Context-Aware Route Planning

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MARP: Multi-Agent Route Planning CARP vs. FPS Algorithms Experiments Conclusions

Context-Aware Route Planning

A. W. ter Mors, C. Witteveen, J. Zutt, F. A. Kuipers

Delft University of Technology

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Problem description

A set of agents, each with their own start and destination location

An infrastructure of limited-capacity resources Find a set of conflict-free, minimum-cost route plans

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Conflict-free routing

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Application domains

(a) Airport taxi routing (b) Automated Guided Vehicles at a container terminal

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Approximations for MARP

Finding an optimal set of route plans is NP-hard Sequential, single-agent approximations:

Context-Aware Route Planning (CARP): find an optimal

single-agent route plan, givenreservationsof other agents

Fixed-Path Scheduling (FPS): find an optimalschedule, along

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CARP and FPS characteristics

CARP:

Worst-case complexity: O(_|A||R_|log(_|A||R_|) +_|A||R_|2) (_A

the set of agents,R the set of resources.)

The CARP algorithm utilizes less-congested (in space and time) areas of the infrastructure

FPS:

Worst-case complexity: O(_|A||R_|log(_|A||R_|))

The FPS approach is limited in the way it can make use of traffic information in the route choice

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Example global plan quality 1

CARP finds an optimal global plan

FPS finds a sub-optimal global plan, in case each agent follows the shortest path

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Example global plan quality 2

FPS finds an optimal global plan, in case each agent follows the shortest path

Depending on the order in which agents plan, CARP may find a sub-optimal global plan

rd,1

rs,2

rs,1

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Route planning in space and time

In sequential routing, each agent plans around the

reservations of other agents

For each infrastructure resource, find the set of free time windows: intervals during which a resource can be entered without creating a conflict

Construct a graphof free time windows, and perform an

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Free time windows

Free time window

A time interval in which the resource load is less than the capacity. The interval should be at least as long as the minimum travel time.

1 2 3 4 5 6 7 8 9

0

λ(ri, t)

cap(ri)

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Free time window graph

Which arcs between free time windows?

1 _{resources must be connected}

2 _{free time windows must overlap}

3 there must be sufficient time to traverse the second resource

A B tt(A)

fB

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CARP algorithm

1: while open6=∅do 2: w _←argmin_w0_∈_openf(w0) 3: mark(w,closed) 4: r ←resource(w) 5: if r =r2 then 6: returnfollowBackPointers(w)

7: texit←g(w) = entryTime(w) +d(resource(w))

8: for allw0 _{∈ {}ρ(r,texit)\closed} do

9: tentry ←max(texit,start(w0))

10: if tentry<entryTime(w0) then

11: entryTime(w0)_←tentry

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CARP algorithm

1: while open6=∅do 2: w _←argmin_w0_∈_openf(w0) . f =g +h 3: mark(w,closed) 4: r ←resource(w) 5: if r =r2 then 6: returnfollowBackPointers(w)

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CARP algorithm

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CARP algorithm

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Experimental setup

Compare global plan quality between CARP and k-shortest

pathscheduling

Infrastructures: model of Amsterdam Airport Schiphol, and randomly generated instances

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k

-shortest paths

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CPU time comparison

100 200 300 400 500 0 100 200 300 400 500 number of agents

computation time (milliseconds)

K = 1 K = 2 K = 3 K = 4 K = 5 CA

Figure: CPU times for CARP and FPS (k = 1, 2,...,5) on random infrastructures of 180 nodes and 300 edges.

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Plan quality comparison: Schiphol

100 200 300 400 500 100 150 200 250 300 350 number of agents

joint plan cost as percentage of lower bound (%)

k = 1 k = 2 k = 3 k = 4 k = 5 CA

(a) Joint plan cost

100 200 300 400 500 1000 2000 3000 4000 number of agents makespan (seconds) k = 1 k = 2 k = 3 k = 4 k = 5 CA (b) Makespan

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Plan quality comparison: random graphs

100 200 300 400 110 120 130 140 150 160 number of agents

k = 1 k = 2 k = 3 k = 4 k = 5 CA

(a) Joint plan cost

100 200 300 400 200 300 400 500 600 number of agents makespan (seconds) k = 1 k = 2 k = 3 k = 4 k = 5 CA (b) Makespan Figure: Random graphs on 180 nodes and 300 edges

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Plan quality comparison: lattice graphs

100 200 300 400 500 100 150 200 250 number of agents

k = 1 k = 2 k = 3 k = 4 k = 5 CA

(a) Joint plan cost

100 200 300 400 500 500 1000 1500 2000 2500 number of agents makespan (seconds) K = 1 K = 2 K = 3 K = 4 K = 5 CA (b) Makespan Figure: Lattice graphs of around 450 resources

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Plan quality comparison: small-world networks

100 200 300 400 500 100 150 200 250 300 number of agents

joint plan cost as a percentage of lower bound (%)

K = 1 K = 2 K = 3 K = 4 K = 5 CA

(a) Joint plan cost

100 200 300 400 500 100 150 200 250 300 350 400 number of agents makespan (seconds) K = 1 K = 2 K = 3 K = 4 K = 5 CA (b) Makespan Figure: Small-world networks of around 450 resources

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Conclusions

Context-aware route planning is fast and performs well on a wide range of infrastructures

The success of fixed-path scheduling depends on finding sufficiently different paths between every pair of locations Future work: can infrastructure agents improve global plan quality of self-interested route planners?

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MARP: Multi-Agent Route Planning CARP vs. FPS Algorithms Experiments Conclusions J. Y. Yen.

Finding the K shortest loopless paths in a network.