Origin
Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi  \pi °\pi },\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }\right)$

Arguments
 $\mathrm{\pi \pi  \pi °\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi },\left[\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi }\right]\right)$ $\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }.\mathrm{\pi \pi \pi \pi }\beta ₯0$ $\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }.\mathrm{\pi \pi \pi \pi }\beta €1$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Make the link between a set variable $\mathrm{\pi \pi  \pi °\pi }$ and those 0-1 variables that are associated with each potential value belonging to $\mathrm{\pi \pi  \pi °\pi }$: The 0-1 variables, which are associated with a value belonging to the set variable $\mathrm{\pi \pi  \pi °\pi }$, are equal to 1, while the remaining 0-1 variables are all equal to 0.

Example
$\left(\begin{array}{c}\left\{1,3,4\right\},\hfill \\ β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi }-0\hfill & \mathrm{\pi \pi \pi }-0,\hfill \\ \mathrm{\pi \pi \pi \pi }-1\hfill & \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi \pi }-0\hfill & \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi \pi }-1\hfill & \mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi \pi }-1\hfill & \mathrm{\pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi \pi }-0\hfill & \mathrm{\pi \pi \pi }-5\hfill \end{array}βͺ\hfill \end{array}\right)$

In the example, the 0-1 variables associated with the values 1, 3 and 4 are all set to 1, while the other 0-1 variables are set to 0. Consequently, the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi \pi  \pi °\pi }$ is set to $\left\{1,3,4\right\}$.

Typical
 $|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }.\mathrm{\pi \pi \pi \pi }\right)>1$
Symmetry

Items of $\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }$ are permutable.

Usage

This constraint is used in order to make the link between a formulation using set variables and a formulation based on linear programming.

Systems
Keywords
Derived Collection
$\mathrm{\pi \pi \pi }\left(\begin{array}{c}\mathrm{\pi \pi ΄\pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right),\hfill \\ \mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-1,\mathrm{\pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi  \pi °\pi }\right)\right]\hfill \end{array}\right)$
Arc input(s)

$\mathrm{\pi \pi ΄\pi }$ $\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }$

Arc generator
$\mathrm{\pi \pi  \pi \pi ·\pi \pi Ά\pi }$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi \pi }=\mathrm{\pi \pi \pi }.\mathrm{\pi \pi \pi }\beta $$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi }.\mathrm{\pi \pi \pi \pi \pi \pi }\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|$

Graph model

The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint is modelled with the following bipartite graph. The first set of vertices corresponds to a single vertex containing the set variable. The second class of vertices contains one vertex for each item of the collection $\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }$. The arc constraint between the set variable $\mathrm{\pi \pi  \pi °\pi }$ and one potential value $v$ of the set variable expresses the following:

• If the 0-1 variable associated with $v$ is equal to 1 then $v$ should belong to $\mathrm{\pi \pi  \pi °\pi }$.

• Otherwise if the 0-1 variable associated with $v$ is equal to 0 then $v$ should not belong to $\mathrm{\pi \pi  \pi °\pi }$.

Since all arc constraints should hold the final graph contains exactly $|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|$ arcs.

PartsΒ (A) andΒ (B) of FigureΒ 5.234.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the arcs of the final graph are stressed in bold. The $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ constraint holds since the final graph contains exactly 6 arcs (one for each 0-1 variable).

Signature

Since the initial graph contains $|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|$ arcs the maximum number of arcs of the final graph is equal to $|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|$. Therefore we can rewrite the graph property $\mathrm{\pi \pi \pi \pi }=|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|$ to $\mathrm{\pi \pi \pi \pi }\beta ₯|\mathrm{\pi ±\pi Ύ\pi Ύ\pi »\pi ΄\pi °\pi ½\pi }|$ and simplify $\underset{Μ²}{\stackrel{Β―}{\mathrm{\pi \pi \pi \pi }}}$ to $\stackrel{Β―}{\mathrm{\pi \pi \pi \pi }}$.