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A comparison of time-based maintenance andcondition-based maintenance
Bram de Jonge Ruud TeunterWarse Klingenberg Tiedo Tinga
University of GroningenThe Netherlands
COPE congres 2013November 1, 2013
Introduction
We consider the following setting:
I A single unit.
I Gradual deterioration, condition monitoring is possible.
I The cost of a breakdown is 1, after which a new unit isputting into use.
I Preventive maintenance can be performed at cost c < 1 andmakes the unit as good as new.
Introduction
We consider the following setting:
I A single unit.
I Gradual deterioration, condition monitoring is possible.
I The cost of a breakdown is 1, after which a new unit isputting into use.
I Preventive maintenance can be performed at cost c < 1 andmakes the unit as good as new.
Introduction
We consider the following setting:
I A single unit.
I Gradual deterioration, condition monitoring is possible.
I The cost of a breakdown is 1, after which a new unit isputting into use.
I Preventive maintenance can be performed at cost c < 1 andmakes the unit as good as new.
Introduction
We consider the following setting:
I A single unit.
I Gradual deterioration, condition monitoring is possible.
I The cost of a breakdown is 1, after which a new unit isputting into use.
I Preventive maintenance can be performed at cost c < 1 andmakes the unit as good as new.
Introduction
We consider the following setting:
I A single unit.
I Gradual deterioration, condition monitoring is possible.
I The cost of a breakdown is 1, after which a new unit isputting into use.
I Preventive maintenance can be performed at cost c < 1 andmakes the unit as good as new.
Introduction
We consider the following setting:
I A single unit.
I Gradual deterioration, condition monitoring is possible.
I The cost of a breakdown is 1, after which a new unit isputting into use.
I Preventive maintenance can be performed at cost c < 1 andmakes the unit as good as new.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Maintenance policies
I Time-based maintenance (TBM)
I Easy to implement, no condition monitoring needed.
I Decision variable: T (maintenance age).
I Maintenance is performed when the unit reaches age T .
I Condition-based maintenance (CBM)
I Condition monitoring required, more appropriate scheduling.
I Decision variable: M (threshold deterioration level).
I Maintenance is performed when the deterioration level M isreached.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The state of the unit is assumed to deteriorate gradually accordingto a stationary/homogeneous gamma process X (t).
Properties:
I Continuous-time process.
I X (0) = 0 with probability 1.
I X (τ) − X (t) ∼ fa(τ−t),b for τ > t ≥ 0 (f is the gammadensity function).
I X (t) has independent increments.
I Jump process with infinitely many jumps in any time interval.
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
The stationary gamma process has two parameters:
I Shape parameter a.
I Scale parameter b.
Properties:
I E (X (t)) = abt.
I Var(X (t)) = ab2t.
A breakdown is assumed to occur if the level of deteriorationexceeds 1.
We choose a and b such that the mean time to failure equals 1(this does not imply ab = 1).
Gamma deterioration process
0
1
X(t)
0 1t
Gamma deterioration process
0
1
X(t)
0 1t
Low a, high b
Gamma deterioration process
0
1
X(t)
0 1t
Low a, high b
High a, low b
Gamma deterioration process
0
1
X(t)
0 1t
Low a, high b
High a, low b
In between
Research methodology
Simulation:
I Exact calculations are practically impossible.
I If the number of iterations is sufficiently high, the outcomesare very accurate.
I Simulation details:I Time step: ∆ = 0.01.I Number of iterations: n = 100,000.
Research methodology
Simulation:
I Exact calculations are practically impossible.
I If the number of iterations is sufficiently high, the outcomesare very accurate.
I Simulation details:I Time step: ∆ = 0.01.I Number of iterations: n = 100,000.
Research methodology
Simulation:
I Exact calculations are practically impossible.
I If the number of iterations is sufficiently high, the outcomesare very accurate.
I Simulation details:I Time step: ∆ = 0.01.I Number of iterations: n = 100,000.
Research methodology
Simulation:
I Exact calculations are practically impossible.
I If the number of iterations is sufficiently high, the outcomesare very accurate.
I Simulation details:I Time step: ∆ = 0.01.I Number of iterations: n = 100,000.
Research methodology
Simulation:
I Exact calculations are practically impossible.
I If the number of iterations is sufficiently high, the outcomesare very accurate.
I Simulation details:I Time step: ∆ = 0.01.I Number of iterations: n = 100,000.
Base case
Gamma deterioration process:
I a = 5, b ≈ 0.2246.
Relative cost preventive maintenance:
I c = 0.2.
Base case
Gamma deterioration process:
I a = 5, b ≈ 0.2246.
Relative cost preventive maintenance:
I c = 0.2.
Base case
Gamma deterioration process:
I a = 5, b ≈ 0.2246.
Relative cost preventive maintenance:
I c = 0.2.
Cost analysis
Failure cost = 1
Cost saving TBM
Cost saving CBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost analysis
Failure cost = 1
Cost saving TBM
Cost saving CBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost analysis
0
1
Cost
0 1T or M
Cost analysis
Failure cost = 1
Cost saving TBM
Cost saving CBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost analysis
0
1
Cost
0 1T or M
Cost analysis
0
1
Cost
0 1T or M
TBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost saving CBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost analysis
0
1
Cost
0 1T or M
TBM
Cost analysis
0
1
Cost
0 1T or M
TBM
CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Effect of deterioration process
0
1
Cost
0 5σ
Effect of deterioration process
0
1
Cost
0 5σ
TBM
Effect of deterioration process
0
1
Cost
0 5σ
TBM
CBM
Cost benefit CBM compared with TBM
0
0.25
Cost
0 5σ
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Planning time
I Preventive maintenance needs to be planned s time units inadvance.
I The cost c of preventive maintenance does not have to bepaid if failure occurs between the moment that maintenance isplanned and the moment that maintenance would have beenperformed.
I Under the CBM policy, maintenance is performed at timet(M) + s, with t(M) the time at which deterioration level Mis reached.
I The planning time s has no influence on the TBM policy.
Planning time
I Preventive maintenance needs to be planned s time units inadvance.
I The cost c of preventive maintenance does not have to bepaid if failure occurs between the moment that maintenance isplanned and the moment that maintenance would have beenperformed.
I Under the CBM policy, maintenance is performed at timet(M) + s, with t(M) the time at which deterioration level Mis reached.
I The planning time s has no influence on the TBM policy.
Planning time
I Preventive maintenance needs to be planned s time units inadvance.
I The cost c of preventive maintenance does not have to bepaid if failure occurs between the moment that maintenance isplanned and the moment that maintenance would have beenperformed.
I Under the CBM policy, maintenance is performed at timet(M) + s, with t(M) the time at which deterioration level Mis reached.
I The planning time s has no influence on the TBM policy.
Planning time
I Preventive maintenance needs to be planned s time units inadvance.
I The cost c of preventive maintenance does not have to bepaid if failure occurs between the moment that maintenance isplanned and the moment that maintenance would have beenperformed.
I Under the CBM policy, maintenance is performed at timet(M) + s, with t(M) the time at which deterioration level Mis reached.
I The planning time s has no influence on the TBM policy.
Planning time
I Preventive maintenance needs to be planned s time units inadvance.
I The cost c of preventive maintenance does not have to bepaid if failure occurs between the moment that maintenance isplanned and the moment that maintenance would have beenperformed.
I Under the CBM policy, maintenance is performed at timet(M) + s, with t(M) the time at which deterioration level Mis reached.
I The planning time s has no influence on the TBM policy.
Planning time
0
1
Cost
0 1T or M
TBM
CBM (s=0)
Planning time
0
1
Cost
0 1T or M
TBM
CBM (s=0)
CBM (s=0.1)
Planning time
0
0.6Cost
0 0.5s
Planning time
0
0.6Cost
0 0.5s
TBM
Planning time
0
0.6Cost
0 0.5s
TBM
CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Imperfect condition monitoring
I The difference between the true level of deterioration and theobserved level of deterioration is modelled by σpW (t).
I W (t) is a standard Brownian motion.
Imperfect condition monitoring
I The difference between the true level of deterioration and theobserved level of deterioration is modelled by σpW (t).
I W (t) is a standard Brownian motion.
Imperfect condition monitoring
I The difference between the true level of deterioration and theobserved level of deterioration is modelled by σpW (t).
I W (t) is a standard Brownian motion.
Imperfect condition monitoring
−0.4
0.0
0.4
Progn
osticerror
0 1t
Low σp
Imperfect condition monitoring
−0.4
0.0
0.4
Progn
osticerror
0 1t
High σp
Imperfect condition monitoring
0
0.6Cost
0 0.5σp
Imperfect condition monitoring
0
0.6Cost
0 0.5σp
TBM
Imperfect condition monitoring
0
0.6Cost
0 0.5σp
TBM
CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Uncertainty deterioration level failure
I The deterioration level at which failure occurs is normallydistributed with mean 1 and standard deviation σf .
I The normal distribution is left truncated at 0 and righttruncated at 2.
Uncertainty deterioration level failure
I The deterioration level at which failure occurs is normallydistributed with mean 1 and standard deviation σf .
I The normal distribution is left truncated at 0 and righttruncated at 2.
Uncertainty deterioration level failure
I The deterioration level at which failure occurs is normallydistributed with mean 1 and standard deviation σf .
I The normal distribution is left truncated at 0 and righttruncated at 2.
Uncertainty deterioration level failure
0
0.8
Cost
0 0.5σf
Uncertainty deterioration level failure
0
0.8
Cost
0 0.5σf
TBM
Uncertainty deterioration level failure
0
0.8
Cost
0 0.5σf
TBM
CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
Cost analysis
Failure cost = 1
Cost saving TBM
Cost due to uncertain failure levelCost due to imperfect condition monitoring
Cost due to maintenance planning
Cost due to noncontinuous deterioration informationCost due to deterioration process with jumps
Minimal cost (PM just before failure) = c
Cost saving CBM
