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Course work: Faulty Phones

Background

The famous technology company, Apricot, is selling phones with a life-time warranty. With this

warranty, Apricot promises to repair faulty phones or replace them free of charge. To maximize their

profits, Apricot has asked you for a report and a set of recommendations on the best strategy for this

warranty scheme. The main consideration that should be addressed in the report is cost of the scheme.

Modelling Assumptions

You are asked to make the following modelling assumptions in your analysis:

• In any given day, assuming a phone has had i faults (for i=0,1,2,...,n−1) the probability of

having another fault is p, independently of all previous days, and the new fault is to be repaired

with probability q

i+1 within the same day.

• If a phone is not repaired, it is replaced instead.

• If a phone has its n’th fault it is immediately replaced.

• The expected cost to Apricot to repair the i’th fault of a phone is £(q×100×2i).

• Replacing a phone costs Apricot £R.

• Apricot’s products fall into one of three classes:

1. Low quality - n=3, R=410, p=5×10−4

for 25% of the production line.

2. Medium quality - n=4, R=850, p=2×10−4

, for 52% of the production line.

3. High quality - n=4, R=950, p=10−4

, for 23% of production line.

• Replacement phones initially have zero faults and are subject to the same warranty.

• Replacement phones always have the same quality class as the originals.

• The probability of experiencing more than one fault within a single day is sufficiently small

and is to be ignored.

Points to address

Apricot is interested in understanding the expected daily cost of the warranty scheme for every phone

and the optimal replacement strategy. In particular, Apricot can invest in its maintenance department

to increase q, which increases the proportion of faults that can be repaired but also increases the

average repair cost.

You are to compute the expected daily costs for each of the three quality classes. First, report these

costs for q=0.8 then produce plots showing the expected total daily costs for all values of q from

0 to 1 and each of the three quality classes. Lastly, recommend the best value of q, up to an error

of 1%, in two cases:

• Different values may be used for different quality classes to optimize the total cost of each class;

• The same value is to be used in all quality classes to optimize the total cost;

and report on the total expected daily costs in each case.

Finally, suppose that the replacement phone is picked randomly from the production line so that

the quality of the replacement is not necessarily the same as the original, write down the transition

probability matrix of the Markov chain that you would use for this case.

HINT: Write down the transition probability matrix for a Markov chain whose current state

represents the number of faults a customer’s phone has experienced (and had repaired) up to the

current day. Compute the stationary distribution of this Markov chain, and then compute the long

term average cost of the scheme.

Further Points to Note

• Collaboration and Plagiarism: Students are allowed to discuss the methods used with

other students, but your submitted project must be all your own work. Particularly please

note the following:

– Coursework reports must be written in a student’s own words and any code in their

coursework must be their own code. If some text or code in the coursework has been

taken from other sources, these sources must be properly referenced.

– Failure to reference work that has been obtained from other sources or to copy the words

and/or code of another student is plagiarism and if detected, this will be reported to the

School’s Discipline Committee. If a student is found guilty of plagiarism, the penalty

could involve voiding the course.

– Students must never give hard or soft copies of their coursework reports or code to another

student. Students must always refuse any request from another student for a copy of

their report and/or code.

– Sharing a coursework report and/or code with another student is collusion, and if detected,

this will be reported to the School’s Discipline Committee. If found guilty of collusion,

the penalty could involve voiding the course.

– Further information: https://www.hw.ac.uk/students/doc/plagiarismguide.pdf.

• Submission: The report should be submitted through Vision using Turnitin. It should not be

submitted by email or handed in as a paper copy.

• Deadline: The deadline for submission is 3:30 pm on Monday, November 4th (local time);

projects may be submitted early. Late projects will be marked in line with the University

Coursework Policy.

• Assessment This coursework will contribute 10% to the final mark for the course. This

coursework will be marked using the marking rubric found in the same folder as this project on

VISION.

• Feedback Marks and feedback for the project will be available from Monday, November 18th.

Report Specification

The report should meet the following specifications:

1. The report must be typed.

2. The report must not be more than three sides of A4 including graphs and tables, but not

including the Appendix.

3. The report must not contain any R code or your calculations.

4. The report should give the results of the analyses that have been carried out and your comments

on and conclusions from these results, but should be as readable as possible by non-specialists.

5. The Appendix must contain all code and workings for all calculations in your analysis. It

should also include the definition of the Markov chain(s) you are using for the model and the

corresponding P matrices.

6. Note this report is for a group of non-specialists and so should be understandable by them.

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