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 ID: 356695
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Slide1
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 UniversitySlide2
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 3Slide3
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-Service
PSlide4
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
4Slide5
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
1
2
3
4
5
6
7
8
5Slide6
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
2
1
8
6
jSlide7
Algorithm of Benavent et al.
The MMKWRPP
0
2
1
7
Depot
1
2
3
4
5
6
7
8Slide8
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
0
2
1
8
3
4
5
6
7
8Slide9
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
1
2
3
4
5
6
7
8
x
xSlide10
Algorithm of Benavent et al.
The MMKWRPP
10
A
B
C
A={red,
yellow
}
B={
black
,
blue
,
teal
}
C={
black
,
yellow,
teal}Slide11
Partitioning Approach
The MMKWRPP
A
B
C
Depot
D
1
2
3
4
1
2
4
3
5
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 endpointSlide12
Partitioning Approach
The MMKWRPP
A
B
C
Depot
D
1
2
3
4
5
6
E
7
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
7Slide13
Partition the transformed graph into k approximately equal parts
Partitioning Approach
The MMKWRPP
A
B
C
Depot
D
1
2
3
4
5
6
E
7
13
Green
Vertex
Green
Edge
1
2
4
3
5
6
7Slide14
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
1
2
3
Depot
4
5
1
2
3
Depot
4
5
1
2
3
Depot
4
5
14
1
2
4
3
5
6
7
1
2
4
3
5
6
7
1
2
4
3
5
6
7Slide15
Visually
appealing
Customers on the same route are close to each otherOther than travel to and from the depot, little overlap
Routes further from depot are smallerCustomers as contiguous as possible
Partitioning Approach
The MMKWRPP
15Slide16
Comparing Partitions
The MMKWRPP
16Slide17
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
Slide18
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
18Slide19
Formula
Motivation
Average Traversal Distance
Pairwise Dist.
Task Pairs
Non-Comp. Routes
Compact Routes
Average Traversal Distance (ATD)
Compactness Metrics
Depot
6
4
2
1
3
7
5
Compact Routes
Depot
6
4
2
1
3
7
5
Non-compact Routes
19
Attempts to measure the compactness of a set of routes. It penalizes pairwise shortest path distances between links requiring service.Slide20
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 arguments
Convex hull of the points comprising the argument
Area of the argument
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.Slide21
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
21Slide22
Computational Results on Real Street Networks
The 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 solveSlide23
Computational Results on Artificial Networks
The 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 averageSlide24
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 Slide25
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