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We develop and experimentally compare policies for the control of a system of $k$ elevators with capacity one in a transport environment with $\ell$ floors, an idealized version of a pallet elevator system in a large distribution center of the Herlitz PBS AG in Falkensee. Each elevator in the idealized system has an individual waiting queue of infinite capacity. On each floor, requests arrive over time in global waiting queues of infinite capacity. The goal is to find a policy that, without any knowledge about future requests, assigns an elevator to each req uest and a schedule to each elevator so that certain expected cost functions (e.g., the average or the maximal flow times) are minimized. We show that a reoptimization policy for minimizing average sq uared waiting times can be implemented to run in real-time ($1\,s$) using dynamic column generation. Moreover, in discrete event simulations with Poisson input it outperforms other commonly used polic ies like multi-server variants of greedy and nearest neighbor.
Oriented matroids are combinatorial structures that encode the combinatorics of point configurations. The set of all triangulations of a point configuration depends only on its oriented matroid. We survey the most important ingredients necessary to exploit oriented matroids as a data structure for computing all triangulations of a point configuration, and report on experience with an implementation of these concepts in the software package TOPCOM. Next, we briefly overview the construction and an application of the secondary polytope of a point configuration, and calculate some examples illustrating how our tools were integrated into the {\sc polymake} framework.
Under high load, the automated dispatching of service vehicles for the German Automobile Association (ADAC) must reoptimize a dispatch for 100--150 vehicles and 400 requests in about ten seconds to near optimality. In the presence of service contractors, this can be achieved by the column generation algorithm ZIBDIP. In metropolitan areas, however, service contractors cannot be dispatched automatically because they may decline. The problem: a model without contractors yields larger optimality gaps within ten seconds. One way-out are simplified reoptimization models. These compute a short-term dispatch containing only some of the requests: unknown future requests will influence future service anyway. The simpler the models the better the gaps, but also the larger the model error. What is more significant: reoptimization gap or reoptimization model error? We answer this question in simulations on real-world ADAC data: only the new model ZIBDIP{\footnotesize dummy} can keep up with ZIBDIP.
Given an affine surjection of polytopes $\pi: P \to Q$, the Generalized Baues Problem asks whether the poset of all proper polyhedral subdivisions of $Q$ which are induced by the map $\pi$ has the homotopy type of a sphere. We extend earlier work of the last two authors on subdivisions of cyclic polytopes to give an affirmative answer to the problem for the natural surjections between cyclic polytopes $\pi: C(n,d') \to C(n,d)$ for all $1 \leq d < d' < n$.
In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of \emph{coherent} mixed subdivisions of a Minkowski sum $\mathcal{A}_1+\cdots+\mathcal{A}_r$ of point configurations and of \emph{coherent} polyhedral subdivisions of the associated Cayley embedding $\mathcal{C}(\mathcal{A}_1,\dots,\mathcal{A}_r)$. In this paper we extend this correspondence in a natural way to cover also \emph{non-coherent} subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.
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 \emph{Online Target Date Assignment Problem} (\textsc{OnlineTDAP}) as a general framework for online problems with this nature. Requests for the \textsc{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 \emph{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 \textsc{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 par ticular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.
The present dissertation deals with the structure of polyhedral subdivisions of point configurations. Of particular interest are the global properties of the set of all subdivisions of a given point configuration. An important open problem in this context is the following: can one always transform any triangulation of a given point configuration to any other triangulation of the same configuration by means of bistellar operations? In other words, is the set of all triangulations of a given point configuration always bistellarly connected? The results presented in this thesis contribute progress from two directions. \begin{itemize} \item The set of all subdivisions that are induced by a polytope projection is in general not bistellarly connected in a generalized sense. This result is obtained by constructing a counterexample to the so-called Generalized Baues Conjecture.'' \item The set of all triangulations of a cyclic polytope forms a bounded poset. The covering relations are given by increasing bistellar operations. Thus we get an affirmative answer to the above question in the case of cyclic polytopes. \end{itemize} In the introduction, the mathematical environment of the structures under consideration is illuminated. The "Generalized Baues Conjecture" has connections to various mathematical concepts, such as combinatorial models for loop spaces, discriminants of polynomials in several variables, etc. The triangulation posets of cyclic polytopes are natural generalizations of the well-studied Tamari lattices in order theory. Moreover, there is a connection to the higher Bruhat orders, which have similar structural properties. As a by-product, the investigations yield the shellability of all triangulations of cyclic polytopes without new vertices. This is in particular interesting because most triangulations of cyclic polytopes are non-regular.
Wo bleibt der Aufzug?
(1999)
Euler is Standing in Line
(1999)
In this paper we study algorithms for ``Dial-a-Ride'' transportation problems. In the basic version of the problem we are given transportation jobs between the vertices of a graph and the goal is to find a shortest transportation that serves all the jobs. This problem is known to be NP-hard even on trees. We consider the extension when precedence relations between the jobs with the same source are given. Our results include a polynomial time algorithm on paths and an approximation algorithm on general graphs with a performance of~$9/4$. For trees we improve the performance to~$5/3$.