Dual Values and Ranges for Linear Models

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 models includes this range information.

A Sensitivity Report for EXAMPLE1.XLS when the Assume Linear Model box is checked is shown below. In addition to the dual values (Reduced Costs for variables, Shadow Prices for constraints), this report provides information about the range over which these values will remain valid.

Note: The formatting of the Final Value, Reduced Cost and Objective Coefficient 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 these entries are formatted the same way. If you select the cell displayed as -2 (the Reduced Cost 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 and Range Information

The dual values (Reduced Costs and Shadow Prices) have the same interpretation for linear models as they have for nonlinear models (see Interpreting Dual Values). In the example above, the Reduced Cost 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 Shadow Price for electronics units is 25, so an extra electronics unit would allow the Solver to increase total profit by $25.

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. In the example below, we'd still build 200 TV sets even if the profitability of TV sets decreased up to $5 per unit (beyond that point, or if the unit profit of speakers increased by more than $2.50, we'd start building speakers).

For each constraint, the report shows the constraint right hand side, and the amount by which the "R.H. Side" could be increased or decreased without changing the dual value. In this example, we could use up to 50 more electronics units, which we'd use to build more TV sets instead of stereos, increasing our profits by $25 per unit. Beyond 650 units, we would switch to building speakers at an incremental profit of $20 per unit (a new dual value).

A value of 1E+30 in these reports represents "infinity:" For example, we are currently not building any speakers, and we would continue to build no speakers regardless of how much the unit profit of speakers was decreased.

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