A Study on Matrices Using Interval Valued Intuitionistic Fuzzy Soft Set and Its Application in Predicting Election Results in India

: Nowadays the concept of matrix is used widely in different fields such as engineering, medical, economics, game theory, geology, computer science etc. Matrices are also used in representing the real world data like the population of people, infant mortality rate etc. In economics very large matrices are used for optimization of problems. Matrices play an important role to represent different types of soft set in concise form by which we can easily perform algebraic operations on them. Classical matrices can’t represent all types of uncertainties present in daily life problems. To tackle those problems related to uncertainties fuzzy matrix is introduced in which every member belongs to the unit interval [0, 1]. By combining soft set and fuzzy matrix a new concept fuzzy soft matrix is introduced. Later it has been extended to intuitionistic fuzzy soft matrix, interval-valued fuzzy soft matrix, interval-valued intuitionistic fuzzy soft matrix etc. In this paper we give a brief discussion on different types of interval valued intuitionistic fuzzy soft matrices and apply some new matrix operations on them. Moreover a new methodology has been developed to solve interval valued intuitionistic fuzzy soft set based real life decision making problems which may contain more than one decision maker and put an effort to apply it to a more relevant way in predicting election results in India by using the concept of choice matrix.


Introduction
Nowadays we dealt with problems which are complex in nature. To tackle those problems which we have experienced in our day to day life becomes complex more and more. And most of them deals with uncertainties i.e. in Mathematics the term 'vague' is usually used for such cases. Introduction of fuzzy set theory in 1965 by L. A. Zadeh [18] handled those problems based on uncertainties quite successfully at some extent. So we need some effective measures or tools which are capable to solve such problems. Some tools such as fuzzy set [18], L-fuzzy set [5], intuitionistic fuzzy set [1], interval valued fuzzy set [16], interval valued intuitionistic fuzzy set [2], rough set [13] etc are introduced earlier.
But due to the more complexity of the modern days problems, sometimes it is difficult to determine the membership or non membership value for each and every case. For this purpose, later on, another mathematical tool known as soft set is introduced by Molodtsov [12] in 1999. Soft set theory basically used for parametrizationin a data.
Many researchers and Mathematicians used soft set theory in multiple directions. Which leads to the notion of fuzzy soft set [10], intuitionistic fuzzy soft set [11], interval valued fuzzy soft set [4], interval valued intuitionistic fuzzy soft set [9] etc.
Moreover to reduce parameter set of a soft set, rough set is used. To store and manipulate data in a computer [3] have introduced the definition of soft matrices which are representations of soft sets. It has several advantages. Hashimoto [7] introduced the concept of fuzzy matrix in which every member belongs to the unit interval. By combining soft set and fuzzy matrix a new concept fuzzy soft matrix [17] is introduced. Then it has been extended to intuitionistic fuzzy soft matrix [15], interval-valued fuzzy soft matrix [19], interval valued intuitionistic fuzzy soft matrix [14] etc.
In this article we have proposed the notion of interval valued intuitionistic fuzzy soft matrix. Then we have defined its types with suitable examples. Here we have also proposed the concept of choice matrix associated with an interval valued intuitionistic fuzzy soft set. Furthermore we have introduced some new operations on interval valued intuitionistic fuzzy soft matrices and choice matrices. Then based on some of these new matrix operations a new efficient solution technique has been developed to solve interval valued intuitionistic fuzzy soft set based real life decision making problem which may contain more than one decision maker. The novelty of the new approach is that it may solve any interval valued intuitionistic fuzzy soft set based decision making problem involving large number of decision makers very easily and the computational procedure is also very simple. Finally to realize this newly proposed methodology we apply it to predict election results.

Preliminaries
Some important basic definitions with examples are discussed in the following: Definition 2.1 [12] Let U be an initial universe and E be a set of parameters.
Let ( )   [16] Let U be an initial universe and E be a set of parameters, a pair(F, E) is called an interval valued-fuzzy soft set over F(U), where F is a mapping given by F:E F(U), where F(U) is the set of all interval-valued fuzzy sets of U.
An interval-valued fuzzy soft set is a parameterized family of interval-valued fuzzy subsets of U, thus, its universe is the set of all interval-valued fuzzy sets of U, i.e. F(U). An interval-valued fuzzy soft set is also a special case of a soft set because it is still a mapping from parameters to F(U), e∈ , F(U) is referred as the interval fuzzy value set of parameters e, it is actually an interval-valued fuzzy set of U where x U ∈ and e E ∈ , it can be written as:   Table 2.
In Table 2, we can see that the precise evaluation for each object on each parameter is unknown while the lower and upper limits of such an evaluation is given. For example, we cannot present the precise degree of how beautiful house h 1 is, however, house h 1 is at least beautiful on the degree of 0.7 and it is at most beautiful on the degree of 0.9.
An interval-valued fuzzy soft set (F, A): intuitionistic fuzzy sets on U and Obviously, we can see that the precise evaluation for each object on each parameter is unknown while the lower and upper limits of such an evaluation are given. For example, we can't present the precise membership degree and non membership degree of how expensive house 1 h is, however, house 1 h is at least expensive on the membership degree of 0.6 and it is at most expansive on the membership degree of 0.8; house 1 h is not at least expensive on the nonmembership degree of 0.1 and it is not at most expansive on the non-membership degree of 0.2.
Definition 2.6 [3] Let ( ) , which is called a relation form of ( ) and then the relation form of ( )

Interval Valued Intuitionistic Fuzzy Soft Matrix
, which is called a relation set   , 11 11 , then we can define a matrix ( ) 11 12 1

Square Interval Valued Intuitionistic Fuzzy Soft Matrix
An interval valued intuitionistic fuzzy soft matrix corresponds to a interval valued intuitionistic fuzzy soft set of order m n × is called a square interval valued intuitionistic fuzzy soft matrix if m n = i.e number of objects= number of parameters otherwise it is called a rectangularinterval valued intuitionistic fuzzy soft matrix. For illustration, see example 3.1.1

Complement of an Interval Valued Intuitionistic Fuzzy Soft Matrix
Let ( )

Trace of an Interval Valued Intuitionistic Fuzzy Soft Matrix
Trace of an interval valued intuitionistic fuzzy soft matrix

Symmetric Interval Valued Intuitionistic Fuzzy Soft Matrix
A square interval valued intuitionistic fuzzy soft matrix ( ) * ij a of order n n × is said to be symmetric if ( ) Note: In our discussion there is no scope to define skew symmetric interval valued intuitionistic fuzzy soft matrix.

Choice Matrix with an Interval Valued Intuitionistic Fuzzy Soft Set
A matrix whose rows and columns both indicate parameters is called a choice matrix. If η is a choice matrix, then its elements ij η is defined as follows: ( ) ( ) The above combined matrix can also be represented in its transpose form.
Suppose we are thinking of another combined matrix associated with three decision makers. For this let Jr. Trump is willing to buy a dress together with Mr. Trump

Product of an Interval Valued Intuitionistic Fuzzy Soft Matrix with a Choice Matrix
Let U be the set of universe and E be the set of parameters. Suppose that A be any interval valued intuitionistic fuzzy soft matrix and C be any choice matrix of a decision maker concerned with the same U and E . Then the product of A and C is denoted by AC .

A New Technique to Solve Interval Valued Intuitionistic Fuzzy Soft Set Based Decision Making Problems
Here we introduce a new technique which is basically based on choice matrices. Choice matrices represent the choice parameters of the decision makers and it also help us to solve the interval valued intuitionistic fuzzy soft matrix based decision making problems with least computational complexity. Now at first we consider a generalized interval valued intuitionistic fuzzy soft set based decision making problem.

The Stepwise Solving Procedure
We consider the following stepwise procedure to solve such type of problems. Algorithm: Step-1: First construct the combined choice matrix with respect to the choice parameters of the decision makers.
Step-2: Compute the product interval valued intuitionistic fuzzy soft matrices by multiplying each given interval valued intuitionistic fuzzy soft matrix with the combined choice matrix as per the rule of multiplication of interval valued intuitionistic fuzzy soft matrices.
Step-3: Compute the sum of these product interval valued intuitionistic fuzzy soft matrices to have the resultant interval valued intuitionistic fuzzy soft matrix ( ) f R . Step Step-5: The object having the highest weight becomes the optimal choice object. If more than one object have the highest weight then go to next step. Step The sum of these products

Predicting Election Results in India
In India there are many factors that influence the voters to vote at the ballot box. One of the key factors is the personal attributes of the candidate, who is the person running for a political purpose, are of the most important characteristics that a person takes into account. Another is the political party to which the candidate belongs. There are also several important factors that how an individual votes. It is these factors and how they influence election results. For the following example, we consider three groups of people namely X, Y and Z to predict election results in India on the basis of one of these factors by using the algorithm discussed in section 4.2.
Example 5.1 Here we consider an example by which we can predict the 2019 West Lok Sabha Election in Tripura,India and it will be held on 11 th April .This election will be contested by the four political parties such as INC, IPFT, BJP and CPIM and their nominated candidates are Subal Bhowmik,Sukhla Chara n Noatia,Pratima Bhowmik and Shankar Prasad Datta Application in Predicting Election Results respectively and they are denoted by the symbols 1 2 3 4 , , and respectively. P P P P Let U be the set of political leaders in India and it is given as { } Now the problem is to find the political leader which is most favourite among these four for winning election from all the groups X, Y and Z.
The combined choice matrices of X, Y and Z in different forms are, [

Conclusion
In this paper first we have proposed the concept of interval valued intuitionistic fuzzy soft matrix and defined various types of matrices in interval valued intuitionistic fuzzy soft set theory with examples. At the end a new relevant solution procedure has been developed to solve interval valued intuitionistic fuzzy soft set based real life decision making problems which may contain more than one decision maker and to realize it in more effective way we also apply it to predict election results.