Topic: Reliability Engineering
Faculty of Engineering & Applied Science
Reliability Engineering, ENGI 9116
Take Home Final Examination
Date: April 11-15, 2016 (on or before 10 AM)
Evaluation: 30% Faisal Khan
Question paper has two sections (A and B) and four pages
Please submit your final exam in Bruneau Centre (IIC 1008)
Section A (12%)
1. Below is the reference of a research paper on risk-based inspection modeling. Please read
this and provide me a summary of what has been done here, how it is done, and how you
could improve this work? [Use Times Roman Font with 12 point size with 1.5 line spacing;
the paper should not exceed two pages]
Shekari, E, Khan, F, and Ahmed, S (2016) A predictive approach to fitness-for-service
assessment of pitting corrosion, Journal of Pressure Vessels and Piping, 137 (2016) 13-21.
2. A photo detector counts photons for 10 nano seconds to decide if a distant light source is
emitting light. When the light source is not emitting light, some photons are still counted by
the detector due to the ambient background radiation. The number of photons counted in 10
nanoseconds is modeled as a random variable X whose parameter ? has value ln(9) if the
light source is not emitting light, and value ln(27) if the source is emitting light. The
maximum-likelihood detector decides that detection of light is the true if and only if the
likelihood ratio = probability of detection when light is present (x)/probability of detection
when light is not present (x) exceeds 1. (a) What value(s) of X result in a decision in favor
of detecting of light? (b) Compute the false alarm probability PFA and the missed detection
or false dismissal probability PMD of the maximum-likelihood decision rule.
3. 100 AIDS patients were given a new drug test. The results were as follows:
Year on drug Number of deaths Number of withdrawals
1 5 2
2 8 4
3 12 2
4 18 10
5 24 12
Withdrawals occurred when patients left the test area or died from causes not related to the
AIDS disease. Construct a life table to estimate the probability (reliability) that a patient
will survive at least five years.
4. Determine the minimum-cost trade-off between the MTBF and the MTTR for a system
requiring 0.99 availability. The MTTR must be between 10 and 30 hr, and the MTBF must
be at least 1000 hr. Consider the cost functions as:
a. C(x) = 0.0002x
; C(y) = 12500/y
b. C(x) = 2x; C(y) = 2000-y
5. The Breakit Company manufactures anti-clock disk brakes (ABS) for use in automobiles.
An essential part of the manufacturing process is the factory assemble and integration
linking (FAIL) system. Unfortunately, this system has experienced numerous failures over
the years under minimal repair concept that is a non-homogeneous Poisson process having
an intensity function ?(t)=1.15×10
with t measured in operating hours. The system
averages 2000 operating hours a year and it currently 5 years old. When the system fails,
the repair time is lognormal with tmed = 6.2 hours and s = 1.2.
a. When the system was new, its manufacturer offered a 500-operating-hr warranty.
What is the system reliability during the warranty period?
b. When the system fails, it takes a crew of two to repair it. If the labor rate is $50 per
hour and based upon the MTTR, what is the maintenance cost of the FAIL system
during its 6
year of operation?
c. If the cost of the system is $30,000, what is its optimal (minimum cost) replacement
d. Management believes that a preventive maintenance (PM) program may be cost
effective. A PM consists primarily of replacing certain parts having a cost of $700
and thus restoring it to as-good-as-new condition. If it takes one technician 4 hours
to do this, what is the optimum (minimum cost) PM interval?
Section B (18%)
Note: Solve the following problems:
(Each 3.0 %)
6. External corrosion is a potential failure mechanism of buried natural gas pipeline. Once the
protective coating and/or the cathodic protection system fail the external corrosion rate
depends largely on the characteristics of the surrounding soil. The corrosion rate is
typically modeled using an empirical power-law relationship as: ? = ??
where y is the
corrosion depth, t is the exposure time, and A and n are site dependent constants, with n
typically less than 1. Below is the data available from a specific site. Estimate the failure
probability of 40 mm pipeline in 20 years. Pipeline failure may be defined as loss of 66%
of the pipeline thickness.
Time (years) Corrosion depth (mm)
7. An offshore platform consists of three large main power generators (MPGs), two operating
and one spare (standby mode). The lifetime of each MPG follows the Weibull distribution
with beta = 2.3 and theta = 40 years. In a given time, two MPGs are used, while third one
can be used to replace a failed unit in 12 days with a cost of $ 5 million. The failure of each
additional unit requires the procurement and installation of a brand new transformer with a
cost of $ 280 million. Estimate failure probability and associated risk with the MPGs over
the next 20 years of operation considering both scenarios (spare is available as good as new
and spare is in failed state).
8. The load (S) and resistance (R) are modeled as log normally distributed random variables
with the following parameters: the mean and coefficient of variation (CoV) of the load are
160 kN and 0.15, respectively, while the CoV of resistance is given as 0.28. a) If the
design criterion is R = S, then what should be the mean resistance? Assume the nominal
resistance (R) is the 5th percentile of the distribution; b) what is the probability of failure
for design criterion R = S?; and c) what is reliability index?
9. A repairable system consists of three non-identical components A, B and C, all of which
must work for system success. When one component fails, no further component failures
can occur. Construct the relevant state space diagram and hence evaluate general
expressions for the individual limiting state probabilities in terms of component failure and
repair rates. Evaluate the unavailability of the system in hour/year if the three components
have the following reliability data.
Component Mean time to Failure (yr) MTTR (hours)
A 1.0 180
B 2.0 12
C 1.0 52
10. The figure below shows the pipe layout in a lawn sprinkler system. Sprayers are located at
every intersection of the pipe and also at the corners of the system. Thus there are 16
sprayers. Two things can happen to a sprayer: it can break off or it can get clogged. If any
one of the sprayers breaks off, a great stream of water will pour forth and the system will
have failed. If one of the sprayers clogs, it will fail to water the lawn in the immediate
vicinity; however, adjacent sprayers can reach the affected areas. The system is judged to
be failed if any two adjacent sprayers are both failed. For one sprayer, let the probability of
breakage be 0.01 and the probability of clogging be 0.05. Find the reliability of the system
(the probability that it does not fail in either mode).
Layout of the sprinkler system
11. A gas-fired furnace is shown below. The hot combustion gases pass through heat
exchanger to heat fresh air for space heating. The gas flow is controlled by an electric
solenoid valve connected to a thermoset. The gas is ignited by a pilot light flame. A hightemperature
switch shuts off all gas in the event of high temperature in the fresh air.
Develop fault tree using step-by-step approach for an event (scenario) “No Space Heating”.
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