OAR Lib: An Open Source Arc Routing Library By Oliver Lum Carmine Cerrone Bruce Golden Edward Wasil

OAR Lib (Motivation) Solution: Problem: A (perceived) barrier to entry that coding experience in a non- modeling language is required No centralized, standardized implementations of many routing algorithms Existing APIs are frequently developed with graph theoretic research in mind Realistic test data procurement Figure generation for papers 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 Design Philosophy: Usability First, Performance Second OSM Integration Gephi toolkit (open source graph visualization) Integration

OAR Lib (Content) Single-Vehicle Solvers Multi-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)

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

Applications Ubiquitous: Various interesting wrinkles: Package Delivery Snow Plowing Military Patrols Various interesting wrinkles: Time-Windows Close-Enough Turn Penalties Asymmetric Costs

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

The Min-Max K WRPP = Required = Included in route = Untraversed Depot

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).

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. 1 2 8

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). 1 2 3 4 5 6 7 8

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 4 4 2 2 3 5 5 2 3 1 3 7 7 1 5 6 1 Depot 6

A Partitioning Scheme Change the vertex weights to account for known dead- heading and distance to the depot. 4 4 2 5 2 3 1 3 7 5 1 Depot 6

A Partitioning Scheme Partition the transformed graph into k approximately equal parts. 4 4 2 5 2 3 1 3 7 5 1 Depot 6

A Partitioning Scheme Route the subgraphs induced by each partition using a single-vehicle solver. 4 4 4 2 2 2 1 3 1 3 1 3 5 5 5 Depot Depot Depot

Cross-Section of Helsinki, Finland Test Instance: Cross-Section of Helsinki, Finland |V|=1230 |E|=1468 = Required Edge = Unrequired Edge = Depot

Existing Approach Route 1

Existing Approach Route 2

Partitioning Approach

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 Calgary, CA 1733 2283 22644

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

Cross-Section of Greenland A Large Instance Test Instance: Cross-Section of Greenland |V|=3047 |E|=3285 Runtime: 328.3 s

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.

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.

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 Moglichkeit, 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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