$6\mathrm{.}6$ Exercises

Exercise $6\mathrm{.}1.$ A campsite owner, Brown, wants to expand his site and therefore needs to persuade a neighbouring small farmer, Smith, to both rent him $30$ acres of land and to work

for him $20$ weeks a year. He wants to determine the prices $($rent of the land in $$ per acre and wages of Smith in $$ per man$-$week, $x_1$ and $x_2,$ respectively$)$ he should offer in order to minimise his total payment to Smith which is $30x_1+20x_2.$ The constraints which limit his cost minimisation are the opportunities which Smith has to use the land and labour himself. In other words, the prices of rent and labour must be such as to ensure that Smith cannot earn more revenue $($by using the land and labour necessary to produce any of the three different crops that are available to him$)$ than by selling them to Brown. The data for the crops available to Smith are shown below.

$\backslash$table$[[$Crop$,\backslash$table$[[$Revenue$],[($ per tonne$)]],\backslash$table$[[$Land requirement$],[($acres per tonne$)]],\backslash$table$[[$Labour requirement$],[($man$-$weeks per tonne$)]]],[1,7,10,3],[2,5,6,2],[3,3,2,2]]$

$($a$)$ Formulate and solve Brown's problem as an LP$.$

$($b$)$ Formulate and solve a problem dual to $($a$).$ How can you interpret this LP$?$