Approximation reductions in an incomplete variable precision multigranulation rough set

This paper deals with approaches to the granular space reductions in the variable precision multigranulation rough set model. The main objective of this study is to extend four kinds of the granular space reductions called a tolerance relations optimistic multigranulation  ower approximation distribution reduction, a tolerance relations optimistic multigranulation  upper approximation distribution reduction, a tolerance relations pessimistic multigranulation  lower approximation distribution reduction and a tolerance relations pessimistic multigranulation  upper approximation distribution reduction ,which preserve the optimistic/pessimistic multigranulation  lower/upper approximation distribution of the decision classes. Some judgement theorems are investigated. The example proves that the new variable precision multigranulation rough set model can effectively deal with incomplete information, from which we can obtain approaches to the granular space reductions of incomplete decision systems in variable precision multigranulation rough theory.