I'm not sure how to determine the range for which a shadow price is valid.
You might be able to skip straight to the question here.
I've been introduced to it using the following approach in 2D.
Context
Given an optimal solution which is at a corner, there are two intersecting lines. Say that these lines represent the following inequalities
\begin{equation} \begin{aligned} 2K + 3S & \leq 10 \\ K + 2S & \leq 6 \\ Z & = 30K + 50S \end{aligned} \end{equation}
Then the shadow price is the change in the objective function, $Z$, when the right hand side of an inequality is changed by one unit.
Computing this for the second equation is done as follows:
\begin{equation} \begin{aligned} 2K + 3S & \leq 10 \\ K + 2S & \leq 6 + \Delta\\ \end{aligned} \end{equation}
Then subtracting the second equation from the first twice
\begin{equation} \begin{aligned} S & = 2 + 2 \Delta \end{aligned} \end{equation}
Therefore
\begin{equation} \begin{aligned} K &= 2 - 3 \Delta \end{aligned} \end{equation}
And the objective function can be written as
\begin{equation} \begin{aligned} 30K + 50S &= 30(2 - 3 \Delta) + 50(2 + 2\Delta) \\ &= 160 + 10 \Delta \end{aligned} \end{equation}
And the shadow price is found from $z(1) - z(0)$, where $z(\Delta) = 160 + 10 \Delta$ which gives $10$.
Therefore the shadow price is 10.
But how do I compute the range for which this is valid? I can get through the algebra here pretty easily, but there's little meaning to it.
Question
How do I determine the range for which I can adjust the second inequality such that the difference made to the objective function is $10 \times \text{adjustment}$.