Sensitivity Range for an Objective Coefficient

The sensitivity range (or range of optimality) for an objective coefficient $c_j$ is the interval of values $c_j$ can take, with all other data held fixed, such that the optimal vertex of the LP does not change.

For a two-variable maximization LP, the optimum sits at a corner where two constraints are binding. The optimal vertex stays optimal as long as the slope of the objective contour line stays between the slopes of those two binding constraints.

If the binding constraints have slopes $m_1$ and $m_2$ with $m_1 \le m_2$, then the optimal vertex remains optimal precisely when

$$m_1 \;\le\; -\dfrac{c_1}{c_2} \;\le\; m_2.$$

Illustration. Suppose the two binding constraints at the optimum have slopes $-3$ and $-\tfrac{1}{2}$, and the objective is $Z = c_1 x_1 + 2 x_2$. With $c_2 = 2$ fixed, the objective slope is $-c_1/2$, so the optimal vertex stays optimal as long as

$$-3 \;\le\; -\dfrac{c_1}{2} \;\le\; -\dfrac{1}{2}.$$

Multiplying through by $-2$ (and flipping the inequalities) gives

$$1 \;\le\; c_1 \;\le\; 6.$$

The sensitivity range for $c_1$ is $[1, 6]$.