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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 46 (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...
Guided Local Search for the Vehicle Routing Problem
, 1997
"... This paper applies GLS to vehicle routing problems with time windows and capacity constraints. Results indicate that GLS can provide excellent results. This paper is organised as follows. In section 2, we introduce a local search algorithm for the vehicle routing problem. We begin first by describin ..."
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Cited by 42 (6 self)
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This paper applies GLS to vehicle routing problems with time windows and capacity constraints. Results indicate that GLS can provide excellent results. This paper is organised as follows. In section 2, we introduce a local search algorithm for the vehicle routing problem. We begin first by describing the move operators and model, and then go on to describe the objective function and search itself. The GLS metaheuristic is then presented in section 3. The application of the metaheuristic to a vehicle routing framework is discussed. Experiments are then performed using GLS on some standard benchmark problems from the literature that involve both capacity constraints on vehicles, and time windows at customers. Conclusions are drawn on the quality of the results in comparison to other methods.
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
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 New Local Search Algorithm Providing High Quality Solutions to Vehicle Routing Problems
, 1997
"... This paper describes a new local search algorithm that provides very high quality solutions to vehicle routing problems. The method uses greedy local search, but avoids local minima by using a large neighbourhood based upon rescheduling selected customer visits using constraint programming technique ..."
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Cited by 17 (0 self)
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This paper describes a new local search algorithm that provides very high quality solutions to vehicle routing problems. The method uses greedy local search, but avoids local minima by using a large neighbourhood based upon rescheduling selected customer visits using constraint programming techniques. The move operator adopted is completely generic, in that virtually any side constraint can be efficiently incorporated into the search process. Computational results show that a naive implementation of the method produces results bettering the best produced by competing techniques using minimaescaping methods. 1 Introduction In recent years, the method of choice for solving vehicle routing problems has been to use a local search technique. These local search methods have been favoured since they quickly provide solutions to problems of practical size that have not been solved by exact methods. However, because local search techniques only make small changes to the solution, they can onl...
Node Ejection Chains for the Vehicle Routing Problem: Sequential and Parallel Algorithms
, 1997
"... this paper is to describe a new tabu search algorithm for the general VRP defined above. Tabu search is a metaheuristic proposed by Glover [13]. The method is generically presented in Glover [14,15] and recent developments may be found in Glover [18,19] (see Glover and de Werra [22], Glover [21] for ..."
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Cited by 12 (3 self)
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this paper is to describe a new tabu search algorithm for the general VRP defined above. Tabu search is a metaheuristic proposed by Glover [13]. The method is generically presented in Glover [14,15] and recent developments may be found in Glover [18,19] (see Glover and de Werra [22], Glover [21] for a survey on tabu search applications and challenges). A number of algorithms based on this approach have already been applied to the VRP, each one using different types of moves leading from one solution to another (see, Pureza and Franca [33], Osman [32], Taillard [40], Gendreau, Hertz and Laporte [11], Rochat and Taillard [38], Rego [36], Xu and Kelly [41]). An important contribution of our method is the use of embedded neighborhood structures based on the idea of ejection chains. Embedded neighborhoods may be conceived as the outcome of compressing a sequence of moves into a single compound move, and ejection chain procedures give a useful way to build these neighborhoods. For a comprehensive description of ejection chain methods we refer to Glover [20,16] and Rego [34]. A number of methods based on this prespective have recently been proposed for various combinatorial problems (see Laguna et al. [28], Dorndorf and Pesch [8], Hubscher and Glover [27], Rego [35,36], Glover, Pesch and Osman [24], Cao and Glover [2]). The remainder of this paper is organized as follows. In section 2, we briefly summarize the ideas underlying ejection chains and we describe their application to the VRP. Section 3 describes the sequential version of the proposed algorithm and a parallel approach is described in section 4. Then, the computational results and a comparative analysis of the algorithms are presented in section 5. Finally, section 6 contains a summary and concluding remarks. 2 New...
A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty
, 2006
"... In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the MillerTuckerZemlin form ..."
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Cited by 5 (3 self)
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In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the MillerTuckerZemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. We present computational results that investigate the tradeoffs of a robust solution for the Augerat et al. suite of capacitated VRP problems and for families of clustered instances. Our computational results show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most profound for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We observe that the robust solution amounts to a clever management of the remaining vehicle capacity.
On Eulerian extensions and their application to nowait flowshop scheduling
 JOURNAL OF SCHEDULING
, 2011
"... ..."
Towards a model of UAVs Navigation in urban canyon through Defeasible Logic ∗
"... This paper shows how a nonmonotonic rule based system (defeasible logic) can be integrated with numerical computation engines, and how this can be applied to solve the Vehicle Routing Problem (VRP). To this end, we have simulated a physical system from which we can obtain numerical information. The ..."
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Cited by 2 (1 self)
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This paper shows how a nonmonotonic rule based system (defeasible logic) can be integrated with numerical computation engines, and how this can be applied to solve the Vehicle Routing Problem (VRP). To this end, we have simulated a physical system from which we can obtain numerical information. The physical system perceives information from its environment and generates predicates that can be reasoned by a defeasible logic engine. The conclusions/decisions derived will then realized by the physical system as it takes actions based on the conclusion derived. Here we consider a scenario where a “flock ” of UAVs have to navigate within an urban canyon environment. The UAVs are selfautonomous without centralized control. The goal of the UAVs is to navigate to their desired destinations without colliding with each other. In case of possible collision, the UAVs concerned will communicate with each other and use their background knowledge or travel guidelines to resolve the conflicts. 1