### 3.7.25. Balanced assignment

• $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$,

• $\mathrm{\pi \pi \pi \pi \pi \pi }$.

A constraint to obtain a balanced assignment over a set of domain variables. Given a set of domain variables $\left\{{x}_{1},{x}_{2},...,{x}_{n}\right\}$, some classical balance criteria reported in

• The maximum value, i.e.,Β the maximum value over ${x}_{i}$ $\left(i\beta \left[1,n\right]\right)$ can be modelled with a $\mathrm{\pi \pi \pi ‘\pi \pi \pi \pi }$ constraint.

• The maximum deviation, i.e.,Β the maximum value over ${x}_{i}-\frac{{\beta }_{j\beta \left[1,n\right]}{x}_{j}}{n}$ $\left(i\beta \left[1,n\right]\right)$.

• The total deviation, i.e.,Β ${\beta }_{i\beta \left[1,n\right]}\left|{x}_{i}-\frac{{\beta }_{j\beta \left[1,n\right]}{x}_{j}}{n}\right|$ can be modelled with a $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint

• The total quadratic deviation, i.e,Β ${\beta }_{i\beta \left[1,n\right]}{\left({x}_{i}-\frac{{\beta }_{j\beta \left[1,n\right]}{x}_{j}}{n}\right)}^{2}$ can be modelled with a $\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint