The Sensitivity Report

The Sensitivity Report provides classical sensitivity analysis information for both linear and nonlinear programming problems, including dual values (in both cases) and range information (for linear problems only). The dual values for (nonbasic) variables are called Reduced Costs in the case of linear programming problems, and Reduced Gradients for nonlinear problems. The dual values for (binding) constraints are called Shadow Prices for linear programming problems, and Lagrange Multipliers for nonlinear problems.

Dual Values for Nonlinear Models

An example of a Sensitivity Report generated for EXAMPLE1.XLS when the Assume Linear Model box is not checked is shown below. Note that it includes only the solution values and the dual values: Reduced Gradients for variables and Lagrange Multipliers for constraints.

Constraints which are simple upper and lower bounds on the variables that you enter in the Constraints list box of the Solver Parameters dialog are treated specially (for efficiency reasons) by both the linear and nonlinear Solver algorithms, and will not appear in the Constraints section of the Sensitivity report.

Note: The formatting of the Final Value, Reduced Gradient and Lagrange Multiplier cells in the report is "inherited" from the corresponding cells in the Solver model. In the example above, the cells in the model were formatted to display as integers (0 decimal places), so the entries in the report are formatted the same way. If you select the cell displayed as -2 (the Reduced Gradient for Speakers), you'll see that the actual value is -2.4999... which has been rounded down to -2. Bear this in mind when designing your model and when reading the report. Since the report is a worksheet, you can always change the cell formatting with the Format menu.

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 above, the dual value for producing speakers is -2, meaning that if we were to build one speaker (and therefore less of something else), our total profit would decrease by $2 (about $2.50 without the rounding due to cell formatting).

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.

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.

Dual Values and Ranges for Linear Models (14K)

Copyright © 1996 Frontline Systems Inc.
Last modified: December 01, 1996