Maximizing Industry Profit w/ Limited Resources

[evinda]

Hello! (Wave)

An industry produces 2 kinds of a product $k_1$ and $k_2$ at its 3 factories $F_1, F_2, F_3$.

At the following matrix we can see the hours that are needed from each factory for the production of one kilogram of $k_1$ and $k_2$.

$\begin{matrix}
& F_1 & F_2 & F_3\\
k_1 & 2 & 3 & 1\\
k_2 & 3 & 2 & 2
\end{matrix}$

The profit per kilogram is $300$ euros for $k_1$ and $250$ for $k_2$.
The factory $F_1$ is able to work till $16$ hours, $F_2$ till $18$ hours, $F_3$ till $10$ hours.
I want to find how many kilograms of each product have to be produced the day so that the profit of the industry is maximized.So do we have to find an objective function with some restrictions or do we use dynamic programming? (Thinking)

 

[I like Serena]

So do we have to find an objective function with some restrictions or do we use dynamic programming? (Thinking)

Hi evinda! (Smile)

We have to find an objective function and some additional restrictions.
And then we have to use linear programming to solve it (the simplex method). (Nerd)

 

[evinda]

Hi evinda! (Smile)

We have to find an objective function and some additional restrictions.
And then we have to use linear programming to solve it (the simplex method). (Nerd)

I think that the problem that we get is the following:

$$\max (300 x_1+ 250x_2) \\ 2x_1+3x_2 \leq 16 \\ 3x_1+2x_2 \leq 18 \\ x_1+ 2 x_2 \leq 10$$

Am I right? (Thinking)

 

[I like Serena]

Yep.

 

[evinda]

Yep.

We suppose that $x_1$ represents the kilograms of the product $k_1$ and $x_2$ represents the kilograms of $k_2$, right? (Thinking)

Do I have to explain the relations that we get further? (Thinking)

 

[evinda]

We get that in order to have the maximum possible profit we have to produce $\frac{22}{5}$ kilograms of $k_1$ and $\frac{12}{5}$ of $k_2$, right?