Compact Routes for the Min-Max K Windy Rural Postman Problem by Oliver Lum 1, Carmine Cerrone 2, Bruce Golden 3, Edward Wasil 4 1. Department of Applied Mathematics and Scientific Computation, University of Maryland, College Park 2. Department of Computer Science, University of Salerno 3. R.H Smith School of Business, University of Maryland, College Park 4 Kogod School of Business, American University

2 Problem Motivation Depot = Required = Included in route = Not traversed The MMKWRPP A natural extension of the Windy Rural Postman Problem Minimize the max route cost Homogenous fleet of 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 Route 1 Route 2 Route 3

3 One of the most appealing features of the Min-Max K Windy Rural Postman Problem is that it has many fundamental arc routing problems as special cases. Generality MMKWRPP MMKURPPMMKDRPP MMKMRPP URPPDRPPMRPP CPPDCPPMCPP WRPP MCPP Graph Transformation Single-Vehicle Full-Service P P

Literature Review 1 Introduction Benavent, Enrique, et al. “Min- max k-vehicles windy rural postman problem.” Networks 54:4 (2009): Metaheuristic 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): Exact Solver Benavent, Enrique, et al. “A branch-price-and-cut method for the min-max k-windy rural postman problem.” Networks 63:1 (2014): The MMKWRPP 2 3 ILP Formulation Polyhedron Characterized Valid Inequalities (Aggregated, Disaggregated, R-odd cut, Honeycomb, etc.) Route-First, Cluster-Second Heuristic Multi-Start, ILS Metaheuristic based on single-vehicle work by same authors Improves on the 2009 work Adds pricing scheme Faster, more scalable method, used to solve larger instances 4

Algorithm of Benavent et al. Step 1: WRPP Solve the single-vehicle variant. Step 2: Compact Route Representation This produces a solution that can be represented as an ordered list of required edges (where any gaps are traversed via shortest paths) Step 3: Split Solve for the optimal split of the route into k distinct routes, by finding k-1 points in the route to return to the depot, preserving ordering The MMKWRPP Depot

Construct 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 Algorithm of Benavent et al. The MMKWRPP 6

Algorithm of Benavent et al. The MMKWRPP 7 Depot

Find a k-edge narrowest path (a path in which the weight of the heaviest edge in the traversal is minimized) from v 0 to v m in the DAG, corresponding to a solution A simple modification to Dijkstra’s single-source shortest path algorithm can produce such a path Algorithm of Benavent et al. The MMKWRPP 8

Algorithm of Benavent et al. The MMKWRPP Step 1: WRPP Solve the single-vehicle variant.. Step 2: Compact Route Representation This produces a solution that can be represented as an ordered list of required edges (where any gaps are traversed via shortest paths). Step 3: Split Solve for the optimal split of the route into k distinct routes, by finding k-1 points in the route to return to the depot, preserving ordering 9 Depot x x

Algorithm of Benavent et al. The MMKWRPP 10 A B C A={red, yellow} B={black, blue, teal} C={black, yellow, teal}

Partitioning Approach The MMKWRPP Depot Transform the graph into a vertex-weighted graph by constructing its edge dual 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

Partitioning Approach The MMKWRPP Depot if link i must be deadheaded otherwise Set the vertex weights to account for known dead-heading and distance to the depot

Partition the transformed graph into k approximately equal parts Partitioning Approach The MMKWRPP Depot 13 Green Vertex Green Edge

For each of the partitions, solve the single-vehicle problem for which only the required edges in the partition are actually required Partitioning Approach The MMKWRPP Depot

Visually appealing Customers on the same route are close to each other Other than travel to and from the depot, little overlap Routes further from depot are smaller Customers as contiguous as possible Partitioning Approach The MMKWRPP 15

Comparing Partitions The MMKWRPP 16

17 Aesthetic Measures In practice, routes often exhibit properties like connectedness and compactness Two metrics (ROI, ATD) proposed in Constantino et al. (European Journal of Operational Research, 2015) are the first to feature interactions between routes We introduce a third metric, Hull Overlap (HO), that incorporates the intuition behind ROI and ATD Average Traversal Distance Route Overlap Index Hull Overlap

Attempts to measure the degree to which a set of routes overlaps. It penalizes each ‘required’ node for every route in which it’s visited, and normalizes based on an ‘ideal’, square instance (shown below on the right) FormulaMotivation Route Overlap IndexNode OverlapSquare InstanceSquare RoutesBorder Compensation Route Overlap Index (ROI) Compactness Metrics 18

FormulaMotivation Average Traversal Distance Pairwise Dist.Task PairsNon-Comp. Routes Compact Routes Average Traversal Distance (ATD) Compactness Metrics Depot Compact Routes Depot Non-compact Routes 19 Attempts to measure the compactness of a set of routes. It penalizes pairwise shortest path distances between links requiring service.

FormulaMotivation First ProcessSecond ProcessThird ProcessFourt ProcessFinal Process Hull Overlap (HO) Compactness Metrics Set of routes in the solution Area of the intersection of arguments Convex hull of the points comprising the argument Area of the argument Depot Non-compact Routes 20 Attempts to measure the degree to which a set of routes overlaps. It calculates the average portion of a route that overlaps with others.

10 real street networks taken from cities using the crowd-sourced Open Street Networks database 10 artificial rectangular networks, with random costs between 1 and 10 Experiments run with 3, 5, and 10 vehicles, with 20%, 50%, and 80% of links required Test Specs: 64-bit PC Intel i5 4690K 3.5 GHz CPU 8 GB RAM Computational Results The MMKWRPP Metrics: Distance of longest route Average Traversal Distance Route Overlap Index Hull Overlap 21

Computational Results on Real Street Networks The MMKWRPP test instances (3 fleet size variations, and 2 depot locations for each of the 10 underlying networks) |V| ranges from 506 to 2027 |E| ranges from 586 to 2588 With respect to max distance, BENAVENT outperforms LUM by 2.36% on average With respect to ROI, LUM outperforms BENAVENT by 81.7% on average With respect to ATD, LUM outperforms BENAVENT by 22.9% on average With respect to HO, LUM outperforms BENAVENT by 26.8% on average BENAVENT runs into memory constraints on the largest two networks. Results only consider the 48 instances both approaches were able to solve

Computational Results on Artificial Networks The MMKWRPP test instances (3 fleet size variations, and 2 depot locations for each of the 10 underlying networks) |V| ranges from 225 to 576 |E| ranges from 420 to 1104 With respect to max distance, BENAVENT outperforms LUM by 4.38% on average With respect to ROI, LUM outperforms BENAVENT by 72.7% on average With respect to ATD, LUM outperforms BENAVENT by 29.6% on average With respect to HO, LUM outperforms BENAVENT by 38.6% on average

Refine the Partitions Route Quality Survey Optimize a Multi-Objective Function Conclusions The MMKWRPP 24 In practice, many routing problems require visually appealing solutions We reviewed previous attempts in the literature to quantify what constitutes a ‘visually appealing’ set of routes and proposed our own metric that captures additional intuition We presented an algorithm to solve a general arc routing variant and compared solutions with the existing state-of-the-art procedure We showed the tradeoff between performance with respect to the objective function and the aesthetic quality of the routes Computational results demonstrate consistent relative performance, robust to network layout, fleet size, and depot position

Refine the Partitions Route Quality Survey Optimize a Multi-Objective Function Build the new metrics into the optimization procedures so that it’s possible to tune a solution technique to the relative importance of having aesthetically pleasing routes Verify and motivate new metric design based on the results of what practitioners actually consider ‘good- looking’ routes Improvement procedures and transformations to iteratively alter the partition Future Work The MMKWRPP 25