Operations Research Models and Methods / Computation / Mathematical Programming /
Linear/Integer Programming

Two Limits for Constraints

The LP Dialog has an alternative model form that provides two limits for the constraint values. That form is illustrated for the example problem in the image below. Note that the upper bound for each constraint is the machine hour limit as before. We use a lower bound of 0 for the constraints. Since all constraint coefficients are nonnnegative, the lower bounds are clearly redundant for this problem. In general, the student should be wary of using 0 for lower bounds that are to be redundant as negative constraint coefficients may make the bounds effective rather than redundant. The default lower bound is a large negative number, while the default upper bound is a large positive number.

All constraint types are possible with this two limit form. A (<=) constraint has a large negative number as the lower limit. A (>=) constraint has a large positive number as the upper limit. An (=) constraint has the same number as both the lower and upper limits. In some situations both lower and upper constraint bounds are relavant and this format is very convenient. One advantage of this model format is the the Excel Solver model always has just four constraint lines, two providing the upper and lower bounds on the variable values and two providing the upper and lower bounds on the constraint values.

Numerically, the two limit form has a disadvantage for the Excel Solver because it provides separate constraints for the lower and upper bound limit. This is shown in the sensitivity analysis obtained from the Excel Solver and shown below. Note that the first four constraints are the upper bound constraints, while the last four are the lower bound constraints. The lower bounds are obviously loose and are redundant for this problem. The Excel Solver also sometimes falsely indicates nonlinearities in the model when large magnitude numbers are used in the limits. For a limit that indicates are redundant bound, use numbers that has the smallest absolute value that will guarantee that the bound will not be effective.

The Jensen LP/IP Solver handles constraints with two limits in a different manner. Rather than using two explicit constraints, it bounds the slack variable for a single constraint. The sensitivity analysis for the two limit problem essentially neglects the lower limits and is the same as the one limit model.