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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 as well as on random instances., Comment: 36 pages, 8 figures, 1 table

In the maximum scatter traveling salesman problem the objective is to find a tour that maximizes the shortest distance between any two consecutive nodes. This model can be applied to manufacturing processes, particularly laser melting processes. We extend an algorithm by Arkin et al. that yields optimal solutions for nodes on a line to a regular $m \times n$-grid. The new algorithm $\textsc{Weave}(m,n)$ takes linear time to compute an optimal tour in some cases. It is asymptotically optimal and a $\frac{\sqrt{10}}{5}$-approximation for the $3\times 4$-grid, which is the worst case., Comment: 6 pages, 2 figures; to appear in OR Proceedings 2015

Opinion dynamics is nowadays a very common field of research. In this article we formulate and then study a novel, namely strategic perspective on such dynamics: There are the usual normal agents that update their opinions, for instance according the well-known bounded confidence mechanism. But, additionally, there is at least one strategic agent. That agent uses opinions as freely selectable strategies to get control on the dynamics: The strategic agent of our benchmark problem tries, during a campaign of a certain length, to influence the ongoing dynamics among normal agents with strategically placed opinions (one per period) in such a way, that, by the end of the campaign, as much as possible normals end up with opinions in a certain interval of the opinion space. Structurally, such a problem is an optimal control problem. That type of problem is ubiquitous. Resorting to advanced and partly non-standard methods for computing optimal controls, we solve some instances of the campaign problem. But even for a very small number of normal agents, just one strategic agent, and a ten-period campaign length, the problem turns out to be extremely difficult. Explicitly we discuss moral and political concerns that immediately arise, if someone starts to analyze the possibilities of an optimal opinion control., Comment: 47 pages, 12 figures, and 11 tables

For several years, students visit us on different occasions at the university. But how to bridge from the school curriculum to the contents of the university mathematics? And how to find a focal point at which an active contribute, despite the lack of knowledge, in view of limited time is possible? Our approach: Translate,under guidance, everyday life optimization problems into the language of mathematics, i.e. using variables, target functions, equations and inequalities. These so-called integer linear programming models are then solved by standard software. In this report we wnat to tell about the lessons we have learned., Comment: 11 pages, 11 figures

We present the Integrated Size and Price Optimization Problem (ISPO) for a fashion discounter with many branches. Based on a two-stage stochastic programming model with recourse, we develop an exact algorithm and a production-compliant heuristic that produces small optimality gaps. In a field study we show that a distribution of supply over branches and sizes based on ISPO solutions is significantly better than a one-stage optimization of the distribution ignoring the possibility of optimal pricing., Comment: 26 pages, 9 tables

We consider a fashion discounter that supplies any of its many branches with an integral multiple of lots whose size assortment structure stems from a set of many applicable lot-types. We design a column generation algorithm for the optimal approximation of the branch and size dependent demand by a supply using a bounded number of lot-types., Comment: 4 pages, 2 tables, presented at ISCO 2012

Fashion discounters face the problem of ordering the right amount of pieces in each size of a product. The product is ordered in pre-packs containing a certain size-mix of a product. For this so-called lot-type design problem, a stochastic mixed integer linear programm was developed, in which price cuts serve as recourse action for oversupply. Our goal is to answer the question, whether the resulting supply strategy leads to a supply that is significantly more consistent with the demand for sizes compared to the original manual planning. Since the total profit is influenced by too many factors unrelated to sizes (like the popularity of the product, the weather or a changing economic situation), we suggest a comparison method which excludes many outer effects by construction. We apply the method to a real-world field study: The improvements in the size distributions of the supply are significant., Comment: 5 pages, 1 table

We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation we provide an appropriate primal heuristic for this problem., Comment: 5 pages, 2 tables

We give an elementary proof of a conjecture by Hegselmann and Krause in opinion dynamics, concerning a symmetric bounded confidence interval model: If there is a truth and all individuals take each other seriously by a positive amount bounded away from zero, then all truth seekers will converge to the truth. Here truth seekers are the individuals which are attracted by the truth by a positive amount. In the absence of truth seekers it was already shown by Hegselmann and Krause that the opinions of the individuals converge., Comment: 22 pages, 8 figures

We propose the new Top-Dog-Index to quantify the historic deviation of the supply data of many small branches for a commodity group from sales data. On the one hand, the common parametric assumptions on the customer demand distribution in the literature could not at all be supported in our real-world data set. On the other hand, a reasonably-looking non-parametric approach to estimate the demand distribution for the different branches directly from the sales distribution could only provide us with statistically weak and unreliable estimates for the future demand. Based on real-world sales data from our industry partner we provide evidence that our Top-Dog-Index is statistically robust. Using the Top-Dog-Index, we propose a heuristics to improve the branch-dependent proportion between supply and demand. Our approach cannot estimate the branch-dependent demand directly. It can, however, classify the branches into a given number of clusters according to an historic oversupply or undersupply. This classification of branches can iteratively be used to adapt the branch distribution of supply and demand in the future., Comment: 6 pages, 7 figures

We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem., Comment: 14 pages

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 branch's 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., Comment: 22 pages, 15 figures

There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension $n-d-3$. Moreover, we resolve positively a new special case of the \emph{Generalized Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension $n-d-2$.

Let $P$ and $Q$ be bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part of $P$ being homotopy equivalent to the suspension of the proper part of~$Q$. An application of this lemma is a unified proof of the sphericity of the higher Bruhat orders under both inclusion order (a known proved earlier by Ziegler) and single step inclusion order (which was not previously known).