What a nice fault tree!
Prompt
Yamaha, a world-famous automotive and electronics conglomerate, is investigating the reliability of their motorcycle’s engines! (Hypothetically.) In this assignment, you will use examine physical acceleration models and fault trees, answering 6 questions using the R functions we have learned to date.
Please work on it on your own.
Format: Please paste your
Rcode into a.docxfile, so that our graders can easily see it. All lines of code must be clearly explained in comments. Please format your submission according to this template.Honor Code: Be sure to add the Honor Code to the top of your assignment!
Other Students: Students may talk to each other about how to approach the problems, but we expect each student to do their own work.
Grade: Assignment graded out of
100points total.Questions: If you have questions, post your question on EdDiscussion pronto!
(All values below are hypothetical, unless otherwise indicated, for educational purposes, but are meant to be plausible examples.)
Can I Handwrite?: Unless otherwise stated, all questions must be done in
R. Questions that can be handwritten are labeled [Handwritten OK].
Q1: Physical Acceleration Models (20 points)
For their standard motorcycle, the Yamaha 1000, Yamaha engineers tested 3 cells of motorcycle engine parts at temperatures of 120, 130, and 140. Each cell had 50 units. Readouts were taken at 24, 48, 96, 150, 500, and 1000 hours. The test ended at the last readout time. See the table below!
| Readout Time<br>(hours) | 120° Cell New Fails | 130° Cell New Fails | 140° Cell New Fails |
|---|---|---|---|
| 24 | 0 | 1 | 2 |
| 48 | 0 | 1 | 1 |
| 96 | 2 | 3 | 3 |
| 150 | 0 | 1 | 1 |
| 500 | 3 | 3 | 10 |
| 1000 | 3 | 8 | 10 |
| Survivors | 42 | 33 | 23 |
Assume their lifetime follows a Weibull distribution, with parameters and , and that the acceleration between stresses can be modeled by the Arrhenius model, since only the thermal stresses are significant.
- (10 points) Estimate the characteristic life for each of our 3 tests (120, 130, and 140 degrees), and estimate their shared, constant shape parameter from the Weibull distribution!
Show the steps of your calculation and estimation in
R. Please use maximum likelihood estimation! Show the steps of your calculation and estimation inR.(You need to setup the problem and estimate parameters “manually”, eg. vector by vector, step by step. Use 1000 as your starting values for , , and when optimizing.)
(5 points) Using your maximum likelihood estimates for , , , and , estimate the Arrhenius model constant . To do so, please use a linear regression model in
Rto estimate . Show your steps!(5 points) Predict
cfor a use temperature of60 degrees C, and then calculate the probability of failure at600 hoursat that temperature. To do so, please use thecandmstatistics andlmmodel you estimated in parts a & b.
- Note: If you could not solve parts a & b, you may use the following approximate estimates: , , , and . Be sure to mention that you’re using the approximation in your comments.
Q2. Boolean Equations and Minimal Cutsets (15 points)
Yamaha is prototyping a new motorcycle engine, which they plan to use for their Yamaha 2000 bike. They have hired you to analyze the likelihood of catastrophic failure in this engine, defined as an engine explosion! You sketched up (1) an initial fault tree visual (shown above) and recorded (2) the data as a set of nodes and edges. Using these resources, please analyze this fault tree, analyzing the following questions below.
(5 points) Find the boolean equation for this fault tree, using the MOCUS algorithm, by hand.[Handwritten OK] Calculate the probability of the top event
Tif events A, B, C, D, E, F, and G are each0.10.(5 points) Find the boolean equation for this fault tree in
R, using thetidyfaultpackage, and then calculate the probability of the top eventTif each event A, B, C, D, E, F, and G have a probability of0.10. Do your answer to part b match your answers to part a? (They should!)(5 points) Yamaha is excited by the boolean equation you produced above, but they want to know which events are most critical to prevent the failure of interest (the top event) in this fault tree, for the Yamaha 2000 bike. Find all the minimal cutsets in the fault tree above, using the
tidyfaultpackage inR. For each, report the total percentage of cutsets that each minimal cutset covers. Show all your steps!
Q3. Visualizing Fault Trees in ggplot2 (5 points)
- (5 points) Import and visualize the fault tree in
ggplot2using thetidyfaultpackage, including informative shapes, colorings, themes, and chart labels. Please label each node by itseventname, eg."G1","A", etc.
- Bonus: +2 points if you use
geom_polygon()to plot theANDandORgates.)
Q4. Estimating Probabilities with Fault Trees (15 points)
Yamaha is so impressed with your work that they hire you again! They have collected data for each of the failure events in your fault tree about the Yamaha 2000, and want you to calculate several probabilities about the main failure of interest (engine explosion!).
Unless otherwise stated, assume that all life distributions of these events (below) are exponentially distributed.
Riders expose their engine to an open flame extremely rarely, just once per year (i.e.
1failure per8760hours).Atmospheric conditions produce particularly dry air below ESD thresholds for an average of
25days every year (i.e.600 / 8760hours). The frequency of dry air conditions is normally distributed, with a standard deviation of5days per year (i.e.120 / 8760hours).Weather changes of 30 degrees or more affect weather for a full 30 days out of every year (
720 / 8760hours), on average. This rate is normally distributed with a standard deviation of 10 days per year (240 / 8760hours).Debris gets stuck in valves occasionally, at about 1 time per year (
1 / 8760hours).Sensor failure occur rarely, at about 2 times per 10 years (
2 / 87600hours).The engine is exposed to air almost constantly, about
0.98times per hour.The fuel pump almost never fails, at a rate of
0.000001times per hour.
(7.5 points) Construct a
functioninRof the boolean equation you estimated in Q2, to calculate cumulative probabilities of overall failure at any time given the failure rates listed above. Estimate the cumulative probability of failure (engine explosion) in the the first 100, 1000, and 5000 hours of use.(7.5 points) Using your
functionfrom part a, calculate the cumulative probability of failure every 100 hours from0to5000hours for 1 motorcycle. Compare that against the probability of failure for an entire fleet of5,10, and100motorcycles. Visualize your resulting 4 curves in a single plot withggplot. (Suggestion: usealpha = 0.2~0.5for visibility.)
Q5. Simulating Probabilities with Fault Trees (20 points)
Yamaha plans to put 5000 more of their original Yamaha 1000 bikes on the market, but before they do, they want to know how often they should expect the top event in their fault tree (engine explosion!) Fortunately, their engineers have already developed the fault tree for the Yamaha 1000, incorporating the events A(Debris), B (Sensor failure), C (fuel pump failure), and D (electric system failure).
A. Debris gets stuck in valves occasionally, at about 12 times per year (12 / 8760 hours). B. Sensor failureoccur occasionally, at about 5 times per 10 years (5 / 87600 hours). C. The fuel pump fails rarely, at a rate of 0.00005 times per hour. D. The electrical system fails occasionally, with a shape parameter of 2 and a characteristic life of 30,000 hours.
The project manager knows that any probabilistic estimate might be subject to some error, so they want you to simulate confidence intervals for the following projections, using two different methods: (1) random binomial simulations and (2) random normal distribution simulations.
- (10 points) Simulate quantities of interest about the top event’s occurrence, using random binomial simulations of your failure events. Please report the (1) total times the top event occurred, (2) the mean occurrence (probability) of the top event, (3) the standard deviation, and (4) the total sample size.
- Hint 1: Calculate the cumulative probability of each failure event (A, B, C, etc.) by
1000hours. - Hint 2: Use those probabilities to randomly simulate event failure for each of 5000 bikes, according to thatprobability. Use a binomial distribution; see Workshop 9.).
- Hint 3: Use the provided fault tree equation to compute whether (1/0) the top event occurred in each of your 5000 bikes’ random simulations.
- (10 points) Using the statistics you generated above, calculate the following statistics, having assumed that the mean occurrence / probability of the top event is normally distributed.
- Calculate the standard error for the probability of the top event.
- Estimate a lower and upper 95% confidence interval for the probability of the top event.
- Estimate a 95% lower and upper confidence interval on the total failures expected among 5000 of these simulated motorcycle engines.
Q6. Estimating Probability of Failure Events (10 points)
Yamaha’s project manager pulls you aside and hires you for a side gig! They are developing an alternative engine for their bike, the Yamaha Z (!!!), to try to reduce the probability of valve failure due to debris (event A). Unfortunately, 1 of their test engines did explode in the lab, such that they no longer have the new valve failure rate collected for the Yamaha Z! They need you to use the fault tree from the Yamaha 1000 model(See Q5) and the new data above to estimate some quantities of interest about the new valve in the Yamaha Z engine.
- Fortunately, this catastrophic failure was pretty hard to miss, so they were still able to calculate the cumulative probability of failure for the top event in our fault tree (engine explosion). After
1000hours, the cumulative probability of failure (engine explosion) was0.008(0.8%). - Assume the lifespan distribution of the top event is exponentially distributed.
- Assume that all other parts of the Yamaha Z engines had the same parameters/failure rates as in the Yamaha 1000.
(10 points) Estimate the cumulative probability of failure for the new valve at 1000 hours. (Hint: You’ll need to estimate the failure rate for the top event.)
Q7. Conditional Reliability (5 points)
Yamaha has 20 old valves from the Yamaha Z tests left over after testing. These valves had an exponential failure function f(t, lambda = 12 / 8760), as you may recall. Estimate the conditional reliability that all 20old valves will remain reliable for the next 100, 150, and 200 hours, given that they survived 1000 hours of testing without failure. (Note: Assume no acceleration factor.)
Q8. Conditional Probability of Failure (10 points)
Yamaha’s team calls you up again. They generated some new failure rates for the Yamaha Z engines as a whole, but these rates were generated under high stress conditions in a lab. They need your help to estimate (a) an acceleration factor and (b) a conditional probability of failure given burn-in. Fortunately, you know the following:
- The engine’s lifespan has an failure function under high stress conditions.
- The engine’s lifespan is assumed to follow an exponential distribution. (Hint: use this assumption to deduce the mean time to failure.)
- Under normal use conditions, it has an cumulative probability of failure of
10%after4000hours. - Acceleration Factors represent the ratio of the an estimate of average/median time to failure under normal use conditions compared to under lab stress conditions.
- For quality control, the manufacturer plans to run engines for the first 100 hours of use to burn-in their lifespans.
(5 points) Estimate the Acceleration Factor for this engine’s lifespan distribution.
(5 points) What is the conditional probability of failure for 1 engine that survived that burn-in period of
100hours, over the next1000,3000, and5000hours?
