Summary 1CK60
Maintenance and service logistics
Information out of lectures and handouts, including figures, tables and formulas
Copyright – SE Stalpers
,Summary Elementary Maintenance Models + Lecture slides Alp Akcay
1. Introduction
Objective: keeping the capital assets running. The cost of maintenance and unavailability of a capital
asset over its lifetime is a multiple of the acquisition price. To perform maintenance, typically a
strategy and several resources are needed, the most important of which are:
- A specialist, mechanic, engineer or other trained professional
- Tools and equipment
- Spare parts
1.1 Maintenance strategies
Maintenance strategies determine when parts
or equipment need to be replaced or
maintained.
Modificative maintenance concerns
interchanging a part with a technically more
advanced part in order to make the equipment
perform better. Under breakdown corrective
maintenance a part is not replaced until it has
failed (ideal for e.g. electronics), while under
preventive maintenance the aim is to replace
parts before failure occurs (ideal for e.g. machine tools). This can be split up into usage/age based
maintenance, where the total usage of a part is measured and maintenance is conducted when a
certain threshold level has been reached. Since usage is usually scheduled, the moment that
maintenance is performed can also be scheduled. If there is large set-up costs, it can be beneficial to
interchange several parts simultaneously (block replacement), otherwise component replacement
can be performed. In condition based maintenance the actual condition of a part is gauged and
maintenance is conducted based on this. The condition of a part can be measured either periodically
during inspections or continuously through a sensor.
1.2 Uncertainty in maintenance operations
For a particular maintenance action there
can be uncertainty both with respect to its
timing and content.
Condition based maintenance is a hybrid
form, in which some but not all uncertainty
is taken away relative to breakdown
corrective maintenance.
2. Short introduction/refresher in reliability theory and renewal processes
Let the random variable T denote the time to failure of some component, T ≥ 0. Let R be the reliability
of the component at time t, is the probability the component survives beyond time t. If T is a
continuous random variable, we denote statistic equations by the following (integrals replaced by
summations when discrete):
Name Formula
Cumulative distribution function CDF 𝑡
𝐹𝑡(𝑡) = 𝑃(𝑇 ≥ 𝑡) = ∫0 𝑓𝑡(𝑥)𝑑𝑥
Probability density function PDF 𝑑
𝑓𝑡(𝑡) = 𝑑𝑡 𝐹𝑡(𝑡)
2
, Reliability 𝑡
𝑅(𝑡) = 𝑃(𝑇 ≥ 𝑡) = 1 − 𝐹𝑡(𝑡) = 1 − ∫0 𝑓𝑡(𝑡 ′ )𝑑𝑡 ′ =
∞
∫𝑡 𝑓𝑡(𝑥)𝑑𝑥
∞
Mean time to failure MTTF 𝑀𝑇𝑇𝐹 = 𝐸[𝑇] = ∫ 𝑡𝑑𝐹𝑡(𝑡) = ∫0 𝑡𝑓𝑡(𝑡)𝑑𝑡 =
∞ ∞
− ∫0 𝑅(𝑡)𝑑𝑡 = ∫0 𝑃(𝑇 > 𝑡)𝑑𝑡
Variability of time to failure 𝑉𝑎𝑟[𝑇] = 𝐸[𝑇 2 ] − (𝐸[𝑇])2
∞
𝐸[𝑇 2 ] = ∫0 𝑡 2 𝑓𝑡(𝑡)𝑑𝑡
Coefficient of variation √𝑉𝑎𝑟[𝑇]
𝑐𝑡 = 𝐸[𝑇]
2.1 Failure rates
Let us consider now a time t at which the component has not failed yet. We ask the following question:
how likely is this component to fail in the next small time interval of length ε relative to the length of
this interval? We would like to know the failure rate, hazard rate. This function can be interpreted as
the instantaneous number of failures per time unit at time t and is known as:
𝑃(𝑇≤𝑡+𝜀 |𝑇≥𝑡) 𝑓𝑡(𝑡) 𝑓𝑡(𝑡)
ℎ(𝑡) = lim 𝜀
= 𝑅(𝑡)
= 1−𝐹𝑡(𝑡)
𝜀↓0
𝑡
− ∫0 ℎ(𝑢)𝑑𝑢
𝑅(𝑡) = 𝑒
When the derivative of this function h(t) is positive, it has an increasing failure rate. Mechanical
devices typically have an IFR, they wear out over time. When it is negative, it has a decreasing failure
rate. Electronic components often have DFR, get more reliable over time. When it is equal to zero, it
has a constant failure rate. Here are random failures that are not caused by a wear out.
There is also a discrete equivalent to the failure rate. If T is a discrete random variable on the
nonnegative integers, hk is then the probability of the component failing after k cycles conditional on
its surviving at least k cycles:
𝑝𝑘 = 𝑃(𝑇 = 𝑘)
𝑅𝑘 = 𝑃(𝑇 ≥ 𝑘)
𝑃(𝑇=𝑘∩𝑇≥𝑘) 𝑝𝑘
ℎ𝑘 = = 0 ≤ ℎ𝑘 ≤ 1
𝑃(𝑇≥𝑘) 𝑅𝑘
2.2 Commonly used distributions
There are a few distributions commonly used in maintenance and reliability engineering.
2.2.1 Exponential distribution
The exponential distribution is important in operations management and reliability engineering
because it has the lack of memory property. This means that the remaining lifetime of a part has the
same distribution as the original lifetime. Another special property is that it has a constant failure rate:
h(t)=𝛌.
2.2.2 Uniform distribution
In this distribution, the probability of the function reaching one value is equally likely as all the other
values in the domain (a, b).
2.2.3 Erlang distribution
The Erlang distribution has a shape parameter k and a scale parameter 𝛌>0. The failure rate of the
Erlang distribution is constant for k=1. (In fact, it reduces to the exponential distribution when k=1.)
for k>1 the Erlang distribution has an increasing failure rate.
3
Maintenance and service logistics
Information out of lectures and handouts, including figures, tables and formulas
Copyright – SE Stalpers
,Summary Elementary Maintenance Models + Lecture slides Alp Akcay
1. Introduction
Objective: keeping the capital assets running. The cost of maintenance and unavailability of a capital
asset over its lifetime is a multiple of the acquisition price. To perform maintenance, typically a
strategy and several resources are needed, the most important of which are:
- A specialist, mechanic, engineer or other trained professional
- Tools and equipment
- Spare parts
1.1 Maintenance strategies
Maintenance strategies determine when parts
or equipment need to be replaced or
maintained.
Modificative maintenance concerns
interchanging a part with a technically more
advanced part in order to make the equipment
perform better. Under breakdown corrective
maintenance a part is not replaced until it has
failed (ideal for e.g. electronics), while under
preventive maintenance the aim is to replace
parts before failure occurs (ideal for e.g. machine tools). This can be split up into usage/age based
maintenance, where the total usage of a part is measured and maintenance is conducted when a
certain threshold level has been reached. Since usage is usually scheduled, the moment that
maintenance is performed can also be scheduled. If there is large set-up costs, it can be beneficial to
interchange several parts simultaneously (block replacement), otherwise component replacement
can be performed. In condition based maintenance the actual condition of a part is gauged and
maintenance is conducted based on this. The condition of a part can be measured either periodically
during inspections or continuously through a sensor.
1.2 Uncertainty in maintenance operations
For a particular maintenance action there
can be uncertainty both with respect to its
timing and content.
Condition based maintenance is a hybrid
form, in which some but not all uncertainty
is taken away relative to breakdown
corrective maintenance.
2. Short introduction/refresher in reliability theory and renewal processes
Let the random variable T denote the time to failure of some component, T ≥ 0. Let R be the reliability
of the component at time t, is the probability the component survives beyond time t. If T is a
continuous random variable, we denote statistic equations by the following (integrals replaced by
summations when discrete):
Name Formula
Cumulative distribution function CDF 𝑡
𝐹𝑡(𝑡) = 𝑃(𝑇 ≥ 𝑡) = ∫0 𝑓𝑡(𝑥)𝑑𝑥
Probability density function PDF 𝑑
𝑓𝑡(𝑡) = 𝑑𝑡 𝐹𝑡(𝑡)
2
, Reliability 𝑡
𝑅(𝑡) = 𝑃(𝑇 ≥ 𝑡) = 1 − 𝐹𝑡(𝑡) = 1 − ∫0 𝑓𝑡(𝑡 ′ )𝑑𝑡 ′ =
∞
∫𝑡 𝑓𝑡(𝑥)𝑑𝑥
∞
Mean time to failure MTTF 𝑀𝑇𝑇𝐹 = 𝐸[𝑇] = ∫ 𝑡𝑑𝐹𝑡(𝑡) = ∫0 𝑡𝑓𝑡(𝑡)𝑑𝑡 =
∞ ∞
− ∫0 𝑅(𝑡)𝑑𝑡 = ∫0 𝑃(𝑇 > 𝑡)𝑑𝑡
Variability of time to failure 𝑉𝑎𝑟[𝑇] = 𝐸[𝑇 2 ] − (𝐸[𝑇])2
∞
𝐸[𝑇 2 ] = ∫0 𝑡 2 𝑓𝑡(𝑡)𝑑𝑡
Coefficient of variation √𝑉𝑎𝑟[𝑇]
𝑐𝑡 = 𝐸[𝑇]
2.1 Failure rates
Let us consider now a time t at which the component has not failed yet. We ask the following question:
how likely is this component to fail in the next small time interval of length ε relative to the length of
this interval? We would like to know the failure rate, hazard rate. This function can be interpreted as
the instantaneous number of failures per time unit at time t and is known as:
𝑃(𝑇≤𝑡+𝜀 |𝑇≥𝑡) 𝑓𝑡(𝑡) 𝑓𝑡(𝑡)
ℎ(𝑡) = lim 𝜀
= 𝑅(𝑡)
= 1−𝐹𝑡(𝑡)
𝜀↓0
𝑡
− ∫0 ℎ(𝑢)𝑑𝑢
𝑅(𝑡) = 𝑒
When the derivative of this function h(t) is positive, it has an increasing failure rate. Mechanical
devices typically have an IFR, they wear out over time. When it is negative, it has a decreasing failure
rate. Electronic components often have DFR, get more reliable over time. When it is equal to zero, it
has a constant failure rate. Here are random failures that are not caused by a wear out.
There is also a discrete equivalent to the failure rate. If T is a discrete random variable on the
nonnegative integers, hk is then the probability of the component failing after k cycles conditional on
its surviving at least k cycles:
𝑝𝑘 = 𝑃(𝑇 = 𝑘)
𝑅𝑘 = 𝑃(𝑇 ≥ 𝑘)
𝑃(𝑇=𝑘∩𝑇≥𝑘) 𝑝𝑘
ℎ𝑘 = = 0 ≤ ℎ𝑘 ≤ 1
𝑃(𝑇≥𝑘) 𝑅𝑘
2.2 Commonly used distributions
There are a few distributions commonly used in maintenance and reliability engineering.
2.2.1 Exponential distribution
The exponential distribution is important in operations management and reliability engineering
because it has the lack of memory property. This means that the remaining lifetime of a part has the
same distribution as the original lifetime. Another special property is that it has a constant failure rate:
h(t)=𝛌.
2.2.2 Uniform distribution
In this distribution, the probability of the function reaching one value is equally likely as all the other
values in the domain (a, b).
2.2.3 Erlang distribution
The Erlang distribution has a shape parameter k and a scale parameter 𝛌>0. The failure rate of the
Erlang distribution is constant for k=1. (In fact, it reduces to the exponential distribution when k=1.)
for k>1 the Erlang distribution has an increasing failure rate.
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