Date post: | 13-Jan-2016 |
Category: |
Documents |
Upload: | norman-copeland |
View: | 213 times |
Download: | 1 times |
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
Algorithm of Benavent et al.The MMKWRPP
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
Algorithm of Benavent et al.The MMKWRPP
0 21 83 4 5 6 7
8
Algorithm of Benavent et al.The MMKWRPP
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
Algorithm of Benavent et al.The MMKWRPP
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
Partitioning ApproachThe MMKWRPP
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
Partitioning ApproachThe MMKWRPP
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
Partitioning ApproachThe MMKWRPP
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
Partitioning ApproachThe MMKWRPP
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
25