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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 20 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. 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