A weighted intuitionistic fuzzy soft set (WIFSS) is an important generalization of intuitionistic fuzzy soft set (IFSS). In this research work, the authors define the basic binary set operations, such as union, restricted union, restricted intersection, extended intersection, restricted difference, sum, AND-product, OR-product, and some concepts on weighted intuitionistic fuzzy soft set. They also present some properties of the operations and established some fundamental results including various De Morgan's type of results in weighted intuitionistic fuzzy soft set context.
Top1. Introduction
Over the years, man has been working tirelessly to find solutions to complicated problems we encountered in various facets of human endeavors such as environmental sciences, engineering, economics, medical sciences, social sciences, management sciences, etc. These types of problems are associated with various degrees of uncertainties and imprecision in them. In search of solution to these problems, many mathematical models were postulated to handle the problems of uncertainties and the imprecision. These theories among others include theory of probability (Prade and Dubois, 1980), theory of fuzzy sets (Zadeh, 1965), theory of interval mathematics (Atanassov, 1994), rough sets (Pawlak, 1982), vague sets (Gau and Buehrer, 1993) which we can consider as mathematical tools for dealing with uncertainties. But it was discovered that all these theories have their short comings in handling the level of uncertainties. The major deficiency associated with these theories is the inadequacies of the parameterization tools. To overcome these limitations, (Molodtsov, 1999) introduced the concept of soft set as a new mathematical tool for dealing with uncertainties and imprecision that is free from the difficulties that have bedeviled the standard existing mathematical models. Molodtsov pointed out the application of soft set in several directions. This theory has proven useful in many different fields such as decision making (Roy and Maji, 2007), data analysis (Zou and Xiao, 2008), forecasting and so on.
Research on soft sets has been progressing rapidly, since its introduction by Molodtsov in 1999 up to the present and several important results have been achieved in theory and practice. Maji et al. (2003) defined many algebraic operations in soft set theory and published a detailed theoretical study on soft sets. Ali et al. (2009) further presented and investigated some new algebraic operations for soft sets. Sezgin and Atagun (2011) proved that certain De Morgan’s law holds in soft set theory with respect to different operations on soft sets and discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Thereafter, it was observed that soft set can be combined with other mathematical models. Maji et al. (2001) were the first to present fuzzy soft set. Maji et al. (2003) established classical soft set to intuitionistic fuzzy soft sets, which were further discussed in (Maji et al., 2003) and (Yin et al., 2012). Since then many researchers have established various hybrid soft sets.
Other applications of soft set and its hybrids in decision making are found in (Cagman and Deli, 2012a, 2012b), (Deli and Cagman, 2015), (Deli, 2015) and (Deli et al., 2018).
Deli and Cagman (2016)
defined some kinds of distance between two intuitionistic fuzzy set and applied it in medical diagnosis of disease.
Smarandache in 2018 further generalized Soft Set theory to HyperSoft Set by transforming the function F in to a multi-attribute function.