3.7.16. Assignment dimension

A constraint for handling placement problems in the broad sense involving an assignment dimension (i.e., one of the attribute of a collection passed as argument indicates the assignment dimension — 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 Figure 3.7.2:

Using constraints like $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$, $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$, $\mathrm{𝚍𝚒𝚏𝚏𝚗}$, $\mathrm{𝚐𝚎𝚘𝚜𝚝}$ or $\mathrm{𝚐𝚎𝚘𝚜𝚝}_\mathrm{𝚝𝚒𝚖𝚎}$ allows to model directly with one single global constraint such problems without knowing in advance to which machine, to which rectangular piece, to which  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  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 $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}$ or $\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚜𝚞𝚖}$ 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., a value in an interval).