Modelling and managing disjunctions in scheduling problems

In this paper is presented a method for modelling and managing various constraints encountered in task scheduling problems. The approach aims at characterizing feasible schedules through the analysis of the set of constraints and their interaction, regardless of any optimization criteria. This analysis is achieved by a constraint propagation process on a constraint graph and produces both restricted domains for the decision variables and an updated formulation of the initial constraints. The graphs usually used to model temporal constraints seem to be limited because they only allow the representation of strict precedence relations between two tasks. In order to model a larger variety of temporal constraints, particularly any constraint that connects two events (start- or finish- time of a task), a model called a time-bound-on-node (TBON) graph is proposed in which each task is featured by two nodes. Then it becomes possible to handle constraints on task durations, due for example to flexibilities in resource utilization. This kind of graph is not new and has already been investigated in related works on project planning and Constraint Satisfaction Problems. But its processing and interpretation deserved to he developed, particularly for the present purpose, which is the search for the necessary conditions of feasibility. With respect to conjunctive temporal constraints, the analysis is achieved with a polynomial algorithm based on the longest path search on a conjunctive TBON graph, yielding the necessary and sufficient conditions of feasibility. Taking account of resource constraints leads to defining disjunctive constraints. To this end, disjunctive sets of arcs are introduced, making the TBON graph nonconjunctive. In this case, a complete characterization of feasibility cannot reasonably be faced, due to the combinatorial feature. Nevertheless, a polynomial algorithm that applies reduction and deletion rules on the nonconjunctive part of the graph is proposed to restart new propagations on the conjunctive part until all deductions have been made.