How to make a variable be of "continuous type" but only allow (positive) integer values as its solution

Hi Paul,

integer variables cannot have a reduced cost, by their nature. In a way the reduced cost measures the change of objective per unit change of the variable, in other words a derivative. That you cannot have for a discrete variable.

The simplest case is when your problem is guaranteed to have integer optimal solutions (TU matrix). In that case you can just drop the integrality and obtain the reduced costs. In general, you can always try to drop the integrality and hope the solutions turn out to be integer. If so, your problem is solved.

Another thing that people commonly do is to solve the integer problem to optimality, then fix the integer variables to their optimal values and then resolve the problem. This way you can at least get reduced costs for the other variables, but still not for the integer ones.

If you want to get a measure similar to a reduced cost for an integer variable you need to perform sensitivity analysis: change the variable by one in each direction and observe how the optimal objective value is changing.

Without knowing your problem my assumption is that your task involves some sensitivity analysis, for which you learned to use the reduced cost in the continuous case, but your problem is integer. In this case you'll have to do the sensitivity analysis manually.

In summary, try to focus on why you think you need the reduced costs and what other ways are there to get the same information without reduced costs.

I hope this helps,

Imre Polik

FICO