5.213. min_index

DESCRIPTIONLINKSGRAPH
Origin

N.Β Beldiceanu

Constraint

πš–πš’πš—_πš’πš—πšπšŽπš‘(𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚)

Arguments
𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡πšπšŸπšŠπš›
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,πšŸπšŠπš›-πšπšŸπšŠπš›)
Restrictions
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>0
𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡β‰₯0
𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡β‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,[πš’πš—πšπšŽπš‘,πšŸπšŠπš›])
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πš’πš—πšπšŽπš‘β‰₯1
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πš’πš—πšπšŽπš‘β‰€|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|
πšπš’πšœπšπš’πš—πšŒπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πš’πš—πšπšŽπš‘)
Purpose

𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡ is the index of the variables corresponding to the minimum value of the collection of variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.

Example
2,πš’πš—πšπšŽπš‘-1πšŸπšŠπš›-3,πš’πš—πšπšŽπš‘-2πšŸπšŠπš›-2,πš’πš—πšπšŽπš‘-3πšŸπšŠπš›-7,πš’πš—πšπšŽπš‘-4πšŸπšŠπš›-2,πš’πš—πšπšŽπš‘-5πšŸπšŠπš›-6
4,πš’πš—πšπšŽπš‘-1πšŸπšŠπš›-3,πš’πš—πšπšŽπš‘-2πšŸπšŠπš›-2,πš’πš—πšπšŽπš‘-3πšŸπšŠπš›-7,πš’πš—πšπšŽπš‘-4πšŸπšŠπš›-2,πš’πš—πšπšŽπš‘-5πšŸπšŠπš›-6

The attribute πšŸπšŠπš›=2 of the second and fourth items of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ is the minimum value over values 3,2,7,2,6. Consequently, both πš–πš’πš—_πš’πš—πšπšŽπš‘ constraints hold since their first arguments 𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡ are respectively set to 2 and 4.

Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable.

  • One and the same constant can be added to the πšŸπšŠπš› attribute of all items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.

Usage

Within the context of scheduling, assume the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection correspond to the starts of a set of tasks. Then 𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡ gives the indexes of those tasks that can be scheduled first.

See also

comparison swapped: πš–πšŠπš‘_πš’πš—πšπšŽπš‘.

Keywords

characteristic of a constraint: minimum.

constraint type: order constraint.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2)

Arc arity
Arc constraint(s)
β‹πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πš”πšŽπš’=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πš”πšŽπš’,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš›<πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›
Graph property(ies)
πŽπ‘πƒπ„π‘(0,0,πš’πš—πšπšŽπš‘)=𝙼𝙸𝙽_π™Έπ™½π™³π™΄πš‡

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.213.1 respectively show the initial and final graph associated with the two examples of the Example slot. Since we use the πŽπ‘πƒπ„π‘ graph property, the vertices of rank 0 (without considering the loops) of the final graph are outlined with a thick circle.

Figure 5.213.1. Initial and final graph of the πš–πš’πš—_πš’πš—πšπšŽπš‘ constraint
ctrs/min_indexActrs/min_indexB
(a) (b)