A Real Time Application of Soft Set in Parameterization Reduction for Decision Making Problem

ABSTRACT


INTRODUCTION
There are many fields like economics, engineering, environment, social science and medical science which involve data of uncertainties. The traditional mathematical tools are not able to handle such type of uncertainties. The researchers used the existing traditional tools such as probability theory, Fuzzy set theory, [9][10][11] theory of mathematics, rough set theory etc. But these theories are not much successful to deal with uncertainty because they all have their own difficulties. One reason for these difficulties may be due to the inadequacy of the parameterization tools [1]. Molodtsov [2] developed the concept of soft set theory as an effective mathematical tool for dealing with uncertainties and non-crisp data. Soft set theory is free from the difficulties that have troubled the usual theoretical approaches. He pointed out several directions for the applications of soft set. Soft Set is called (binary, basic, elementary) neighborhood systems [3]. The soft set [10], [11] is a mapping from parameter to the crisp subset of universe. The structure of soft set can classify the objects into two classes such as yes/1/true or no/0/false [4]. This statement shows that a standard soft set deals with Boolean valued systemsMethod" to describe the step of research and used in the chapter "Results and Discussion" to support the analysis of the results [2]. If the manuscript was written really have high originality, which proposed a new method or algorithm, the additional chapter after the "Introduction" chapter and before the "Research Method" chapter can be added to explain briefly the theory and/or the proposed method/algorithm [4].

SOFT SET FUNDAMENTALS
In this section soft set theory fundamentals are explained.

Information System in Soft Set
In relational database relations are known as tables and a relation is the combination of rows and columns. In the same way information systems are similar to the relations on relational database. An information systems of a soft set is a quadruple i.e. 4-tuples [1], [4], [5] S = (U, A, V, f) [1] where, U-Non empty finite set of objects Non empty finite set of attributes V-Value set of attributes f-Information function An information System is also known as knowledge representation system or an attribute valued system.

Introduction to Softset Theory
Soft set is a simple mathematical tool that deals with a collection of approximate behavior or description of objects. It is a parameterized tool to deal with approximate value and uncertainty. Each approximate description can divide in two parts. First one is a predicate part and second one is an approximate value set [1]. In classical mathematics, a mathematical model of an object is constructed and defines the notion of an exact solution of this model. A classical model is generally used to find the exact solutions i.e. this type of model is used for precise value. In mathematical model it is a tough task to obtain the exact solution for the imprecise values. So, it is more reliable and easy to define an approximate solution first and then the solution is calculated. The soft set theory has given an opposite approach to solve these types of problems. In the initial description of an object it has an approximate value and it does not required to introduce the notion of exact solution. These types of conditions on approximate behavior are not present in soft set theory. This property makes soft set theory reliable, simple and comfortable to apply in any real world applications. With the help of words, sentences, numbers, functions the parameterization of soft set can be expressed.
In the whole paper U indicates to the universe, E is defined by a set of parameters and P (U) is the power set of U. Definition 1. ( [1], [4]). A pair (F, E) is called a soft set over U, where F is a mapping given by

( )
In other words U is universe in soft set and it is a parameterized family of subsets of universe U. for , F(e) may be considered as the set of e elements of the soft set (F, E) as the set of e approximate elements of soft set [1]. Ultimately, soft set is a non-crisp set Definition 2. (See [1]). Let R be a family of equivalence relations and let . It is say that [4]. Otherwise A is indispensable in R. The family R is Independent if each is indispensable in R. otherwise R is dependent. Q subset of P is a reduction of P if Q is independent and IND (Q) = IND (P), that is to say Q is the minimal subset of P that keeps the classification ability [7]. The set of all indispensable relations in P is called the core of P, and is denoted as CORE (P). Clearly, CORE (P) = ∩RED (P), where RED (P) is the family of all reductions of P [2], [7], [10].
Definition 3. (See [1], [8]). Let (F, E) be a soft set over the universe U and . A parameter co-occurrence set of an object u can be defined as [8]: Support of an object u is defined by:

EXAMPLE OF REDUCTION IN SOFT SET
In this example there is soft set (F, E) which describes the "nature of Bikes to purchase" that a customer is supposed to make a necessary action as a decision to buy. In the showroom there are five types of bikes are ready to sale. The nature of different types of bikes is Small bike, Medium bike, Expensive bike, Second-hand bike, and imported bike. Suppose we have six bikes which are under consideration, U = {b1, b2, b3, b4, b5, b6} where, b1, b2, b3, b4, b5, b6 are bikes under consideration and E is a set of decision parameters, E = {e1, e2, e3, e4, e5} [1] where, For the parameter "small bike", we have e1 For the parameter "medium bike", we have e2 For the parameter "expensive bike", we have e3 For the parameter "Second-hand bike", we have e4 For the parameter "imported bike", we have e5 Consider the mapping ( )given by "bikes (.)", Here (.) is to be filled in by one of parameters [1] Suppose that F (e1) = {b2, b3, b4, b5}, F (e2) = {b1, b6}, F (e3) = {b1, b2, b6}, F (e4) = {b1, b2, b3, b4, b5, b6}, F (e5) = {b1, b2, b3, b4, b5, b6}. Therefore, F (e1) means "the size of bikes are small", its functional value is the set is F (e1) = {b2, b3, b4, b5,}. It is the soft set (F, E) as a collection of approximations as: The soft set (F, P) is can be expressed as a binary table, as shown Table 1, to solve this problem [1]. For this hij= 1 if hi F (ej) then hij= 1, otherwise hij= 0, where hij are the entries in Table 1.
So a soft set can now be known as a knowledge representation system. In this system we generally use set of parameters instead of set of attributes. Every approximation can divided into two parts, first a predicate part e and second one is approximate value set p [1]. for example, for the approximation small bike={b2,b3,b4,b5}, have the predicate name of bikes with small size and the value set is {b2,b3,b4,b5} [1].

Result and Discussion
In this step we identify that how parameters are dispensable [1] so after that we work out to reduce the dimension of data for this task parameters can be removed without affected the original decisions [1].
Let us consider the representation of soft set (F, P) in tabular form. Suppose Q is a reduction of P, then the new soft set (F, Q) is called the reduct soft set of the soft set (F, P) [1]. Algorithm: Suppose Mr. XYZ selects bike: The process of selecting a bike may has three basic steps a. Soft set theory can be used to transformed the dataset into a Boolean-valued information (as Table 1) system as S= (U, A, V (0, 1), f) soft set theory is used for this conversion [1]. b. The next process is Input Input the soft set (F, E), Input the set P of choice parameters of Mr. XYZ which is a subset of E. c. At first the data set must be reduced by removing dispensable items before make a decision.
We used the soft set theory reduce parameters [1] [4]. Find all reduct-soft-sets of (F, P), choose one reduct-soft-set say (F, Q) of (F, P).

Explanation
It  Table 1 it is proved that e4 and e5 are not relevant and is affect the choices of the bike since they take the same values for every bike.
So the Mr. XYZ is ready to make a decision to purchase bike based on the parameters {e1, e2, e3}. We can apply Definition 3 to partition the objects based on the parameter co-occurrence and the support value [1].
For the small sample set both methods produced the same reducts. So this task is important when dealing with a large data set. This method or algorithm is also beneficial to measures different types of performances of a data set. Feature selection is one of the important measures by using this method. IJECE ISSN: 2088-8708 

PERFORMANCE OF SOFT SET OVER OTHER APPROACHES
We can not use classical approaches to solve complicated problems in engineering, economics because of various uncertainties in these area. There are three theories which can consider as mathematical tools for dealing with uncertainties [2]. These theories are probability, fuzzyset and interval mathematics. Each theory has its own difficulty. Theory of probability can deal only with stochastically stable problems [2]. Interval mathematics deals the errors of calculations by constructing an interval estimate for the exact solution of a problem. This method is useful in many cases but this theory is not sufficiently adaptablefor problem with different uncertainties [2]. Fuzzy set is ver powerful tool to deal with complicated problems but there exists a difficulty how to set the membership function in each particular case [2].
The concept of soft theory is a mathematical tool for dealing with uncertainties which is free from the above difficulties. It is a parameterized tool to deal with approximate value and uncertainty. Each approximate description can divide in two parts. First one is a predicate part and second one is an approximate value set [1].

CONCLUSION
In this paper, we discussed data uncertainty and non-crisp data. We used Soft set theory to convert a small data set to binary valued data set. We also discussed how to reduce the unnecessary parameters as reducts from the data set. The loss of these parameters does not affect the original information of the used data set. So Soft set theory is also used in dimensionality reduction and generally used to provide better and quick decision making in compare of previous theories.