$($a$)$ Adam, a farmer, is wondering which crops he should plant in the upcoming

season. He can grow wheat, rye and barley on his $4000$ acres of farmland.

He uses only organic fertilizers on his farm. He estimates that a maximum of

$10000kg$ of organic fertilizers could be procured for the upcoming season.

Comparison of wheat, rye and barley is as follows:

Write down a linear programming model to maximise Adam's profit.

$($b$)$ Find the optimal value of the objective function of the following LP problem

by directly solving the dual.

maxz$=,3x_1-12x_2-2x_3,$

$s.t.-x_1+2x_2-2x_3,-20$

$,x_1,x_2,x_3,0.$

$($c$)$ Write down an LP in standard form in vector$-$matrix notation. Are the entries

of $b,$ the right$-$hand side of the constraints, allowed to be negative?

$($d$)$ Explain in words how does the memetic algorithm given in the lecture notes

and lecture slides work. What is the difference between genetic and memetic

algorithms?