Decompositions of soft sets and soft matrices with applications in group decision making

The decompositions of soft sets and soft matrices are important tools for theoretical and practical studies. In this paper, firstly, we study the decomposition of soft sets in detail. Later, we introduce the concepts of $\alpha$-upper, $\alpha$-lower, $\alpha$-intersection and $\alpha$-union for soft matrices and present some decomposition theorems. Some of these operations are set-restricted types of existing operations of soft sets/matrices, others are $\alpha$-oriented operations that provide functionality in some cases. Moreover, some relations of decompositions of soft sets and soft matrices are investigated and the newfound relations are supported with numerical examples. Finally, two new group decision making algorithms based on soft sets/matrices are constructed, and then their efficiency and practicality are demonstrated by dealing with real life problems and comparison analysis. By using these proposed approaches, solutions can be presented to soft set-based multi-criteria decision making problems, both ordinary and involving primary assessments. These allow to handle soft set-based multi-criteria decision making from different perspectives.