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We have to give a mathematical model for the following problem.

There is a company with two locations. For the production of two different products, it uses one base material where only $20$ units are available per day. The first product is partly produced at the first location, and partly produced at the second location. The second product is only produced at the first location. The machines at the first location cannot work more than $12$ hours, and the machines of the second location cannot work more than $16$ hours.

For the production of the first product, $2$ units of the base material are used, for the second product $4$ units.

At the first location, you need $2$ hours for the first product and for the second product $2$ hours. At the second location, $4$ hours for first product and $0$ hours for second (thus it's not produced here).

For every piece of the first product, you get $2000$ USD and for every piece of the second product, you get $3000$ USD. How many pieces of the different products does the company have to produce each day, in order to achieve the maximum profit (by respecting all the constraints)?

I need to find the linear system of inequalities and the other constraints. Here is my solution but I'm not sure if it's alright?

$$ \begin{align} 2x+4y &\leq 12 && \mbox{Time constraint location 1}\\ 2x+0y &\leq 16 && \mbox{Time constraint location 2}\\ x,y &\geq 0 && \mbox{Non-negativity constraints} \end{align} $$ And the prices were also given, so here is the inequality for the profit:

$z(x,y) = 2000x + 3000y$

Is there anything I did wrong / missed?

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  • $\begingroup$ You forgot to include the constraint that models the limited availability of base material. The first product uses 2 units, the second product uses 4 units and there are 20 units available, thus $2x+4y \leq 20$. Furthermore, the time constraint for the first location is incorrect since the second product only takes 2 hours instead of 4. There is a similar issue with the time constraint for the second location. $\endgroup$ – Michiel Jun 23 '17 at 15:25

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