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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 24 (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 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 eld of VRPTW
both with respect to heuristic solution methods and exact (optimal) methods.
Through a series of experimental tests we seek to dene and examine a number
of structural characteristics.
The 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 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 signicant 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
benet for the solution process. These columns are subsequently removed from
the master problem. Experiments demonstrate a signicant 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
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 UNIC 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 reactive variable neighborhood search for the vehicle routing problem with time windows
 INFORMS Journal on Computing
, 2003
"... The purpose of this paper is to present a new deterministic metaheuristic based on a modification of Variable Neighborhood Search of Mladenovic and Hansen (1997) for solving the vehicle routing problem with time windows. Results are reported for the standard 100, 200 and 400 customer data sets by So ..."
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Cited by 21 (0 self)
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The purpose of this paper is to present a new deterministic metaheuristic based on a modification of Variable Neighborhood Search of Mladenovic and Hansen (1997) for solving the vehicle routing problem with time windows. Results are reported for the standard 100, 200 and 400 customer data sets by Solomon (1987) and Gehring and Homberger (1999) and two reallife problems by Russell (1995). The findings indicate that the proposed procedure outperforms other recent local searches and metaheuristics. In addition four new bestknown solutions were obtained. The proposed procedure is based on a new fourphase approach. In this approach an initial solution is first created using new route construction heuristics followed by route elimination procedure to improve the solutions regarding the number of vehicles. In the third phase the solutions are improved in terms of total traveled distance using four new local search procedures proposed in this paper. Finally in phase four the best solution obtained is improved by modifying the objective function to escape from a local minimum. (Metaheuristics; Vehicle Routing; Time Windows) 1.
Strategies for the parallel implementation of metaheuristics
 Essays and Surveys in Metaheuristics
, 2002
"... Abstract. Parallel implementationsof metaheuristicsappear quite naturally asan effective alternative to speed up the search for approximate solutions of combinatorial optimization problems. They not only allow solving larger problems or finding improved solutions with respect to their sequential cou ..."
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Cited by 15 (5 self)
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Abstract. Parallel implementationsof metaheuristicsappear quite naturally asan effective alternative to speed up the search for approximate solutions of combinatorial optimization problems. They not only allow solving larger problems or finding improved solutions with respect to their sequential counterparts, but they also lead to more robust algorithms. We review some trends in parallel computing and report recent results about linear speedups that can be obtained with parallel implementations using multiple independent processors. Parallel implementations of tabu search, GRASP, genetic algorithms, simulated annealing, and ant colonies are reviewed and discussed to illustrate the main strategies used in the parallelization of different metaheuristics and their hybrids. 1. Introduction. Although
Parallel Strategies for Metaheuristics
"... We present a stateoftheart survey of parallel metaheuristic developments and results, discuss general design and implementation principles that apply to most metaheuristic classes, instantiate these principles for the three metaheuristic classes currently most extensively used  genetic metho ..."
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Cited by 14 (4 self)
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We present a stateoftheart survey of parallel metaheuristic developments and results, discuss general design and implementation principles that apply to most metaheuristic classes, instantiate these principles for the three metaheuristic classes currently most extensively used  genetic methods, simulated annealing, and tabu search, and identify a number of trends and promising research directions.
RealTime Decision Problems: an Operations Research Perspective
"... This paper is concerned with realtime decision problems. These constitute a generic class of dynamic and stochastic problems. The objective is to provide responses of a required quality in a continuously evolving environment, within a prescribed time frame, using limited resources and information t ..."
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Cited by 9 (3 self)
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This paper is concerned with realtime decision problems. These constitute a generic class of dynamic and stochastic problems. The objective is to provide responses of a required quality in a continuously evolving environment, within a prescribed time frame, using limited resources and information that is often incomplete or uncertain. Furthermore, the outcome of any particular decision may also be uncertain. This paper provides an overview of this class of problems, reviews the relevant Artificial Intelligence literature, proposes a dynamic programming framework, and assesses the potential usefulness of Operations Research approaches for their solution. Throughout the paper, a vehicle dispatching application illustrates the relevant concepts.
Parallel Metaheuristics
, 1997
"... Metaheuristic parallel search methods  tabu search, simulated annealing and genetic algorithms, essentially  are reviewed, classified and examined not according to particular methodological characteristics, but following the unifying approach of the level of parallelization. It is hoped that by ..."
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Cited by 4 (2 self)
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Metaheuristic parallel search methods  tabu search, simulated annealing and genetic algorithms, essentially  are reviewed, classified and examined not according to particular methodological characteristics, but following the unifying approach of the level of parallelization. It is hoped that by examining the commonalities among parallel implementations across the field of metaheuristics, insights may be gained, trends may be discovered, and research challenges may be identified. Particular attention is paid to applications of parallel metaheuristics to transportation problems.
A method for vehicle routing problems with multiple vehicle types and time windows
 Proceedings of Natural Science Council
, 1999
"... ..."
A Parallel Tabu Search Algorithm for Optimizing Multiobjective VLSI Placement
"... Abstract. In this paper, we present a parallel tabu search (TS) algorithm for efficient optimization of a constrained multiobjective VLSI standard cell placement problem. The primary purpose is to accelerate TS algorithm to reach near optimal placement solutions for large circuits. The proposed tech ..."
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Abstract. In this paper, we present a parallel tabu search (TS) algorithm for efficient optimization of a constrained multiobjective VLSI standard cell placement problem. The primary purpose is to accelerate TS algorithm to reach near optimal placement solutions for large circuits. The proposed technique employs a candidate list partitioning strategy based on distribution of mutually disjoint set of moves among the slave processes. The implementation is carried out on a dedicated cluster of workstations. Experimental results using ISCAS85/89 benchmark circuits illustrating quality and speedup trends are presented. A comparison of the obtained results is made with the results of a parallel genetic algorithm (GA) implementation. 1 Introduction and Related Work General iterative heuristics such as tabu search and genetic algorithms (GAs) have been widely used to solve numerous hard problems [1]. This interest is attributed to their generality, ease of implementation, and ability to reach near
Parallelism in Combinatorial Optimisation
, 1995
"... This report addresses the issues arising from the use of parallel machines and considers the various techniques used by members of the consortium in this context. Before considering the algorithms in detail, we describe, in section 2, the main types of parallel architecture and survey various attemp ..."
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This report addresses the issues arising from the use of parallel machines and considers the various techniques used by members of the consortium in this context. Before considering the algorithms in detail, we describe, in section 2, the main types of parallel architecture and survey various attempts at providing a taxonomy. Then, in section 3, we address the difficult issue of the measurement of processor performance in order to quantify any enhancement obtained by implementing an algorithm in parallel. Section 4 presents the main features of in general, and PVM ( ) in particular. The latter is an application that is used to generate distributed versions of sequential algorithms for use on networks of workstations. The parallel implementations of the GA toolkit, , and the associated simulated annealing toolkit, , both developed at UEA, have been produced using PVM. Exact algorithms will always find the optimal solution to a problem given enough time and space. Subject to these constraints, they must always be the preferred method of solution. In practice, the time and space constraints can prevent the use of an exact algorithm and thus the potential of parallelism to reduce these factors becomes an important factor. Total enumeration is embarassingly parallel. With processors it is reasonable to expect anfold reduction in time to undertake such a thorough search. Such a saving is seldom sufficient to make the method viable so we will concentrate on other exact methods here. In section 5, we review parallel branchand bound, reprinting a survey paper written by the UEA partners in the consortium and previously published in [1]. Because of the interest in interior point methods for the CALMA project and its widely cited potential for parallelisation, this provides the ...
Advanced MultiStage Local Search Applications to Vehicle Routing Problem with Time Windows: A Review
"... ... This paper presents a survey of the latest research motivated by this recognition. The presentation is focused on multistage applications of advanced local search techniques on the VRPTW. Multistage algorithms optimize the number of vehicles and travel time independently in order to ensure tha ..."
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... This paper presents a survey of the latest research motivated by this recognition. The presentation is focused on multistage applications of advanced local search techniques on the VRPTW. Multistage algorithms optimize the number of vehicles and travel time independently in order to ensure that the search is directed towards the achievement of the primary objective. Basic features of these algorithms, as well as hybridization strategies are described. For most algorithms, experimental results on Solomon's benchmark test problems are provided and analyzed