Model a Rounded Variable

There might be cases where you need to model a variable var1 as the closest integer to another variable or parameter value. Essentially, a constraint like:

var1 = Round(value)

However, using the Round() function in a constraint is not allowed in a mixed integer program and results in an error message, especially if value is also a variable.

Constraint programming constraints cannot be used in combination with real valued variables, only with integer valued variables,
element valued variables, and activities.

We can avoid such errors by declaring var1 as an integer variable and using two auxiliary non-negative variables in a target constraint as shown below:

var1 = value + aux1 - aux2

Depending on the direction of your math program, you add (minimize) or subtract (maximize) the sum aux1+aux2 to the objective function. This ensures that only one of aux1 or aux2 will take a non-zero value.

The possible situations are:

  1. value is integer: aux1 = aux2 = 0

  2. Round(value) = Ceil(value): aux2 = 0 and aux1 = Ceil(value) - value

  3. Round(value) = Floor(value): aux1 = 0 and aux2 = value - Floor(value)


Let value = 2.3, whose closest integer value is 2. So we want var1 = 2. This is situation 3 from above, so our constraint will result in:

var1 = 2.3 + 0 - 0.3 = 2