Question $2.($a$-$c$)$

Maggie's Farm is a dairy farm that supplies milk to the Edinburgh region. One of

the issues that they face is that as cows become older they produce less milk.

The milk revenue per cow of a particular age is given below. Milking age starts

when the cow reaches maturity and can start producing milk.

The value of a cow is as follows:

These values can be used to represent the asset price at the end of the planning

horizon. The purchase price of a new cow $($i$.$e$.$ of milking age $0)$ is $1000.$

British Law requires that $5$ year old cows be sold for beef as older cows can

produce milk infected by various cow based diseases.

The concern for Maggie's Farm is when to sell a cow and replace it with a newly

mature cow $($i$.$e$.$ of milking age $0).$ The new cows are purchased from a supplier

so Maggie's Farm does not need to worry about raising them to maturity.

a$)$ Formulate this as a DP model and use this model to determine the

purchase$/$replacement policy that will maximise the revenue of both a $2$ and $3$

year old $($in milking age$)$ cow over a period of $4$ years.

$(60$ Marks$)$

b$)$ Define the principle of optimality and explain how it has allowed you to find a

solution in part $($a$).$

$(20$ marks$)$

c$)$ Define this problem mathematically with an expression for the base case $($i$.$e$.$

the final stage$)$ and a recursive case $($i$.$e$.$ a general expression for all other

stages$).$

$(20$ marks$)$