All Chapters Covered
j j
SOLUTIONS
, RELIABILITYjENGINEERINGj–jAjLIFEjCYCLEjAPPROACH
INSTRUCTOR’Sj MANUAL
CHAPTERj 1
ThejMontyjHalljProblem
Thejtruthjisjthatjonejincreasesjone’sjprobabilityjofjwinningjbyjchangingjone’sjchoice.jThejeasiestj
wayjtojlookjatjthisjfromjajprobabilityjpointjofjviewjisjtojsayjthatjoriginallyjtherejisjajprobabilityjofj
⅓joverjeveryjdoor.jSojtherejisjajprobabilityjofj⅓joverjthejdoorjoriginallyjchosen,jandjajcombinedjpro
babilityjofj⅔joverjthejremainingjtwojdoors.jOncejonejofjthosejtwojdoorsjisjopened,jtherejremainsj
ajprobabilityjofj⅓joverjthejdoorjoriginallyjchosen,jandjthejotherjunopened jdoorjnowjhasjthejproba
bilityj⅔.jHencejitjincreasesjone’sjprobabilityjofjwinningjthejcarjbyjchangingjone’sjchoicejofjdoor.
Thisjdoesjnotjmeanjthatjthejcarjisjnotjbehindjthejdoorjoriginallyjchosen,jonlyjthatjifjonejwerejtojrep
eatjthejexercisejsayj100jtimes,jthenjthejcarjwouldjbejbehindjthejfirstjdoorjchosenjaboutj33jtimesjan
djbehindjthejalternativejchoicejaboutj66jtimes.jProvejforjyourselfjusingjExcel!
AnotherjwayjtojprovejthisjresultjisjtojusejBayesjTheorem,jwhichjthejreaderjcanjsourcejforjhimselfj
onjthejinternet.
Assignmentj 1.2:j Failurej Freej Operatingj Period
ThejFFOPj(FailurejFreejOperatingjPeriod)jisjthejtimejforjwhichjthejdevicejwilljrunjwithoutjfailur
ejandjthereforejwithoutjthejneedjforjmaintenance.jItjisjthejGammajvaluejforjthejdistribution.jFromj
thejlistjofjfailurejtimesj150,j190,j220,j275,j300,j350,j425,j475,jthejOffsetjisjcalculatedjasj97.42jhour
s j–
jsayj 100j hours.j Thisjisjthejtimej forj which jtherejshould jbejnoj probability jofjfailure.j Itj willjbejseenjfr
omjthejgraphjinjthejsoftwarejwithjBetaj=j2jthatjthejdistributionjisjofjalmostjperfectjnormaljshapeja
ndjthatjthejdistributionjdoesjnotjbeginjatjthejorigin.jThejgapjisjthej100jhoursjthatjthejsoftwarejcalc
ulatesjwhenjasked.
WhenjthejgraphjisjstudiedjforjBetaj=j2jitjwilljbejseenjthatjtherejisjajdownwardjtrajectoryjinjthejthreejl
eftjhandjpoints.jIfjthisjtrajectoryjisjtakenjdownjtojthejhorizontaljaxisjitjisjseenjtojintersectjitjatjaboutj1
20jhours.jThisjisjthejestimationjofjGamma.jInjthejdaysjbeforejsoftwarejthisjwasjalwaysjthejmostju
nreliablejestimatejofjajWeibulljparameterjandjthejmostjdifficultjtojobtainjgraphically.
Assignmentj 1.3
Whenjthejoffsetjisjcalculatedjitjisjseenjtojbejnegativejatj–
j185.59 j(say j180). jThis jindicates jthatjthejdistribution j starts jbeforejzero jonjthejhorizontal jaxis.jThisj
isjthejphenomenonjofjshelfjlife.jSomejitemsjhavejfailedjbeforejbeingjputjintojservice.jThisjcanjapp
lyjinjpracticejtojrubberjcomponentsjandjpaints,jforjexample.
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,Assignmentj1.4:jThejChoicejbetweenjTwojDesignsjofjSpring
DESIGNj A DESIGNjB
Number CyclesjtojFailure Number CyclesjtojFailure
1 726044 1 529082
2 615432 2 729000
3 807863 3 650000
4 755000 4 445834
5 508000 5 343280
6 848953 6 959900
7 384558 7 730049
8 666600 8 973224
9 555201 9 258006
10 483337 10 730008
UsingjthejWEIBULL-DRjsoftwarejforjDESIGNjAjabovejwejget
βj=j4
Correlationj=j0.9943
F400kj=j8%j(measuredjfromjthejgraphjinjthejWeibulljprintoutjbelowjFigjM1.4jSetjA)jHencejR40
0kj=j92%
ForjDESIGNjBjwejgetjfromjthejWEIBULL-DRjsoftwarej(notjshownjhere)
Βj=j2
Correlationj=j0.9867
F400kj=j20%
HencejR400kj=j80%
HencejDESIGNjAjisjbetter
FromjFigj1.4.1jSetjAjwejcanjreadjinjthejtablejthatjforjFj=j1%jatj90%jconfidence,jthejRjvaluejisj1269
22jcycles.jForjanjaveragejusejofj8000jcyclesjperjyearjwejgetj126922/8000j=j15.86jyearsj Ajconservati
vejguaranteejwouldjthereforejbyj15jyears.
NOTE:jThejabovejcalculationsjignorejthejγjvalue.jIfjthisjisjcalculated,jthejfollowingjfiguresjemer
gejasjshownjinjFigj1.4.2j(thejobscurationjofjsomejofjthejfiguresjisjthejwayjthejcurrentjversionjofjth
ejsoftwarejprintsjout)
DESIGNjA
βj=j3
γj=j101j828.6jsayj100j000
ForjFj=j1%jatj90%jconfidence,jFj=j176149jDividingjb
yj8000jwejgetj176149/8000j=j22jyears
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, Figj1.4.1jSetjA
Ajfigurejofj22jyearsjorjevenj15jyearsjforjanyjguaranteejisjveryjlongjindeed.jCompanyjpolicyjwoul
djhavejtojbejinvokedj–
jtherej arej matters jtoj considerjinjthejdetermination jofj guarantees j otherjthan jthejtestjdataj provided.j
Thesejmattersjcouldjincludejcorrosion,juserjabusejetc.jSuchjfactorsjarejmorejlikelyjtojoccur,jthejlo
ngerjthejoperatingjperiod.jQuestionsjneedjtojbejaskedjsuchjasjisjtherejanjindustryjstandardjforjsuc
hjguarantees,jwhatjarejcompetitorsjofferingjasjguarantees,jetc.
AjfurtherjpointjtojnotejisjthatjDESIGNjBjexhibitsjveryjpeculiarjcharacteristicsjifjthejγjvaluejisjtakenj
intojaccount.jThejβjvaluejremainsjatj2jbutjthejγjvaluejisjnegativejatjoverj50j000jcycles!jThisjimpliesj
thatjtherejisjajprobabilityjofjfailurejbeforejenteringjservice.jThisjdatajlooksjsuspectjandjfurtherjtestsjsh
ouldjbejdonejtojconfirmjthejreliabilityjcharacteristicsjofjDESIGNjB.
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j j
SOLUTIONS
, RELIABILITYjENGINEERINGj–jAjLIFEjCYCLEjAPPROACH
INSTRUCTOR’Sj MANUAL
CHAPTERj 1
ThejMontyjHalljProblem
Thejtruthjisjthatjonejincreasesjone’sjprobabilityjofjwinningjbyjchangingjone’sjchoice.jThejeasiestj
wayjtojlookjatjthisjfromjajprobabilityjpointjofjviewjisjtojsayjthatjoriginallyjtherejisjajprobabilityjofj
⅓joverjeveryjdoor.jSojtherejisjajprobabilityjofj⅓joverjthejdoorjoriginallyjchosen,jandjajcombinedjpro
babilityjofj⅔joverjthejremainingjtwojdoors.jOncejonejofjthosejtwojdoorsjisjopened,jtherejremainsj
ajprobabilityjofj⅓joverjthejdoorjoriginallyjchosen,jandjthejotherjunopened jdoorjnowjhasjthejproba
bilityj⅔.jHencejitjincreasesjone’sjprobabilityjofjwinningjthejcarjbyjchangingjone’sjchoicejofjdoor.
Thisjdoesjnotjmeanjthatjthejcarjisjnotjbehindjthejdoorjoriginallyjchosen,jonlyjthatjifjonejwerejtojrep
eatjthejexercisejsayj100jtimes,jthenjthejcarjwouldjbejbehindjthejfirstjdoorjchosenjaboutj33jtimesjan
djbehindjthejalternativejchoicejaboutj66jtimes.jProvejforjyourselfjusingjExcel!
AnotherjwayjtojprovejthisjresultjisjtojusejBayesjTheorem,jwhichjthejreaderjcanjsourcejforjhimselfj
onjthejinternet.
Assignmentj 1.2:j Failurej Freej Operatingj Period
ThejFFOPj(FailurejFreejOperatingjPeriod)jisjthejtimejforjwhichjthejdevicejwilljrunjwithoutjfailur
ejandjthereforejwithoutjthejneedjforjmaintenance.jItjisjthejGammajvaluejforjthejdistribution.jFromj
thejlistjofjfailurejtimesj150,j190,j220,j275,j300,j350,j425,j475,jthejOffsetjisjcalculatedjasj97.42jhour
s j–
jsayj 100j hours.j Thisjisjthejtimej forj which jtherejshould jbejnoj probability jofjfailure.j Itj willjbejseenjfr
omjthejgraphjinjthejsoftwarejwithjBetaj=j2jthatjthejdistributionjisjofjalmostjperfectjnormaljshapeja
ndjthatjthejdistributionjdoesjnotjbeginjatjthejorigin.jThejgapjisjthej100jhoursjthatjthejsoftwarejcalc
ulatesjwhenjasked.
WhenjthejgraphjisjstudiedjforjBetaj=j2jitjwilljbejseenjthatjtherejisjajdownwardjtrajectoryjinjthejthreejl
eftjhandjpoints.jIfjthisjtrajectoryjisjtakenjdownjtojthejhorizontaljaxisjitjisjseenjtojintersectjitjatjaboutj1
20jhours.jThisjisjthejestimationjofjGamma.jInjthejdaysjbeforejsoftwarejthisjwasjalwaysjthejmostju
nreliablejestimatejofjajWeibulljparameterjandjthejmostjdifficultjtojobtainjgraphically.
Assignmentj 1.3
Whenjthejoffsetjisjcalculatedjitjisjseenjtojbejnegativejatj–
j185.59 j(say j180). jThis jindicates jthatjthejdistribution j starts jbeforejzero jonjthejhorizontal jaxis.jThisj
isjthejphenomenonjofjshelfjlife.jSomejitemsjhavejfailedjbeforejbeingjputjintojservice.jThisjcanjapp
lyjinjpracticejtojrubberjcomponentsjandjpaints,jforjexample.
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,Assignmentj1.4:jThejChoicejbetweenjTwojDesignsjofjSpring
DESIGNj A DESIGNjB
Number CyclesjtojFailure Number CyclesjtojFailure
1 726044 1 529082
2 615432 2 729000
3 807863 3 650000
4 755000 4 445834
5 508000 5 343280
6 848953 6 959900
7 384558 7 730049
8 666600 8 973224
9 555201 9 258006
10 483337 10 730008
UsingjthejWEIBULL-DRjsoftwarejforjDESIGNjAjabovejwejget
βj=j4
Correlationj=j0.9943
F400kj=j8%j(measuredjfromjthejgraphjinjthejWeibulljprintoutjbelowjFigjM1.4jSetjA)jHencejR40
0kj=j92%
ForjDESIGNjBjwejgetjfromjthejWEIBULL-DRjsoftwarej(notjshownjhere)
Βj=j2
Correlationj=j0.9867
F400kj=j20%
HencejR400kj=j80%
HencejDESIGNjAjisjbetter
FromjFigj1.4.1jSetjAjwejcanjreadjinjthejtablejthatjforjFj=j1%jatj90%jconfidence,jthejRjvaluejisj1269
22jcycles.jForjanjaveragejusejofj8000jcyclesjperjyearjwejgetj126922/8000j=j15.86jyearsj Ajconservati
vejguaranteejwouldjthereforejbyj15jyears.
NOTE:jThejabovejcalculationsjignorejthejγjvalue.jIfjthisjisjcalculated,jthejfollowingjfiguresjemer
gejasjshownjinjFigj1.4.2j(thejobscurationjofjsomejofjthejfiguresjisjthejwayjthejcurrentjversionjofjth
ejsoftwarejprintsjout)
DESIGNjA
βj=j3
γj=j101j828.6jsayj100j000
ForjFj=j1%jatj90%jconfidence,jFj=j176149jDividingjb
yj8000jwejgetj176149/8000j=j22jyears
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, Figj1.4.1jSetjA
Ajfigurejofj22jyearsjorjevenj15jyearsjforjanyjguaranteejisjveryjlongjindeed.jCompanyjpolicyjwoul
djhavejtojbejinvokedj–
jtherej arej matters jtoj considerjinjthejdetermination jofj guarantees j otherjthan jthejtestjdataj provided.j
Thesejmattersjcouldjincludejcorrosion,juserjabusejetc.jSuchjfactorsjarejmorejlikelyjtojoccur,jthejlo
ngerjthejoperatingjperiod.jQuestionsjneedjtojbejaskedjsuchjasjisjtherejanjindustryjstandardjforjsuc
hjguarantees,jwhatjarejcompetitorsjofferingjasjguarantees,jetc.
AjfurtherjpointjtojnotejisjthatjDESIGNjBjexhibitsjveryjpeculiarjcharacteristicsjifjthejγjvaluejisjtakenj
intojaccount.jThejβjvaluejremainsjatj2jbutjthejγjvaluejisjnegativejatjoverj50j000jcycles!jThisjimpliesj
thatjtherejisjajprobabilityjofjfailurejbeforejenteringjservice.jThisjdatajlooksjsuspectjandjfurtherjtestsjsh
ouldjbejdonejtojconfirmjthejreliabilityjcharacteristicsjofjDESIGNjB.
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