1. Iterative Flattening in Cumulative Scheduling

2. Cumulative Scheduling Problem Set of Jobs Each job consists of a sequence of activities Each activity has –a duration –a machine on which it executes –a demand for the capacity of this machine

3. Cumulative Scheduling Problem Each machine has a capacity Precedence constraint between activities of a single job objective: –satisfy all constraints –finish all the jobs as soon as possible

4. Problem Modeling Cumulative constraints –considered as soft constraints precedence constraints –considered as hard constraints

5. Iterative Flattening Iteratively –Iteratively FLATTEN Overcoming over-utilization –RELAX Trying to overcome under-utilization Return the schedule with the earliest finish time

6. Precedence Graph Nodes –the activities, a source and a sink Edges –precedence constraints between activities

8. Scheduling with Precedence Graphs The Makespan of an schedule translates to the length of the longest path of the graph Earliest start time of an activity –the length of the longest path from the source to that node Latest start time of an activity –The makespan minus its earliest start time A precedence graph defines an schedule –assigns each activity to its earliest start time

9. Flattening and Relaxation in Precedence Graphs Add precedence to satisfy cumulative constraints –this flattens the graph Remove precedence to make the critical path shorter –This relaxes the graph

10. FLATTEN Choose the time with the maximum over- utilization Find minimal critical sets (MCS) of the activities running on that time for that machine (If you take one activity out of a MCS, over- utilization will be removed) Choose a pair of activities in each MCS with the least impact on the makespan and introduce precedence between them

11. Critical Set A critical set of a given time for a given machine –A set of activities running on that time –sum of their demands exceeds the capacity of the machine Minimal Critical Set is a critical set whose every proper subset is not a critical set

12. Maximum Flexibility In flattening –Choose a pair from the critical set with smallest impact on the makespan –Introduce a precedence between them This pair has to have maximum flexibility –Latest start time of the second minus earliest finish time of the first

13. RELAX trying to overcome under-utilization

14. Relaxation Considers each precedence constraint on the critical path that were introduced in some flattening step and removes each precedence with some probability Only removing edges on the longest path may reduce the length of the longest path

15. Limitations of Iterative Flattening There is a 10% gap between the quality of solutions found by IFLAT and best known solutions In the example below removing all critical arcs does not affect the early parts of the algorithm

17. Solution: Iterative Relaxation It packs the schedule and thus the profiles of each machine Changes the critical path, exposing bottlenecks that were previously unknown Relaxing arcs on a new critical path makes it possible to schedule some of the activities earlier

20. Experimental Results It usually takes about 100,000’s iterations It has been tested for scheduling as many as 900 activities It delivers solutions that are within 1% of the best available upper bounds in the average

21. References Cesta, A.; Oddi, A.; and Smith, S. F. 1999b. An iterative sampling procedure for resource constrained project scheduling with time windows. In IJCAI, 1022–1033. Michel, L., and Van Hentenryck, P. 2004. Iterative Relaxations for Iterative Flattening in Cumulative Scheduling. In ICAPS