Introduction to Sensitivity Analysis (Post-Optimality) in Linear Programming
Discover how to perform sensitivity analysis to determine how changes in constraints and objective functions affect your optimal solution.
What is Sensitivity Analysis?
Sensitivity Analysis (or Post-Optimality Analysis) is the study of how changes in the parameters of a linear programming model affect the optimal solution. Instead of resolving the entire problem from scratch every time a parameter changes, sensitivity analysis allows us to use the final Simplex tableau to determine the impact.
1. Shadow Prices (Dual Values)
The Shadow Price of a constraint is the rate of change in the objective function value per unit increase in the right-hand side (RHS) of that constraint.
For example, if the shadow price of a "labor hours" constraint is $15, it means acquiring one additional hour of labor will increase total profit by $15. In the final Simplex tableau, shadow prices are found in the objective row under the slack/surplus variables corresponding to the constraints.
2. Changing the Objective Function Coefficients
Sensitivity analysis helps us determine the range of optimality for an objective function coefficient. This is the range of values for which the current optimal basis remains optimal.
If the profit of a product changes, the quantities to produce (the basic variables) might remain the same as long as the change falls within this allowable range.
3. Changing the Right-Hand Side (RHS)
Similarly, we can calculate the range of feasibility for a constraint's RHS. This defines how much the availability of a resource can change before the current basis becomes infeasible (i.e., a basic variable becomes negative).
Conclusion
In real-world business scenarios, parameters like costs, prices, and resource availability are constantly fluctuating. Sensitivity analysis transforms a static optimal solution into a dynamic tool for strategic decision-making.