MS-GIS colloquium: 9/28/05 Least Cost Path Problem in the Presence of Congestion* # Avijit Sarkar...

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

2 of 32MS-GIS colloquium: 9/28/05

2005 Urban Mobility Study http://mobility.tamu.edu/


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


Traffic Mobility Data for Riverside-San Bernardino, CA 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

67 32

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/


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


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

(4,3)

P (9,10)


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


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


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


Alternative ApproachMemory-based Probing Algorithm

Turning step


Why Convexity Restriction?

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


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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Exponential Number of Staircase Paths


At most Two Entry-Exit Points

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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


Memory-based Probing Algorithm

O

D


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


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

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• Algorithm coded in C


Number of CRs Intersected vs Number of Nodes Generated


Number of CRs Intersectedvs CPU seconds


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


Now the Paradox

Optimal path for α=0.30


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


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


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


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

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MS-GIS colloquium: 9/28/05 Least Cost Path Problem in the Presence of Congestion* # Avijit Sarkar...

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