finding rectilinear least cost paths in the presence of convex polygonal congested regions #

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/

Traffic Mobility Data for 2003 http://mobility.tamu.edu/

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

Percentage of major road system congestion 59 34

# of hours each day when congestion is encountered

7.1 4.5

Travel Time Index Trends http://mobility.tamu.edu/

Traffic Mobility Data for Riverside-San Bernardino, CA http://mobility.tamu.edu/

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

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

Least Cost PathsEfficient route => determine rectilinear least cost paths in the presence of

congested regions

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

Mixed Integer Linear Programming (MILP) Approach to Determine Entry/Exit Points

P (9,10)

MILP Formulation

otherwise

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Mzyyxxyyxx

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,,,1,,

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4,2,11

4,2,1)1(

4,2,10)1(

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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

Results

33.1 (z = 20)

Entry=(5,4)

Exit=(5,10)

Example: For α=0.30, cost = 2 + 6(1+0.30) + 4 = 13.80

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

Alternative ApproachMemory-based Probing AlgorithmMotivation from Larson and Sadiq (Operations Research, 1983)

Turning step

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

iODii l

Exponential Number of Staircase Paths

At most Two Entry-Exit Points

61.0,60.0

,59.00

XE1E2E3E4P

XCBP (bypass)

XCE3E4P

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

Results until now

Potentially exponential number of staircase paths exist Any one of them could be least cost

Maximum 2 entries and 2 exits

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

Numerical Results (Sarkar, Batta, Nagi: Submitted to European Journal of Operational Research)

condsseCPU

generatednodesofnumber

ctedterseinCRsofnumber

CRsofnumberp

• Algorithm coded in C

Number of CRs Intersected vs Number of Nodes Generated

Number of CRs Intersectedvs CPU seconds

Number of CRs intersected vs log2ρ

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

Now the Paradox

Optimal path for α=0.30

Why Convexity Restriction?

Approach Determine an upper bound on the number of entry/exit points Associate memory with probes => eliminate turning steps

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

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

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

Future Research

Integration within a GIS framework

Incorporate barriers to travel

Facility location models in congested urban areas

UAV routing problem

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

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

Questions

finding rectilinear least cost paths in the presence of convex polygonal congested regions #

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