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- 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.

- The Integrated Size and Price Optimization problem (2012)
- 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.

- 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.

- On the characteristic of integral point sets in $\mathbb{E}^m$ (2005)
- We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.

- Maximal integral point sets over Z^2 (2008)
- Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 if all pairwise distances are integral and every additional point p(n+1) destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.

- Lotsize optimization leading to a p-median problem with cardinalities (2007)
- 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.

- Integral point sets over Z_n^m (2007)
- There are many papers studying properties of point sets in the Euclidean space or on integer grids, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of Z_n, and study the properties of the resulting combinatorial structures.

- Integral point sets over finite fields (2007)
- We consider point sets in the affine plane GF(q)^2 where each Euclidean distance of two points is an element of GF(q). These sets are called integral point sets and were originally defined in m-dimensional Euclidean spaces. We determine their maximal cardinality I(GF(q),2). For arbitrary commutative rings R instead of GF(q) or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.

- Inclusion-maximal integral point sets over finite fields (2007)
- We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in $GF(q)^2$ where the formally defined Euclidean distance of every pair of points is an element of $GF(q)$. From another point of view we consider point sets over $GF(q)^2$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integral point sets over $GF(q)^2$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.

- Enumeration of integral tetrahedra (2007)
- We determine the numbers of integral tetrahedra with diameter d up to isomorphism for all d<=1000 via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most d in O(d^5) time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral 4x4 matrices with diameter d fulfilling the triangle inequalities we derive an exact formula.