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- A homotopy argument and its applications to the transformation rule for bi-Lipschitz mappings, the Brouwer fixed point theorem and the Brouwer degree (2005)
- The main purpose of the paper is to present an elementary self-contained proof of the change of variables formula for injective, locally bi-Lipschitz mappings. The proof is based on a homotopy argument. Various properties of bi-Lipschitz mappings are studied. As a by-product Lipschitz variants of the classical implicit function theorem and the local diffeomorphism theorem are proved. With the help of the homotopy argument a simple proof is given of Brouwer’s fixed point theorem and the main properties of Brouwer’s degree of mapping.

- A generalized job-shop problem with more than one resource demand per task (2011)
- We study a generalized job-shop problem called the Laser Sharing Problem with fixed tours (LSP-T) where the tasks may need more than one resource simultaneously. This fact will be used to model possible collisions between industrial robots. For three robots we will show that the special case where only one resource is used by more than one robot is already NP-hard. This also implies that one machine scheduling with chained min delay precedence constraints is NP-hard for at least three chains. On the positive side, we present a polynomial algorithm for the two robot case and a pseudo-polynomial algorithm together with an FPTAS for an arbitrary but constant number of robots. This gives a sharp boundary of the complexity status for a constant number of robots.

- Zwei auf einen Streich: Optimierte dynamische Einsatzplanung für Gelbe Engel und Lastenaufzüge (2007)
- Wir modellieren zwei verschiedene dynamische Einsatzplanungsprobleme: die dynamische Einsatzplanung Gelber Engel beim ADAC und die Steuerung von Lastenaufzügen in einem Versandlager der Herlitz PBS AG. Wir benutzen eine Reoptimierungspolitik, die die Steuerung des Systems mit Hilfe der Lösung von statischen Schnappschussproblemen durchführt. Für die auftretenden Schnappschussprobleme vergleichen wir zwei Modellierungsansätze (Flussmodell versus Tourenmodell), von denen nur einer echtzeittauglich ist. Das Verfahren zur dynamischen Einsatzplanung Gelber Engel ist beim ADAC in Betrieb.

- How to avoid collisions in scheduling industrial robots? (2010)
- In modern production facilities industrial robots play an important role. When two ore more of them are moving in the same area, care must be taken to avoid collisions between them. Due to expensive equipment costs our approach to handle this is very conservative: Each critical area is modeled as a shared resource where only one robot is allowed to use it at a time. We studied collision avoidance in the context of arc welding robots in car manufacture industry. Here another shared resource comes into place. When using laser welding technology every robot needs to be connected to a laser source supplying it with the necessary energy. Each laser source can be connected to up to six robots but serve only one at a time. An instance of the problem consists of a set of robots, a set of welding task, a number of laser sources, a distance table, collision information and a production cycle time. The goal is to design robot tours covering all task and schedule them resource conflict free such that the makespan does not exceed the cycle time. We propose a general model for integrated routing and scheduling including collision avoidance as well as a branch-and-bound algorithm for it. Computational results on data generated with the robot simulation software KuKa Sim Pro are also provided showing that our algorithm outperforms standard mixed-integer models for our application.

- The Stochastic Guaranteed Service Model with Recourse for Multi-Echelon Warehouse Management (2012)
- The Guaranteed Service Model (GSM) computes optimal order-points in multi-echelon inventory control under the assumptions that delivery times can be guaranteed and the demand is bounded. Our new Stochastic Guaranteed Service Model (SGSM) with Recourse covers also scenarios that violate these assumptions. Simulation experiments on real-world data of a large German car manufacturer show that policies based on the SGSM dominate GSM-policies.

- A survey of the higher Stasheff-Tamari orders (2012)
- The Tamari lattice, thought as a poset on the set of triangulations of a convex polygon with n vertices, generalizes to the higher Stasheff-Tamari orders on the set of triangulations of a cyclic d-dimensional polytope having n vertices. This survey discusses what is known about these orders, and what one would like to know about them.

- On the minimum diameter of plane integral point sets (2007)
- Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2,n) of a plane integral point set consisting of n points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2,n) achieving the known upper bound n^{c_2loglog n} up to a constant in the exponent.

- The Top-Dog Index: A New Measurement for the Demand Consistency of the Size Distribution in Pre-Pack Orders for a Fashion Discounter with Many Small Branches (2008)
- We propose the new Top-Dog-Index, a measure for the branch-dependent historic deviation of the supply data of apparel sizes from the sales data of a fashion discounter. A common approach is to estimate demand for sizes directly from the sales data. This approach may yield information for the demand for sizes if aggregated over all branches and products. However, as we will show in a real-world business case, this direct approach is in general not capable to provide information about each branchs individual demand for sizes: the supply per branch is so small that either the number of sales is statistically too small for a good estimate (early measurement) or there will be too much unsatisfied demand neglected in the sales data (late measurement). Moreover, in our real-world data we could not verify any of the demand distribution assumptions suggested in the literature. Our approach cannot estimate the demand for sizes directly. It can, however, individually measure for each branch the scarcest and the amplest sizes, aggregated over all products. This measurement can iteratively be used to adapt the size distributions in the pre-pack orders for the future. A real-world blind study shows the potential of this distribution free heuristic optimization approach: The gross yield measured in percent of gross value was almost one percentage point higher in the test-group branches than in the control-group branches.

- Bounds for the minimum oriented diameter (2008)
- We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound.

- An exact column-generation approach for the lot-type design problem (2012)
- We consider a fashion discounter distributing its many branches with integral multiples from a set of available lot-types. For the problem of approximating the branch and size dependent demand using those lots we propose a tailored exact column generation approach assisted by fast algorithms for intrinsic subproblems, which turns out to be very efficient on our real-world instances.