2013 ARS, North America, Minneapolis Red Room, Begins at 10:30 AM, Thursday, June 6th Repairable Systems: Data Analysis and Modeling Athanasios Gerokostopoulos
The following presentation was delivered at the: PRESENTATION SLIDES International Applied Reliability Symposium, North America June 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.
Introduction Agenda 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. Slide Number: 2 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. Slide Number: 3 Definitions Background 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 In Reliability Engineering analysis we divide Items into two categories: Repairable and Non-Repairable. 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 Slide Number: 4 The analysis differs based on the type of Item under consideration. Slide Number: 5
Background Questions of Interest in Repairable Systems Analysis Repairable system is a system that can be restored to an operating condition following a failure. How many failures over a fixed time interval? What is the probability of a failure in the next time interval? 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. Slide Number: 6 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? Slide Number: 7 Repairable Systems Analysis Common Mistake in the Analysis of Repairable Systems 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. Slide Number: 8 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. System T s =0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T i is the Cumulative Time To Failure t i is the Interarrival time = T i T i-1 Slide Number: 9 T E
Why is this a Mistake? Why is this a Mistake? When fitting a distribution we assume that the events are Statistically Independent and Identically Distributed (s.i.i.d.). When a Failure occurs in a repairable system the Remaining components have a current age. In a repairable system the events (failures) are Not Independent and in most cases Not Identically Distributed. Slide Number: 10 The Next Failure Event depends on this current age. Thus the Failure Events at the System Level are DEPENDENT. Slide Number: 11 Why is this a Mistake? 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 T E and that the Rate of Occurrence of Failures is Increasing with Time. System T s =0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 Slide Number: 12 T E 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 t 1 hours of operation, System 2 after t 2,, etc. T s =0 System 1 System 2 System 3... System 9 t 2 t 3 t 1 t 9 (suspension) Slide Number: 13
Why is this a Mistake? Will the Driver Finish the Race? 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. Changing the Starter of a Car. Slide Number: 14 Slide Number: 15 The Data Data is Collected from Three Vehicles in the Field: Assumptions Each race is 200Km. The Only components that are Changed after each race are the Brakes. +Data could be from the field or from Testing + PM: Preventive Maintenance Slide Number: 16 All Other components Stay on the car for the Next race. All three Systems Operate under Similar Conditions. Slide Number: 17
Common Mistake Weibull Analysis Common Mistake Take the Time-Between-Failures for Each System and Fit a Distribution: = 584.232 249.85 Notice that the PM data is removed. The Time Between the Last Failure and the Current Age is a Suspension. Slide Number: 18 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. Slide Number: 19 Results are NOT Valid What is the Probability that the Driver will Complete the Race? Correct Approach Remember: Distribution analysis is OK for Non-Repairable Systems and Components. WRONG RESULTS!!! Slide Number: 20 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. Slide Number: 21
Correct Application of Weibull Analysis Unanswered Questions This is the Probability that the Car will NOT fail in the First 200Km Notice that the Confidence interval is very Wide. What is the Probability that there will be No Failures after 10 Races? Slide Number: 22 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? Slide Number: 23 NHPP Model NHPP Model We need a model that will take into account the fact that when a failure occurs the system has a Current Age. The NHPP with a Power Law Failure Intensity is such a model: 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 Slide Number: 24 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). Slide Number: 25
NHPP Model Parameters NHPP Model Results The Expected Number of Failures after 10 races is 6. Slide Number: 26 Slide Number: 27 NHPP Model Results In other words, we expect 6 Failures per System. NHPP Model Results The probability that the Driver will finish the Race is 87%. 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 Slide Number: 28 Slide Number: 29
NHPP Model Results Overhaul The probability that the Driver will finish the 3 rd Race given that his car has run the first 2 races is 67%. If we decide to Overhaul the System, when is the Optimum time? Slide Number: 30 In order to find the Optimum Overhaul Time we need to consider Costs: Average Repair Cost = $192,000 Overhaul Cost = $500,000 Slide Number: 31 Overhaul Review The Optimum Overhaul Time is Calculated to be every 1560 km. This is Approximately every 8 Races per System. Slide Number: 32 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? Slide Number: 33
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. The System Rear Assembly Front Assembly In this example we have data on 6 Items, which we assume are Replaceable: Engine Transmission Front & Rear Brakes Front & Rear Suspension Slide Number: 34 Slide Number: 35 Reliability Block Diagrams Failure Distributions For each Component find the Times-To-Failure from each System and then Combine the Data. Engine Data: Slide Number: 36 Suspension = 2500-2186.9 Failure = 1470-872 Slide Number: 37
Engine Analysis Component Analysis Slide Number: 38 Slide Number: 39 Component Properties Additional Properties Enter Failure and Repair Information for each Block. Slide Number: 40 For the Brakes, enter the Preventive Maintenance Policies: Every 200Km all 4 brakes are replaced. When one brake fails, the other brakes are replaced. Slide Number: 41
Simulating the RBD System Results Slide Number: 42 Slide Number: 43 Component Results RS FCI: Percentage of System Failures Caused by a Component. RBD Analysis Conclusions Advantages Criticality and Sensitivity analysis can be performed. Identify weak components in the system Number of Spares Slide Number: 44 Perform optimization and reliability allocation Obtain Availability, Downtime, Expected Failures, etc., at the component level as well as the system level. Slide Number: 45
RBD Analysis Conclusions Disadvantages Detailed information is required, such as: o Failure Data at the LRU level o Repair Data at the LRU level NHPP Model Conclusions Advantages Quickly obtain system results No detailed information required Slide Number: 46 Disadvantages Limited results No availability, downtime, etc., estimations No sensitivity/criticality results Slide Number: 47 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). Slide Number: 48 Questions and Discussion Slide Number: 49
Additional Information Presenter Information 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 Slide Number: 50 Athanasios Gerokostopoulos Reliability Engineer ReliaSoft Corporation Athanasios.Gerokostopoulos@ReliaSoft.com Slide Number: 51