On Generalized Interval Valued Fuzzy Soft Matrices

: Interval valued fuzzy soft set was a combination of the interval valued fuzzy set and soft set, while in generalized interval valued fuzzy soft set a degree was attached with the parameterization of fuzzy sets in defining an interval valued fuzzy soft set. In this paper we introduced the concept of generalized interval valued fuzzy soft matrices. We discussed some of its types and some operations. We also discussed about the similarity of two generalized interval valued fuzzy soft matrices


Introduction
A lot of problems in our real life in economics, social sciences, medical sciences, environmental sciences and engineering etc. involve various uncertainties. Many theories have been developed to deal with these uncertainties. Some of these theories are probability theory [1], fuzzy set theory (FST) [2], rough set theory (RST) [3], interval mathematics [4] and intuitionistic fuzzy set theory (IFST) [5] etc. Molodtsov [6] pointed out that all these theories have some inherent difficulties. He proposed soft set theory (SST) to overcome these difficulties. It was a generic mathematical tool for dealing problems having uncertainty. Later Maji and Biswas [7] defined soft subset and soft super set. They also defined absolute soft set and null soft set. They introduced some operations on soft set and De Morgan's laws are also verified by them. Ali et al [8] pointed out some errors of the previous work and introduced some new operations. They further studied more and discussed some algebraic structures of soft sets. Maji et al. [9] proposed fuzzy soft set (FSS), an improvement of the SST by combining (FST) and (SST). Roy and Maji [10] gave an application of fuzzy soft set in decision making. Yang et al. [11] introduced the intervalvalued fuzzy soft set (IVFSS) which was a combination of the IVFS and SST.
Majumdar and Samanta [12] introduced the concept of generalized fuzzy soft sets (GFSS). B. K. Saikia et al. [13] defined generalized fuzzy soft matrix (GFSM) and applied it to a decision making (DM) problem. Shawkat Alkhazaleh and Abdul Razak Salleh [14] introduced generalized interval valued fuzzy soft set (GIVFSS). In their generalization of FSS, they attached a degree with the parameterization of fuzzy sets in defining an IVFSS. They discussed various operations and properties of GIVFSS. Some of these are GIVFS subset, GIVFS equal set, generalized null interval valued fuzzy soft set (GNIVFS), generalized absolute interval valued fuzzy soft set (GAIVFS), compliment of GIVFSS, union of GIVFSS's and intersection of GIVFSS's. They defined AND and OR operations on GIVFSS and similarity measure of two GIVFSS's. They also give some applications of GIVFSS in DM problem and medical diagnosis.
Mi Jung Son [15] introduced interval valued fuzzy soft set and defined some of its types. P. Rajarajeswari and P. Dhanalakshmi [16] developed interval-valued fuzzy soft matrix theory. Zulqarnain. M and M. Saeed [17] defined some new types of interval valued fuzzy soft matrix and gave an application of IVFSM in a decision making problem. Anjan Mukherjee and Sadhan Sarkar [18,19] introduced Similarity measures for interval-valued intuitionistic fuzzy soft sets and gave applications in medical diagnosis problems. B. Chetia and P. K. Das [20] used interval-valued fuzzy soft sets and Sanchez's approach for medical diagnosis. In recent years many researchers [21][22][23][24][25] have been worked on applications of interval valued fuzzy soft sets.
In this paper we extended the concept of GIVFSS and introduced generalized interval valued fuzzy soft matrix (GIVFSM). We defined different types of GIVFSM's and studied some properties. We also discussed some operators on the basis of weights and some of their properties.

Some Basic Definitions
Here µ j (p i ) denotes the membership of P i in the fuzzy soft set F (p j ).
Definition 2. 4. [13] Let X be the universal set, P be the set of parameters and A ⊆ P. Let (F & , P) be a GFSS over (X, P). A subset of X × P, R A = {(x, p), p ∈ P, x ∈ F & (p)} is a relation form of (F & , P), where ' ( : X × P → [0,1] and )R A : X × P → [0,1], such that ' ( : (x, p) ∈ [0, 1], ∀ x ∈ X, p ∈ P and ) (x i , p j )), then we can define an m × n generalized fuzzy soft matrix (GFSM) of GFSS over (X, P) as where M 7 x is an interval value and is called degree of membership of an element x to F(e) and 7 is called the degree of possibility of this belongness. Then F K is a GIVFSS.

Generalized Interval Valued Fuzzy Soft Matrices
Definition 3.1 Let U be the Universal set and E be the set of parameters. Let A ⊆ E and be a fuzzy subset of A. Let F: A → Ƥ U and : A → I = [0, 1] where Ƥ U is the collection of all Interval valued fuzzy subsets of U. A function F K : A → Ƥ U × I defined as Then the generalized interval valued fuzzy soft set F K can be expressed in matrix form as where MO7 P = MO7 A P , MO7 a P represents the membership of 7 in the GIVFSS F K 7 , such that 0 ≤ MO7 A P ≤ MO7 a P ≤ 1.
If MO7 P = 7 then the GIVFSM reduces to a GFSM.

Conclusion
We have introduced the concept of GIVFSM's in this paper. Some of its types are defined. Some basic operations like union, intersection, compliment, AND operation and OR operation have been defined and exemplified. Arithmetic mean, geometric mean, harmonic mean and their weighted means are also defined and some properties of these operators are discussed. Furthermore, similarity between two GIVFSM's is discussed. To future concern, GIVFSM's can be used to solve decision making problems in situations where uncertainty involved.