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 $\left\{1,3,4\right\}$.

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 one 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 5.202.1 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 $\stackrel{Β―}{\mathrm{\pi \pi \pi \pi }}$.