SB Research Presentation – 12/2/05 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

SB Research Presentation – 12/2/05 2 of Urban Mobility Study

SB Research Presentation – 12/2/05 3 of 36 Traffic Mobility Data for

SB Research Presentation – 12/2/05 4 of 36 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

SB Research Presentation – 12/2/05 5 of 36 Travel Time Index Trends

SB Research Presentation – 12/2/05 6 of 36 Traffic Mobility Data for Riverside-San Bernardino, CA

SB Research Presentation – 12/2/05 7 of 36 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

SB Research Presentation – 12/2/05 8 of 36 Example For α = 1.6, cost inside = 14.4 For α = 1.6, cost outside = 14 Hence bypass Threshold: α = 1.5 for α= (1+0.3) + 3 = 9.2

SB Research Presentation – 12/2/05 9 of 36 Least Cost Paths Efficient route => determine rectilinear least cost paths in the presence of congested regions

SB Research Presentation – 12/2/05 10 of 36 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

SB Research Presentation – 12/2/05 11 of 36 Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points (4,3) P (9,10)

SB Research Presentation – 12/2/05 12 of 36 MILP Formulation 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

SB Research Presentation – 12/2/05 13 of 36 Results (z = 20) Entry=(5,4) Exit=(5,10) Example: For α=0.30, cost = 2 + 6(1+0.30) + 4 = 13.80

SB Research Presentation – 12/2/05 14 of 36 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

SB Research Presentation – 12/2/05 15 of 36 Alternative Approach Memory-based Probing Algorithm Motivation from Larson and Sadiq (Operations Research, 1983) Turning step

SB Research Presentation – 12/2/05 16 of 36 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 )

SB Research Presentation – 12/2/05 17 of 36 Exponential Number of Staircase Paths

SB Research Presentation – 12/2/05 18 of 36 At most Two Entry-Exit Points XE 1 E 2 E 3 E 4 P XCBP (bypass) XCE 3 E 4 P

SB Research Presentation – 12/2/05 19 of 36 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

SB Research Presentation – 12/2/05 20 of 36

SB Research Presentation – 12/2/05 21 of 36 Results until now Potentially exponential number of staircase paths exist Any one of them could be least cost Maximum 2 entries and 2 exits

SB Research Presentation – 12/2/05 22 of 36 Memory-based Probing Algorithm O D

SB Research Presentation – 12/2/05 23 of 36 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

SB Research Presentation – 12/2/05 24 of 36 Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research) Algorithm coded in C

SB Research Presentation – 12/2/05 25 of 36 Number of CRs Intersected vs Number of Nodes Generated

SB Research Presentation – 12/2/05 26 of 36 Number of CRs Intersected vs CPU seconds

SB Research Presentation – 12/2/05 27 of 36 Number of CRs intersected vs log 2 ρ

SB Research Presentation – 12/2/05 28 of 36 Summary of Results O(2 0.5φ ), i.e., O(1.414 φ ) entry/exit points rather than O(4 p ) in worst case Works well up to CRs Heuristic approaches for larger problem instances

SB Research Presentation – 12/2/05 29 of 36 Now the Paradox Optimal path for α=0.30

SB Research Presentation – 12/2/05 30 of 36 Why Convexity Restriction? Approach Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps

SB Research Presentation – 12/2/05 31 of 36 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

SB Research Presentation – 12/2/05 32 of 36 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

SB Research Presentation – 12/2/05 33 of 36 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

SB Research Presentation – 12/2/05 34 of 36 Future Research Integration within a GIS framework Incorporate barriers to travel Facility location models in congested urban areas UAV routing problem

SB Research Presentation – 12/2/05 35 of 36 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

SB Research Presentation – 12/2/05 36 of 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: 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

SB Research Presentation – 12/2/05 37 of 36 Questions