ms-gis colloquium: 9/28/05 least cost path problem in the presence of congestion* # avijit sarkar...
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MS-GIS colloquium: 9/28/05
Least Cost Path Problem in the Presence of Congestion*#
Avijit SarkarAssistant ProfessorSchool 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
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2005 Urban Mobility Study http://mobility.tamu.edu/
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Traffic Mobility Data for 2003 http://mobility.tamu.edu/
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Traffic Mobility Data for Riverside-San Bernardino, CA http://mobility.tamu.edu/
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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
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Travel Time Index Trends http://mobility.tamu.edu/
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Congested Regions – Definition and Details
Urban zones where travel times are greatly increasedClosed and bounded area in the planeApproximated by convex polygonsPenalizes 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
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Least Cost PathsEfficient route => determine rectilinear least cost paths in the presence of
congested regions
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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
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Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points
(4,3)
P (9,10)
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MILP Formulation (Sarkar, Batta, Nagi: Socio Economic Planning Sciences: 38(4), Dec 04)
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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 4Takes care of additional distance
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12 of 32MS-GIS colloquium: 9/28/05
Results
33.10
33.1 (z = 20)
Entry=(5,4)
Exit=(5,10)
Example: For α=0.30, cost = 2+6(1+0.30)+4 = 13.80
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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
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Alternative ApproachMemory-based Probing Algorithm
Turning step
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Why Convexity Restriction?
Approach Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps
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16 of 32MS-GIS colloquium: 9/28/05
Observation 1: Exponential Number of Staircase Paths may ExistStaircase path:Length of staircase path through p CRs
No a priori elimination possible22p+1 (O(4p)) staircase paths between O and D
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17 of 32MS-GIS colloquium: 9/28/05
Exponential Number of Staircase Paths
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At most Two Entry-Exit Points
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XCBP (bypass)
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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
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20 of 32MS-GIS colloquium: 9/28/05
Memory-based Probing Algorithm
O
D
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Memory-based Probing Algorithm
Each probe has associated memory what were the directions of two previous probes?
Eliminates turning stepsUses previous result: upper bound of entry/exit pointsNecessary to probe from O to D and backGenerate network of entry/exit pointsTwo types of arcs: (i) inside CRs (ii) outside CRsSolve shortest path problem on generated network
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Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research)
condsseCPU
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• Algorithm coded in C
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Number of CRs Intersected vs Number of Nodes Generated
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Number of CRs Intersectedvs CPU seconds
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Summary of Results
O(1.414p) entry/exit points rather than O(4p) in worst case
Works well up to 12-15 CRs
Heuristic approaches for larger problem instances
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Now the Paradox
Optimal path for α=0.30
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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
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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
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Some Issues
Congestion factor has been assumed to be constantIn urban transportation settings α will be time-dependent
Time-dependent shortest path algorithms α will be stochastic
Convex polygonal restrictionCannot determine threshold values of α
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30 of 32MS-GIS colloquium: 9/28/05
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
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Some Insight into the UAV Routing Problem
Threat zones and threat levels are surrogates for congested regions and congestion factorsDifference: Euclidean distancesObjective: minimize probability of detection in the presence of multiple threat zonesCan assume the probability of escape to be a Poisson random variableBasic 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
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Questions