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Exercise. A forest is in flames and the government is planning a firefighter operation. The fire is of small dimensions and is progressing slowly and it must be extinct after $3$ hours of operations. There will be helicopters and planes mobilized to the forest. They obey the following table:

\begin{array}{|c|c|c|} \hline \text{Device} & \text{Efficiency $(m^2/hour)$} & \text{Cost (Euros/hour)} & \text{Personal Necessity}\\ \hline \text{Helicopter AH1} & 15000 & 2000 & 2 \text{ pilots}\\ \hline \text{Tank Airplane} & 40000 & 4000 & \text{2 pilots + 1 operator}\\ \hline \text{B67 Airplane} & 85000 & 10000 & \text{2 pilots + 3 operators}\\ \end{array} The area of the forest covered by fire is $3 000 000$ $m^2$. In the support areas, we have $14$ Airplane pilots available, $10$ Helicopter pilots available and $22$ operators available. Model the problem of minimization of the costs of this operation.

My attempt and doubts. I defined the following: \begin{equation*} x_1 \rightarrow \text{ number of activity hours of AH1} \\ x_2 \rightarrow \text{ number of activity hours of Tank} \\ x_3 \rightarrow \text{ number of activity hours of B67} \end{equation*} And I defined the following restrictions: \begin{equation*} 1500x_1 +40000x_2 + 85000x_3 \leq 3000000 \\[.25cm] 0 \leq x_{1,2,3} \leq 3 \end{equation*} The first one is related to the forest area covered by the fire and the second one is related to the hours the operation must hold. I think these are the only restrictions I should take related to these variables. I defined the objective function as the follwing: \begin{equation*} z(x_1,x_2,x_3) = 2000x_1 + 4000x_2 + 10000x_3 \end{equation*} And the objective would be to minimize this function. Now come my problems:

I wasn't able to define restrictions realted to the number of pilots/operators needed. I have thought about adding new variables to do so, but been unable to (somehow everytime I try it doesn't make sense to me). So this is basically why I am posting this, to know how to model the pilots/operators restrictions. Thanks for all the help in advance.

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In my view we have to fix the operation time at $3$ hours in this model. I don´t see any other way. Then I define the variables in a different way:

\begin{equation*} x_1 \rightarrow \text{ number Helicopters AH1} \\ x_2 \rightarrow \text{ number of Tank Airplanes} \\ x_3 \rightarrow \text{ number of B67 Airplanes} \end{equation*}

Then in 3 hours each of the 3 different aircraft types can cover the following areas:

\begin{equation*} \text{ Helicopter AH1: 15000*3=45000} \\ \text{ Tank Airplane: 40000*3=120000} \\ \text{ B67 Airplane: 85000*3=255000} \end{equation*}

The the area-restriction is

$$45000\cdot x_1+120000\cdot x_2+255000\cdot x_3=3000000$$

The cost per hour has to be multiplied by $3$ as well to obtain the cost per aircraft.

And the restriction for the airplane pilots would be $2x_2+2x_3\leq 14$, for instance. Btw, defining variables is the best what you can do in linear programming. Good job, especially in this regard.

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    $\begingroup$ Hi @callculus42 . Sorry for the delay, I have solved the problem using your solution. Thank you for the help, made a lot of sense to me fixing the $3$ hours! I guess that's what I was missing :D $\endgroup$ Commented Nov 12, 2021 at 18:25
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    $\begingroup$ @Seabourn I can comprehend your confusion. In my view the exsercise isn´t very clear formulated.. $\endgroup$ Commented Nov 12, 2021 at 18:32

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