Oversubscription Planning with Numeric Goals

1
Over-subscription Planning with Numeric Goals
Minh Do Palo Alto Research Center (PARC) Palo
Alto, CA

• J. Benton
• Computer Sci. Eng. Dept.
• Arizona State University
• Tempe, AZ

Subbarao Kambhampati Computer Sci. Eng.
Dept. Arizona State University Tempe, AZ
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Over-subscription Planning
300
300
Util 500
B
200
cost 200
cost 300

• Goals optional have utility
• Actions have cost
• Maximize utility-cost
• Benefit

Util 200
A
C
cost 500
-100
Initial At A Goals Soil_Sample _at_ B C
Rovers Example
The Mystery Talk, Smith 2003
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Motivation

• Numeric goals also have utility
• More soil gives better instrument reading
• More packages give more profit
• Cost for achieving varying values differs
• More soil requires more weight
• More packages require more deliveries

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Objective
Satisfy numeric goals at different values to
give varying utility

• Want more/less
• G soil-sample ? 2,4
• U(G) ( (soil-sample) 2)
• Challenge A measurable level of numeric goal
  achievement degree of satisfaction

action cost
soil collected
1 gram
Collect Cost1
1 gram
Collect Cost2
cost3
util224
1 gram
Collect Cost3
Benefit4-31
cost6
util326
Benefit6-60
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Modeling Numeric Goal Over-subscription
Infinity on range OK
1. Fixed utility for satisfying level

• Achieve with a given utility
• Specify a goal range

2. Linear
U t i l i t y
8
6
4
2
G soil-sample ? 2,4
0
1
2
3
4
Sample
U(G) ( (soil-sample) 2)
4. Model as a separate goal
3. Hard bounds
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SapaMps Architecture
Based on SapaPS
Over-subscribed Planning Planning Problem
Select state with best f-value
Input Initial State
Queue of Time-Stamped States
Better benefit plan?
Output Plan
Yes
No
Generate States by Applying Actions

Build RTPG Propagate Cost Find Utility
Anytime A Search
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Challenge Heuristic Support

• Heuristic needs to
• Estimate cost of achieving variable values
• Find the utility of the values
• Extend current state-of-the-art techniques
• Planning graph structure
• Reachability estimation
• Cost propagation

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Challenge Find Goal Achievement Cost

• Propagate reachable values with cost

Move(Waypoint1)
Sample_Soil
Sample_Soil
Communicate
A range of possible values
2
0
1
2.5
Cost of achieving each value bound
v1 0,0 0,1 0,2
cost( ) 0 1
2
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Cost Propagation on Variable Bounds
Sample_Soil Effect v11

• Bound cost dependent upon
• action cost
• previous bound cost- current bound cost adds to
  the next
• Cost of all bounds in expressions

Sample_Soil
Sample_Soil
v1 0,0 0,1 0,2
Cost(v12)
C(Sample_Soil)Cost(v11)
Sample_Soil Effect v1v2
Sample_Soil
Sample_Soil
v1 0,0 0,3 0,6
v2 0,3
Cost(v16)
C(Sample_Soil)Cost(v23)Cost(v13)
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Extracting Relaxed Plan with Numeric Info

• Start with best benefit bounds
• Relaxed plan includes
• Actions
• Supporting bounds

11
Dur 1
Dur 1.25
Dur 1.5
Sample_Soil 1 (Sa1)
Sample_Soil 2 (Sa2)
Communicate (Com)
(at start) V2 V1
Cost 1
(at end) V1 1
Cost 2
(at end) V1 2
Cost 3
(at start) V1 1
Sa1
C1
upper bound _at_ time point
0
1
1.25
2
2.5
3
3.75
4
t
value
v1
cost
value
v2
cost
v1 soil sample in rovers store
Goal v2 ? 5,8, U(v2 ? 5,8) v2 3
v2 soil sample communicated
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Dur 1
Dur 1.25
Dur 1.5
Sample_Soil 1 (Sa1)
Sample_Soil 2 (Sa2)
Communicate (Com)
Cost 1
(at end) V1 1
Cost 2
(at end) V1 2
Cost 3
(at start) V2 V1
(at start) V1 1
Sa1
C1
Sa2
C2
Com
C4
0
1
1.25
2
2.5
3
3.75
4
t
value
v1
satisfies goal
cost
value
v2
cost
h(S) U(G) - (cost of actions cost of bounds)
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Results Modified Rovers

• Added numeric variables
• Soil and rock sample amount in rover store
• More communicated soil/rock - greater utility

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Results Modified Rovers
Average improvement 3.06
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Anytime A Search Behavior
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Results Modified Logistics

• Added numeric variables
• Number of packages at location
• More packages - greater utility

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Results Modified Logistics
Average improvement 2.88
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Summary

• Over-subscription planning in the presence of
• Numeric goals
• Durative actions
• Propagating cost over numeric values

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Future Work

• Delayed satisfaction of goals
• Goal utility dependency

late -10
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• Questions.