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How can I linearize $\min(x_1,x_2,x_3)$ in a maximization linear programming problem? Please help me. I've tried many things but I didn't solve.. My LP equations are as follows:

Objective function is: maximize $z=\min(x_1,x_2,x_3)$

Constraints:

$0.6 x_1 + 0.8 x_2 \leq 4500 \times 20$

$0.2 x_2 \leq 4500 \times 3$

$0.3 x_2 \leq 4500 \times 6$

$0.4 x_1 + 0.6 x_2 \leq 4500 \times 10$

$0.3 x_1 \leq 4500 \times 5$

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    Don't you know how to linearize z=min(x1,x2,x3) ? I do not need the solution of the problem but the linearization of z=min(x1,x2,x3) my friend ? Thank you for your care.2012-12-09

2 Answers 2

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How you implement min($x_1, x_2, x_3$) in an LP solver depends on what you are trying to do with it. Since you are maximizing it you can do the following

$ {\rm maximize } \ z $ subject to $ z \le x_1 \\ z \le x_2 \\ z \le x_3 $ $z$ will take the value of min($x_1, x_2, x_3$) in any optimal solution.

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    Shouldn't z consist of x1,x2 or x3 in an LP? I thought that situation but I didn't implement this solution's code.2012-12-09
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assume $y=\min\{x_1,x_2,x_3\}$

Now we need to maximize $y$

since value of $y$ can either be $x1$, $x2$ or $x3$, we can safely add the following constraints:

$x_1\geq y \qquad x_2 \geq y \qquad x_3\geq y$

or

$x_1-y\geq 0 \qquad x_2-y \geq 0 \qquad x_3-y\geq 0$

now solve the problem.

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    Viraj, welcome to Math.SE. You will find that [MathJax](http://math.stackexchange.com/help/notation) and $\LaTeX$ can be used to post mathematical expressions.2015-11-21