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11
SALSA: A Language for Search Algorithms
 In CP’98
, 1998
"... Constraint Programming is a technique of choice for solving hard combinatorial optimization problems. However, it is best used in conjunction with other optimization paradigms such as local search, yielding hybrid algorithms with constraints. Such combinations lack a language supporting an elegant d ..."
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Constraint Programming is a technique of choice for solving hard combinatorial optimization problems. However, it is best used in conjunction with other optimization paradigms such as local search, yielding hybrid algorithms with constraints. Such combinations lack a language supporting an elegant description and retaining the original declarativity of Constraint Logic Programming. We propose a language, SALSA, dedicated to specifying (local, global or hybrid) search algorithms. We illustrate its use on a few examples from combinatorial optimization for which we specify complex optimization procedures with a few simple lines of code of high ion level. We report preliminary experiments showing that such a language can be implemented on top of CP systems, yielding a powerful environment for combinatorial optimization.
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 25 (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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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.
Multiobjective Genetic Algorithms for Vehicle Routing Problem with Time Windows
 APPLIED INTELLIGENCE
, 2006
"... The Vehicle Routing Problem with Time windows (VRPTW) is an extension of the capacity constrained Vehicle Routing Problem (VRP). The VRPTW is NPComplete and instances with 100 customers or more are very hard to solve optimally. We represent the VRPTW as a multiobjective problem and present a genet ..."
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Cited by 18 (1 self)
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The Vehicle Routing Problem with Time windows (VRPTW) is an extension of the capacity constrained Vehicle Routing Problem (VRP). The VRPTW is NPComplete and instances with 100 customers or more are very hard to solve optimally. We represent the VRPTW as a multiobjective problem and present a genetic algorithm solution using the Pareto ranking technique. We use a direct interpretation of the VRPTW as a multiobjective problem, in which the two objective dimensions are number of vehicles and total cost (distance). An advantage of this approach is that it is unnecessary to derive weights for a weighted sum scoring formula. This prevents the introduction of solution bias towards either of the problem dimensions. We argue that the VRPTW is most naturally viewed as a multimodal problem, in which both vehicles and cost are of equal value, depending on the needs of the user. A result of our research is that the multiobjective optimization genetic algorithm returns a set of solutions that fairly consider both of these dimensions. Our approach is quite effective, as it provides solutions competitive with the best known in the literature, as well as new solutions that are not biased toward the number of vehicles. A set of wellknown benchmark data are used to compare the effectiveness of the proposed method for solving the VRPTW.
Efficient Local Search Algorithms for the Vehicle Routing Problem with Time Windows
 EUR. J. OPER. RES
, 2001
"... this paper is to present new methods for solving the vehicle routing problem with time windows (VRPTW). VRPTW can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visite ..."
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Cited by 12 (2 self)
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this paper is to present new methods for solving the vehicle routing problem with time windows (VRPTW). VRPTW can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visited only once by exactly one vehicle within a given time interval; all routes start and end at the depot, and the total demands of all points on one particular route must not exceed the capacity of the vehicle
A method for vehicle routing problems with multiple vehicle types and time windows
 Proceedings of Natural Science Council
, 1999
"... ..."
Incremental Local Search in Ant Colony Optimization: Why It Fails for the Quadratic Assignment Problem
"... Abstract. Ant colony optimization algorithms are currently among the best performing algorithms for the quadratic assignment problem. These algorithms contain two main search procedures: solution construction by artificial ants and local search to improve the solutions constructed by the ants. Incre ..."
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Abstract. Ant colony optimization algorithms are currently among the best performing algorithms for the quadratic assignment problem. These algorithms contain two main search procedures: solution construction by artificial ants and local search to improve the solutions constructed by the ants. Incremental local search is an approach that consists in reoptimizing partial solutions by a local search algorithm at regular intervals while constructing a complete solution. In this paper, we investigate the impact of adopting incremental local search in ant colony optimization to solve the quadratic assignment problem. Notwithstanding the promising results of incremental local search reported in the literature in a different context, the computational results of our new ACO algorithm are rather negative. We provide an empirical analysis that explains this failure. 1
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
Route Construction and Local . . .
, 2001
"... This report presents a survey of research on the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visit ..."
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This report presents a survey of research on the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visited only once by exactly one vehicle within a given time interval, all routes start and end at the depot, and the total demands of all points on one particular route must not exceed the capacity of the vehicle. Both traditional heuristic route construction methods and recent local search algorithms are examined. The basic features of each method are described, and experimental results for Solomon’s benchmark test problems are presented and analyzed. Moreover, we discuss how heuristic methods should be evaluated and propose using the concept of Pareto optimality in the comparison of different heuristic