Question

A company operates a solar installation in the desert in Western Australia. It is reviewing its operating practices with a view to making them more efficient

. a) The solar installation generates electric power from sunlight and incurs operating costs for cleaning the solar modules (sometimes called solar panels) and replacing solar modules that have failed. The annual revenue from the electric power is variable due to variable cloudiness and solar module failure and has a mean of $2.78m and a standard deviation of $0.32m. The annual operating costs have a mean of $0.51m and a standard deviation of $0.12m. Calculate the mean and standard deviation of the annual profit = annual revenue – annual operating costs.

b) Expected revenue varies systematically from one month to another, being higher in the summer when there is more sunshine. Monthly operating costs follow the same probability model regardless of the month (same mean and standard deviation apply to all months). Calculate, if possible, the mean and standard deviation of (i) monthly operating costs (ii) monthly profits. If a calculation is not possible, give the reason.

c) The solar installation is located in the desert 100 km from the nearest office of the company that operates it and the company sends a maintenance crew out quarterly (once every 3 months) to clean dust and sand off the solar modules and check for mechanical or electrical problems. Each solar module is also monitored electronically over the Internet so that the operating company is alerted immediately when a solar module fails. On average 1.3 modules fail per month and the maintenance crew replaces any failed modules on their quarterly visits. Module failures are independent of each other and occur at random. The loss of a few solar modules does not impact revenue enough to justify the cost of sending the maintenance crew before the next quarterly visit. However the operating company decides that if more than 7 modules have failed they should send the maintenance crew out immediately to replace the failed modules. What is the probability of the maintenance crew having to go to the solar installation before the end of the regular 3-month period?

d) If 8 modules fail, the maintenance crew loads 9 replacement modules into their truck in case one is smashed during the 100 km drive, much of which is over uneven dirt tracks through the desert. Past experience shows that the probability of any individual module being smashed on this journey is 0.043. The operations manager wants the probability that the crew arrives with less than 8 working modules to be < 0.05. How many replacement modules should the maintenance crew load into their truck so as to achieve this objective? Answer this question, stating your assumptions clearly, and comment on whether the assumptions are likely to be true.

e) The solar modules are covered by a 25-year warranty which covers the cost of the replacement module itself but not the cost of driving 100 km and installing it. The operating company plans on visiting the site only once every 3 months and is therefore considering purchasing “business continuity insurance” which would cover the loss of revenue from failed solar modules for an annual premium of $5000. In order to decide whether it is worth paying this premium the company needs to calculate its expected revenue loss from failed modules. The average loss of revenue from one failed module is $200 per month. If one module fails during a 3-month period, we assume it fails in the middle of that period so that it has failed for a total of 1.5 months and the loss of revenue is 1.5*200 = $300. We make similar assumptions if 2,3,4, … modules fail during the 3 month period. Considering the probabilities of 0,1,2, …,10 modules failing during a 3-month period, what is the expected revenue loss during a 3-month period? Based on this expected loss, should the company purchase business continuity insurance?

Answer 1