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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 83 (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).
Robust branchandcutandprice for the capacitated vehicle routing problem
 IN PROCEEDINGS OF THE INTERNATIONAL NETWORK OPTIMIZATION CONFERENCE
, 2003
"... During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branchandcut algorithms giving bett ..."
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Cited by 55 (14 self)
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During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branchandcut algorithms giving better results. However, several instances in the range of 50–80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over qroutes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branchandcutandprice could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.
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 34 (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
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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.
Solving capacitated arc routing problems using a transformation to the CVRP
 Computers & Operations Research
, 2006
"... A well known transformation by Pearn, Assad and Golden reduces a Capacitated Arc Routing Problem (CARP) into an equivalent Capacitated Vehicle Routing Problem (CVRP). However, that transformation is regarded as unpractical, since an original instance with r required edges is turned into a CVRP over ..."
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Cited by 21 (3 self)
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A well known transformation by Pearn, Assad and Golden reduces a Capacitated Arc Routing Problem (CARP) into an equivalent Capacitated Vehicle Routing Problem (CVRP). However, that transformation is regarded as unpractical, since an original instance with r required edges is turned into a CVRP over a complete graph with 3r + 1 vertices. We propose a similar transformation that reduces this graph to 2r + 1 vertices, with the additional restriction that r edges are already fixed to 1. Using a recent branchandcutandprice algorithm for the CVRP, we observed that it yields an effective way of attacking the CARP, being significantly better than the exact methods created specifically for that problem. Computational experiments obtained improved lower bounds for almost all open instances from the literature. Several such instances could be solved to optimality.
A new ILPbased refinement heuristic for Vehicle Routing Problems
, 2004
"... ... Problem (DCVRP), where k minimumcost routes through a central depot have to be constructed so as to cover all customers while satisfying, for each route, both a capacity and a totaldistancetravelled limit. Our starting point is the following refinement procedure proposed in 1981 by Sarvanov ..."
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Cited by 10 (0 self)
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... Problem (DCVRP), where k minimumcost routes through a central depot have to be constructed so as to cover all customers while satisfying, for each route, both a capacity and a totaldistancetravelled limit. Our starting point is the following refinement procedure proposed in 1981 by Sarvanov and Doroshko for the pure Travelling Salesman Problem (TSP): given a starting tour, (a) remove all the nodes in even position, thus leaving an equal number of “empty holes ” in the tour; (b) optimally reassign the removed nodes to the empty holes through the efficient solution of a minsum assignment (weighted bipartite matching) problem. We first extend the SarvanovDoroshko method to DCVRP, and then generalize it. Our generalization involves a procedure to generate a large number of new sequences through the extracted nodes, as well as a more sophisticated ILP model for the reallocation of some of these sequences. An important feature of our method is that it does not rely on any specialized ILP code, as any generalpurpose ILP solver can be used to solve the reallocation model. We report computational results on a large set of capacitated VRP instances from the literature (with symmetric/asymmetric costs and with/without distance constraints), along with an analysis of the performance of the new method and of its features. Interestingly, in 12 cases the new method was able to improve the bestknow solution available from the literature.
Decomposition and dynamic cut generation in integer linear programming
, 2004
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Robust branchcutandprice algorithms for vehicle routing problems
 The Vehicle Routing Problem: Latest Advances and New Challenges
, 2008
"... Summary. This article presents techniques for constructing robust BranchCutandPrice algorithms on a number of Vehicle Routing Problem variants. The word “robust ” stress the effort of controlling the worstcase complexity of the pricing subproblem, keeping it pseudopolynomial. Besides summarizing ..."
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Cited by 6 (3 self)
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Summary. This article presents techniques for constructing robust BranchCutandPrice algorithms on a number of Vehicle Routing Problem variants. The word “robust ” stress the effort of controlling the worstcase complexity of the pricing subproblem, keeping it pseudopolynomial. Besides summarizing older research on the topic, some promising new lines of investigation are also presented, specially the development of new families of cuts over large extended formulations. Computational experiments over benchmark instances from ACVRP, COVRP, CVRP and HFVRP variants are provided. Key words: Column generation; cutting plane algorithms; extended formulations; branchandbound. 1
Robust branchcutandprice for the capacitated minimum spanning tree problem over a large extended formulation
 Universidade Federal Fluminense
, 2008
"... This paper presents a robust branchcutandprice algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to qarbs, a structure that arises from a relaxation of the capacitated prizecollecting arborescence problem in order to make it solvable in pseudopoly ..."
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Cited by 5 (1 self)
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This paper presents a robust branchcutandprice algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to qarbs, a structure that arises from a relaxation of the capacitated prizecollecting arborescence problem in order to make it solvable in pseudopolynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the ORLibrary show very significant improvements over previous algorithms. Several open instances could be solved to optimality. 1 1