Philippe Baptiste

Constraint-based scheduling and planning

Publisher Summary This chapter describes constraint-based scheduling as the discipline that studies how to solve scheduling problems by using constraint programming (CP). Constraint-based planning in turn is the discipline that studies how to solve planning problems by CP. The chapter discusses that constraint-based scheduling is one of the most successful application areas of CP. One of the key factors of this success lies in the fact that a combination was found of the best of two fields of research that pay attention to scheduling—namely, operations research (OR) and artificial intelligence (AI). The chapter reviews that OR approach aims at achieving a high level of efficiency in its algorithms whereas AI research tends to investigate more general scheduling models and tries to solve the problems by using general problem-solving paradigms. The use of CP in planning is because of the problem complexity, which is less mature than its use in scheduling. Constraint-based planning thus follows the same pattern as constraint-based scheduling where CP is used as a framework for integrating efficient special purpose algorithms into a flexible and expressive paradigm. It also presents CP models for scheduling together with descriptions of propagation techniques for constraints used in these models.

Comparison of Propagation Techniques

Many deductive rules and algorithms have been presented. The aim of this chapter is to compare the deductive power of these rules and algorithms, thereby providing a much clearer overview of the state of the art in the field.

Min-Sum Scheduling Problems

We illustrate and compare the efficiency of the constraint propagation algorithms described in previous chapters on two well-known scheduling problems. In Section 8.1 we study the problem of minimizing the weighted number of late jobs on m parallel identical machines and in Section 8.2 we consider a general-shop scheduling problem with sequence-dependent setup times and alternative machines where the optimization criteria are both makespan and sum of setup times.

Cumulative Scheduling Problems

Many industrial scheduling problems are variants, extensions or restrictions of the “Resource-Constrained Project Scheduling Problem”. Given (i) a set of resources with given capacities, (ii) a set of non-interruptible activities of given processing times, (iii) a network of precedence constraints between the activities, and (iv) for each activity and each resource the amount of the resource required by the activity over its execution, the goal of the RCPSP is to find a schedule meeting all the constraints whose makespan (i.e., the time at which all activities are finished) is minimal. The decision variant of the RCPSP, i.e., the problem of determining whether there exists a schedule of makespan smaller than a given deadline, is NP-hard in the strong sense [77].

Propagation of the One-Machine Resource Constraint

In this chapter we study several methods to propagate a One-Machine resource constraint: A set of n activities A1,…, A n require the same resource of capacity 1. The propagation of resource constraints is a purely deductive process that allows to deduce inconsistencies and to tighten the temporal characteristics of activities and resources. In the non-preemptive case (Section 2.1), the earliest start times and the latest end times of activities are updated. When preemption is allowed (Section 2.2), modifications of earliest end times and latest start times also apply.

Propagation of Objective Functions

In our model, a variable criterion represents the value taken by the objective function.

Propagation of Cumulative Constraints

criterion = F(end({A_1}),...,\;end({A_n}))

Constraint-Based Scheduling: Applying Constraint Programming to Scheduling Problems

(5.1) Considering the objective constraint and the resource constraints independently is not a problem when F is a “maximum” such as Cmax or Tmax. Indeed, the upper bound on criterion is directly propagated on the completion time of each activity, i.e., deadlines are efficiently tightened. The situation is much more complex for sum functions such as Σw i C i , Σw i T i or Σw i U i. For these functions, the constraint (5.1) has to be taken into account at each step of the search tree. An efficient constraint propagation technique must consider the resource constraints and the objective constraint simultaneously. In the following sections, we propose two efficient constraint-propagation techniques for the Σw i U i criterion and for the minimization of setup times.