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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
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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.
25
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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