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48
Metaheuristics in combinatorial optimization: Overview and conceptual comparison
 ACM COMPUTING SURVEYS
, 2003
"... The field of metaheuristics for the application to combinatorial optimization problems is a rapidly growing field of research. This is due to the importance of combinatorial optimization problems for the scientific as well as the industrial world. We give a survey of the nowadays most important meta ..."
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Cited by 314 (17 self)
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The field of metaheuristics for the application to combinatorial optimization problems is a rapidly growing field of research. This is due to the importance of combinatorial optimization problems for the scientific as well as the industrial world. We give a survey of the nowadays most important metaheuristics from a conceptual point of view. We outline the different components and concepts that are used in the different metaheuristics in order to analyze their similarities and differences. Two very important concepts in metaheuristics are intensification and diversification. These are the two forces that largely determine the behaviour of a metaheuristic. They are in some way contrary but also complementary to each other. We introduce a framework, that we call the I&D frame, in order to put different intensification and diversification components into relation with each other. Outlining the advantages and disadvantages of different metaheuristic approaches we conclude by pointing out the importance of hybridization of metaheuristics as well as the integration of metaheuristics and other methods for optimization.
Guided Local Search
, 2010
"... Combinatorial explosion problem is a well known phenomenon that prevents complete algorithms from solving many reallife combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter describes the principles of Guided Local Search (GLS) and Fast Local Sea ..."
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Cited by 64 (5 self)
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Combinatorial explosion problem is a well known phenomenon that prevents complete algorithms from solving many reallife combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter describes the principles of Guided Local Search (GLS) and Fast Local Search (FLS) and surveys their applications. GLS is a penaltybased metaheuristic algorithm that sits on top of other local search algorithms, with the aim to improve their efficiency and robustness. FLS is a way of reducing the size of the neighbourhood to improve the efficiency of local search. The chapter also provides guidance for implementing and using GLS and FLS. Four problems, representative of general application categories, are examined with detailed information provided on how to build a GLSbased method in each case.
Guided local search and its application to the traveling salesman problem
, 1998
"... The Traveling Salesman Problem (TSP) is one of the most famous problems in combinatorial optimization. In this paper, we are going to examine how the techniques of Guided Local Search (GLS) and Fast Local Search (FLS) can be applied to the problem. Guided Local Search sits on top of local search heu ..."
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Cited by 59 (16 self)
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The Traveling Salesman Problem (TSP) is one of the most famous problems in combinatorial optimization. In this paper, we are going to examine how the techniques of Guided Local Search (GLS) and Fast Local Search (FLS) can be applied to the problem. Guided Local Search sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems. Guided Local Search can be combined with the neighborhood reduction scheme of Fast Local Search which significantly speeds up the operations of the algorithm. The combination of GLS and FLS with TSP local search heuristics of different efficiency and effectiveness is studied in an effort to determine the dependence of GLS on the underlying local search heuristic used. Comparisons are made with some of the best TSP heuristic algorithms and general optimization techniques which demonstrate the advantages of GLS over alternative heuristic approaches suggested for the problem.
Solving Vehicle Routing Problems using Constraint Programming and Metaheuristics
 Journal of Heuristics
, 1997
"... . Constraint Programming typically uses the technique of depthfirst branch and bound as the method of solving optimisation problems. Although this method can give the optimal solution, for large problems, the time needed to find the optimal can be prohibitive. This paper introduces a method for usi ..."
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Cited by 50 (4 self)
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. Constraint Programming typically uses the technique of depthfirst branch and bound as the method of solving optimisation problems. Although this method can give the optimal solution, for large problems, the time needed to find the optimal can be prohibitive. This paper introduces a method for using iterative improvement techniques within a Constraint Programming framework, and applies this technique to vehicle routing problems. We introduce a Constraint Programming model for vehicle routing, after which we describe a system for integrating Constraint Programming and iterative improvement techniques. We then describe how the method can be greatly accelerated by handling core constraints using fast local checks, while other more complex constraints are left to the constraint propagation system. We have coupled our iterative improvement technique with a metaheuristic to avoid the search being trapped in local minima. Two metaheuristics are investigated: a simple Tabu Search procedur...
HumanGuided Simple Search
 IN PROCEED OF AAAI 2000, JULY 30–AUGUST 3
, 2000
"... Scheduling, routing, and layout tasks are examples of hard operationsresearch problems that have broad application in industry. Typical algorithms for these problems combine some form of gradient descent to find local minima with some strategy for escaping nonoptimal local minima. Our idea is to di ..."
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Cited by 46 (10 self)
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Scheduling, routing, and layout tasks are examples of hard operationsresearch problems that have broad application in industry. Typical algorithms for these problems combine some form of gradient descent to find local minima with some strategy for escaping nonoptimal local minima. Our idea is to divide these two subtasks cleanly between human and computer: in our paradigm of humanguided simple search the computer is responsible only for finding local minima using a simple hillclimbing search; using visualization and interaction techniques, the human user identifies promising regions of the search space for the computer to explore, and intervenes to help it escape nonoptimal local minima. We have applied our approach to the problem of capacitated vehicle routing with time windows, a commercially important problem with a rich research history. Despite its simplicity, our prototype system is competitive with the majority of previously reported systems on benchmark academic problems, and has the advantage of keeping a human tightly in the loop to handle the complexities of realworld applications.
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 38 (1 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.
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 38 (2 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
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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 #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.
Guided local search for solving SAT and weighted MAXSAT problems
 Journal of Automated Reasoning
, 2000
"... Abstract. In this paper, we show how Guided Local Search (GLS) can be applied to the SAT problem and show how the resulting algorithm can be naturally extended to solve the weighted MAXSAT problem. GLS is a general, penaltybased metaheuristic, which sits on top of local search algorithms to help g ..."
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Cited by 37 (7 self)
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Abstract. In this paper, we show how Guided Local Search (GLS) can be applied to the SAT problem and show how the resulting algorithm can be naturally extended to solve the weighted MAXSAT problem. GLS is a general, penaltybased metaheuristic, which sits on top of local search algorithms to help guide them out of local minima. GLS has been shown to be successful in solving a number of practical real life problems, such as the travelling salesman problem, BT's workforce scheduling problem, the radio link frequency assignment problem and the vehicle routing problem. We present empirical results of applying GLS to instances of the SAT problem from the DIMACS archive and also a small set of weighted MAXSAT problem instances and compare them against the results of other local search algorithms for the SAT problem. Keywords: SAT problem, Local Search, Metaheuristics, Optimisation 1.
VRP with Time Windows
 The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications
, 2002
"... Abstract This paper presents a survey of the research on the Vehicle Routing Problem with Time Windows (VRPTW), an extension of the Capacitated Vehicle Routing Problem. In the VRPTW, the service at each customer must start within an associated time window and the vehicle must remain at the customer ..."
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Cited by 32 (3 self)
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Abstract This paper presents a survey of the research on the Vehicle Routing Problem with Time Windows (VRPTW), an extension of the Capacitated Vehicle Routing Problem. In the VRPTW, the service at each customer must start within an associated time window and the vehicle must remain at the customer location during service. Soft time windows can be violated at a cost while hard time windows do not allow for a vehicle to arrive at a customer after the latest time to begin service. We first present a multicommodity network flow formulation with time and capacity constraints for the VRPTW. Approximation methods proposed in the literature to derive upper bounds are then reviewed. Then we explain how lower bounds can be obtained using optimal approaches, namely, Lagrangean relaxation and column generation. Next, we provide branching and cutting strategies that can be embedded within these optimal approaches to produce integer solutions. Special cases and extensions to the VRPTW follow as well as our conclusions. Résumé Cet article synthèse porte sur les récents développements concernant le problème du routage de véhicules sous des contraintes de fenêtres de temps. Dans ce problème, le serviceà un client doit débuterà l'intérieur d'un intervalle de temps. Celuici peutêtre, soit relaché au prix d'une certaine pénalité, soit rigide, auquel cas, il n'est pas permis de dépasser la limite supérieure. Nous présentons un modèle de réseau multiflots avec des contraintes de temps et de capacité. Les méthodes heuristiques permettant de calculer des bornes supérieures sont d'abord présentées. Suivent les modèles d'optimisation basés sur la relaxation lagrangienne et la génération de colonnes pourévaluer des bornes inférieures. Enfin, on présente les stratégies de coupes et de branchements liéesà ces méthodes afin de déterminer des solutions entières. L'article se termine par l'étude de cas particuliers et d'extensions ainsi que nos conclusions.
A Comparison of Traditional and Constraintbased Heuristic Methods on Vehicle Routing Problems with Side Constraints
 CONSTRAINTS
, 1998
"... The vehicle routing problem (VRP) is a variant of the familiar travelling salesman problem (TSP). In the VRP we are to perform a number of visits, using a limited number of vehicles, while minimizing the distance travelled. The VRP can be further complicated by associating time windows on visits, c ..."
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Cited by 27 (4 self)
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The vehicle routing problem (VRP) is a variant of the familiar travelling salesman problem (TSP). In the VRP we are to perform a number of visits, using a limited number of vehicles, while minimizing the distance travelled. The VRP can be further complicated by associating time windows on visits, capacity constraints on vehicles, sequencing constraints between visits, and so on. In this paper we introduce a constraintbased model of the capacitated VRP with time windows and side constraints. The model is implemented using a constraint programming toolkit. We investigate the performance of a number of construction and improvement techniques, and show that as problems become richer and more constrained conventional techniques fail while constraint directed techniques continue to perform acceptably. This suggests that constraint programming is an appropriate technology for real world VRP's.