Intuitionistic Fuzzy Soft Matrix Theory

The purpose of this paper is to put forward the notion of intuitionistic fuzzy soft matrix theory and some basic results. In this paper, we define intuitionistic fuzzy soft matrices and have introduced some new operators with weights, some properties and their proofs and examples which make theoretical studies in intuitionistic fuzzy soft matrix theory more functional. Moreover, we have given one example on weighted arithmetic mean for decision making problem.


Introduction
Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, in real life, there are many complicated problems in engineering, economics, environment, social sciences medical sciences etc. that involve data which are not all always crisp, precise and deterministic in character because of various uncertainties typical problems. Such uncertainties are being dealing with the help of the theories, like theory of probability, theory of fuzzy sets, theory of intuitionistic fuzzy sets, theory of interval mathematics and theory of rough sets etc. Molodtsov [1] also described the concept of "Soft Set Theory" having parameterization tools for dealing with uncertainties. Researchers on soft set theory have received much attention in recent years. Maji and Roy [3] first introduced soft set into decision making problems. Maji et al. [6] introduced the concept of fuzzy soft sets by combining soft sets and fuzzy sets. Cagman and Enginoglu [4] defined soft matrices which were a matrix representation of the soft sets and constructed a soft max-min decision making method . Borah et al. [8] extended fuzzy soft matrix theory and its application. Deli and Cagmam[9] introduced intuitionistic fuzzy parameterized soft sets. They have also applied to the problems that contain uncertainties based on intuitionistic fuzzy parameterized soft sets. Babitha and John [11] described generalized intuitionistic fuzzy soft sets and solved multi criteria decision making problem in generalized intuitionistic fuzzy soft sets . Rajarajeswari and Dhanalakshmi[10] described intuitionistic fuzzy soft matrix with some traditional operations.
In this paper , we have introduced some operators on intuitionistic fuzzy soft matrix on the basis of weights . We have also discussed their properties . Finally we have given an elementary application on decision making problem on the basis of weighted arithmetic mean.

Soft Set [1]
Let U be an initial universe , P(U) be the power set of U , E be the set of all parameters and A⊆ E. A soft set ( ,E) on the universe U is defined by the set of order pairs ( ,E) = {(e , (e)) : e∈ E , (e) ∈ P(U) } where : E → P(U) such that (e) = if e ∉ A. Here is called an approximate function of the soft set ( ,E). The set (e) is called e-approximate value set or e-approximate set which consists of related objects of the parameter e ∈ E.
2 , 3 })} over U which describe the "colour of the shirts" which Mr. X is going to buy.
We may represent the soft set in the following form:

Fuzzy set [6]
Let U be an initial universe , E be the set of all parameters  We may represent the fuzzy soft set in the following form :

Fuzzy Soft Matrices (FSM) [5]
Let ( , E) be fuzzy soft set over U. Then a subset of U x E is uniquely defined by = { ( u , e) : e∈ A , u ∈ (e) }, which is called relation form of ( , E) . The characteristic function of is written by µ : U x E → [ 0 , 1] , where µ (u , e ) ∈ [ 0,1] is the membership value of u∈U for each e ∈U.
Therefore we can say that a fuzzy soft set ( ,E) is uniquely characterized by the matrix [µ ] and both concepts are interchangeable. Hence the fuzzy soft matrix [µ ] can be written as

Intuitionistic Fuzzy Soft Matrix ( IFSM) [5]
Let U be an initial universe , E be the set of parameters and A⊆ E and both concepts are interchangeable. The set of all m x n IFS matrices will be denoted by IFSM .

Intuitionistic Fuzzy Soft µ-Universal Matrix
An Intuitionistic fuzzy soft matrix of order m x n is said to be an Intuitionistic Fuzzy Soft µ-Universal Matrix if � = 1 and � = 0 for all i and j. It is denoted by ̃ .

Intuitionistic Fuzzy Soft ν -Universal Matrix
An Intuitionistic fuzzy soft matrix of order m x n is said to be an Intuitionistic Fuzzy Soft ν -Universal Matrix if � = 0 and � =1 for all i and j. It is denoted by ⏟ .

Intuitionistic Fuzzy Soft Super Matrix
for all i and j .

Intuitionistic Fuzzy Soft Equal Matrix
Then ̃ is said to be equal � denoted by ̃ = � if � = � and � = � for all i and j .

Union of Intuitionistic Fuzzy Soft Matrices
for all i and j .

Intersection of Intuitionistic Fuzzy Soft Matrices
for all i and j .
∈ IFSM . Then De Morgan's type results are true which can be written as: Proof: a) For all i and j we have, The result b) can be proved in similar way.

Intuitionistic Fuzzy Soft Square Matrix
Then ̃ is said to be Intuitionistic Fuzzy Soft Square Matrix if m=n for all i and j .

Intuitionistic Fuzzy Soft Row Matrix
Let ̃= [( � , � )] ∈ IFSM . Then ̃ is said to be Intuitionistic Fuzzy Soft Row Matrix if n=1 for all i and j.

Intuitionistic Fuzzy Soft Column Matrix
Then ̃ is said to be Intuitionistic Fuzzy Soft Column matrix if m=1for all i and j .

Intuitionistic Fuzzy Soft Diagonal Matrix
Then ̃ is said to be intuitionistic Fuzzy Soft Diagonal Matrix if m=n and i= j.

Operators of Intuitionistic Fuzzy Soft Matrices
= ̃ @ � □ Similar proof for others.

Proposition5.( Commutative law)
= � ∩ � ̃ Validity : Same as (i) Validity: Same as viii) when 1 = 2 Similarly , we can prove the other results and also the validity.

Proposition6.( Commutative laws on the base of weight)
Let = � @ ̃ □ Similar proof for others.

Application of Weighted Arithmetic Mean ( ) of Intuitionistic Fuzzy Soft Matrix in Decision Making
In this section , we define arithmetic mean and weighted arithmetic mean of intuitionistic fuzzy soft matrix.

Weighted Arithmetic Mean of Intuitionistic Fuzzy
Soft Matrix ( ) )] when weights are equal .
Input : Intuitionistic fuzzy soft sets with m objects , each of which has n parameters.
Output : An optimum result.

Algorithm:
Step-1: Choose the set of parameters.
Step -2: Construct the intuitionistic fuzzy soft matrix for the set of parameters.
Step-3: Compute the weighted arithmetic mean of membership and non-membership value of intuitionistic fuzzy soft matrix as .
Step-4: Choose the object with highest membership value. In case of tie i.e. when more than one object with same highest membership value , choose the object with highest membership value as well as lowest non-membership value.
Example 7. Suppose a company A produces five different types of cars 1 , 2 , 3 , 4 , 5 such that U = { 1 , 2 , 3 , 4 , 5 } and E = { 1 (comfort) , 2 ( good mileage ) , 3 ( good power steering ) } be a set of parameters . Suppose Mr. X is going to buy a car and an intuitionistic fuzzy soft matrix is constructed on the basis of the parameters as follows : If we prefer to "comfort" of the cars and weights .8 , .1 , .1 are given on the parameters "comfort" , "good mileage" , "good power steering" respectively, then From the above results (1) and (2) , it is clear that if we give equal preference , we have .667 is the highest membership value of (1) . So 4 car is most suitable for Mr. X . But if we give more preference on comfort than other parameters(good mileage, good power steering) ,then 2 car is the most suitable for Mr. X .

Conclusion
In this paper, we have introduced some new operators on the base of weights (weighted A.M, weighted G.M , weighted H.M ) and properties on intuitionistic fuzzy soft matrix. Some of the associative laws and distributive properties have been proved and verified with examples. Some of the commutative laws and idempotent laws on the base of weight have been proved. Lastly we have given one elementary application for decision making problem on the basis weighted arithmetic mean. This method can be applied on other decision making problem with uncertain parameters.