### 3.7.16. Assignment dimension

A constraint for handlingΒ placement problems in the broad sense involving an assignment dimension (i.e.,Β one of the attribute of a collection passed as argument indicates the assignment dimension β the attribute is shown in parenthesis for each constraint). In order to illustrate the notion of assignment dimension let us first introduce three typical examples described in FigureΒ 3.7.2:

Using constraints like $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi \pi \pi }$ or $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ allows to model directly with one single global constraint such problems without knowing in advance to which machine, to which rectangular piece, to which Β container, a task, a rectangle, a box will be assigned. For each object the potential values of its assignment variable provide the machines, the rectangular pieces, the Β containers to which the object can possibly be assigned. Note that this allows to avoid 0-1 variables for modelling such problems.

Within constraints like $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ or $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ the concept of assignment dimension is extended from the fact that a variable is assigned a value to the fact that a variable is assigned an interval (i.e.,Β a value in an interval).