Distinguishing Real Sensitivity Analysis from Re-Solving Under Perturbations

Real sensitivity analysis uses the data in a solver's sensitivity report to predict how the optimum changes — without re-solving the LP. This shortcut is only valid when the input change stays within the allowable range given by the report. Push beyond that range and the optimal basis can change; the report's predictions no longer apply, and you must re-solve the LP from scratch.

For an objective coefficient $c_j$ on decision variable $x_j$, the report lists an allowable increase $\Delta c_j^+$ and an allowable decrease $\Delta c_j^-$. As long as the new coefficient

$$c_j' \in [\, c_j - \Delta c_j^-,\ c_j + \Delta c_j^+ \,],$$

the optimal decision values $x_1^*, x_2^*, \ldots$ stay the same, and the new objective value is

$$Z' = Z + (c_j' - c_j)\, x_j^*.$$

If $c_j'$ leaves this range, the optimal basis may shift. Shadow prices, reduced costs, and the optimum itself may all change — you must re-solve.

For example, if $c_1 = 8$ has allowable increase $2$ and allowable decrease $3$, the allowable range is $[5, 10]$. A change to $c_1' = 9$ is within range; a change to $c_1' = 12$ is not.