5.163. indexed_sum

DESCRIPTIONLINKSGRAPH
Origin

N.Β Beldiceanu

Constraint

πš’πš—πšπšŽπš‘πšŽπš_πšœπšžπš–(π™Έπšƒπ™΄π™Όπš‚,πšƒπ™°π™±π™»π™΄)

Arguments
π™Έπšƒπ™΄π™Όπ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šŸπšŠπš›,πš πšŽπš’πšπš‘πš-πšπšŸπšŠπš›)
πšƒπ™°π™±π™»π™΄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šœπšžπš–πš–πšŠπšπš’πš˜πš—-πšπšŸπšŠπš›)
Restrictions
|π™Έπšƒπ™΄π™Όπš‚|>0
|πšƒπ™°π™±π™»π™΄|>0
πš›πšŽπššπšžπš’πš›πšŽπš(π™Έπšƒπ™΄π™Όπš‚,[πš’πš—πšπšŽπš‘,πš πšŽπš’πšπš‘πš])
π™Έπšƒπ™΄π™Όπš‚.πš’πš—πšπšŽπš‘β‰₯1
π™Έπšƒπ™΄π™Όπš‚.πš’πš—πšπšŽπš‘β‰€|πšƒπ™°π™±π™»π™΄|
πš›πšŽπššπšžπš’πš›πšŽπš(πšƒπ™°π™±π™»π™΄,[πš’πš—πšπšŽπš‘,πšœπšžπš–πš–πšŠπšπš’πš˜πš—])
πšƒπ™°π™±π™»π™΄.πš’πš—πšπšŽπš‘β‰₯1
πšƒπ™°π™±π™»π™΄.πš’πš—πšπšŽπš‘β‰€|πšƒπ™°π™±π™»π™΄|
πš’πš—πšŒπš›πšŽπšŠπšœπš’πš—πš_𝚜𝚎𝚚(πšƒπ™°π™±π™»π™΄,πš’πš—πšπšŽπš‘)
Purpose

Given several items of the collection π™Έπšƒπ™΄π™Όπš‚ (each of them having a specific fixed πš’πš—πšπšŽπš‘ as well as a πš πšŽπš’πšπš‘πš that may be negative or positive), and a table πšƒπ™°π™±π™»π™΄ (each entry of πšƒπ™°π™±π™»π™΄ corresponding to a πšœπšžπš–πš–πšŠπšπš’πš˜πš— variable), assign each item to an entry of πšƒπ™°π™±π™»π™΄ so that the sum of the weights of the items assigned to that entry is equal to the corresponding πšœπšžπš–πš–πšŠπšπš’πš˜πš— variable.

Example
πš’πš—πšπšŽπš‘-3πš πšŽπš’πšπš‘πš--4,πš’πš—πšπšŽπš‘-1πš πšŽπš’πšπš‘πš-6,πš’πš—πšπšŽπš‘-3πš πšŽπš’πšπš‘πš-1,πš’πš—πšπšŽπš‘-1πšœπšžπš–πš–πšŠπšπš’πš˜πš—-6,πš’πš—πšπšŽπš‘-2πšœπšžπš–πš–πšŠπšπš’πš˜πš—-0,πš’πš—πšπšŽπš‘-3πšœπšžπš–πš–πšŠπšπš’πš˜πš—--3

The πš’πš—πšπšŽπš‘πšŽπš_πšœπšžπš– constraint holds since the summation variables associated with each entry of πšƒπ™°π™±π™»π™΄ are equal to the sum of the weights of the items assigned to the corresponding entry:

  • πšƒπ™°π™±π™»π™΄[1].πšœπšžπš–πš–πšŠπšπš’πš˜πš—=π™Έπšƒπ™΄π™Όπš‚[2].πš πšŽπš’πšπš‘πš=6 (since πšƒπ™°π™±π™»π™΄[1].πš’πš—πšπšŽπš‘=π™Έπšƒπ™΄π™Όπš‚[2].πš’πš—πšπšŽπš‘=1),

  • πšƒπ™°π™±π™»π™΄[2].πšœπšžπš–πš–πšŠπšπš’πš˜πš—=0 (since πšƒπ™°π™±π™»π™΄[2].πš’πš—πšπšŽπš‘=2 does not occur as a value of the πš’πš—πšπšŽπš‘ attribute of an item of π™Έπšƒπ™΄π™Όπš‚),

  • πšƒπ™°π™±π™»π™΄[3].πšœπšžπš–πš–πšŠπšπš’πš˜πš—=π™Έπšƒπ™΄π™Όπš‚[1].πš πšŽπš’πšπš‘πš+π™Έπšƒπ™΄π™Όπš‚[3].πš πšŽπš’πšπš‘πš=-4+1=-3 (since πšƒπ™°π™±π™»π™΄[3].πš’πš—πšπšŽπš‘=π™Έπšƒπ™΄π™Όπš‚[1].πš’πš—πšπšŽπš‘=π™Έπšƒπ™΄π™Όπš‚[3].πš’πš—πšπšŽπš‘=3).

Typical
|π™Έπšƒπ™΄π™Όπš‚|>1
πš›πšŠπš—πšπšŽ(π™Έπšƒπ™΄π™Όπš‚.πš’πš—πšπšŽπš‘)>1
|πšƒπ™°π™±π™»π™΄|>1
πš›πšŠπš—πšπšŽ(πšƒπ™°π™±π™»π™΄.πšœπšžπš–πš–πšŠπšπš’πš˜πš—)>1
Symmetries
  • Items of π™Έπšƒπ™΄π™Όπš‚ are permutable.

  • Items of πšƒπ™°π™±π™»π™΄ are permutable.

Reformulation

The πš’πš—πšπšŽπš‘πšŽπš_πšœπšžπš–(π™Έπšƒπ™΄π™Όπš‚,πšƒπ™°π™±π™»π™΄) constraint can be expressed in term of a set of reified constraints and of |πšƒπ™°π™±π™»π™΄| arithmetic constraints (i.e.,Β πšœπšŒπšŠπš•πšŠπš›_πš™πš›πš˜πšπšžπšŒπš constraints).

  1. For each item π™Έπšƒπ™΄π™Όπš‚[i] (i∈[1,|π™Έπšƒπ™΄π™Όπš‚|]) and for each table entry j (j∈[1,|πšƒπ™°π™±π™»π™΄|]) of πšƒπ™°π™±π™»π™΄ we create a 0-1 variable B ij that will be set to 1 if and only if π™Έπšƒπ™΄π™Όπš‚[i].πš’πš—πšπšŽπš‘ is fixed to j (i.e.,Β B ij β‡”π™Έπšƒπ™΄π™Όπš‚[i].πš’πš—πšπšŽπš‘=j).

  2. For each entry j of the table πšƒπ™°π™±π™»π™΄, we impose the sum π™Έπšƒπ™΄π™Όπš‚[1].πš πšŽπš’πšπš‘πšΒ·B 1j +π™Έπšƒπ™΄π™Όπš‚[2].πš πšŽπš’πšπš‘πšΒ·B 2j +...+π™Έπšƒπ™΄π™Όπš‚[|π™Έπšƒπ™΄π™Όπš‚|].πš πšŽπš’πšπš‘πšΒ·B |π™Έπšƒπ™΄π™Όπš‚|j to be equal to πšƒπ™°π™±π™»π™΄[j].πšœπšžπš–πš–πšŠπšπš’πš˜πš—.

See also

specialisation: πš‹πš’πš—_πš™πšŠπšŒπš”πš’πš—πšΒ (negative contribution not allowed, effective use variable for each bin replaced by an overall fixed capacity), πš‹πš’πš—_πš™πšŠπšŒπš”πš’πš—πš_πšŒπšŠπš™πšŠΒ (negative contribution not allowed, effective use variable for each bin replaced by a fixed capacity for each bin).

used in graph description: πšœπšžπš–_πšŒπšπš›.

Keywords

application area: assignment.

modelling: variable indexing, variable subscript.

For all items of πšƒπ™°π™±π™»π™΄:

Arc input(s)

π™Έπšƒπ™΄π™Όπš‚ πšƒπ™°π™±π™»π™΄

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—(πš’πšπšŽπš–πšœ,πšπšŠπš‹πš•πšŽ)

Arc arity
Arc constraint(s)
πš’πšπšŽπš–πšœ.πš’πš—πšπšŽπš‘=πšπšŠπš‹πš•πšŽ.πš’πš—πšπšŽπš‘
Sets
π–²π–΄π–’π–’β†¦πšœπš˜πšžπš›πšŒπšŽ,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ-πšŒπš˜π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šŸπšŠπš›-πšπšŸπšŠπš›),πš’πšπšŽπš–(πšŸπšŠπš›-π™Έπšƒπ™΄π™Όπš‚.πš πšŽπš’πšπš‘πš)]

Constraint(s) on sets
πšœπšžπš–_πšŒπšπš›(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ,=,πšƒπ™°π™±π™»π™΄.πšœπšžπš–πš–πšŠπšπš’πš˜πš—)
Graph model

We enforce the πšœπšžπš–_πšŒπšπš› constraint on the weight of the items that are assigned to the same entry. Within the context of the Example slot, partΒ (A) of FigureΒ 5.163.1 shows the initial graphs associated with entries 1, 2 and 3 (i.e.,Β one initial graph for each item of the πšƒπ™°π™±π™»π™΄ collection). PartΒ (B) of FigureΒ 5.163.1 shows the corresponding final graphs associated with entries 1 and 3. Each source vertex of the final graph can be interpreted as an item assigned to a specific entry of πšƒπ™°π™±π™»π™΄.

Figure 5.163.1. Initial and final graph of the πš’πš—πšπšŽπš‘πšŽπš_πšœπšžπš– constraint
ctrs/indexed_sumActrs/indexed_sumB
(a) (b)