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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
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 eĆcient 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.
Integrating Constraint Logic Programming and Operations Research Techniques for the Crew Rostering Problem
, 1998
"... In this paper, we investigate the possibility of integrating Artificial Intelligence (AI) and Operations Research (OR) techniques for solving the Crew Rostering Problem (CRP). CRP calls for the optimal sequencing of a given set of duties into rosters satisfying a set of constraints. The optimality c ..."
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Cited by 11 (0 self)
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In this paper, we investigate the possibility of integrating Artificial Intelligence (AI) and Operations Research (OR) techniques for solving the Crew Rostering Problem (CRP). CRP calls for the optimal sequencing of a given set of duties into rosters satisfying a set of constraints. The optimality criterion requires the minimization of the number of crews needed to cover the duties. This kind of problem has been traditionally solved with OR techniques. In recent years, a new programming paradigm based on Logic Programming, named Constraint Logic Programming (CLP), has been successfully used for solving hard combinatorial optimization problems. CLP maintains all the advantages of Logic Programming such as declarativeness, nondeterminism and an incremental style of programming, while overcoming its limitations, mainly due to the inefficiency in exploring the search space. CLP achieves good results on hard combinatorial optimization problems which, however, are not comparable with those ...
Informationtheoretic approaches to branching in search
, 2006
"... Deciding what to branch on at each node is a key element of search algorithms. We present four families of methods for selecting what question to branch on. They are all informationtheoretically motivated to reduce uncertainty in remaining subproblems. In the first family, a good variable to branch ..."
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Cited by 11 (4 self)
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Deciding what to branch on at each node is a key element of search algorithms. We present four families of methods for selecting what question to branch on. They are all informationtheoretically motivated to reduce uncertainty in remaining subproblems. In the first family, a good variable to branch on is selected based on lookahead. In realworld procurement optimization, this entropic branching method outperforms default CPLEX and strong branching. The second family combines this idea with strong branching. The third family does not use lookahead, but instead exploits features of the underlying structure of the problem. Experiments show that this family significantly outperforms the stateoftheart branching strategy when the problem includes indicator variables as the key driver of complexity. The fourth family is about branching using carefully constructed linear inequality constraints over sets of variables. 1
Rostering By Iterating Integer Programming And Simulation
 Proceedings of the 1998 Winter Simulation Conference
, 1998
"... We present a new technique (RIIPS) for solving rostering problems in the presence of service uncertainty. RIIPS stands for "Rostering by Iterating Integer Programming and Simulation". RIIPS allows great complexity of the stochastic system being rostered. This modelling freedom comes at a price, as t ..."
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Cited by 11 (2 self)
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We present a new technique (RIIPS) for solving rostering problems in the presence of service uncertainty. RIIPS stands for "Rostering by Iterating Integer Programming and Simulation". RIIPS allows great complexity of the stochastic system being rostered. This modelling freedom comes at a price, as the approach can be extremely computationally intensive. Therefore any reduction in computational effort using, for example, efficiency improvement techniques, is of great interest. We specify several ways in which these may be applied. 1 INTRODUCTION In rostering applications, one wishes to minimise the costs of staffing, subject to the constraint of maintaining reasonable customer service. The prototypical example is the call centre. In these systems, customers dial a number, possibly join a queue of waiting customers, and then receive service from a service agent. The quality of a customer's service is usually defined in terms of a measure called "customer grade of service" (CGOS), which...
An AbductiveBased Scheduler for AirCrew Assignment
, 1998
"... This paper presents the design and implementation of an aircrew assignment system for producing and refining a solution to this problem based on the Artificial Intelligence principles and techniques of abductive reasoning as captured by the framework of Abductive Logic Programming (ALP). The system ..."
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Cited by 10 (3 self)
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This paper presents the design and implementation of an aircrew assignment system for producing and refining a solution to this problem based on the Artificial Intelligence principles and techniques of abductive reasoning as captured by the framework of Abductive Logic Programming (ALP). The system offers a highlevel of flexibility in addressing both the tasks of crew scheduling and rescheduling. It can be used to generate a valid and good quality initial solution and then to help the human operators adjust and refine further this solution in order to meet extra requirements of the problem. These additional needs can arise either due to new foreseen requirements that the company wants to have or experiment with for a particular period in time or due to unexpected events that have occurred while the solution (crewroster) is in operation. Our work shows the ability and flexibility of abduction, and more specifically of ALP, in tackling problems of this type with complex and changing r...
Integrated Simulation, Heuristic and Optimisation Approaches to Staff Scheduling
"... This paper details a new simulation and optimisation based system for personnel scheduling (rostering) of Customs staff at the Auckland International Airport, New Zealand. An integrated approach using simulation, heuristic descent and integer programming techniques has been developed to determine ne ..."
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Cited by 8 (2 self)
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This paper details a new simulation and optimisation based system for personnel scheduling (rostering) of Customs staff at the Auckland International Airport, New Zealand. An integrated approach using simulation, heuristic descent and integer programming techniques has been developed to determine nearoptimal staffing levels. The system begins by using a new simulation system embedded within a heuristic search to determine minimum staffing levels for arrival and departure work areas. These staffing requirements are then used as the input to an integer programming model which optimally allocates full and parttime staff to each period of the working day. These shifts are then assigned to daily work schedules having a sixdayon, threedayoff structure. The application of these techniques has resulted in significantly lower staffing levels, while at the same time creating both high quality rosters and ensuring that all passenger processing targets are met. This paper charts the development of this system, outlines failures where they have occurred, and summarises the ongoing impacts of this work on the organisation.
Crew Assignment via Constraint Programming: Integrating Column Generation and Heuristic Tree Search
, 2002
"... The Airline Crew Assignment Problem (ACA) consists of assigning lines of work to a set of crew members such that a set of activities is partitioned and the costs for that assignment are minimized. Especially for European airline companies, complex constraints defining the feasibility of a line of wo ..."
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Cited by 8 (0 self)
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The Airline Crew Assignment Problem (ACA) consists of assigning lines of work to a set of crew members such that a set of activities is partitioned and the costs for that assignment are minimized. Especially for European airline companies, complex constraints defining the feasibility of a line of work have to be respected. We developed two different algorithms to tackle the large scale optimization problem of Airline Crew Assignment. The first is an application of the Constraint Programming (CP) based Column Generation Framework. The second approach performs a CP based heuristic tree search. We present how both algorithms can be coupled to overcome their inherent weaknesses by integrating methods from Constraint Programming and Operations Research. Numerical results show the superiority of the hybrid algorithm in comparison to CP based tree search and column generation alone.
Crew Pairing Optimization Based on CLP
, 1996
"... The crew pairing optimization problem is faced by airline companies as an intensive part of the crew scheduling process. Crew scheduling is the assignment of cockpit and cabin crews to the flight legs that a company has to carry out during a predefined period of time. Due to the significant contribu ..."
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Cited by 4 (2 self)
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The crew pairing optimization problem is faced by airline companies as an intensive part of the crew scheduling process. Crew scheduling is the assignment of cockpit and cabin crews to the flight legs that a company has to carry out during a predefined period of time. Due to the significant contribution of the crew cost to the overall operating cost of an airline company, the automation of the crew scheduling procedure is highly desirable. However, the crew pairing optimization subproblem of crew scheduling is extremely difficult and combinatorial in nature due to the large number and complexity of the involved constraints. The requirement for optimality makes it even more difficult. Many attempts have been made in the past 40 years to tackle the crew pairing optimization problem using methods from Operations Research. In this paper, an approach based on pure Constraint Logic Programming is presented, which leads to an elegant and flexible modeling of the problem. The whole process is ...
Integrating Direct CP Search and CPbased Column Generation for the Airline Crew Assignment Problem
, 2000
"... . We introduce a hybrid approach integrating CPbased Column Generation and pure CP methods. It is being applied to the large scale optimization problem of Airline Crew Assignment. The combination of methods from CP and OR results in an algorithm that overcomes the inherent weaknesses of each ap ..."
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Cited by 3 (2 self)
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. We introduce a hybrid approach integrating CPbased Column Generation and pure CP methods. It is being applied to the large scale optimization problem of Airline Crew Assignment. The combination of methods from CP and OR results in an algorithm that overcomes the inherent weaknesses of each approach. First numerical results show the superiority of the hybrid algorithm in comparison to direct CP and CPbased Column Generation alone. Keywords: Airline Crew Assignment, hybrid ORCP method, Constrained Based Column Generation 1 Introduction Scheduling ying crews of airline companies is a hard combinatorial problem, given the complexity of the constraints that have to be satised and the huge search space that has to be explored. The problem is often tackled by breaking it down into the crew pairing subproblem which consists of constructing pairings from ight legs, and the crew rostering (or assignment) subproblem which consists of assigning the constructed pairings to crew me...
Solving the Train Driver Recovery Problem Extended Abstract
"... The daily operations of the Danish railway operator DSB Stog suffer from disruptions of various magnitude almost every day. Disruptions initiated by e.g. signalling problems or rolling stock failures cause train delays and cancellations. Changes in the train schedule affect the train driver duties. ..."
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Cited by 2 (0 self)
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The daily operations of the Danish railway operator DSB Stog suffer from disruptions of various magnitude almost every day. Disruptions initiated by e.g. signalling problems or rolling stock failures cause train delays and cancellations. Changes in the train schedule affect the train driver duties. If a train is cancelled or delayed, the driver assigned to the train task might not be able to reach the station of his next train departure in time. In practice, if a driver is not available for the train departure, another driver, for instance, a reserve, is assigned to the task. If there are no drivers available to cover the task on time, the train is delayed or cancelled, causing further disruptions. The train driver recovery is performed by dispatchers, who often work under tremendous pressure. The size of the schedule (more than 2 000 train tasks, which are covered by approximately 270 drivers on a weekday) and a high frequency of the train departures with headways down to 2 minutes at certain network segments makes it difficult to find a good recovery solution fast.