Filtering for a Continuous Multi-Resources cumulative Constraint with Resource Consumption and Production

Within the framework of continuous and multiresources cumulative constraints, a task T expresses a piecewise linear resource function and is represented by a sequence of p contiguous trapezoid sub-tasks with variable durations and heights. In this context, this paper provides an algorithm in O(p) for filtering the resource assignment and the temporal attributes of such a task, according to a trapezoid of a minimum cumulated resource profile, in order to avoid an overflow of the resource capacity. Introduction The vast majority of existing models concerning resourceconstrained scheduling deal with tasks (activities) that consume or produce a constant amount of resource during their execution (Herroelen, Demeulemeester and De Reyck, 1998). But tasks that consume or produce resource in a continuous way are ubiquitous in applications areas such as electricity or chemical production (Muscettola 2002), (Sourd and Rogerie 2005) and (Maravelias and Grossmann 2004). Recently, in (Beldiceanu and Poder 2007) we have extended the task model proposed in (Poder, Beldiceanu and Sanlaville 2004) for modelling piecewise linear consumptions or productions and introduced the corresponding cumulatives pwl constraint: a task is represented by a sequence of contiguous trapezoid sub-tasks with variable durations and heights (positive or negative) and to a task corresponds a set of possible resource assignments. They have given a sweep algorithm to compute for each resource its minimum and maximum cumulated resource’s profile. In this paper, we focus on the central sub-problem of filtering the resource assignment and the temporal attributes (start, duration, end of a task and duration of its sub-tasks) of a task according to one trapezoid of a minimum cumulated resource profile in order to avoid an overflow of the resource capacity. This filtering is absolutely required for implementing the cumulatives pwl (Beldiceanu and Poder 2007) or the cumulative trapeze (CHIP 2001) (Poder 2002) constraints and no result about filtering a sequence of trapezoid sub-tasks has been yet published. The main challenge Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. comes from the fact that all attributes of a task are variable. This filtering may be done while computing, with the sweep algorithm and for a given resource, the successive trapezoids building its minimum cumulated resource profile. In this paper, we first recall the cumulatives pwl constraint and the notion of minimum cumulated resource’s profile. Then, we give the main applications of this constraint and we describe the contribution of this paper that is, given a task and a trapezoid of a minimum cumulated resource’s profile, how to filter the task’s attributes according to this trapezoid in order to avoid an overflow of the resource’s capacity.1 Finally, we conclude. Background We consider a set of q resources where the k resource has a maximum capacity Ck ≥ 0 and a set of n non–preemptive tasks. Each task needs for its execution to be assigned to exactly one resource within a given subset (that depends on the considered task) of the q resources. Finally, each task is composed of a sequence of contiguous trapezoid sub-tasks which expresses a piecewise linear function of resource. Definition 1 A task Ti is defined by the quintuple (sTi , tdTi , eTi ,SeqTi , aTi) where: • The variables2 sTi , tdTi , and eTi represent respectively the start, the total duration and the end of task Ti. • SeqTi is a sequence of pi contiguous trapezoid subtasks 〈T 1 i , T 2 i , · · · , T pi i 〉 where T j i is defined by the triple (shT j i , dT j i , ehT j i ) which represents its start height (i.e., the resource requirement at its start), duration and end height (i.e., the resource requirement at its end). An height may be positive or negative. Moreover, dT j i ≥ 0 and w.l.o.g., we assume that the two heights of a sub-tasks have the same signs. In the context of both non negative heights and one single resource, a detailed version of the filtering of each attribute of a task (start, total duration, end) and of its sub-tasks (duration, heights) is available in (Poder 2002)). A variable v may range over the interval of consecutive integer [v..v] or over the rational interval [v, v]. The results given in this paper may be used in the two cases. [a, b] (resp. [a, b[) denotes the rational interval between a and b included (resp. b not included). 264 Proceedings of the Eighteenth International Conference on Automated Planning and Scheduling (ICAPS 2008)