Least Cost Path Problem in the Presence of
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
2005 Urban Mobility Study http: //mobility. tamu. edu/ MS-GIS colloquium: 9/28/05 2
Traffic Mobility Data for 2003 http: //mobility. tamu. edu/ MS-GIS colloquium: 9/28/05 3
Traffic Mobility Data for Riverside-San Bernardino, CA http: //mobility. tamu. edu/ MS-GIS colloquium: 9/28/05 4
How far has congestion spread? http: //mobility. tamu. edu/ Some Results 2003 1982 # of urban areas with TTI > 1. 30 28 1 Percentage of traffic experiencing peak period travel congestion 67 32 Percentage of major road system congestion 59 34 # of hours each day when congestion is encountered 7. 1 4. 5 MS-GIS colloquium: 9/28/05 5
Travel Time Index Trends MS-GIS colloquium: 9/28/05 http: //mobility. tamu. edu/ 6
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 n n n Congestion factor α Cost inside = (1+α)x(Cost Outside) 0< α<∞ Shortest path ≠ Least Cost Path Entry/exit point n n Point at which least cost path enters/exits a congested region Not known a priori MS-GIS colloquium: 9/28/05 7
Least Cost Paths Efficient route => determine rectilinear least cost paths in the presence of congested regions MS-GIS colloquium: 9/28/05 8
Previous Results (Butt and Cavalier, Socio-Economic Planning Sciences, 1997) Planar p-median problem in the presence of congested regions Least 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 9
Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points P (9, 10) (4, 3) MS-GIS colloquium: 9/28/05 10
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 11
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 12
Discussion Formulation outputs n n Entry/exit points Length of least cost path Advantages n n Models multiple entry/exit points Automatic choice of number of entry/exit points Automatic edge selection Break point of α Disadvantages n n Generic problem formulation very difficult: due to combinatorics Complexity increases with w Number of sides w Number of congested regions MS-GIS colloquium: 9/28/05 13
Alternative Approach Memory-based Probing Algorithm Turning step MS-GIS colloquium: 9/28/05 14
Why Convexity Restriction? Approach n n Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps MS-GIS colloquium: 9/28/05 15
Observation 1: Exponential Number of Staircase Paths may Exist Staircase path: Length of staircase path through p CRs No a priori elimination possible 22 p+1 (O(4 p)) staircase paths between O and D O(4 p) MS-GIS colloquium: 9/28/05 16
Exponential Number of Staircase Paths MS-GIS colloquium: 9/28/05 17
At most Two Entry-Exit Points XE 1 E 2 E 3 E 4 P XCBP (bypass) MS-GIS colloquium: 9/28/05 18
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 19
Memory-based Probing Algorithm D O MS-GIS colloquium: 9/28/05 20
Memory-based Probing Algorithm Each probe has associated memory n 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 21
Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research) • Algorithm coded in C MS-GIS colloquium: 9/28/05 22
Number of CRs Intersected vs Number of Nodes Generated MS-GIS colloquium: 9/28/05 23
Number of CRs Intersected vs CPU seconds MS-GIS colloquium: 9/28/05 24
Summary of Results O(1. 414 p) entry/exit points rather than O(4 p) in worst case Works well up to 12 -15 CRs Heuristic approaches for larger problem instances MS-GIS colloquium: 9/28/05 25
Now the Paradox Optimal path for α=0. 30 MS-GIS colloquium: 9/28/05 26
Known Entry-Exit Heuristic Entry-exit points are known a priori Ø Least cost path coincides with an easily identifiable finite grid Ø Convex polygonal restriction no longer necessary MS-GIS colloquium: 9/28/05 27
Potential Benefits Refine distance calculation in routing algorithms Large scale disaster n n n Land parcels (polygons) may be destroyed De-congested routes may become congested Can help w Identify entry/exit points w 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 28
Some Issues Congestion factor has been assumed to be constant In urban transportation settings n α will be time-dependent w Time-dependent shortest path algorithms n α will be stochastic Convex polygonal restriction Cannot determine threshold values of α MS-GIS colloquium: 9/28/05 29
OR-GIS Models for US Military UAV routing problem n n n UAVs employed by US military worldwide Missions are extremely dynamic UAV flight plans consider w Time windows w Threat level of hostile forces w Time required to image a site w Bad weather n Surface-to-air threats exist enroute and may increase at certain sites MS-GIS colloquium: 9/28/05 30
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 n n n 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 31
Questions MS-GIS colloquium: 9/28/05 32
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