I'm doing a linear program to a game and I don't know how to express $\forall$ in linear programing (or if I had the right intuition to do it).
Here is the problem:
I have several vessels that are going to attack several rocket launchers: I want to minimize the fuel consumption but still have enough weaponry capacity to destroy the rocket launchers.
\begin{array} & & \mbox{Light Fighter} & \mbox{Heavy Fighter} & \mbox{Cruiser} & \mbox{Battleship}\\ \mbox{Weaponry} & 65 & 195 & 520 & 1300\\ \mbox{Defence} & 6015& 15037,5 & 40575 & 90300\\ \mbox{Consumption} & 20 & 75 & 300 & 500\\ \end{array}
$d_i$ being the number of rocket launchers.
- I want that for each rocket launcher it exists a vessel (or set of vessels) that have individually more weaponry than the sum of hull and the shield of the rocket launcher.
That is to say:
$$\forall y\in RL,\exists x_i\in V, \underbrace {\sum\alpha_i x_i}_\mbox{One or several vessels} >(H+S),\alpha_i\in W$$
$S=30, H=3000$
- I also want that for each vessel or set of vessels, the sum of hull and the shield be greater or equal than the sum of the weaponry of the worst case: each rocket launchers attacks the set of vessels (in reality the worst case would have been that each rocket launcher attacks one vessel alone, I think).
Therefore would the linear program be:
\begin{cases} \min & 20x_1 &+ 75x_2 & +300x_3&+500x_4\\ &65x_1&+195x_2&+520x_3&+1300x_4&>3030\\ & 6015 x_1 &+ 15037,5x_2 &+ 40575 x_3 &+ 90300x_4 &> 104*d_i \end{cases}
Therefore, I think I'm not quite sure of my gait, I'm trying to find a set of vessels minimizing their fuel conumption, able to destroy at least one rocket launcher but also able to support concentrated fire on it in order to be able to find an optimal set for all rocket launcher I would had to face.
Am I not missing something when trying to generalize with the last line of the linear program?