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 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.

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.

If you are not accustomed to analyzing sensitivity information for nonlinear problems, 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 a range (see below).

Interpreting Range Information

In linear programming problems (unlike nonlinear problems), the dual values are constant over a range of possible changes in the objective function coefficients and the constraint right hand sides. The Sensitivity Report for linear programming problems includes this range information.

For each decision variable, the report shows its coefficient in the objective function, and the amount by which this coefficient could be increased or decreased without changing the dual value. For each constraint, the report shows the constraint right hand side, and the amount by which the RHS could be increased or decreased without changing the dual value.

Interpreting the Limits Report

The Limits Report provides a specialized kind of “sensitivity analysis” information. It is created by re-running the Solver model with each decision variable in turn as the objective (both maximizing and minimizing), and all other variables held fixed. Hence, it shows a “lower limit” for each variable, which is the smallest value that a variable can take while satisfying the constraints and holding all of the other variables constant, and an “upper limit,” which is the largest value the variable can take under these circumstances.

Understanding the Linearity Report

The purpose of the Linearity Report is to help you pinpoint nonlinear formulas in your model. The format of the Linearity Report is similar to that of the Answer Report: It lists each decision variable and constraint on a separate row, with its cell reference, a “name” as described for the Answer Report, the cell’s original and final values, and a column containing “Yes” (the objective or constraint is a linear function, or the variable occurs linearly throughout the model) or “No” (the function is nonlinear, or the variable occurs nonlinearly). Since you are normally interested in the nonlinearities, any “No” entries appear in boldface.

If your objective or constraints are computed through a chain of formulas in different cells that ultimately depend on the decision variable cells, you may want to use Excel’s auditing features to trace the dependents of your formula cells and find the point where you’ve introduced a nonlinear dependence. If you multiply or divide two quantities that both depend on decision variables, the result is nonlinear. Excel functions other than SUM, SUMPRODUCT and selected other cases will compute a nonlinear or non-smooth function of the variables. For more information, see the topic Linear Functions.

Once you identify specific formulas that are nonlinear, you should determine whether they are correct for your problem, and decide whether they can be rewritten as linear functions, or whether there is an alternative, linear formulation of your problem. If you can formulate the model as a linear programming problem, you’ll have the benefit of faster and more reliable solutions – especially if you also have integer restrictions on decision variables. Otherwise, you’ll have to select the GRG Nonlinear or Evolutionary Solving method to solve your problem.