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Warranty Analysis of Repairable Systems
David Trindade and Jeff Glosup, Sun MicrosystemsBill Heavlin, Google
Quality and Productivity Research Conference June 2007 Santa Fe, New Mexico
Page 2
Repairable Systems• Main focus is repairable systems
> A system is repairable if by replacing or repairing system components, the system can be returned to service.
> After the repair, the system may be > As good as new (renewal process)> Worse than new ☹> Better than new ☺
> Behavior of system after repair depends on repair actions and effects of replaced components.
> A key characteristic of a repairable system is that an individual system may fail multiple times for the same or different causes.
Page 3
Warranty and Reliability Analysis• Warranty analysis and reliability analysis of field
data are closely related areas.• The authors have used graphical Time Dependent
Reliability (TDR) methods extensively for analyzing product field reliability data.> TDR is for use with repairable systems.
• TDR methods are also very useful for the analysis of warranty data.> Can isolate specific failure modes and associated cost
data easily
Page 4
Time Dependent Reliability• Graphical technique based on studying the
cumulative occurrence of events of interest over time on individual systems> The overall behavior is represented by the average of
events across systems at fixed times – the mean cumulative function (MCF).
> Can change definition of events of interest to suit study> Can be all field failures for field reliability> Can be warranty claims for warranty data analysis> Can be costs of repairs or claims
> TDR approach is consistent with warranty analysis techniques such as suggested by Kalbfleisch, Lawless and Robinson.
Page 5
Estimating the MCF
• Consider a collection of M observed times of events tk for a set of units sold since a start time t
0
> Assume the event times are set in increasing order so t0≤ t
1 ≤ t
2 ≤ … ≤ t
M
> Note times may be calendar dates or may represent time in operation, e.g., system age
> The number of units active at any time is variable due to manufacture and sale at different times.> Let N(t
k) be the number of units active at the time represented
by tk
Page 6
Estimating the MCF - 2• The Mean Cumulative Function (MCF) is defined
successively as
where w(tk) is an incremental weight
• Incremental weight w(tk) may be
> Number of failures at time tk
> Costs of failures at time tk
MCF t k =MCF t k−1w t k N tk
MCF t 0=0
Page 7
Extending the Analysis• The TDR technique can be used to graphically
explore the modes of failure by re-defining the events of interest.> For example, we can produce a MCF for each failure
mode or failing component.> The resulting chart is a dynamic Pareto chart of failing
components. > Each of these curves should fall beneath the overall MCF curve.> These component curves should add up to produce the overall
MCF curve.> Again, cost data can be incorporated by weighting the individual
component fails by their repair cost – scaled in dollars.
Page 8
Example• This example contains roughly 1500 systems
manufactured and installed over approximately a two year period.
• The data represent all systems manufactured over this period.
• Each system consists of many components, but four of these components are responsible for a large fraction of the failures.
• Adjustments were made for missing data. (See Glosup, 2002)
Page 9
Age Versus Date Dependency
• In TDR analysis, we typically view the data in two forms.
• Age dependency looks at failure causes related to the age of the system.
• Date dependency shows causes related to events associated with a specific date. Possible causes include software updates, general maintenance actions, physical relocation of systems, sudden environmental changes, and so on.
Page 10
Active Systems PlotsActive Systems Versus Date
0
200
400
600
800
1000
1200
1400
1600
1800
11/5
/200
1
2/13
/200
2
5/24
/200
2
9/1/
2002
12/1
0/20
02
3/20
/200
3
6/28
/200
3
10/6
/200
3
Date
Num
ber o
f Sys
tem
s
Active Systems Versus Age
0
200
400
600
800
1000
1200
1400
1600
1800
0 100 200 300 400 500 600 700
AGE (Days)
Num
ber o
f Sys
tem
s
Page 11
MCF Plots by AgeTotal MCF and Failure Mode Specific MCF Plots
0
2
4
6
8
10
12
14
0 100 200 300 400 500 600 700
Age (Days)
Aver
age N
umbe
r of F
ailur
es
MCF.TotalMCF.AMCF.BMCF.CMCF.D
Total MCF and Failure Mode Specific MCF Plots
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 100 200 300 400 500 600 700
Age (Days)
Aver
age N
umbe
r of F
ailur
es
MCF.TotalMCF.AMCF.BMCF.CMCF.D
Page 12
MCF Plots by Date
05
1015
20
Date
Ave
rage
Num
ber o
f Eve
nts
per S
yste
m
04/02 11/02 05/03
Overall AverageComponent AComponent BComponent CComponent D
Page 13
MCF Plot with Reversed Time• The MCF plot can also be reportrayed with the time
axis reversed and the MCF curves shifted.> Shows most recent events to the left side of the graph.> Graph emphasizes to a greater extent the differences in
failure modes in the later ages associated with the most recent events.
> Reference time zero point used is oldest age.
Page 14
Reverse Time Plots
0 100 200 300 400 500 600
02
46
810
12
Time Since End of Observation Period (days)
Aver
age
Num
ber o
f Eve
nts p
er S
yste
m
Overall AverageComponent AComponent BComponent CComponent D
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
Time Since End of Observation Period (days)
Aver
age
Num
ber o
f Eve
nts p
er S
yste
m
Overall AverageComponent AComponent BComponent CComponent D
Page 15
NHPP Power Law Model for MCF Vs. Age
• Power law model for MCF:M(t) = αtβ
• Intensity function (theoretical recurrence rate)λ(t) = dM(t)/dt = αβtβ-1
Page 16
Parameter Estimation for Power Law Model• Although MLE methods exist for estimating the
parameters of the Power Law model (see Crow, 1974), a simple and direct approach is to estimate the parameters by minimizing the sum of squares of the residuals between the MCF and the model.
• The non-linear least squares Solver routine in the spreadsheet program EXCEL easily facilitates the estimation.
Page 17
Power Law Model FitsTotal MCF and Failure Mode Specific MCFs with Model
Fits
0
2
4
6
8
10
12
14
0 100 200 300 400 500 600 700
Age (Days)
Ave
rage
Num
ber
of F
ailu
res
MCF.TotalMCF.AMCF.BMCF.CMCF.DModel.TotalModel.AModel.BModel.CModel.D
Page 18
Parameters of Power Law Model
α β
• Overall MCF .0207 .991• Component A .00175 1.23• Component B .00887 .910• Component C .00455 .868• Component D .00148 .940
Page 19
Projections Using the Power Law Model• With the estimated parameters, it is possible to use
models to do overall or failure mode specific projections for future ages or dates.
• Such projections are a key component of warranty analysis, for example, to estimate spare parts inventories and staffing levels for field support.
Page 20
Empirical Recurrence Rates (RR)• It is possible to numerically differentiate the MCF
curve and determine recurrence rates as a function of the system age or calendar date.
• A simple approach to the recurrence rate calculations is possible using the built-in spreadsheet SLOPE function.
• A slope is found for a window (odd) number of consecutive MCF points and plotted at the median age of the points. Then, the first point is dropped and another consecutive point added, with the process repeated. (See Trindade, 1975)
Page 21
ARR (Empirical) Vs. AgeAnnualized Recurrence Rate (ARR) Vs. Age
(31 Day Window)
0
2
4
6
8
10
12
0 100 200 300 400 500 600 700
Age (Days)
AR
R (E
vent
s pe
r Ye
ar)
RR TotalRR.ARR.BRR.CRR.D
Page 22
Annualized Empirical Recurrence Rates (ARR) Vs. Date
Annulaized Recurrence Rate (ARR) Vs. Date31 Day Window
0
5
10
15
20
25
30
35
40
4511
/5/2
001
2/13
/200
2
5/24
/200
2
9/1/
2002
12/1
0/20
02
3/20
/200
3
6/28
/200
3
10/6
/200
3
Date
AR
R
RR.TotalRR.ARR.BRR.CRR.D
Page 23
Annualized Recurrence Rates and Power Law Model Fits
Total ARR and Failure Mode Specific ARRs with Model Fits
-2
0
2
4
6
8
10
12
0 100 200 300 400 500 600 700
Age (Days)
Ann
ualiz
ed R
ecur
renc
e R
ate
(Eve
nts
per
Yea
r)
RR TotalRR.ARR.BRR.CRR.DModel.TotalModel.AModel.BModel.CModel.D
Page 24
Total RR Vs. Failure Mode RR SumRecurrence Rate Total and Sum of Four Modes
0
2
4
6
8
10
12
0 100 200 300 400 500 600 700
Age (Days)
Ann
ualiz
ed R
ecur
renc
e R
ate
(Eve
nts
per Y
ear)
RR TotalSum
Page 25
Dynamic Pareto• The idea of the dynamic Pareto can be made
easier to interpret by rescaling the curves.• For the components, take the ratio of the
component MCF to the overall MCF, e.g, for component A
R Atk =MCF Atk
MCF Total tk
Page 26
Dynamic MCF Pareto by Age
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 100 200 300 400 500 600 700
System Age (Days)
Pro
port
ion
of T
otal
MC
F
Comp.AComp.BComp.CComp.D
Page 27
Dynamic MCF Pareto by Date
0
0.05
0.1
0.15
0.2
0.25
0.3
11/5
/200
1
2/13
/200
2
5/24
/200
2
9/1/
2002
12/1
0/20
02
3/20
/200
3
6/28
/200
3
10/6
/200
3
Date
Pro
port
ion
of T
otal
MC
F
ABCD
Page 28
Dynamic RR Pareto by AgeDynamic Pareto Empirical Recurrence Rate Plots
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 100 200 300 400 500 600 700
Age (Days)
Pro
port
ion
of T
otal
RR
comp.Acomp.Bcomp.Ccomp.D
Page 29
Dynamic RR Pareto by Date
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
11/5
/200
1
2/13
/200
2
5/24
/200
2
9/1/
2002
12/1
0/20
02
3/20
/200
3
6/28
/200
3
10/6
/200
3
Date
Pro
port
ion
of T
otal
RR
ABCD
Page 30
Average Cost per Unit Time• The Cost per Unit Time computed from the MCF:
Cost per Unit Time = where
n(m,tk) = number of fails of failure mode m at time t
k,
cm = (marginal) cost of repairing failure mode m.
MCF t k =MCF t k−1∑m
cm n m ,t k N tk
MCF t 0=0
MCF T /T−t 0 ,
Page 31
Cost Per Unit Time PlotsCosts Per Unit Time
0
0.5
1
1.5
2
2.5
0 200 400 600 800
Age (Days)
Cos
t/Uni
t TIm
e
MCF Cost/TimeModel Cost/TimeDOA Adder
Costs Per Unit Time
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 200 400 600 800
Age (Days)
Cos
t/Uni
t TIm
e
MCF Cost/TimeModel Cost/TimeDOA Adder
Assigned Cost Weights: A(10), B(1), C(1), D(1)
Page 32
MCF Analysis Benefits• MCFs are close to the data and so one can do
data-sensitive calculations. • Benefits shows up with:
> (a) the failure-mode-specific MCFs> (b) calendar MCFs> (c) reverse-time/recent failure plotting> (d) the dynamic Pareto plots
• MCFs work with large populations & MCFs work with small populations > The empirical approach is revealing with cost-
per-unit-time plots, too, and goodness-of-fit plots of power law models.
Page 33
Dynamic Pareto and Visualization• The dynamic Pareto concept is very useful in the
analysis of warranty claims.• Both MCFs and RRs can reveal interesting aspects
of the data using the dynamic Pareto approach. • These visualization methods are easy to grasp and
provide excellent insight into specific failure causes that vary with age or date.
• In most cases, the analysis can be easily accomplished using simple spreadsheet programs.
• Power law model complements MCF & RR analysis.
Page 34
References● Ascher and Feingold (1984), Repairable Systems Reliability: Modeling,
Inference, Misconceptions and Their Causes, M. Dekker● Crow (1974) Reliability Analysis for Complex, Repairable Systems, in
Reliability and Biometry, ed. By Proschan & Serfing, SIAM● Glosup (2002) "Estimation and Inference for Failure Rate in the Presence of
Incomplete Failure Data", JSM Proceedings● Lawless (1997), “Statistical Analysis of Product Warranty Data”, IIQP
Technical Report RR-97-05, University of Waterloo● Kalbfleisch and Lawless (1993), “Statistical Analysis of Warranty Data”, IIQP
Technical Report RR-93-03, University of Waterloo ● Kalbfleisch, Lawless and Robinson (1991), “Methods for the Analysis and
Prediction of Warranty Claims”, Technometrics, Vol 33 ● Nelson (2004), Recurrent Events Data: Analysis for Product Repairs, Disease
recurrences and Other Applications, SIAM● Tobias and Trindade (1995), Applied Reliability, Chapman & Hall ● Trindade (1975), “An APL Program to Differentiate Data by the Ruler
Method”, IBM Technical Report, TR19.0361
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