1. 1 Fundamentals of Reliability Engineering and Applications Dr. E. A. Elsayed Department of Industrial and Systems Engineering Rutgers University (elsayed@rci.rutgers.edu) Systems Engineering Department King Fahd University of Petroleum and Minerals KFUPM, Dhahran, Saudi Arabia April 20, 2009

2. 2 Reliability Engineering Outline Reliability definition Reliability estimation System reliability calculations 2

3. 3 Reliability Importance One of the most important characteristics of a product, it is a measure of its performance with time (Transatlantic and Transpacific cables) Products’ recalls are common (only after time elapses). In October 2006, the Sony Corporation recalled up to 9.6 million of its personal computer batteries Products are discontinued because of fatal accidents (Pinto, Concord) Medical devices and organs (reliability of artificial organs) 3

4. 4 Reliability Importance Business data 4 Warranty costs measured in million dollars for several large American manufacturers in 2006 and 2005. (www.warrantyweek.com)

5. Some Initial Thoughts Repairable and Non-Repairable Another measure of reliability is availability (probability that the system provides its functions when needed). 5

6. Some Initial Thoughts Warranty Will you buy additional warranty? Burn in and removal of early failures. (Lemon Law). 6

7. 7 Reliability Definitions Reliability is a time dependent characteristic.  It can only be determined after an elapsed time but can be predicted at any time.  It is the probability that a product or service will operate properly for a specified period of time (design life) under the design operating conditions without failure. 7

8. 8 Other Measures of Reliability Availability is used for repairable systems  It is the probability that the system is operational at any random time t.  It can also be specified as a proportion of time that the system is available for use in a given interval (0,T). 8

9. 9 Other Measures of Reliability Mean Time To Failure (MTTF): It is the average time that elapses until a failure occurs. It does not provide information about the distribution of the TTF, hence we need to estimate the variance of the TTF. Mean Time Between Failure (MTBF): It is the average time between successive failures. It is used for repairable systems. 9

10. 10 Mean Time to Failure: MTTF Time t R(t) 1 0 1 2 2 is better than 1? 10

11. 11 Mean Time Between Failure: MTBF 11

12. 12 Other Measures of Reliability Mean Residual Life (MRL): It is the expected remaining life, T-t, given that the product, component, or a system has survived to time t. Failure Rate (FITs failures in 10 9 hours): The failure rate in a time interval [ ] is the probability that a failure per unit time occurs in the interval given that no failure has occurred prior to the beginning of the interval. Hazard Function: It is the limit of the failure rate as the length of the interval approaches zero. 12

13. 13 Basic Calculations Suppose n 0 identical units are subjected to a test. During the interval (t, t+∆t), we observed n f (t) failed components. Let n s (t) be the surviving components at time t, then the MTTF, failure density, hazard rate, and reliability at time t are: 13

14. 14 Basic Definitions Cont’d Time Interval (Hours)Failures in the interval 0-1000 1001-2000 2001-3000 3001-4000 4001-5000 5001-6000 6001-7000 100 40 20 15 10 8 7 Total200 14

15. 15 Calculations Time Interval (Hours) Failures in the interval 0-1000 1001-2000 2001-3000 3001-4000 4001-5000 5001-6000 6001-7000 100 40 20 15 10 8 7 Total200 15

16. 16 Failure Density vs. Time 1 2 3 4 5 6 7 x 10 3 Time in hours 16 ×10 -4

17. 17 Hazard Rate vs. Time 1 2 3 4 5 6 7 × 10 3 Time in Hours 17 ×10 -4

18. 18 Calculations Time Interval (Hours) Failures in the interval 0-1000 1001-2000 2001-3000 3001-4000 4001-5000 5001-6000 6001-7000 100 40 20 15 10 8 7 Total200 18

19. 19 Reliability vs. Time 1 2 3 4 5 6 7 x 10 3 Time in hours 19

20. 20 Exponential Distribution Definition (t) Time 20

21. 21 Exponential Model Cont’d Statistical Properties 21 MTTF=200,000 hrs or 20 years Median life =138,626 hrs or 14 years

22. 22 Empirical Estimate of F(t) and R(t) When the exact failure times of units is known, we use an empirical approach to estimate the reliability metrics. The most common approach is the Rank Estimator. Order the failure time observations (failure times) in an ascending order:

23. 23 Empirical Estimate of F(t) and R(t) is obtained by several methods 1.Uniform “naive” estimator 2.Mean rank estimator 3.Median rank estimator (Bernard) 4.Median rank estimator (Blom)

24. 24 Empirical Estimate of F(t) and R(t) Assume that we use the mean rank estimator 24 Since f(t) is the derivative of F(t), then

25. 25 Empirical Estimate of F(t) and R(t) 25 Example: Recorded failure times for a sample of 9 units are observed at t=70, 150, 250, 360, 485, 650, 855, 1130, 1540. Determine F(t), R(t), f(t),,H(t)

26. 26 Calculations 26 it (i)t(i+1)F=i/10R=(10-i)/10 f=0.1/  t  =1/(  t.(10-i)) H(t) 0070010.001429 0 1701500.10.90.0012500.0013890.10536052 21502500.20.80.0010000.0012500.22314355 32503600.30.70.0009090.0012990.35667494 43604850.40.60.0008000.0013330.51082562 54856500.5 0.0006060.0012120.69314718 66508550.60.40.0004880.0012200.91629073 785511300.70.30.0003640.0012121.2039728 8113015400.80.20.0002440.0012201.60943791 91540-0.90.1 2.30258509

27. 27 Reliability Function 27

28. 28 Probability Density Function 28

29. 29 Failure Rate Constant 29

30. 30 Exponential Distribution: Another Example Given failure data: Plot the hazard rate, if constant then use the exponential distribution with f(t), R(t) and h(t) as defined before. We use a software to demonstrate these steps. 30

31. 31 Input Data 31

32. 32 Plot of the Data 32

33. 33 Exponential Fit 33

34. Exponential Analysis

35. 35 Go Beyond Constant Failure Rate -Weibull Distribution (Model) and Others 35

36. 36 The General Failure Curve Time t 1 Early Life Region 2 Constant Failure Rate Region 3 Wear-Out Region Failure Rate 0 ABC Module 36

37. 37 Related Topics (1) Time t 1 Early Life Region Failure Rate 0 Burn-in: According to MIL-STD-883C, burn-in is a test performed to screen or eliminate marginal components with inherent defects or defects resulting from manufacturing process. 37

38. 38 21 Motivation – Simple Example Suppose the life times (in hours) of several units are: 1 2 3 5 10 15 22 28 3-2=1 5-2=3 10-2=8 15-2=13 22-2=20 28-2=26 After 2 hours of burn-in

39. 39 Motivation - Use of Burn-in Improve reliability using “cull eliminator” 1 2 MTTF=5000 hours Company After burn-in Before burn-in 39

40. 40 Related Topics (2) Time t 3 Wear-Out Region Hazard Rate 0 Maintenance: An important assumption for effective maintenance is that component has an increasing failure rate. Why? 40

41. 41 Weibull Model Definition 41

42. 42 Weibull Model Cont. Statistical properties 42

43. 43 Weibull Model 43

44. 44 Weibull Analysis: Shape Parameter 44

45. 45 Weibull Analysis: Shape Parameter 45

46. 46 Weibull Analysis: Shape Parameter 46

47. 47 Normal Distribution 47

48. Weibull Model

49. Input Data

50. Plots of the Data

51. Weibull Fit

52. Test for Weibull Fit

53. Parameters for Weibull

54. Weibull Analysis

55. Example 2: Input Data

56. Example 2: Plots of the Data

57. Example 2: Weibull Fit

58. Example 2:Test for Weibull Fit

59. Example 2: Parameters for Weibull

60. Weibull Analysis

61. 61 Versatility of Weibull Model Hazard rate: Time t Constant Failure Rate Region Hazard Rate 0 Early Life Region Wear-Out Region 1  61

62. 62 Graphical Model Validation Weibull Plot is linear function of ln (time). Estimate at t i using Bernard’s Formula For n observed failure time data 62

63. 63 Example - Weibull Plot T~Weibull(1, 4000) Generate 50 data 0.632   If the straight line fits the data, Weibull distribution is a good model for the data 63