Application of TOPSIS Method in Decision Making Via Soft Set

Application of TOPSIS Method in Decision Making Via Soft Set. Abstract Molodtsov introduced a general mathematical theory, to addressing uncertain and undefined objects which is known as soft set theory. In this work, we study soft set with some definitions and operations, also discuss a soft set technique for order preference by similarity to ideal solution (TOPSIS) method and use this method for decision making.


Introduction
In recent time, classical methods are used to solve different problems which are faced in medical sciences, engineering, social sciences, etc. Many theories like fuzzy set theory, probability theory, rough set theory, etc. are used to solve these uncertainties, but these theories have their own problems which were attensioned by Zadeh. First of all, (Zadeh) in [1] initiated the concept of fuzzy sets by the extension of classical notion on sets. Nowadays fuzzy set theory is rapidly progressing, but cases, we face some limitations such as how to adjust the membership functions in this theory.
Molodtsov proposed the concept of soft set to solve those problems which contain uncertainty in 1999 and defines a soft set as a parameterized family of subsets of universe set where each element is considered as a set of approximate elements of the soft set in the past few years of fundamentals of soft set theory studied by various researchers [2]. Various potential application in soft set in many areas like in smoothness of functions game theory operation research Riemann integration theory of measurement and so on are highlighted in Molodtsov. In these days, mathematician plays a vital role in the soft set fuzzification. After the fuzzification of soft set, a new theory introduced which is known as a fuzzy soft set with different types and properties Maji et al. [3]. The soft set theory was reviewed [4] and used this theory for decision making, they also introduced different types of soft set with examples and defined some operations such as And-operation, Or-operation, union, intersection, complement, etc. on soft set [5]. In Chen et al. [6] redefine the idea of parameterization reduction in a soft set and compare it with a rough set, they also improve the applications of soft set by using a new definition in the decision-making problem. Some new notions on soft set Ali et al. [7] are introduced, such as restricted union, restricted intersection, restricted difference and extended intersection with examples and properties. Yeşim et al. used the fuzzy TOPSIS method [8] to select a supplier out of three suppliers in a company of garment operating in Turkey. Jahanshahloo et al. extended the approach of the TOPSIS method for decision making by the help of fuzzy data [9].
They also introduced a new concept of normalized fuzzy numbers with the help of α-cuts and applied this proposed concept on numerical problems. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) introduced in [10] to solve multi criteria decision making problems with different alternatives.
In [11] Chen & Hwang extend the idea of the TOPSIS method and presented a new model for TOPSIS. He also used the newly proposed method for decision making to solve uncertain data in [12]. The authors used this method for the prediction of Diabetes patients in medical diagnosis. In [13][14][15], the authors presented the applications of fuzzy TOPSIS in different areas and used for decision making. Zulqarnain M and Saeed M [16] used the interval-valued fuzzy soft matrix in decision making. They also introduced a new decision-making method in [17,18] on interval-valued fuzzy soft matrix. In this paper, we discuss about soft set TOPSIS method and used this method for the selection of best candidates for bank jobs.

Definition: 2.1 [2]
A pair (F, E) is called a soft set over a given universal set U, if and only if F is mapping of a set of parameters E, into the power set of U. i.e.

F E P U →
Clearly, an soft set over U is a parameterized family of subsets of a given universe U.
Also, for any p ϵ E, F (p) is considered the set of p -approximate element of the soft set (F, P)

Definition 2.2 [2]
Let (F, A) and (G, B) are two soft sets over the same universe U, For all t A ∈ , F(t) and G(t) are an identical approximation

Definition 2.3 [5]
(F, A) and (G, B) are called equal soft sets if ( ) ( ) . It can be written as

Definition 2.4 [19]
Let (F, A) and (G, B) are two soft sets over U, then their union is defined as

Definition 2.5 [19]
Let (F, A) and (G, B) are two soft sets over U, then their intersection is defined as

Definition 2.6 [5]
Let (F, A) and (G, B) are two soft sets over the same universe U, then their restricted difference is defined as

Definition 2.7 [19,20]
A soft set (F, A) over a universe U is said to be a null soft set

Definition 2.8 [2]
A soft set (F, A) over a universe U is called absolute soft set

Step 2: Calculation of the Normalized Decision Matrix
The normalized decision matrix (NDM) denotes the normalized values which represent the relative performance of the alternatives.   Where J ′ is associated with the non-beneficial attributes and J is associated with beneficial attributes.
Step Step 6: Relative Closeness to the Ideal Solution.
The relative closeness of the ideal solution is computed as

Step 7: Ranking of Preference Order
Ranking is done based on the values of l C the higher value of the relative closeness has a high rank and hence the better performance of the alternative. Rank the preference in descending order to compare the better performances of alternatives" .    where ϒ (Y i ) decide the values of y i , so the decision matrix is ii

A Decision-Making Method on Soft Set by TOPSIS
Step 5: Ranking the preference order Ranking is done based on the values of ϒ (Y i ) the higher the value of the relative closeness has high rank and hence the better performance of the candidates. Rank the preference in descending order to compare the better performances of candidates.

Application of soft set by TOPSIS method a) Step 1
A bank manager wants to select two candidates out of from five which is our universal set  Table 2) Now we construct the soft set for selectors P i (Table 3) Then decision matrix is    Ranking among the candidates would be created in the order in descending order of the values ϒ (Y j ) calculated in the fifth step.
So, when the fifth step in the calculation of the evaluation of the candidate from small to large ϒ ( the order form is realized in the form of ranking Y 1 ˃ Y 5 ˃ Y 2 ˃ Y 4 ˃ Y 3 . In other words, we can say that Y 1 and Y 5 are the best candidates for the bank job.

Conclusion
In this paper, we discuss the soft set and TOPSIS method along with some definitions and examples. Secondly, we discuss the soft set TOPSIS method. Finally, we apply the soft set TOPSIS method for the selection of two best candidates for the bank job.