In Linear Programming (LP), a shadow price represents the marginal change in the objective function value per unit increase in a constraint's right-hand side (RHS), assuming all other input parameters stay constant.

When multiple constraints change simultaneously—or when changes exceed allowable ranges—standard single-variable sensitivity analysis breaks down. Operations Research addresses these scenarios through specific analytical tools and limits.

Sensitivity analysis is useful because real-world conditions are rarely fixed. Costs, profits, demand, and available resources can change.

When multiple constraints change at the same time, we can use the 100% rule to check whether the combined changes stay within the allowable ranges. The rule is reliable only when the total percentage of changes is 100% or less; if it exceeds 100%, the current optimal solution may change, so the problem should be solved again to get the new optimum.

You handle simultaneous constraint changes using the 100% rule, which sums the percentage changes of each resource to ensure the total does not exceed one. This rule stops being reliable when individual changes exceed their allowable limits, which can cause the optimal basis to shift or introduce non-linear behavior.

When multiple constraints change at the same time, you can use the 100% rule to check whether the combined changes remain within the range where the current optimal basis and shadow prices are expected to remain valid. For each RHS change, calculate the fraction of its allowable increase or decrease that is being used, then add the fractions; if the total is ≤ 100%, the current basis is generally expected to remain valid. If the total exceeds 100%, the 100% rule cannot guarantee that the current basis or shadow prices remain valid, so the LP should be re-solved. In practice, re-solving is also preferable whenever the changes are large, the model is highly sensitive, or the assumptions behind the sensitivity report no longer hold.

To handle real-world situations where multiple constraints change at once, you add up each constraint's relative change calculated by dividing the actual change by its maximum allowable change to see if the total percentage stays at or below 100%, which lets you safely sum their individual shadow price impacts to find the new total profit or cost. The 100% Rule stops being reliable as soon as this combined total exceeds 100%, because such large shifts reshape the feasible region and force the optimal solution to a new corner point (basis), altering the shadow prices and requiring you to re-solve the linear programming model.