1. A Two-Stage Partitioning Approach for the Min-Max K Windy Rural Postman Problem Oliver Lum Carmine Cerrone Bruce Golden Edward Wasil 1

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) 2

3. OAR Lib Content  Well-Known Algorithms  Single-Source Shortest Paths  All-Pairs Shortest Paths  Min-Cost Matching  Min-Cost Flow  Hierholzer’s Algorithm  Minimum Spanning Tree  Minimum Spanning Arborescence  Connectivity Tests 3

4. Applications  Well-established  Package Delivery  Snow Plowing  Military Patrols  Variants  Time-Windows  Close-Enough  Turn Penalties  Asymmetric Costs 4

5. 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 5

6. Min-Max K WRPP 6 Depot = Required = Included in route = Not traversed

7. Min-Max K WRPP  Literature review  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 63:1 (2014): 34-45. 7

8. Benavent’s 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). 8 8 Depot 1 2 3 4 5 6 7 8

9. Benavent’s Algorithm  Set up a directed, acyclic graph (DAG) 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 9

10. Benavent’s Algorithm  Calculate a k-edge narrowest path from v 0 to v m in the DAG, corresponding to a solution (a simple modification to Dijkstra’s single-source shortest path algorithm) 10

11. Compactness Metrics 11  In practice, usable routes must often exhibit intuitive properties like connectedness and compactness. Two metrics proposed in Constantino et al. “The mixed capacitated arc routing poblem with non-overlapping routes.” European Journal of Operational Research (2015, under review)  Route Overlap Index (ROI)  Average Traversal Distance (ATD)

12. Route Overlap Index 12

13. Average Traversal Distance 13 Depot Compact Routes Non-compact Routes

14. 14 Benavent’s Approach

15. 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 15 Depot 1 2 4 3 5 6 7

16. Partitioning Scheme  Set the vertex weights to account for known dead- heading and distance to the depot 16 Depot if link i must be deadheaded oth.

17. Partitioning Scheme  Linear weights, chosen empirically 17

18. Partitioning Scheme  Linear weights, chosen empirically, evenly spaced in [.01,.03] 18

19. Partitioning Scheme  Partition the transformed graph into k approximately equal parts. 19 Depot

20. Partitioning Scheme  Route the subgraphs induced by each partition using a single-vehicle solver. 20 Depot

21. 21 Partitioning Approach

22. 22 Benavent’s Approach

23. Results  Tested on a 64-bit PC running an Intel i5 4690K 3.5 GHz CPU, with 8 GB RAM  Two sets of benchmark instances  Real street networks taken from cities using the crowd- sourced Open Street Networks database  Trimmed to largest connected component  ~50% of links randomly assigned to be required  Artificial rectangular networks  ~50% of links randomly assigned to be required 23

24. Results: Street Networks 24 Instance|V||E|Partition Obj. ROIATDRuntime (s) Benavent Obj. ROIATDRuntime (s) % Diff San Francisco 7068459875.15920.24 759309 1.381311.7 1746.0 Washington D.C. 59266210694.091121.14 5210103 3.421963.2 1085.8 London, UK8489966162.19544.95 1285941 2.11737.82 2943.7 Istanbul, TR6317827391.50569.37 847195 2.78769.59 2692.7 Perth, AUS5325926776.12721.77 437039 2.01919.86 77-3.8 Auckland, AUS 1149123412785.091166.11 20313372 2.011537.3 664-4.6 Helsinki, FI129315316756.32566.30 4206509 1.78730.70 8583.8 Vienna, AU4905713662.12459.73 483522 1.31533.63 564.0 Paris, FR1949227414468.13899.04 1090N/A Calgary, CA1733228220676.241177.77 840N/A

25. Results: Rectangular Networks 25 Instance|V||E|Partition Obj. ROIATDRuntime (s) Benavent Obj. ROIATDRuntime (s) % Diff Random 15761104841.3749.92 94833 3.0483.05 188.96 Random 25291012774.5050.98 68766 2.9575.36 1391.04 Random 3484924685.3545.94 60688 2.5570.75 99-.44 Random 4441840639.5942.54 52635 2.9767.28 88.63 Random 5400760583.4739.18 45570 2.9962.08 692.28 Random 6361684527.5139.67 37518 2.4557.13 461.73 Random 7324612500.5237.31 32483 2.6055.85 383.51 Random 8289544472.5537.24 28454 2.4765.62 313.96 Random 9256480381.4633.16 25391 2.4353.75 21-2.62 Random 10225420360.4331.08 22361 2.6149.43 16-.28

26. Conclusions and Future Work  Advantages of partitioning heuristic  Can solve 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 )  Speed – each perturbation takes considerable time  Future Work  Exploring relationship between number of vehicles, and tuning parameters 26

27. Large Instance 27 Test Instance: Cross-Section of Greenland |V|=3047 |E|=3285 Runtime: 328.3 s

28. References  Ahr, Dino, and Gerhard Reinelt. "New heuristics and lower bounds for the Min-Max k-Chinese Postman Problem." 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." Computational Optimization and Applications 4:1 (1995): 67-77.  Derigs, Ulrich. Optimization and operations research. Eolss Publishers Company Limited, 2009.  Dussault, Benjamin, et al. "Plowing with precedence: A variant of the windy postman problem."Computers & Operations Research (2012).  Edmonds, Jack, and Ellis L. Johnson. "Matching, Euler tours and the Chinese postman." Mathematical Programming 5:1 (1973): 88-124. 28

29. References  Eiselt, Horst A., Michel Gendreau, and Gilbert Laporte. "Arc routing problems, part II: The rural postman problem." Operations Research 43:3 (1995): 399-414.  Frederickson, Greg N. "Approximation algorithms for some postman problems." Journal of the ACM(JACM) 26:3 (1979): 538-554.  Grotschel, Martin, and Zaw Win. "A cutting plane algorithm for the windy postman problem." Mathematical 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.  http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=minimumCo stFlow2 http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=minimumCo stFlow2 29

30. References  http://en.wikipedia.org/wiki/Dijkstra's_algorithm http://en.wikipedia.org/wiki/Dijkstra's_algorithm  http://en.wikipedia.org/wiki/Floyd\OT1\textendashWarshall_algorithm http://en.wikipedia.org/wiki/Floyd\OT1\textendashWarshall_algorithm  http://en.wikipedia.org/wiki/Prim%27s_algorithm http://en.wikipedia.org/wiki/Prim%27s_algorithm  Karypis, George, and Vipin Kumar. "A fast and high quality multilevel scheme for partitioning irregulargraphs." SIAM Journal on Scientific Computing 20:1 (1998): 359-392.  Kolmogorov, Vladimir. "Blossom V: a new implementation of a minimum cost perfect matching algorithm." Mathematical Programming Computation 1:1 (2009): 43-67.  Lau, Hang T. A Java library of graph algorithms and optimization. CRC Press, 2010. 30

31. References  Letchford, Adam N., Gerhard Reinelt, and Dirk Oliver Theis. "A faster exact separation algorithm for blossom inequalities." Integer Programming and Combinatorial Optimization. Springer Berlin Heidelberg, 2004. 196-205.  Padberg, Manfred W., and M. Ram Rao. "Odd minimum cut-sets and b- matchings." Mathematics of Operations 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 the mixed Chinese postman problem." Optimization and Engineering 3:2 (2002): 157-187 31

32. Backup 32

33. OAR Lib Motivation  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  Open Street Maps Integration  Gephi toolkit (open source graph visualization) Integration 33  A (perceived) barrier to entry that coding experience in a non- modeling language is required  No centralized, standardized implementations of many routing algorithms  Existing Application Programming Interfaces (APIs) are frequently developed with graph theoretic research in mind  Realistic test data procurement  Figure generation for papers Problem: Solution:

34. Interchange  Two-Interchange and Or-Interchange move a (string of) required link(s) to a different position in the route 34 12 21 Two-Interchange

35. Swap  Change 1-to-1, 1-to-0, and 2-to-0 swap or move edges off of a route 35 Change 1-to-0

36. Compact Representation  A route may be represented simply as an ordered list of the required links it traverses, with implied shortest paths taken between them 36 1 23 4 5 2 4

37. A New Objective Function  Attempt to incorporate compactness into the measure of solution quality 37