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KNE240 Reliability Engineering 2019 : Assignment 1

Due end of Week 7 (5pm Friday 30th August); worth 10%

Your brief

A friend of yours (Alex) is planning to buy a repair shop and has hired you to work out how many mechanics she should

employ. Alex has obtained data collected from the previous one year’s working days (250) and has given you those data

to help you.

Currently the business determines which jobs to do and who does them according to the following rules:

1. Jobs come in every working day.

2. Many jobs are short and the rest are long.

3. There are 1st and 2nd class mechanics.

4. Short jobs can be completed in one hour by any mechanic.

5. Eight short jobs can be completed in one day.

6. Long jobs take all day for one mechanic (even if they start later).

7. Only 1st Class Mechanics can do long jobs.

8. Unfinished jobs carry over until the next day.

9. Short jobs get priority – they are done before any long jobs.

10. A mechanic doing short jobs isn’t available to do any long jobs on the same day (long jobs take all day, remember!).

11. Any job carried over to the next day costs $30.

12. Short jobs earn $50 on completion.

13. Long jobs earn $500 on completion.

14. 1st class mechanics cost $80000 per 250 working days.

15. 2nd class mechanics cost $35000 per 250 working days.

16. The business pays $24000 in rent per 250 working days.

Example: Suppose there are three 1st class and three 2nd class mechanics employed, and that 30 jobs arrive, of which

26 are short and 4 are long. All the 26 short jobs are completed first, using all three 2nd class mechanics and leaving two

other short jobs, which a 1st class mechanic has to do (and therefore won’t be able to do anything else that day!) The 4

long jobs can only be done by the remaining two 1st class mechanics, who can only do 1 each, leaving 2 incomplete to be

carried over to the next day.

What is expected

You are expected to turn in a report that includes

1. A two (or more) page report of your findings and recommendations, written in a way that would be understandable

to your friend, who is not a statistician, and to her bank manager, who may be asked

2. An appendix that gives details of the mathematics and commands (Excel or other spreadsheet or R) used to solve

each of the tasks below. Include all your working: spreadsheets, R scripts, etc.

3. If you do not include the working then it will not be possible to verify your analyses.

4. A suggested structure is as follows:

12

(a) Introduction for Alex, with summary of recommendations;

(b) Analysis of the observed data, with figures and summary statistics;

(c) Description of what distributions you choose to represent the observed data;

(d) Description of the simulation process you used;

(e) Detailed recommendation for how many 1st and 2nd class mechanics Alex should employ;

(f) Any other recommendations you have (perhaps a change in the way jobs are assigned);

(g) Appendix with all working, in such a way that your analysis is repeatable and can be checked.

Submission

Submit your assignment online only, using the electronic cover sheet available on MyLO.

Your report must be in pdf format, not Word, Rich Text, HTML, or raw text. If you submit a report that is not in pdf

format then it will be returned and incur an automatic 10% penalty.

Here are your tasks

Task 1

The file mechanicsOfClass.xlsx records the number of jobs coming in each day, and of those, the number of jobs classed

as “long” and the number classed as “short”. The file mechanicsOfClass.csv has the same data but without the suggested

columns to help you with your work (but is readable in R). Use the spreadsheet or csv file as a basis for your work.

Create an empirical probability distribution of the number of jobs arriving per day, and plot it.

Task 2

Choose which distribution of those you have studied so far best fits the number of arriving jobs that are short. State why

you have chosen this distribution and estimate all the parameters involved. It may help to plot the proportion of short

jobs received each day.

Task 3

Choose which distribution of those you have studied so far best fits the observed pattern of total number of jobs arriving

per day. State clearly the justification for choosing this distribution.

Use the method of moments to fit the distribution to the number of jobs arriving per day, with full working.

Plot the fitted theoretical distribution on the same figure as the observed data (Task 1).

Task 4

Use the inverse transform method to simulate the number of jobs (both short and long) arriving on any given day. (Note

that jobs come in integer amounts.)

Simulate what might happen over a 250 (working) day period. Include in your simulation the number of 1st and 2nd

class mechanics (from 0 to 10 each).

You may use the template file mechanicsOfClass.xlsx to help get started.

Estimate using your simulations how many 1st and 2nd class mechanics Alex should employ.3

Style Guide

Write your report in clear English, with correct spelling and grammar.

Use full sentences for all answers: instead of writing “mean: 5” write “The mean value was 5” or similar.

Use clear figures, fully labelled, to illustrate your answers. - to a job as it comes in.

Lay your report out logically and clearly so information is easy to find.

Assessment

You will be graded according to

1. Creating and plotting of empirical distribution function of number of jobs arriving per day 2

2. Choice and justification of distribution for short jobs 2

3. Choice and justification of distribution for total number of jobs arriving per day 4

4. Using the method of moments to create a fitted distribution to observed number of jobs arriving 4

5. Simulating the number of jobs, including short and long, arriving each day 5

6. Calculating the profit / loss from the rules given 3

7. Overall recommendation based on the simulations 5

8. Presentation: clarity, organization, and quality of communication 5

Total marks: 30

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