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 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
The MMKWRPP
A natural extension of the Windy
Rural Postman Problem
Route 2
= Required
= Included in route
= Not traversed
Route 1
•
•
•
•
•
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
Depot
Route 3
3
Generality
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.
MMKWRPP
Graph Transformation
MMKURPP
MMKDRPP
MMKMRPP
Single-Vehicle
URPP
DRPP
MRPP
WRPP
Full-Service
P
CPP
DCPP
MCPP
MCPP
4
Literature Review
The MMKWRPP
1
2
3
Introduction
Metaheuristic
Exact Solver
Benavent, Enrique, et al. “Minmax 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.
.
• 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
5
Algorithm of Benavent et al.
The MMKWRPP
Step 1: WRPP
4
Solve the single-vehicle variant.
Step 2: Compact Route
Representation
2
5
3
This produces a solution that can be
represented as an ordered list of required
edges (where any gaps are traversed via
shortest paths)
6
1
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
Depot
8
7
6
Algorithm of Benavent et al.
The MMKWRPP
• 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
7
Algorithm of Benavent et al.
The MMKWRPP
4
2
5
3
6
1
Depot
8
7
8
Algorithm of Benavent et al.
The MMKWRPP
• 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
9
Algorithm of Benavent et al.
The MMKWRPP
Step 1: WRPP
4
Solve the single-vehicle variant..
Step 2: Compact Route
Representation
2
This produces a solution that can be
represented as an ordered list of required
edges (where any gaps are traversed via
shortest paths).
5
3
x
x
6
1
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
Depot
8
7
10
Algorithm of Benavent et al.
The MMKWRPP
B
C
A
A={red, yellow}
B={black, blue, teal}
C={black, yellow, teal}
11
Partitioning Approach
The MMKWRPP
• 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
4
2
3
5
7
1
6
Depot
12
Partitioning Approach
The MMKWRPP
• Set the vertex weights to account for known dead-heading and distance to the
depot
4
2
5
3
6
7
1
Depot
dist (i )
wi  c(i ) * (1  d i )  2 *
 nearest (i )
|E|
|R|
ì0
di = í
î1
if link i must be deadheaded
otherwise
13
Partitioning Approach
The MMKWRPP
• Partition the transformed graph into k approximately equal parts
4
2
5
3
7
6
1
Depot
Green Edge
Green Vertex
14
Partitioning Approach
The MMKWRPP
• For each of the partitions, solve the single-vehicle problem for which only the
required edges in the partition are actually required
4
4
4
2
2
2
5
3
1
Depot
5
3
7
7
6
5
3
6
1
Depot
7
6
1
Depot
15
Partitioning Approach
The MMKWRPP
• 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
16
Comparing Partitions
The MMKWRPP
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
Hull Overlap
Route Overlap
Index
NO | N |
ROI 
2
( | R |  | N |  1)  | N |
| R|
NO   n
iN r 1
r
i
Average Traversal
Distance
| R|
ATD 
  Dist
r 1 a ,bS r
taskpairs
ab
18
Route Overlap Index (ROI)
Compactness Metrics
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
NO | N |
ROI 
2
( | R |  | N |  1)  | N |
| R|
NO   n
iN r 1
r
i
Route Overlap Index
Node Overlap
Square Instance
Square Routes
Border Compensation
19
Average Traversal Distance (ATD)
Compactness Metrics
Attempts to measure the compactness of a set of routes. It penalizes pairwise shortest path distances between links
requiring service.
Formula
Motivation
| R|
ATD 
  Dist
r 1 a ,bS r
ab
Depot
taskpairs
taskpairs 
Non-compact Routes
tasks * ( tasks  R )
Average Traversal Distance
2* R
Pairwise Dist.
Depot
Compact Routes
Task Pairs
Non-Comp. Routes
Compact Routes
20
Hull Overlap (HO)
Compactness Metrics
Attempts to measure the degree to which a set of routes overlaps. It calculates the average portion of a route that
overlaps with others.
Formula
Motivation
Set of routes in the solution
Area of the intersection of arguments
Convex hull of the points comprising the argument
Depot
Area of the argument
First Process
Second Process
Non-compact Routes
Third Process
Fourt Process
Final Process
21
Computational Results
The MMKWRPP
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
Metrics:
• Distance of longest route
• Average Traversal Distance
• Route Overlap Index
• Hull Overlap
22
Computational Results on Real Street Networks
The MMKWRPP
• 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
23
Computational Results on Artificial Networks
The MMKWRPP
• 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
24
Conclusions
The MMKWRPP
• In practice, many routing problems require visually appealing
Route
Optimize a
solutions
Refine the
Quality
Multi-Objective
Partitions
• We reviewed previous attempts
in the literature to Function
quantify what
Survey
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
25
Future Work
The MMKWRPP
Refine the
Partitions
Improvement procedures and
transformations to iteratively
alter the partition
Route
Quality
Survey
Optimize a
Multi-Objective
Function
Verify and motivate new
metric design based on the
results of what practitioners
actually consider ‘goodlooking’ routes
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