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Let's consider a simple linear programming problem:

$\min \text{cost} = ax+by+dz$

subject to:

$a+b+c \leq K$

My question is: what if the I have to optimize the cost function several times, each time with a different values of $K$?

The value of $K$ cannot be adjusted, and it varies with the time, so it's not actually a variable that can be put in the cost function. The only values which can be controlled are the variables $a$, $b$ and $c$.

I usually optimize the problem many times, each time changing the value of $K$, but I was wondering if there a specific method to use when the constraints change during time. Is there anyone who know that?

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Yes, there are methods for solving linear programming problems with coefficients that change according to a varying parameter. See "parametric linear programming" in many textbooks on LP.

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