repairable systems: data analysis and · pdf file2013 ars, north america, minneapolis red...

2013 ARS, North America, MinneapolisRed Room, Session 9

Repairable Systems: Data Analysis and Modeling

Athanasios Gerokostopoulos

Begins at 10:30 AM, Thursday, June 6th

PRESENTATION SLIDESThe following presentation was delivered at the:

International Applied Reliability Symposium, North AmericaJune 5 - 7, 2013: Minneapolis, Minnesota

http://www.ARSymposium.org/2013/

The International Applied Reliability Symposium (ARS) is intended to be a forum for reliability and maintainability practitioners within industry and government to discuss their success stories and lessons learned regarding the application of reliability techniques to meet real world challenges. Each year, the ARS issues an open

"Call for Presentations" at http://www.arsymposium.org/present.htm and the presentations delivered at the Symposium are selected on the basis of the presentation proposals received.

Although the ARS may edit the presentation materials as needed to make them ready to print, the content of the presentation is solely the responsibility of the author. Publication of these presentation materials in the

ARS Proceedings does not imply that the information and methods described in the presentation have been verified or endorsed by the ARS and/or its organizers.

The publication of these materials in the ARS presentation format is Copyright © 2013 by the ARS, All Rights Reserved.

Gerokostopoulos, ReliaSoft Corporation Slide Number: 2Session 9Red Room App

lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Introduction

The Purpose of this Presentation is to Explore the Different Methods Available for Analyzing Repairable Systems.

Repairable System Analysis Differs from the Analysis of Non-Repairable Systems/Items.

Mistakes are very Common in the Analysis of Repairable Systems.

In this Presentation the Most Common Mistakes will be Identified, and two Correct Approaches will be Presented.

The Purpose of this Presentation is to Explore the Different Methods Available for Analyzing Repairable Systems.

Repairable System Analysis Differs from the Analysis of Non-Repairable Systems/Items.

Mistakes are very Common in the Analysis of Repairable Systems.

In this Presentation the Most Common Mistakes will be Identified, and two Correct Approaches will be Presented.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Agenda Background.

Common Mistake in the Analysis of Repairable Systems.

Using the Non-Homogeneous Poisson Process for the Analysis of Repairable Systems.

Using Reliability Block Diagrams and Simulation for the Analysis of Repairable Systems.

Summary/Conclusions.

Background.

Common Mistake in the Analysis of Repairable Systems.

Using the Non-Homogeneous Poisson Process for the Analysis of Repairable Systems.

Using Reliability Block Diagrams and Simulation for the Analysis of Repairable Systems.

Summary/Conclusions.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Definitions LRU – Lowest Replaceable Unit

It is a unit (i.e., component), that when it fails it is replaced with a new and identical unit.

RBD – Reliability Block Diagram

NHPP – Non-Homogeneous Poisson Process

Repair – An Action that brings the System to an Operating Condition

Item – Can be a System, a Subsystem, an Assembly a Subassembly, or a Component

Overhaul – A Maintenance activity that brings the System to its New Condition

LRU – Lowest Replaceable Unit It is a unit (i.e., component), that when it fails it is replaced with a

new and identical unit.

RBD – Reliability Block Diagram

NHPP – Non-Homogeneous Poisson Process

Repair – An Action that brings the System to an Operating Condition

Item – Can be a System, a Subsystem, an Assembly a Subassembly, or a Component

Overhaul – A Maintenance activity that brings the System to its New Condition


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Background

In Reliability Engineering analysis we divide Items into two categories: Repairable and Non-Repairable.

The analysis differs based on the type of Item under consideration.

In Reliability Engineering analysis we divide Items into two categories: Repairable and Non-Repairable.

The analysis differs based on the type of Item under consideration.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Background

Repairable system is a system that can be restored to an operating condition following a failure.

This definition allows us to make a distinction between models for life lengths prior to failure (i.e., failure distributions), and the models/methods that will be used in this presentation to represent periods of operation that might extend across several failures over the life length of the system.

Repairable system is a system that can be restored to an operating condition following a failure.

This definition allows us to make a distinction between models for life lengths prior to failure (i.e., failure distributions), and the models/methods that will be used in this presentation to represent periods of operation that might extend across several failures over the life length of the system.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Questions of Interest in Repairable Systems Analysis How many failures over a fixed time interval?

What is the probability of a failure in the next time interval?

What is the availability of the system?

How many spare parts should be purchased?

What is the cost of maintaining the system?

What is the optimum overhaul time?

How many failures over a fixed time interval?

What is the probability of a failure in the next time interval?

What is the availability of the system?

How many spare parts should be purchased?

What is the cost of maintaining the system?

What is the optimum overhaul time?


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Repairable Systems Analysis

There are two methods available for analyzing Repairable Systems By collecting and analyzing the data at the

system level, and using a Stochastic Process model such as the NHPP. By collecting and analyzing the data at the

component level (Lowest Replaceable Unit).

There are advantages and disadvantages in each method.

There are two methods available for analyzing Repairable Systems By collecting and analyzing the data at the

system level, and using a Stochastic Process model such as the NHPP. By collecting and analyzing the data at the

component level (Lowest Replaceable Unit).

There are advantages and disadvantages in each method.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Common Mistake in the Analysis of Repairable Systems One of the most Common Mistakes in

analyzing repairable systems is fitting a distribution to the system’s Interarrival data.

Interarrival data is the Time Between Failures of a Repairable System.

Ti is the Cumulative Time To Failure

ti is the Interarrival time = Ti – Ti-1

One of the most Common Mistakes in analyzing repairable systems is fitting a distribution to the system’s Interarrival data.

Interarrival data is the Time Between Failures of a Repairable System.

Ti is the Cumulative Time To Failure

ti is the Interarrival time = Ti – Ti-1

SystemT1 T2 T3 T4T5T6 T7 T8

t1 t2 t3 t4 t5 t6 t7 t8

Ts=0 TE


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Why is this a Mistake?

When fitting a distribution we assume that the events are Statistically Independent and Identically Distributed (s.i.i.d.).

In a repairable system the events (failures) are Not Independent and in most cases Not Identically Distributed.

When fitting a distribution we assume that the events are Statistically Independent and Identically Distributed (s.i.i.d.).

In a repairable system the events (failures) are Not Independent and in most cases Not Identically Distributed.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13


When a Failure occurs in a repairable system the Remaining components have a current age.

The Next Failure Event depends on this current age.

Thus the Failure Events at the System Level are DEPENDENT.

When a Failure occurs in a repairable system the Remaining components have a current age.

The Next Failure Event depends on this current age.

Thus the Failure Events at the System Level are DEPENDENT.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Why is this a Mistake? What we need to model is the Rate of Occurrence of

Failures and the Number of Failures within a given time.

For example, we need a model that will tells us that we expect 8 Failures by TE and that the Rate of Occurrence of Failures is Increasing with Time.

What we need to model is the Rate of Occurrence of Failures and the Number of Failures within a given time.

For example, we need a model that will tells us that we expect 8 Failures by TE and that the Rate of Occurrence of Failures is Increasing with Time.

SystemT1 T2 T3 T4T5T6 T7 T8

t1 t2 t3 t4 t5 t6 t7 t8

Ts=0 TE


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13


If we perform a Distribution Analysis on the Time-Between-Failures, then this is equivalent to saying that we have 9 different systems, and System 1 failed after t1 hours of operation, System 2 after t2,…, etc.

If we perform a Distribution Analysis on the Time-Between-Failures, then this is equivalent to saying that we have 9 different systems, and System 1 failed after t1 hours of operation, System 2 after t2,…, etc.

System 1t1

Ts=0

System 2t2

System 3t3

.

.

.System 9

t9 (suspension)


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Why is this a Mistake? This is the same as assuming that the System is AS-

GOOD-AS-NEW after the repair, which is NOT true in Repairable Systems in general.

In most cases the System is AS-BAD-AS-OLD after the repair.

This is particularly true for Large Systems, where replacing a component does not have a great impact on the Reliability of the system.

Example: Changing the Starter of a Car.

This is the same as assuming that the System is AS-GOOD-AS-NEW after the repair, which is NOT true in Repairable Systems in general.

In most cases the System is AS-BAD-AS-OLD after the repair.

This is particularly true for Large Systems, where replacing a component does not have a great impact on the Reliability of the system.

Example: Changing the Starter of a Car.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:Will the Driver Finish the Race?


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:The Data Data is Collected from Three Vehicles in

the Field: Data is Collected from Three Vehicles in

the Field:

+Data could be from the field or from Testing

+ PM: Preventive Maintenance


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:Assumptions Each race is 200Km.

The Only components that are Changed after each race are the Brakes.

All Other components Stay on the car for the Next race.

All three Systems Operate under Similar Conditions.

Each race is 200Km.

The Only components that are Changed after each race are the Brakes.

All Other components Stay on the car for the Next race.

All three Systems Operate under Similar Conditions.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13Example:Common Mistake Take the Time-Between-Failures for Each

System and Fit a Distribution:

Notice that the PM data is removed.

The Time Between the Last Failure and the Current Age is a Suspension.

Take the Time-Between-Failures for Each System and Fit a Distribution:

Notice that the PM data is removed.

The Time Between the Last Failure and the Current Age is a Suspension.

= 584.232 – 249.85


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:Weibull Analysis – Common Mistake

This Analysis Assumes that

we have a Sample of 19 Systems, and one System

Failed at 7.2Km, the other Failed at 27.3Km,

etc.

This Analysis Assumes that

we have a Sample of 19 Systems, and one System

Failed at 7.2Km, the other Failed at 27.3Km,

etc.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:Results are NOT ValidWhat is the Probability that the Driver will

Complete the Race?What is the Probability that the Driver will

Complete the Race?

WRONG RESULTS!!!WRONG RESULTS!!!


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:Correct Approach Remember: Distribution analysis is OK for Non-Repairable

Systems and Components.

In Repairable Systems events are Dependent, and other Methods should be used.

However, it is Correct to fit a Distribution on the First-Time-to-Failure of each System.

Remember: Distribution analysis is OK for Non-Repairable

Systems and Components.

In Repairable Systems events are Dependent, and other Methods should be used.

However, it is Correct to fit a Distribution on the First-Time-to-Failure of each System.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13Example:Correct Application of Weibull Analysis

This is the Probability that the Car will NOT fail in the First200Km

Notice that the Confidence interval is very Wide.

What is the Probability that there will be No Failures after 10 Races?

This is the Probability that the Car will NOT fail in the First200Km

Notice that the Confidence interval is very Wide.

What is the Probability that there will be No Failures after 10 Races?


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:Unanswered Questions The Probability of No Failure in

the first 10 Races (2000Km) is Zero!

In other words, we know that the System will Fail At Least Once.

How Many Times will it Fail?

Should we Overhaul the System?

When?

The Probability of No Failure in the first 10 Races (2000Km) is Zero!

In other words, we know that the System will Fail At Least Once.

How Many Times will it Fail?

Should we Overhaul the System?

When?


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:NHPP Model We need a model that will take into account

the fact that when a failure occurs the system has a “Current Age.”

For example in System 1, the System has an Age of 249.86 km after the Engine is replaced.

In other words, all other components in the system have an Age of 249.86 km, and the Next Failure event is based on this fact.

The Engine is Less Likely to fail anytime soon, since it was just replaced.

We need a model that will take into account the fact that when a failure occurs the system has a “Current Age.”

For example in System 1, the System has an Age of 249.86 km after the Engine is replaced.

In other words, all other components in the system have an Age of 249.86 km, and the Next Failure event is based on this fact.

The Engine is Less Likely to fail anytime soon, since it was just replaced.

The System is As-Bad-As-Old after

each Repair


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:NHPP Model The NHPP with a Power Law Failure Intensity

is such a model:

Where: Pr[N(T)=n] is the probability that n failures will be observed

by time, T. (T) is the Failure Intensity Function (Rate of Occurrence of

Failures).

The NHPP with a Power Law Failure Intensity is such a model:

Where: Pr[N(T)=n] is the probability that n failures will be observed

by time, T. (T) is the Failure Intensity Function (Rate of Occurrence of

Failures).


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13Example:NHPP Model Parameters


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:NHPP Model Results

The Expected Number of Failures after 10 races is 6.

The Expected Number of Failures after 10 races is 6.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:NHPP Model Results In other words, we expect 6 Failures per

System.

There are two cars in each race 12 failures in the fleet If the Average Cost per failure is $192,000,

then the total Maintenance Cost for the Fleet is estimated to be:

12 Failures * $192,000/failure = $2,304,000

In other words, we expect 6 Failures per System.

There are two cars in each race 12 failures in the fleet If the Average Cost per failure is $192,000,

then the total Maintenance Cost for the Fleet is estimated to be:

12 Failures * $192,000/failure = $2,304,000


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Example:NHPP Model Results The probability that the Driver will finish

the Race is 87%. The probability that the Driver will finish

the Race is 87%.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13Example:NHPP Model Results The probability that the Driver will finish the 3rd Race

given that his car has run the first 2 races is 67%. The probability that the Driver will finish the 3rd Race

given that his car has run the first 2 races is 67%.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Overhaul

If we decide to Overhaul the System, when is the Optimum time?

In order to find the Optimum Overhaul Time we need to consider Costs: Average Repair Cost = $192,000 Overhaul Cost = $500,000

If we decide to Overhaul the System, when is the Optimum time?

In order to find the Optimum Overhaul Time we need to consider Costs: Average Repair Cost = $192,000 Overhaul Cost = $500,000


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Overhaul

The Optimum Overhaul Time is Calculated to be every 1560 km.

This is Approximately every 8 Races per System.

The Optimum Overhaul Time is Calculated to be every 1560 km.

This is Approximately every 8 Races per System.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Review

The NHPP model allowed us to: Estimate the Reliability of the System in the next time

interval. Estimate the Number of Failures over a fixed time

interval. Estimate the Cost of Maintaining the System. Estimate the Optimum Overhaul time.

Unanswered Questions: How Many Spare Parts should we purchase? Which components cause most of the failures? Can we get a more accurate cost estimate?

The NHPP model allowed us to: Estimate the Reliability of the System in the next time

interval. Estimate the Number of Failures over a fixed time

interval. Estimate the Cost of Maintaining the System. Estimate the Optimum Overhaul time.

Unanswered Questions: How Many Spare Parts should we purchase? Which components cause most of the failures? Can we get a more accurate cost estimate?


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

RBD Approach

This approach is based on creating a Reliability Block Diagram of the System Components.

The Failure Distribution of each Component in the System needs to be estimated first.

In this example we have data on 6 Items, which we assume are Replaceable: Engine Transmission Front & Rear Brakes Front & Rear Suspension

This approach is based on creating a Reliability Block Diagram of the System Components.

The Failure Distribution of each Component in the System needs to be estimated first.

In this example we have data on 6 Items, which we assume are Replaceable: Engine Transmission Front & Rear Brakes Front & Rear Suspension


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

The SystemRear AssemblyRear Assembly Front AssemblyFront Assembly


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Reliability Block Diagrams


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Failure Distributions

For each Component find the Times-To-Failure from each System and then Combine the Data.

Engine Data:

For each Component find the Times-To-Failure from each System and then Combine the Data.

Engine Data:

Suspension = 2500 - 2186.9

Failure = 1470 - 872


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Engine Analysis


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Component Analysis


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Component Properties

Enter Failure and Repair Information for each Block.

Enter Failure and Repair Information for each Block.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Additional Properties

For the Brakes, enter the Preventive Maintenance Policies: Every 200Km all

4 brakes are replaced. When one brake fails,

the other brakes are replaced.

For the Brakes, enter the Preventive Maintenance Policies: Every 200Km all

4 brakes are replaced. When one brake fails,

the other brakes are replaced.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Simulating the RBD


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

System Results


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Component Results RS FCI: Percentage of System Failures Caused by a Component.

Number of SparesNumber of Spares


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

RBD AnalysisConclusions Advantages Criticality and Sensitivity analysis can be

performed. Identify weak components in the system Perform optimization and reliability allocation Obtain Availability, Downtime, Expected

Failures, etc., at the component level as well as the system level.


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13RBD AnalysisConclusions Disadvantages Detailed information is required, such as:

o Failure Data at the LRU levelo Repair Data at the LRU level

Disadvantages Detailed information is required, such as:

o Failure Data at the LRU levelo Repair Data at the LRU level


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

NHPP ModelConclusions Advantages Quickly obtain system results No detailed information required

Disadvantages Limited results No availability, downtime, etc., estimations No sensitivity/criticality results

Advantages Quickly obtain system results No detailed information required

Disadvantages Limited results No availability, downtime, etc., estimations No sensitivity/criticality results


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Summary

Two Different Methods of Analyzing Repairable System Data were presented: NHPP RBD

The analysis method chosen will depend on the available data: For a small amount of data with little detail, NHPP

can easily be applied. For detailed data with enough information at the

component level both methods can be used, but the RBD approach is preferred (more detailed analysis).

Two Different Methods of Analyzing Repairable System Data were presented: NHPP RBD

The analysis method chosen will depend on the available data: For a small amount of data with little detail, NHPP

can easily be applied. For detailed data with enough information at the

component level both methods can be used, but the RBD approach is preferred (more detailed analysis).


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Questions and Discussion


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

ReliaSoft’s Reliability Growth and Repairable System Analysis Reference.

ReliaSoft’s Life Data Analysis Reference. ReliaSoft’s System Reliability Reference. www.reliawiki.org www.weibull.com Software:Weibull++ 8 BlockSim 8 RGA 7

Additional Information


lied

Rel

iabi

lity

Sym

posi

um, N

orth

Am

eric

a 20

13

Athanasios Gerokostopoulos Reliability Engineer ReliaSoft Corporation [email protected]

Presenter Information

repairable systems: data analysis and · pdf file2013 ars, north america, minneapolis red...

Documents