Read the following case study based on the Applegold Cider Company. You should

then apply decision analysis to the problem facing Applegold. This will involve:

a$)$ Formulating the problem as a decision tree$-$

b$)$ using Bayesian analysis to update prior probabilities and find the correct proba$-$

bilities in the tree;

c$)$ discuss the strengths and limitations of your analysis.

Applegold is a major cider producer, producing draught cider for pubs and clubs,

as well as bottles and cans, which has recently seen production and the number of

outlets selling its products increase significantly.

The growth in dranght cider has created some problems for the company's managers.

In particular, there is concern that when sales reach their peak in August, there

might not be enough kegs $($steel re$-$usable cider containers$)$ available to meet demand.

Applegold own about $100000$kegs, but it is felt by some managers that the stock

sholud be increased.

The Operations Manager has propoased that $8000$ new kegs be ordered immediately.

The Accountant was not convinced. Kegs coet $260$ each, so this would lead to an

expenditure of $480000.$ They would be usable next year, but assuming $5\%$ interest

on capital, buying now rather than waiting until next year would cost $24000.$

The Sales Manager proposes that the company wait until an accurate long range

weather forecast is available for August, since demand depends heavily on the

weather, with hot dry months leading to high demand. Such a forecast will be

available in July. One problem is that it might be the case that other brewers had

bought all of the available kegs by this time and the Operations Manager estimates

a probability of $0\mathrm{.}8$ of getting the kegs if they wait until July.

The Sales Manager produces an interim forecast, with no knowledge of the weather,

of how good sales are likely to be in August. She estimates that they will be at

least $10\%$ higher with probability $0\mathrm{.}5,$ they will increase by a lower amount with

probability $0\mathrm{.}4,$ and there will be no increase with probability $0\mathrm{.}1.$

The Data Processing Manager suggests three possible strategies; baying $0,4000$ or

$8000$ kegs immediately. The associated change in profit from these three strategies

were estimated, based upon the assumption that with a $10\%$ sales increase $8000$

extra kegs will be used $($if available$),$ and for a lower increase $4000$ extra kegs will be

used $($if available$),$ with an associated profit of $f7$ per keg. These are summarised

in Table $1.$

The Sales Manager suggests that it might still be better to wait for the weather

forecast in July, and run the risk of the kegs not being available. Whether it is best

to do so depends upon how accurate the forecasts are. The data in Table $2$ give

some data on the recent performance of the forecasts.

Table $1$: Predicted changes in profit $($in f$)$ for combinations of different immediate

purchasing strategies and sales increases.

Table $2$: Sales over the last $84$ months.