All Chapters Covered
Q Q
SOLUTIONS
, RELIABILITY ENGINEERING – A LIFE CYCLE APPROACH Q Q Q Q Q Q
INSTRUCTOR’S MANUAL Q
CHAPTER 1 Q
The Monty Hall Problem Q Q Q
The truth is that one increases one’s probability of winning by changing one’s choice. The easiest
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
way to look at this from a probability point of view is to say that originally there is a probability of
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
⅓ over every door. So there is a probability of ⅓ over the door originally chosen, and a combined pro
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
bability of ⅔ over the remaining two doors. Once one of those two doors is opened, there remains
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
a probability of ⅓ over the door originally chosen, and the other unopened door now has the proba
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
bility ⅔. Hence it increases one’s probability of winning the car by changing one’s choice of door.
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
This does not mean that the car is not behind the door originally chosen, only that if one were to rep
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
eat the exercise say 100 times, then the car would be behind the first door chosen about 33 times an
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
d behind the alternative choice about 66 times. Prove for yourself using Excel!
Q Q Q Q Q Q Q Q Q Q Q Q
Another way to prove this result is to use Bayes Theorem, which the reader can source for himself
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
on the internet.
Q Q
Assignment 1.2: Failure Free Operating Period Q Q Q Q Q
The FFOP (Failure Free Operating Period) is the time for which the device will run without failur
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
e and therefore without the need for maintenance. It is the Gamma value for the distribution. From
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
the list of failure times 150, 190, 220, 275, 300, 350, 425, 475, the Offset is calculated as 97.42 hou
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
rs – Q
say 100 hours. This is the time for which there should be no probability of failure. It will be seen fr
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
om the graph in the software with Beta = 2 that the distribution is of almost perfect normal shape a
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
nd that the distribution does not begin at the origin. The gap is the 100 hours that the software calc
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ulates when asked. Q Q
When the graph is studied for Beta = 2 it will be seen that there is a downward trajectory in the three l
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
eft hand points. If this trajectory is taken down to the horizontal axis it is seen to intersect it at about 1
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
20 hours. This is the estimation of Gamma. In the days before software this was always the most u
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
nreliable estimate of a Weibull parameter and the most difficult to obtain graphically.
Q Q Q Q Q Q Q Q Q Q Q Q
Assignment 1.3 Q
When the offset is calculated it is seen to be negative at –
Q Q Q Q Q Q Q Q Q Q Q Q
185.59 (say 180). This indicates that the distribution starts before zero on the horizontal axis. This
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
is the phenomenon of shelf life. Some items have failed before being put into service. This can app
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ly in practice to rubber components and paints, for example.
Q Q Q Q Q Q Q Q Q
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,Assignment 1.4: The Choice between Two Designs of Spring Q Q Q Q Q Q Q Q
DESIGN A Q DESIGN B Q
Number Cycles to Failure Q Q Number Cycles to Failure Q Q
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
Using the WEIBULL-DR software for DESIGN A above we get
Q Q Q Q Q Q Q Q Q
β=4
Q Q
Correlation = 0.9943 Q Q
F400k = 8% (measured from the graph in the Weibull printout below Fig M1.4 Set A) Hence R40
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
0k = 92%
Q Q
For DESIGN B we get from the WEIBULL-DR software (not shown here)
Q Q Q Q Q Q Q Q Q Q Q
Β=2 Q Q
Correlation = 0.9867 Q Q
F400k = 20% Q Q
Hence R400k = 80% Q Q Q
Hence DESIGN A is better Q Q Q Q
From Fig 1.4.1 Set A we can read in the table that for F = 1% at 90% confidence, the R value is 1269
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
22 cycles. For an average use of 8000 cycles per year we get 126922/8000 = 15.86 years A conservati
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ve guarantee would therefore by 15 years.
Q Q Q Q Q Q
NOTE: The above calculations ignore the γ value. If this is calculated, the following figures emer
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ge as shown in Fig 1.4.2 (the obscuration of some of the figures is the way the current version of th
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
e software prints out)
Q Q Q
DESIGN A Q
β=3
Q Q
γ = 101 828.6 say 100 000
Q Q Q Q Q Q
For F = 1% at 90% confidence, F = 176149 Dividing b
Q Q Q Q Q Q Q Q Q Q Q
y 8000 we get 176149/8000 = 22 years
Q Q Q Q Q Q Q
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, Fig 1.4.1 Set A Q Q Q
A figure of 22 years or even 15 years for any guarantee is very long indeed. Company policy woul
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
d have to be invoked –
Q Q Q Q Q
there are matters to consider in the determination of guarantees other than the test data provided.
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
These matters could include corrosion, user abuse etc. Such factors are more likely to occur, the lo
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
nger the operating period. Questions need to be asked such as is there an industry standard for suc
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
h guarantees, what are competitors offering as guarantees, etc.
Q Q Q Q Q Q Q Q
A further point to note is that DESIGN B exhibits very peculiar characteristics if the γ value is taken
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
into account. The β value remains at 2 but the γ value is negative at over 50 000 cycles! This implies
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
that there is a probability of failure before entering service. This data looks suspect and further tests s
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
hould be done to confirm the reliability characteristics of DESIGN B.
Q Q Q Q Q Q Q Q Q Q
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Q Q
SOLUTIONS
, RELIABILITY ENGINEERING – A LIFE CYCLE APPROACH Q Q Q Q Q Q
INSTRUCTOR’S MANUAL Q
CHAPTER 1 Q
The Monty Hall Problem Q Q Q
The truth is that one increases one’s probability of winning by changing one’s choice. The easiest
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
way to look at this from a probability point of view is to say that originally there is a probability of
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
⅓ over every door. So there is a probability of ⅓ over the door originally chosen, and a combined pro
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
bability of ⅔ over the remaining two doors. Once one of those two doors is opened, there remains
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
a probability of ⅓ over the door originally chosen, and the other unopened door now has the proba
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
bility ⅔. Hence it increases one’s probability of winning the car by changing one’s choice of door.
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
This does not mean that the car is not behind the door originally chosen, only that if one were to rep
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
eat the exercise say 100 times, then the car would be behind the first door chosen about 33 times an
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
d behind the alternative choice about 66 times. Prove for yourself using Excel!
Q Q Q Q Q Q Q Q Q Q Q Q
Another way to prove this result is to use Bayes Theorem, which the reader can source for himself
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
on the internet.
Q Q
Assignment 1.2: Failure Free Operating Period Q Q Q Q Q
The FFOP (Failure Free Operating Period) is the time for which the device will run without failur
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
e and therefore without the need for maintenance. It is the Gamma value for the distribution. From
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
the list of failure times 150, 190, 220, 275, 300, 350, 425, 475, the Offset is calculated as 97.42 hou
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
rs – Q
say 100 hours. This is the time for which there should be no probability of failure. It will be seen fr
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
om the graph in the software with Beta = 2 that the distribution is of almost perfect normal shape a
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
nd that the distribution does not begin at the origin. The gap is the 100 hours that the software calc
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ulates when asked. Q Q
When the graph is studied for Beta = 2 it will be seen that there is a downward trajectory in the three l
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
eft hand points. If this trajectory is taken down to the horizontal axis it is seen to intersect it at about 1
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
20 hours. This is the estimation of Gamma. In the days before software this was always the most u
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
nreliable estimate of a Weibull parameter and the most difficult to obtain graphically.
Q Q Q Q Q Q Q Q Q Q Q Q
Assignment 1.3 Q
When the offset is calculated it is seen to be negative at –
Q Q Q Q Q Q Q Q Q Q Q Q
185.59 (say 180). This indicates that the distribution starts before zero on the horizontal axis. This
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
is the phenomenon of shelf life. Some items have failed before being put into service. This can app
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ly in practice to rubber components and paints, for example.
Q Q Q Q Q Q Q Q Q
1
@@SSee
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titoionn
,Assignment 1.4: The Choice between Two Designs of Spring Q Q Q Q Q Q Q Q
DESIGN A Q DESIGN B Q
Number Cycles to Failure Q Q Number Cycles to Failure Q Q
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
Using the WEIBULL-DR software for DESIGN A above we get
Q Q Q Q Q Q Q Q Q
β=4
Q Q
Correlation = 0.9943 Q Q
F400k = 8% (measured from the graph in the Weibull printout below Fig M1.4 Set A) Hence R40
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
0k = 92%
Q Q
For DESIGN B we get from the WEIBULL-DR software (not shown here)
Q Q Q Q Q Q Q Q Q Q Q
Β=2 Q Q
Correlation = 0.9867 Q Q
F400k = 20% Q Q
Hence R400k = 80% Q Q Q
Hence DESIGN A is better Q Q Q Q
From Fig 1.4.1 Set A we can read in the table that for F = 1% at 90% confidence, the R value is 1269
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
22 cycles. For an average use of 8000 cycles per year we get 126922/8000 = 15.86 years A conservati
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ve guarantee would therefore by 15 years.
Q Q Q Q Q Q
NOTE: The above calculations ignore the γ value. If this is calculated, the following figures emer
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
ge as shown in Fig 1.4.2 (the obscuration of some of the figures is the way the current version of th
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
e software prints out)
Q Q Q
DESIGN A Q
β=3
Q Q
γ = 101 828.6 say 100 000
Q Q Q Q Q Q
For F = 1% at 90% confidence, F = 176149 Dividing b
Q Q Q Q Q Q Q Q Q Q Q
y 8000 we get 176149/8000 = 22 years
Q Q Q Q Q Q Q
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, Fig 1.4.1 Set A Q Q Q
A figure of 22 years or even 15 years for any guarantee is very long indeed. Company policy woul
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
d have to be invoked –
Q Q Q Q Q
there are matters to consider in the determination of guarantees other than the test data provided.
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
These matters could include corrosion, user abuse etc. Such factors are more likely to occur, the lo
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
nger the operating period. Questions need to be asked such as is there an industry standard for suc
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
h guarantees, what are competitors offering as guarantees, etc.
Q Q Q Q Q Q Q Q
A further point to note is that DESIGN B exhibits very peculiar characteristics if the γ value is taken
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
into account. The β value remains at 2 but the γ value is negative at over 50 000 cycles! This implies
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
that there is a probability of failure before entering service. This data looks suspect and further tests s
Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q
hould be done to confirm the reliability characteristics of DESIGN B.
Q Q Q Q Q Q Q Q Q Q
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