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21
The general pickup and delivery problem
 TRANSPORTATION SCIENCE
, 1995
"... In pickup and delivery problems vehicles have to transport loads from origins to destinations without transshipment atintermediate locations. In this paper, we discuss several characteristics that distinguish them from standard vehicle routing problems and present a survey of the problem types and s ..."
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Cited by 90 (3 self)
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In pickup and delivery problems vehicles have to transport loads from origins to destinations without transshipment atintermediate locations. In this paper, we discuss several characteristics that distinguish them from standard vehicle routing problems and present a survey of the problem types and solution methods found in the literature.
A New Generation of Vehicle Routing Research: Robust Algorithms, Addressing Uncertainty
 Operations Research
, 1993
"... In recent years new insights and algorithms have been obtained for the classical, deterministic, vehicle routing problem as well as for natural stochastic and dynamic variations of it. These new developments are based on theoretical analysis, combine probabilistic and combinatorial modelling and ..."
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Cited by 45 (0 self)
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In recent years new insights and algorithms have been obtained for the classical, deterministic, vehicle routing problem as well as for natural stochastic and dynamic variations of it. These new developments are based on theoretical analysis, combine probabilistic and combinatorial modelling and lead to (1) new algorithms that produce near optimal solutions and (2) a deeper understanding of uncertainty issues in vehicle routing. In this paper we survey these new developments with an emphasis on the insights gained and on the algorithms proposed. Research supported in part by ONR contract N0001490J1649, NSF contracts DDM8922712, DDM9014751, and by a Presidential Young Investigator award DDM9158118 with matching funds from Draper Laboratory. y Sloan School of Management, MIT, Cambridge, MA 02139. z Dept. of Industrial Engineering and Operations Research, Columbia University, NY, NY, 10027 and Department of Operations Research and Management Sciences, Northwestern Universi...
The capacitated vehicle routing problem
"... We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Travelin ..."
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Cited by 34 (5 self)
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We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two wellstudied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decompositionbased separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.
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 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.
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 ..."
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Cited by 18 (1 self)
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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.
On the effectiveness of set covering formulations for the vehicle routing problem with time windows
 Operations Research
, 1997
"... The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem. The method starts by ..."
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Cited by 16 (0 self)
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The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem. The method starts by solving its linear programming relaxation, via column generation, and then uses a branch and bound strategy to find an integer solution to the set covering problem: a solution to the VRPTW. An empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient. In this paper, we explain this behavior by demonstrating that for any distribution of service times, time windows, customer loads and locations, the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases.
Static Pickup and Delivery Problems: A Classification Scheme and Survey
, 2007
"... Pickup and delivery problems constitute an important class of vehicle routing problems in which objects or people have to be collected and distributed. This paper introduces a general framework to model a large collection of pickup and delivery problems, as well as a threefield classification schem ..."
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Cited by 15 (2 self)
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Pickup and delivery problems constitute an important class of vehicle routing problems in which objects or people have to be collected and distributed. This paper introduces a general framework to model a large collection of pickup and delivery problems, as well as a threefield classification scheme for these problems. It surveys the methods used for solving them.
Transportation on demand
 In Transportation
"... Département d’informatique et de recherche opérationnelle and Centre de recherche sur les transports, Université de Montréal ..."
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Cited by 13 (6 self)
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Département d’informatique et de recherche opérationnelle and Centre de recherche sur les transports, Université de Montréal
A SetPartitioningBased Model for the Stochastic Vehicle Routing Problem
, 2006
"... The objective of the Vehicle Routing Problem (VRP) is to construct a minimum cost set of vehicle routes that visits all customers and satisfies demands without violating the vehicle capacity constraints. The Stochastic Vehicle Routing Problem (SVRP) results when one or more elements of the VRP are m ..."
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Cited by 3 (0 self)
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The objective of the Vehicle Routing Problem (VRP) is to construct a minimum cost set of vehicle routes that visits all customers and satisfies demands without violating the vehicle capacity constraints. The Stochastic Vehicle Routing Problem (SVRP) results when one or more elements of the VRP are modeled as random variables. In this paper, we present a setpartitioningbased modeling framework for the VRP with stochastic demands (VRPSD). The framework can be adapted easily for routing problems with randomness in other problem elements, such as random customers and random travel times. We formulate the VRPSD as a twostage stochastic program and introduce an extended recourse strategy in which vehicles are allowed to serve additional customers from failed routes prior to returning to the depot or to serve customers from failed routes on a new route after returning to the depot. Computational experiments show that route plans generated using the new recourse function perform quite well, especially for problems with few customers per route, where cost savings of roughly 5 % are possible. 1