Complex Neutrosophic Soft Set

2017·0.14 MB·English

by  Said Broumi, Assia Bakali, Florentin Smarandache, Mohamed Talea, Mumtaz Ali, Ganeshsree Selvachandran;

2017 FUZZ-IEEE Conference on Fuzzy Systems, Naples, Italy, 9-12 July 2017

Complex Neutrosophic Soft Set
Said Broumi Assia Bakali
Laboratory of Information Processing, Faculty of Science Ecole Royale Navale, Boulevard Sour Jdid, B.P 16303
Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco
Casablanca, Morocco
[email protected]
[email protected]
Mohamed Talea
Florentin Smarandache
Laboratory of Information processing, Faculty of Science
Department of Mathematics, University of New Mexico, Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman,
705 Gurley Avenue, Gallup, NM 87301, USA Casablanca, Morocco
[email protected] [email protected]
Mumtaz Ali Ganeshsree Selvachandran
Faculty of Business and Information Science,UCSI
University of Southern Queensland 4300, QLD
University, Jalan Menara Gading, 56000 Cheras, Kuala
Australia
Lumpur, Malaysia
[email protected], [email protected]
[email protected]
Abstract—In this paper, we propose the complex neutrosophic The study of complex fuzzy sets were initiated by Ramot et al.
soft set model, which is a hybrid of complex fuzzy sets, neutrosophic [28]. Among the well-known complex fuzzy based models in
sets and soft sets. The basic set theoretic operations and some literature are complex intuitionistic fuzzy sets (CIFSs) [29, 30],
concepts related to the structure of this model are introduced, and
complex vague soft sets (CVSSs) [31, 32] and complex
illustrated. An example related to a decision making problem
intuitionistic fuzzy soft sets (CIFSSs) [33].These models have
involving uncertain and subjective information is presented, to
been used to represent the uncertainty and periodicity aspects of
demonstrate the utility of this model.
an object simultaneously, in a single set. The complex-valued
Keywords—complex fuzzy sets; soft sets; complex neutrosophic sets; membership and non-membership functions in these models have
complex neutrosophic soft sets; decision making the potential to be used to represent uncertainty in instances such
as the wave function in quantum mechanics, impedance in
I. INTRODUCTION electrical engineering, the changes in meteorological activities,
The neutrosophic set model (NS) proposed by Smarandache [1, and time-periodic decision making problems.
2]is a powerful tool to deal with incomplete, indeterminate and Recently, Ali and Smarandache [34] developed a hybrid model of
inconsistent information in the real world. It is a generalization of complex fuzzy sets and neutrosophic sets, called complex
the theory of fuzzy sets [3], intuitionistic fuzzy sets [4, 5], neutrosophic sets. This model has the capability of handling the
interval-valued fuzzy sets [6] and interval-valued intuitionistic different aspects of uncertainty, such as incompleteness,
fuzzy sets [7]. A neutrosophic set is characterized by a truth- indeterminacy and inconsistency, whilst simultaneously handling
membership degree (t), an indeterminacy-membership degree (i) the periodicity aspect of the objects, all in a single set. The
and a falsity-membership degree (f), all defined independently, complex neutrosophic set is defined by complex-valued truth,
and all of which lie in the real standard or nonstandard unit indeterminacy and falsity membership functions. The complex-
interval ]−0, 1+[. Since this interval is difficult to be used in real- valued truth membership function consists of a truth amplitude
life situations, Wang et al. [8] introduced single-valued term (truth membership) and a phase term which represents the
neutrosophic sets (SVNSs) whose functions of truth, periodicity of the object. Similarly, the complex-valued
indeterminacy and falsity all lie in [0, 1]. Neutrosophic sets and indeterminacy and falsity membership functions consists of an
its extensions such as single valued neutrosophic sets, interval indeterminacy amplitude and a phase term, and a falsity
neutrosophic sets, and intuitionistic neutrosophic sets have been amplitude and a phase term, respectively. The complex
applied in a wide variety of fields including decision making, neutrosophic set is a generalized framework of all the other
computer science, engineering, and medicine [1-2, 8-27, 36-37]. existing models in literature.
978-1-5090-6034-4/17/$31.00 @2017 IEEE
However, as the CNS model is an extension of ordinary fuzzy
sets, it lacks adequate parameterization qualities. Adequate Definition 3. [8] The complement of a neutrosophic set
parameterization refers to the ability of a model to define the denoted by is as defined below for all
(cid:1),
parameters in a more comprehensive manner, without any
(cid:1) , (cid:5) ∈(cid:16):
restrictions. Soft set theory works by defining the initial
(cid:7)(cid:8)(cid:9)(cid:5)(cid:10)=(cid:12)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10)=1−(cid:11)(cid:8)(cid:9)(cid:5)(cid:10),(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)=(cid:7)(cid:8)(cid:9)(cid:5)(cid:10).
description of the parameters in an approximate manner, and Definition 4. [10] Let be an initial set and be a set of
allows for any form of parameterization that is preferred by the parameters. Let denote the power set of and let
" #
users. This includes using words and sentences, real numbers, A pair is called a soft set over where is a mapping
$(cid:9)"(cid:10) ", (cid:1)→#.
functions and mappings, among others to describe the parameters. given by In other words, a soft set is a
(cid:9)(cid:12),(cid:1)(cid:10) ", (cid:12)
The absence of any restrictions on the approximate description in parameterized family of subsets of the set Every where
(cid:12):(cid:1)→$(cid:9)"(cid:10).
soft set theory makes it very convenient to be used and easily from this family may be considered as the set of -
". (cid:12)(cid:9)%(cid:10),
applicable in practice. The adequate parameterization capabilities elements of the soft set
%∈#, %
of soft set theory and the lack of such capabilities in the existing
(cid:9)(cid:12),(cid:1)(cid:10).
CNS model served as the motivation to introduce the CNSS Definition 5. [34] A complex neutrosophic set defined on a
model in this paper. This is achieved by defining the complex universe of discourse is characterized by a truth membership
(cid:1)
neutrosophic set in a soft set setting. function an indeterminacy membership function
(cid:16),
The rest of the paper is organized as follows. In section 2, we
and a falsity membership function that assigns a complex-
(cid:7)(cid:8)(cid:9)(cid:5)(cid:10), (cid:11)(cid:8)(cid:9)(cid:5)(cid:10),
present an overview of some basic definitions and properties
valued grade for each of these membership function in for any
(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)
which serves as the background to our work in this paper. In
The values of and and their sum may
(cid:1)
section 3, the main definition of the CNSS and some related
assume any values within a unit circle in the complex plane, and
(cid:5) ∈(cid:16). (cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10) (cid:12)(cid:8)(cid:9)(cid:5)(cid:10),
concepts are presented. In section 4, the basic set theoretic
is of the form and
operations for this model are defined. The utility of this model is All the ’a(m)(cid:9)p*l(cid:10)itude and phase ’t,e)r(cid:9)m*(cid:10)s are
demonstrated by applying it in a decision making problem in (cid:7)(cid:8)(cid:9)(cid:5)(cid:10)=&(cid:8)(cid:9)(cid:5)(cid:10)% ,(cid:11)(cid:8)(cid:9)(cid:5)(cid:10)=+(cid:8)(cid:9)(cid:5)(cid:10)% ,
real-valued a’.n)d(cid:9) *(cid:10) whereas
section 5. Concluding remarks are given in section 6. (cid:12)(cid:8)(cid:9)(cid:5)(cid:10)=-(cid:8)(cid:9)(cid:5)(cid:10)% .
such that the condition
&(cid:8)(cid:9)(cid:5)(cid:10),+(cid:8)(cid:9)(cid:5)(cid:10),-(cid:8)(cid:9)(cid:5)(cid:10)∈[0,1],
/(cid:8)(cid:9)(cid:5)(cid:10),0(cid:8)(cid:9)(cid:5)(cid:10),1(cid:8)(cid:9)(cid:5)(cid:10)∈(cid:9)0,22],
II. PRELIMINARIES is satisfied. A 0c≤om&p(cid:8)l(cid:9)e(cid:5)x(cid:10) +ne+u(cid:8)t(cid:9)ro(cid:5)s(cid:10)o+ph-i(cid:8)c(cid:9) (cid:5)s(cid:10)et≤ 3 c a n t h u s (cid:9)3be(cid:10)
represented in set form as:
In this section, we recapitulate some important concepts (cid:1)
related to neutrosophic sets (NSs), and complex neutrosophic sets
where
(CNSs). We refer the readers to [1, 8, 10, 34] for further details (cid:1)=(cid:3)〈(cid:5),(cid:7)(cid:8)(cid:9)(cid:5)(cid:10)=34,(cid:11)(cid:8)(cid:9)(cid:5)(cid:10)=35,(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)=36〉:(cid:5) ∈(cid:16)(cid:17),
pertaining to these models. (cid:7)(cid:8): (cid:16)→(cid:3)34:34 ∈7,|34|≤1(cid:17), and(cid:11)(cid:8) a:l(cid:16)so→(cid:3)35:35 ∈
(4)
7,|35|≤1(cid:17),(cid:12)(cid:8):(cid:16)→(cid:3)36:36 ∈7,|36|≤1(cid:17),
Let X be a space of points (objects) with generic elements in
The interval is chosen for the phase term to be in line with
X denoted by x. |(cid:7)(cid:8)(cid:9)(cid:5)(cid:10)+(cid:11)(cid:8)(cid:9)(cid:5)(cid:10)+(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)|≤3.
the original definition of a complex fuzzy set in which the
(cid:9)0,22]
Definition 1. [1] A neutrosophic set A is an object having the amplitude terms lie in an interval of and the phase terms
form where the functions lie in an interval of
(cid:9)0,1(cid:10),
denote the truth, indeterminacy, and falsity
(cid:1)=(cid:3)〈(cid:5),(cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10),(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)〉:(cid:5) ∈(cid:16)(cid:17), (cid:9)0,22].
membership f(cid:21)uncti(cid:24)ons, respectively, of the element with Remark: In the definition above, denotes the imaginary number
(cid:7),(cid:11),(cid:12) ∶(cid:16)→] 0,1 [ ,
respect to set These membership functions must satisfy the and it is this imaginary number that makes the CNS
(cid:5) ∈(cid:16) 9
condition have complex-valued membership grades. The term denotes
(cid:1). 9 =√−1 9
The funct(cid:21)io0n≤s (cid:7)(cid:8)(cid:9)(cid:5)(cid:10)+ (cid:11)(cid:8)(cid:9)(cid:5) (cid:10)an+d (cid:12)(cid:8)(cid:9)(cid:5)(cid:10)≤ ar3e(cid:24) . r e a l s t a n d a r d (cid:9)1o(cid:10)r the exponential form of a complex number and repre%se’;nts
nonstandard subsets of the interval ]−0,1+[. However, these ’;
(cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10), (cid:12)(cid:8)(cid:9)(cid:5)(cid:10) % =
intervals make it difficult to apply NSs to practical problems, and
cos?+9sin?.
this led to the introduction of a single-valued neutrosophic set Definition 6. [34] Let be a
(SVNS) in [12]. This model is a special case of NSs and is better complex neutrosophic set over Then the complement of
(cid:1)=(cid:3)〈(cid:5),(cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10),(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)〉:(cid:5) ∈(cid:16)(cid:17)
suited to handle real-life problems and applications. denoted by is defined as:
(cid:16). (cid:1),

(cid:1) ,
Definition 2. [8] An SVNS is a neutrosophic set that is
where (cid:1) =(cid:3)〈(cid:5),(cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10),(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)〉:(cid:5) ∈(cid:16)(cid:17),
characterized by a truth-me(cid:1)mbership function an (cid:7)(cid:8) (cid:9)(cid:5)(cid:10)=-(cid:8)(cid:9)(cid:5)(cid:10)%’BCD(cid:21)()(cid:9)*(cid:10)E, and
indeterminacy-membership function and a falsity-
(cid:7)(cid:8)(cid:9)(cid:5)(cid:10), ’BCD(cid:21),)(cid:9)*(cid:10)E
membership function where (cid:11)(cid:8)(cid:9)(cid:5)(cid:10)=B1−+(cid:8)(cid:9)(cid:5)(cid:10)E% ,
(cid:11)(cid:8)(cid:9)(cid:5)(cid:10), ’BCD(cid:21).)(cid:9)*(cid:10)E
A SVNS can thus be(cid:12) w(cid:8)(cid:9)ri(cid:5)tt(cid:10)e,n as (cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10),(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)∈[0,1]. (cid:12)(cid:8)(cid:9)(cid:5)(cid:10)=&(cid:8)(cid:9)(cid:5)(cid:10)% .
(cid:1)
(cid:1)=(cid:3)〈(cid:5),(cid:7)(cid:8)(cid:9)(cid:5)(cid:10),(cid:11)(cid:8)(cid:9)(cid:5)(cid:10),(cid:12)(cid:8)(cid:9)(cid:5)(cid:10)〉:(cid:5) ∈(cid:16)(cid:17) (cid:9)2(cid:10)
III. COMPLEX NEUTROSOPHIC SOFT SETS y (e )
A 3
( ) ( )
In this section, we introduce the complex neutrosophic soft set  0.4.ej0.4p ,0.1.ejp4,0.2.ej0.p1 0.3ej0.2p ,0.3ej0.p4 ,0.2ejp3 
(CNSS) model which is a hybrid of the CNS and soft set models.  
, ,
The formal definition of this model as well as some concepts  u u 
= 1 2 ,
related to this model are as given below: ( ) ( )
 0.2ej0.1p ,0.4.ej0.p5 ,0.5ej0p.2 0.5ejp0.2 ,0.4pej2 ,0.6pej0.1 
 , 
Definition 7. Let be universal set, be a set of parameters  u u 
3 4
under consideratio"n, and be# a complex neutrosophic and
set over for all (cid:1)⊆ T#h,en a Gco(cid:8)mplex neutrosophic soft set y (e )
Hde(cid:8)noovteers "t"h ies dseetf ionfe (cid:5)dc oa∈ms "ap lm.exa pnpeiuntgr oHs(cid:8)o:p#hi→c s7etIs (cid:9)in" (cid:10), w ahnedr e 7I(cid:9)"(cid:10) A(04.3ej0.2p ,0.5ejp4,0.5ej0.p1 ) (0.1ej0p ,0.6ej0.p4 ,0.4ejp3) 
if Here is called a comple"x , neutJro(cid:8)s(cid:9)o(cid:5)p(cid:10)h=ic  u , u , 
a∅pprox(cid:5)im∉a(cid:1)te. functionJ (cid:8)o(cid:9)f(cid:5) (cid:10) and the values of is called the =( 1 ) ( 2 ).
-elements of the CNSS for all Thus, can be  0.1ej0.1p ,0.2ej0.p2 ,0.4ej0p.2 0.2ejp0.2 ,0.5epj0.3 ,0.3pej0.1 
H(cid:8) J(cid:8)(cid:9)(cid:5)(cid:10)
represented by the set of ordered pairs of the following form:  , 
(cid:5) (cid:5) ∈". H(cid:8)  u u 
3 4
Then the complex neutrosophic soft set can be written as a
where H(cid:8) =MB(cid:5),J(cid:8)(cid:9)(cid:5)(cid:10)E:(cid:5) ∈#,J(cid:8)(cid:9)(cid:5)(cid:10)∈7I(cid:9)"(cid:10)N, collection of CNSs of the form:
are real-valued an’d( )(cid:9)l*ie(cid:10) in ’,)(cid:9) *a(cid:10)nd ’.)(cid:9)*(cid:10) H(cid:8)
J(cid:8)(cid:9)(cid:5)(cid:10)=B&(cid:8)(cid:9)(cid:5)(cid:10)% ,+(cid:8)(cid:9)(cid:5)(cid:10)% ,-(cid:8)(cid:9)(cid:5)(cid:10)% E,
This is done to ensure that the definition of the CNSS
&(cid:8),+(cid:8),-(cid:8) [0,1], /(cid:8),0(cid:8),1(cid:8) ∈
model is line with the original structure of the complex fuzzy set, H(cid:8) =(cid:3)J(cid:8)(cid:9)%O(cid:10),J(cid:8)(cid:9)%C(cid:10),J(cid:8)(cid:9)%P(cid:10),J(cid:8)(cid:9)%Q(cid:10)(cid:17).
(cid:9)0,22]. Definition 8. Let and be two CNSSs over a universe
on which the CNSS model is based on.
Then we have the following:
H(cid:8) Hl ".
(i) is said to be an empty CNSS, denoted by if
Example 1. Let be a set of developing countries in the
Southeast Asian (SEA) region, that are under consideration, be H(cid:8) for all H(cid:8)∅,
" (ii) is said to be an absolute CNSS, denoted by if
a set of parameters that describe a country’s economic indicators, J(cid:8)(cid:9)(cid:5)(cid:10)=∅, (cid:5) ∈";
#
and where these sets are as defined H(cid:8) for all H(cid:8)n,
below(cid:1): =(cid:3)%O,%C,%P,%Q(cid:17)⊆#, J(ii(cid:8)i(cid:9)) (cid:5)(cid:10)= is "said to be(cid:5) a∈ C"N;S-subset of denoted by
if for all that is the following
H(cid:8) Hl, H(cid:8) ⊆Hl,
conditions are satisfied:
" = (cid:3) RO =Republic of Philippines,RC = Vietnam, (cid:5) ∈", J(cid:8)(cid:9)%(cid:10)⊆Jl(cid:9)%(cid:10),

RP =Myanmar,RQ =Indonesia(cid:17),
and
g# ro=w(cid:3)t%hO r=atien,f%lQat=ionu nreamtep,%loCy=mpeonpt uralateti,o%ni g=roewxptho,r %t Pvo=luGmDeP(cid:17). (iv) &/is(cid:8)(cid:8) (cid:9)(cid:9)s%a%i(cid:10)(cid:10)d≤ ≤to& /blle(cid:9)(cid:9) e%%q(cid:10)(cid:10)u,,+a0l(cid:8)(cid:8) t(cid:9)(cid:9)o%% (cid:10)(cid:10)≤≤ d+0elln(cid:9)(cid:9)o%%t(cid:10)(cid:10)e,,d1- (cid:8)b(cid:8)(cid:9)y(cid:9)% %(cid:10)(cid:10)≤≤-1l(cid:9)l%(cid:9)(cid:10) %i,f(cid:10) .for all
The CNS and are defined as: H(cid:8) the following conditioHnls, are satisfied:H (cid:8) =Hl,
y A((e01.)6ejJ0.8(cid:8)p(cid:9),%0O.3(cid:10),eJj3(cid:8)4p ,(cid:9)0%.C5(cid:10)e,Jj0.(cid:8)p3(cid:9))%P(cid:10)(0.7ejJ0p(cid:8),(cid:9)0%.Q2(cid:10)ej0.p9 ,0.1ej23p )  a(cid:5) n ∈d " /&(cid:8)(cid:8)(cid:9)(cid:9)%%(cid:10)(cid:10)==/&ll(cid:9)(cid:9)%%(cid:10)(cid:10),,+0(cid:8)(cid:8)(cid:9)(cid:9)%%(cid:10)(cid:10)==0+ll(cid:9)(cid:9)%%(cid:10)(cid:10),,1-(cid:8)(cid:8)(cid:9)(cid:9)%%(cid:10)(cid:10)==-1l(cid:9)l%(cid:9)(cid:10)%,(cid:10) .
 u , u ,  Proposition 1. Let Then the following hold:
= 1 2 , (i)
( ) ( ) H(cid:8) ∈7I(cid:9)"(cid:10).
 0.9ej0.1p ,0.4epj ,0.7ej0p.7 0.3ejp0.4 ,0.2epj0.6 ,0.7pej0.5  (ii)
 ,  (cid:9)H (cid:8)(cid:10) =H(cid:8);
 u u  ProoHf(cid:8). ∅T=he Hp(cid:8)rnoo.fs are straightforward from Definition 8.
3 4

y (e )
A 2
(0.2ej0.2p ,0.5ej34p ,0.6ej0.p3 ) (0.4ej0.3p ,0.4ej0.p4 ,0.2ej23p )  IV. OPERATIONS SOONF CTO SMETPSL EX NEUTROSOPHIC
, ,
= u1 u2 , In this section we define the basic set theoretic operations on
(0.3ej0.1p ,0.2ej0.p1 ,0.3ej0p.7 ) (0.1ejp0.4 ,0.5epj1.2 ,0.3pej0.1 ) CNSSs, namely the complement, union and intersection.
 , 
 u u  Let and be two CNSSs over a universe
3 4

H(cid:8) Hl ".
Definition 9. The complement of denoted by is a CNSS
defined by H(cid:8), where H(cid:8) , is the
complex neutro sophic com plement of
H(cid:8) =MB(cid:5),G(cid:8)(cid:9)(cid:5)(cid:10)E:(cid:5) ∈"N, G(cid:8)(cid:9)(cid:5)(cid:10)
G(cid:8).
Example 2. Consider Example 1. The complement of is given Definition 11. The intersection of and denoted by
by For the sake of is defined as:
H(cid:8) H(cid:8) Hl,
brevit y, we o nly give the com plement for below:
H(cid:8) =(cid:3)G(cid:8)(cid:9)%O(cid:10),G(cid:8)(cid:9)%C(cid:10),G(cid:8)(cid:9)%P(cid:10),G(cid:8)(cid:9)%Q(cid:10)(cid:17). H(cid:8)∩pHl,
y Ac(e1) G(cid:8) (cid:9)%O(cid:10) H} =H(cid:8)∩pHl =(cid:3)(cid:9)(cid:5),G(cid:8)(cid:9)(cid:5)(cid:10)∩pGl(cid:9)(cid:5)(cid:10)(cid:10)∶(cid:5) ∈"(cid:17),
( ) ( )
 0.5ej1.2p ,0.7ej54p ,0.6ej1.p7 0.1ej2p ,0.8ej1.p1 ,0.7ej43p  B(cid:5),G(cid:8)(cid:9)(cid:5)(cid:10)E if %∈(cid:1)−s,
 
=( u1 ) ,( u2 ,). H}(cid:9)%(cid:10)=rB(cid:9)(cid:5)(cid:5),,GGl(cid:8)(cid:9)(cid:9)(cid:5)(cid:5)(cid:10)(cid:10)E∩p G l (cid:9) (cid:5) (cid:10) (cid:10) iiff %%∈∈(cid:1)s∩−s(cid:1),,
 0.7ej1.9p ,0.6epj ,0.9ej1p.3 0.7epj1.6 ,0.8epj1.4 ,0.3pej1.5  where and
 , 
 u u  ~ =(cid:1)∪s,(cid:5) ∈",
3 4 ’B()(cid:9)*(cid:10)∪(w(cid:9)*(cid:10)E
The complements for the rest of the CNSs can be found in a B&(cid:8)(cid:9)(cid:5)(cid:10)∧&l(cid:9)(cid:5)(cid:10)E%
’B,)(cid:9)*(cid:10)∪,w(cid:9)*(cid:10)E
similar manner. G(cid:8)(cid:9)(cid:5)(cid:10)∩pGl(cid:9)(cid:5)(cid:10)=uB+(cid:8)(cid:9)(cid:5)(cid:10)∨+l(cid:9)(cid:5)(cid:10)E% y,
where and denote the maximum and’ Bm.)in(cid:9)*im(cid:10)∪u.mw(cid:9) *(cid:10)oEperators
B-(cid:8)(cid:9)(cid:5)(cid:10)∨-l(cid:9)(cid:5)(cid:10)E%
Definition10. The union of and denoted by is respectively, whereas the phase terms of the truth, indeterminacy
∨ ∧
defined as: H(cid:8) Hl, H(cid:8)∪pHl, and falsity functions lie in the interval and can be
calculated using any one of the following operators that were
(cid:9)0,22],
Hq =H(cid:8)∪pHl =(cid:3)(cid:9)(cid:5),G(cid:8) (cid:9)(cid:5)(cid:10)∪pGl(cid:9)(cid:5)(cid:10)(cid:10)∶(cid:5) ∈"(cid:17), defined in Definition 10.

B(cid:5),G(cid:8)(cid:9)(cid:5)(cid:10)E if %∈(cid:1)−s, V. APPLICATION OF THE CNSS MODEL IN A
Hq(cid:9)%(cid:10)=rB(cid:5),Gl(cid:9)(cid:5)(cid:10)E if %∈s−(cid:1), DECISION MAKING PROBLEM

(cid:9)(cid:5),G(cid:8)(cid:9)(cid:5)(cid:10)∪pGl(cid:9)(cid:5)(cid:10)(cid:10) if %∈(cid:1)∩s, In Example 1, we presented an example related to the economic
where and
indicators of four countries. In this section, we use the same
7 =(cid:1)∪s,(cid:5) ∈", information to determine which one of the four countries that are
’B()(cid:9)*(cid:10)∪(w(cid:9)*(cid:10)E
B&(cid:8)(cid:9)(cid:5)(cid:10)∨&l(cid:9)(cid:5)(cid:10)E% studied has the strongest economic indicators. To achieve this, a
’B,)(cid:9)*(cid:10)∪,w(cid:9)*(cid:10)E modified algorithm and an accompanying score function is
G(cid:8)(cid:9)(cid:5)(cid:10)∪pGl(cid:9)(cid:5)(cid:10)=uB+(cid:8)(cid:9)(cid:5)(cid:10)∧+l(cid:9)(cid:5)(cid:10)E% y,
presented in Definition 12 and 13. This algorithm and score
where and denote the maximum and’ Bm.)in(cid:9)*im(cid:10)∪u.mw(cid:9) *(cid:10)oEperators
B-(cid:8)(cid:9)(cid:5)(cid:10)∧-l(cid:9)(cid:5)(cid:10)E% function are an adaptation of the corresponding concepts
respectively, whereas the phase terms of the truth, indeterminacy
∨ ∧ introduced in [35], which was then made compatible with the
and falsity functions lie in the interval and can be
structure of the CNSS model. The steps involved in the decision
calculated using any one of the following operators:
(cid:9)0,22], making process, in the context of this example, until a final
(i) Sum:
decision is reached, is as given below.

and
/(cid:8)∪l(cid:9)(cid:5)(cid:10)=/(cid:8)(cid:9)(cid:5)(cid:10)+/l(cid:9)(cid:5)(cid:10),0(cid:8)∪l(cid:9)(cid:5)(cid:10)=0(cid:8)(cid:9)(cid:5)(cid:10)+0l(cid:9)(cid:5)(cid:10),
(ii) Max: Definition 12. A comparison matrix is a matrix whose rows
1(cid:8)∪l(cid:9)(cid:5)(cid:10)=1(cid:8)(cid:9)(cid:5)(cid:10)+1l(cid:9)(cid:5)(cid:10).
consists of the elements of the universal set
whereas the columns consists of the
a/n(cid:8)d∪ l(cid:9)(cid:5)(cid:10)=maxB/(cid:8)(cid:9)(cid:5)(cid:10),/l(cid:9)(cid:5)(cid:10)E,0(cid:8)∪l (cid:9)(cid:5)(cid:10)=maxB0(cid:8)(cid:9)(cid:5)(cid:10),0l(cid:9)(cid:5)(cid:10)E,
corresponding parameters that are being
(iii) Min: "=(cid:3)RO,RC,…,R(cid:128)(cid:17),
1(cid:8)∪l(cid:9)(cid:5)(cid:10)=maxB1(cid:8)(cid:9)(cid:5)(cid:10),1l(cid:9)(cid:5)(cid:10)E. considered in the problem. The entries of this matrix are
#=(cid:3)%O,%C,…,%(cid:129)(cid:17)
such that

/an(cid:8)d∩ l(cid:9)(cid:5)(cid:10)=minB/(cid:8)(cid:9)(cid:5)(cid:10),/l(cid:9)(cid:5)(cid:10)E,0(cid:8)∩l (cid:9)(cid:5)(cid:10)=minB0(cid:8)(cid:9)(cid:5)(cid:10),0l(cid:9)(cid:5)(cid:10)E, (cid:130)’(cid:131),
(iv) “The game of winner, neutral, and loser”:
1(cid:8)∩l(cid:9)(cid:5)(cid:10)=minB1(cid:8)(cid:9)(cid:5)(cid:10),1l(cid:9)(cid:5)(cid:10)E. where the components of this formula are as defined below for all
(cid:130)’(cid:131) =B(cid:132)(cid:133)(cid:128)(cid:134)+(cid:135)(cid:133)(cid:128)(cid:134)−(cid:136)(cid:133)(cid:128)(cid:134)E+B(cid:132)(cid:134)(cid:137)(cid:133)(cid:138)(cid:139) +(cid:135)(cid:134)(cid:137)(cid:133)(cid:138)(cid:139)−(cid:136)(cid:134)(cid:137)(cid:133)(cid:138)(cid:139)E,
such that
/(cid:8)(cid:9)(cid:5)(cid:10) if &(cid:8)(cid:9)(cid:5)(cid:10)>&l(cid:9)(cid:5)(cid:10) the number of times the value of the amplitude term of
/(cid:8)∪l(cid:9)(cid:5)(cid:10)=z , (cid:140)(cid:141) ∈", (cid:140)’ ≠(cid:140)(cid:141):
/l(cid:9)(cid:5)(cid:10) if &l(cid:9)(cid:5)(cid:10)>&(cid:8)(cid:9)(cid:5)(cid:10) and (cid:132)(cid:133)(cid:128)(cid:134) =
0(cid:8)(cid:9)(cid:5)(cid:10) if +(cid:8)(cid:9)(cid:5)(cid:10)<+l(cid:9)(cid:5)(cid:10)
0(cid:8)∪l(cid:9)(cid:5)(cid:10)=z , (cid:7)(cid:143)(cid:144)B%(cid:131)E ≥th(cid:7)e (cid:143)n(cid:146)uBm%(cid:131)bE,er of times the value of the amplitude term of
0l(cid:9)(cid:5)(cid:10) if +l(cid:9)(cid:5)(cid:10)<+(cid:8)(cid:9)(cid:5)(cid:10)
1(cid:8)(cid:9)(cid:5)(cid:10) if -(cid:8)(cid:9)(cid:5)(cid:10)<-l(cid:9)(cid:5)(cid:10) (cid:135)(cid:133)(cid:128)(cid:134) =
1A(cid:8)ll∪ lo(cid:9)f(cid:5) (cid:10)t=hez 1olp(cid:9)e(cid:5)ra(cid:10)t o r s ipf r -else(cid:9)n(cid:5)t(cid:10)ed< -a(cid:8)b(cid:9)o(cid:5)v(cid:10)e. are straightforward (cid:11)(cid:143)(cid:144)B%(cid:131)E≥ th(cid:11)e(cid:143) (cid:146)nBu%m(cid:131)Eb,er of times the value of the amplitude term of
generalizations of the corresponding operators that were (cid:136)(cid:133)(cid:128)(cid:134) =
originally defined in [28]. The intersection between CNSSs are
(cid:12)an(cid:143)(cid:144)dB % (cid:131)E≥(cid:12)(cid:143)(cid:146)B%(cid:131)E,
defined in a similar manner in Definition 11.
the number of times the value of the phase term of computing the number of times the value of the phase term of
element exceeds the value of the phase term of element
(cid:132)(cid:134)(cid:137)(cid:133)(cid:138)(cid:139) =
(cid:7)(cid:143)(cid:144)B%(cid:131)E≥ t(cid:7)h(cid:143)e(cid:146) Bn%u(cid:131)mE,ber of times the value of the phase term of Step 3: C(cid:140)a’l(cid:131)culate the score function (cid:140)(cid:141)(cid:131).
(cid:135)(cid:134)(cid:137)(cid:133)(cid:138)(cid:139) =
Compute the scores for each element using
(cid:11)(cid:143)(cid:144)B%(cid:131)E≥ t(cid:11)h(cid:143)e(cid:146) Bn%(cid:131)uEm,ber of times the value of the phase term of Definition 13. The score values obtained are given in Table 2.
(cid:147)’ R’(cid:9)9=1,2,… ,(cid:149)(cid:10)
(cid:136)(cid:134)(cid:137)(cid:133)(cid:138)(cid:139) =
Table 2. Score function for
(cid:12)(cid:143)(cid:144)B%(cid:131)E≥(cid:12)(cid:143)(cid:146)B%(cid:131)E.
Definition 13. The score of an element u can be calculated by H(cid:8)
i
the score function which is defined as " (cid:147)917
(cid:147)’ (cid:147)’ =∑(cid:131)(cid:130)’(cid:131). R1 11
Next, we apply the algorithm and score function in a decision
making problem. The steps are as given below: R2 1
Step 1: Define a CNSS R3 19
Construct a CNSS for the problem that is being studied, which R4
includes the elements and the set of Step 4: Conclusion and discussion
parameters R’(cid:9)9 th=at1 a,r2e, b…e i,n(cid:149)g (cid:10)c,onsidered. The values of the score function are compared and the element
In the con%te(cid:131)(cid:9)x(cid:150)t =of1 ,t2h,is… ,e(cid:151)x(cid:10)a,mple, the universal set set of with the maximum score will be chosen as the optimal alternative.
In the event that there are more than one element with the
parameters and the CNSS that were defined in Example 1
", maximum score, any of the elements may be chosen as the
will be used.
(cid:1), H(cid:8) optimal alternative.
Step 2: Construct and compute the comparison matrix In the context of this example, As such, it can be
A comparison matrix is constructed, and the values of for each concluded that country i.e. Im(cid:153)n(cid:144)d∈ao(cid:154)xn(cid:3)e(cid:147)s’i(cid:17)a= isR tQh.e country with the
element and the corresponding parameter is calcu(cid:130)l’a(cid:131)ted using strongest economic indiRcaQtors, followed closely by the Republic
the formula given in Definition 12. For this example, the of Phillipines and Vietnam, whereas Myanmar is identified as the
R’ %(cid:131)
comparison matrix is given in Table 1. country with the weakest and slowest growing economy, among
the four South East Asian countries that were considered.
Table 1.Comparison matrix for
H(cid:8)
VI. CONCLUSION
" %O %C %P %Q In this paper, we introduced the complex neutrosophic soft set
6 3 3 5
model which is a hybrid between the complex neutrosophic set and
R1 soft set models. This model has a more generalized framework
3 5 1 2
than the fuzzy soft set, neutrosophic set, complex fuzzy set models
R2 and their respective generalizations. The basic set theoretic
4 -3 1 -1
operations were defined. The CNSS model was then applied in a
R3 decision making problem involving to demonstrate its utility in
-1 7 5 8 representing the uncertainty and indeterminacy that exists when
dealing with uncertain and subjective data.
R4
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