# DISPO

DISPO – how should various sizes of a fashion article be distributed over many branchesl?

# DISPO – A Decision Support System for the Integrated Size and Price Optimization

Project Leader: Prof. Dr. Jörg Rambau
Prof. Dr. Jörg Schlüchtermann
Contact: Prof. Dr. Jörg Rambau

Project start: 2007/11/01 Project end: 2010/10/31

Project team: External partners:

NKD Vertriebs GmbH

Support:

Project description:

## Application Background

The company NKD Vertriebs GmbH sells apparel via a large network of branches. Since all goods have to be ordered three months in advance to the sales process, ususally all pieces of an article have to be sold with no option for replenishment orders. Since apparel comes in various sizes, it is important which assortments of sizes should be ordered for which branch. A bad distribution causes damage to the company by the need for heavier price reductions in order to get all pieces sold in all branches. Because the price reduction is under the control of the company as well, we face a two-stage optimization problem: Order and distribute among branches quantities for all sizes of a fashion article so that, given an optimal reaction by mark-downs during the sales process,  the expected profit is maximized.  In order to use appropriate key performance indicators for optimization,  the Lehrstuhl Wirtschaftsmathematik (Prof. Dr. Jörg Rambau) cooperates with the Lehrstuhl BWL V, Produktionswirtschaft und Industriebetriebslehre (Prof. Dr. Jörg Schlüchtermann).

## Contribution to the Mission of MODUS

The two-stage structure of the problem is formalized by a two-stage stochastic program with recourse. In this model, the here-and-now decisions (i.e., those that must be taken without knowing uncertain data) are the ordering and distribution decisions. The wait-and-see decisions (i.e., those that react to a realization of uncertain data) are the mark-down decisions. Uncertainty is formalized by a finite set of scenarios that describe the economic overall succes of an article. The resulting stochastic programming model can be transformed to a deterministic equivalent program (i.e., a formuation that formally does not contain random variables anymore). Formally, this program (called ISPO – Integrated Size and Price Optimization) is a large-scale mixed-integer linear program (MILP) that cannot be solved by standard software in reasonable time.

## Results

The project team was able to develop an exact algorithm (via the branch-and-bound technique with variable-strength dual bounds) and a fast heuristics for operational use. A blind field study with randomly selected test and control groups of branches provided evidence for a significant improvement over a more basic optimization method. A core part of our method (though not the most complicated final model) was put into practice at NKD.

University of Bayreuth -