Variable Precision Intuitionistic Fuzzy Rough Set Model And Fuzzy Soft Rough Set Model Based On Residual Implicators

Fuzzy set theory, rough set theory, intuitionistic fuzzy set theory and soft set the-ory are powerful mathematical tools for modeling various types of uncertainties and in-complete information data. Moreover, these theories have much closer to each other for practical needs to use these theories complementarily in some cases for managing uncer-tainties. Considering their advantages in dealing with uncertainty problems, by employing an intuitionistic fuzzy residual implicator I and an intuitionistic fuzzy triangle norm T intuitionistic fuzzy inclusion sets and intuitionistic fuzzy inclusion ratio are defined, and variable precision intuitionistic fuzzy rough set models are constructed and their proper-ties and attribute reduction are examined. Moreover, considering the flexibility of soft set parameters, a new extension of the rough set theory is proposed by means of replacing a binary fuzzy relation with a fuzzy soft set, i.e.,(I, J)-fuzzy soft rough set model is p-resented based on a pair of border implicators (I, J).Therefore,(I, J)-fuzzy soft rough set model extends the existing fuzzy rough set model, because, employing a fuzzy soft set instead of a binary fuzzy relation to construct (I, J)-fuzzy soft rough set can reduce the errors which generated by constructing a binary fuzzy relation. Furthermore, using a fuzzy soft set to construct the lower and upper approximation operators of a fuzzy set make all information values in decision table involve the computation of approximation operators, thus the boundary region is reduced largely. Especially, we prove that (I, J)-soft fuzzy rough sets in our work are equivalent to (I, J)-fuzzy rough sets by using a T-equivalence fuzzy relation. If we take α=1,β=0,then variable precision intuitionistic fuzzy rough set model is also equivalent to (I, J)-fuzzy soft rough set model. Simultaneously, the re-lationships between (I, J)-fuzzy soft rough set model and the others rough set models are also investigated. Finally, an example is given to illustrate our methods and results.