## 5.229. ninterval

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi  \pi °\pi »},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}\right)$

Arguments
 $\mathrm{\pi ½\pi  \pi °\pi »}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\mathrm{\pi ½\pi  \pi °\pi »}\beta ₯\mathrm{\pi \pi \pi }\left(1,|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\right)$ $\mathrm{\pi ½\pi  \pi °\pi »}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}>0$
Purpose

Consider the intervals of the form $\left[\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·k,\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}Β·k+\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}-1\right]$ where $k$ is an integer. $\mathrm{\pi ½\pi  \pi °\pi »}$ is the number of intervals for which at least one value is assigned to at least one variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Example
$\left(2,β©3,1,9,1,9βͺ,4\right)$

In the example, the third argument $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}=4$ defines the following family of intervals $\left[4Β·k,4Β·k+3\right]$, where $k$ is an integer. Values 3, 1, 9, 1 and 9 are respectively located within intervals $\left[0,3\right]$, $\left[0,3\right]$, $\left[8,11\right]$, $\left[0,3\right]$ and $\left[8,11\right]$. Since we only use the two intervals $\left[0,3\right]$ and $\left[8,11\right]$ the first argument of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint is set to value 2.

Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ that belongs to the $k$-th interval, of size $\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$, can be replaced by any other value of the same interval.

Usage

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ constraint is useful for counting the number of effectively used periods, no matter how many time each period is used. A period can for example stand for a hour or for a day.

Algorithm

related: $\mathrm{\pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$), $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$), $\mathrm{\pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi }$ of $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$).

specialisation: $\mathrm{\pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }/\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$).

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }/\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi Έ\pi ½\pi \pi ΄\pi \pi  \pi °\pi »}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½\pi  \pi °\pi »}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.229.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to those values of an interval that are assigned to some variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection. The values 1, 3 and the value 9, which respectively correspond to intervals $\left[0,3\right]$ and $\left[8,11\right]$, are assigned to the variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection.