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SB Research Presentation – 12/2/05
Finding Rectilinear Least Cost Paths in the Presence of Convex Polygonal
Congested Regions#
Avijit SarkarSchool of Business
University of Redlands
# Submitted to European Journal of Operations Research
2 of 36SB Research Presentation – 12/2/05
2005 Urban Mobility Study http://mobility.tamu.edu/
3 of 36SB Research Presentation – 12/2/05
Traffic Mobility Data for 2003 http://mobility.tamu.edu/
4 of 36SB Research Presentation – 12/2/05
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
5 of 36SB Research Presentation – 12/2/05
Travel Time Index Trends http://mobility.tamu.edu/
6 of 36SB Research Presentation – 12/2/05
Traffic Mobility Data for Riverside-San Bernardino, CA http://mobility.tamu.edu/
7 of 36SB Research Presentation – 12/2/05
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
8 of 36SB Research Presentation – 12/2/05
Example
• For α = 1.6, cost inside = 14.4
• For α = 1.6, cost outside = 14
• Hence bypass
• Threshold: α = 1.5
for α=0.3 1 + 4(1+0.3) + 3 = 9.2
9 of 36SB Research Presentation – 12/2/05
Least Cost PathsEfficient route => determine rectilinear least cost paths in the presence of
congested regions
10 of 36SB Research Presentation – 12/2/05
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
11 of 36SB Research Presentation – 12/2/05
Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points
(4,3)
P (9,10)
12 of 36SB Research Presentation – 12/2/05
MILP Formulation
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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
13 of 36SB Research Presentation – 12/2/05
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
14 of 36SB Research Presentation – 12/2/05
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 of 36SB Research Presentation – 12/2/05
Alternative ApproachMemory-based Probing AlgorithmMotivation from Larson and Sadiq (Operations Research, 1983)
Turning step
16 of 36SB Research Presentation – 12/2/05
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
|||| DoDoOD yyxxl
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O(4p)
17 of 36SB Research Presentation – 12/2/05
Exponential Number of Staircase Paths
18 of 36SB Research Presentation – 12/2/05
At most Two Entry-Exit Points
61.0
61.0,60.0
,59.00
XE1E2E3E4P
XCBP (bypass)
XCE3E4P
19 of 36SB Research Presentation – 12/2/05
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
20 of 36SB Research Presentation – 12/2/05
21 of 36SB Research Presentation – 12/2/05
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 of 36SB Research Presentation – 12/2/05
Memory-based Probing Algorithm
O
D
23 of 36SB Research Presentation – 12/2/05
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 back: why?Generate network of entry/exit pointsTwo types of arcs: (i) inside CRs (ii) outside CRsSolve shortest path problem on generated network
24 of 36SB Research Presentation – 12/2/05
Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research)
condsseCPU
generatednodesofnumber
ctedterseinCRsofnumber
CRsofnumberp
• Algorithm coded in C
25 of 36SB Research Presentation – 12/2/05
Number of CRs Intersected vs Number of Nodes Generated
26 of 36SB Research Presentation – 12/2/05
Number of CRs Intersectedvs CPU seconds
27 of 36SB Research Presentation – 12/2/05
Number of CRs intersected vs log2ρ
28 of 36SB Research Presentation – 12/2/05
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 of 36SB Research Presentation – 12/2/05
Now the Paradox
Optimal path for α=0.30
30 of 36SB Research Presentation – 12/2/05
Why Convexity Restriction?
Approach Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps
31 of 36SB Research Presentation – 12/2/05
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 of 36SB Research Presentation – 12/2/05
Contribution of this workIncorporates 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
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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
Convexity restrictionCannot determine threshold values of α
34 of 36SB Research Presentation – 12/2/05
Future Research
Integration within a GIS framework
Incorporate barriers to travel
Facility location models in congested urban areas
UAV routing problem
35 of 36SB Research Presentation – 12/2/05
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
36 of 36SB Research Presentation – 12/2/05
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
37 of 36SB Research Presentation – 12/2/05
Questions