# Constraint-Based Scheduling: Applying Constraint Programming by Philippe Baptiste

By Philippe Baptiste

Constraint Programming is a problem-solving paradigm that establishes a transparent contrast among pivotal features of an issue: (1) an actual definition of the restrictions that outline the matter to be solved and (2) the algorithms and heuristics allowing the choice of choices to unravel the matter.

this is why of those services that Constraint Programming is more and more being hired as a problem-solving software to unravel scheduling difficulties. therefore the advance of Constraint-Based Scheduling as a box of analysis.

the purpose of this ebook is to supply an outline of the main customary Constraint-Based Scheduling recommendations. Following the rules of Constraint Programming, the booklet includes 3 designated elements:

- the 1st bankruptcy introduces the elemental rules of Constraint Programming and gives a version of the limitations which are the main usually encountered in scheduling difficulties.
- Chapters 2, three, four, and five are thinking about the propagation of source constraints, which generally are answerable for the "hardness" of the scheduling challenge.
- Chapters 6, 7, and eight are devoted to the answer of numerous scheduling difficulties. those examples illustrate the use and the sensible potency of the constraint propagation equipment of the former chapters. additionally they exhibit that in addition to constraint propagation, the exploration of the hunt house needs to be rigorously designed, bearing in mind particular houses of the thought of challenge (e.g., dominance family members, symmetries, attainable use of decomposition rules).

bankruptcy nine mentions a number of extensions of the version and offers promising learn directions.

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**Example text**

Notice that tyld allows to keep the previous value of tl. • Algorithm 5 describes the procedure update(Lst, tyld, tl) that reorders the array Lst in increasing order of di - pi(tl). This procedure will be described later on. • Before starting the inner loop, a variable slope is initialized (line 7). It corresponds to the slope of the function t --+ W PE(tl, t) immediately 53 Propagation of Cumulative Constraints Algorithm 6 Computation of W PE( tl, t2) 1: 2: 3: 4: D:= activities sorted in increasing order of di Lst:= activities sorted in increasing order of di - Pi tyld := mini(r;) for tl in the set of earliest start times (sorted in inc.

An algorithm relying on Time-Table, network flow, disjunctive constraint, or Edge-Finding) on activities A~, ... , A~ of the instance F(I). 2, for each activity A~, four time bounds can be sharpened: the earliest start time ~, the latest possible start time ls~, the earliest possible end time ee~, and the latest end time <. 3 Update the four time bounds of each Ai. 2. Note that again the Edge-Finding technique provides the best possible bounds. Proposition 9. , the lower and upper bounds for the start and end time of activities can be reached by some feasible fully elastic schedules.

Propagation of the Not-First Not-Last rule. 4, the Not-First rule deduces that Ai cannot start before 2 while the other deductive rules seen up to now deduce nothing. The problem which consists of performing all the time-bound adjustments corresponding to the first and third rules can be called the "NotFirst" problem, since it consists of updating the earliest start time of every activity Ai which cannot be first to execute in a set 0 U {Ad. Similarly, the problem which consists of performing all the time-bound adjustments corresponding to the second and fourth rules can be called the "Not-Last" problem.