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OAR Lib: An Open Source Arc
Routing Library
By Oliver Lum
Carmine Cerrone
Bruce Golden
Edward Wasil
1
OAR Lib (Motivation)
Problem:
A (perceived) barrier to entry that
coding experience in a nonmodeling language is required
No centralized, standardized
implementations of many routing
algorithms
Solution:
An open-source java library aimed at
new operations researchers in the
field of arc-routing
An architecture for future software
development in routing and
scheduling
Existing APIs are frequently
developed with graph theoretic
research in mind
Design Philosophy: Usability First,
Performance Second
Realistic test data procurement
OSM Integration
Figure generation for papers
Gephi toolkit (open source graph
visualization) Integration
2
OAR Lib (Content)
Single-Vehicle Solvers
Un/Directed Chinese Postman
(UCPP/DCPP)
Mixed Chinese Postman (MCPP)
Windy Chinese Postman (WPP)
Directed Rural Postman Problem
(DRPP)
Windy Rural Postman Problem (WRPP)
Multi-Vehicle Solvers
Min-Max K Windy Rural Postman
Problem (MM-k WRPP)
3
OAR Lib (Content)
Common Algorithms:
Single-Source Shortest Paths
All-Pairs Shortest Paths
Min-Cost Matching (JNI)
Min-Cost Flow
Hierholzer’s Algorithm
(Stochastic) Minimum Spanning Tree
Minimum Spanning Arborescence
(JNI)
Connectivity Tests
4
Applications
Ubiquitous:
Package Delivery
Snow Plowing
Military Patrols
Various interesting wrinkles:
Time-Windows
Close-Enough
Turn Penalties
Asymmetric Costs
5
The Min-Max K WRPP
A natural extension of the WRPP
Objective: Minimize the max route cost
Homogenous fleet, K vehicles
Asymmetric Traversal Costs
Required and unrequired edges
Generalization of the directed, undirected, and mixed
variants
Takes into account route balance, and customer
satisfaction
6
The Min-Max K WRPP
= Required
= Included in route
= Untraversed
Depot
7
The Min-Max K WRPP
Existing Literature:
Benavent, Enrique, et al. “Min-Max K-vehicles windy rural
postman problem.” Networks 54.4 (2009): 216-226.
Benavent, Enrique, Angel Corberan, Jose M. Sanchis. “A
metaheuristic for the min-max windy rural postman problem
with K vehicles.” Computational Management Science 7.3
(2010): 269-287.
Benavent, Enrique, et al. “A branch-price-and-cut method
for the min-max k-windy rural postman problem.” Networks
(2014, to appear).
8
Existing Algorithm
Solve the
single-vehicle
variant. This
produces a
solution that
can be
represented as
an ordered list
of required
edges (where
any gaps are
traversed via
shortest paths).
Depot
9
Existing Algorithm
Set up a directed, acyclic graph with m+1 vertices,
(0,1,…m) where the cost of the arc (i-1,j) is the cost of the
tour starting at the depot, going to the tail of edge i,
continuing along the single-vehicle solution through
edge j, and then returning to the depot.
10
Existing Algorithm
Calculate a k-edge narrowest path from v0 to vm in the
DAG, corresponding to a solution (a simple modification
to Dijkstra’s single-source shortest path algorithm suffices).
11
A Partitioning Scheme
Transform the graph into a vertex-weighted graph in the
following way:
Create a vertex for each edge in the original graph
Connect two vertices i and j if, in the original graph, edge I
and edge j shared an endpoint
4
2
3
1
5
7
6
Depot
12
A Partitioning Scheme
Change the vertex weights to account for known deadheading and distance to the depot.
Depot
13
A Partitioning Scheme
Partition the transformed graph into k approximately
equal parts.
Depot
14
A Partitioning Scheme
Route the subgraphs induced by each partition using a
single-vehicle solver.
Depot
Depot
Depot
15
Test Instance:
Cross-Section of Helsinki,
Finland
|V|=1230
|E|=1468
= Required Edge
= Unrequired Edge
= Depot
16
Existing Approach Route 1
17
Existing Approach Route 2
18
Partitioning Approach
19
Results
Instance
|V|
|E|
Partition
Obj.
Benavent
Obj.
% Diff
San Francisco
705
844
11645
10188
14.3
Washington D.C.
592
663
10742
10649
.87
London, UK
845
994
6116
5866
4.2
Istanbul, TR
629
780
7388
7516
-1.7
Perth, AUS
535
597
7379
6970
5.8
Auckland, AUS
1045
1130
13655
13368
2.1
Helsinki, FI
1230
1468
6640
6866
-2.5
Vienna, AU
483
557
3875
3947
-1.9
Paris, FR
1931
2256
14862
N/A
N/A
Calgary, CA
1733
2283
22644
N/A
N/A
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Conclusions / Future Work
Advantages:
Capable of solving large instances
Service contiguity – adjacent links are more likely to be
serviced by the same vehicle
Memory usage – the widest path calculation in the existing
algorithm is extremely memory intensive ( order )
Future Work:
Incorporate / apply existing improvement procedures to
both procedures and compare.
Exploring relationship between number of vehicles, and
tuning parameters
21
A Large Instance
Test Instance:
Cross-Section of
Greenland
|V|=3047
|E|=3285
Runtime: 328.3 s
22
References
Ahr, Dino, and Gerhard Reinelt. "New heuristics and lower bounds for
the Min-Max k-Chinese PostmanProblem." Algorithms|ESA 2002.
Springer Berlin Heidelberg, 2002. 64-74.
Benavent, Enrique, et al. "New heuristic algorithms for the windy rural
postman problem." Computers& operations research 32.12 (2005):
3111-3128.
Campos, V., and J. V. Savall. "A computational study of several
heuristics for the DRPP." ComputationalOptimization and Applications
4.1 (1995): 67-77.
http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=
minimumCostFlow2
Edmonds, Jack, and Ellis L. Johnson. "Matching, Euler tours and the
Chinese postman." Mathematicalprogramming 5.1 (1973): 88-124.
23
References
Eiselt, Horst A., Michel Gendreau, and Gilbert Laporte. "Arc routing problems,
part II: The ruralpostman problem." Operations Research 43.3 (1995): 399-414.
Derigs, Ulrich. Optimization and operations research. Eolss Publishers Company
Limited, 2009.
http://en.wikipedia.org/wiki/Dijkstra's_algorithm
http://en.wikipedia.org/wiki/Floyd\OT1\textendashWarshall_algorithm
http://en.wikipedia.org/wiki/Prim%27s_algorithm
Dussault, Benjamin, et al. "Plowing with precedence: A variant of the windy
postman problem."Computers & Operations Research (2012).
Frederickson, Greg N. "Approximation algorithms for some postman problems."
Journal of the ACM(JACM) 26.3 (1979): 538-554.
24
References
Grotschel, Martin, and Zaw Win. "A cutting plane algorithm for the windy postman
problem." Math-ematical Programming 55.1-3 (1992): 339-358.
Hierholzer, Carl, and Chr Wiener. "Uber die M•oglichkeit, einen Linienzug ohne
Wiederholung und ohneUnterbrechung zu umfahren." Mathematische Annalen 6.1
(1873): 30-32.
Karypis, George, and Vipin Kumar. "A fast and high quality multilevel scheme for
partitioning irregulargraphs." SIAM Journal on scientic Computing 20.1 (1998): 359-392.
Kolmogorov, Vladimir. "Blossom V: a new implementation of a minimum cost perfect
matching algo-rithm." Mathematical Programming Computation 1.1 (2009): 43-67.
Lau, Hang T. A Java library of graph algorithms and optimization. CRC Press, 2010.
Letchford, Adam N., Gerhard Reinelt, and Dirk Oliver Theis. "A faster exact separation
algorithm forblossom inequalities." Integer programming and combinatorial optimization.
Springer Berlin Heidelberg,2004. 196-205.
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References
Padberg, Manfred W., and M. Ram Rao. "Odd minimum cutsets and b-matchings." Mathematics ofOperations Research
7.1 (1982): 67-80
Thimbleby, Harold. "The directed chinese postman problem."
Software: Practice and Experience 33.11(2003): 1081-1096.
Win, Zaw. "On the windy postman problem on Eulerian
graphs." Mathematical Programming 44.1-3(1989): 97-112.
Yaoyuenyong, Kriangchai, Peerayuth Charnsethikul, and
Vira Chankong. "A heuristic algorithm for themixed Chinese
postman problem." Optimization and Engineering 3.2 (2002):
157-187
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