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Cyclic timetabling for public transportation companies is usually modeled by the periodic
event scheduling problem. To deduce a mixed-integer programming formulation, artificial integer
variables have to be introduced. There are many ways to define these integer variables.
We show that the minimal number of integer variables required to encode an instance is
achieved by introducing an integer variable for each element of some integral cycle basis. An
integral cycle basis consists of |A|-|V|+1 oriented cycles of a directed graph D = (V;A) that
enable any oriented cycle of the directed graph to be expressed as an integer linear combination.
The solution times for the originating application vary extremely with different integral
cycle bases. However, our computational studies show that the width of integral cycle bases
is a good empirical measure for the solution time of the MIP. Clearly, integral cycle bases
permit a much wider choice than the former standard approach, in which integer variables are
associated with the co-tree arcs of some spanning tree. Hence, to formulate better solvable
integer programs, we present algorithms that construct integral cycle bases of small width.
To that end, we investigate classes of directed cycle bases that are closely related to integral
cycle bases, namely (generalized) fundamental and undirected cycle bases. This gives rise to
both, a compact classification of directed cycle bases and notable reductions of running times
for cyclic timetabling.
Periodic timetabling for railway networks is usually modeled by the Periodic Event Scheduling
Problem (PESP). This model permits to express many requirements that practitioners impose
on periodic railway timetables. We discuss a requirement practitioners are asking for, but which,
so far, has not been the topic of mathematical studies: the concept of symmetry.
Several motivations why symmetric timetables might seem promising will be given. However,
we provide examples showing that symmetry leads to suboptimality.
To integrate symmetry into the graph model of the PESP, there are many obstacles to overcome.
Nevertheless, adding symmetry requirements to mixed-integer programming formulations
explicitly, enables MIP solvers such as CPLEX
to terminate earlier with good solutions.
We discuss the problem to count, or, more modestly, to estimate
the number f(m; n) of unimodular triangulations of the planar grid of
size m * n.
Among other tools, we employ recursions that allow one to compute
the (huge) number of triangulations for small m and rather large n by
dynamic programming; we show that this computation can be done in
polynomial time if m is fixed, and present computational results from
our implementation of this approach.
We also present new upper and lower bounds for large m and n,
and we report about results obtained from a computer simulation of
the random walk that is generated by
ips.
We consider the scheduling problem of minimizing the average-weighted completion time on identical parallel machines when jobs are arriving over time. For both the preemptive and the nonpreemptive setting, we show that straightforward extensions of Smith's ratio rule yield smaller competitive ratios than the previously best-known deterministic on-line algorithms.
Today's telecommunication networks are configured statically. Whenever a connection
is established, the customer has permanent access to it. However, it is
observed that usually the connection is not used continuously. At this point, dynamic
provisioning could increase the utilization of network resources. WDM
based Optical Transport Networks (OTNs) will shortly allow for fast dynamic
network reconfiguration. This enables optical broadband leased line services on
demand. Since service requests competing for network resources may lead to service
blocking, it is vital to use appropriate strategies for routing and wavelength
assignment in transparent optical networks. We simulate the service blocking
probabilities of various dynamic algorithms for this problem using a well-founded
traffic model for two realistic networks. One of the algorithms using shortest path
routings performs best on all instances. Surprisingly, the tie-breaking rule between
equally short paths in different wavelengths decides between success or
failure.
Given a set of service requests (events), a set of guided servers (units),
and a set of unguided service contractors (conts), the vehicle dispatching problem
VDP is the task to find an assignment of events to units and conts as well as tours
for all units starting at their current positions and ending at their home positions
(dispatch) such that the total cost of the dispatch is minimized.
The cost of a dispatch is the sum of unit costs, cont costs, and event costs. Unit
costs consist of driving costs, service costs and overtime costs; cont costs consist of
a fixed cost per service; event costs consist of late costs linear in the late time, which
occur whenever the service of the event starts later than its deadline.
The program ZIBDIP based on dynamic column generation and set partitioning
yields solutions on heavy-load real-world instances (215 events, 95 units) in less
than a minute that are no worse than 1% from optimum on state-of-the-art personal
computers.
We consider a model for scheduling under uncertainty. In this model, we combine the main characteristics of online and stochastic scheduling in a simple and natural way. Job processing times are assumed to be stochastic, but in contrast to traditional stochastic scheduling models, we assume that jobs arrive online, and there is no knowledge about the jobs that will arrive in the future. The model incorporates both, stochastic scheduling and online scheduling
as a special case. The particular setting we consider is non-preemptive parallel machine scheduling, with the objective to
minimize the total weighted completion times of jobs. We analyze
simple, combinatorial online scheduling policies for that model, and
derive performance guarantees that match performance guarantees previously
known for stochastic and online parallel machine scheduling, respectively.
For processing times that follow NBUE distributions, we
improve upon previously best known performance bounds from
stochastic scheduling, even though we consider a more general
setting.
Many online problems encountered in real-life involve a two-stage decision process: upon arrival of a new request, an irrevocable
first-stage decision (the assignment of a specific resource to the request) must be made immediately, while in a second stage process, certain ``subinstances'' (that is, the instances of all requests assigned to a particular resource) can be solved to optimality (offline) later.
We introduce the novel concept of an Online Target Date Assignment Problem (OnlineTDAP) as a general framework for online problems with this nature. Requests for the OnlineTDAP become known at certain dates. An online algorithm has to assign a target date to each request, specifying on which date the request should be processed (e.g., an appointment with a customer for a
washing machine repair). The cost at a target date is given by the downstream cost, the optimal cost of processing all requests
at that date w.r.t. some fixed downstream offline optimization problem (e.g., the cost of an optimal dispatch for service
technicians). We provide general competitive algorithms for the OnlineTDAP independently of the particular downstream problem,
when the overall objective is to minimize either the sum or the maximum of all downstream costs. As the first basic examples, we analyze the competitive ratios of our algorithms for the particular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.
A multistage stochastic programming approach to airline network revenue management is presented. The objective is to determine seat protection levels for all itineraries, fare classes, point of sales of the airline network and all data collection points of the booking horizon such that the expected revenue is maximized. While the passenger demand and cancelation rate processes are the stochastic inputs of the model, the stochastic protection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processes are approximated by a nite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursive scenario reduction scheme. Numerical results for a small hub-and-spoke network are reported.