 # Some Hybrid Soft Sets and Their Application in Decision Making

Pinaki Majumdar (M. U. C. Women's College, India)
DOI: 10.4018/978-1-4666-9798-0.ch006

## Abstract

In this chapter the notions of two hybrid soft sets, namely fuzzy parameterized soft sets and vague soft sets has been discussed and some operations on them are defined. Some properties of these two hybrid soft sets under these operations are also studied. Application of fuzzy parameterized soft sets and vague soft sets in decision making has also been shown. Then the notion of similarity measure between two hybrid soft sets has been introduced and their basic properties are studied. Applications of these similarity measures in various problems like medical diagnosis and decision making have been shown. The idea of entropy of a fuzzy parameterized soft set has been introduced.
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## Introduction

There are several techniques to handle various types of uncertainties prevailing in this physical world. There are theories namely theory of probability, theory of fuzzy sets (Zadeh, 1965), theory of multisets, theory of rough sets (Pawlak, 1982), theory of vague sets (Gau & Buehrer,1993) etc. which can handle uncertainties of various types. In 1999, Molodtsov (1999) initiated the theory soft sets as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, etc. Later other authors like Maji, Biswas and Roy (2003) have further studied the theory of soft sets and used this theory to solve some decision making problems (Maji & Roy, 2002). They have also introduced the concept of fuzzy soft set, a more generalised concept, which is a combination of fuzzy set and soft set and studied its properties (Maji, Biswas & Roy, 2001). Recently, Wei, Xu., et. al. (2010) has coined the idea of vague soft sets which are combinations of vague set and soft sets. Vague soft sets are capable of having more strength to handle uncertain information. Feng et. al. (2010), Kong et. al. (2008, 2009) and many other authors has defined several hybrid versions of soft sets and applied them in decision making. Currently research work in soft set theory is progressing very rapidly and works in many directions like algebraic structures (Aktas & Cagman, (2007), Aygunoglu & Aygun, (2009)), Jun & Park, (2008)), topology (Muhammad Shabir & Naz, (2011), Hazra, Majumdar & Samanta (2012)), soft relation (Babitha & Sunil, (2010), Chaudhuri, De, & Chatterjee, (2009)), soft mapping (Majumdar & Samanta, 2010), similarity measure (Majumdar & Samanta (2008, 2011), Majumdar (2011), Pappis & Karacapilidis, (1993)) etc. are being done. Here we have presented two types of hybrid soft sets namely fuzzy parameter set and vague soft set. On the other hand similarity has been a key concept in a number of domains, such as linguistics, physiology, computational intelligence, pattern recognition, decision making etc. In several problems it is often needed to compare two sets. The sets may be fuzzy, may be vague etc. We often interested to know whether two patterns or images are identical or approximately identical or at least to what degree they are identical. Several researchers like Chen (1995, 1997), Li and Xu (2001), Hong and Kim (1999), C.P. Pappis (1991, 1993) etc. has studied the problem of similarity measurement between fuzzy sets, fuzzy numbers and vague sets. In the paper by Grzegorzewski (2004), several types of distances between intuitionistic fuzzy sets have been given. Majumdar & Samanta (2008, 2010, 2011) have studied the techniques of similarity measurement between two soft sets, fuzzy soft sets and intuitionistic fuzzy soft sets. We have extended this concept of similarity in vague soft sets. Again decision making is a problem involving the determination of optimal strategies when a decision maker is faced with many alternatives regarding the uncertainty of some future event. Recently many authors have used soft sets and fuzzy soft sets for decision making. In 2009, Chaudhuri et al. have used fuzzy soft relations in decision making. Feng Feng et. al. (2010) has also studied fuzzy soft sets based decision making. Also fuzzy soft set based forecasting method and certain flood alarm models (Kalayathankal & Singh, (2010), Maji & Roy, (2002)) etc. has been designed by many authors. Majumdar & Samanta (2008, 2010, 2011) have used similarity measurement techniques for detection of diseases.

## Key Terms in this Chapter

Decision Making: The technique of selecting a logical choice from the available options.

Similarity Measure of Soft Sets: Amount of similarity between two soft sets.

Entropy: It is a measure of uncertainty expressed by a set.

Fuzzy Set: A fuzzy set F of U is defined by the mapping F : U ? I = [0, 1].

Hausdorff metric: For any two subsets U and W of a Banach space Z the Hausdorff metric is:

Fuzzy Soft Set: Let IU denote the power set of all fuzzy subsets of U. Let A pair ( F , A ) is called a fuzzy soft set over U, where F is a mapping given by F : U ? I U .

Fuzzy Parameterized Soft Set: Let U be an initial universal set and let E be a set of parameters. Then the pair ( U , E ) will be called a soft universe. Let P ( U ) denote the power set of U and µ E be a fuzzy set over E . A triplet ( U , µ E , F µ ) is called a fuzzy parameterized soft set (FPS set in short) over ( U , E ) iff F µ is a mapping given by and is defined as follows: For where

Level Set: Let ( U , µ E , F µ ) be a FPS set over ( U , E ). Let L a : I ' = [0, 1) ? P ( U ) be a mapping be defined as follows: Then are called level sets of ( U , µ E , F µ ).

Vague Soft Sets: Let U be an initial universe set. A vague set A over U can be expressed as: where t v , f v : U ? [0, 1], here t v ( u i ) is a lower bound on the grade of membership of u i and f v ( u i ) is a lower bound on the negation of u i with t v ( u i ) + f v ( u i ) = 1. The grade of membership of u i in the vague set A is bounded by a subinterval [ t A ( u i ), 1 – f A ( u i )] of [0, 1] called the vague value. The vague value [ t A ( u i ), 1 – f A ( u i )] means that the exact grade of membership µ A ( u i ) is bounded by t A ( u i ) = µ A ( u i ) = 1 – f A ( u i ).

Soft Set: A soft set is a parametrized family of subsets of U . A soft set over the soft universe ( U , E ) is denoted by ( F , A ), where F : A ? P ( U ), where and P ( U ) is the power set of U .

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