On Complex Intuitionistic Fuzzy Soft Sets with Distance Measures and Entropies

We introduce the concept of complex intuitionistic fuzzy soft sets which is parametric in nature. However, the theory of complex fuzzy sets and complex intuitionistic fuzzy sets are independent of the parametrization tools. Some real life problems, for example, multicriteria decision making problems, involve the parametrization tools. In order to get their new entropies, some important properties and operations on the complex intuitionistic fuzzy soft sets have also been discussed. On the basis of some well-known distance measures, some new distance measures for the complex intuitionistic fuzzy soft sets have also been obtained. Further, we have established correspondence between the proposed entropies and the distance measures of complex intuitionistic fuzzy soft sets. 1. Introduction Ramot et al. [1, 2] introduced a new innovative concept of complex fuzzy set (CFS), where the membership function instead of being a real valued function with the range of is replaced by a complex-valued function of the form (), where is a real valued function such that and is a periodic function. The key feature of complex fuzzy sets is the presence of phase and its membership. This gives those complex fuzzy sets wavelike properties which could result in constructive and destructive interference depending on the phase value. Several examples are given in [2], which demonstrate the utility of these complex fuzzy sets. They also defined several important operations such as complement, union, and intersection and discussed fuzzy relations for such complex fuzzy sets. On the other hand, Ma et al. [3] used the complex fuzzy set to represent the information with uncertainty and periodicity, where they introduced a product-sum aggregation operator based prediction (PSAOP) method to find the solution of the multiple periodic factor prediction (MPFP) problems. Further, Chen et al. [4] proposed a neurofuzzy system architecture to implement the complex fuzzy rule as a practical application of the concept of complex fuzzy logic. Intuitionistic fuzzy set (IFS), introduced by Atanassov [5], is a controlling tool to deal with vagueness and uncertainty. A prominent characteristic of IFS is that it assigns to each element a membership degree and a nonmembership degree with certain amount of hesitation degree. Atanassov [6, 7] and many other researchers [8, 9] studied different properties of IFSs in decision making problems, particularly in the case of medical diagnosis, sales analysis, new product marketing, financial services, and so forth. Alkouri and Salleh [10] introduced the concept