1. MODULE 3: CASE STUDIES Professor D.N.P. Murthy The University of Queensland Brisbane, Australia

2. CASE STUDY - 1 Source: Warranty Cost Analysis [Chapter 13] Item: Aircraft component Problem:  Item supplied without warranty  Customer requests two-year warranty  Select warranty terms  Predict costs

3. Data and Analysis  Operational data 88 failure times 65 service times  Repairable item: Repaired back to new  Special analysis required  ( “incomplete data”)  Weibull distribution fits the data (Increasing failure rate: Shape parameter > 1)

4. Data and Analysis [Cont.] Summary Failed items: MTTF = 2580 flight hours Failed items: MTTF = 2580 flight hours Service times: Mean = 2081 flight hours Service times: Mean = 2081 flight hours Estimate of overall MTTF: 3061 fl. hrs. Estimate of overall MTTF: 3061 fl. hrs. (Based on Weibull distribution.) (Based on Weibull distribution.)

5. Policies Considered 1. Nonrenewing FRW, W = 5000 fl. hrs. 2. Nonrenewing FRW, W = 2 years, calendar time 3. Rebate PRW, W = 5000 4. Rebate PRW, W = 2 years (Average usage rate: 3061 flight hours per year)

6. Results Costs: cs = $9000 cb = $17500 cr = $5400 Policy Estimated Cost 1$15,669 2 18,978 3 13,300 4 16,098

7. CASE STUDY - 2 Product: Microwave Links Major components  Crystal Receiver  Crystal Transmitter  2Mb card  2Mb PCM card

8. CASE STUDY - 2  Sold in lots (size varying from 1 - 100)  Sold with 3 year FRW policy  Failed items returned in batches  No information about –the time at which the item was put in use –the time at which the item failed

14. ESTIMATES BASED ON DATA ANALYSIS  Manufacturing Cost per Item, Cs = $7316.18  Repair Cost per Item, Cr = $143.94  Weibull scale parameter, = 0.43233  Weibull shape parameter,  = 1.57479

17. REPAIR COST PER ITEM VERSUS NUMBER OF ITEMS RETURNED

18. WARRANTY SERVICING COST FOR DIFFERENT WARRANTY PERIODS.

19. WARRANTY SERVICING COSTS FOR DIFFERING REPAIR COSTS

20. WARRANTY SERVICING COSTS FOR VARYING SCALE PARAMETER WARRANTY SERVICING COSTS FOR VARYING SCALE PARAMETER

21. WARRANTY SERVICING COSTS FOR VARYING SHAPE PARAMETER 

22. CASE STUDY: PHOTOCOPIER [Service Agent Perspective]

23. DATA FOR MODELLING  Supplied by the service agent  Single machine: Failures over a 5 year period  Part of the data is shown on the left side

24. MODELLING  One can either use number of copies (count) or time (age) as the variable in modelling at both component and system level  The count and time between failures are correlated (correlation coefficient 0.753)

25. COMPONENT FAILURES  Photocopier has several components  Frequency distribution of component failures is given on the left

26. COMPONENT FAILURES

27. SYSTEM LEVEL MODELLING

28. SERVICE CALLS  Service calls modelled as a point process through rate of occurrence of failure (ROCOF) which defines probability of service call in a short interval as a function of age (time)  ROCOF: Weibull intensity function  : Scale parameter  : Shape parameter

29. SERVICE CALLS  The shape parameter  > 1 implies that service call frequency increases (due to reliability decreasing) with time (age)  Data indicates that this is indeed the case. The next slide verifies this where TTF denotes the time between service calls.

31. MODELLING ROCOF

32.  Time used as the variable in the modelling  = 157.5 days,  = 1.55  Estimated average number of service calls per year:

33. COMPONENT LEVEL MODELLING COMPONENT: CLEANING WEB

34. MODELLING  Failed components replaced by new ones  Time to failure modelled by a failure distribution function F(t)  The form of the distribution function determined using the failure data available (black-box modelling)

35. MODELLING Several distribution function were examined for modelling at the component level. Some of them were:  2- and 3-parameter (delayed) Weibull  Mixture Weibull  Competing risk Weibull  Multiplicative Weibull  Sectional Weibull

36. COMPONENT LEVEL  A list of the different distributions considered can in found in the following book: Murthy, D.N.P., Xie, M. and Jiang, R. (2003), Weibull Models, Wiley, New York  We consider modelling based on both “counts” and “age”

37. HISTOGRAM (COUNTS)

38. HISTOGRAM (AGE)

39. WPP PLOT  WPP plot allows one to decide if one of the Weibull models is appropriate for modelling a given data set  For 2-parameter Weibull: WPP is a straight line  For more on WPP plot, see Weibull Models by Murthy et al (cited earlier)

40. NOTATION  Two sub-populations  Scale parameters  Shape parameters  Location parameter  Mixing parameter p  error: (square of the error between model and data on the WPP)

41. MODEL FIT

42. MIXTURE MODEL (COUNT)

43. SPARES NEEDED  The average number of spares needed each year can be obtained by solving the renewal integral equation. See, the book on Reliability by Blischke and Murthy (cited earlier) for details. It is as follows:

44. REFERENCE  For further details of this case study, see, Bulmer M. and Eccleston J.E. (1992), Photocopier Reliability Modeling Using Evolutionary Algorithms, Chapter 18 in Case Studies in Reliability and Maintenance, Blischke, W.R. and Murthy, D.N.P. (eds) (1992), Wiley, New York

45. Thank you