Modelling and managing disjunctions in scheduling problems

Abstract

__________________________________________________________________________In this paper we present a method for modeling and managing various constraints encountered in task scheduling problems. Our approach aims at characterizing feasible schedules through the analysis of the set of constraints and their interaction, regardless to 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), we propose a model called a Time-Bound-On-Node graph 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 be developed, particularly for our 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 non conjunctive. 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 non conjunctive part of the graph is proposed to restart new propagations on the conjunctive part until all deductions have been made.

Generalized disjunctive constraint propagation for solving the job shop problem with time lags

“…To represent a broader variety of time constraints in scheduling problems, the disjunctive time-bound-on-node (TBON) graph has been introduced [11]. The disjunctive TBON representation is equivalent to the disjunctive AON graph, but we choose to use it in the Table 1 sequel since it allows the distinct visualization of the different components of the problem: duration time lags, start and finish times, although it yields a larger number of nodes.…”

Anais do XX Simpósio Brasileiro de Engenharia de Software (SBES 2006)

“…It is considered that the maximum allowed duration of a case for the "Handle Complaint Process" is 105 time units. Knowing the beginning date of a case, it is possible to calculate the previsional visibility intervals associated to the tokens in the waiting places using constraint propagation techniques classically used in scheduling problems based on activity-on-arc graphs [Esquirol and Lopez 1995]. On the Petri net of figure 4, two cases represented by two tokens in place E 0 are considered.…”

“…The dynamic behavior of a system imposes a scheduling of control flow. The scheduling problem [Esquirol and Lopez 1995] consists of organizing in time the sequence of activities considering time constraints (time intervals) and constraints of shared resources utilization necessary for activity execution. From the traditional point of view of Software Engineering [Pressman 1995], the scheduling problem is similar to the activity of scenario execution.…”

Characterization of all ρ-approximated sequences for some scheduling problems

“…Among the possible dynamic approaches, some of them are based on the characterization of solutions [7,8,4,2,1]. Characterization means that several solutions are implicitly proposed, without enumeration, instead of only one, and a system is designed to exploit this flexibility for reacting in real time.…”