | k-1) - \rho V(C[k] | k-1)up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]).rhois the corresponding maintenance effect parameter vector, it is of size 1 and represents\rho.ARA1(rho)for Arithmetic Reduction of Age model with memory one. This also equivalent to a Kijima type I model with deterministic constant effect. The effect of such maintenance is to reduce the virtual age from a quantity proportional to the supplement of age accumulated since the last maintenance. That is to say, after an ARA1 maintenance done at time
C[k], the virtual age is equal toV(t | k) = V(t | k-1) - \rho ( V(C[k] | k-1) - V(C[k-1] | k-1) )up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]).rhois the corresponding maintenance effect parameter vector, it is of size 1 and represents\rho.ARAm(rho) or ARAm(rho | m)for Arithmetic Reduction of Age model with memory
m. In the ARAInf model, maintenance is supposed to reduce the whole virtual age, whereas in the ARA1 model, it is supposed to reduce only the supplement of age accumulated since the last maintenance. The idea of the ARAm model is to proposed an intermediate effect on themlast times between maintenances. Consequently, an ARAm model withm=1corresponds to an ARA1 model, and an ARAm model withmbigger than the number of maintenance time observations corresponds to an ARAInf model. But, the biggermis, in the ARAm model, the less efficient (from an algorithmic point of view) are the different algorithms of the VAM package. The default value formis 1.rhois the corresponding maintenance effect parameter vector, it is of size 1 and has the same meaning as for ARAInf and ARA1 models.QR(rho)for Quasi Renewal maintenance model. This also equivalent to the geometric model. Let
Y_1, Y_2, Y_3, ...be iid random variables. The classical geometric process assumes that the successive times between failures follows the same distribution asY_1, a Y_2, a^2 Y_3, .... Consequently, after a QR maintenance done at timeC[k], the virtual age is equal toV(t | k) = \rho ( V(t | k-1) - V(C[k] | k-1) )up to the next maintenance timeC[k+1](forC[k] < t <= C[k+1]). A QR maintenance modifies the virtual age time speed.rhois the corresponding maintenance effect parameter vector, it is of size 1 and represents\rho = 1/a.GQR(rho)orGQR(rho | fun)for Generalized Quasi Renewal maintenance model. This model is similar to the extended geometric process. The geometric evolution of the times between maintenances in the QR model often seems to much explosive to be realistic from a practical point of view. The idea of the GQR model is simply to replace
a^nbya^(f(n))wheref()stands for a non decreasing function that can be specified by thefunargument. The possible values forfunareidentity: (default value)f(x)=x, in this case the GQR model corresponds to a QR model,sqrt:f(x)=x^(0.5),log:f(x)=log(x+1).
A GQR maintenance modifies the virtual age time speed.
rhois the corresponding maintenance effect parameter vector, it is of size 1.GQR_ARAInf(rho)orGQR_ARAInf(rho | fun)-
A GQR maintenance puts back the virtual age to zero and modifies the virtual age time speed. An ARAInf maintenance does not modify the virtual age time speed but restores the virtual age into an intermediate condition between its value before maintenance and zero. The GQR_ARAInf model mixes both effects.
rhois the corresponding maintenance effect parameter vector, it is of size 2 :rho[1]characterizes the GQR effect andrho[2]characterizes the ARAInf effect. GQR_ARA1(rho)orGQR_ARA1(rho | fun)-
A GQR maintenance puts back the virtual age to zero and modifies the virtual age time speed. An ARA1 maintenance does not modify the virtual age time speed but restores the virtual age into an intermediate condition between its value before maintenance and zero. The GQR_ARA1 model mixes both effects.
rhois the corresponding maintenance effect parameter vector, it is of size 2 :rho[1]characterizes the GQR effect andrho[2]characterizes the ARA1 effect. GQR_ARAm(rho)orGQR_ARAm(rho | fun)orGQR_ARAm(rho | m)orGQR_ARAm(rho | m, fun)orGQR_ARAm(rho | fun, m)-
The GQR_ARAm model mixes simultaneously GQR effect and ARAm effect.
rhois the corresponding maintenance effect parameter vector, it is of size 2 :rho[1]characterizes the GQR effect andrho[2]characterizes the ARAm effect.
Time to failure distribution of the new unmaintained system
InitDistr characterizes the hazard rate of the new unmaintained system h(.). The available distributions are:
Weibull(alpha,beta)for a Weibull initial hazard rate with parametrization
h(t) = \alpha \beta t^(\beta-1)with\alpha > 0and\beta > 0.LogLinear(alpha,beta)for a log-linear hazard rate with parametrization
h(t) = \alpha exp(\beta t)with\alpha > 0.Weibull3(alpha,beta,c)for a three parameter Weibull initial hazard rate with parametrization
h(t) = \alpha \beta (t+c)^(\beta-1)with\alpha > 0,\beta > 0andc > 0.
PM policies
PMpol or PMpol_1, ...., PMpol_k characterize the PM policy. Some of the PM policies available can manage simultaneously several PM effect types. That is why the number of PM effect arguments (PMeff_1, ...., PMeff_n) in formula is not necessary the same as the number of PM policy arguments (PMpol_1, ...., PMpol_k). Whatever the first PM policy argument refers to the corresponding first PM effect arguments and so on. The different PM policies available are:
Periodic(by,from=0,prob=1)a PM is done every
byunits of times from timefrom. This PM policy can managek>1different PM effect types ifprobis a vector of lengthk. In this case, each PM time as a probabilityprob[j]to be ofjth type (for the different PM effect considered) for everyjbetween 1 andk.AtTimes(times,cycle=TRUE)-
a PM is done every at every successive times of vector
times. Ifcycle==TRUE, possible PM times are extended after the last times of vectortimesby recycling, that is to say by reusing the times between events of vectortimes. AtIntensity(level,model=NULL)a PM is done as soon as the failure intensity reaches
level. By defaultlevelrefers to the failure intensity computed with the model defined byformula. But, it can also refer to another model denotedmodel.modelis asim.vamobject depending of another formula denotedformula_bis. The number of different PM effects has to be the same informulaandformula_bis. The grammar used to writeformula_bisis the same as this offormula. But informula_bisthe PM policy is useless, so it has not to be necessarily defined. For example in the case where only one type of PM effect is considered informula, thenformula_biscan equivalently follows the form :~(CMeff_bis|InitDist_bis)&(PMeff_bis|PMpolicy_bis)or~(CMeff_bis|InitDist_bis)&(PMeff_bis).AtVirtualAge(level,model=NULL)a PM is done as soon as the virtual age reaches
level. By defaultlevelrefers to the virtual age computed with the model defined byformula. But, it can also refers to another model similarly toAtIntensityPM policy.AtFailureProbability(level,model=NULL)a PM is done as soon as the conditional probability of failure reaches
level. By defaultlevelrefers to the conditional failure probability computed with the model defined byformula. But, it can also refers to another model similarly toAtIntensityPM policy.
Value
The function produces an object of class sim.vam which contains the virtual age model considered.
Author(s)
L. Doyen and R. Drouilhet
References
Classes of imperfect repair models based on reduction of failure intensity or virtual age: L. Doyen, O. Gaudoin, Reliability Engineering and System Safety, Elsevier, 2004, 84 (1), pp.45-56.
Modelling and assessment of aging and efficiency of corrective and planned preventive maintenance: L. Doyen, O. Gaudoin, IEEE Transactions on Reliability, 2011, 60 (4), pp.759-769.
See Also
simulate.sim.vam for simulation.
plot.sim.vam for plotting characteristics of the model.
Examples
##########
# Simulation model:
# ARA Infinite CM with \rho=0.4
# Weibull initial intensity h(t)=0.001*2.5*t^(1.5)
simCM<-sim.vam( T & U ~ (ARAInf(.4) | Weibull(.001,2.5)))
(simulate(simCM,30))
plot(simCM,'i')
##########
# Simulation model:
# Weibull h(t)=0.001*2.5*t^(1.5)
# CM ARA1 (rhoMC=0.6)
# PM ARA Infinite (rhoMP=0.4)
# PM at fixed failure intensity level
simCMPM<-sim.vam( ~ (ARA1(.9) | Weibull(.001,2.5)) & (ARAInf(.4) | AtIntensity(0.2)))
(simulate(simCMPM,50))
plot(simCMPM,'i')
###########
# Simulation model:
# Weibull h(t)=0.001*2.5*t^(1.5)
# CM ARA infinite (rhoMC=0.3)
# periodic PM randomly chosen upon two types
# PM ARA Infinite (rhoMP=0.6) with probability 0.6
# PM ARA Infinite (rhoMP=-0.2) with probability 0.4
simCMPM<-sim.vam( ~ (ARAInf(.3) | Weibull(.001,2.5)) & (ARAInf(.6)+ARAInf(-.2) | Periodic(12,prob=c(0.6,0.4))))
(simulate(simCMPM,20,nb.system=3))
plot(simCMPM,'i',system.index=3)
##########
# Combined PM policies:
# Weibull h(t)=0.001*2.6*t^(1.6)
# CM ABAO
# periodic PM randomly chosen upon two types
# PM ARA Infinite (rhoMP=0.6) with probability 0.6
# PM ARA Infinite (rhoMP=-0.2) with probability 0.4
# and AGAN PM renew the system as soon as the failure intensity reaches the level 0.3
simCMPM<-sim.vam( ~ (ABAO() | Weibull(.001,2.6)) & (ARAInf(.6)+ARAInf(-.2)+AGAN() | Periodic(12,prob=c(0.6,0.4))*AtIntensity(0.3)))
(simulate(simCMPM,15))
plot(simCMPM,'i')
###########
# PM policy not using the same model as for simulation:
# The model used for PM planning
modMPplan <- model.vam( ~ (ARA1(.87) | Weibull(.0015,2.7)) & (ARAInf(.44)))
# The model used for failure time simulation
# a PM is done as soon as the failure intensity of the previous model reaches level 0.3
simCMPM<-sim.vam( ~ (ARA1(.9) | Weibull(.001,2.5)) & (ARAInf(.35) | AtIntensity(0.3,modMPplan)) )
simData<-simulate(simCMPM,20)
modMPplan_plot<-model.vam(Time & Type ~ (ARA1(.87) | Weibull(.0015,2.7)) & (ARAInf(.44)),data=simData)
# The failure intensity of the model used for PM planning
# a PM is done when this failure intensity reaches the level 0.3
plot(modMPplan_plot,'i',col='blue')
# The failure intensity of the model used for failure simulation
plot(simCMPM,'i-cm-pm',col='darkblue',add=TRUE)
###########
# Simulation under the estimated model
simARAInf<-sim.vam( ~ (ARAInf(.4) | Weibull(.001,2.5)))
simData<-simulate(simARAInf,30)
mleARAInf <- mle.vam(Time & Type ~ (ARAInf(.5) | Weibull(1,3)),data=simData)
coef(mleARAInf)
# The following instruction creates a simulator using the plug in estimated model
sim_mle<-sim.vam(mleARAInf)
(simulate(sim_mle,30))
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.