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means that only values of $x$ that are greater than or equal to the $y$ values are allowed.
means that only $x$ values which are less than or equal to 20 are allowed.
Once we have determined the feasible region the solution of our problem will be the feasible point where the objective function is a maximum / minimum. Sometimes there will be more than one feasible point where the objective function is a maximum/minimum — in this case we have more than one solution.
A simple problem that can be solved with linear programming involves Mrs Nkosi and her farm.
Mrs Nkosi grows mielies and potatoes on a farm of 100 m ${}^{2}$ . She has accepted orders that will need her to grow at least 40 m ${}^{2}$ of mielies and at least 30 m ${}^{2}$ of potatoes. Market research shows that the demand this year will be at least twice as much for mielies as for potatoes and so she wants to use at least twice as much area for mielies as for potatoes. She expects to make a profit of R650 per m ${}^{2}$ for her mielies and R1 500 per m ${}^{2}$ on her potatoes. How should she divide her land so that she can earn the most profit?
Let $m$ represent the area of mielies grown and let $p$ be the area of potatoes grown.
We shall see how we can solve this problem.
You will need to be comfortable with converting a word description to a mathematical description for linear programming. Some of the words that are used is summarised in [link] .
Words | Mathematical description |
$x$ equals $a$ | $x=a$ |
$x$ is greater than $a$ | $x>a$ |
$x$ is greater than or equal to $a$ | $x\ge a$ |
$x$ is less than $a$ | $x<a$ |
$x$ is less than or equal to $a$ | $x\le a$ |
$x$ must be at least $a$ | $x\ge a$ |
$x$ must be at most $a$ | $x\le a$ |
Mrs Nkosi grows mielies and potatoes on a farm of 100 m ${}^{2}$ . She has accepted orders that will need her to grow at least 40 m ${}^{2}$ of mielies and at least 30 m ${}^{2}$ of potatoes. Market research shows that the demand this year will be at least twice as much for mielies as for potatoes and so she wants to use at least twice as much area for mielies as for potatoes.
There are two decision variables: the area used to plant mielies ( $m$ ) and the area used to plant potatoes ( $p$ ).
Write the following constraints as equations:
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