5.98. cumulative_product
DESCRIPTION | LINKS | GRAPH |
- Origin
- Constraint
- Arguments
- Restrictions
- Purpose
Consider a set of tasks described by the collection. The constraint forces that at each point in time, the product of the heights of the set of tasks that overlap that point, does not exceed a given limit. A task overlaps a point if and only if (1)ย its origin is less than or equal to , and (2)ย its end is strictly greater than . It also imposes for each task of the constraint .
- Example
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Figureย 5.98.1 shows the solution associated with the example. To each task of the constraint corresponds a set of rectangles coloured with the same colour: the sum of the lengths of the rectangles corresponds to the duration of the task, while the height of the rectangles (i.e.,ย all the rectangles associated with a task have the same height) corresponds to the height of the task. The profile corresponding to the product of the heights of the tasks that overlap a given point is depicted by a thick red line. The constraint holds since at each point in time the product of the heights of the tasks that overlap that point is not strictly greater than the upper limit 6 enforced by the last argument of the constraint.
Figure 5.98.1. Resource consumption profile in red corresponding to the product of the heights of the five tasks of the Example slot
- Typical
- Symmetries
Items of are permutable.
can be decreased to any value .
One and the same constant can be added to the and attributes of all items of .
can be increased.
- Arg. properties
Contractible wrt. .
- Reformulation
The constraint can be expressed in term of a set of reified constraints and of constraints of the form :
For each pair of tasks of the collection we create a variable which is set to the height of task if task overlaps the origin attribute of task , and to 1 otherwise:
If :
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If :
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ย ย ย
ย ย ย
For each task we impose a constraint of the form .
- See also
- Keywords
characteristic of a constraint: product.
constraint type: scheduling constraint, resource constraint, temporal constraint.
- Arc input(s)
- Arc generator
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- Arc arity
- Arc constraint(s)
- Graph property(ies)
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- Arc input(s)
- Arc generator
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- Arc arity
- Arc constraint(s)
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- Graph class
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- Sets
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- Constraint(s) on sets
- Graph model
Partsย (A) andย (B) of Figureย 5.98.2 respectively show the initial and final graph associated with the second graph constraint of the Example slot. On the one hand, each source vertex of the final graph can be interpreted as a time point. On the other hand the successors of a source vertex correspond to those tasks that overlap that time point. The constraint holds since for each successor set of the final graph the product of the heights of the tasks in does not exceed the limit .
Figure 5.98.2. Initial and final graph of the constraint
(a) (b) - Signature
Since is the maximum number of vertices of the final graph of the first graph constraint we can rewrite to . This leads to simplify to .