Let $$z_1 = 3x_1 + x_2 + x_3$$
$$z_2 = 2x_1 + 5x_2 + 2x_3$$
$$z_3 = x_2 + 7x_3$$
There exists a variable $w$ defined as: $$w = \min(z_1, z_2, z_3)$$
Also, the $x$-variables are subject to the following restrictions:
$$x_1 + 0.35x_2 + .5x_3 \leq 100$$
$$3x_1 + 2x_2 + 0.35x_3 \leq 360$$
$$2x_1 + 5x_2 + 35x_3 \geq 500$$
$x_1$, $x_2$, and $x_3$ are real and positive
(i.e. Find the values that minimize the maximum of the three linear expressions ($z_1$, $z_2$, and $z_3$) which are subject to three additional restrictions)
a) Transform the problem stated above into one of linear programming. (Hint: We are looking for the smallest w that is greater than $z_1$, $z_2$, and $z_3$)
I am really lost. Can someone please explain this problem to me? I am sorry that I don't know how the format for the subscripts
Since this looks like a homework question, I can give you hints, and if you still don't get it, I can give more hints...
Based on the first 3 equations: $${z_1 = 3x_1 + x_2 + x_3}$$ $${z_2 = 2x_1 + 5x_2 + 2x_3}$$ $${z_3 = x_2 + 7x_3}$$ Can you rewrite the equations to get $x_i$ in terms of $z_i$? Something along the lines of $${x_1 = az_1 + bz_2 + cz_3}$$ where $a, b, c$ are real constants for all three $x_i$.
Once you do that, you can just substitute $x_i$ in terms of $z_i$ in the inequalities. Something along the lines of: $${k_1z_1 + k_2z_2 + k_3z_3 \leq 100}$$ Where $k_1,k_2,k_3$ are real constants.
Does that look similar to a linear programming question where you are minimizing $w$?