Questions There are a series of machines in a manufacturing line to complete a required manufacture



  There are a series of machines in a manufacturing line to complete a required manufacture process, see Table 1. The manufacture line is controlled by the Control Centre.

Table 1 Machines in a manufacturing line


Machine & Process Control

Number of machines in the line

Minimum No. required

Distribution of time to failure (in hours)



Cutting machines  3                             2                      Weibull (b = 2.15,  q = 20990)

Drilling machines  4                             3                      Weibull (b = 2.38, q = 17760)

Milling machines   2                             2                      Lognormal (s = 0.40, t med

= 17600)


Multi-axis Machining

Centre                    2                            2

Normal (m = 10958, s = 1250)


Polishing machines                              4                      3    Lognormal (s = 0.55, t med

= 19260)

Painting machines 5                             4                      Weibull (b = 2.0, q = 31500)

Control Centre      1                             1                      Exponential with l =



Note: b is shape parameter and q is scale parameter of Weibull distribution; s is shape parameter and t med is location parameter of Lognormal distribution (see, page 81); m is mean and s is standard deviation of Normal distribution.


Assume all machines on the manufacturing line keep operation status unless one fails. Answer the following questions: (total 23 marks)


1)       Construct the reliability block diagram to calculate the probability (reliability) that the manufacturing line can fulfil the required process without failure in a periodical maintenance interval of 6 months?    (8 marks)

2)       What is the time at which a periodical maintenance is carried out when the probability of the manufacturing line to fulfil the required process without causing the process failure is at 85%? (7 marks)

3)       If the probability (reliability) that the manufacturing line can perform well to fulfil the required process without failure in 6 months is required to be 87.5%, what measure should be taken? (8 marks)


  Compare three alternative designs by completing the following table where time is measured in days:

(15 marks)



Alternative              Distribution                   Rsystem(2days)   MTTFsystem (in days)


Load sharing system with two different units

Standby system with 3 identical units and no failure in standby; at least 2 units are working, the system works

A 2-out-of-3 system having identical units

Exponential with l1 =0.30,

l2 = 0.40,  l1+= 0.55,  and

l2+ = 0.65



Exponential with l = 0.75



Lognormal (s = 0.30, t med = 3.5)




Note: The distribution refers to time to failure distribution of each unit.

  The pilots of Green Airlines which flies the Boeing 797 exclusively are known for their “rough†landings. When the force of the landing exceeds the strength of a wheel strut, a fracture (failure) will occur. The strength of wheel strut is affected by number of landings, n, and described as 650exp(- 0.00132n) psi. Measurements on the force of each landing have determined that the distribution is Weibull with a shape parameter of 1.85 and a scale parameter of 209 psi. Each aircraft averages 3 landings per day.


Answer the following questions: (total 16 marks)


1)       Determine the time (in days) to initial inspection of the landing gear struts to insure no more than a 0.5%  chance of  a strut  fracture (failure).  Any fractured struts are then  replaced and the inspection interval starts over.  (8 marks)

2)       If no fracture is found, the struts are inspected again after 60 days. What is the probability of detecting a fracture at the next inspection?   (8 marks)

  Given the following fault tree:






AND                            OR                             OR


OR                              OR



AND                            OR



Answer the following questions (total 24 marks)


1)       Express the system failure in terms of the basic events using Boolean logic. (6 marks)

2)       Find the minimum cut sets. From this analysis, what are the top four basic events that should be the focus of the reliability design effort?  (6 marks)

3)       If the probability of each basic event is 0.03, estimate the probability of the system failure. (6 marks)

4)       If the probability of the system failure is required to be reduced 50 percent based on the result obtained in 3), consider using redundancy of basic events for achieving the design target. It is required to keep the total number of basic events minimum in all possible solutions. Then, modify the fault tree diagram accordingly to represent your design.  (6 marks)


  A consumer decides to perform an economic analysis to compare two different automobiles intended for use over the next 10 years. The first auto named Auto_1 averages 23 km per gallon and costs

$25,000. Consumer Reports says that the time between failures is gamma with a shape parameter of

2.58 and a scale parameter of 5000 km with an average repair cost of $508. Routine servicing is required every 6000 km at a cost of $130. Trade-in value after 10 years is estimated to be $11,000. The second auto named Auto_2 averages 21 km per gallon and costs $23,000. It has a time between failure distribution that Consumer Reports says is minimum extreme value with a scale parameter of 805 and a location parameter of 9350 km. Average repair cost as advertised is $400. Routine


servicing is required every 6500 km at a cost of $145. Trade-in value after 10 years is estimated to be

$9500. The consumer averages 18,000 km per year and has an available investment rate of return 4.5 percent. Inflation is expected to average 3.25% in every year over the next decade.


Questions  (total 22 marks)

1)       Given the cost of gasoline will be $3.0 per gallon in the first year and it will be increased year by year based on the inflation rate, which automobile should be purchased based only on the economic data provided? Give a short analysis report.  (11 marks)

2)       How would the life cycle costs compare if the cost of gasoline averages 3.50 (inflation has been considered) a gallon and the automobile is driven 16,000 km/year?  (11 marks)

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