Subject: ENGG961: Systems Reliability Engineering

Assignment 2: Reliability Data Modelling

Purpose:

To complete failure data modelling by applying the knowledge learnt.

Learning objectives:

1. Understand and apply reliability concepts and terminology.

2. Interpret the principles of reliability engineering and reliability engineering processes.

3. Identify and use mathematical tools and techniques commonly used in systems reliability analysis and how they can be applied to different situations.

4. Collect and analyse reliability data (times to failure and times to repair) using empirical and parametric methods (exponential, Weibull, normal and lognormal are in syllabus); conduct Goodness- of-fit test for model selection.

Tasks

1. Generator is a critical component in power drive-train system in a wind turbine. The generators must pass the required reliability testing before they can be applied to wind turbines. Three suppliers are providing generators for wind turbine manufacturers. Extensive reliability testing has resulted in the determination of the failure distribution for each vendor’s generator, see below:

Vendor Failure Distribution

Wind Power Generator Weibull distribution with scale parameter ? = 6,200 operating days and shape parameter ? = 1.20.

GE Generator Lognormal distribution with median time tmed = 4,000 operating days and s = 0.65 (s is shape parameter, see Page 81 of the prescribed textbook)

Siemens Generator F(t) = 1 ? a2/(a+t) 2 where 0 ? t (measured in operating days, a = 6,500)

Compare each vendor’s product by finding (total 20 marks)

1. R(10 years) (1 year =330 days) (2 marks)

2. The MTTF and median time to life (2 marks)

3. The mode of each distribution model by plotting pdf (5 marks)

4. The 95-percent design life (2 marks)

5. The reliability for the next 5 years if it has survived the first 10 years (4 marks)

6. Plotting the hazard function (3 marks)

7. Whether the hazard function is DFR, CFR, or IFR (2 marks)

Your report on the questions is required to present in the following template table:

No. Item Wind Power Generator GE Generator Siemens Generator

1 R(10 years)

2 MTTF =

tmed =

3 tmod determined by plotting

4 t0.95

5 Reliability in next 5 years

6 Hazard function plotting

7 Hazard function type

2. Fifty automobiles using a new type of motor oil were monitored over a period of several months to determine when the oil is needed to be replaced due to the level of contaminants. These times were recorded in kilometres. Several units were censored from the study as a result of vehicle losses. Motor-oil failures are believed to follow a Weibull distribution.

1770 2034 2876 3200+ 2390 5751 553 1450 2319 682

2220 2207+ 654+ 1855 1393 480 1526 4006 3069 2100

1230 5050+ 2019 2622 3675 1714 810+ 2146 1819 1793

1187 2300 2859 2038 2180 2330 2110 2550 1980 890

1500 2750 2450 1110 1220+ 1250 4000 3150 850 3200

+ means censored in testing

Answer the following questions (total 24 marks)

1. Give expression of Weibull model parameter estimate using maximum likelihood approach and determine a replacement interval in kilometres based upon a 95-percent design life. Compare this to the mean time to failure (MTTF) and median time to failure. (8 marks)

2. Use Least Squares estimate to find the model parameters and give R2 value of the distribution model

obtained for the given data set. (8 marks)

3. Plot the data points and the fitted model plots using maximum likelihood estimates and the Least Square estimate on the Weibull Probability Paper (WPP). (8 marks)

3. A company manufactures various household products. Of concern to the company is its relatively low production rate on its powdered detergent production line because of the limited availability of the line itself. The line fails frequently generating considerable downtime. The line has two primary failure modes: Mode A reflects operation failures such as jams, breaks, spills, and overflows on the line and Mode B represents mechanical and electrical failures of motors, glue guns, rollers, belts, etc. Over the last 65 line start-ups, the following times in hours until the line shut down were recorded:

Time mode 0.1

A 0.3

A 0.3

A 0.4

A 0.5

A 0.6

A 0.8

A 1.1

A 1.1

A 1.2

A 1.5

A 1.8

A 1.9

A

Time mode 2.1

A 2.3

A 2.8

A 4.0

A 5.7

A 8.7

A 9.4

A 10.0

A 10.1

A 11.7

A 13.7

A 15.1

A 15.3

A

Time mode 19.1

A 19.3

A 19.6

B 21.3

A 23.2

A 24.9

A 25.2

A 32.7

A 34.4

B 47.3

A 59.9

B 64.8

A 65.5

A

Time mode 73.1

B 86.8

B 93.0

B 99.0

B 103.1

B 115.5

B 118.3

B 118.8

B 122.7

B 125.0

B 134.4

B 135.4

B 147.6

B

Time 149.1 160.4 161.5 163.4 165.4 180.5 181.1 192.6 196.2 199.6 204.1 205.2 211.4

mode B B B B B B B B B B B B B

Answer the following questions (total 40 marks)

1) From among the Weibull and lognormal distributions find the best fit for each failure mode based

upon the Least Squares R2 value. (10 marks)

2) Find a best fit model from among Weibull, gamma and lognormal distributions by comparison of AICc and BIC values calculated for each of the three distribution models selected for Mode A and

Mode B failure. (10 marks)

3) Use maximum likelihood estimates (MLE) to compute the reliability that the line will operate for one hour (a) without a Mode A failure, and (b) without either Mode A or B failure. (5 marks)

4) Referring to the results obtained in 3), can you find a better fit model for the test data including Mode A and Mode B failure? If not, explain why? (10 marks)

5) What conclusion can be reached concerning the operational failures? (5 marks)

4. Fifty units are placed on test with test terminated after 100 days (2400 hours). The following failure times in hours were observed: (total 10 marks)

19 23 145 196 210 220 240 244 318 322 405 440 455 480 528

540 550 554 588 642 645 658 726 765 780 811 911 1059 1063 1115

1298 1313 1416 1531 1570 1750 1970 2058 2110 2285

1) From among Weibull and lognormal distributions find a better fit for the test data. (5 marks)

2) Test the selected model using Chi-Square Goodness-of-Fit test. (5 marks)

5. A reliability engineer is asked to answer the following questions. Can you help answer them? (total 6 marks)

1) If two distributions have the same MTTF, can it be concluded that the reliabilities given by the two distributions are of the same? Give a short discussion. (3 marks) 2) Is reliability lower if failure rate is higher? Give a short discussion. (3 marks)

Learning Guides

In order to complete the assignment tasks, you need to read Chapter 4, Chapter 12 and Chapter 15 given that you have learnt Chapter 1 to Chapter 3; and you are required to have knowledge of Statistical Tests in Chapter 16 in the prescribed textbook. While you are reading, you need to understand or answer the following questions:

1) Be familiar with terminologies used in reliability engineering such as MTTF, median time to failure, CDF, pdf, hazard function, failure rate, mode of a distribution, confidence interval, Quantiles, Least Square estimate, maximum likelihood estimate (MLE), Goodness-of-Fit test.

2) Be familiar with distribution models such as exponential, normal, lognormal, Weibull, gamma distribution.

3) Know how to calculate reliability, failure rate, pdf.

4) Understand definitions of MTTF and MTBF, how to calculate each of them.

5) Understand why MTBF was popularly applied since World War II? Need to consider if it should be utilised?

6) Is reliability lower if failure rate is higher?

7) How to calculate reliability if a distribution model is given?

8) Understand Type-I and Type-II tests and data characteristics obtained under each test.

9) Understand Least-Squares estimate of model parameters.

10) Understand maximum likelihood estimates of model parameters.

11) Get known of Weibull plot and Normal plot, i.e., plot data on Weibull probability paper (WPP) and Normal probability paper.

12) How to select an optimal model for a test data set? To be familiar with Goodness-of-Fit test (Chapter 16).

13) Learn to know what are AIC, AICc and BIC, and how to use them.

(http://en.wikipedia.org/wiki/Akaike_information_criterion; http://en.wikipedia.org/wiki/Bayesian_information_criterion )

14) Learn how to calculate 5% and 95% confidence level given a data point.

15) Learn to know how to calculate the lower and upper bound for a 95% confidence interval in parameter estimate.

16) Learn to use and become very familiar with JMP and Excel.

Assignment 2: Reliability Data Modelling

Purpose:

To complete failure data modelling by applying the knowledge learnt.

Learning objectives:

1. Understand and apply reliability concepts and terminology.

2. Interpret the principles of reliability engineering and reliability engineering processes.

3. Identify and use mathematical tools and techniques commonly used in systems reliability analysis and how they can be applied to different situations.

4. Collect and analyse reliability data (times to failure and times to repair) using empirical and parametric methods (exponential, Weibull, normal and lognormal are in syllabus); conduct Goodness- of-fit test for model selection.

Tasks

1. Generator is a critical component in power drive-train system in a wind turbine. The generators must pass the required reliability testing before they can be applied to wind turbines. Three suppliers are providing generators for wind turbine manufacturers. Extensive reliability testing has resulted in the determination of the failure distribution for each vendor’s generator, see below:

Vendor Failure Distribution

Wind Power Generator Weibull distribution with scale parameter ? = 6,200 operating days and shape parameter ? = 1.20.

GE Generator Lognormal distribution with median time tmed = 4,000 operating days and s = 0.65 (s is shape parameter, see Page 81 of the prescribed textbook)

Siemens Generator F(t) = 1 ? a2/(a+t) 2 where 0 ? t (measured in operating days, a = 6,500)

Compare each vendor’s product by finding (total 20 marks)

1. R(10 years) (1 year =330 days) (2 marks)

2. The MTTF and median time to life (2 marks)

3. The mode of each distribution model by plotting pdf (5 marks)

4. The 95-percent design life (2 marks)

5. The reliability for the next 5 years if it has survived the first 10 years (4 marks)

6. Plotting the hazard function (3 marks)

7. Whether the hazard function is DFR, CFR, or IFR (2 marks)

Your report on the questions is required to present in the following template table:

No. Item Wind Power Generator GE Generator Siemens Generator

1 R(10 years)

2 MTTF =

tmed =

3 tmod determined by plotting

4 t0.95

5 Reliability in next 5 years

6 Hazard function plotting

7 Hazard function type

2. Fifty automobiles using a new type of motor oil were monitored over a period of several months to determine when the oil is needed to be replaced due to the level of contaminants. These times were recorded in kilometres. Several units were censored from the study as a result of vehicle losses. Motor-oil failures are believed to follow a Weibull distribution.

1770 2034 2876 3200+ 2390 5751 553 1450 2319 682

2220 2207+ 654+ 1855 1393 480 1526 4006 3069 2100

1230 5050+ 2019 2622 3675 1714 810+ 2146 1819 1793

1187 2300 2859 2038 2180 2330 2110 2550 1980 890

1500 2750 2450 1110 1220+ 1250 4000 3150 850 3200

+ means censored in testing

Answer the following questions (total 24 marks)

1. Give expression of Weibull model parameter estimate using maximum likelihood approach and determine a replacement interval in kilometres based upon a 95-percent design life. Compare this to the mean time to failure (MTTF) and median time to failure. (8 marks)

2. Use Least Squares estimate to find the model parameters and give R2 value of the distribution model

obtained for the given data set. (8 marks)

3. Plot the data points and the fitted model plots using maximum likelihood estimates and the Least Square estimate on the Weibull Probability Paper (WPP). (8 marks)

3. A company manufactures various household products. Of concern to the company is its relatively low production rate on its powdered detergent production line because of the limited availability of the line itself. The line fails frequently generating considerable downtime. The line has two primary failure modes: Mode A reflects operation failures such as jams, breaks, spills, and overflows on the line and Mode B represents mechanical and electrical failures of motors, glue guns, rollers, belts, etc. Over the last 65 line start-ups, the following times in hours until the line shut down were recorded:

Time mode 0.1

A 0.3

A 0.3

A 0.4

A 0.5

A 0.6

A 0.8

A 1.1

A 1.1

A 1.2

A 1.5

A 1.8

A 1.9

A

Time mode 2.1

A 2.3

A 2.8

A 4.0

A 5.7

A 8.7

A 9.4

A 10.0

A 10.1

A 11.7

A 13.7

A 15.1

A 15.3

A

Time mode 19.1

A 19.3

A 19.6

B 21.3

A 23.2

A 24.9

A 25.2

A 32.7

A 34.4

B 47.3

A 59.9

B 64.8

A 65.5

A

Time mode 73.1

B 86.8

B 93.0

B 99.0

B 103.1

B 115.5

B 118.3

B 118.8

B 122.7

B 125.0

B 134.4

B 135.4

B 147.6

B

Time 149.1 160.4 161.5 163.4 165.4 180.5 181.1 192.6 196.2 199.6 204.1 205.2 211.4

mode B B B B B B B B B B B B B

Answer the following questions (total 40 marks)

1) From among the Weibull and lognormal distributions find the best fit for each failure mode based

upon the Least Squares R2 value. (10 marks)

2) Find a best fit model from among Weibull, gamma and lognormal distributions by comparison of AICc and BIC values calculated for each of the three distribution models selected for Mode A and

Mode B failure. (10 marks)

3) Use maximum likelihood estimates (MLE) to compute the reliability that the line will operate for one hour (a) without a Mode A failure, and (b) without either Mode A or B failure. (5 marks)

4) Referring to the results obtained in 3), can you find a better fit model for the test data including Mode A and Mode B failure? If not, explain why? (10 marks)

5) What conclusion can be reached concerning the operational failures? (5 marks)

4. Fifty units are placed on test with test terminated after 100 days (2400 hours). The following failure times in hours were observed: (total 10 marks)

19 23 145 196 210 220 240 244 318 322 405 440 455 480 528

540 550 554 588 642 645 658 726 765 780 811 911 1059 1063 1115

1298 1313 1416 1531 1570 1750 1970 2058 2110 2285

1) From among Weibull and lognormal distributions find a better fit for the test data. (5 marks)

2) Test the selected model using Chi-Square Goodness-of-Fit test. (5 marks)

5. A reliability engineer is asked to answer the following questions. Can you help answer them? (total 6 marks)

1) If two distributions have the same MTTF, can it be concluded that the reliabilities given by the two distributions are of the same? Give a short discussion. (3 marks) 2) Is reliability lower if failure rate is higher? Give a short discussion. (3 marks)

Learning Guides

In order to complete the assignment tasks, you need to read Chapter 4, Chapter 12 and Chapter 15 given that you have learnt Chapter 1 to Chapter 3; and you are required to have knowledge of Statistical Tests in Chapter 16 in the prescribed textbook. While you are reading, you need to understand or answer the following questions:

1) Be familiar with terminologies used in reliability engineering such as MTTF, median time to failure, CDF, pdf, hazard function, failure rate, mode of a distribution, confidence interval, Quantiles, Least Square estimate, maximum likelihood estimate (MLE), Goodness-of-Fit test.

2) Be familiar with distribution models such as exponential, normal, lognormal, Weibull, gamma distribution.

3) Know how to calculate reliability, failure rate, pdf.

4) Understand definitions of MTTF and MTBF, how to calculate each of them.

5) Understand why MTBF was popularly applied since World War II? Need to consider if it should be utilised?

6) Is reliability lower if failure rate is higher?

7) How to calculate reliability if a distribution model is given?

8) Understand Type-I and Type-II tests and data characteristics obtained under each test.

9) Understand Least-Squares estimate of model parameters.

10) Understand maximum likelihood estimates of model parameters.

11) Get known of Weibull plot and Normal plot, i.e., plot data on Weibull probability paper (WPP) and Normal probability paper.

12) How to select an optimal model for a test data set? To be familiar with Goodness-of-Fit test (Chapter 16).

13) Learn to know what are AIC, AICc and BIC, and how to use them.

(http://en.wikipedia.org/wiki/Akaike_information_criterion; http://en.wikipedia.org/wiki/Bayesian_information_criterion )

14) Learn how to calculate 5% and 95% confidence level given a data point.

15) Learn to know how to calculate the lower and upper bound for a 95% confidence interval in parameter estimate.

16) Learn to use and become very familiar with JMP and Excel.

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