Results 1  10
of
40
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 ..."
Abstract

Cited by 139 (2 self)
 Add to MetaCart
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...
Probabilistic Diversification And Intensification In Local Search For Vehicle Routing
 Journal of Heuristics
, 1995
"... : This paper presents a probabilistic technique to diversify, intensify and parallelize a local search adapted for solving vehicle routing problems. This technique may be applied to a very wide variety of vehicle routing problems and local searches. It is shown that efficient first level taboo sear ..."
Abstract

Cited by 108 (3 self)
 Add to MetaCart
: This paper presents a probabilistic technique to diversify, intensify and parallelize a local search adapted for solving vehicle routing problems. This technique may be applied to a very wide variety of vehicle routing problems and local searches. It is shown that efficient first level taboo searches for vehicle routing problems may be significantly improved with this technique. Moreover, the solutions produced by this technique may often be improved by a postoptimization technique presented in this paper too. The solutions of nearly 40 problem instances of the literature have been improved. Key words : Vehicle routing, local searches, parallel algorithms. 1. INTRODUCTION More and more, local search methods are used to find good solutions to combinatorial optimization problems. Throughout the paper, we use the term local search as a synonym of neighbourhood search. Local searches are sometimes restricted to steepest descent algorithms but we also include taboo search and simulated...
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 ..."
Abstract

Cited by 35 (10 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 30 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
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 Location Based Heuristic for General Routing Problems
 Operations Research
, 1993
"... We present a general framework for modeling routing problems based on formulating them as a traditional location problem called the Capacitated Concentrator Location Problem. We apply this framework to two classical routing problems: the Capacitated Vehicle Routing Problem and InventoryRouting Prob ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
We present a general framework for modeling routing problems based on formulating them as a traditional location problem called the Capacitated Concentrator Location Problem. We apply this framework to two classical routing problems: the Capacitated Vehicle Routing Problem and InventoryRouting Problem. In the former case, the heuristic is proven to be asymptotically optimal for any distribution of customer demands and locations. Computational experiments show that the heuristic performs well for both problems and in most cases outperforms all published heuristics on a set of standard test problems. 1 Introduction Vehicle routing problems have received much attention in recent years due to the increased importance of determining efficient distribution strategies to reduce operational costs in distribution systems. A typical routing problem consists of a fleet of vehicles located at a central depot or warehouse that must be scheduled to provide some type of service to customers geograp...
Modeling and Solving the Train Timetabling Problem
, 2000
"... The Train Timetabling Problem aims at determining a periodic timetable for a set of trains which does not violate track capacities and satises some operational constraints. In particular, we concentrate on the problem of a single, oneway track linking two major stations, with a number of interme ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
The Train Timetabling Problem aims at determining a periodic timetable for a set of trains which does not violate track capacities and satises some operational constraints. In particular, we concentrate on the problem of a single, oneway track linking two major stations, with a number of intermediate stations in between. Each train connects two given stations along the track (possibly dierent from the two major stations) and may have to stop for a minimum time in some of the intermediate stations. Trains can overtake each other only in correspondence of an intermediate station, and a minimum time interval between two consecutive departures and arrivals of trains in each station is specied. In this paper, we propose a graph theoretic formulation for the problem using a directed multigraph in which nodes correspond to departures/arrivals at a certain station at a given time instant. This formulation is used to derive an integer linear programming model which is relaxed in a Lagrangian way. A novel feature of our model is that the variables in the relaxed constraints are associated only with nodes (as opposed to arcs) of the aforementioned graph. This allows a considerable speedup in the solution of the relaxation. The relaxation is embedded within a heuristic algorithm which makes extensive use of the dual information associated with the Lagrangian multipliers. We report extensive computational results on realworld instances provided from Ferrovie dello Stato SpA, the Italian railway company, and from Ansaldo Segnalamento Ferroviario SpA.
A Network FlowBased Tabu Search Heuristic For The Vehicle Routing Problem
 TRANSPORTATION SCIENCE
, 1996
"... We develop a new local search approach based on a network flow model that is used to simultaneously evaluate several customer ejection and insertion moves. We use this approach and a direct customer swap procedure to solve the wellknown Vehicle Routing Problem. The capacity constraints are relaxe ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
We develop a new local search approach based on a network flow model that is used to simultaneously evaluate several customer ejection and insertion moves. We use this approach and a direct customer swap procedure to solve the wellknown Vehicle Routing Problem. The capacity constraints are relaxed using penalty terms whose parameter values are adjusted according to time and search feedback. Tabu Search is incorporated into the procedure to overcome local optimality. More advanced issues such as intensification and diversification strategies are developed to provide effective enhancements to the basic tabu search algorithm. Computational experience on standard test problems is discussed and comparisons with bestknown solutions are provided.
A New BranchandCut Algorithm for the Capacitated Vehicle Routing Problem
 Mathematical Programming
, 2003
"... We present a new branchandcut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomo ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
We present a new branchandcut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomory mixedinteger cuts. For each of these classes of inequalities we descrine our separation algorithms in detail......