I'm practicing for a linear programming test and here is a task I like to see if I did it correct and if not maybe how to do it correctly? Need to create a mathematical model whose requirements are represented by linear relationships.
A company wants produce two different kind of cars, $A$ and $B$. The production process is set up by these two things: installation and finishing. For the installation of $A$, the company needs $4$ hours and for the installation of $B$ they need $6$ hours. The finishing requires $6$ hours for car $A$ and $3$ hours for $B$. The profit for each car $A$ is $4000$ USD and for $B$ is $3000$ USD.
Dependent on other projects, the company has $720$ hours time for the installation and $480$ hours for the finishing within a production-cycle. The management requests for the duration of a cycle at least $20$ cars of kind $A$ and at least $30$ cars of kind $B$.
How many cars of each kind the company needs to produce within one cycle such that every (production-)condition is unhurt, the requirements of the management satisfied and the maximum profit is achieved?
Due to the long text, I like to keep it as short as possible. I have the following model:
$4000x_1 +3000x_2 \rightarrow \min$
$4x_1+6x_2 \geq 720$
$6x_1+3x_2 \geq 480$
$x_1 \geq 20$
$x_2 \geq 30$
The company wants to maximize the profit: $4000x_1 +3000x_2 \rightarrow \color{red}{\max}$
$4x_1+6x_2 \color{red}{\leq} 720$
$6x_1+3x_2 \color{red}{\leq} 480$
The other two constraints look O.K.
$x_1 \geq 20$
$x_2 \geq 30$
The linear programming can be solved graphically or with Simplex algorithm.