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Article

Applications of Neutrosophic Bipolar Fuzzy Sets in HOPE Foundation for Planning to Build a Children Hospital with Different Types of Similarity Measures

by
Raja Muhammad Hashim
1,
Muhammad Gulistan
1,* and
Florentin Smarandache
2
1
Department of Mathematics, Hazara University, Mansehra 21120, Pakistan
2
Department of Mathematics, University of New Mexico, Albuquerque, NM 87301, USA
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(8), 331; https://doi.org/10.3390/sym10080331
Submission received: 8 July 2018 / Revised: 24 July 2018 / Accepted: 3 August 2018 / Published: 9 August 2018

Abstract

:
In this paper we provide an application of neutrosophic bipolar fuzzy sets in daily life’s problem related with HOPE foundation that is planning to build a children hospital, which is the main theme of this paper. For it we first develop the theory of neutrosophic bipolar fuzzy sets which is a generalization of bipolar fuzzy sets. After giving the definition we introduce some basic operation of neutrosophic bipolar fuzzy sets and focus on weighted aggregation operators in terms of neutrosophic bipolar fuzzy sets. We define neutrosophic bipolar fuzzy weighted averaging ( N B FWA ) and neutrosophic bipolar fuzzy ordered weighted averaging ( N B FOWA ) operators. Next we introduce different kinds of similarity measures of neutrosophic bipolar fuzzy sets. Finally as an application we give an algorithm for the multiple attribute decision making problems under the neutrosophic bipolar fuzzy environment by using the different kinds of neutrosophic bipolar fuzzy weighted/fuzzy ordered weighted aggregation operators with a numerical example related with HOPE foundation.
MSC:
(2010 Mathematics Subject Classifications) 62C05; 62C86; 03B52; 03E72; 90B50; 91B06; 91B10; 46S40; 47H99

1. Introduction

Zadeh [1] started the theory of fuzzy set and since then it has been a significant tool in learning logical subjects. It is applied in many fields, see [2]. There are numbers of over simplifications/generalization of Zadeh’s fuzzy set idea to interval-valued fuzzy notion [3], intuitionistic fuzzy set [4], L-fuzzy notion [5], probabilistic fuzzy notion [6] and many others. Zhang [7,8], provided the generality of fuzzy sets as bipolar fuzzy sets. The extensions of fuzzy sets with membership grades from [ 1 , 1 ] , are the bipolar fuzzy sets. The membership grade [ 1 , 0 ) of a section directs in bipolar fuzzy set that the section fairly fulfils the couched stand-property, the membership grade ] 0 , 1 ] of a section shows that the section fairly fulfils the matter and the membership grade 0 of a section resources that the section is unrelated to the parallel property. While bipolar fuzzy sets and intuitionistic fuzzy sets aspect parallel to one another, they are really distinct sets (see [3]). When we calculate the place of an objective in a universe, positive material conveyed for a collection of thinkable spaces and negative material conveyed for a collection of difficult spaces [9]. Naveed et al. [10,11,12], discussed theoretical aspects of bipolar fuzzy sets in detail. Smarandache [13], gave the notion of neutrosophic sets as a generalization of intutionistic fuzzy sets. The applications of Neutrosophic set theory are found in many fields (see http://fs.gallup.unm.edu/neutrosophy.htm). Recently Zhang et al. [14], Majumdar et al. [15], Liu et al. [16,17], Peng et al. [18] and Sahin et al. [19] have discussed various uses of neutrosophic set theory in deciding problems. Now a days, neutrosophic sets are very actively used in applications and MCGM problems. Bausys and Juodagalviene [20], Qun et al. [21], Zavadskas et al. [22], Chan and Tan [23], Hong and Choi [24], Zhan et al. [25] studied the applications of neutrosophic cubic sets in multi-criteria decision making in different directions. Anyhow, these approaches use the maximum, minimum operations to workout the aggregation procedure. This leads to subsequent loss of data and, therefore, inaccurate last results. How ever this restriction can be dealt by using famous weighted averaging (WA) operator [26] and the ordered weighted averaging (OWA) operator [27]. Medina and Ojeda-Aciego [28], gave t-notion lattice as a set of triples related to graded tabular information explained in a non-commutative fuzzy logic. Medina et al. [28] introduces a new frame work for the symbolic representation of informations which is called to as signatures and given a very useful technique in fuzzy modelling. In [29], Nowaková et al., studied a novel technique for fuzzy medical image retrieval (FMIR) by vector quantization (VQ) with fuzzy signatures in conjunction with fuzzy S-trees. In [30] Kumar et al., discussed data clustering technique, Fuzzy C-Mean algorithem and moreover Artificial Bee Colony (ABC) algorithm. In [31] Scellato et al.,discuss the rush of vehicles in urban street networks. Recently Gulistan et al. [32], combined neutrosophic cubic sets and graphs and gave the concept of neutrosophic cubic graphs with practical life applications in different areas. For more application of neutrosophic sets, we refer the reader to [33,34,35,36,37]. Since, the models presented in literature have different limitations in different situations. We mainly concern with the following tools:
(1)
Neutrosophic sets are the more summed up class by which one can deal with uncertain informations in a more successful way when contrasted with fuzzy sets and all other versions of fuzzy sets. Neutrosophic sets have the greater adaptability, accuracy and similarity to the framework when contrasted with past existing fuzzy models.
(2)
And bipolar fuzzy sets are proved to very affective in uncertain problems which can characterized not only the positive characteristics but also the negative characteristics of a certain problem.
We try to blend these two concepts together and try to develop a more powerful tool in the form of neutrosophic bipolar fuzzy sets. In this work we initiate the study of neutrosophic bipolar fuzzy sets which are the generalization of bipolar fuzzy sets and neutrosophic sets. After introducing the definition we give some basic operations, properties and applications of neutrosophic bipolar fuzzy sets. And the rest of the paper is structured as follows; Section 2 provides basic material from the existing literature to understand our proposal. Section 3 consists of the basic notion and properties of neutrosophic bipolar fuzzy set. Section 4 gives the role of weighted aggregation operator in terms of neutrosophic bipolar fuzzy sets. We define neutrosophic bipolar fuzzy weighted averaging operator ( N B FWA ) and neutrosophic bipolar fuzzy ordered weighted averaging N B FOWA operators. Section 5 includes different kinds of similarity measures. In Section 6, an algorithm for the multiple attribute decision making problems under the neutrosophic bipolar fuzzy environment by using the different kinds of similarity measures of neutrosophic bipolar fuzzy sets and neutrosophic bipolar fuzzy weighted/fuzzy ordered weighted aggregation operators is proposed. In Section 7, we provide a daily life example related with HOPE foundation, which shows the applicability of the algorithm provided in Section 6. In Section 8, we provide a comparison with the previous existing methods. In Section 9, we discuss conclusion and some future research directions.

2. Preliminaries

Here we provide some basic material from the literature for subsequent use.
Definition 1.
Let Y be any nonempty set. Then a bipolar fuzzy set [7,8], is an object of the form
B = u , μ + ( u ) , μ ( u ) : u Y ,
and μ + u : Y 0 , 1 and μ u : Y 1 , 0 , μ + ( u ) is a positive material and μ ( u ) is a negative material of u Y . For simplicity, we donate the bipolar fuzzy set as B = μ + , μ in its place of B = u , μ + ( u ) , μ ( u ) : u Y .
Definition 2.
Let B 1 = μ 1 + , μ 1 and B 2 = μ 2 + , μ 2 be two bipolar fuzzy sets [7,8], on Y . Then we define the following operations.
(1) 
B 1 = 1 μ 1 + u , 1 μ 1 u ;
(2) 
B 1 B 2 = max ( μ 1 + u , μ 2 + u ) , min ( μ 1 u , μ 2 u ) ;
(3) 
B 1 B 2 = min ( μ 1 + u , μ 1 + u ) , max ( μ 1 u , μ 2 u ) .
Definition 3.
A neutrosophic set [13], is define as:
L = x , Tru L ( x ) , Ind L ( x ) , Fal L ( x ) : x X ,
where X is a universe of discoveries and L is characterized by a truth-membership function Tru L : X ] 0 , 1 + [ , an indtermency-membership function Ind L : X ] 0 , 1 + [ and a falsity-membership function Fal L : X ] 0 , 1 + [ such that 0 Tru L ( x ) + Ind L ( x ) + Fal L ( x ) 3 .
Definition 4.
A single valued neutrosophic set [16], is define as:
L = x , Tru L ( x ) , Ind L ( x ) , Fal L ( x ) : x X ,
where X is a universe of discoveries and L is characterized by a truth-membership function Tru L : X [ 0 , 1 ] , an indtermency-membership function Ind L : X [ 0 , 1 ] and a falsity-membership function Fal L : X [ 0 , 1 ] such that 0 Tru L ( x ) + Ind L ( x ) + Fal L ( x ) 3 .
Definition 5.
Let [16]
L = x , Tru L ( x ) , Ind L ( x ) , Fal L ( x ) : x X ,
and
B = x , Tru B ( x ) , Ind B ( x ) , Fal B ( x ) : x X ,
be two single valued neutrosophic sets. Then
(1) 
L B if and only if Tru L ( x ) Tru B ( x ) , Ind L ( x ) Ind B ( x ) , Fal L ( x ) Fal B ( x ) .
(2) 
L = B if and only if Tru L ( x ) = Tru B ( x ) , Ind L ( x ) = Ind B ( x ) , Fal L ( x ) = Fal B ( x ) , for any x X .
(3) 
The complement of L is denoted by L c and is defined by
L c = x , Fal L ( x ) , 1 Ind L ( x ) , Tru L ( x ) / x X .
(4) 
The intersection
L B = x , min Tru L ( x ) , Tru B ( x ) , max Ind L ( x ) , Ind B ( x ) , max Fal L ( x ) , Fal B ( x ) : x X .
(5) 
The Union
L B = x , max Tru L ( x ) , Tru B ( x ) , min Ind L ( x ) , Ind B ( x ) , min Fal L ( x ) , Fal B ( x ) : x X .
Definition 6.
Let A ˜ 1 = Tru 1 , Ind 1 , Fal 1 and A ˜ 2 = Tru 2 , Ind 2 , Fal 2 be two single valued neutrosophic number [16]. Then, the operations for NNs are defined as below:
(1) 
λ A ˜ = 1 ( 1 Tru 1 ) λ , Ind 1 λ , Fal 1 λ ;
(2) 
A ˜ 1 λ = Tru 1 λ , 1 ( 1 Ind 1 ) λ , 1 ( 1 Fal 1 ) λ ;
(3) 
A ˜ 1 + A ˜ 2 = Tru 1 + Tru 2 Tru 1 Tru 2 , Ind 1 Ind 2 , Fal 1 Fal 2 ;
(4) 
A ˜ 1 A ˜ 2 = Tru 1 Tru 2 , Ind 1 + Ind 2 Ind 1 Ind 2 , Fal 1 + Fal 2 Fal 1 Fal 2 where λ > 0 .
Definition 7.
Let A ˜ 1 = Tru 1 , Ind 1 , Fal 1 be a single valued neutrosophic number [16]. Then, the score function s ( A ˜ 1 ) , accuracy function L ( A ˜ 1 ) , and certainty function c ( A ˜ 1 ) , of an NNs are define as under:
(1) 
s ( A ˜ 1 ) = ( Tru 1 + 1 Ind 1 + 1 Fal 1 ) 3 ;
(2) 
L ( A ˜ 1 ) = Tru 1 Fal 1 ;
(3) 
c ( A ˜ 1 ) = Tru 1 .

3. Neutrosophic Bipolar Fuzzy Sets and Operations

In this section we apply bipolarity on neutrosophic sets and initiate the notion of neutrosophic bipolar fuzzy set with the help of Section 2, which is the generalization of bipolar fuzzy set. We also study some basic operation on neutrosophic bipolar fuzzy sets.
Definition 8.
A neutrosophic bipolar fuzzy set is an object of the form N B = ( N B + , N B ) where
N B + = u , Tru N B + , Ind N B + , Fal N B + : u Y , N B = u , Tru N B , Ind N B , Fal N B : u Y ,
where Tru N B + , Ind N B + , Fal N B + : Y 0 , 1 and Tru N B , Ind N B , Fal N B : Y 1 , 0 .
Note: In the Definition 8, we see that a neutrosophic bipolar fuzzy sets N B = ( N B + , N B ) , consists of two parts, positive membership functions N B + and negative membership functions N B . Where positive membership function N B + denotes what is desirable and negative membership function N B denotes what is unacceptable. Desirable characteristics are further characterize as: Tru N B + denotes what is desirable in past, Ind N B + denotes what is desirable in future and Fal N B + denotes what is desirable in present time. Similarly Tru N B denotes what is unacceptable in past, Ind N B denotes what is unacceptable in future and Fal N B denotes what is unacceptable in present time.
Definition 9.
Let N 1 B = ( N 1 B + , N 1 B ) and N 2 B = ( N 2 B + , N 2 B ) be two neutrosophic bipolar fuzzy sets. Then we define the following operations:
(1) 
N 1 B c = 1 Tru N 1 B + , 1 Ind N 1 B + , 1 Fal N 1 B + and 1 Tru N 1 B , 1 Ind N 1 B , 1 Fal N 1 B ;
(2) 
N 1 B N 2 B = max ( Tru N 1 B + , Tru N 2 B + ) , max ( Ind N 1 B + , Ind N 2 B + ) , min ( Fal N 1 B + , Fal N 2 B + ) , max ( Tru N 1 B , Tru N 2 B ) , max ( Ind N 1 B , Ind N 2 B ) , min ( Fal N 1 B , Fal N 2 B ) ;
(3) 
N 1 B N 2 B = min ( Tru N 1 B + , Tru N 2 B + ) , min ( Ind N 1 B + , Ind N 2 B + ) , max ( Fal N 1 B + , Fal N 2 B + ) , min ( Tru N 1 B , Tru N 2 B ) , min ( Ind N 1 B , Ind N 2 B ) , max ( Fal N 1 B , Fal N 2 B ) . .
Definition 10.
Let N 1 B = ( N 1 B + , N 1 B ) and N 2 B = ( N 2 B + , N 2 B ) be two neutrosophic bipolar fuzzy sets. Then we define the following operations:
(1) 
N 1 B + N 2 B + = Tru N 1 B + + Tru N 2 B + Tru N 1 B + · Tru N 2 B + , Ind N 1 B + + Ind N 2 B + Ind N 1 B + · Ind N 2 B + , ( Fal N 1 B + · Fal N 2 B + ) ,
and
N 1 B N 2 B = Tru N 1 B + Tru N 2 B Tru N 1 B · Tru N 2 B , Ind N 1 B + Ind N 2 B Ind N 1 B · Ind N 2 B , ( Fal N 1 B · Fal N 2 B ) ;
(2) 
N 1 B + N 2 B + = Tru N 1 B + · Tru N 2 B + , Ind N 1 B + · Ind N 2 B + , Fal N 1 B + + Fal N 2 B + ( Fal N 1 B + · Fal N 2 B + ) ,
and
N 1 B N 2 B = Tru N 1 B · Tru N 2 B , Ind N 1 B · Ind N 2 B , Fal N 1 B + Fal N 2 B ( Fal N 1 B · Fal N 2 B ) ;
(3) 
N 1 B + N 2 B + = min ( Tru N 1 B + , Tru N 2 B + ) , min ( Ind N 1 B + , Ind N 2 B + ) , max ( Fal N 1 B + , Fal N 2 B + ) ,
and
N 1 B N 2 B = min ( Tru N 1 B , Tru N 2 B ) , min ( Ind N 1 B , Ind N 2 B ) , max ( Fal N 1 B , Fal N 2 B ) .
Definition 11.
Let N B = ( N B + , N B ) be a neutrosophic bipolar fuzzy set and λ 0 . Then,
(1) 
λ N B + = 1 ( 1 Tru N B + ) λ , 1 ( 1 Ind N B + ) λ , Fal N B + λ , λ N B = 1 ( 1 Tru N B ) λ , 1 ( 1 Ind N B ) λ , Fal N B λ .
(2) 
N B + λ = ( Tru N B + ) λ , ( Ind N B + ) λ , 1 + 1 + Fal N B + λ , N B λ = ( Tru N B ) λ , ( Ind N B ) λ , 1 + 1 + Fal N B ( u ) λ .
Theorem 1.
Let N 1 B = ( N 1 B + , N 1 B ) , N 2 B = ( N 2 B + , N 2 B ) and N 3 B = ( N 3 B + , N 3 B ) be neutrosophic bipolar fuzzy sets. Then, the following properties hold:
(1) 
Complementary law: ( N 1 B c ) c = N 1 B .
(2) 
Idempotent law:
( i ) N 1 B N 1 B = N 1 B , i i N 1 B N 1 B = N 1 B .
(3) 
Commutative law:
i N 1 B N 2 B = N 2 B N 1 B , i i N 1 B N 2 B = N 2 B N 1 B , i i i N 1 B N 2 B = N 2 B N 1 B , i v N 1 B N 2 B = N 2 B N 1 B .
(4) 
Associative law:
i ( N 1 B N 2 B ) N 3 B = N 1 B ( N 2 B N 3 B ) , i i ( N 1 B N 2 B ) N 3 B = N 1 B N 2 B N 3 B , i i i ( N 1 B N 2 B ) N 3 B = N 1 B ( N 2 B N 3 B ) , i v ( N 1 B N 2 B ) N 3 B = N 1 B ( N 2 B N 3 B ) .
(5) 
Distributive law:
i N 1 B ( N 2 B N 3 B ) = ( N 1 B N 2 B ) ( N 1 B N 3 B ) , i i N 1 B ( N 2 B N 3 B ) = ( N 1 B N 2 B ) ( N 1 B N 3 B ) , i i i N 1 B ( N 2 B N 3 B ) = ( N 1 B N 2 B ) ( N 1 B N 3 B ) , i v N 1 B ( N 2 B N 3 B ) = ( N 1 B N 2 B ) ( N 1 B N 3 B ) , ( v ) N 1 B ( N 2 B N 3 B ) = ( N 1 B N 2 B ) ( N 1 B N 3 B ) , v i N 1 B ( N 2 B N 3 B ) = ( N 1 B N 2 B ) ( N 1 B N 3 B ) .
(6) 
De Morgan’s laws:
i ( N 1 B N 2 B ) c = N 1 B c N 2 B c , i i N 1 B N 2 B c = N 1 B c N 2 B c , i i i ( N 1 B N 2 B ) c N 1 B c N 2 B c , i v N 1 B N 2 B c N 1 B c N 2 B c .
Proof. 
Straightforward. ☐
Theorem 2.
Let N 1 B = ( N 1 B + , N 1 B ) and N 2 B = ( N 2 B + , N 2 B ) be two neutrosophic bipolar fuzzy sets and let N 3 B = N 1 B N 2 B and N 4 B = λ N 1 B λ > 0 . Then both N 3 B and N 4 B are also neutrosophic bipolar fuzzy sets.
Proof. 
Straightforward. ☐
Theorem 3.
Let N 1 B = ( N 1 B + , N 1 B ) and N 2 B = ( N 2 B + , N 2 B ) be two neutrosophic bipolar fuzzy sets, λ , λ 1 , λ 2 > 0 . Then, we have:
( i ) λ ( N 1 B N 2 B ) = λ N 1 B λ N 2 B , ( i i ) λ 1 N 1 B λ 2 N 2 B = ( λ 1 λ 2 ) N 1 B .
Proof. 
Straightforward. ☐

4. Neutrosophic Bipolar Fuzzy Weighted/Fuzzy Ordered Weighted Aggregation Operators

After defining neutrosophic bipolar fuzzy sets and some basic operations in Section 3. We in this section as applications point of view we focus on weighted aggregation operator in terms of neutrosophic bipolar fuzzy sets. We define ( N B FWA ) and N B FOWA operators.
Definition 12.
Let N j B = ( N j B + , N j B ) be the collection of neutrosophic bipolar fuzzy values. Then we define N B FWA as a mapping N B FWA k : Ω n Ω by
N B FWA k N 1 B , N 2 B , , N n B = k 1 N 1 B k 2 N 2 B , , k n N n B .
If k = 1 n , 1 n , , 1 n then the N B FWA operator is reduced to
N B FA N 1 B , N 2 B , , N n B = 1 n N 1 B N 2 B , , N n B .
Theorem 4.
Let N j B = ( N j B + , N j B ) be the collection of neutrosophic bipolar fuzzy values. Then
N B FWA k N 1 B + , N 2 B + , , N j B + = 1 Π j = 1 n 1 Tru N j B + k j , 1 Π j = 1 n 1 Ind N j B + k j , Π j = 1 n Fal N j B + k j N B FWA k N 1 B , N 2 B , , N j B = 1 Π j = 1 n 1 Tru N j B k j , 1 Π j = 1 n 1 Ind N j B k j , Π j = 1 n Fal N j B k j .
Proof. 
Let N j B = ( N j B + , N j B ) be a collection of neutrosophic bipolar fuzzy values. We first prove the result for n = 2 . Since
k 1 N L B + = 1 1 Tru N L B + k 1 , 1 1 Ind N L B + k 1 , ( Fal N L B + ) k 1 , k 1 N L B = 1 1 Tru N L B k 1 , 1 1 Ind N L B k 1 , ( Fal N L B ) k 1 , k 1 N b B + = 1 1 Tru N b B + k 2 , 1 1 Ind N b B + k 2 , ( Fal N b B + ) k 2 , k 1 N b B + = 1 1 Tru N b B k 2 , 1 1 Ind N b B k 2 , ( Fal N b B ) k 2 ,
then
N B FWA k N L B , N b B = k 1 N 1 B k 2 N 2 B , N B FWA k N L B + , N b B + = k 1 N 1 B + k 2 N 2 B + , N B FWA k N L B , N b B = k 1 N 1 B k 2 N 2 B , N B FWA k N L B + , N b B + = 2 1 Tru N L B + k 1 1 Tru N b B + k 2 1 1 Tru N L B + k 1 × 1 1 Tru N b B + k 2 , 2 1 Ind N L B + k 1 1 Ind N b B + k 2 1 1 Ind N L B + k 1 × 1 1 Ind N b B + k 2 , ( Fal N L B + ) k 1 ( Fal N b B + ) k 2 , N B FWA k N L B + , N b B + = 1 1 Tru N L B + k 1 1 Tru N b B + k 2 , 1 1 Ind N L B + k 1 1 Ind N b B + k 2 , ( Fal N L B + ) k 1 ( Fal N b B + ) k 2 ,
N B FWA k N L B , N b B = k 1 N 1 B k 2 N 2 B , N B FWA k N L B , N b B = 2 1 Tru N L B k 1 1 Tru N b B k 2 1 1 Tru N L B k 1 × 1 1 Tru N b B k 2 , 2 1 Ind N L B k 1 1 Ind N b B k 2 1 1 Ind N L B k 1 × 1 1 Ind N b B k 2 , ( Fal N L B ) k 1 ( Fal N b B ) k 2 , N B FWA k N L B , N b B = 1 1 Tru N L B k 1 1 Tru N b B k 2 , 1 1 Ind N L B k 1 1 Ind N b B k 2 , ( Fal N L B ) k 1 ( Fal N b B ) k 2 .
So N B FWA k N L B , N b B = k 1 N 1 B k 2 N 2 B . If result is true for n = k , that is
N B FWA k N 1 B + , N 2 B + , , N j B + = 1 Π j = 1 k 1 Tru N J B + k j , 1 Π j = 1 k 1 Ind N J B + k j , Π j = 1 k Fal N j B + k j , N B FWA k N 1 B , N 2 B , , N j B = 1 Π j = 1 k 1 Tru N J B k j , 1 Π j = 1 k 1 Ind N J B k j , Π j = 1 k Fal N j B k j ,
then, when k + 1 , we have
N B FWA k N 1 B + , N 2 B + , , N j B + = 1 Π j = 1 k 1 Tru N j B + k j + 1 1 Tru N k + 1 B + k k + 1 ( 1 Π j = 1 k 1 Tru N j B + k j ) × 1 1 Tru N k + 1 B + k k + 1 , 1 Π j = 1 k 1 Ind N j B + k j + 1 1 Ind N k + 1 B + k k + 1 ( 1 Π j = 1 k 1 Ind N j B + k j ) × 1 1 Ind N k + 1 B + k k + 1 , Π j = 1 k + 1 Fal N j B + k j = 1 Π j = 1 k + 1 1 Tru N j B + k j , 1 Π j = 1 k + 1 1 Ind N j B + k j , Π j = 1 k + 1 Fal N j B + k j ,
N B FWA k N 1 B , N 2 B , , N j B = 1 Π j = 1 k 1 Tru N j B k j + 1 1 Tru N k + 1 B k k + 1 ( 1 Π j = 1 k 1 Tru N j B + k j ) × 1 1 Tru N k + 1 B k k + 1 , 1 Π j = 1 k 1 Ind N j B k j + 1 1 Ind N k + 1 B k k + 1 ( 1 Π j = 1 k 1 Ind N j B + k j ) × 1 1 Ind N k + 1 B k k + 1 , Π j = 1 k + 1 Fal N j B k j = 1 Π j = 1 k + 1 1 Tru N j B k j , 1 Π j = 1 k + 1 1 Ind N j B k j , Π j = 1 k + 1 Fal N j B k j .
So result holds for n = k + 1 .  ☐
Theorem 5.
Let N j B = ( N j B + , N j B ) be the collection of neutrosophic bipolar fuzzy values and k = k 1 , k 2 , , k n T is the weight vector of N j B j = 1 , 2 , , n , with k j 0 , 1 and Σ j = 1 n k j = 1 . Then we have the following:
(1) 
(Idempotency): If all N j B j = 1 , 2 , , n are equal, i.e., N j B = N j B , for all j, then
N B FWA k N 1 B , N 2 B , , N n B = N B .
(2) 
(Boundary):
N B N B FWA k N 1 B , N 2 B , , N n B N B + , for every k .
(3) 
(Monotonicity) If Tru N j B + Tru N j B + * , Ind N j B + Ind N j B + * and Fal N j B Fal N j B * , for all j, then
N B FWA k N 1 B , N 2 B , , N n B N B FWA k N 1 * B , N 2 * B , , N n * B , for every k .
Definition 13.
Let N j B = ( N j B + , N j B ) be the N B FWA be a collection of neutrosophic bipolar fuzzy values. An neutrosophic bipolar fuzzy O W A ( N B FOWA ) operator of dimension is a mapping N B FOWA : Ω n Ω defined by
N B FOWA k N 1 B + , N 2 B + , , N n B + = k 1 N σ 1 B + k 2 N σ 2 B + , , k n N σ n B + , N B FOWA k N 1 B , N 2 B , , N n B = k 1 N σ 1 B k 2 N σ 2 B , , k n N σ n B ,
where σ 1 , σ 2 , , σ n is a permutation of 1 , 2 , , n such that N σ j 1 B N σ j B for all j. If k = 1 n , 1 n , , 1 n T then B F O W A operator is reduced to B F A operator having dimension n.
Theorem 6.
Let N j B = ( N j B + , N j B ) be the collection of neutrosophic bipolar fuzzy values. Then
N B FOWA k N 1 B + , N 2 B + , , N n B + = 1 Π j = 1 n 1 Tru N σ j B + k j , 1 Π j = 1 n 1 Ind N σ j B + k j , Π j = 1 n Tru N σ j B + k j , N B FOWA k N 1 B + , N 2 B + , , N n B + = 1 Π j = 1 n 1 Tru N σ j B k j , 1 Π j = 1 n 1 Ind N σ j B k j , Π j = 1 n Tru N σ j B k j , ,
where
k = k 1 , k 2 , , k n T ,
is the weight vector of N B FOWA operator with k j 0 , 1 and Σ j = 1 n k j = 1 , for all j = 1 , 2 , , n , i.e., all N j B j = 1 , 2 , , n , are reduced to the following form:
N B FOWA k N 1 B + , N 2 B + , , N n B + = 1 Π j = 1 n 1 Tru N σ j B + k j , N B FOWA k N 1 B , N 2 B , , N n B = 1 Π j = 1 n 1 Tru N σ j B k j .
Theorem 7.
Let N j B = N N j B B + , N N j B B j = 1 , 2 , , n be a collection of neutrosophic bipolar fuzzy values and
k = k 1 , k 2 , , k n T ,
is the weighting vector of N B FOWA operator with k j 0 , 1 and Σ j = 1 n k j = 1 ; then we have the following.
(1) 
Idempotency: If all N j B j = 1 , 2 , , n are equal, i.e., N j B = N B , for all j, then
N B FOWA k N 1 B , N 2 B , , N n B = N B .
(2) 
Boundary:
N B N B FOWA k N 1 B , N 2 B , , N n B N B + ,
for where k , where N j B = ( N j B + , N j B ) be the N B FWA N j B + = Tru N j B + , Ind N j B + , Fal N j B + j = 1 , 2 , , n and N j B = Tru N j B , Ind N j B , Fal N j B j = 1 , 2 , , n be a collection of neutrosophic bipolar fuzzy values
N B = min j Tru N j B , min j Ind N j B , max j Fal N j B , N B + = max j Tru N j B + , max j Ind N j B + , min j Fal N j B + .
(3) 
Monotonicity: Let N j B + * and N j B * j = 1 , 2 , , n be a collection of neutrosophic bipolar fuzzy values. If Tru N j B + Tru N j B + * , Ind N j B + Ind N j B + * and Fal N j B Fal N j B * , for all j, then
N B FOWA k N 1 B , N 2 B , , N n B B F W L k N 1 * B , N 2 * B , , N n * B , for every k .
(4) 
Commutativity: Let N j B = ( N j B + , N j B ) be a collection of neutrosophic bipolar fuzzy values. Then
B F O W L k N 1 B , N 2 B , , N n B = B F O W L k N 1 B , N 2 B , , N n B ,
for every w , where N 1 B , N 2 B , , N n B is any permutation of N 1 B , N 2 B , , N n B .
Theorem 8.
Let N j B = ( N j B + , N j B ) be a collection of neutrosophic bipolar fuzzy values
k = k 1 , k 2 , , k n T ,
is the weighting vector of N B FOWA operator with
k j 0 , 1 and Σ j = 1 n k j = 1 ;
then we have the following:
(1) 
If k = 1 , 0 , , 0 T , then
N B FOWA k N 1 B , N 2 B , , N n B = max j N j B .
(2) 
If k = 0 , 0 , , 1 T , then
N B FOWA k N 1 B , N 2 B , , N n B = min j N j B .
(3) 
If k j = 1 , k i = 0 , and i j , then
BFOWA k N 1 B , N 2 B , , N n B = N σ ( j ) B ,
where N σ ( j ) B is the largest of N i B i = 1 , 2 , , n .

5. Similarity Measures of Neutrosophic Bipolar Fuzzy Sets

In Section 4 we define different aggregation operators with the help of operations defined in Section 3. Next in this section we are aiming to define some similarity measures which will be used in the next Section 6. A comparisons of several different fuzzy similarity measures as well as their aggregations have been studied by Beg and Ashraf [38,39]. Theoretical and computational properties of the measures was further investigated with the relationships between them [15,40,41,42]. A review, or even a listing of all these similarity measures is impossible. Here in this section we define different kinds of similarity measures of neutrosophic bipolar fuzzy sets.

5.1. Neutrosophic Bipolar Fuzzy Distance Measures

Definition 14.
A function E : N B F S s X 0 , 1 is called an entropy for N B F S s X ,
(1) 
E N B = 1 N B is a crisp set.
(2) 
E N B = 0
Tru N 1 B + ( x ) = Tru N 1 B ( x ) , Ind N 1 B + ( x ) = Ind N 1 B ( x ) , Fal N 1 B + ( x ) = Fal N 1 B ( x ) x X .
(3) 
E N B = E N B c for each N B B F S s X .
(4) 
E N 1 B E N 2 B if N 1 B is less than N 2 B , that is,
Tru N 1 B + ( x ) Tru N 2 B + ( x ) , Ind N 1 B + ( x ) Ind N 2 B + ( x ) , Fal N 1 B + ( x ) Fal N 2 B + ( x ) , Tru N 1 B ( x ) Tru N 2 B ( x ) , Ind N 1 B ( x ) Ind N 2 B ( x ) , Fal N 1 B ( x ) Fal N 2 B ( x ) ,
for Tru N 2 B + ( x ) Tru N 2 B ( x )
or Tru N 1 B + ( x ) Tru N 2 B + ( x ) , Ind N 1 B + ( x ) Ind N 2 B + ( x ) ,
and
Fal N 1 B ( x ) Fal N 2 B ( x ) N B 2 B ( x ) for Tru N 1 B + ( x ) Fal N 2 B ( x ) .
Definition 15.
Let X = { x 1 , x 2 , , x n } and N B = ( N B + , N B ) be an N B F S . The entropy of N B F S is denoted by E ( N B + , N B ) and given by
E ( N B + ) = 1 n i = 1 n min ( ( Tru N 1 B + ´ ( x ) ) , min ( Ind N 1 B + ´ ( x ) ) , Fal N 1 B + ´ ( x ) ) max ( ( Tru N 1 B + ´ ( x ) ) , max ( Ind N 1 B + ´ ( x ) ) , Fal N 1 B + ( x ) ) E ( N B ) = 1 n i = 1 n min ( ( Tru N 1 B ´ ( x ) ) , min ( Ind N 1 B ´ ( x ) ) , Fal N 1 B ´ ( x ) ) max ( ( Tru N 1 B ´ ( x ) ) , max ( Ind N 1 B ´ ( x ) ) , Fal N 1 B ( x ) ) ,
and for a neutrosophic bipolar fuzzy number N B = N L B + , N L B , the bipolar fuzzy entropy is given by
E ( N L B + ) = min ( ( Tru L 1 + ´ ( x ) , min ( Ind L 1 + ( x ) ´ ) , Fal L 1 + ( x ) ´ ) max ( Tru L 1 + ( x ) ´ ) , max ( Ind L 1 + ( x ) ´ ) , Fal L 1 + ´ ( x ) ) E ( N L B ) = min ( ( Tru L 1 ´ ( x ) , min ( Ind L 1 ( x ) ´ ) , Fal L 1 ( x ) ´ ) max ( Tru L 1 ( x ) ´ ) , max ( Ind L 1 ( x ) ´ ) , Fal L 1 ´ ( x ) ) .
Definition 16.
Let X = x 1 , x 2 , , x n . We define the Hamming distance between N 1 B and N 2 B belonging to N B F S s ( X ) defined as follows:
(1) 
The Hamming distance:
d ( N 1 B + , N 2 B + ) = 1 2 j = 1 n ( | Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) | + | Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) | + | | Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) | | ) H a m m i n g d i s t a n c e f o r p o s i t i v e n e u t r o s o p h i c b i p o l a r s e t s d ( N 1 B , N 2 B ) = 1 2 j = 1 n ( | Tru N 1 B ( x j ) Tru N 2 B ( x j ) | + | Ind N 1 B ( x j ) Ind N 2 B ( x j ) | + | | Fal N 1 B ( x j ) Fal N 1 B ( x j ) | | ) H a m m i n g d i s t a n c e f o r n e g a t i v e n e u t r o s o p h i c b i p o l a r s e t s .
(2) 
The normalized Hamming distance:
d ( N 1 B + , N 2 B + ) = 1 2 n j = 1 n ( | Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) | + | Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) | + | | Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) | | ) n o r m a l i z e d H a m m i n g d i s t a n c e f o r p o s i t i v e n e u t r o s o p h i c b i p o l a r s e t s d ( N 1 B , N 2 B ) = 1 2 n j = 1 n ( | Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) | + | Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) | + | | Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) | | ) n o r m a l i z e d H a m m i n g d i s t a n c e f o r n e g a t i v e n e u t r o s o p h i c b i p o l a r s e t s .
(3) 
The Euclidean distance:
d ( N 1 B + , N 2 B + ) = 1 2 j = 1 n ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) 2 + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) 2 + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) 2 d ( N 1 B , N 2 B ) = 1 2 j = 1 n ( Tru N 1 B ( x j ) Tru N 2 B ( x j ) ) 2 + ( Ind N 1 B ( x j ) Ind N 2 B ( x j ) ) 2 + ( Fal N 1 B ( x j ) Fal N 1 B ( x j ) ) 2 .
(4) 
The normalized Euclidean distance:
d ( N 1 B + , N 2 B + ) = 1 2 n j = 1 n ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) 2 + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) 2 + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) 2 d ( N 1 B , N 2 B ) = 1 2 n j = 1 n ( Tru N 1 B ( x j ) Tru N 2 B ( x j ) ) 2 + ( Ind N 1 B ( x j ) Ind N 2 B ( x j ) ) 2 + ( Fal N 1 B ( x j ) Fal N 1 B ( x j ) ) 2 .
(5) 
Based on the geometric distance formula, we have
d ( N 1 B + , N 2 B + ) = 1 2 j = 1 n ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) L + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) L + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) L 1 α d ( N 1 B , N 2 B ) = 1 2 j = 1 n ( Tru N 1 B ( x j ) Tru N 2 B ( x j ) ) L + ( Ind N 1 B ( x j ) Ind N 2 B ( x j ) ) L + ( Fal N 1 B ( x j ) Fal N 1 B ( x j ) ) L 1 α .
(6) 
Normalized geometric distance formula:
d ( N 1 B + , N 2 B + ) = 1 2 n j = 1 n ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) L + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) L + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) L 1 α d ( N 1 B , N 2 B ) = 1 2 n j = 1 n ( Tru N 1 B ( x j ) Tru N 2 B ( x j ) ) L + ( Ind N 1 B ( x j ) Ind N 2 B ( x j ) ) L + ( Fal N 1 B ( x j ) Fal N 1 B ( x j ) ) L 1 α ,
where α > 0 .
(i) 
If α = 1 , then Equations (9) and (10), reduce to Equations (5) and (6).
(ii) 
If α = 2 , then Equations (9) and (10), reduce to Equations (7) and (8).
(iii) 
We define a weighted distance as follows:
d ( N 1 B + , N 2 B + ) = 1 2 j = 1 n k j ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) L + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) L + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) L 1 α d ( N 1 B , N 2 B ) = 1 2 j = 1 n k j ( Tru N 1 B ( x j ) Tru N 2 B ( x j ) ) L + ( Ind N 1 B ( x j ) Ind N 2 B ( x j ) ) L + ( Fal N 1 B ( x j ) Fal N 1 B ( x j ) ) L 1 α ,
where k = ( k 1 , k 2 , , k n ) T is the weight vector of x j ( j = 1 , 2 , , n ) , and α > 0 .
(i) 
Especially, if α = 1 , then Equation (11) is reduced as
d ( N 1 B + , N 2 B + ) = 1 2 j = 1 n k j ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) d ( N 1 B , N 2 B ) = 1 2 j = 1 n k j ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) .
If k = ( 1 n , 1 n , , 1 n ) T , then Equation (11) goes to Equation (10), and Equation (12) goes to Equation (6).
(ii) 
If α = 2 , then Equation (11) is reduced to the as:
d ( N 1 B + , N 2 B + ) = 1 2 j = 1 n ( Tru N 1 B + ( x j ) Tru N 2 B + ( x j ) ) 2 + ( Ind N 1 B + ( x j ) Ind N 2 B + ( x j ) ) 2 + ( Fal N 1 B + ( x j ) Fal N 1 B + ( x j ) ) 2 d ( N 1 B , N 2 B ) = 1 2 j = 1 n ( Tru N 1 B ( x j ) Tru N 2 B ( x j ) ) 2 + ( Ind N 1 B ( x j ) Ind N 2 B ( x j ) ) 2 + ( Fal N 1 B ( x j ) Fal N 1 B ( x j ) ) 2 .
If k = ( 1 n , 1 n , , 1 n ) T , then Equation (13) is reduced to Equation (8).

5.2. Similarity Measures of Neutrosophic Bipolar Fuzzy Set

Definition 17.
Let s ^ be a mapping s ^ : Ω ( X ) 2 [ 0 , 1 ] , then the degree of similarity between N 1 B Ω ( X ) and N 2 B Ω ( X ) is defined as