5.325. sum

DESCRIPTIONLINKSGRAPH
Origin

[Yunes02].

Constraint

πšœπšžπš–(π™Έπ™½π™³π™΄πš‡,πš‚π™΄πšƒπš‚,π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚,πš‚)

Arguments
π™Έπ™½π™³π™΄πš‡πšπšŸπšŠπš›
πš‚π™΄πšƒπ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š’πš—πš,𝚜𝚎𝚝-πšœπš’πš—πš)
π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚌𝚜𝚝-πš’πš—πš)
πš‚πšπšŸπšŠπš›
Restrictions
|πš‚π™΄πšƒπš‚|β‰₯1
πš›πšŽπššπšžπš’πš›πšŽπš(πš‚π™΄πšƒπš‚,[πš’πš—πš,𝚜𝚎𝚝])
πšπš’πšœπšπš’πš—πšŒπš(πš‚π™΄πšƒπš‚,πš’πš—πš)
|π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚|β‰₯1
πš›πšŽπššπšžπš’πš›πšŽπš(π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚,𝚌𝚜𝚝)
Purpose

πš‚ is equal to the sum of the constants corresponding to the π™Έπ™½π™³π™΄πš‡ th set of the πš‚π™΄πšƒπš‚ collection.

Example
8,πš’πš—πš-8𝚜𝚎𝚝-{2,3},πš’πš—πš-1𝚜𝚎𝚝-{3},πš’πš—πš-3𝚜𝚎𝚝-{1,4,5},πš’πš—πš-6𝚜𝚎𝚝-{2,4},4,9,1,3,1,10

The πšœπšžπš– constraint holds since its last argument πš‚=10 is equal to the sum of the 2th and 3th items of the collection 〈4,9,1,3,1βŒͺ. As illustrated by FigureΒ 5.325.1, this stems from the fact that its first argument π™Έπ™½π™³π™΄πš‡=8 corresponds to the value of the πš’πš—πš attribute of the first item of the πš‚π™΄πšƒπš‚ collection. Consequently the corresponding set {2,3} is used for summing the 2th and 3th items of the π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚ collection.

Figure 5.325.1. Illustration of the correspondence between the arguments of the πšœπšžπš– constraint in the context of the Example slot
ctrs/sum1
Symmetry

Items of πš‚π™΄πšƒπš‚ are permutable.

Usage

In his article introducing the πšœπšžπš– constraint, TallysΒ H.Β Yunes mentions the Sequence Dependent Cumulative Cost Problem as the subproblem that originally motivates this constraint.

Algorithm

The articleΒ [Yunes02] gives the convex hull relaxation of the πšœπšžπš– constraint.

See also

common keyword: πšŽπš•πšŽπš–πšŽπš—πšΒ (data constraint), πšœπšžπš–_πšŒπšπš›, πšœπšžπš–_𝚜𝚎𝚝 (sum).

used in graph description: πš’πš—_𝚜𝚎𝚝.

Keywords

characteristic of a constraint: convex hull relaxation, sum.

constraint type: data constraint.

filtering: linear programming.

Arc input(s)

πš‚π™΄πšƒπš‚ π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚

Arc generator
π‘ƒπ‘…π‘‚π·π‘ˆπΆπ‘‡β†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚜𝚎𝚝𝚜,πšŒπš˜πš—πšœπšπšŠπš—πšπšœ)

Arc arity
Arc constraint(s)
β€’ π™Έπ™½π™³π™΄πš‡=𝚜𝚎𝚝𝚜.πš’πš—πš
β€’ πš’πš—_𝚜𝚎𝚝(πšŒπš˜πš—πšœπšπšŠπš—πšπšœ.πš”πšŽπš’,𝚜𝚎𝚝𝚜.𝚜𝚎𝚝)
Graph property(ies)
π’π”πŒ(π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚,𝚌𝚜𝚝)=πš‚

Graph model

According to the value assigned to π™Έπ™½π™³π™΄πš‡ the arc constraint selects for the final graph:

  • The π™Έπ™½π™³π™΄πš‡ th item of the πš‚π™΄πšƒπš‚ collection,

  • The items of the π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚ collection for which the key correspond to the indices of the π™Έπ™½π™³π™΄πš‡ th set of the πš‚π™΄πšƒπš‚ collection.

Finally, since we use the π’π”πŒ graph property on the 𝚌𝚜𝚝 attribute of the π™²π™Ύπ™½πš‚πšƒπ™°π™½πšƒπš‚ collection, the last argument πš‚ of the πšœπšžπš– constraint is equal to the sum of the constants associated with the vertices of the final graph.

PartsΒ (A) andΒ (B) of FigureΒ 5.325.2 respectively show the initial and final graph associated with the Example slot. Since we use the π’π”πŒ graph property we show the vertices from which we compute πš‚ in a box.

Figure 5.325.2. Initial and final graph of the πšœπšžπš– constraint
ctrs/sumActrs/sumB
(a) (b)