Q:

Sensitivity analysis in linear programming

Could someone please explain in detailed steps how to apply a sensitivity analysis to such problem:
$$maximize \ \ 2x_1 + 3x_2 \\
s.t. \ \ 4x_1+3x_2≤600 \\ 2x_1+2x_2≤320 \\ 3x_1+7x_2≤840 \\ x_i≥0$$
The goal is it to determine the boundaries of $x_2$.

A:

To conduct a sensitivity analysis the final tableau is needed. 
Basis   x1     x2     s1      s2      s3     RHS
x3      0       0      1     -2.375   0.25   50
x1      1       0      0      0.875  -­0.25   70
x2      0       1      0     -­0.375   0.25   90
z       0       0      0      0.625   0.25   410

The variables $x_1,x_2$ and $x_3$ are in the basis. Now it can be analyzed under which condition $x_2$ remains in the basic.
Lower bound for the parameter of $x_2$:
We take the reciprocal of every positive value of the non-basic variables in the $x_2$-row at multiply them by the corresponding objective function values. Then we take the minimum of it. In this case we have only one fraction:
$min\left( \frac{0.25}{0.25} \right)=\frac{0.25}{0.25}=1$
If there would be no positive values the lower bound would be $-\infty$.
Upper bound for the parameter of $x_2$:
We take the reciprocal of every negative value of the non-basic variables in the $x_2$-row at multiply them by the corresponding objective function values. Then we take the maximum of it. In this case we have only one fraction as well:
$max\left( \frac{0.625}{-0.375} \right)=\frac{0.625}{-0.375}=-\frac53$
If there would be no negative values the upper bound would be $\infty$.
To get the final lower/upper bound we take the value of the parameter of the initial tableau (objective function) and substract the identified values above. 
Lower bound $3-1=2$
Upper bound $3-(-\frac53)=3+\frac53=4\frac23$
For further information see here, especially page 10 and 11. If you have any question about it feel free to ask.

