## 5.128. disjunctive_or_same_start

Origin

Scheduling.

Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi °\pi \pi Ί\pi }\right)$

Synonyms

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$.

Argument
 $\mathrm{\pi \pi °\pi \pi Ί\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi °\pi \pi Ί\pi },\left[\mathrm{\pi \pi \pi \pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\right]\right)$ $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta ₯0$
Purpose

All pairs of tasks of the collection $\mathrm{\pi \pi °\pi \pi Ί\pi }$ that have a duration strictly greater than 0 should either not overlap either have the same start, i.e.Β $\beta i\beta \left[1,|\mathrm{\pi \pi °\pi \pi Ί\pi }|\right],\beta j\beta \left[i+1,|\mathrm{\pi \pi °\pi \pi Ί\pi }|\right]:$ $\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }=0\beta ¨\mathrm{\pi \pi °\pi \pi Ί\pi }\left[j\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }=0\beta ¨\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi }+\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi °\pi \pi Ί\pi }\left[j\right].\mathrm{\pi \pi \pi \pi \pi \pi }\beta ¨\mathrm{\pi \pi °\pi \pi Ί\pi }\left[j\right].\mathrm{\pi \pi \pi \pi \pi \pi }+\mathrm{\pi \pi °\pi \pi Ί\pi }\left[j\right].\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi }\beta ¨\mathrm{\pi \pi °\pi \pi Ί\pi }\left[i\right].\mathrm{\pi \pi \pi \pi \pi \pi }=\mathrm{\pi \pi °\pi \pi Ί\pi }\left[j\right].\mathrm{\pi \pi \pi \pi \pi \pi }$.

Example
$\left(\begin{array}{c}β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi \pi }-4\hfill & \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi \pi \pi \pi }-7\hfill & \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi \pi \pi \pi }-4\hfill & \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }-1\hfill \end{array}βͺ\hfill \end{array}\right)$

Since the starts of the first and third tasks coincide, and since the second task does neither overlap the first task nor the third task, the $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds.

Typical
 $|\mathrm{\pi \pi °\pi \pi Ί\pi }|>2$ $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta ₯1$
Symmetries
• Items of $\mathrm{\pi \pi °\pi \pi Ί\pi }$ are permutable.

• $\mathrm{\pi \pi °\pi \pi Ί\pi }.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ can be decreased to any value $\beta ₯0$.

• One and the same constant can be added to the $\mathrm{\pi \pi \pi \pi \pi \pi }$ attribute of all items of $\mathrm{\pi \pi °\pi \pi Ί\pi }$.

Arg. properties

Contractible wrt. $\mathrm{\pi \pi °\pi \pi Ί\pi }$.

Keywords
Arc input(s)

$\mathrm{\pi \pi °\pi \pi Ί\pi }$

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

Arc arity
Arc constraint(s)
$\beta \left(\begin{array}{c}\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }=0,\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }=0,\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi \pi \pi }+\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi \pi }+\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi \pi }\hfill \end{array}\right)$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi \pi °\pi \pi Ί\pi }|*\left(|\mathrm{\pi \pi °\pi \pi Ί\pi }|-1\right)/2$

Graph model

We generate a clique with a non-overlapping constraint or a same start constraint between each pair of distinct tasks and state that the number of arcs of the final graph should be equal to the number of arcs of the initial graph.

PartsΒ (A) andΒ (B) of FigureΒ 5.128.1 respectively show the initial and final graph associated with the Example slot. The $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since all the arcs of the initial graph belong to the final graph.