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Parallelization of the Vehicle Routing Problem with Time Windows
, 2001
"... Routing with time windows (VRPTW) has been an area of research that have
attracted many researchers within the last 10 { 15 years. In this period a number
of papers and technical reports have been published on the exact solution of the
VRPTW.
The VRPTW is a generalization of the wellknown capacitat ..."
Abstract

Cited by 27 (1 self)
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Routing with time windows (VRPTW) has been an area of research that have
attracted many researchers within the last 10 { 15 years. In this period a number
of papers and technical reports have been published on the exact solution of the
VRPTW.
The VRPTW is a generalization of the wellknown capacitated routing problem
(VRP or CVRP). In the VRP a
eet of vehicles must visit (service) a number
of customers. All vehicles start and end at the depot. For each pair of customers
or customer and depot there is a cost. The cost denotes how much is costs a
vehicle to drive from one customer to another. Every customer must be visited
exactly ones. Additionally each customer demands a certain quantity of goods
delivered (know as the customer demand). For the vehicles we have an upper
limit on the amount of goods that can be carried (known as the capacity). In
the most basic case all vehicles are of the same type and hence have the same
capacity. The problem is now for a given scenario to plan routes for the vehicles
in accordance with the mentioned constraints such that the cost accumulated
on the routes, the #12;xed costs (how much does it cost to maintain a vehicle) or
a combination hereof is minimized.
In the more general VRPTW each customer has a time window, and between
all pairs of customers or a customer and the depot we have a travel time. The
vehicles now have to comply with the additional constraint that servicing of the
customers can only be started within the time windows of the customers. It
is legal to arrive before a time window \opens" but the vehicle must wait and
service will not start until the time window of the customer actually opens.
For solving the problem exactly 4 general types of solution methods have
evolved in the literature: dynamic programming, DantzigWolfe (column generation),
Lagrange decomposition and solving the classical model formulation
directly.
Presently the algorithms that uses DantzigWolfe given the best results
(Desrochers, Desrosiers and Solomon, and Kohl), but the Ph.D. thesis of Kontoravdis
shows promising results for using the classical model formulation directly.
In this Ph.D. project we have used the DantzigWolfe method. In the
DantzigWolfe method the problem is split into two problems: a \master problem"
and a \subproblem". The master problem is a relaxed set partitioning
v
vi
problem that guarantees that each customer is visited exactly ones, while the
subproblem is a shortest path problem with additional constraints (capacity and
time window). Using the master problem the reduced costs are computed for
each arc, and these costs are then used in the subproblem in order to generate
routes from the depot and back to the depot again. The best (improving) routes
are then returned to the master problem and entered into the relaxed set partitioning
problem. As the set partitioning problem is relaxed by removing the
integer constraints the solution is seldomly integral therefore the DantzigWolfe
method is embedded in a separationbased solutiontechnique.
In this Ph.D. project we have been trying to exploit structural properties in
order to speed up execution times, and we have been using parallel computers
to be able to solve problems faster or solve larger problems.
The thesis starts with a review of previous work within the #12;eld of VRPTW
both with respect to heuristic solution methods and exact (optimal) methods.
Through a series of experimental tests we seek to de#12;ne and examine a number
of structural characteristics.
The #12;rst series of tests examine the use of dividing time windows as the
branching principle in the separationbased solutiontechnique. Instead of using
the methods previously described in the literature for dividing a problem into
smaller problems we use a methods developed for a variant of the VRPTW. The
results are unfortunately not positive.
Instead of dividing a problem into two smaller problems and try to solve
these we can try to get an integer solution without having to branch. A cut is an
inequality that separates the (nonintegral) optimal solution from all the integer
solutions. By #12;nding and inserting cuts we can try to avoid branching. For the
VRPTW Kohl has developed the 2path cuts. In the separationalgorithm for
detecting 2path cuts a number of test are made. By structuring the order in
which we try to generate cuts we achieved very positive results.
In the DantzigWolfe process a large number of columns may be generated,
but a signi#12;cant fraction of the columns introduced will not be interesting with
respect to the master problem. It is a priori not possible to determine which
columns are attractive and which are not, but if a column does not become part
of the basis of the relaxed set partitioning problem we consider it to be of no
bene#12;t for the solution process. These columns are subsequently removed from
the master problem. Experiments demonstrate a signi#12;cant cut of the running
time.
Positive results were also achieved by stopping the routegeneration process
prematurely in the case of timeconsuming shortest path computations. Often
this leads to stopping the shortest path subroutine in cases where the information
(from the dual variables) leads to \bad" routes. The premature exit
from the shortest path subroutine restricts the generation of \bad" routes signi
#12;cantly. This produces very good results and has made it possible to solve
problem instances not solved to optimality before.
The parallel algorithm is based upon the sequential DantzigWolfe based
algorithm developed earlier in the project. In an initial (sequential) phase unsolved
problems are generated and when there are unsolved problems enough
vii
to start work on every processor the parallel solution phase is initiated. In the
parallel phase each processor runs the sequential algorithm. To get a good workload
a strategy based on balancing the load between neighbouring processors is
implemented. The resulting algorithm is eÆcient and capable of attaining good
speedup values. The loadbalancing strategy shows an even distribution of work
among the processors. Due to the large demand for using the IBM SP2 parallel
computer at UNI#15;C it has unfortunately not be possible to run as many tests
as we would have liked. We have although managed to solve one problem not
solved before using our parallel algorithm.
A.: Duty scheduling in public transit
 MATHEMATICS—Key Technology for the Future
, 2003
"... Abstract. This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward rel ..."
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Abstract. This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant parts of 9. large problem. Computational results for three European scenarios are reported. 1
Engine Routing and Scheduling at Industrial InPlant Railroads
, 2003
"... Inplant railroad engine scheduling involves routing and scheduling decisions for a heterogeneous fleet of switching engines in order to serve a set of time window and capacity constrained transportation requests. The current planning is purely by pencil and paper. Facing an ever increasing competit ..."
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Cited by 5 (3 self)
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Inplant railroad engine scheduling involves routing and scheduling decisions for a heterogeneous fleet of switching engines in order to serve a set of time window and capacity constrained transportation requests. The current planning is purely by pencil and paper. Facing an ever increasing competition, industrial railroads look for an active decision support.
A column generation approach to airline crew scheduling
, 2005
"... The airline crew scheduling problem deals with the construction of crew rotations in order to cover the flights of a given schedule at minimum cost. The problem involves complex rules for the legality and costs of individual pairings and base constraints for the availability of crews at home bases. ..."
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Cited by 4 (0 self)
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The airline crew scheduling problem deals with the construction of crew rotations in order to cover the flights of a given schedule at minimum cost. The problem involves complex rules for the legality and costs of individual pairings and base constraints for the availability of crews at home bases. A typical instance considers a planning horizon of one month and several thousand flights. We propose a column generation approach for solving airline crew scheduling problems that is based on a set partitioning model. We discuss algorithmic aspects such as the use of bundle techniques for the fast, approximate solution of linear programs, a pairing generator that combines Lagrangean shortest path and callback techniques, and a novel “rapid branching” IP heuristic. Computational results for a number of industrial instances are reported. Our approach has been implemented within the commercial crew scheduling system NetLine/Crew of Lufthansa Systems Berlin GmbH.
Scheduling Duties by Adaptive Column Generation
, 2001
"... This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant part ..."
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Cited by 3 (0 self)
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This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant parts of a large problem. Computational results for three European scenarios are reported.
A Generalized Threshold Algorithm for the Shortest Path Problem with Time Windows
"... Abstract: In this paper, we present a new labeling algorithm for the shortest path problem with time windows (SPPTW). It is generalized from the threshold algorithm for the unconstrained shortest path problem. Our computational experiments show that this generalized threshold algorithm outperforms a ..."
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Abstract: In this paper, we present a new labeling algorithm for the shortest path problem with time windows (SPPTW). It is generalized from the threshold algorithm for the unconstrained shortest path problem. Our computational experiments show that this generalized threshold algorithm outperforms a label setting algorithm for the SPPTW on a set of randomly generated test problems. The average running time of the new algorithm is about 40 % less than the label setting algorithm, which istoday the best algorithm based on published experimental evidence. 1 The shortest path problem with time windows (SPPTW) is a generalization of the classical (unconstrained) shortest path problem (SPP) involving the added complexity of time windows. The SPPTW can be described as follows. Let G =(V�A) be a directed graph where V = N [fp � qg is the set of nodes with source node p and sink node q, A is the set of arcs. Each nodei2V has a time window [ai�bi] within which nodeican be visited. Each arc (i � j) has a positive duration tij
An Exact Algorithm for the Vehicle and Crew Scheduling Problem
 COMPUTERAIDED TRANSIT SCHEDULING. VOLUME 471 OF LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SYSTEMS
, 1999
"... We present a model for the vehicle and crew scheduling problem in urban public transport systems by combining models for vehicle and crew scheduling that cover a great variety of real world aspects, especially constraints for crews resulting from wage agreements and internal regulations. The main pa ..."
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Cited by 2 (0 self)
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We present a model for the vehicle and crew scheduling problem in urban public transport systems by combining models for vehicle and crew scheduling that cover a great variety of real world aspects, especially constraints for crews resulting from wage agreements and internal regulations. The main part of the model consists of a set partitioning formulation to cover the desired trips of the schedule. Because of the great number of columns, e.g. more than 5 million for a problem with 30 trips, a column generation algorithm is implemented to use all columns implicitly for the calculation of the continuous relaxation of the set partitioning problem. The column generation algorithm is embedded in a branch and bound approach to generate an exact solution for the problem. To generate even better lower bound, polyhedral cuts basing on clique detection and a variant of the column generation algorithm that suits the cuts were tested.
Freight flow consolidation in presence of time windows”, accepted for publication
 in Proceedings of the Operations Research 2004 International Conference
"... Abstract. This contribution addresses the consideration of time windows in the optimization of multicommodity network flows. For each node, one interval is specified in which the visitation is allowed. Applications in freight flow consolidation let this problem become interesting. An optimization m ..."
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Abstract. This contribution addresses the consideration of time windows in the optimization of multicommodity network flows. For each node, one interval is specified in which the visitation is allowed. Applications in freight flow consolidation let this problem become interesting. An optimization model is proposed and a construction heuristic is presented. For improving the generated solutions, a genetic algorithm framework including several hill climbing procedures for local optimization, is configured. 1
Solution Methods for the Multitrip Elementary Shortest Path Problem with Resource Constraints
, 2010
"... We investigate the multitrip elementary shortest path problem (MESPPRC) with resource constraints in which the objective is to find a shortest path between a source node and a sink node such that nodes other than the specified replenishment node are visited at most once and resource constraints are ..."
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We investigate the multitrip elementary shortest path problem (MESPPRC) with resource constraints in which the objective is to find a shortest path between a source node and a sink node such that nodes other than the specified replenishment node are visited at most once and resource constraints are not violated. After each visit to the replenishment node, which we take to be the sink node in our study, resource consumption levels can be reset to a certain initial value. As this problem arises primarily as a subproblem in decompositionbased algorithms for a wide variety of practical applications, we illustrate it in the context of an integrated routing and scheduling problem with capacitated and time restricted vehicles. We propose exact and heuristic algorithms and evaluate the performance of these in a branchandprice framework. 1