Possibility neutrosophic soft sets with applications in decision making and similarity measure

In this paper, concept of possibility neutrosophic soft set and its operations are defined, and their properties are studied. An application of this theory in decision making is investigated. Also a similarity measure of two possibility neutrosophic soft sets is introduced and discussed. Finally an application of this similarity measure is given to select suitable person for position in a firm.

Authors

• 3 publications
• Bipolar Fuzzy Soft sets and its applications in decision making problem

In this article, we combine the concept of a bipolar fuzzy set and a sof...
03/23/2013 ∙ by Muhammad Aslam, et al. ∙ 0

• Generalized Neutrosophic Soft Set

In this paper we present a new concept called generalized neutrosophic s...
05/13/2013 ∙ by Said Broumi, et al. ∙ 0

• Neutrosophic soft sets with applications in decision making

We firstly present definitions and properties in study of Maji maji-2013...
05/30/2014 ∙ by Faruk Karaaslan, et al. ∙ 0

• Relations on FP-Soft Sets Applied to Decision Making Problems

In this work, we first define relations on the fuzzy parametrized soft s...
02/13/2014 ∙ by Irfan Deli, et al. ∙ 0

• New Trends in Neutrosophic Theory and Applications

Neutrosophic theory and applications have been expanding in all directio...
11/23/2016 ∙ by Florentin Smarandache, et al. ∙ 0

• Soft Neutrosophic Algebraic Structures and Their Generalization

Study of soft sets was first proposed by Molodtsov in 1999 to deal with ...
08/23/2014 ∙ by Florentin Smarandache, et al. ∙ 0

• Formal Model of Uncertainty for Possibilistic Rules

Given a universe of discourse X-a domain of possible outcomes-an experim...
03/20/2013 ∙ by Arthur Ramer, et al. ∙ 0

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1 Introduction

In this physical world problems in engineering, medical sciences, economics and social sciences the information involved are uncertainty in nature. To cope with these problems, researchers proposed some theories such as the theory of fuzzy set zadeh-1965 , the theory of intuitionistic fuzzy set atanassov-1986 , the theory of rough set paw-82 , the theory of vague set gau-1993 . In 1999, Molodtsov molodtsov-1999

initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties as different from these theories. A wide range of applications of soft sets have been developed in many different fields, including the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory. Maji et al.

maji-2002 ; maj-03sst applied soft set theory to decision making problem and in 2003, they introduced some new operations of soft sets. After Maji’s work, works on soft set theory and its applications have been progressing rapidly. see ali-09osnop ; aktas-2007 ; cagman-2010 ; cagman-2014 ; feng-2010 ; feng-2012 ; feng-2013 ; neo-2011 ; pei-2005 ; sezgin-2011 ; xia-2012 ; yang-08 ; zhu-2013 .

Neutrosophy has been introduced by Smarandache smarandache-2005a ; smarandache-2005b as a new branch of philosophy and generalization of fuzzy logic, intuitionistic fuzzy logic, paraconsistent logic. Fuzzy sets and intuitionistic fuzzy sets are characterized by membership functions, membership and non-membership functions, respectively. In some real life problems for proper description of an object in uncertain and ambiguous environment, we need to handle the indeterminate and incomplete information. But fuzzy sets and intuitionistic fuzzy sets don’t handle the indeterminant and inconsistent information. Thus neutrosophic set is defined by Samarandache smarandache-2005b , as a new mathematical tool for dealing with problems involving incomplete, indeterminacy, inconsistent knowledge.

Maji maji-2013 introduced concept of neutrosophic soft set and some operations of neutrosophic soft sets. Karaaslan karaaslan-2014 redefined concept and operations of neutrosophic soft sets as different from Maji’s neutrosophic soft set definition and operations. Recently, the properties and applications on the neutrosophic soft sets have been studied increasingly bro-13gns ; bro-13ins ; bro-2014 ; deli-14ivnss ; deli-14nsm ; sahin-2014 .

Alkhazaleh et al alkhazaleh-2011 were firstly introduced concept of possibility fuzzy soft set and its operations. They gave applications of this theory in solving a decision making problem and they also introduced a similarity measure of two possibility fuzzy soft sets and discussed their application in a medical diagnosis problem. In 2012, Bashir et al. bashir-2012 introduced concept of possibility intuitionistic fuzzy soft set and its operations and discussed similarity measure of two possibility intuitionistic fuzzy sets. They also gave an application of this similarity measure.

This paper is organized as follow: in Section 2, some basic definitions and operations are given regarding neutrosophic soft set required in this paper. In Section 3, possibility neutrosophic soft set is defined as a generalization of possibility fuzzy soft set and possibility intuitionistic fuzzy soft set introduced by Alkhazaleh alkhazaleh-2011 and Bashir bashir-2012 , respectively. In Section 4, a decision making method is given using the possibility neutrosophic soft sets. In Section 5, similarity measure is introduced between two possibility neutrosophic soft sets and in Section 6, an application related personal selection for a firm is given as regarding this similarity measure method.

2 Preliminary

In this paper, we recall some definitions, operation and their properties related to the neutrosophic soft set karaaslan-2014 ; maji-2013 required in this paper.

Definition 0.

A neutrosophic soft set (or namely ns-set) over is a neutrosophic set valued function from to . It can be written as

 f={(e,{⟨u,tf(e)(u),if(e)(u),ff(e)(u)⟩:u∈U}):e∈E}

where, denotes set of all neutrosophic sets over . Note that if , the element is not appeared in the neutrosophic soft set . Set of all ns-sets over is denoted by .

Definition 0.

karaaslan-2014 Let . is said to be neutrosophic soft subset of , if , , , . We denote it by . is said to be neutrosophic soft super set of if is a neutrosophic soft subset of . We denote it by .

If is neutrosophic soft subset of and is neutrosophic soft subset of . We denote it

Definition 0.

karaaslan-2014 Let . If and for all and for all , then is called null ns-set and denoted by .

Definition 0.

karaaslan-2014 Let . If and for all and for all , then is called universal ns-set and denoted by .

Definition 0.

karaaslan-2014 Let . Then union and intersection of ns-sets and denoted by and respectively, are defined by as follow

 f⊔g = {(e,{⟨u,tf(e)(u)∨tg(e)(u),if(e)(u)∧ig(e)(u),

and ns-intersection of and is defined as

 f⊓g = {(e,{⟨u,tf(e)(u)∧tg(e)(u),if(e)(u)∨ig(e)(u), ff(e)(u)∨fg(e)(u)⟩:u∈U}):e∈E}.
Definition 0.

karaaslan-2014 Let . Then complement of ns-set , denoted by , is defined as follow

 f~c={(e,{⟨u,ff(e)(u),1−if(e)(u),tf(e)(u)⟩:u∈U}):e∈E}.
Proposition 0.

karaaslan-2014 Let . Then,

1. and

Proposition 0.

karaaslan-2014 Let . Then

1. .

Proposition 0.

karaaslan-2014 Let . Then,

1. and

2. and

3. and

4. and

5. and

6. and

Proof.

The proof is clear from definition and operations of neutrosophic soft sets.

Theorem 10.

karaaslan-2014 Let . Then, De Morgan’s law is valid.

Definition 0.

karaaslan-2014 Let . Then ’OR’ product of ns-sets and denoted by , is defined as follow

 f⋁g = {((e,e′),{⟨u,tf(e)(u)∨tg(e)(u),if(e)(u)∧ig(e)(u), ff(e)(u)∧fg(e)(u)⟩:u∈U}):(e,e′)∈E×E}.
Definition 0.

karaaslan-2014 Let . Then ’AND’ product of ns-sets and denoted by , is defined as follow

 f⋀g = {((e,e′),{⟨u,tf(e)(x)∧tg(e)(u),if(e)(u)∨ig(e)(u), ff(e)(u)∨fg(e)(u)⟩:u∈U}):(e,e′)∈E×E}.
Proposition 0.

karaaslan-2014 Let . Then,

Proof.

The proof is clear from Definition 11 and 12. ∎

Definition 0.

alkhazaleh-2011 Let be the universal set of elements and 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:

 Fμ(e)=(F(e)(u),μ(e)(u)),∀u∈U.

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 .

Definition 0.

bashir-2012 Let be the universal set of elements and be the universal set of parameters. The pair will be called a soft universe. Let where is the collection of all intuitionistic fuzzy subsets of and is the collection of all intuitionistic fuzzy subsets of . Let be a fuzzy subset of , that is, and let be a function defined as follows:

 Fp(e)=(F(e)(u),p(e)(u)),F(e)(u)=(μ(u),ν(u))∀u∈U.

Then is called a possibility intuitionistic fuzzy soft set (PIFSS 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 .

3 Possibility neutrosophic soft sets

In this section, we introduced the concept of possibility neutrosophic soft set, possibility neutrosophic soft subset, possibility neutrosophic soft null set, possibility neutrosophic soft universal set and possibility neutrosophic soft set operations such as union, intersection and complement.

Throughout paper is an initial universe, is a set of parameters and is an index set.

Definition 0.

Let be an initial universe, be a parameter set, be the collection of all neutrosophic sets of and is collection of all fuzzy subset of . A possibility neutrosophic soft set -set over

is defined by the set of ordered pairs

 fμ={(ei,{(ujf(ei)(uj),μ(ei)(uj)):uj∈U}):ei∈E}

where, , is a mapping given by and is a fuzzy set such that . Here, is a mapping defined by .

For each parameter , indicates neutrosophic value set of parameter and where are the membership functions of truth, indeterminacy and falsity respectively of the element . For each and , . Also , degrees of possibility of belongingness of elements of in . So we can write

From now on, we will show set of all possibility neutrosophic soft sets over with such that is parameter set.

Example 0.

Let be a set of three cars. Let be a set of qualities where cheap, equipment, fuel consumption and let . We define a function as follows:

also we can define a function as follows:

For the purpose of storing a possibility neutrosophic soft set in a computer, we can use matrix notation of possibility neutrosophic soft set . For example, matrix notation of possibility neutrosophic soft set can be written as follows: for ,

 fμ=⎛⎜⎝(⟨0.5,0.2,0.6⟩,0.8)(⟨0.7,0.3,0.5⟩,0.4)(⟨0.4,0.5,0.8⟩,0.7)(⟨0.8,0.4,0.5⟩,0.6)(⟨0.5,0.7,0.2⟩,0.8)(⟨0.7,0.3,0.9⟩,0.4)(⟨0.6,0.7,0.5⟩,0.2)(⟨0.5,0.3,0.7⟩,0.6)(⟨0.6,0.5,0.4⟩,0.5)⎞⎟⎠

where the

th row vector shows

and th column vector shows .

Definition 0.

Let , . Then, is said to be a possibility neutrosophic soft subset -subset of , and denoted by , if

1. is a fuzzy subset of , for all

2. is a neutrosophic subset of ,

Example 0.

Let be a set of tree houses, and let be a set of parameters where modern, big and cheap. Let be a -set defined as follows:

be another -set defined as follows:

it is clear that is of

Definition 0.

Let . Then, and are called possibility neutrosophic soft equal set and denoted by , if and .

Definition 0.

Let . Then, is said to be possibility neutrosophic soft null set denoted by , if , such that , where and .

Definition 0.

Let . Then, is said to be possibility neutrosophic soft universal set denoted by , if , such that , where and .

Let and . Then,

1. and

Proof.

It is clear from Definition 20, 21 and 22. ∎

Definition 0.

Let , where and for all , . Then for and ,

1. is said to be truth-membership part of ,
and ,

2. is said to be indeterminacy-membership part of ,
and ,

3. is said to be truth-membership part of ,
and ,

We can write a possibility neutrosophic soft set in form .

If considered the possibility neutrosophic soft set in Example 17, can be expressed in matrix form as follow:

 ftμ=⎛⎜⎝(0.5,0.8)(0.7,0.4)(0.4,0.7)(0.8,0.6)(0.5,0.8)(0.7,0.4)(0.6,0.2)(0.5,0.6)(0.6,0.5)⎞⎟⎠
 fiμ=⎛⎜⎝(0.2,0.8)(0.3,0.4)(0.5,0.7)(0.4,0.6)(0.7,0.8)(0.3,0.4)(0.7,0.2)(0.3,0.6)(0.5,0.5)⎞⎟⎠
 ffμ=⎛⎜⎝(0.6,0.8)(0.5,0.4)(0.8,0.7)(0.5,0.6)(0.2,0.8)(0.9,0.4)(0.5,0.2)(0.7,0.6)(0.4,0.5)⎞⎟⎠
Definition 0.

schweirer-1960 A binary operation is continuous norm if satisfies the following conditions

1. is commutative and associative,

2. is continuous,

3. , ,

4. whenever and .

Definition 0.

schweirer-1960 A binary operation is continuous conorm (s-norm) if satisfies the following conditions

1. is commutative and associative,

2. is continuous,

3. , ,

4. whenever and .

Definition 0.

Let and . Then be a lattices together with partial ordered relation , where order relation on can be defined by for

 (a,b,c)⪯(e,f,g)⇔a≤e,b≥f,c≥g
Definition 0.

A binary operation

 ~⊗:([0,1]×[0,1]×[0,1])2→[0,1]×[0,1]×[0,1]

is continuous norm if satisfies the following conditions

1. is commutative and associative,

2. is continuous,

3. , , ,

4. whenever and .

Here,

 a~⊗b=~⊗(⟨t(a),i(a),f(a)⟩,⟨t(b),i(b),f(b)⟩)=⟨t(a)⊗t(b),i(a)⊕i(b),f(a)⊕f(b)
Definition 0.

A binary operation

 ~⊕:([0,1]×[0,1]×[0,1])2→[0,1]×[0,1]×[0,1]

is continuous conorm if satisfies the following conditions

1. is commutative and associative,

2. is continuous,

3. , , ,

4. whenever and .

Here,

 a~⊕b=~⊕(⟨t(a),i(a),f(a)⟩,⟨t(b),i(b),f(b)⟩)=⟨t(a)⊕t(b),i(a)⊗i(b),f(a)⊗f(b)
Definition 0.

Let The union of two possibility neutrosophic soft sets and over , denoted by is defined by as follow:

 fμ∪gν={(ei,{(uj(ftij(ei)⊕gtij(ei),fiij(ei)⊗giij(ei),ffij(ei)⊗gfij(ei)),μij(ei)⊕νij(ei)):uj∈U}):ei∈E}
Definition 0.

Let The intersection of two possibility neutrosophic soft sets and over , denoted by is defined by as follow:

 fμ∩gν={(ei,{(uj(ftij(ei)⊗gtij(ei),fiij(ei)⊕giij(ei),ffij(ei)⊕gfij(ei)),μij(ei)⊗νij(ei)):uj∈U}):ei∈E}
Example 0.

Let us consider the possibility neutrosophic soft sets and defined as in Example 17. Let us suppose that norm is defined by and the conorm is defined by for . Then,

and

Let . Then,

1. and

2. and

3. and

4. and

5. and

6. and

Proof.

The proof can be obtained from Definitions 30. and 31. ∎

Definition 0.

fodor-1994 ; trillas-1979 A function is called a negation if , and is non-increasing . A negation is called a strict negation if it is strictly decreasing and continuous. A strict negation is said to be a strong negation if it is also involutive, i.e.

Definition 0.

smarandache-2005a A function is called a negation if ,