1. Scheduling with uncertain resources Search for a near-optimal solution Eugene Fink, Matthew Jennings, Ulaş Bardak, Jean Oh, Stephen Smith, and Jaime Carbonell Carnegie Mellon University

2. Problem Scheduling a conference under uncertainty Uncertain room properties Uncertain equipment needs Uncertain speaker preferences We need to build a schedule with high expected quality.

3. Representation Available rooms Conference events Schedule

4. Available rooms Room name Availability AuditoriumConf. room Properties Size: 1200 Stations: 10 Mikes: 5 Size: 700 Stations: 5 Mikes: 1 Size: 500 Stations: 5 Mikes: 2 Audit- orium Class- room Conf. room 11:00 11:30 12:00 12:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 Classroom Distances Dist: 400 Dist: 50 Dist: 400

5. Available rooms Classroom Size: 700 Stations: 5 Mikes: 1 Dist: 50..70 Dist: 400 Audit- orium Class- room Conf. room 11:00 11:30 12:00 12:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 We represent uncertain properties and distances by intervals of possible values. AuditoriumConf. room Size: 1200 Stations: 10 Mikes: 5 Size: 500..750 Stations: 5 Mikes: 2

6. Conference events For every other event, the distance to that event For every other event, the start time w.r.t. that event We specify the name and numeric importance of an event. We also specify acceptable and preferred ranges for the following parameters: Every room property Start time and duration

7. Conference events Constraints on times and room properties Constraints on distances and relative times

8. Conference events We represent uncertain importances and range boundaries by intervals of possible values. Demo Importance: 4..6 Minimal duration: 60..90 Preferred duration: 90..120...

9. Schedule For every event, we need to select: Room Start time Duration Audit- orium Class- room Conf. room 11:00 11:30 12:00 12:30 1:00 1:30 2:00 2:30 3:00 3:30 4:00 Demo Tutorial Work- shop Discus- sion Comm- ittee

10. Schedule quality If start time, duration, room properties, distances, or relative times are outside their acceptable ranges, the quality is 0.0 If all these values are within their preferred ranges, the quality is 1.0 If all these values are acceptable, but some are not preferred, the quality is between 0.0 and 1.0 We compute the quality for each event.

11. Schedule quality We compute the quality for each event. The schedule quality is the weighted sum of event quality values: Quality = Importance 1 ∙ Quality 1 + Importance 2 ∙ Quality 2 + … If the specification of rooms and events includes uncertainty, we compute the expected quality: Quality = E(Importance 1 ) ∙ E(Quality 1 ) + E(Importance 2 ) ∙ E(Quality 2 ) + … The schedule quality is the weighted sum of event quality values.

12. At each step, reschedule one event Use randomized hill-climbing Search Stop after finding a local maximum

13. For each event: - Consider all possible placements, i.e. rooms, start times, and durations - Select the placement with the highest expected quality Sort events in the decreasing order of their importances Search If found any new placements, repeat from the beginning

14. Experiments Scheduling of a large conference Eighty-four events Four days, fourteen rooms 2500 numeric values

15. Experiments: W/o uncertainty 14 rooms 84 events 5 rooms 32 events 9 rooms 62 events 0.6 0.7 0.8 0.9 1.0 Schedule Quality 0.61 0.92 Manual Automatic 0.94 Manual Automatic 0.83 0.94 Automatic 0.93 problem size

16. Experiments: With uncertainty 0.5 0.6 0.7 0.8 0.9 Schedule Quality 0.63 0.78 Manual Automatic 0.8 Manual Automatic 0.72 Manual 0.83 Automatic 0.83 problem size 14 rooms 84 events 5 rooms 32 events 9 rooms 62 events

17. without uncertainty with uncertainty 10 0.8 0.9 0.7 0.6 1 2 3 4 5678 9 Schedule Quality Time (seconds) 14 rooms 84 events Experiments: Search time

18. Conclusions Optimization based on uncertain knowledge of available resources and scheduling constraints Fast high-quality solutions for large real-life problems