Cumulative scheduling constraint or scheduling under resource constraints. Consider a set of tasks described by the collection. The constraint enforces that at each point in time, the cumulated height 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) its origin is less than or equal to , and (2) its end is strictly greater than . It also imposes for each task of the constraint .
Figure 5.82.1 shows the cumulated profile 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., all the rectangles associated with a task have the same height) corresponds to the resource consumption of the task. The constraint holds since at each point in time we do not have a cumulated resource consumption strictly greater than the upper limit 8 enforced by the last argument of the constraint.
Figure 5.82.1. Resource consumption profile
In the original constraint of CHIP the parameter was a domain variable corresponding to the maximum peak of the resource consumption profile. Given a fixed time frame, this variable could be used as a cost in order to directly minimise the maximum resource consumption peak.
Some systems like Ilog CP Optimizer also assume that a zero -duration task overlaps a point if and only if (1) its origin is less than or equal to , and (2) its end is greater than or equal to . Under this definition, the height of a zero -duration task is also taken into account in the resource consumption profile.
Note that the concept of cumulative is different from the concept of rectangles non -overlapping even if, most of the time, each task of a ground solution of a constraint is simply drawn as a single rectangle. As illustrated by Figure 5.100.5, this is in fact not always possible (i.e., some rectangles may need to be broken apart). In fact the constraint is only a necessary condition for rectangles non -overlapping (see Figure 5.100.4 and the corresponding explanation in the Algorithm slot of the constraint).
The first filtering algorithms were related to the notion of compulsory part of a task [Lahrichi82]. They compute a cumulated resource profile of all the compulsory parts of the tasks and prune the origins of the tasks with respect to this profile in order to not exceed the resource capacity. These methods are sometimes called time tabling. Even if these methods are quite local, i.e., a task has a non -empty compulsory part only when the difference between its latest start and its earliest start is strictly less than its duration, it scales well and is therefore widely used. Later on, more global algorithmsEven if these more global algorithms usually can prune more early in the search tree, these algorithms do not catch all deductions derived from the cumulated resource profile of compulsory parts. based on the resource consumption of the tasks on specific intervals were introduced [ErschlerLopez90], [CaseauLaburthe96b], [Lock96]. A popular variant, called edge finding, considers only specific intervals [MercierVanHentenryck08]. An efficient implementation of edge finding in , where is the number of distinct task heights and is the number of tasks, based on a specific data structure, so called a cumulative -tree [Vilim09a], is provided in [Vilim09b]. A filtering algorithm based on tasks that can not be the earliest (or not be the latest) is described in [SchuttWolf10].
Within the context of linear programming, the reference [HookerYan02] provides a relaxation of the constraint.
A necessary condition for the constraint is obtained by stating a constraint on a subset of tasks such that, for each pair of tasks of , the sum of the two corresponding minimum heights is strictly greater than . This can be done by applying the following procedure:
Let denote the set of tasks of the constraint for which the minimum height is greater than or equal to . By construction, the tasks of cannot overlap. But we can eventually add one more task as shown by the next step.
When it exists, we can add one task that does not belong to and such that its minimum height is strictly greater than . Again, by construction, this task cannot overlap all the tasks of .
In the context where, both the duration and height of all the tasks are fixed, [BeldiceanuCarlssonPoder08] provides two kinds of additional filtering algorithms that are specially useful when the slack (i.e., the difference between the available space and the sum of the surfaces of the tasks) is very small:
The first one introduces bounds for the so called cumulative longest hole problem. Given an integer that does not exceed the resource limit, and a subset of tasks for which the resource consumption is a most , the cumulative longest hole problem is to find the largest integer such that there is a cumulative placement of maximum height involving a subset of tasks of where, on one interval of the cumulative profile, the area of the empty space does not exceed .
The second one used dynamic programming for filtering so called balancing knapsack constraints. When the slack is 0, such constraints express the fact that the total height of tasks ending at instant must equal the total height of tasks starting at instant . Such constraints can be generalized to non -zero slack.
- See also
common keyword: (scheduling constraint), (resource constraint, sum of replaced by number of distinct values), (resource constraint), (resource constraint, defined by a set of ), (resource constraint, sum of replaced by product of ), (resource constraint, a constraint for each set of having a priority less than or equal to a given threshold).
generalisation: ( replaced by with a ).
implied by: ( is a neccessary condition for each dimension of the constraint).
related: , (lexicographic ordering on the origins of , , ), (controlling the shape of the cumulative profile for breaking symmetry).
specialisation: ( replaced by ), (all have a of 1 and a fixed ), (all have a of 1).
- Arc input(s)
- Arc generator
- Arc arity
- Arc constraint(s)
- Graph property(ies)
- Arc input(s)
- Arc generator
- Arc arity
- Arc constraint(s)
- Graph class
- Constraint(s) on sets
- Graph model
The first graph constraint enforces for each task the link between its origin, its duration and its end. The second graph constraint makes sure, for each time point corresponding to the start of a task, that the cumulated heights of the tasks that overlap does not exceed the limit of the resource.
Parts (A) and (B) of Figure 5.82.2 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 sum of the heights of the tasks in does not exceed the limit .
Figure 5.82.2. Initial and final graph of the constraint
Figure 5.82.3 depicts the automaton associated with the constraint. To each item of the collection corresponds a signature variable that is equal to 1.
Figure 5.82.3. Automaton of the constraint