MS-GIS colloquium: 9/28/05 Least Cost Path Problem in the Presence of Congestion* # Avijit Sarkar Assistant Professor School of Business University of Redlands * This is joint work with Drs. Rajan Batta & Rakesh Nagi, Department of Industrial Engineering, SUNY at Buffalo # Submitted to European Journal of Operations Research

MS-GIS colloquium: 9/28/05 2 of Urban Mobility Study

MS-GIS colloquium: 9/28/05 3 of 32 Traffic Mobility Data for

MS-GIS colloquium: 9/28/05 4 of 32 Traffic Mobility Data for Riverside-San Bernardino, CA

MS-GIS colloquium: 9/28/05 5 of 32 How far has congestion spread? Some Results # of urban areas with TTI > Percentage of traffic experiencing peak period travel congestion 6732 Percentage of major road system congestion 5934 # of hours each day when congestion is encountered

MS-GIS colloquium: 9/28/05 6 of 32 Travel Time Index Trends

MS-GIS colloquium: 9/28/05 7 of 32 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

MS-GIS colloquium: 9/28/05 8 of 32 Least Cost Paths Efficient route => determine rectilinear least cost paths in the presence of congested regions

MS-GIS colloquium: 9/28/05 9 of 32 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 For α=0.30, cost=13.8

MS-GIS colloquium: 9/28/05 10 of 32 Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points (4,3) P (9,10)

MS-GIS colloquium: 9/28/05 11 of 32 MILP Formulation (Sarkar, Batta, Nagi: Socio Economic Planning Sciences: 38(4), Dec 04) Entry point E 1 lies on exactly one edge Exit point E 2 lies on exactly one edge Entry point E 3 lies on exactly one edge Provide bounds on x-coordinates of E 1, E 2, E 3 Final exit point E 4 lies on edge 4 Takes care of additional distance

MS-GIS colloquium: 9/28/05 12 of 32 Results (z = 20) Entry=(5,4) Exit=(5,10) Example: For α=0.30, cost = 2+6(1+0.30)+4 = 13.80

MS-GIS colloquium: 9/28/05 13 of 32 Discussion Formulation outputs Entry/exit points 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

MS-GIS colloquium: 9/28/05 14 of 32 Alternative Approach Memory-based Probing Algorithm Turning step

MS-GIS colloquium: 9/28/05 15 of 32 Why Convexity Restriction? Approach Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps

MS-GIS colloquium: 9/28/05 16 of 32 Observation 1: Exponential Number of Staircase Paths may Exist Staircase path: Length of staircase path through p CRs No a priori elimination possible 2 2p+1 (O(4 p )) staircase paths between O and D O(4 p )

MS-GIS colloquium: 9/28/05 17 of 32 Exponential Number of Staircase Paths

MS-GIS colloquium: 9/28/05 18 of 32 At most Two Entry-Exit Points XE 1 E 2 E 3 E 4 P XCBP (bypass) XCE 3 E 4 P

MS-GIS colloquium: 9/28/05 19 of 32 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

MS-GIS colloquium: 9/28/05 20 of 32 Memory-based Probing Algorithm O D

MS-GIS colloquium: 9/28/05 21 of 32 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 Generate network of entry/exit points Two types of arcs: (i) inside CRs (ii) outside CRs Solve shortest path problem on generated network

MS-GIS colloquium: 9/28/05 22 of 32 Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research) Algorithm coded in C

MS-GIS colloquium: 9/28/05 23 of 32 Number of CRs Intersected vs Number of Nodes Generated

MS-GIS colloquium: 9/28/05 24 of 32 Number of CRs Intersected vs CPU seconds

MS-GIS colloquium: 9/28/05 25 of 32 Summary of Results O(1.414 p ) entry/exit points rather than O(4 p ) in worst case Works well up to CRs Heuristic approaches for larger problem instances

MS-GIS colloquium: 9/28/05 26 of 32 Now the Paradox Optimal path for α=0.30

MS-GIS colloquium: 9/28/05 27 of 32 Known Entry-Exit Heuristic Entry-exit points are known a priori  Least cost path coincides with an easily identifiable finite grid  Convex polygonal restriction no longer necessary

MS-GIS colloquium: 9/28/05 28 of 32 Potential Benefits Refine distance calculation in routing algorithms 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 Form the basis to solve facility location problems in the presence of congestion

MS-GIS colloquium: 9/28/05 29 of 32 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 Convex polygonal restriction Cannot determine threshold values of α

MS-GIS colloquium: 9/28/05 30 of 32 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

MS-GIS colloquium: 9/28/05 31 of 32 Some Insight into the UAV Routing Problem Threat zones and threat levels are surrogates for congested regions and congestion factors Difference: Euclidean distances Objective: minimize probability 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

MS-GIS colloquium: 9/28/05 32 of 32 Questions