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The Vehicle Routing Problem with Time Windows  Part II: Genetic Search
, 1996
"... This paper is the second part of a work on the application of new search techniques for the vehicle routing problem with time windows. It describes GENEROUS, the GENEtic ROUting System, which is based on the natural evolution paradigm. Under this paradigm, a population of solutions evolves from one ..."
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Cited by 44 (1 self)
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This paper is the second part of a work on the application of new search techniques for the vehicle routing problem with time windows. It describes GENEROUS, the GENEtic ROUting System, which is based on the natural evolution paradigm. Under this paradigm, a population of solutions evolves from one generation to the next by "mating" parent solutions to form new offspring solutions that exhibit characteristics inherited from their parents. For this vehicle routing application, a specialized methodology is devised for merging two vehicle routing solutions into a single solution that is likely to be feasible with respect to the time window constraints. Computational results on a standard set of test problems are reported, and comparisons are provided with other heuristics.
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 42 (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).
Hybrid Genetic Algorithm, Simulated Annealing and Tabu Search Methods for Vehicle Routing Problems with Time Windows
, 1993
"... The Vehicle Routing Problem with Time Windows (VRPTW) involves servicing a set of customers, with earliest and latest time deadlines, with varying demands using capacitated vehicles with limited travel times. The objective of the problem is to service all customers while minimizing the number of veh ..."
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Cited by 32 (1 self)
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The Vehicle Routing Problem with Time Windows (VRPTW) involves servicing a set of customers, with earliest and latest time deadlines, with varying demands using capacitated vehicles with limited travel times. The objective of the problem is to service all customers while minimizing the number of vehicles and travel distance without violating the capacity and travel time of the vehicles and customer time constraints. In this paper we describe a λinterchange mechanism that moves customers between routes to generate neighborhood solutions for the VRPTW. The λinterchange neighborhood is searched using Simulated Annealing and Tabu Search strategies. The initial solutions to the VRPTW are obtained using the PushForward Insertion heuristic and a Genetic Algorithm based sectoring heuristic. The hybrid combination of the implemented heuristics, collectively known as the GenSAT system, were used to solve 60 problems from the literature with customer sizes varying from 100 to 417 customers. The computational results of GenSAT obtained new best solutions for 40 test problems. For the remaining 20 test problems, 11 solutions obtained by the GenSAT system equal previously known best solutions. The average performance of GenSAT is significantly better than known competing heuristics. For known optimal solutions to the VRPTW problems, the GenSAT system obtained the optimal number of vehicles.
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 26 (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 #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
v
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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.
An Adaptive Large Neighborhood Search Heuristic for the Pickup and Delivery Problem with Time Windows
 TRANSPORTATION SCIENCE
, 2006
"... The pickup and delivery problem with time windows is the problem of serving a number of transportation requests using a limited amount of vehicles. Each request involves moving a number of goods from a pickup location to a delivery location. Our task is to construct routes that visit all locations s ..."
Abstract

Cited by 22 (5 self)
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The pickup and delivery problem with time windows is the problem of serving a number of transportation requests using a limited amount of vehicles. Each request involves moving a number of goods from a pickup location to a delivery location. Our task is to construct routes that visit all locations such that corresponding pickups and deliveries are placed on the same route and such that a pickup is performed before the corresponding delivery. The routes must also satisfy time window and capacity constraints. This paper presents a heuristic for the problem based on an extension of the Large Neighborhood Search heuristic previously suggested for solving the vehicle routing problem with time windows. The proposed heuristic is composed of a number of competing subheuristics which are used with a frequency corresponding to their historic performance. This general framework is denoted Adaptive Large Neighborhood Search. The heuristic is tested on more than 350 benchmark instances with up to 500 requests. It is able to improve the best known solutions from the literature for more than 50 % of the problems. The computational experiments indicate that it is advantageous to use several competing subheuristics instead of just one. We believe that the proposed heuristic is very robust and is able to adapt to various instance characteristics.
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 21 (0 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.
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 21 (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.
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.
Heuristic Approaches to Vehicle Routing with Backhauls and Time Windows
, 1996
"... The vehicle routing problem with backhauls and time windows (VRPBTW) involves the pickup and delivery of goods at different customer locations, with earliest and latest time deadlines, and varying demands. The demands are serviced using capacitated vehicles with limited route time. Moreover, all del ..."
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Cited by 13 (0 self)
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The vehicle routing problem with backhauls and time windows (VRPBTW) involves the pickup and delivery of goods at different customer locations, with earliest and latest time deadlines, and varying demands. The demands are serviced using capacitated vehicles with limited route time. Moreover, all deliveries (linehauls) must be done before the pickups (backhauls). The objective of the problem is to service all customers while minimizing the number of vehicles and distance travelled while not violating the capacity and route time constraints of the vehicles, and the time window constraint at each customer. In this paper, we describe a route construction heuristic for the VRPBTW, as well as different local search heuristics to improve the initial solution. The heuristics were tested on 45 problems obtained from the literature, consisting of 25, 50 and 100customer problems for which the optimal solutions are known in most cases. The solutions produced by the heuristics are within 2.5% of the optimal solutions on average.
InTime AgentBased Vehicle Routing with a Stochastic Improvement Heuristic
 11th Conference on Innovative Applications of Artificial Intelligence
, 1999
"... Vehicle routing problems (VRP's) involve assigning a fleet of limited capacity service vehicles to service a set of customers. This paper describes an innovative, agentbased approach to solving a realworld vehiclerouting problem embedded in a highly dynamic, unpredictable domain. Most V ..."
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Cited by 12 (0 self)
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Vehicle routing problems (VRP's) involve assigning a fleet of limited capacity service vehicles to service a set of customers. This paper describes an innovative, agentbased approach to solving a realworld vehiclerouting problem embedded in a highly dynamic, unpredictable domain. Most VRP research, and all commercial products for solving VRP's, make a staticworld assumption, ignoring the dynamism in the real world. Our system is explicitly designed to address dynamism, and employs an intime algorithm that quickly finds partial solutions to a problem, and improves these as time allows. Our fundamental innovation is a stochastic improvement mechanism that enables a distributed, agentbased system to achieve highquality solutions in the absence of a centralized dispatcher. This solutionimprovement technology overcomes inherent weaknesses in the distributed problemsolving approach that make it difficult to find highquality solutions to complex optimization problems...