module 3: case studies professor d.n.p. murthy the university of queensland brisbane, australia

45
MODULE 3: CASE STUDIES MODULE 3: CASE STUDIES Professor D.N.P. Murthy Professor D.N.P. Murthy The University of The University of Queensland Queensland Brisbane, Australia Brisbane, Australia

Upload: roberta-sanders

Post on 22-Dec-2015

227 views

Category:

Documents


2 download

TRANSCRIPT

MODULE 3: CASE STUDIESMODULE 3: CASE STUDIES

Professor D.N.P. MurthyProfessor D.N.P. Murthy

The University of QueenslandThe University of Queensland

Brisbane, AustraliaBrisbane, Australia

CASE STUDY - 1CASE STUDY - 1

Source: Source: Warranty Cost Analysis Warranty Cost Analysis

[Chapter 13][Chapter 13]

Item: Item: Aircraft componentAircraft component

Problem:Problem: Item supplied without warrantyItem supplied without warranty Customer requests two-year warrantyCustomer requests two-year warranty Select warranty termsSelect warranty terms Predict costsPredict costs

Data and AnalysisData and Analysis

Operational dataOperational data88 failure times 88 failure times

65 service times65 service times

Repairable item: Repaired back to newRepairable item: Repaired back to new Special analysis required Special analysis required

((“incomplete data”)“incomplete data”)

Weibull distribution fits the data Weibull distribution fits the data (Increasing failure rate: Shape parameter > 1)(Increasing failure rate: Shape parameter > 1)

Data and Analysis [Cont.]Data and Analysis [Cont.]

SummarySummary

Failed items: MTTF = 2580 flight hoursFailed items: MTTF = 2580 flight hours

Service times: Mean = 2081 flight hoursService 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.)

Policies ConsideredPolicies Considered

1. Nonrenewing FRW, W = 5000 fl. hrs.1. Nonrenewing FRW, W = 5000 fl. hrs.

2. Nonrenewing FRW, W = 2 years, 2. Nonrenewing FRW, W = 2 years, calendar time calendar time

3. Rebate PRW, W = 50003. Rebate PRW, W = 5000

4. Rebate PRW, W = 2 years4. Rebate PRW, W = 2 years

(Average usage rate: 3061 flight hours per (Average usage rate: 3061 flight hours per year)year)

ResultsResults

Costs: cs = $9000 cb = $17500 cr = $5400 Costs: cs = $9000 cb = $17500 cr = $5400

Policy Estimated CostPolicy Estimated Cost

11 $15,669$15,669

22 18,978 18,978

33 13,300 13,300

44 16,098 16,098

CASE STUDY - 2CASE STUDY - 2

Product: Microwave LinksProduct: Microwave LinksMajor componentsMajor components

Crystal ReceiverCrystal Receiver Crystal Transmitter Crystal Transmitter 2Mb card 2Mb card 2Mb PCM card 2Mb PCM card

CASE STUDY - 2CASE STUDY - 2 Sold in lots (size varying from 1 - 100)Sold in lots (size varying from 1 - 100) Sold with 3 year FRW policySold with 3 year FRW policy Failed items returned in batchesFailed items returned in batches No information about No information about

– the time at which the item was put in usethe time at which the item was put in use– the time at which the item failedthe time at which the item failed

Manufacturing Cost Data

Customer

Order

Number

Number

of Systems

in Batch

Labour

Cost

($)

Material

Cost

($)

Overhead

Recovered

($)

Direct

Expenses

($)

Total

Cost

($)

Average

Cost per

System ($)

034-1605 2 1329 11474 929 0 13732 6866.00

034-1616 24 18155 131811 12355 1035 163356 6806.50

034-1758 2 2665 11940 1865 0 16470 8235.00

034-1809 38 30600 178243 21415 0 230258 6059.42

034-1899 2 1800 10250 1260 0 13310 6655.00

056-1976 4 2812 29448 1966 0 34226 8556.50

068-1838 4 3966 27528 2776 0 34270 8567.50

072-1955 2 1238 18782 867 0 20887 10443.50

099-1429 100 72900 529900 51100 7100 661000 6610.00

106-1682 16 14180 132379 9928 1386 157873 9867.06

Repair Cost Data

Repair

Job

Number

Number

of

Failed Parts

Labour

Cost

($)

Material

Cost

($)

Overhead

Recovered

($)

Direct

Expenses

($)

Total

Cost

($)

Average

Cost per

System ($)

034-W348 4 230 435 161 0 826 206.50

034-W355 1 83 0 62 0 145 145.00

034-W364 2 108 2 75 0 185 92.50

034-W378 1 54 0 37 0 91 91.00

040-R215 1 63 0 44 0 107 107.00

040-W241 1 63 0 44 0 107 107.00

040-W252 3 111 2 78 0 191 63.67

Failure Records

Repair

Job #

Failure

Date

Customer

Job #

Despatch

Date

Age

(Days)

Serial No

040-W285 9/16/94 040-1145 7/14/93 429 87

196-R344 5/3/95 196-1306 7/30/93 642 1376

040-W376 3/8/95 040-1168 9/20/93 534 1396

040-W376 3/8/95 040-1168 9/20/93 534 1397

040-W376 3/8/95 040-1168 9/20/93 534 1398

040-W376 3/8/95 040-1168 9/20/93 534 1300

040-W252 7/11/94 040-1359 10/29/93 255 1559

040-W252 7/11/94 040-1360 10/29/93 255 1498

Survival Records

Customer

Job

Number

Despatch

Date

Number

of

Systems

Date of

Data

Collection

Age (Days) at

Date of Data

Collection

196-1306 7/30/93 2 6/9/95 679

196-1322 9/11/93 2 6/9/95 636

355-1336 9/24/93 2 6/9/95 623

040-1358 10/29/93 2 6/9/95 588

040-1359 10/29/93 2 6/9/95 588

040-1360 10/29/93 2 6/9/95 588

040-1361 10/29/93 2 6/9/95 588

034-1393 11/12/93 10 6/9/95 574

Data used in MATLAB program for analysis

Batch

Number

j

Batch

Size

Nj

Age of

Batch

Tj

Number of

Failures in Batch

nj

Failure

Times

xji

1 2 679 1 642

2 2 636 0

3 2 623 0

4 2 588 0

5 2 588 2 255

350

6 2 588 3 255

354

ESTIMATES BASED ON ESTIMATES BASED ON DATA ANALYSISDATA ANALYSIS

Manufacturing Cost per Item, Cs = Manufacturing Cost per Item, Cs = $7316.18$7316.18

Repair Cost per Item, Cr = $143.94Repair Cost per Item, Cr = $143.94 Weibull scale parameter, Weibull scale parameter, = 0.43233 = 0.43233 Weibull shape parameter, Weibull shape parameter, = 1.57479 = 1.57479

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 1 2 3 4 5 6 7

Time (Years)

Fai

lure

Rat

e r(

t)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5 6 7

Time (Years)

f(t)

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0 5 10 15 20 25

Number of Items

Rep

air

Co

st p

er I

tem

($)

REPAIR COST PER ITEM VERSUS REPAIR COST PER ITEM VERSUS NUMBER OF ITEMS RETURNEDNUMBER OF ITEMS RETURNED

WARRANTY SERVICING COST FOR WARRANTY SERVICING COST FOR DIFFERENT WARRANTY PERIODSDIFFERENT WARRANTY PERIODS..

Warranty

Period

(Years)

W

Expected

Number of Failures

in Warranty Period

M(W)

Warranty

Servicing

Cost ($)

cr*M(W)

Warranty Servicing Cost

as a percentage of

the manufacture cost

1 0.27 38.43 0.53

2 0.80 114.48 1.56

3 1.51 216.78 2.96

4 2.37 341.02 4.66

5 3.37 484.61 6.62

6 4.49 645.78 8.83

WARRANTY SERVICING COSTS FOR WARRANTY SERVICING COSTS FOR DIFFERING REPAIR COSTSDIFFERING REPAIR COSTS

Warranty

Period

(Years)

Repair

Cost

$143.94

Repair Cost

plus 1 Standard

Deviation

$238.87

Repair Cost

plus 2 Standard

Deviations

$333.81

Repair Cost

plus 3 Standard

Deviations

$428.75

1 38.43 63.78 89.12 114.47

2 114.48 189.99 265.49 341.00

3 216.78 359.77 502.76 645.75

4 341.02 565.95 790.89 1015.82

5 484.61 804.25 1123.90 1443.55

6 645.78 1071.73 1497.69 1923.64

WARRANTY SERVICING COSTS FOR WARRANTY SERVICING COSTS FOR VARYING SCALE PARAMETERVARYING SCALE PARAMETER

Warranty

Period

(Years)

99.9%

Confidence

Interval for

Lower Limit

0.42295

99%

Confidence

Interval for

Lower Limit

0.42499

95%

Confidence

Interval for

Lower Limit

0.42674

Point

Estimate

for

0.43233

95%

Confidence

Interval for

Upper Limit

0.43793

99%

Confidence

Interval for

Upper Limit

0.43968

99.9%

Confidence

Interval for

Upper Limit

0.44172

1 37.12 37.41 37.65 38.43 39.21 39.46 39.75

2 110.59 111.43 112.15 114.48 116.82 117.56 118.42

3 209.42 211.01 212.38 216.78 221.21 222.61 224.24

4 329.43 331.93 334.10 341.02 347.99 350.19 352.75

5 468.14 471.70 474.77 484.61 494.52 497.64 501.28

6 623.84 628.58 632.68 645.78 658.98 663.15 668.00

WARRANTY SERVICING COSTS FOR WARRANTY SERVICING COSTS FOR VARYING SHAPE PARAMETER VARYING SHAPE PARAMETER

Warranty

Period

(Years)

99.9%

Confidence

Interval for

Lower Limit

1.56540

99%

Confidence

Interval for

Lower Limit

1.56744

95%

Confidence

Interval for

Lower Limit

1.56920

Point

Estimate

for

1.57479

95%

Confidence

Interval for

Upper Limit

1.58038

99%

Confidence

Interval for

Upper Limit

1.58214

99.9%

Confidence

Interval for

Upper Limit

1.58418

1 38.73 38.67 38.61 38.43 38.25 38.19 38.13

2 114.63 114.60 114.57 114.48 114.38 114.35 114.32

3 216.25 216.37 216.47 216.78 217.10 217.20 217.31

4 339.27 339.65 339.98 341.02 342.06 342.39 342.78

5 481.11 481.87 482.52 484.61 486.70 487.36 488.13

6 640.03 641.27 642.35 645.78 649.23 650.32 651.58

CASE STUDY: CASE STUDY: PHOTOCOPIERPHOTOCOPIER

[Service Agent Perspective][Service Agent Perspective]

DATA FOR MODELLINGDATA FOR MODELLING

Supplied by the Supplied by the service agent service agent

Single machine: Single machine: Failures over a 5 year Failures over a 5 year periodperiod

Part of the data is Part of the data is shown on the left shown on the left sideside

Count Day Component 60152 29 Cleaning Web 60152 29 Toner Filter 60152 29 Feed Rollers

132079 128 Cleaning Web 132079 128 Drum Cleaning Blade 132079 128 Toner Guide 220832 227 Toner Filter 220832 227 Cleaning Blade 220832 227 Dust Filter 220832 227 Drum Claws 252491 276 Drum Cleaning Blade 252491 276 Cleaning Blade 252491 276 Drum 252491 276 Toner Guide 365075 397 Cleaning Web 365075 397 Toner Filter

MODELLINGMODELLING

One can either use number of copies One can either use number of copies (count) or time (age) as the variable in (count) or time (age) as the variable in modelling at both component and modelling at both component and system levelsystem level

The count and time between failures are The count and time between failures are correlated (correlation coefficient 0.753)correlated (correlation coefficient 0.753)

COMPONENT FAILURESCOMPONENT FAILURES

Photocopier has Photocopier has several componentsseveral components

Frequency Frequency distribution of distribution of component failures component failures is given on the left is given on the left

Failed Component Frequency Cleaning web 15 Toner filter 6 Feed rollers 11 Drum blade 2 Toner guide 7 Cleaning blade 7 Dust filter 6 Drum claws 5 Crum 6 Ozone filter 8 Upper fuser roller 5 Upper roller claws 5 TS block front 2 Charging wire 6 Lower roller 2 Optics PS felt 3 Drive gear D 2

COMPONENT FAILURESCOMPONENT FAILURES

SYSTEMSYSTEM LEVEL MODELLINGLEVEL MODELLING

SERVICE CALLSSERVICE CALLS

Service calls modelled as a point Service calls modelled as a point process through rate of occurrence process through rate of occurrence of failure (ROCOF) which defines of failure (ROCOF) which defines probability of service call in a short probability of service call in a short interval as a function of age (time)interval as a function of age (time)

ROCOF: Weibull intensity function ROCOF: Weibull intensity function : Scale parameter : Scale parameter : Shape parameter: Shape parameter

( 1)( ) ( / )t t

SERVICE CALLSSERVICE CALLS

The shape parameter The shape parameter > 1 implies that > 1 implies that service call frequency increases (due to service call frequency increases (due to reliability decreasing) with time (age) reliability decreasing) with time (age)

Data indicates that this is indeed the Data indicates that this is indeed the case. The next slide verifies this where case. The next slide verifies this where TTF denotes the time between service TTF denotes the time between service calls. calls.

Actual

Fits

Actual

Fits

403020100

100

50

0

TT

F-S

yste

m

Time

Yt = 75.6950 - 1.70911*t

MSD:MAD:MAPE:

344.096 14.479 44.989

Trend Analysis for TTF-SystemLinear Trend Model

MODELLING ROCOFMODELLING ROCOF

250 500 750 1000 1250 1500

10

20

30

40

MODELLING ROCOFMODELLING ROCOF

Time used as the variable in the Time used as the variable in the modellingmodelling

= 157.5 days, = 157.5 days, = 1.55 = 1.55 Estimated average number of service Estimated average number of service

calls per year: calls per year:

Year

Estimated Service Calls 3.7 7.1 9.4 11.3 13.0 14.6 16.0 17.3 18.5 19.7

1.

COMPONENT LEVEL MODELLINGCOMPONENT LEVEL MODELLING

COMPONENT: CLEANING WEBCOMPONENT: CLEANING WEB

MODELLINGMODELLING

Failed components replaced by new Failed components replaced by new onesones

Time to failure modelled by a failure Time to failure modelled by a failure distribution function distribution function FF(t)(t)

The form of the distribution function The form of the distribution function determined using the failure data determined using the failure data available (black-box modelling)available (black-box modelling)

MODELLINGMODELLING

Several distribution function were Several distribution function were examined for modelling at the examined for modelling at the component level. Some of them were: component level. Some of them were:

2- and 3-parameter (delayed) Weibull2- and 3-parameter (delayed) Weibull Mixture WeibullMixture Weibull Competing risk WeibullCompeting risk Weibull Multiplicative WeibullMultiplicative Weibull Sectional WeibullSectional Weibull

COMPONENT LEVELCOMPONENT LEVEL

A list of the different distributions A list of the different distributions considered can in found in the following considered can in found in the following book: Murthy, D.N.P., Xie, M. and Jiang, book: Murthy, D.N.P., Xie, M. and Jiang, R. (2003), R. (2003), Weibull ModelsWeibull Models, Wiley, New , Wiley, New YorkYork

We consider modelling based on both We consider modelling based on both “counts” and “age” “counts” and “age”

HISTOGRAM (COUNTS)HISTOGRAM (COUNTS)

24000020000016000012000080000400000

5

4

3

2

1

0

cleaning web counter

Fre

que

ncy

Cleaning W eb Counter

HISTOGRAM (AGE)HISTOGRAM (AGE)

25020015010050

6

5

4

3

2

1

0

cleaning web day

Fre

que

ncy

Cleaning Web Day

WPP PLOTWPP PLOT

WPP plot allows one to decide if one of WPP plot allows one to decide if one of the Weibull models is appropriate for the Weibull models is appropriate for modelling a given data setmodelling a given data set

For 2-parameter Weibull: WPP is a For 2-parameter Weibull: WPP is a straight linestraight line

For more on WPP plot, see For more on WPP plot, see Weibull Weibull Models Models by Murthy et al (cited earlier)by Murthy et al (cited earlier)

NOTATIONNOTATION

Two sub-populationsTwo sub-populations Scale parametersScale parameters Shape parameters Shape parameters Location parameter Location parameter Mixing parameter Mixing parameter pp error: error: (square of the error between model (square of the error between model

and data on the WPP)and data on the WPP)

1 2, 1 2,

MODEL FITMODEL FIT

Model t0 p error

Weibull(2) 1.060 * 80035 * * * * 0.543

Delayed Weibull 1.060 * 80035 * 0 * * 0.543

Mixture 0.851 5.230 79377 67923 * * 0.674 0.092

Multiplicative 11.013 1.060 390693 80090 * * * 0.543

Competing Risk 0.630 1.266 1046923 96110 * * * 0.502

Sectional 0.926 1.199 107232 84078 3483.4 11832.9 * 0.484

MIXTURE MODEL (COUNT)MIXTURE MODEL (COUNT)

SPARES NEEDEDSPARES NEEDED

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

Year

Estimated Cleaning Webs 2.83 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04

REFERENCEREFERENCE

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

Thank youThank you