An airline company wants to install an air connection between two cities. Their goal is to transport $16000$ people and $96$ tons of cargo in a specific interval of time. At the moment, two types of airplanes are available: $11$ airplanes of type $A$ and $8$ airplanes of type $B$. Type $A$ costs $4000$ € and can carry $200$ people, and $6$ tons cargo per flight. Type $B$ costs $1000$ € and can carry $100$ people, and $15$ tons cargo per flight.
How many airplanes of each type will the airline company use, in order to minimize their costs (of course, by also respecting the constraints)?
We are supposed to determine the system of linear inequalities (and then continue calculating the best possible solution). But I would like to know only if the inequalities I found are correct, and maybe some other important things that need to be taken from the question?
I have started by creating a table:
| type A | type B | total
------------------------------------------
people | 200 | 100 | 1600
------------------------------------------
cargo | 6 | 15 | 96
------------------------------------------
costs | 4000 | 1000 |
So we have as constraints:
$I: 200x + 100y \geq 1600$
$II: 6x + 15y \geq 96$
(Also we might need to keep in mind that we don't want negative solution, so we make sure we also set $x,y \geq 0 ?)$
I think that's all. Or did I forget something?
You might want to add the constraints $$0 \le x \le 11 \mbox{ and } 0 \le y \le 8$$ because you cannot use more planes than available.
The objective function is the total cost, but you asked only about constraints (and I'd like to respect your request).