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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 31 (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
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 #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 hybrid exact algorithm for the TSPTW
 INFORMS Journal on Computing
, 2002
"... The Traveling Salesman Problem with Time Windows (TSPTW) is the problem of finding a minimumcost path visiting a set of cities exactly once, where each city must be visited within a specific time window. We propose a hybrid approach for solving the TSPTW that merges Constraint Programming propagati ..."
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Cited by 27 (5 self)
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The Traveling Salesman Problem with Time Windows (TSPTW) is the problem of finding a minimumcost path visiting a set of cities exactly once, where each city must be visited within a specific time window. We propose a hybrid approach for solving the TSPTW that merges Constraint Programming propagation algorithms for the feasibility viewpoint (find a path), and Operations Research techniques for coping with the optimization perspective (find the best path). We show with extensive computational results that the synergy between Operations Research optimization techniques embedded in global constraints, and Constraint Programming constraint solving techniques, makes the resulting framework effective in the TSPTW context also if these results are compared with stateoftheart algorithms from the literature.
A branchandcut algorithm for the pickup and delivery traveling salesman problem with LIFO loading
 Networks
, 2006
"... In the pickup and delivery problem with time windows (PDPTW), capacitated vehicles must be routed to satisfy a set of transportation requests between given origins and destinations. In addition to capacity and time window constraints, vehicle routes must also satisfy pairing and precedence constrain ..."
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Cited by 25 (3 self)
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In the pickup and delivery problem with time windows (PDPTW), capacitated vehicles must be routed to satisfy a set of transportation requests between given origins and destinations. In addition to capacity and time window constraints, vehicle routes must also satisfy pairing and precedence constraints on pickups and deliveries. This paper introduces two new formulations for the PDPTW and the closely related dialaride problem (DARP) in which a limit is imposed on the elapsed time between the pickup and the delivery of a request. Several families of valid inequalities are introduced to strengthen these two formulations. These inequalities are used within branchandcut algorithms which have been tested on several instance sets for both the PDPTW and the DARP. Instances with up to eight vehicles and 96 requests (194 nodes) have been solved to optimality.
Combinatorial Benders’ Cuts for MixedInteger Linear Programming
 Operations Research
"... MixedInteger Programs (MIP’s) involving logical implications modelled through bigM coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependenc ..."
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Cited by 16 (1 self)
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MixedInteger Programs (MIP’s) involving logical implications modelled through bigM coefficients, are notoriously among the hardest to solve. In this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the bigM coefficients. Our solution scheme defines a master Integer Linear Problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled ” from the original MIP model. The master solutions are sent to a slave Linear Program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism resembles closely the Benders ’ one, but the cuts we produce are purely combinatorial and do not depend on the bigM values used in the MIP formulation. This produces an LP relaxation of the master problem which can be considerably tighter than the one associated with original MIP formulation. Computational results on two specific classes of hardtosolve MIP’s indicate the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.
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 15 (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...
Container movement by trucks in metropolitan networks: modeling and optimization
 TRANSPORT. RES
, 2005
"... Container movement by trucks with time constraints at origins and destinations is modeled as an asymmetric “multiTraveling Salesmen Problem with Time Windows” (mTSPTW) with social constraints. A twophase exact algorithm based on dynamic programming (DP) is proposed that finds the best routes for ..."
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Cited by 12 (1 self)
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Container movement by trucks with time constraints at origins and destinations is modeled as an asymmetric “multiTraveling Salesmen Problem with Time Windows” (mTSPTW) with social constraints. A twophase exact algorithm based on dynamic programming (DP) is proposed that finds the best routes for a fleet of trucks. Since the mTSPTW problem is NPhard, the computational time for optimally solving large size problems becomes prohibitive. For large size problems, we develop a hybrid methodology consisting of DP in conjunction with genetic algorithms. The developed algorithms are compared with an insertion heuristic method. Computational results demonstrate the efficiency of the developed algorithms.
Projected ChvátalGomory cuts for Mixed Integer Linear Programs
, 2006
"... Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure Integer Linear Program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the ChvátalGomory (CG) separation problem as a Mixed Int ..."
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Cited by 10 (6 self)
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Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure Integer Linear Program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the ChvátalGomory (CG) separation problem as a Mixed Integer Linear Program (MILP) which is then solved by a generalpurpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory Mixed Integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive ChvátalGomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, whose solution can typically be carried out in a reasonable amount of computing time.
Duty scheduling in public transit
 MATHEMATICS—KEY TECHNOLOGY FOR THE FUTURE
, 2003
"... This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant pa ..."
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Cited by 9 (2 self)
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This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant parts of 9. large problem. Computational results for three European scenarios are reported.
Scheduling Duties by Adaptive Column Generation
, 2001
"... This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant part ..."
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Cited by 3 (0 self)
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This article is about adaptive column generation techniques for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant parts of a large problem. Computational results for three European scenarios are reported.