Optimal scale combination selection for multi-scale decision tables based on three-way decision

Optimal scale combination selection plays a critical role for knowledge discovery in multi-scale decision tables (MDTs) and has attracted considerable attention. However, searching for all optimal scale combinations from the scale collection may result in a combinatorial explosion, and the existing approaches are time-consuming. The main goal of this study is to improve the efficiency of searching for all optimal scale combinations. To this end, a sequential three-way decision model of the scale collection and an extended stepwise optimal scale selection method are proposed to quickly search for all optimal scale combinations. First, a sequential three-way decision model of the scale collection is proposed, and it can be proved that a local optimal scale combination on the boundary region is also a global optimal scale combination on the scale collection. Therefore, all optimal scale combinations of a MDT can be obtained by searching for a single local optimal scale combination on the boundary regions in a step-by-step manner. Second, an extended stepwise optimal scale selection method is introduced to quickly search for a single local optimal scale combination on the boundary region. Moreover, a necessary and sufficient condition under which a MDT has a unique optimal scale combination is given, and two efficient methods for computing the maximal elements of the boundary region are provided. Finally, an efficient optimal scale combination selection algorithm based on sequential three-way decision is presented to search for all optimal scale combinations. Experimental results demonstrate that the proposed algorithms can significantly reduce overall computational time.