The farm family possess 640-acre farm. They can produce a total of 4,000 person-hours’ worth of labour during the winter

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The farm family possess 640-acre farm. They can produce a total
of 4,000 person-hours’ worth of labour during the winter and spring
months and 4,500 person hours during the summer and autumn months.
If they do not need all this time, they can work on neighbouring
farms for £5 per hour during the winter and spring months and £5.50
during the summer and autumn months.
The farm supports two types of livestock, dairy cows, and hens
and three types of cash crops – rape, corn, and wheat. All three
are cash crops, but the corn also is a feed crop for the cows and
the wheat also is used for chicken feed. The crops are harvested
during the late summer and fall. During the winter months, they
need to decide what mix of livestock and crops they should aim for
in the coming year. Their last harvest which was very successful
has provided them with a sum of £20,000 which they feel they can
use to buy more livestock. The cost of seed is sufficiently small
to be considered part of the routine expenses. The farm at present
has 30 cows valued at £35,000 and 2,000 hens valued at £5,000. They
wish to keep all this livestock and perhaps purchase more. Each cow
bought would cost £1,500 and each hen would cost £3. Over a year
the value of a herd of cows decreases by about 10% and the value of
a flock of hens will decrease by about 25% due to aging.
Each cow requires 2 acres of land for grazing and 10
person-hours of work per month while producing an annual net cash
income of £850. Hens do not require any acreage but need 0.05
person-hours a month and produce an annual net cash income of
£4.25. The chicken house can accommodate a maximum of 5,000 hens
and the milking parlour puts a limit of 42 on the number of
cows.
For each acre of crops planted, the following table gives the
number of person hours of work required in the first and second
halves of the year as well as an estimate of the crops net value in
either income or savings in livestock feed.
Table 5
Soyabean
Corn
Wheat
Winter/spring person hours
1
0.9
0.6
Summer/autumn person hours
1.4
1.2
0.7
Net value
£70
£60
£40
To provide feed, there should be at least 1 acre of corn for
each cow and at least 0.05 acres of wheat for each hen in the next
year’s flock. They estimate that they need £40,000 each year for
their living expenses and the basic running of the farm.
a) Formulate the equation of this linear programming.
b) Advise the farm family on how many livestock they should buy
and what acreage of the different crops they should plant to
maximise their net wealth at the end of the next year.
c) There are several issues they are seeking advice on. For
example:
(i) They are interested in how much they should be willing to
pay for any farmland that may become available to rent over the
coming winter.