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

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

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

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

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

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

Min-Max K WRPP  Literature review  Benavent, Enrique, et al. “Min-Max K-vehicles windy rural postman problem.” Networks 54:4 (2009):  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):  Benavent, Enrique, et al. “A branch-price-and-cut method for the min-max k-windy rural postman problem.” Networks 63:1 (2014):

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

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

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

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)

Route Overlap Index 12

Average Traversal Distance 13 Depot Compact Routes Non-compact Routes

14 Benavent’s Approach

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

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.

Partitioning Scheme  Linear weights, chosen empirically 17

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

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

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

21 Partitioning Approach

22 Benavent’s Approach

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

Results: Street Networks 24 Instance|V||E|Partition Obj. ROIATDRuntime (s) Benavent Obj. ROIATDRuntime (s) % Diff San Francisco Washington D.C London, UK Istanbul, TR Perth, AUS Auckland, AUS Helsinki, FI Vienna, AU Paris, FR N/A Calgary, CA N/A

Results: Rectangular Networks 25 Instance|V||E|Partition Obj. ROIATDRuntime (s) Benavent Obj. ROIATDRuntime (s) % Diff Random Random Random Random Random Random Random Random Random Random

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

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

References  Ahr, Dino, and Gerhard Reinelt. "New heuristics and lower bounds for the Min-Max k-Chinese Postman Problem." Algorithms|ESA Springer Berlin Heidelberg,  Benavent, Enrique, et al. "New heuristic algorithms for the windy rural postman problem." Computers & Operations Research 32:12 (2005):  Campos, V., and J. V. Savall. "A computational study of several heuristics for the DRPP." Computational Optimization and Applications 4:1 (1995):  Derigs, Ulrich. Optimization and operations research. Eolss Publishers Company Limited,  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):

References  Eiselt, Horst A., Michel Gendreau, and Gilbert Laporte. "Arc routing problems, part II: The rural postman problem." Operations Research 43:3 (1995):  Frederickson, Greg N. "Approximation algorithms for some postman problems." Journal of the ACM(JACM) 26:3 (1979):  Grotschel, Martin, and Zaw Win. "A cutting plane algorithm for the windy postman problem." Mathematical Programming 55:1-3 (1992):  Hierholzer, Carl, and Chr Wiener. "Uber die Moglichkeit, einen Linienzug ohne Wiederholung und ohneUnterbrechung zu umfahren." Mathematische Annalen 6:1 (1873):  stFlow2 stFlow2 29

References     Karypis, George, and Vipin Kumar. "A fast and high quality multilevel scheme for partitioning irregulargraphs." SIAM Journal on Scientific Computing 20:1 (1998):  Kolmogorov, Vladimir. "Blossom V: a new implementation of a minimum cost perfect matching algorithm." Mathematical Programming Computation 1:1 (2009):  Lau, Hang T. A Java library of graph algorithms and optimization. CRC Press,

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,  Padberg, Manfred W., and M. Ram Rao. "Odd minimum cut-sets and b- matchings." Mathematics of Operations Research 7:1 (1982):  Thimbleby, Harold. "The directed chinese postman problem." Software: Practice and Experience 33:11(2003):  Win, Zaw. "On the windy postman problem on Eulerian graphs." Mathematical Programming 44:1-3(1989):  Yaoyuenyong, Kriangchai, Peerayuth Charnsethikul, and Vira Chankong. "A heuristic algorithm for the mixed Chinese postman problem." Optimization and Engineering 3:2 (2002):

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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:

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

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

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

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