5.61. coloured_cumulatives

DESCRIPTIONLINKSGRAPH
Origin

Derived from ๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ and ๐š—๐šŸ๐šŠ๐š•๐šž๐šŽ๐šœ.

Constraint

๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ(๐šƒ๐™ฐ๐š‚๐™บ๐š‚,๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚)

Synonym

๐šŒ๐š˜๐š•๐š˜๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ.

Arguments
๐šƒ๐™ฐ๐š‚๐™บ๐š‚๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-๐š๐šŸ๐šŠ๐š›,๐š˜๐š›๐š’๐š๐š’๐š—-๐š๐šŸ๐šŠ๐š›,๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-๐š๐šŸ๐šŠ๐š›,๐šŽ๐š—๐š-๐š๐šŸ๐šŠ๐š›,๐šŒ๐š˜๐š•๐š˜๐šž๐š›-๐š๐šŸ๐šŠ๐š›
๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—(๐š’๐š-๐š’๐š—๐š,๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข-๐š’๐š—๐š)
Restrictions
๐š›๐šŽ๐šš๐šž๐š’๐š›๐šŽ๐š(๐šƒ๐™ฐ๐š‚๐™บ๐š‚,[๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ,๐šŒ๐š˜๐š•๐š˜๐šž๐š›])
๐š›๐šŽ๐šš๐šž๐š’๐š›๐šŽ_๐šŠ๐š_๐š•๐šŽ๐šŠ๐šœ๐š(2,๐šƒ๐™ฐ๐š‚๐™บ๐š‚,[๐š˜๐š›๐š’๐š๐š’๐š—,๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—,๐šŽ๐š—๐š])
๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—โ‰ฅ0
๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š˜๐š›๐š’๐š๐š’๐š—โ‰ค๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐šŽ๐š—๐š
๐š›๐šŽ๐šš๐šž๐š’๐š›๐šŽ๐š(๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚,[๐š’๐š,๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข])
๐š๐š’๐šœ๐š๐š’๐š—๐šŒ๐š(๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚,๐š’๐š)
๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šขโ‰ฅ0
Purpose

Consider a set ๐’ฏ of tasks described by the ๐šƒ๐™ฐ๐š‚๐™บ๐š‚ collection. The ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ constraint enforces for each machine m of the ๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚ collection the following condition: at each point in time p, the numbers of distinct colours of the set of tasks that both overlap that point p and are assigned to machine m does not exceed the capacity of machine m. A task overlaps a point i if and only if (1)ย its origin is less than or equal to i, and (2)ย its end is strictly greater than i. It also imposes for each task of ๐’ฏ the constraint ๐š˜๐š›๐š’๐š๐š’๐š—+๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—=๐šŽ๐š—๐š.

Example
๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-1๐š˜๐š›๐š’๐š๐š’๐š—-6๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-6๐šŽ๐š—๐š-12๐šŒ๐š˜๐š•๐š˜๐šž๐š›-1,๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-1๐š˜๐š›๐š’๐š๐š’๐š—-2๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-9๐šŽ๐š—๐š-11๐šŒ๐š˜๐š•๐š˜๐šž๐š›-2,๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-2๐š˜๐š›๐š’๐š๐š’๐š—-7๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-3๐šŽ๐š—๐š-10๐šŒ๐š˜๐š•๐š˜๐šž๐š›-2,๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-1๐š˜๐š›๐š’๐š๐š’๐š—-1๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-2๐šŽ๐š—๐š-3๐šŒ๐š˜๐š•๐š˜๐šž๐š›-1,๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-2๐š˜๐š›๐š’๐š๐š’๐š—-4๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-5๐šŽ๐š—๐š-9๐šŒ๐š˜๐š•๐š˜๐šž๐š›-2,๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ-1๐š˜๐š›๐š’๐š๐š’๐š—-3๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—-10๐šŽ๐š—๐š-13๐šŒ๐š˜๐š•๐š˜๐šž๐š›-1,๐š’๐š-1 ๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข-2,๐š’๐š-2 ๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข-1

Figureย 5.61.1 shows the solution associated with the example. Each rectangle of the figure corresponds to a task of the ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ constraint. Tasks that have their colour attribute set to 1 and 2 are respectively coloured in blue and pink. The ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ constraint holds since for machineย 1 we have at most two distinct colours in parallel (which is the maximum capacity for machineย 1), while for machineย 2 we have no more than one single colour in parallel (which is actually the maximum capacity for machineย 2).

Figure 5.61.1. Assignment of the tasks on the two machines
ctrs/coloured_cumulatives1
Typical
|๐šƒ๐™ฐ๐š‚๐™บ๐š‚|>1
๐š›๐šŠ๐š—๐š๐šŽ(๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ)>1
๐š›๐šŠ๐š—๐š๐šŽ(๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š˜๐š›๐š’๐š๐š’๐š—)>1
๐š›๐šŠ๐š—๐š๐šŽ(๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—)>1
๐š›๐šŠ๐š—๐š๐šŽ(๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐šŽ๐š—๐š)>1
๐š›๐šŠ๐š—๐š๐šŽ(๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐šŒ๐š˜๐š•๐š˜๐šž๐š›)>1
๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—>0
|๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚|>1
๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข>0
๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข<๐š—๐šŸ๐šŠ๐š•(๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐šŒ๐š˜๐š•๐š˜๐šž๐š›)
|๐šƒ๐™ฐ๐š‚๐™บ๐š‚|>|๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚|
Symmetries
  • Items of ๐šƒ๐™ฐ๐š‚๐™บ๐š‚ are permutable.

  • Items of ๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚ are permutable.

  • ๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข can be increased.

  • All occurrences of two distinct values in ๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ or ๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐š’๐š can be swapped; all occurrences of a value in ๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ or ๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐š’๐š can be renamed to any unused value.

Usage

Useful for scheduling problems where several machines are available and where you have to assign each task to a specific machine. In addition each machine can only proceed in parallel a maximum number of tasks of distinct types.

Reformulation

The ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ constraint can be expressed in term of a set of reified constraints and of |๐šƒ๐™ฐ๐š‚๐™บ๐š‚| ๐š—๐šŸ๐šŠ๐š•๐šž๐šŽ constraints:

  1. For each pair of tasks ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i],๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j] (i,jโˆˆ[1,|๐šƒ๐™ฐ๐š‚๐™บ๐š‚|]) of the ๐šƒ๐™ฐ๐š‚๐™บ๐š‚ collection we create a variable C ij which is set to the colour of task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j] if both tasks are assigned to the same machine and if task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j] overlaps the origin attribute of task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i], and to the colour of task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i] otherwise:

    • If i=j:

      • C ij =๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐šŒ๐š˜๐š•๐š˜๐šž๐š›.

    • If iโ‰ j:

      • C ij =๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐šŒ๐š˜๐š•๐š˜๐šž๐š›โˆจC ij =๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐šŒ๐š˜๐š•๐š˜๐šž๐š›.

      • ((๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ=๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ โˆง

        ย ย ย ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐š˜๐š›๐š’๐š๐š’๐š—โ‰ค๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐š˜๐š›๐š’๐š๐š’๐š— โˆง

        ย ย ย ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐šŽ๐š—๐š>๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐š˜๐š›๐š’๐š๐š’๐š—)โˆง(C ij =๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐šŒ๐š˜๐š•๐š˜๐šž๐š›)) โˆจ

        ((๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽโ‰ ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ โˆจ

        ย ย ย ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐š˜๐š›๐š’๐š๐š’๐š—>๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐š˜๐š›๐š’๐š๐š’๐š— โˆจ

        ย ย ย ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[j].๐šŽ๐š—๐šโ‰ค๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐š˜๐š›๐š’๐š๐š’๐š—)โˆง(C ij =๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i].๐šŒ๐š˜๐š•๐š˜๐šž๐š›))

  2. For each task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i] (iโˆˆ[1,|๐šƒ๐™ฐ๐š‚๐™บ๐š‚|]) we create a variable N i which gives the number of distinct colours associated to the tasks that both are assigned to the same machine as task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i] and overlap the origin of task ๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i] (๐šƒ๐™ฐ๐š‚๐™บ๐š‚[i] overlaps its own origin) and we impose N i to not exceed the maximum number of distinct colours ๐™ป๐™ธ๐™ผ๐™ธ๐šƒ allowed at each instant:

See also

assignment dimension removed: ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽย (๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ attribute removed), ๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽย (๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ attribute removed and number of distinct ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šœ replaced by sum of ๐š๐šŠ๐šœ๐š” ๐š‘๐šŽ๐š’๐š๐š‘๐š๐šœ).

common keyword: ๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ, ๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœย (resource constraint).

related: ๐š—๐šŸ๐šŠ๐š•๐šž๐šŽ.

used in graph description: ๐š—๐šŸ๐šŠ๐š•๐šž๐šŽ๐šœ.

Keywords

characteristic of a constraint: coloured.

constraint type: scheduling constraint, resource constraint, temporal constraint.

filtering: compulsory part.

modelling: number of distinct values, assignment dimension, zero-duration task.

Arc input(s)

๐šƒ๐™ฐ๐š‚๐™บ๐š‚

Arc generator
๐‘†๐ธ๐ฟ๐นโ†ฆ๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—(๐š๐šŠ๐šœ๐š”๐šœ)

Arc arity
Arc constraint(s)
๐š๐šŠ๐šœ๐š”๐šœ.๐š˜๐š›๐š’๐š๐š’๐š—+๐š๐šŠ๐šœ๐š”๐šœ.๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—=๐š๐šŠ๐šœ๐š”๐šœ.๐šŽ๐š—๐š
Graph property(ies)
๐๐€๐‘๐‚=|๐šƒ๐™ฐ๐š‚๐™บ๐š‚|

For all items of ๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚:

Arc input(s)

๐šƒ๐™ฐ๐š‚๐™บ๐š‚ ๐šƒ๐™ฐ๐š‚๐™บ๐š‚

Arc generator
๐‘ƒ๐‘…๐‘‚๐ท๐‘ˆ๐ถ๐‘‡โ†ฆ๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—(๐š๐šŠ๐šœ๐š”๐šœ1,๐š๐šŠ๐šœ๐š”๐šœ2)

Arc arity
Arc constraint(s)
โ€ข ๐š๐šŠ๐šœ๐š”๐šœ1.๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ=๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐š’๐š
โ€ข ๐š๐šŠ๐šœ๐š”๐šœ1.๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ=๐š๐šŠ๐šœ๐š”๐šœ2.๐š–๐šŠ๐šŒ๐š‘๐š’๐š—๐šŽ
โ€ข ๐š๐šŠ๐šœ๐š”๐šœ1.๐š๐šž๐š›๐šŠ๐š๐š’๐š˜๐š—>0
โ€ข ๐š๐šŠ๐šœ๐š”๐šœ2.๐š˜๐š›๐š’๐š๐š’๐š—โ‰ค๐š๐šŠ๐šœ๐š”๐šœ1.๐š˜๐š›๐š’๐š๐š’๐š—
โ€ข ๐š๐šŠ๐šœ๐š”๐šœ1.๐š˜๐š›๐š’๐š๐š’๐š—<๐š๐šŠ๐šœ๐š”๐šœ2.๐šŽ๐š—๐š
Graph class
โ€ข ๐™ฐ๐™ฒ๐šˆ๐™ฒ๐™ป๐™ธ๐™ฒ
โ€ข ๐™ฑ๐™ธ๐™ฟ๐™ฐ๐š๐šƒ๐™ธ๐šƒ๐™ด
โ€ข ๐™ฝ๐™พ_๐™ป๐™พ๐™พ๐™ฟ

Sets
๐–ฒ๐–ด๐–ข๐–ขโ†ฆ๐šœ๐š˜๐šž๐š›๐šŒ๐šŽ,๐šŸ๐šŠ๐š›๐š’๐šŠ๐š‹๐š•๐šŽ๐šœ-๐šŒ๐š˜๐š•๐š…๐™ฐ๐š๐™ธ๐™ฐ๐™ฑ๐™ป๐™ด๐š‚-๐šŒ๐š˜๐š•๐š•๐šŽ๐šŒ๐š๐š’๐š˜๐š—(๐šŸ๐šŠ๐š›-๐š๐šŸ๐šŠ๐š›),๐š’๐š๐šŽ๐š–(๐šŸ๐šŠ๐š›-๐šƒ๐™ฐ๐š‚๐™บ๐š‚.๐šŒ๐š˜๐š•๐š˜๐šž๐š›)]

Constraint(s) on sets
๐š—๐šŸ๐šŠ๐š•๐šž๐šŽ๐šœ(๐šŸ๐šŠ๐š›๐š’๐šŠ๐š‹๐š•๐šŽ๐šœ,โ‰ค,๐™ผ๐™ฐ๐™ฒ๐™ท๐™ธ๐™ฝ๐™ด๐š‚.๐šŒ๐šŠ๐š™๐šŠ๐šŒ๐š’๐š๐šข)
Graph model

Partsย (A) andย (B) of Figureย 5.61.2 respectively shows the initial and final graph associated with machinesย 1 and 2 involved in the Example slot. On the one hand, each source vertex of the final graph can be interpreted as a time point p on a specific machine m. On the other hand the successors of a source vertex correspond to those tasks that both overlap that time point p and are assigned to machine m. The ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ constraint holds since for each successor set ๐’ฎ of the final graph the number of distinct colours in ๐’ฎ does not exceed the capacity of the machine corresponding to the time point associated with ๐’ฎ.

Figure 5.61.2. Initial and final graph of the ๐šŒ๐š˜๐š•๐š˜๐šž๐š›๐šŽ๐š_๐šŒ๐šž๐š–๐šž๐š•๐šŠ๐š๐š’๐šŸ๐šŽ๐šœ constraint
ctrs/coloured_cumulativesA
(a)
ctrs/coloured_cumulativesB
(b)
Signature

Since ๐šƒ๐™ฐ๐š‚๐™บ๐š‚ is the maximum number of vertices of the final graph of the first graph constraint we can rewrite ๐๐€๐‘๐‚ = |๐šƒ๐™ฐ๐š‚๐™บ๐š‚| to ๐๐€๐‘๐‚ โ‰ฅ |๐šƒ๐™ฐ๐š‚๐™บ๐š‚|. This leads to simplify ๐๐€๐‘๐‚ ยฏ ฬฒ to ๐๐€๐‘๐‚ ยฏ.