1. Over-subscription Planning with Numeric Goals J. Benton Computer Sci. & Eng. Dept. Arizona State University Tempe, AZ Minh Do Palo Alto Research Center (PARC) Palo Alto, CA Subbarao Kambhampati Computer Sci. & Eng. Dept. Arizona State University Tempe, AZ

2. Over-subscription Planning  Goals optional & have utility  Actions have cost  Maximize utility-cost  “Benefit” cost = 200 cost = 500 cost = 300 Util = 500 Util = 200 B C A Initial: At A Goals: Soil_Sample @ B & C [“The Mystery Talk”, Smith 2003] -100 300 200 Rovers Example 300

3. 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

4. Objective  Want more/less G = soil-sample ∈ [2,4] U(G) = (* (soil-sample) 2)  Challenge – A measurable level of numeric goal achievement: degree of satisfaction Collect Cost=1 Collect Cost=2 1 gram cost=3 soil collected util=2*2=4 Collect Cost=3 1 gram action cost cost=6 util=3*2=6 Benefit=4- 3=1 Benefit=6-6=0 Satisfy numeric goals at different values to give varying utility BenefitBenefit v a l u e best benefit

5. Modeling Numeric Goal Over-subscription  Achieve with a given utility  Specify a goal range U(G) = (* (soil-sample) 2) G = soil-sample ∈ [2,4] 4 2 8 1 23 4 0 6 Sample UtilityUtility 1. Fixed utility for satisfying level 2. Linear 3. Hard bounds Infinity on range OK 4. Model as a separate goal

6. Sapa Mps Architecture Over-subscribed Planning Planning Problem Input Initial State Select state with best f-value Queue of Time-Stamped States Better benefit plan? Yes Output Plan Generate States by Applying Actions Build RTPG Propagate Cost Find Utility No Anytime A* Search Based on Sapa PS

7. 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

8. Challenge – Find Goal Achievement Cost  Propagate reachable values with cost Sample_Soil Communicate 01 2 2.5 Move(Waypoint1) Sample_Soil cost( ): 0 1 2 Cost of achieving each value bound v 1 : [0,0] [0,1] [0,2] A range of possible values

9. Cost Propagation on Variable Bounds  Bound cost dependent upon  action cost  previous bound cost - current bound cost adds to the next  Cost of all bounds in expressions Sample_Soil Cost(v 1 =2) Sample_Soil C(Sample_Soil)+Cost(v 1 =1) v 1 : [0,0] [0,1] [0,2] Sample_Soil Cost(v 1 =6) Sample_Soil C(Sample_Soil)+Cost(v 2 =3)+Cost(v 1 =3) v 1 : [0,0] [0,3] [0,6] v 2 : [0,3] Sample_Soil Effect: v 1 +=1 Sample_Soil Effect: v 1 +=v 2

10. Extracting Relaxed Plan with Numeric Info  Start with best benefit bounds  Relaxed plan includes  Actions  Supporting bounds BenefitBenefit v a l u e best benefit

11. Sample_Soil 1 (Sa1) Dur = 1 Cost: 1 (at end) V 1 += 1 Sample_Soil 2 (Sa2) Dur = 1.25 Cost: 2 (at end) V 1 += 2 Communicate (Com) Dur = 1.5 Cost: 3 (at start) V 1 ≥ 1 Sa1 t 011.2522.533.75 C:1 Sa1 C:1 Sa1 C:1 Sa2 C:2 Sa2 C:2 Sa2 C:2 Com C:4 Com C:4 4 Goal: v2 ∈ [5,∞], U(v2 ∈ [5,∞]) = v2 * 3 (at start) V 2 := V 1 v1v1 value cost value cost v2v2 upper bound @ time point v 1 – soil sample in rover’s store v 2 – soil sample communicated

12. Sample_Soil 1 (Sa1) Dur = 1 Cost: 1 (at end) V 1 += 1 Sample_Soil 2 (Sa2) Dur = 1.25 Cost: 2 (at end) V 1 += 2 Communicate (Com) Dur = 1.5 Cost: 3 (at start) V 2 := V 1 (at start) V 1 ≥ 1 Sa1 t 011.2522.533.75 C:1 Sa1 C:1 Sa1 C:1 Sa2 C:2 Sa2 C:2 Sa2 C:2 Com C:4 4 v1v1 value cost value cost Com C:4 satisfies goal h(S) = U(G) - (cost of actions + cost of bounds) v2v2

13. Results – Modified Rovers  Added numeric variables:  Soil and rock sample amount in rover store  More communicated soil/rock - greater utility

14. Average improvement: 3.06 Results – Modified Rovers

15. Anytime A* Search Behavior

16. Results – Modified Logistics  Added numeric variables:  Number of packages at location  More packages - greater utility

17. Results – Modified Logistics Average improvement: 2.88

18. Summary  Over-subscription planning in the presence of  Numeric goals  Durative actions  Propagating cost over numeric values

19. Future Work  Delayed satisfaction of goals  Goal utility dependency late -10 late -10

20. Questions.