Lexicographic Min-Max Optimization in Fair Scheduling

Mariusz Kaleta and Tomasz Śliwiński*

Classical forecasting of sales data for companies on competitive markets based on statistical methods works well in case of periodical or seasonal data. However, it can hardly take into account special events or circumstances. In this case specialized methods have been developed, algorithms based on neural networks being an example. Unfortunately, for many of the special factors, there is no historical data that could be used for tuning any forecasting method. Moreover, making decisions on sales schedules is not only based on forecasting the market, but also is a part of creating the market and stimulating the demand. In the paper we consider methods for supporting the experts with interactive procedure of adjusting the sale schedules. The work is based on our experience coming from cooperation with one of the big FMCG companies. The expert constraints, that can be included during the scheduling proces, introduce some deviations to the most preferred basic schedule derived from the historical data. The optimization objective is to keep the deviations as small as possible, but preserving some fairness among the affected entities. The fairness criteria mean that more uniform distribution of deviations is preferred, that is, no sales representative or sale point is burdened relatively more than others. Such a solution can be achieved by solving proposed multi-criteria problem with lexicographic min-max optimization. However, because of the enormous number of criteria, the problem is computationally complex. We have proposed the approximation of the lexicographic min-max based the ordered values approach which allowes some tradeoff between absolute fairness and the computational performance. The computational times for the resulting algorithm are tractable even for large problems. Our approach has been implemented and deployed for a big FMCG company, where it is used for monthly sales planning. Comprehensive, expressive and low-budget tool was developed by combining common spreadsheet software and linear programming optimization solver.

Mathematics Subject Classification: 90B50

Keywords: lexicographic min-max; fairness; sales scheduling

Minisymposion: Integration of Optimization, Modeling and Data Analysis for Solving Real World Problems