Scheduling with uncertain resources Search for a near-optimal solution

1. Scheduling with uncertain resourcesSearch 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. Properties • Distances Dist:400 Size: 1200 Stations: 10 Mikes: 5 Size: 500 Stations: 5 Mikes: 2 Dist:50 Dist:400 Size: 700 Stations: 5 Mikes: 1 Available rooms • Room name Unavailable • Availability Auditorium Conf. room Unavailable Classroom Unavailable

5. Available rooms We represent uncertain properties and distances by intervals of possible values. Unavailable Auditorium Conf. room Size: 1200 Stations: 10 Mikes: 5 Size: 500..750 Stations: 5 Mikes: 2 Unavailable Classroom Dist:50..70 Dist:400 Size: 700 Stations: 5 Mikes: 1 Unavailable

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

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

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

9. Schedule For every event, we need to select: • Room • Start time • Duration Demo Tutorial Unavailable Work-shop Unavailable Discus-sion Unavailable Comm- ittee

10. Schedule quality We compute the quality for each event. • 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

11. The schedule quality is the weighted sum of event quality values. If the specification of rooms and events includes uncertainty, we compute the expected quality: Quality = E(Importance1) ∙ E(Quality1) + E(Importance2) ∙ E(Quality2) + … Schedule quality We compute the quality for each event. The schedule quality is the weighted sum of event quality values: Quality = Importance1 ∙ Quality1 + Importance2 ∙ Quality2 + …

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

13. Search • Sort events in the decreasingorder of their importances • For each event:- Consider all possible placements, i.e. rooms, start times, and durations- Select the placement with the highest expected quality • 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. 0.94 0.94 0.93 0.92 Automatic Automatic Manual 0.83 Automatic Manual 0.61 Experiments: W/o uncertainty Schedule Quality 1.0 0.9 0.8 0.7 0.6 9 rooms62 events 5 rooms32 events 14 rooms84 events problem size

16. 0.83 0.83 0.8 0.78 Automatic Automatic Manual 0.72 Automatic Manual 0.63 Manual Experiments: With uncertainty Schedule Quality 0.9 0.8 0.7 0.6 0.5 9 rooms62 events 5 rooms32 events 14 rooms84 events problem size

17. Experiments: Search time ScheduleQuality without uncertainty 0.9 0.8 with uncertainty 0.7 0.6 4 1 3 9 2 5 6 7 8 10 Time (seconds) 14 rooms 84 events

18. Conclusions • Optimization based on uncertainknowledge of available resourcesand scheduling constraints • Fast high-quality solutions forlarge real-life problems