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## Quantitative Methods for Risk Analysis

QUESTION 1
Auto Safe Tech (AST) is considering marketing, at the start of next year, a new anti-theft device for motor vehicles. Based on GPS technology, the device would be supplied free to customers. Customers would then pay an annual subscription for AST’s monitoring services. AST wants to estimate the accumulated profit at the end of the next 5 years.
Assume that
• there are 610,000 vehicles which could potentially use the device at the start of next year, and that this figure will grow each year.
• the annual percentage growth in the number of vehicles which could potentially use the device is Normally distributed with a mean of 5.4% and a standard deviation of 1.1%.
• the proportion (market share) of the 610,000 vehicles that AST will have as subscribers next year (ie, the first year) is modelled by the Triangular distribution, with a minimum of 25%, a most likely value of 40% and a maximum of 75%. If no competitors enter the market, AST expect to retain that market share.
• there are 3 potential competitors.
o At the start of each year (including the first year), there is a 40% chance that any competitor who has not yet entered the market will enter during that year.
o The year after it enters the market, a competitor will have impact on AST’s market share.
Any competitor entering the market will capture 20% of the market share that AST had prior to that competitor’s entry. eg, if AST had 60% market share, the entry of
* 1 competitor would reduce that 60% share to 60(1 – 0.2) = 48%;
* 2 competitors in the same year would reduce that 60% share to 60(1 – 2*0.2) = 36%.
o Once a competitor has entered the market, that competitor will remain there.
o If all 3 competitors have entered the market, no more competitors will enter.
o The competitors behave independently.
• each annual subscription yields \$220 revenue to AST.
• each subscriber costs AST \$40 per year to service (including provision of the device).
(a) (i) Use simulation, with 1,000 iterations, to estimate the accumulated profit after 5 years for
AST. (You may ignore the use of Net Present Value.)
(ii) Provide a histogram and summary statistics for each of your output variable(s).

(b) From your simulation, report your findings on the estimated accumulated profit(loss) after 5 years for AST. Your responses do not need to be set out as a formal report, but should be succinct and directed to an imagined target audience of colleagues with a similar level of statistical knowledge to yours. Substantiate your findings with suitable figures/tables/graphs – as appropriate. Explain any additional assumptions that you have made.

QUESTION 2
Car-Go has developed a petrol additive, called Sparky. It is designed to limit damage to a car’s fuel system by protecting the spark plugs.
Car-Go has to decide whether or not to market this product. Furthermore, before deciding whether to market it, Car-Go can choose to engage engineering consultants to comprehensively test the effectiveness of the product. This cost of this testing is \$225,000.
Car-Go expects a net result (taking into account development and marketing costs) of
• \$5,000,000 if it markets this product and the marketing is successful.
• -\$2,600,000 if it markets this product and the marketing is unsuccessful.
In the past, similar motoring products have succeeded for Car-Go 60% of the time.
Car-Go believes the chance that the marketing of Sparky will be successful is
• 80%, if the test results indicate that the product is very effective.
• 30%, if the test results indicate that the product is not very effective.
Car-Go believes there is a 0.60 probability that the test results will indicate that the product is very effective.
(a) (i) In Excel, construct a table of input values for this problem. Provide a printout.
(ii) Construct a well-labelled decision tree for this problem. Provide a printout.
(iii) On the basis of this decision tree analysis, what decision(s) should Car-Go make?
(b) Perform sensitivity analysis on the optimal decision making
(i) by varying the estimate of the cost of testing.
(ii) by simultaneously varying the estimate of the profit from successful marketing and the
loss from unsuccessful marketing.

QUESTION 3
Beyond the Paleo (BtP) is a large and busy restaurant. It has an excellent reputation for its food and service. To maintain this reputation, the owners have established key measures of product and service quality. One such measure is the amount of time that customers wait – from the time they are seated until the time they are served.
Every day, each 20 minutes that the business is open, four tables are randomly selected. The elapsed time from when customers are seated at these tables until their orders arrive is recorded. Data for one 12 hour day is shown in the file BtP.xlsx.

(a) (i) Generate X-bar and R charts for the waiting time data in BtP.xlsx.

(b) For each “other out-of-control-indication” (if there are any),
• state which “other out-of-control-indication” has occurred,
• list all the subsamples that are involved in that “other out-of-control-indication”.

(c) On the basis of your analysis to date, what are the important aspects of the control charts that should be brought to the attention of management. Refer to any relevant calculations, tables, graphs, etc. (You are not being asked to make recommendations for improvements.)

(d) Discuss any further points which can be made about the waiting time data in BtP.xlsx
and its analysis. For example, discuss any features, limitations or further information. (You are not being asked to make recommendations for improvements.)

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