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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 ..."
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Cited by 25 (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 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.
A Genetic Algorithm for the Set Partitioning Problem
, 1995
"... In this paper we present a genetic algorithmbased heuristic for solving the set partitioning problem. The set partitioning problem is an important combinatorial optimisation problem used by many airlines as a mathematical model for flight crew scheduling. We develop a steadystate genetic algori ..."
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Cited by 17 (0 self)
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In this paper we present a genetic algorithmbased heuristic for solving the set partitioning problem. The set partitioning problem is an important combinatorial optimisation problem used by many airlines as a mathematical model for flight crew scheduling. We develop a steadystate genetic algorithm in conjunction with a specialised heuristic feasibility operator for solving the set partitioning problem. Some basic genetic algorithm components, such as fitness definition, parent selection and population replacement are modified. The performance of our algorithm is evaluated on a large set of realworld set partitioning problems provided by the airline industry. Computational results show that the genetic algorithmbased heuristic is capable of producing highquality solutions. In addition a number of the ideas presented (separate fitness, unfitness scores and subgroup population replacement) are applicable to any genetic algorithm for constrained problems. Keywords: combinator...
Semidefinite Programming for Assignment and Partitioning Problems
, 1996
"... Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simpl ..."
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Cited by 13 (2 self)
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Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simple structure such as the maxcut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems. A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the socalled "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primaldual inter...
Column Generation and the Airline Crew Pairing Problem
, 1998
"... . The cost of flight crews is the second largest operating cost of an airline. Minimizing it is a fundamental problem in airline planning and operations, and one which has leant itself to mathematical optimization. We discuss several recent advances in the methods used to solve these problems. A ..."
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Cited by 12 (0 self)
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. The cost of flight crews is the second largest operating cost of an airline. Minimizing it is a fundamental problem in airline planning and operations, and one which has leant itself to mathematical optimization. We discuss several recent advances in the methods used to solve these problems. After describing the general approach taken, we discuss a new method which can be used to obtain approximate solutions to linear programs, dramatically improving the solution time of these problems. This is the socalled volume algorithm. We also describe several other ideas used to make it routinely possible to get very good solutions to these large mixed integer programs. 1991 Mathematics Subject Classification: 90B35, 90C09, 90C10 Keywords and Phrases: linear programming, integer programming, volume algorithm, crew pairing, column generation 1 Airline Operations and the Crew Pairing Problem The Airline Crew Pairing problem has become well known as a prototypical applied problem w...
Recent Advances in Exact Optimization of Airline Scheduling Problems
, 1995
"... We discuss the formulation and solution of large scale integer optimization problems that arise in the scheduling of transport related services. We first set the context for these problems within the airline industry by discussing the scheduling process. We then discuss the two key activities of fle ..."
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Cited by 6 (0 self)
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We discuss the formulation and solution of large scale integer optimization problems that arise in the scheduling of transport related services. We first set the context for these problems within the airline industry by discussing the scheduling process. We then discuss the two key activities of fleet assignment and crew scheduling that turn a schedule into an operational plan. We provide current formulations in terms of key objectives and constraints for both the fleet and the crew assignment problems. This is followed by a discussion of the state of the art in solution methodology for each problem. We conclude with ideas about promising areas for further work in the application of combinatorial optimization to airline scheduling.
Optimized Crew Scheduling at Air New Zealand
"... The aircrewscheduling problem consists of two important subproblems: the toursofduty planning problem to generate minimumcost tours of duty (sequences of duty periods and rest periods) to cover all scheduled flights, and the rostering problem to assign tours of duty to individual crew members. B ..."
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The aircrewscheduling problem consists of two important subproblems: the toursofduty planning problem to generate minimumcost tours of duty (sequences of duty periods and rest periods) to cover all scheduled flights, and the rostering problem to assign tours of duty to individual crew members. Between 1986 and 1999, Air New Zealand staff and consultants in collaboration with the University of Auckland have developed eight applicationspecific optimizationbased computer systems to solve all aspects of the toursofduty planning and rostering processes for Air New Zealandâs national and international operations. These systems have saved NZ$15,655,000 per year while providing crew rosters that better respect crew membersâ preferences.
DantzigWolfe Decomposition for Solving MultiStage Stochastic CapacityPlanning Problems
, 2008
"... We describe a multistage, stochastic, mixedintegerprogramming model for planning discrete capacity expansion of production facilities. A scenario tree represents uncertainty in the model; a general mixedinteger program defines the operational submodel at each scenariotree node; and capacityexp ..."
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We describe a multistage, stochastic, mixedintegerprogramming model for planning discrete capacity expansion of production facilities. A scenario tree represents uncertainty in the model; a general mixedinteger program defines the operational submodel at each scenariotree node; and capacityexpansion decisions link the stages. We apply “variable splitting ” to two model variants, and solve those variants using DantzigWolfe decomposition. The DantzigWolfe master problem can have a much stronger linearprogramming relaxation than is possible without variable splitting, over 700 % stronger in one case. The master problem solves easily and tends to yield integer solutions, obviating the need for a full branchandprice solution procedure. For each scenariotree node, the decomposition defines a subproblem that may be viewed as a singleperiod, deterministic, capacityplanning problem. An effective solution procedure results as long as the subproblems solve efficiently, and the procedure incorporates a good “duals stabilization scheme.” We present computational results for a model to plan the capacity expansion of an electricity distribution network in New Zealand, given uncertain future demand. The largest problem we solve to optimality has 6 stages and 243 scenarios, and corresponds to a deterministic equivalent with a quarter of a million binary variables.
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.
PETRA: A Programmable Optimisation Engine and Toolbox for Personnel Rostering Applications
 Presented at the 15th Triennial International Federation of Operational Research Societies Conference IFORS 99
, 1999
"... This paper presents a general optimisationbased rostering engine developed by the University of Auckland. This engine is designed to solve a wide range of personnel scheduling (staff rostering) problems. The engine is currently being used to roster over 700 staff at an Australian call centre and al ..."
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This paper presents a general optimisationbased rostering engine developed by the University of Auckland. This engine is designed to solve a wide range of personnel scheduling (staff rostering) problems. The engine is currently being used to roster over 700 staff at an Australian call centre and also for the rostering of 50 immigration staff at Auckland International Airport. It is being trialed for use at several nurse rostering sites.
Generating crew pairings for very large flight networks
, 2006
"... Airline operations depend on the complex control of many expensive, tightlyconstrained, interdependent resources such as crews and aircraft. Operations research (OR) has played an important role in developing planning tools for this complex system. In recent years, however, the already intense pres ..."
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Cited by 1 (1 self)
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Airline operations depend on the complex control of many expensive, tightlyconstrained, interdependent resources such as crews and aircraft. Operations research (OR) has played an important role in developing planning tools for this complex system. In recent years, however, the already intense pressure on the airline industry has dramatically increased due to rising fuel costs, competitive pricing, increased congestion, and security concerns. Important research opportunities still exist in the areas of integrated planning, robust planning, and optimizationbased recovery tools. One outstanding challenge is that all of these require the simultaneous consideration of aircrafts and crews, which are typically treated independently in current systems. The result is far larger instances of the crew scheduling problem than are currently being solved, because this problem can no longer be decomposed by fleet type. Existing methods, which typically rely on enumerationbased approaches to solve the imbedded crew pairing problem, become intractable for these larger problem instances. In this paper we introduce an alternative approach, based on integer programming, to generate crew pairings over very large flight networks. Computational results based on a major U.S. carrier are provided. 1