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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.

- 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.

- Local Approximation of Discounted Markov Decision Problems by Mathematical Programming Methods (2011)
- We develop a method to approximate the value vector of discounted Markov decision problems (MDP) with guaranteed error bounds. It is based on the linear programming characterization of the optimal expected cost. The new idea is to use column generation to dynamically generate only such states that are most relevant for the bounds by incorporating the reduced cost information. The number of states that is sufficient in general and necessary in the worst case to prove such bounds is independent of the cardinality of the state space. Still, in many instances, the column generation algorithm can prove bounds using much fewer states. In this paper, we explain the foundations of the method. Moreover, the method is used to improve the well-known nearest-neighbor policy for the elevator control problem.