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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 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
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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 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.
Parallel simulated annealing for the vehicle routing problem with time windows
 10th Euromicro Workshop on Parallel, Distributed and Networkbased Processing, Canary Islands–Spain
, 2002
"... A parallel simulated annealing algorithm to solve the vehicle routing problem with time windows is presented. The objective is to find the best possible solutions to some wellknown instances of the problem by using parallelism. The empirical evidence indicate that parallel simulated annealing can be ..."
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Cited by 15 (0 self)
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A parallel simulated annealing algorithm to solve the vehicle routing problem with time windows is presented. The objective is to find the best possible solutions to some wellknown instances of the problem by using parallelism. The empirical evidence indicate that parallel simulated annealing can be applied with success to bicriterion optimization problems. Key words. Parallel simulated annealing, message passing model of parallel computation, vehicle routing problem with time windows, bicriterion optimization 1
Solving a Practical Pickup and Delivery Problem
 Transportation Science
, 2001
"... We consider a pickup and delivery vehicle routing problem commonly encountered in realworld logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple ve ..."
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Cited by 15 (0 self)
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We consider a pickup and delivery vehicle routing problem commonly encountered in realworld logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple vehicle types available to cover a set of pickup and delivery orders, each of which has multiple pickup time windows and multiple delivery time windows. Orders and carrier/vehicle types must satisfy a set of compatibility constraints that specify which orders cannot be covered by which carrier/vehicle types and which orders cannot be shipped together. Order loading and unloading sequence must satisfy the nested precedence constraint that requires that an order cannot be unloaded until all the orders loaded into the truck later than this order are unloaded. Each vehicle trip must satisfy the driver's work rules prescribed by the Department of Transportation which specify legal working hours of ...
A twostage hybrid algorithm for the pickup and delivery vehicle routing problem with time windows
 Comput. Oper. Res
, 2005
"... Abstract. This paper presents a twostage hybrid algorithm for pickup and delivery vehicle routing problems with time windows and multiple vehicles (PDPTW). The first stage uses a simple simulated annealing algorithm to decrease the number of routes, while the second stage uses LNS to decrease total ..."
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Cited by 13 (2 self)
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Abstract. This paper presents a twostage hybrid algorithm for pickup and delivery vehicle routing problems with time windows and multiple vehicles (PDPTW). The first stage uses a simple simulated annealing algorithm to decrease the number of routes, while the second stage uses LNS to decrease total travel cost. Experimental results show the effectiveness of the algorithm which has produced many new best solutions on problems with 100, 200, and 600 customers. In particular, it has improved 47 % and 76 % of the best solutions on the 200 and 600customer benchmarks, sometimes by as much as 3 vehicles. These results further confirm the benefits of twostage approaches in vehicle routing. They also answer positively the open issue in the original LNS paper, which advocated the use of LNS for the PDPTW and argue for the robustness of LNS with respect to sideconstraints. 1
A Parallel CuttingPlane Algorithm for the Vehicle Routing Problem With Time Windows
, 1999
"... In the vehicle routing problem with time windows a number of identical vehicles must be routed to and from a depot to cover a given set of customers, each of whom has a specified time interval indicating when they are available for service. Each customer also has a known demand, and a vehicle may on ..."
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Cited by 11 (1 self)
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In the vehicle routing problem with time windows a number of identical vehicles must be routed to and from a depot to cover a given set of customers, each of whom has a specified time interval indicating when they are available for service. Each customer also has a known demand, and a vehicle may only serve the customers on a route if the total demand does not exceed the capacity of the vehicle. The most effective solution method proposed to date for this problem is due to Kohl, Desrosiers, Madsen, Solomon, and Soumis. Their algorithm uses a cuttingplane approach followed by a branchand bound search with column generation, where the columns of the LP relaxation represent routes of individual vehicles. We describe a new implementation of their method, using Karger's randomized minimumcut algorithm to generate cutting planes. The standard benchmark in this area is a set of 87 problem instances generated in 1984 by M. Solomon; making using of parallel processing in both the cuttingpla...
A Heuristic for the Vehicle Routing Problem with Time Windows
 Journal of Heuristics
, 2001
"... Editor: Abstract. In this paper we propose a heuristic algorithm to solve theVehicle Routing Problem with Time Windows. Its framework is a smart combination of three simple procedures: the classical kopt exchanges improve the solution, an ad hoc procedure reduces the number of vehicles and a second ..."
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Cited by 9 (0 self)
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Editor: Abstract. In this paper we propose a heuristic algorithm to solve theVehicle Routing Problem with Time Windows. Its framework is a smart combination of three simple procedures: the classical kopt exchanges improve the solution, an ad hoc procedure reduces the number of vehicles and a second objective function drives the search out of local optima. No parameter tuning is required and no random choice is made: these are the distinguishing features with respect to the recent literature. The algorithm has been tested on benchmark problems which prove ittobe more e ective than comparable algorithms.
A Dynamic LotSizing Model with Demand Time Windows
, 1999
"... : One of the basic assumptions of the classical dynamic lotsizing model is that the aggregate demand of a given period must be satisfied in that period. Under this assumption, if backlogging is not allowed then the demand of a given period cannot be delivered earlier or later than the period. If b ..."
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Cited by 9 (0 self)
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: One of the basic assumptions of the classical dynamic lotsizing model is that the aggregate demand of a given period must be satisfied in that period. Under this assumption, if backlogging is not allowed then the demand of a given period cannot be delivered earlier or later than the period. If backlogging is allowed, the demand of a given period cannot be delivered earlier than the period, but can be delivered later at the expense of a backordering cost. Like most mathematical models, the classical dynamic lotsizing model is a simplified paraphrase of what might actually happen in real life. In most real life applications, the customer offers a grace period  we call it a demand time window  during which a particular demand can be satisfied with no penalty. That is, in association with each demand, the customer specifies an earliest and a latest delivery time. The time interval characterized by the earliest and latest delivery dates of a demand represents the corresponding time w...
Exploring bounded optimal coordination for heterogeneous teams with crossschedule dependencies
, 2011
"... ..."
Vehicle Routing and Scheduling with Fuzzy Time Windows and Fuzzy Goal
"... The NPhard problem of vehicle routing and scheduling problem with fuzzy time windows and fuzzy goal is formulated and two heuristic algorithms for solving this problem are presented. ..."
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The NPhard problem of vehicle routing and scheduling problem with fuzzy time windows and fuzzy goal is formulated and two heuristic algorithms for solving this problem are presented.