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Algorithms for the vehicle routing and scheduling problems with time window constraints
 Operations Research
, 1987
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Cited by 225 (0 self)
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Operations Research is currently published by INFORMS. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
Using Constraint Programming and Local Search Methods to Solve Vehicle Routing Problems
, 1998
"... We use a local search method we term Large Neighbourhood Search (LNS) for solving vehicle routing problems. LNS meshes well with constraint programming technology and is analogous to the shuffling technique of jobshop scheduling. The technique explores a large neighbourhood of the current solution ..."
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Cited by 136 (2 self)
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We use a local search method we term Large Neighbourhood Search (LNS) for solving vehicle routing problems. LNS meshes well with constraint programming technology and is analogous to the shuffling technique of jobshop scheduling. The technique explores a large neighbourhood of the current solution by selecting a number of customer visits to remove from the routing plan, and reinserting these visits using a constraintbased tree search. We analyse the performance of LNS on a number of vehicle routing benchmark problems. Unlike related methods, we use Limited Discrepancy Search during the tree search to reinsert visits. We also maintain diversity during search by dynamically altering the number of visits to be removed, and by using a randomised choice method for selecting visits to remove. We analyse the performance of our method for various parameter settings controlling the discrepancy limit, the dynamicity of the size of the removal set, and the randomness of the choice. We demonst...
An Improved Ant System Algorithm for the Vehicle Routing Problem
 Annals of Operations Research
, 1997
"... this paper an improved ant system algorithm for the Vehicle Routing Problem with one central depot and identical vehicles. Computational results on fourteen benchmark problems from the literature are reported and a comparison with five other metaheuristic approaches to solve vehicle routing problems ..."
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Cited by 84 (6 self)
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this paper an improved ant system algorithm for the Vehicle Routing Problem with one central depot and identical vehicles. Computational results on fourteen benchmark problems from the literature are reported and a comparison with five other metaheuristic approaches to solve vehicle routing problems is made.
A unified Tabu Search heuristic for vehicle routing problems with time windows
 JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY
, 2001
"... This article reviews ten of the most important tabu search heuristics for the vehicle routing problem. Some of the main tabu search features are first described: neighborhood structures, short term memory, long term memory, intensification. The tabu search algorithms are then described, followed by ..."
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Cited by 60 (9 self)
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This article reviews ten of the most important tabu search heuristics for the vehicle routing problem. Some of the main tabu search features are first described: neighborhood structures, short term memory, long term memory, intensification. The tabu search algorithms are then described, followed by computational results and the conclusion.
Applying the Ant System to the Vehicle Routing Problem
, 1997
"... In this paper we use a recently proposed metaheuristic, the Ant System, to solve the Vehicle Routing Problem (VRP) in its basic form, i.e. with capacity and distance restrictions, one central depot and identical vehicles. A "hybrid" Ant System algorithm is first presented and then improved using pro ..."
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Cited by 59 (7 self)
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In this paper we use a recently proposed metaheuristic, the Ant System, to solve the Vehicle Routing Problem (VRP) in its basic form, i.e. with capacity and distance restrictions, one central depot and identical vehicles. A "hybrid" Ant System algorithm is first presented and then improved using problem specific information (savings, capacity utilization). Experiments on various aspects of the algorithm and computational results for fourteen benchmark problems are reported and compared to those of other metaheuristic approaches such as Tabu Search, Simulated Annealing and Neural Networks. 1 Introduction The Ant System, introduced by Colorni, Dorigo and Maniezzo [6], [10], [12] is a new distributed metaheuristic for hard combinatorial optimization problems and was first used on the well known Traveling Salesman Problem (TSP). It has further been applied to the Job Shop Scheduling Problem in [7], to the Graph Colouring Problem in [8] and to the Quadratic Assignment Problem in [18]. Obse...
A general heuristic for vehicle routing problems
 Computers & Operations Research
, 2007
"... We present a unified heuristic, which is able to solve five different variants of the vehicle routing problem: the vehicle routing problem with time windows (VRPTW), the capacitated vehicle routing problem (CVRP), the multidepot vehicle routing problem (MDVRP), the site dependent vehicle routing pr ..."
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Cited by 35 (3 self)
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We present a unified heuristic, which is able to solve five different variants of the vehicle routing problem: the vehicle routing problem with time windows (VRPTW), the capacitated vehicle routing problem (CVRP), the multidepot vehicle routing problem (MDVRP), the site dependent vehicle routing problem (SDVRP) and the open vehicle routing problem (OVRP). All problem variants are transformed to a rich pickup and delivery model and solved using the Adaptive Large Neighborhood Search (ALNS) framework presented in Ropke and Pisinger (2004). The ALNS framework is an extension of the Large Neighborhood Search framework by Shaw (1998) with an adaptive layer. This layer adaptively chooses among a number of insertion and removal heuristics, to intensify and diversify the search. The presented approach has a number of advantages: ALNS provides solutions of very high quality, the algorithm is robust, and to some extent selfcalibrating. Moreover, the unified model allows the dispatcher to mix various variants of VRP problems for individual customers or vehicles. As we believe that the ALNS framework can be applied to a large number of tightly constrained optimization problems, a general description of the framework is given, and it is discussed how the various components can be designed in a particular setting. The paper is concluded with a computational study, in which the five different variants of the vehicle routing problem are considered on standard benchmark tests from the literature. The outcome of the tests is promising as the algorithm is able to improve 183 best known solutions out of 486 benchmark tests. The heuristic has also shown promising results for a large class of vehicle routing problems with backhauls, as demonstrated in Ropke and Pisinger (2005).
Using Experimental Design to Find Effective Parameter Settings for Heuristics
 Journal of Heuristics
, 2001
"... In this paper, we propose a procedure, based on statistical design of experiments and gradient descent, that finds effective settings for parameters found in heuristics. We develop our procedure using four experiments. We use our procedure and a small subset of problems to find parameter settings fo ..."
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Cited by 31 (1 self)
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In this paper, we propose a procedure, based on statistical design of experiments and gradient descent, that finds effective settings for parameters found in heuristics. We develop our procedure using four experiments. We use our procedure and a small subset of problems to find parameter settings for two new vehicle routing heuristics. We then set the parameters of each heuristic and solve 19 capacityconstrained and 15 capacityconstrained and routelengthconstrained vehicle routing problems ranging in size from 50 to 483 customers. We conclude that our procedure is an effective method that deserves serious consideration by both researchers and operations research practitioners. Key Words: statistical design of experiments, heuristics, vehicle routing 1.
A Subpath Ejection method for the Vehicle Routing Problem
, 1996
"... Generically, ejection chains are methods conceived to allow solution transformations to be efficiently carried out by modifying a variable number of their components at each step of a local search algorithm. We consider a subpath ejection chain method for the vehicle routing problem (VRP) under capa ..."
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Cited by 29 (5 self)
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Generically, ejection chains are methods conceived to allow solution transformations to be efficiently carried out by modifying a variable number of their components at each step of a local search algorithm. We consider a subpath ejection chain method for the vehicle routing problem (VRP) under capacity and route length restrictions. The method undertakes the identification of a substructure named the flower reference structure which besides coordinating moves during an ejection chain construction allows the creation of neighborhood structures with interesting combinatorial characteristics. Specifically, we base the method on a fundamental property of creating alternating paths and cycles during an ejection chain construction. A new algorithm based on a Tabu search framework is proposed and computational results on a set of academic and realworld problems indicate that the algorithm may be a good alternative to the best heuristic algorithms for the VRP. 1 Introduction We consider t...
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 ..."
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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
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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.