Scheduling with Uncertain Resources

1. Scheduling with Uncertain Resources Eugene Fink, Jaime G. Carbonell, Ulas Bardak, Alex Carpentier, Steven Gardiner,Andrew Faulring, Blaze Iliev, P. Matthew Jennings,Brandon Rothrock, Mehrbod Sharifi,Konstantin Salomatin, Peter Smatana

2. Motivation The available knowledgeis inherently uncertain. We usually make decisionsbased on incomplete and partially inaccurate data.

3. Challenges • Representation of uncertainty • Fast reasoning based on uncertain knowledge • Elicitation of criticaladditional data • Learning of reasonable defaults

4. RADAR project Scheduling and resource allocation under uncertainty. Part of the RADAR project,aimed at building an intelligent assistant for an office manager.

5. Demo Planning a conferencebased on uncertain dataabout available resources and scheduling constraints.

6. Outline • Representation of uncertainty • Reasoning based on uncertain knowledge • Elicitation of missing data • Default assumptions Representation of uncertainty

7. Probability distributions Weight: Phone location: 95% purse 2% home 2% office 1% car probabilitydensity 140 160 Alternative representations • Approximations • Mary’s weight is about 150. Mary’s cell phone is probably in her purse. • Ranges or sets of possible values • Mary’s weight is between 140 and 160. Mary’s cell phone may be in her purse, office, home, or car.

8. BUT… DEFAULT APPROACH • We assume that small input changes do not cause large output changes WHICH MAY NOT WORK FOR SOME CASES Approximations Simple and intuitive approach, which usually does not require changes to standard algorithms.

9. amount ofmedication patient weight Approximations Example: Selecting an amount of medication. Since small input changes translate intosmall output changes, we can use anapproximate weight value.

10. chance ofoverloading 155 LB 140 LB load weight Approximations Example: Loading an elevator. We can adapt this procedure to the useof approximate weights by subtracting asafety margin from the weight limit.

11. If your weight isexactly 150 lb,you are a winner! prize player weight Approximations Example: Playing the “exact weight” game. If we use approximate weight values, we cannot determine the chances of winning.

12. BUT… We may lose the accuracy of computation, and we cannot evaluate the probabilities of different possible values. Ranges or sets of possible values • Explicit representation of a margin of error • Moderate changes to standard algorithms

13. chance ofoverloading load weight Ranges or sets of possible values Example: Loading an elevator. We identify the danger of overloading, but we cannot determine its probability.

14. prize player weight Ranges or sets of possible values Example: Playing the “exact weight” game. We still cannot determine the chances of winning.

15. BUT… • Major changes to standard algorithms • Major increase of the running time Probability distributions Accurate analysis of possible values and their probabilities.

16. Probability distributions Example: Playing the “exact weight” game. prize player weight We can determine possible outcomes and evaluate their probabilities.

17. probabilitydensity 140 150 160 weight Proposed approach ranges or sets of values ranges or setswith probabilities probability distributions We approximate a probability density function by a set of uniform distributions, and represent it as a set of ranges with probabilities. Weight: 0.1 chance: [140..145] 0.8 chance: [145..155] 0.1 chance: [155..160]

18. Uncertain data • Nominal values An uncertain nominal value is a set of possible values and their probabilities. Phone location: 0.95 chance: purse 0.02 chance: home 0.02 chance: office 0.01 chance: car

19. Uncertain data • Nominal values • Integers and reals An uncertain numeric value is a probability-density function represented by a set of uniform distributions. Weight: 0.1 chance: [140..145] 0.8 chance: [145..155] 0.1 chance: [155..160] probabilitydensity 140 150 160 weight

20. 0.2 chance 0.8 chance or a set of possible functions and their probabilities. Uncertain data • Nominal values • Integers and reals • Functions An uncertain function is apiecewise-linear function with uncertain coordinates amount ofmedication patient weight

21. Outline • Representation of uncertainty • Reasoning based on uncertain knowledge • Elicitation of missing data • Default assumptions

22. Arithmetic operations + - x ≤ ≠ ¬ • Logical operations • Function application μσ • Analysis of distributions Uncertainty arithmetic We have developed a library of basic operations on uncertain data, which input and output uncertain values.

23. BUT… • Approximate and relatively slow • Assumes that all probability distributions are independent Uncertainty arithmetic • Allows extension of standard algorithms to reasoning with uncertain values

24. RADAR application Scheduling and resource allocation based on uncertain knowledge of resources and constraints. • Uncertain room and event properties • Uncertain resource availability and prices We use an optimizer that searches for a schedule with the greatest expected quality.

25. Manual and auto scheduling Search time ScheduleQuality ScheduleQuality 0.83 0.83 0.80 0.78 0.72 Auto Auto Auto 0.63 Manual 0.9 Manual Manual 0.8 0.7 0.6 4 1 3 9 2 5 6 7 8 10 13 rooms 84 events 5 rooms 32 events 9 rooms 62 events Time (seconds) 13 rooms 84 events problem size RADAR results Scheduling of conference events. without uncertainty with uncertainty

26. Outline • Representation of uncertainty • Reasoning based on uncertain knowledge • Elicitation of missing data • Default assumptions

27. Elicitation challenge • Identification of critical missing data • Analysis of the trade-off between the cost of data acquisition and the expected performance improvements

28. Proposed approach • For each candidate question, estimate the probabilities of possible answers • For each possible answer, compute its cost, as well as its impact on the optimization utility • For each question, compute its expected impact on the overall utility, and select questions with highest expected impacts

29. Initial schedule: Posters Talk RADAR example: Initial schedule Available rooms: 2 1 3 • Assumptions: • Invited talk: – Needs a projector • Poster session: – Small room is OK – Needs no projector • Events: • Invited talk, 9–10am: Needs a large room • Poster session, 9–11am: Needs a room • Missing info: • Invited talk: – Projector need • Poster session: – Room size – Projector need

30. Useless info: There are no large rooms w/o a projector × Useless info: There are no unoccupied larger rooms × √ Potentially useful info RADAR example: Choice of questions Initial schedule: 2 1 Posters 3 Talk • Candidate questions: • Invited talk: Needs a projector? • Poster session:Needs a larger room? Needs a projector? • Events: • Invited talk, 9–10am: Needs a large room • Poster session, 9–11am: Needs a room

31. Initial schedule: 2 1 Posters 3 Talk New schedule: 2 1 3 Talk RADAR example: Improved schedule • Events: • Invited talk, 9–10am: Needs a large room • Poster session, 9–11am: Needs a room Info elicitation: System: Does the poster sessionneed a projector? Posters User:A projector may be useful,but not really necessary.

32. Dependency of the qualityon the number of questions Manual and auto repair ScheduleQuality ScheduleQuality 0.72 0.68 0.72 0.61 Auto withElicitation 0.50 Auto w/oElicitation ManualRepair After Crisis 0.68 10 30 40 50 20 Number of Questions RADAR results Repairing a conference schedule after a “crisis” loss of rooms.

33. Outline • Representation of uncertainty • Reasoning based on uncertain knowledge • Elicitation of missing data • Default assumptions

34. Example assumptions: • Almost all people weigh less than 500 lb • Tall people usually weigh more than short people Defaults assumptions Learning to make reasonable common-sense assumptions in the absence of specific data.

35. Defaults assumptions Learning to make reasonable common-sense assumptions in the absence of specific data. • Representation of general and context-specific assumptions • Dynamic learning of assumptions • Integration with data elicitation

36. ScheduleQuality 0.72 with default learning without learning 0.67 20 60 80 100 40 Number of Questions RADAR results Dependency of the schedule quality on the number of questions.

37. Reasoning under Uncertainty