scheduling with uncertain resources search for a near-optimal solution eugene fink, matthew...
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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
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.
Representation
Available rooms Conference events Schedule
Available rooms Room name
Availability
Auditorium Conf. room
Properties
Size: 1200Stations: 10Mikes: 5
Size: 700Stations: 5Mikes: 1
Size: 500Stations: 5Mikes: 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
Available rooms
ClassroomSize: 700Stations: 5Mikes: 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.
Auditorium Conf. roomSize: 1200Stations: 10Mikes: 5
Size: 500..750Stations: 5Mikes: 2
Conference events
For every other event, thedistance to that event
For every other event, thestart 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
Conference eventsConstraints on times and room properties
Constraints on distances and relative times
Conference events We represent uncertain importances and range
boundaries by intervals of possible values.
DemoImportance: 4..6Minimal duration: 60..90Preferred duration: 90..120 ...
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-sionComm-
ittee
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.
Schedule qualityWe compute the quality for each event.
The schedule quality is the weighted sum of event quality values:
Quality = Importance1 ∙ Quality1
+ Importance2 ∙ Quality2
+ …
If the specification of rooms and events includes uncertainty, we compute the expected quality:
Quality = E(Importance1) ∙ E(Quality1)
+ E(Importance2) ∙ E(Quality2)
+ …
The schedule quality is the weighted sum of event quality values.
At each step, reschedule one event
Use randomized hill-climbing
Search
Stop after finding a local maximum
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 decreasingorder of their importances
Search
If found any new placements,repeat from the beginning
ExperimentsScheduling of a large conference Eighty-four events Four days, fourteen rooms 2500 numeric values
Experiments: W/o uncertainty
14 rooms84 events
5 rooms32 events
9 rooms62 events
0.6
0.7
0.8
0.9
1.0
ScheduleQuality
0.61
0.92
Manual
Auto
matic
0.94
Manual
Auto
matic
0.83
0.94
Auto
matic
0.93
problem size
Experiments: With uncertainty
0.5
0.6
0.7
0.8
0.9
ScheduleQuality
0.63
0.78
Manual
Auto
matic
0.8
Manual
Auto
matic
0.72
Manu
al0.83
Auto
matic
0.83
problem size
14 rooms84 events
5 rooms32 events
9 rooms62 events
withoutuncertainty
withuncertainty
10
0.8
0.9
0.7
0.61 2 3 4 5 6 7 8 9
ScheduleQuality
Time (seconds)14 rooms84 events
Experiments: Search time
Conclusions
Optimization based on uncertainknowledge of available resourcesand scheduling constraints
Fast high-quality solutions forlarge real-life problems