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Valentin Polishchuk Joint work with Joseph S.B. Mitchell, Ph.D. and Jimmy Krozel, Ph.D.

Estimating Capacity and Routing Flows Among Known and Probabilistic Constraints. Valentin Polishchuk Joint work with Joseph S.B. Mitchell, Ph.D. and Jimmy Krozel, Ph.D. Problem Statement. Given airspace constraints (“obstacles”) Estimate capacity Find air routes. Airspace. 2D

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Valentin Polishchuk Joint work with Joseph S.B. Mitchell, Ph.D. and Jimmy Krozel, Ph.D.

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  1. Estimating Capacity and Routing Flows Among Known and Probabilistic Constraints Valentin Polishchuk Joint work with Joseph S.B. Mitchell, Ph.D. and Jimmy Krozel, Ph.D.

  2. Problem Statement • Given • airspace • constraints (“obstacles”) • Estimate • capacity • Find • air routes

  3. Airspace • 2D • en-route • A simple polygon • Polygonal obstacles • no-fly zones • always there • weather • stochastic • moving • Source and Sink • boundary edges

  4. Versions • Deterministic vs. stochastic • Static vs. dynamic • Depends on look-ahead time

  5. Overview • Network Flows • Flows in 2D • Airspace capacity estimation • deterministic case • stochastic case • Route planning • deterministic case (many routes) • stochastic case (one route) • Future work • dynamic setting • stochastic case • route planning (many routes)

  6. Network Flows

  7. 1 4 1 3 1 4 2 1 Network / Graph • Nodes • source s and sink t • Arcs (i,j) • capacities uij • integers s t

  8. 1 4 1 3 1 4 2 1 Cut • Partition nodes into sets S and T, with • s in S • t in T • Arc (i,j) crosses the cut if • i in S • j in T • Capacity of the cut: • sum of capacities • of arcs that cross the cut s t 6

  9. 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 Flow • Assignment • integer fij • for each arc (i,j) • Respect capacities • fij≤ uij • Conserve at the nodes • node i ≠ s,t • flow in = flow out • ∑(j,i) fij = ∑(i,k) fik • Value • ∑(s,i) fsi = ∑(j,t) fjt 1 1 s t 1 1 1

  10. Two Basic Problems • Max Flow • maximize value • Min-Cost Flow • costs cij • minimize cost ∑ cij fij Efficient algorithms known

  11. 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 Maxflow-Mincut Theorem maxflow = mincut 2 = 2 1 1 1 s t 1 1 1

  12. 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 Flow Decomposition Theorem flow = union of paths “carrying” pieces of the flow 1 1 1 s t 1 1 1

  13. 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 Network Flows (Summary) • Given • network • capacities • costs • Find • maxflow • min-cost flow • Maxflow/Mincut Theorem • Flow Decomposition Theorem 1 1 1 s t 1 1 1 s t

  14. 1 4 1 3 1 4 2 1 Shortest Path • Given • network • costs on edges • Find • shortest s-t path s t

  15. Flows in 2D Geometry

  16. Problem Statement • Polygonal domain P • source and sink edges • Cut • partition domain into subsets S and T • capacity • length of the boundary shared by S and T • Flow • vector field

  17. 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 Discrete Network 2D Domain • Source and sink nodes • Cut • partition nodes • capacity • edges that cross • Flow • integers on arcs • Source and sink edges • Cut • partition domain • capacity • length of the boundary • Flow • vector field 1 1 1 s t 1 1 1

  18. Flow Decomposition 1 1 4 4 1 1 3 3 1 1 4 4 2 2 1 1 Maxflow = Mincut 1 1 1 s s t t 1 1 1 [Mitchell’90] [MP’06] How do we use all this in ATM?

  19. Capacity Estimation

  20. Map the Theory to ATM Domain Important issueNot a continuous fluid flow Maximum integer number of air lanes

  21. Algorithm: Mincut (Deterministic) 1. Airspace with Hazardous Weather Constraints

  22. Algorithm: Mincut (Deterministic) 2. Critical Graph: nodes for each obstacle, B, T

  23. Algorithm: Mincut (Deterministic) 3. Cost: integer number of air lanes cij = | dij / w |

  24. Algorithm: Mincut (Deterministic) 4. Search for Shortest B-T Path within Critical Graph

  25. Algorithm: Mincut (Deterministic) 5. Shortest B-T Pathdefines the mincut → defines capacity

  26. Overall: Known Constraints Given: Airspace with Hazardous Weather Constraints

  27. Overall: Known Constraints Output: capacity

  28. Stochastic Case • Given: • N forecasts (scenarios) • probability pk that the kth forecast is the final outcome • Output: • Capacity X (a random variable) • distribution of X

  29. Algorithm: Mincut (Probability Distribution) • For each Scenario k (k=1,2,…,N): • Create critical graph • Search for mincut • Record mincut length xk • Output (Capacity = X): • - FX(t) = Sk:xk≤tpk Cumulative Distribution Function • - E[X] = Skxkpk Expected Value • - Var[X] = E[X2] – E[X]2 = Skxk2pk – E[X]2 Variance

  30. Example: Mincut (Probability Distribution) NWP Model 1 NWP Model 2 NWP Model 3 4 air lanes 4 air lanes 7 air lanes Output (Capacity = X): - E(X) = Skxkpk = 70%(4) + 10%(4) + 20%(7) = 4.6 air lanes - Var(X) = Skxk2pk - [E(X)]2 = 1.44 - FX(t) = Sk:xk ≤tpk  P(X=4) = 80% P(X=7) = 20%

  31. Matlab algorithm for Capacity Estimation • Identifies flow “Bottleneck” • “Projects into future” by growing obstacles • for how long is this good? • how bad a weather will change the picture? • For growing weather cases, the mincut moves around over time For simplicity of implementation/random generation in experiments: Constraints modeled as a union of disks (though method allows any shape of constraints)

  32. Movie Time! • RNP 10 • RNP 20

  33. Popcorn Convection vs Squall Line Organization flow flow 44800 Synthetic Weather Data Experiments

  34. Real World Weather Scenario RNP-3 RNP-5 RNP-10 Linear  looks like Squall Line Convection

  35. Probability x Air Lanes will get through? • How many air lanes can we get through the FCA when we have a probabilistic weather forecast model? • Probability that x air lanes get through decreases with increasing RNP requirements 1 1 2 2 3 4 3 5 6 4 5

  36. Route Planning

  37. Known Constraints Given: Airspace with Hazardous Weather Constraints

  38. Known Constraints Output: capacity

  39. Routes too! Find the air lanes

  40. Geometric Shortest Paths • Given: • (Outer) Polygon P • Polygonal obstacles • Points s and t • Find: • shortest s-t path • avoiding obstacles • Efficient Algorithms Known

  41. Visibility Graph Shortest Path lies on VG search VG worst-case size/time to compute is O(n2)

  42. Shortest Path Map Decomposition of the free space Shortest Paths go through same vertices worst-case size/time is O(n) / O(n log n)

  43. Multiple Paths • Given • K pairs of terminals (sk,tk) • Find • shortest sk-tk paths • minsum • minmax • non-crossing • NP-hard in general • Terminals on the boundary of P • linear time OK Not allowed

  44. Thick Paths Thick path = reference_path + unit_circle Minkowski sum Place center everywhere on the reference path Length = length(reference path)

  45. Finding 1 Shortest Thick Path Inflate by 1 Find shortest reference path Inflate the reference path

  46. K Thick Non-Crossing Paths • Shortest sk-tk paths • minsum • minmax • Thick • Non-intersecting OK Not allowed

  47. Our Solution: K = 2 Inflate by 1 Route red path Inflate by 2 Find shortest reference path Same on the other side Inflate reference paths

  48. K > 2 Map to circle For k = 1…K Inflate correspondingly Find shortest reference path Inflate reference path

  49. Small Number of Holes For every “threading” For k = 1…K Inflate correspondingly Find shortest reference path Inflate reference path Large number of holes NP-hard

  50. Extensions • Different widths • 3 RNP-20, 5 RNP-10, 2 RNP-3 • Curvature Constraint • shortest paths • curvature < 1

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