Research Article | Open Access

# Possibility Fuzzy Soft Set

**Academic Editor:**C. D. Lai

#### Abstract

We introduce the concept of possibility fuzzy soft set and its operation and study some of its properties. We give applications of this theory in solving a decision-making problem. We also introduce a similarity measure of two possibility fuzzy soft sets and discuss their application in a medical diagnosis problem.

#### 1. Introduction

Fuzzy set was introduced by Zadeh in [1] as a mathematical way to represent and deal with vagueness in everyday life. After that many authors have studied the applications of fuzzy sets in different areas (see Klir and Yuan [2]). Molodtsov [3] initiated the theory of 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, and so forth. Maji et al. [4, 5] have further studied the theory of soft sets and used this theory to solve some decision-making problems. They have also introduced the concept of fuzzy soft set, a more general concept, which is a combination of fuzzy set and soft set and studied its properties [6], and also Roy and Maji used this theory to solve some decision-making problems [7]. Alkhazaleh et al. [8] introduced soft multiset as a generalization of Molodtsov’s soft set. They also introduced in [9] the concept of fuzzy parameterized interval-valued fuzzy soft set and gave its application in decision making. Zhu and Wen in [10] incorporated Molodtsov’s soft set theory with the probability theory and proposed the notion of probabilistic soft sets. In [11] Chaudhuri et al. defined the concepts of soft relation and fuzzy soft relation and then applied them to solve a number of decision-making problems. Majumdar and Samanta [12] defined and studied the generalised fuzzy soft sets where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. In this paper, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [6] to the possibility fuzzy soft set. In our generalisation of fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Also we give some applications of the possibility fuzzy soft set in decision-making problem and medical diagnosis.

#### 2. Preliminaries

In this section, we recall some definitions and properties regarding fuzzy soft set and generalised fuzzy soft set required in this paper.

Let be a universe set, and let be a set of parameters. Let denote the power set of and .

*Definition 2.1 (see [3]). * pair is called a *soft set* over , where is a mapping given by . In other words, a soft set over is a parameterized family of subsets of the universe .

*Definition 2.2 (see [6]). *Let be an initial universal set, and let be a set of parameters. Let denote the power set of all fuzzy subsets of . Let pair is called a *fuzzy soft set* over where is a mapping given by .

The following definitions and propositions are due to Majumdar and Samanta [12].

*Definition 2.3. *Let be the universal set of elements, and let be the universal set of parameters. The pair will be called a soft universe. Let and be a fuzzy subset of , that is, , where is the collection of all fuzzy subsets of . Let be a function defined as follows:
Then is called a *generalized fuzzy soft set* (GFSS in short) over the soft universe . Here for each parameter indicates not only the degree of belongingness of the elements of in but also the degree of possibility of such belongingness which is represented by . So we can write as follows:
where are the degrees of belongingness and is the degree of possibility of such belongingness.

*Definition 2.4. *Let and be two GFSSs over . is said to be a generalised fuzzy soft subset of if (i) is a fuzzy subset of ;(ii) is also a fuzzy subset of , forall .In this case, we write .

*Definition 2.5. **Union* of two GFSSs and , denoted by , is a GFSS , defined as such that
where , , and is any norm.

*Definition 2.6. **Intersection* of two GFSSs and , denoted by , is a GFSS , defined as such that
where , , and is any norm.

*Definition 2.7. * GFSS is said to be a *generalised null fuzzy soft set*, denoted by , if such that , where , forall and forall .

*Definition 2.8. * GFSS is said to be a *generalised absolute fuzzy soft set*, denoted by , if such that , where , forall and forall .

Proposition 2.9. *Let be a GFSS over . Then the following holds:*(i)*,*(ii)*,*(iii)*,*(iv)*,*(v)*,*(vi)*.*

Proposition 2.10. *Let , and be any three GFSSs over . Then the following holds:*(i)*, *(ii)*,*(iii)*, *(iv)*.*

*Definition 2.11. **Similarity* between the two GFSSs and , denoted by , is defined as follows:
such that , where

*Definition 2.12. *Let and be two GFSSs over the same universe *.* We call the two GFSSs to be *significantly similar* if .

Proposition 2.13. *Let and be any two GFSSs over . Then the following holds:*(i)*,*(ii)*,*(iii)*,*(iv)*,*(v)*. *

#### 3. Possibility Fuzzy Soft Sets

In this section, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [6]. In our generalisation of fuzzy soft set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft set.

*Definition 3.1. *Let be the universal set of elements and let be the universal set of parameters. The pair will be called a soft universe. Let and be a fuzzy subset of , that is, , where is the collection of all fuzzy subsets of . Let be a function defined as follows:
Then is called a *possibility fuzzy soft set* (PFSS in short) over the soft universe . For each parameter indicates not only the degree of belongingness of the elements of in but also the degree of possibility of belongingness of the elements of in , which is represented by . So we can write as follows:
Sometime we write as . If , we can also have a PFSS .

*Example 3.2. *Let be a set of three blouses. Let be a set of qualities where = bright, = cheap, and = colourful, and let . We define a function as follows:
Then is a PFSS over . In matrix notation, we write

*Definition 3.3. *Let and be two PFSSs over . is said to be a *possibility fuzzy soft subset* (PFS subset) of , and one writes if(i) is a fuzzy subset of , forall,(ii) is a fuzzy subset of .

*Example 3.4. *Let be a set of three cars, and let be a set of parameters where = cheap, = expensive, and = red. Let be a PFSS over defined as follows:
Let be another PFSS over defined as follows:
It is clear that is a PFS subset of .

*Definition 3.5. *Let and be two PFSSs over . Then and are said to be *equal,* and one writes if is a PFS subset of and is a PFS subset of .

In other words, if the following conditions are satisfied:

(i) is equal to ,(ii)is equal to .

*Definition 3.6. * PFSS is said to be a *possibility null fuzzy soft set*, denoted by , if such that
where , and .

*Definition 3.7. * PFSS is said to be a *possibility absolute fuzzy soft set*, denoted by , if such that
where and .

*Example 3.8. *Let be a set of three blouses. Let be a set of qualities where = bright, = cheap, and = colorful, and let . We define a function which is a PFSS over defined as follows:
Then is a possibility null fuzzy soft set.

Let , and we define the function which is a PFSS over as follows:
Then is a possibility absolute fuzzy soft set.

*Definition 3.9. *Let be a PFSS over (. Then the *complement* of , denoted by , is defined by such that and , where is a fuzzy complement.

*Example 3.10. *Consider the matrix notation in Example 3.2:
By using the basic fuzzy complement, we have where

#### 4. Union and Intersection of PFSS

In this section, we introduce the definitions of union and intersection of PFSS, derive some properties, and give some examples.

*Definition 4.1. **Union* of two PFSSs and , denoted by , is a PFSS defined by
such that and where is an norm.

*Example 4.2. *Let and . Let be a PFSS defined as follows:
Let be another PFSS over defined as follows:
By using the basic fuzzy union, we have , where
Similarly we get
In matrix notation, we write

*Definition 4.3. **Intersection* of two PFSSs and , denoted by , is a PFSS defined by
such that and where is a fuzzy norm.

*Example 4.4. *Consider the Example 4.2 where and are PFSSs defined as follows:
By using the basic fuzzy intersection, we have , where
Similarly we get
In matrix notation, we write

Proposition 4.5. *Let , and be any three PFSSs over . Then the following results hold:*(i)*,*(ii)*,*(iii)*,
*(iv)*. *

*Proof. *The proof is straightforward by using the fact that fuzzy sets are commutative and associative.

Proposition 4.6. *Let be a PFSS over . Then the following results hold:*(i)*,*(ii)*,*(iii)*,
*(iv)*,
*(v)*,
*(vi)*. *

*Proof. *The proof is straightforward by using the definitions of union and intersection.

Proposition 4.7. *Let , and be any three PFSSs over . Then the following results hold:*(i)*,
*(ii)*. *

*Proof. *For all ,
We can use the same method in (i).

#### 5. AND and OR Operations on PFSS with Applications in Decision Making

In this section, we introduce the definitions of AND and OR operations on possibility fuzzy soft sets. Applications of possibility fuzzy soft sets in decision-making problem are given.

*Definition 5.1. *If and are two PFSSs then “ AND ”, denoted by is defined by
where , such that and .

*Example 5.2. *Suppose the universe consists of three machines , and there are three parameters which describe their performances according to certain specific task. Suppose a firm wants to buy one such machine depending on any two of the parameters only. Let there be two observations and by two experts defined as follows:
Then where
Similarly we get
In matrix notation, we have
Now to determine the best machine, we first mark the highest numerical grade (values with underline mark) in each row. Now the score of each of such machines is calculated by taking the sum of the products of these numerical grades with the corresponding possibility . The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs , as both the parameters are the same. Then the firm will select the machine with the highest score. Hence, they will buy machine (see Table 1).

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Score. Score. Score. |

*Definition 5.3. *If and are two PFSSs then “ OR ”, denoted by , is defined by
where , such that and .

*Example 5.4. *Let ; consider and as in Example 5.2. suppose now the firm wants to buy a machine depending on any one of two parameters. Then we have where
Similarly we get
In matrix notation, we have
Now to determine the best machine, we first mark the highest numerical grade (value with underline mark) in each row. Now the score of each of such machines is calculated by taking the sum of the products of these numerical grades with the corresponding possibility . The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs , as both the parameters are the same. Then the firm will select the machine with the highest score. Hence, they will buy the machine (see Table 2).

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Score |