Finding Rectilinear Least Cost Paths in the Presence of Convex Polygonal Congested Regions #

1. Finding Rectilinear Least Cost Paths in the Presence of Convex Polygonal Congested Regions# Avijit Sarkar School of Business University of Redlands # Submitted to European Journal of Operations Research

2. 2005 Urban Mobility Study http://mobility.tamu.edu/

3. Traffic Mobility Data for 2003 http://mobility.tamu.edu/

4. How far has congestion spread?http://mobility.tamu.edu/

5. Travel Time Index Trends http://mobility.tamu.edu/

6. Traffic Mobility Data for Riverside-San Bernardino, CAhttp://mobility.tamu.edu/

7. Congested Regions – Definition and Details • Urban zones where travel times are greatly increased • Closed and bounded area in the plane • Approximated by convex polygons • Penalizes travel through the interior • Congestion factor α • Cost inside = (1+α)x(Cost Outside) • 0 < α < ∞ • Shortest path ≠ Least Cost Path • Entry/exit point • Point at which least cost path enters/exits a congested region • Not known a priori

8. Example for α=0.3 1 + 4(1+0.3) + 3 = 9.2 • For α = 1.6, cost inside = 14.4 • For α = 1.6, cost outside = 14 • Hence bypass • Threshold: α = 1.5

9. Least Cost Paths • Efficient route => determine rectilinear least cost paths in the presence of congested regions

10. For α=0.30, cost=13.8 Previous Results (Butt and Cavalier, Socio-Economic Planning Sciences, 1997) • Planar p-median problem in the presence of congested regions • Least cost coincides with easily identifiable grid • Imprecise result: holds for rectangular congested regions For α=0.30, cost=14

11. Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points P (9,10) (4,3)

12. MILP Formulation Entry point E1 lies on exactly one edge Exit point E2 lies on exactly one edge Entry point E3 lies on exactly one edge Provide bounds on x-coordinates of E1, E2, E3 Final exit point E4 lies on edge 4 Takes care of additional distance

13. (z = 20) Results Entry=(5,4) Exit=(5,10) Example: For α=0.30, cost = 2 + 6(1+0.30) + 4 = 13.80

14. Advantages and Disadvantages of MILP Approach • Formulation outputs • Coordinates of entry/exit points • Edges on which entry/exit points lie • Length of least cost path • Advantages • Models multiple entry/exit points • Automatic choice of number of entry/exit points • Automatic edge selection • Break point of α • Disadvantages • Generic problem formulation very difficult: due to combinatorics • Complexity increases with • Number of sides • Number of congested regions

15. Turning step Alternative Approach • Memory-based Probing Algorithm • Motivation from Larson and Sadiq (Operations Research, 1983)

16. Observation 1: Exponential Number of Staircase Paths may Exist • Staircase path: • Length of staircase path through p CRs • No a priori elimination possible • 22p+1 (O(4p)) staircase paths between O and D O(4p)

17. Exponential Number of Staircase Paths

18. XE1E2E3E4P XCE3E4P XCBP (bypass) At most Two Entry-Exit Points

19. 3-entry 3-exit does not exist • Compare 3-entry/exit path with 2-entry/exit and 1-entry/exit paths • Proof based on contradiction • Use convexity and polygonal properties

21. Results until now • Potentially exponential number of staircase paths exist • Any one of them could be least cost • Maximum 2 entries and 2 exits

22. Memory-based Probing Algorithm D O

23. Memory-based Probing Algorithm • Each probe has associated memory • what were the directions of two previous probes? • Eliminates turning steps • Uses previous result: upper bound of entry/exit points • Necessary to probe from O to D and back: why? • Generate network of entry/exit points • Two types of arcs: (i) inside CRs (ii) outside CRs • Solve shortest path problem on generated network

24. Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research) • Algorithm coded in C

25. Number of CRs Intersected vs Number of Nodes Generated

26. Number of CRs Intersectedvs CPU seconds

27. Number of CRs intersected vs log2ρ

28. Summary of Results • O(20.5φ), i.e., O(1.414φ) entry/exit points rather than O(4p) in worst case • Works well up to 12-15 CRs • Heuristic approaches for larger problem instances

29. Now the Paradox Optimal path forα=0.30

30. Why Convexity Restriction? • Approach • Determine an upper bound on the number of entry/exit points • Associate memory with probes => eliminate turning steps

31. Known Entry-Exit Heuristic – Urban Commuting • Entry-exit points are known a priori • Least cost path coincides with an easily identifiable finite grid • Convex polygonal restriction no longer necessary

32. Contribution of this work • Incorporates congestion in Corridor Location Problem • Identify the best route across a landscape that connects two points • Planar problem converted to a network representation • Lack of such models (R. Church, Computers & OR, 2002) • Application 1: Large scale disaster • Land parcels (polygons) may be destroyed • De-congested routes may become congested • Can help • Identify entry/exit points • Determine least cost path for rescue teams • Application 2: Routing AGVs in congested facilities • Accurate representation of travel distances in the presence of congestion • Memory based probing algorithm provides framework for distance measurement • Refine distance calculation in vehicle routing applications

33. Some Issues • Congestion factor has been assumed to be constant • In urban transportation settings • α will be time-dependent • Time-dependent shortest path algorithms • α will be stochastic • Convexity restriction • Cannot determine threshold values of α

34. Future Research • Integration within a GIS framework • Incorporate barriers to travel • Facility location models in congested urban areas • UAV routing problem

35. OR-GIS Models for US Military • UAV routing problem • UAVs employed by US military worldwide • Missions are extremely dynamic • UAV flight plans consider • Time windows • Threat level of hostile forces • Time required to image a site • Bad weather • Surface-to-air threats exist enroute and may increase at certain sites

36. Some Insight into the UAV Routing Problem • Threat zones and threat levels are surrogates for congested regions and congestion factors • Difference: Euclidean distances • Objective: minimizeprobability of detection in the presence of multiple threat zones • Can assume the probability of escape to be a Poisson random variable • Basic result • One threat zone: reduces to solving a shortest path problem • Result extends or not for multiple threat zones? • Potential application to combine GIS network analysis tools with OR algorithms

37. Questions