Scheduling-Location (ScheLoc) Problems integrate the separate fields of
scheduling and location problems. In ScheLoc Problems the objective is to
find locations for the machines and a schedule for each machine subject to
some production and location constraints such that some scheduling object-
ive is minimized. In this paper we consider the Discrete Parallel Machine
Makespan (DPMM) ScheLoc Problem where the set of possible machine loc-
ations is discrete and a set of n jobs has to be taken to the machines and
processed such that the makespan is minimized. Since the separate location
and scheduling problem are both NP-hard, so is the corresponding ScheLoc
Problem. Therefore, we propose an integer programming formulation and
different versions of clustering heuristics, where jobs are split into clusters
and each cluster is assigned to one of the possible machine locations. Since
the IP formulation can only be solved for small scale instances we propose
several lower bounds to measure the quality of the clustering heuristics. Ex-
tensive computational tests show the efficiency of the heuristics.
Motivated by the time-dependent location problem over T time-periods introduced in
Maier and Hamacher (2015) we consider the special case of two time-steps, which was shown
to be equivalent to the static 2-facility location problem in the plane. Geometric optimality
conditions are stated for the median objective. When using block norms, these conditions
are used to derive a polygon grid inducing a subdivision of the plane based on normal cones,
yielding a new approach to solve the 2-facility location problem in polynomial time. Combinatorial algorithms for the 2-facility location problem based on geometric properties are
deduced and their complexities are analyzed. These methods differ from others as they are
completely working on geometric objects to derive the optimal solution set.
In this paper we give an overview on the system of rehabilitation clinics in Germany in general and the literature on patient scheduling applied to rehabilitation facilities in particular.
We apply a class-teacher model developed to this environment and then generalize it to meet some of the specific constraints of inpatient rehabilitation clinics. To this end we introduce a restricted edge coloring on undirected bipartite graphs which is called group-wise balanced. The problem considered is called patient-therapist-timetable problem with group-wise balanced constraints (PTTPgb). In order to specify weekly schedules further such that they produce a reasonable allocation to morning/afternoon (second level decision) and to the single periods (third level decision) we introduce (hierarchical PTTPgb). For the corresponding model, the hierarchical edge coloring problem, we present some first feasibility results.
A new algorithm for optimization problems with three objective functions is presented which computes a representation for the set of nondominated points. This representation is guaranteed to have a desired coverage error and a bound on the number of iterations needed by the algorithm to meet this coverage error is derived. Since the representation does not necessarily contain nondominated points only, ideas to calculate bounds for the representation error are given. Moreover, the incorporation of domination during the algorithm and other quality measures are discussed.
We argue that the concepts of resilience in engineering science and robustness in mathematical optimization are strongly related. Using evacuation planning as an example application, we demonstrate optimization techniques to improve solution resilience. These include a direct modelling of the uncertainty for stochastic or robust optimization, as well as taking multiple objective functions into account.
A large class of estimators including maximum likelihood, least squares and M-estimators are based on estimating functions. In sequential change point detection related monitoring functions can be used to monitor new incoming observations based on an initial estimator, which is computationally efficient because possible numeric optimization is restricted to the initial estimation. In this work, we give general regularity conditions under which we derive the asymptotic null behavior of the corresponding tests in addition to their behavior under alternatives, where conditions become particularly simple for sufficiently smooth estimating and monitoring functions. These regularity conditions unify and even extend a large amount of existing procedures in the literature, while they also allow us to derive monitoring schemes in time series that have not yet been considered in the literature including non-linear autoregressive time series and certain count time series such as binary or Poisson autoregressive models. We do not assume that the estimating and monitoring function are equal or even of the same dimension, allowing for example to combine a non-robust but more precise initial estimator with a robust monitoring scheme. Some simulations and data examples illustrate the usefulness of the described procedures.
Edgeworth expansions have been introduced as a generalization of the central limit theorem and allow to investigate the convergence properties of sums of i.i.d. random variables. We consider triangular arrays of lattice random vectors and obtain a valid Edgeworth expansion for this case. The presented results can be used, for example, to study the convergence behavior of lattice models.
We consider the problem of finding efficient locations of surveillance cameras, where we distinguish
between two different problems. In the first, the whole area must be monitored and the number of cameras
should be as small as possible. In the second, the goal is to maximize the monitored area for a fixed number of
cameras. In both of these problems, restrictions on the ability of the cameras, like limited depth of view or range
of vision are taken into account. We present solution approaches for these problems and report on results of
their implementations applied to an authentic problem. We also consider a bicriteria problem with two objectives:
maximizing the monitored area and minimizing the number of cameras, and solve it for our study case.
A single facility problem in the plane is considered, where an optimal location has to be
identified for each of finitely many time-steps with respect to time-dependent weights and
demand points. It is shown that the median objective can be reduced to a special case of the
static multifacility median problem such that results from the latter can be used to tackle the
dynamic location problem. When using block norms as distance measure between facilities,
a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the
resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation
between dynamic location problems for T time periods and T-facility problems, this algorithm
can also be applied to the static 2-facility location problem.
It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions.