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

Compact Routes for the Min-Max K Windy Rural Postman Problem

by Oliver Lum1, Carmine Cerrone2, Bruce Golden3, Edward Wasil4

1. Department of Applied Mathematics and Scientific Computation, University of Maryland, College Park2. Department of Computer Science, University of Salerno3. R.H Smith School of Business, University of Maryland, College Park4 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

MMKURPP MMKDRPP MMKMRPP

URPP DRPP MRPP

CPP DCPP MCPP

WRPP

MCPP

Graph Transformation

Single-Vehicle

Full-ServiceP

Literature Review

1

Introduction

Benavent, Enrique, et al. “Min-max k-vehicles windy rural postman problem.”

Networks 54:4 (2009): 216-226.

.

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): 269-287.

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): 34-45.

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: WRPPSolve the single-vehicle variant.

Step 2: Compact Route RepresentationThis produces a solution that can be represented as an ordered list of required edges (where any gaps are traversed via shortest paths)

Step 3: SplitSolve 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

1

2

3

4

5

67

8

5

• 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

0 21 8

6

j


0 21

7

Depot

1

23

4

5

67

8

• Find a k-edge narrowest path (a path in which the weight of the heaviest

edge in the traversal is minimized) from v0 to vm in the DAG, corresponding

to a solution

• A simple modification to Dijkstra’s single-source shortest path algorithm can produce such a path


0 21 83 4 5 6 7

8


Step 1: WRPPSolve the single-vehicle variant..

Step 2: Compact Route RepresentationThis produces a solution that can be represented as an ordered list of required edges (where any gaps are traversed via shortest paths).

Step 3: SplitSolve 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

1

2

3

4

5

67

8

x x


10

A

B

C A={red, yellow}B={black, blue, teal}

C={black, yellow, teal}

Partitioning ApproachThe MMKWRPP

A

B

C

Depot

D

1

2 3

4

1

2

4

35

6

5

6

E

7 7

11

• 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


A

B

C

Depot

D

1

2 3

4

5

6

E

7

)(

||||)(

*2)1(*)( inearest

REidist

dicw ii

if link i must be deadheaded

otherwise

• Set the vertex weights to account for known dead-heading and distance to the depot

12

1

2

4

3 5

6 7

• Partition the transformed graph into k approximately equal parts


A

B

C

Depot

D

1

2 3

4

5

6

E

7

13

Green VertexGreen Edge

1

2

4

35

67

• For each of the partitions, solve the single-vehicle problem for which only the required edges in the partition are actually required


1

2

3

Depot

4

5

1

2

3

Depot

4

5

1

2

3

Depot

4

5

14

1

2

4

35

67

1

2

4

35

67

1

2

4

35

67

• 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


15

Comparing PartitionsThe 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

taskpairs

Dist

ATD

R

r Sbaab

r

||

1 ,

Route Overlap Index

||)1||||(

||2 NNR

NNOROI

Ni

R

r

rinNO

||

1

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)

Formula Motivation

Route Overlap Index Node Overlap Square Instance Square Routes Border Compensation

Route Overlap Index (ROI)Compactness Metrics

||)1||||(

||2 NNR

NNOROI

Ni

R

r

rinNO

||

1

18

Formula Motivation

Average Traversal Distance Pairwise Dist. Task Pairs Non-Comp. Routes Compact Routes

Average Traversal Distance (ATD)Compactness Metrics

Depot

64

2

1

3

7

5

Compact Routes

Depot

64

2

1

3

7

5

Non-compact Routes

R

Rtaskstaskstaskpairs

*2

)(*

taskpairs

Dist

ATD

R

r Sbaab

r

||

1 ,

19

Attempts to measure the compactness of a set of routes. It penalizes pairwise shortest path distances between links requiring service.

Formula Motivation

First Process Second Process Third Process Fourt Process Final Process

Hull Overlap (HO)Compactness Metrics

Set of routes in the solution

Area of the intersection of argu-

ments

Convex hull of the points comprising the argu-

ment

Area of the argu-

ment

Depot

6

4

2

1

3

7

5

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 ResultsThe MMKWRPP

Metrics:• Distance of longest route • Average Traversal

Distance• Route Overlap Index• Hull Overlap

21

Computational Results on Real Street NetworksThe MMKWRPP

22

• 60 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 NetworksThe MMKWRPP

23

• 60 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

ConclusionsThe 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 WorkThe MMKWRPP

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Documents

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