3.4 Application Modelling of Reliability and Performance in Engineering Design 283 Fig. 3.83 Design specification FMECA—final absorption tower Total failures + suspensions = 72 Mean time to failure (MTTF) = 2.35 (days) The Kolmogorov–Smirnovgoodness-of-fittest The Kolmogorov–Smirnov(K–S) test is used to decide if a sample comes from a population with a specific distribu- tion. The K–S test is based o n the empirical distribution function (e.c.d.f.) whereby, given N ordered data points Y 1 ,Y 2 , Y N , the e.c.d.f. is defined as: EN = n(i)/N , (3.212) where n(i) is the number of points less than Y i ,andtheY i are ordered from smallest to largest value. This is a step function that increases by 1/N at the value of each ordered data point. An attractive feature of this test is that the distribution of the K–S test statistic itself does not depend on the cumulativedistribution function being tested. Another advantage is that it is an exact test; however, the goodness-of-fit test depends on an adequate sample size for the approximations to be valid. The K–S test h as several important limitations, specifically: • It applies only to continuous distributions. • It tends to be more sensitive near the centre o f the distribution than at the tails. 284 3 Reliability and Performance in Engineering Design Table 3.26 Acid plant failure data (repair time RT and time before failure TBF) Failure time RT (min) TBF (day) Failure time RT (min) TBF (day) Failure time RT (min) TBF (day) 7/28/01 0:00 38 0 9/25/01 0:00 31 5 11/9/01 0:00 360 1 7/30/01 0:00 35 2 9/27/01 0:00 79 2 11/10/01 0:00 430 1 7/31/01 0:00 148 1 9/29/01 0:00 346 2 11/20/01 0:00 336 10 8/1/01 0:00 20 1 9/30/01 0:00 80 1 11/26/01 0:00 175 6 8/5/01 0:00 27 4 10/1/01 0:00 220 1 11/28/01 0:00 118 2 8/7/01 0:00 15 2 10/4/01 0:00 63 3 12/1/01 0:00 35 3 8/11/01 0:00 5 4 10/7/01 0:00 176 3 12/2/01 0:00 556 1 8/12/01 0:00 62 1 10/8/01 0:00 45 1 12/5/01 0:00 998 3 8/13/01 0:00 580 1 10/10/01 0:00 52 2 12/6/01 0:00 124 1 8/14/01 0:00 897 1 10/10/01 0:00 39 0 12/11/01 0:00 25 5 8/15/01 0:00 895 1 10/11/01 0:00 55 1 12/12/01 0:00 120 1 8/16/01 0:00 498 1 10/12/01 0:00 36 1 12/17/01 0:00 35 5 8/17/01 0:00 308 1 10/14/01 0:00 10 2 12/26/01 0:00 10 9 8/19/01 0:00 21 2 10/18/01 0:00 1,440 4 1/2/02 0:00 42 7 8/21/01 0:00 207 2 10/19/01 0:00 590 1 1/18/02 0:00 196 16 8/22/01 0:00 346 1 10/22/01 0:00 43 3 1/29/02 0:00 22 11 8/23/01 0:00 110 1 10/24/01 0:00 107 2 2/9/02 0:00 455 11 8/25/01 0:00 26 2 10/29/01 0:00 495 5 2/10/02 0:00 435 1 8/28/01 0:00 15 3 10/30/01 0:00 392 1 2/13/02 0:00 60 3 9/4/01 0:00 41 7 10/31/01 0:00 115 1 2/13/02 0:00 30 0 9/9/01 0:00 73 5 11/1/01 0:00 63 1 2/17/02 0:00 34 4 9/12/01 0:00 134 3 11/2/01 0:00 245 1 2/24/02 0:00 71 7 9/19/01 0:00 175 7 11/4/01 0:00 40 2 3/4/02 0:00 18 8 9/20/01 0:00 273 1 11/8/01 0:00 50 4 3/9/02 0:00 23 5 • The distribution must be fully specified—that is, if location, scale, and sha pe parameters are estimated from the data, the critical region of the K–S test is no longer valid, and must be determined by Monte Carlo (MC) simulation. Goodness-of-fit results The K–S test r esult of the acid plant data given in Table 3.26 is the following: Kolmogorov-Smirnov(D) statistic = 347 Modified D statistic = 2.514 Critical value of modified D = 1.094 Confidence levels = 90% 95% 97.5% 99% Tabled values of K–S statistic = 0.113 0.122 0.132 0.141 Observed K–S statistic = 325 Mean absolute prob. error = 0.1058 Model accuracy = 89.42% (poor) The hypothesis that the data fit the two-Weibull distribution is rejected with 99% confidence. 3.4 Application Modelling of Reliability and Performance in Engineering Design 285 Fig. 3.84 Weibull distribution chart for failure data Three-parameter Weibull fit—ungrouped data (Fig. 3.84): Minimum life = 0.47 (days) Shape parameter BETA = 1.63 Scale parameter ETA = 1.74 (days) Mean life = 2.03 (days) Characteristic life = 2.21 (days) Standard deviation = 0.98 (days) Test for random failures The hypothesis that failures are random is rejected at 5% level. 3.4.3 Application Modelling Outcome The acid plant failure data do not suitably fit the Weibull distribution, with 89% model accuracy. However, the failures are not random (i.e. the failure rate is not constant), and it is essential to determine whether failures are in the early phase or in the wear-out phase of the plant’s life cycle—especially so soon af ter its installa- 286 3 Reliability and Performance in Engineering Design tion (less than 24 mon ths). The distribution must be fully specified—that is, the K–S test is no longer valid, and must be determined by Monte Carlo (MC) simulation. However, prior to simulation, a closer definition of the source of most of the failures of the critical systems (determined through the case study FMECA) is necessary. Table 3.27 shows the total downtime of the acid plant’s critical systems. The down- time failure data grouping indicates that the highest downtime is due to the hot gas feed induced draft fan, then the reverse jet scrubber, the drying tower blowers, and final absorption. Engineered Installation Downtime Table 3.27 Total downtime of the environmental plant critical systems Downtime reason description Total hours Direct hours Indirect hours Hot gas feed, hot gas fan total 1,514 1,388 126 Gas cleaning, RJS total 680 581 99 Drying to wer, SO2 blowers total 496 248 248 Gas absorption, final absorption total 195 100 95 Total 2,885 2,317 568 Monte Carlo simulation With the K–S test, the distribution of the f ailure data must be fully specified—that is, if location,scale and shapeparameters are estimated from the data, the critical regionof the K–S test is no longer valid,and must be determined by Monte Carlo (MC) simulation. MC simulation emulates the chance variations in the critical systems’ time be- fore failure (TBF) by generating random numbers that form a uniform distribution that is used to select values from the samp le TBF data, and for which various TBF values are established to develop a large population of representative sample data. The model then determines if the representative sample data come from a popula- tion with a specific distribution (i.e. exponential, Weibull or gamma distributions). The outcome of the M C simulation gives the following distribution parameters (Tables 3.28 and 3.29): Time Between Failure Distribution Table 3.28 Values of distribution models for time between failure Distribution model Parameter Parameter value 1. Exponential model Gamma 4.409E-03 2. Weibull model Gamma 1.548E+00 Theta 3.069E+02 3. Gamma model Gamma 7.181E-01 Theta 3.276E+02 3.4 Application Modelling of Reliability and Performance in Engineering Design 287 Repair Time Distribution Table 3.29 Values of distribution models for repair time Distribution model Parameter Parameter value 1. Exponential model Gamma 2.583E-01 2. Weibull model Gamma 8.324E-01 Theta 3.623E+00 3. Gamma model Gamma 4.579E-01 Theta 8.720E+00 The results of the MC simulation are depicted in Fig. 3.85. The representative sam- ple data come from a population with a gamma distribution, as illustrated. The me- dian (MTTF) of the representative data is given as approximately 2.3, which does not differ greatly from the MTTF for the three-parameter Weibull distribution for ungrouped data, which equals 2.35 (days). This Weibull distribution has a shape pa- rameter, BETA, o f 1.63, which is greater than 1 , indicating a wear-out condition in the plant’s life cycle. Fig. 3.85 Monte Carlo simulation spreadsheet results for a gamma distribution best fit of TBF data 288 3 Reliability and Performance in Engineering Design Conclusion From the case stu dy data, the assumption can be made that the critical systems’ specific high-ranking critical components are inadequately designed from a design integrity point of view, as they indicate wear-out too early in the plant’s life cycle. This is with reference to the items listed in Table 3.25, particularly the drying tower blowers’ shafts, bearings (PLF) and scroll housings (TLF), the hot gas feed induced draft fan (PFC), the reverse jet scrubber’s acid spray nozzles (TLF), the final absorption tower vessel and cooling fan guide vanes (TLF), and the IPAT SO3 cooler’s cooling fan control vanes (TLF). Figure 3.85 shows a typical Monte Carlo simulation spreadsheet of the critical systems’ time before failure and MC results for a gamma distribution best fit of TBF data. 3.5 Review Exercises and References Review Exercises 1. Discuss total cost models for design reliability with regard to risk cost estimation and project cost estimation. 2. Give a brief account of interference theory and reliability modelling. 3. Discuss system reliability modelling based on system per formance. 4. Compare functional failure and functional performance. 5. Consider the significance of functional failure and reliability. 6. Describe the benefits of a system breakdown structure (SBS). 7. Give reasons for the application and benefit of Markov modelling (continuous- time an d discrete states) in designing for reliability. 8. Discuss the binomial method with regard to series networks and parallel net- works. 9. Give a brief account of the principal steps in failure modes and effects analysis (FMEA). 10. Discuss the different types of FMEA and their associated benefits. 11. Discuss the advantages and disadvantages of FMEA. 12. Compare the significant differences between failure modes and effects analysis (FMEA) and failure modes and effects criticality analysis (FMECA) . 13. Compare the advantages and disadvantages of the RPN technique with those of the military standard technique. 14. Discuss the relevance of FMECA data sources and users. 15. Consider the significance of fault-tree analysis (FTA) in reliability, safety and risk assessment. 16. Describe the fundamental fault-tree analysis steps. 17. Explain the basic properties of the hazard rate function and give a brief descrip- tion of the main elements of the hazard rate curve. 18. Discuss component reliability and failure distributions. 3.5 Review Exercises and References 289 19. Define the application of the exponential failure distribution in reliability anal- ysis and discuss the distribution’s statistical properties. 20. Define the application of the Weibull failure distribution in reliability analysis and discuss the distribution’s statistical properties. 21. Explain the Weibull shape parameter and its use. 22. Discuss the significance of the Weibull distribution function in hazards analysis. 23. Describe the principal properties and use of the Weibull graph chart. 24. Consider the a pplication o f reliability evaluation of two-state device networks. 25. Describe the fundamental differences between two-state device series networks, parallel networks, and k-out-of-m unit networks. 26. Consider the application of reliability evaluation of three-state device networks. 27. Briefly describe three-state device parallel networksand three-state device series networks. 28. Discuss system performance measures in designing for r eliability. 29. Consider pertinent approaches to determination of the most reliable design in conceptual design. 30. Discuss conceptual design optimisation. 31. Describe the basic comparisons of conceptual designs. 32. Define labelled interval calculus (LIC) with regard to constraint labels, set la- bels, and labelled interval inferences. 33. Consider the application of labelled interval calculus in designing for reliability. 34. Give a brief description with supporting examples of the methods for: a. Determination of a data point: two sets of limit intervals. b. Determination of a data point: one upper limit interval. c. Determination of a data point: one lower limit interval. d. Analysis of the interval matrix. 35. Give reasons for the application of FMEA and FMECA in engineering design analysis. 36. Define reliability-critical items. 37. Describe algorithmic modelling in failure mod es and effects analysis with re- gard to numerical analysis, order of magnitude, qualitative simulation, and fuzzy techniques. 38. Discuss qualitative reasoning in failure modes and effects analysis. 39. Give a brief accountof the conceptof uncertaintyin engineeringdesign analysis. 40. Discuss uncertainty and incompleteness in knowledge. 41. Give a brief overview of fuzziness in engineering design analysis. 42. Describe fuzzy logic and fuzzy reasoning in engineering design. 43. Define the theory of approximate reasoning. 44. Consider uncertainty and incompleteness in design analysis. 45. Give a brief account of modelling uncertainty in FMEA and FMECA. 46. In the development o f the qualitative FMECA, describ e the concepts of lo gical expression and expression of uncertainty in FMECA. 47. Give an example of uncertainty in the extended FMECA. 48. Describe the typical results expected of a qualitative FMECA. 290 3 Reliability and Performance in Engineering Design 49. Define the proportionalhazards model with regard to non-parametric model for- mulation and parametric model formulation. 50. Define the maximum likelihood estimation parameter. 51. Briefly describe the characteristics of the one-parameter exponential distribu- tion. 52. Explain the process of estimating the parameter of the exponential distribution. 53. Consider the approach to determining the maximum likelihood estimation (MLE) parameter. 54. Compare the characteristics of the two-parameter Weibull distribution with those of the three-parameter Weibull model. 55. Give a brief account of the procedures to calculate the Weibull parameters β , μ and γ . 56. Describe the procedure to derive the mean time between failures (MTBF) μ from the Weibull distribution model. 57. Describe the procedure to obtain the standard deviation σ from the Weibull distribution model. 58. Give a brief account of the method of qualitative analysis of the Weibull distri- bution model. 59. Consider expert judgment as data. 60. Discuss uncertainty, probability theory and fuzzy logic in design ing f or reliabil- ity. 61. Describe the application of fu zzy logic in reliability evaluation. 62. Describe the application of fuzzy judgment in reliability evaluation. 63. Give a brief account of elicitation and analysis of expert judgment in designing for reliability. 64. Explain initial reliability calculation using Monte Carlo simulation. 65. Give an example of fuzzy judgment in reliability evaluation. References Abernethy RB (1992) New methods for Weibull and log normal analysis. ASME Pap no 92- WA/DE-14, ASME, New York Agarwala AS (1990) Shortcomings in MIL-STD-1629A: guidelines for criticality analysis. 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ICS Industrial Consulting Services, Gold Coast City, Queensland IEEE Std 323-1974 (1974) IEEE Standard for Qualifying Class IE Equipment for Nuclear Power Generating Stations. Institute of Electrical and Electronics Engineers, New York . fuzziness in engineering design analysis. 42. Describe fuzzy logic and fuzzy reasoning in engineering design. 43. Define the theory of approximate reasoning. 44. Consider uncertainty and incompleteness. York Agarwala AS (1990) Shortcomings in MIL-STD-1629A: guidelines for criticality analysis. In: Re- liability Maintainability Symp, pp 494–496 AMCP 70 6-1 96 (1976) Engineering design handbook: development. Test and evaluation of system reliability, availability and maintain- ability. Of ce of the Under Secretary of Defense for Research and Engineering, DoD 3235.1-H Cox DR (1972) Regression models and