MODULE 3: CASE STUDIES

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MODULE 3 CASE STUDIES

• Professor D.N.P. Murthy
• The University of Queensland
• Brisbane, Australia

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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

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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 gt 1)

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Data and Analysis Cont.

• Summary
• Failed items MTTF 2580 flight hours
• Service times Mean 2081 flight hours
• Estimate of overall MTTF 3061 fl. hrs.
• (Based on Weibull distribution.)

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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)

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Results

• Costs cs 9000 cb 17500 cr 5400
• Policy Estimated Cost
• 1 15,669
• 2 18,978
• 3 13,300
• 4 16,098

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CASE STUDY - 2

• Product Microwave Links
• Major components
• Crystal Receiver
• Crystal Transmitter
• 2Mb card
• 2Mb PCM card

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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

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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

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REPAIR COST PER ITEM VERSUS NUMBER OF ITEMS
RETURNED
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WARRANTY SERVICING COST FOR DIFFERENT WARRANTY
PERIODS.
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WARRANTY SERVICING COSTS FOR DIFFERING REPAIR
COSTS
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WARRANTY SERVICING COSTS FOR VARYING SCALE
PARAMETER ?
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WARRANTY SERVICING COSTS FOR VARYING SHAPE
PARAMETER ?
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CASE STUDY PHOTOCOPIER

• Service Agent Perspective

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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

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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)

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COMPONENT FAILURES

• Photocopier has several components
• Frequency distribution of component failures is
  given on the left

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COMPONENT FAILURES
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SYSTEM LEVEL MODELLING
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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

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SERVICE CALLS

• The shape parameter ? gt 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.

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MODELLING ROCOF
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MODELLING ROCOF

• Time used as the variable in the modelling
• ? 157.5 days, ? 1.55
• Estimated average number of service calls per
  year

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COMPONENT LEVEL MODELLING

• COMPONENT CLEANING WEB

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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)

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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

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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

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HISTOGRAM (COUNTS)
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HISTOGRAM (AGE)
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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)

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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)

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MODEL FIT
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MIXTURE MODEL (COUNT)
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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

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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

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Thank you