Interpreting Dual Values

Dual values are the most basic form of sensitivity analysis information. The dual value for a variable is nonzero only when the variable's value is equal to its upper or lower bound (usually zero) at the optimal solution. This is called a nonbasic variable, and its value was driven to the bound during the optimization process. Moving the variable's value away from the bound will worsen the objective function's value; conversely, "loosening" the bound will improve the objective. The dual value measures the increase in the objective function's value per unit increase in the variable's value. In the example Sensitivity Report below, the dual value for producing speakers is -2.499, meaning that if we were to build one speaker (and therefore less of something else), our total profit would decrease by $2.50.

The dual value for a constraint is nonzero only when the constraint is equal to its bound. This is called a binding constraint, and its value was driven to the bound during the optimization process. Moving the constraint left hand side's value away from the bound will worsen the objective function's value; conversely, "loosening" the bound will improve the objective. The dual value measures the increase in the objective function's value per unit increase in the constraint's bound. In the example Sensitivity Report below, increasing the number of electronics units from 600 to 601 will allow the Solver to increase total profit by $25.

An example of a Sensitivity Report generated for the Product Mix problem when the Solver "engine" is the nonlinear GRG solver is shown below. Note that it includes only the solution values and the dual values: Reduced Gradients for variables and Lagrange Multipliers for constraints. If you solve a quadratic programming problem with the LP/Quadratic "engine" in either the Quadratic or Large-Scale LP Solvers, the report will also appear in this format.

If you are not accustomed to analyzing sensitivity information for nonlinear problems, you should bear in mind that the dual values are valid only at the single point of the optimal solution -- if there is any curvature involved, the dual values begin to change (along with the constraint gradients) as soon as you move away from the optimal solution. In the case of linear problems, the dual values remain constant over the range of Allowable Increases and Decreases in the variables' objective coefficients and the constraints' right hand sides, respectively.