On statistical concepts of intuitionistic fuzzy soft set theory via utility

: In this paper, we establish the foundation of intuitionistic fuzzy soft statistics with the help of utility theory of mathematical economics. We use ideas of ( 𝛼, 𝛽 ) -cut with respect to utility theory to prove results related to intuitionistic fuzzy soft mean, intutionistic fuzzy soft covariance, intuitionistic fuzzy soft attribute correlation coefﬁcients, etc. Suitable examples are provided in each case. Concepts of utility-wise representation of intuitionistic fuzzy soft have been discussed. Here, we also discuss the generating process of a new intuitonistic fuzzy soft set from the old one with respect to utility theory and prove some important theorems.


Introduction
Mathematical theories are based on various abstract ideas. Here, one has freedom to develop certain abstract environments by neglecting many facts; for example in physics, the frictional effect of air on a free falling body is often neglected to make the calculations easier, but this fact is fully impossible in real life. Similarly, medical science, economics, engineering, social sciences, etc., are full of uncertainties. Zadeh [19] initiated the study of uncertainties with the introduction of fuzzy sets in 1965. Later, Atanassov introduced intuitionistic fuzzy set theory [5]. Molodtsov [14] introduced the concept of soft set theory in the year 1999 and investigated various applications in game theory, smoothness of functions, operation researches, Perron integration, probability theory, theory of measurement, etc.
Later Maji et al. [10] defined fundamental operations of soft sets. Pei and Miao [16], Chen [7] pointed out errors in some of the results of Maji et al. [11] and introduced some new notions and properties. At present, investigations of different properties and applications of soft set theory have attracted many researchers from various backgrounds. Since then many applications of soft set theory can be found in other branches of science and social science. Fuzzy soft set was introduced by Maji et al. [10] as a hybrid structure of soft set with fuzzy set. Later, Intuitionistic fuzzy soft sets were introduced by Maji et al. [12]. One may refer to Mitchell [13], Szmidt and Kacprzyk [17], Huang [8] for researches related to correlation coefficients on intuitionistic fuzzy sets. One may refer to [9,18] for some works on fuzzy soft sets.
Applications of uncertainty-based statistical ideas related to sociological issues viz. human trafficking and illegal immigration can found in Acharjee and Mordeson [3], Mordeson et al. [15], Acharjee et al. [4]. Moreover, hybrid structures related to soft set can be found in Acharjee [1], Acharjee and Tripathy [2], and many others.
In this paper, we establish the foundation of intuitionistic fuzzy soft statistics. Here, we try to connect two different domains of nature, i.e., uncertainties, which are present in large scale data, and preference (i.e., utility based on human choice behavior) by developing statistical ideas based on intuitionistic fuzzy soft set theory. Utility theory is applicable in various domains of computational social sciences and information systems. One may find uses of utility theory in computational social choice theory, mathematical psychology, robotics, decision making, etc. It is to be understood that our statistical ideas connect attributes, linguistic variables, [0,1], etc. and, thus, it may have potentiality of applications in various areas of science and social science.

Preliminaries
The following definitions are due to Ç agman et al. [6].
Definition 2.1. [6] A soft set on the universe is defined by the set of ordered pairs is called an approximate function of the soft set . The value of ( ) may be arbitrary. Some of them may be empty, some may have nonempty intersection. We will denote the set of all soft sets over as ( ).

Definition 2.2. [6] Let
∈ ( ). If ( ) = ∅ for all ∈ , then is called a soft empty set, denoted by ∅ . ( ) = ∅ means there is no element in related to the parameter ∈ . Therefore, we do not display such elements in the soft sets, as it is meaningless to consider such parameters.
[6] Let us consider a universe = { , , } and Then, the representation of ( , ) in tabular form is shown in Table 1:  Then, 1 − ( ) − ( ) is called the hesitancy degree of the element ∈ to the set ; denoted by ( ). ( ) is called the intuitionistic index of to . Greater ( ) indicates more vagueness on . Obviously, when ( ) = 0 ∀ ∈ , the IFS generates into an ordinary fuzzy set. In the sequel, all IFSs of is denoted by IFSs( ). Definition 2.9. [5] For ∈ IFSs( ) and ∈ IFSs( ), some relations between them are defined as: where is the complement of .

Main results
In this section, we introduce utility based statistical concepts in intuitionistic fuzzy soft sets. Throughout this paper, we shall write IFSS and IFS in short to represent "intuitionistic fuzzy soft set" and "intuitionistic fuzzy set", respectively. We shall denote = {1, 2, 3, . . . , }

Some new definitions
Definition 3.1. If ( , ) be an IFSS over a universe , where ( ) is an IFS for the attribute ∈ , ∈ , then the intuitionistic fuzzy soft mean of ( , Here, and are membership value and non-membership value of , respectively, for the attribute , where ∈ . Here, 1 ( )( ) indicates the membership value of in ℎ place of ( ) ∀ ∈ Δ. Similarly, 2 ( )( ) indicates the non-membership value of in ℎ place of ( ) ∀ ∈ Δ.
Definition 3.4. Let be a universe and be the set of attributes, where ⊆ and | | = . If ( , ) be an IFSS over , then ( , )-cut IFSS standard deviation of ( , ) is denoted by ) and it is defined as Thus, The above result indicates that standard deviation of (0.

Concept of utility-wise representation of intuitionistic fuzzy soft set
Consider an IFSS ( , ) over a universe and R is the set of real numbers. We define ancut level utility function : → R as ⪰ ⇐⇒ ( ) ≥ ( ) for , ∈ and so on with fundamental notions of utility theory. Similarly, we define a -cut level utility function : → R as ⪰ ⇐⇒ ( ) ≥ ( ) for , ∈ . Together we call them as ( , )-cut level utility function.

Generating process of a new intuitionistic fuzzy soft set
from the old one with respect to utility based ( , )-cut (2) Let the scale of ( , ) be ℎ. Then, similarly as discussed above, we can construct  We define -cut level utility function as Similarly, we define -cut level utility function as : → R as ( ) = , such that ∈ ( ) with non-membership value min {non-membership value of in ( ), } if Thus, Similarly, Hence, proved. Thus,