Solving Dynamic Rcpsp Using Explanation-based Constraint Programming

The Resource Constrained Project Scheduling Problem (rcpsp) is a general scheduling problem. It consists of a set of activities and a set of renewable resources. Each resource is available in a given constant amount. Each activity has a duration and requires a constant amount of resource to be processed. Preemption is not allowed. Activities are related by two sets of constraints: temporal constraints modelled through precedence constraints, and resource constraints that state that for each time period and for each resource, the total demand cannot exceed the resource capacity. The objective considered here is the minimization of the makespan (total duration) of the project. This problem is NP-hard [3]. Most work about rcpsp consider static problems in which activities are known in advance and constraints are fixed. However, every schedule is subject to unexpected events (consider for example a new activity to schedule, or a resource failure – eg. machine breakdown). When such a situation arises, a new solution, taking these events into account, is needed in a preferably short time. Two classical methods used to solve such problems are: recomputing a new schedule from scratch each time an event occurs (a quite time consuming technique) and constructing a partial schedule and completing it progressively as time goes by (like in on-line scheduling problems – this is not compatible with planning purposes). Recently, Artigues et al. [1] developed a polynomial algorithm based on a flow network model to update an initial static schedule when considering the insertion of an unexpected activity. Constraint Satisfaction Problems (csp) are increasingly used for solving scheduling problems. However, dynamic csp (an extension of the csp framework where the set of variables or/and constraints evolves throughout computation [7]) are not as spread. We introduce here such a new method applied to dynamic rcpsp. Efficiently solving dynamic problems requires incremental addition and retraction of constraints. Even though incremental constraint addition is naturally handled by modern constraint solvers, incremental retraction of constraints is often performed with recording trace/undo information [2]. Such an information is used to determine past effects of removed constraint that need to be undone. Explanations are a generalization of that information. An explanation is a set of constraints that justifies an action of the solver (classically value removals) i.e. as long as each constraint that appears in the explanation remains active (not removed) the value removal is valid; thus, no valid solution can be build from this partial assignment [10]. Providing explanations for temporal binary constraints is straightforward (as they are binary mathematical relations – see [10]). Explanations for resource management constraints are not that easy. It is necessary to study the algorithms used for propagation (resource limitations maintenance): core-times [11], task-interval [6, 5] and resource-histogram [6, 5]. We added explanation capabilities to these techniques [8]. Moreover, we developed a branch and bound search using explanation-based techniques (as in [9]) and inspired from a branch and bound algorithm from [4]. Our interactive system accepts several types of modification on the scheduling problem: temporal events (adding/removing precedence/overlapping/ disjunctive relations, modifying time-windows), activity related events (adding/removing),