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.

最佳规模组合选择对于多尺度决策表（MDT）中的知识发现起着至关重要的作用，并引起了广泛的关注。但是，从比例尺集合中搜索所有最佳比例尺组合可能会导致组合爆炸，并且现有方法非常耗时。这项研究的主要目标是提高搜索所有最佳尺度组合的效率。为此，提出了一个连续的三向标尺集合决策模型和一种扩展的逐步最优标尺选择方法，以快速搜索所有最优标尺组合。首先，提出了规模收集的顺序三向决策模型，可以证明边界区域上的局部最优尺度组合也是尺度集合上的全局最优尺度组合。因此，可以通过在边界区域上逐步搜索单个局部最佳比例组合来获得MDT的所有最佳比例组合。其次，引入了扩展的逐步最优尺度选择方法，以快速搜索边界区域上的单个局部最优尺度组合。此外，给出了MDT具有唯一的最佳比例组合的必要和充分条件，并提供了两种计算边界区域的最大元素的有效方法。最后，提出了一种基于顺序三路决策的高效最优尺度组合选择算法，以搜索所有最优尺度组合。实验结果表明，提出的算法可以显着减少总体计算时间。