Assignment Questions on Learning Units 2-4

This assignment consists of 4 questions assessing the material covered in learning units 2-4 (weeks 2-5). You should attempt all questions.

The marks allocated for each question, or for its parts, are shown on the right.

Hand-written work is expected (due to the mathematical calculations involved) and there is no need for wordprocessing. Hand-in Details:

You must submit your work electronically via Blackboard before the deadline of midnight Friday 2nd July 2021 (U time). You will need to scan your written solutions and save as /convert to pdf format (not jpeg). You can then submit the pdf via the Assignments link from the Assessments tab in Blackboard. Click on the Submit Assignment link, attach the file, and submit. You will get a message of completion that will be your receipt of submission. You are advised to:

• Ensure that you have access to a scanner and find out how to produce a pdf well before it is needed.
• Note that although you can submit your file more than once before the deadline, only the most recent submission will be accepted.

Diameters of bolts produced by a particular machine are normally distributed with mean 0.66cm and standard deviation 0.014cm. Specifications call for the diameters to range from 0.63cm to 0.68cm

1. a) What proportion of bolts will be smaller than 0.64cm?
2. b) What value of the diameter is exceeded by 15% of the bolts?
3. c) What proportion of bolts will meet the specification?
4. d) What value of the standard deviation is required so that less than 2% of the bolts have a diameter greater than 0.68cm?

a) A certain electronic component has a constant failure rate of 3.4 × 10−4 /hour. A system requires the use of 37 units of this component and all the components must function for the system to work. What is the system failure rate?

b) A system consists of four units, an X unit and three identical Y units as shown below. If the X unit has reliability 0.995 and the system reliability goal is 0.985, what is the minimum reliability that each of the Y units must have?

c) The diagram below shows the layout of a system which consists of two parallel production lines, L1 and L2.

1. Assuming a constant failure rate, calculate the mean time to failure for component A.
2. Find the reliability for each production line and hence the reliability of the system.
3. Calculate the probability that the system has no output.
4. Explain the term redundancy and briefly discuss the advantages and disadvantages of incorporating redundancy into a system.
5. Consider the following times to failure of a sample of five items: Failure Number

a) Plot the data using Weibull probability paper. Do you consider it appropriate to continue to fit a Weibull distribution to this data?

b) Estimate the Weibull parameters (β, η and γ) from your chart and give an interpretation of their values.

c) Use your chart to estimate the following:

1. i) The reliability of the item after 1200 hours.
2. ii) The time at which the item is 70% reliable.
3. For each of the following cases, describe the pattern that is likely to be seen in a Weibull plot and describe how you would continue your analysis after the initial plot.
4. i) multiple failure modes exist
5. ii) there exists an amount of failure free time
6. A company is considering to purchase a piece of equipment for a specific task. Two alternative machines are available on the market – one a standard machine (A) and the other a semi-automatic one (B). The comparative data of the two machines is given below:

Increase in maintenance cost after 1st year £0 £200

Labour and maintenance costs incurred during a year can be assumed to occur at the end of the year. The prevailing interest rate for money is 5% per annum.

1. a) For each machine, calculate the total cost (excluding salvage value), in present terms, associated with a single estimated lifetime cycle.
2. b) Based on capitalised costs, which machine should the company choose? You must show all your working in arriving at your answer.