Neutrosophic Sets and Systems. Book series, vol. 7

2015·5.5 MB·English

by  Florentin Smarandache

Neutrosophic Sets and Systems
An International Book Series in Information Science and Engineering
Quarterly
Editor-in-Chief: Associate Editor:
Mumtaz Ali
Prof. Florentin Smarandache
Address: Quaid-i-azam University Islamabad, Pakistan
Address:
Associate Editors:
“Neutrosophic Sets and Systems”
Dmitri Rabounski and Larissa Borissova, independent researchers.
(An International Journal in Information
W. B. Vasantha Kandasamy, IIT, Chennai, Tamil Nadu, India.
Science and Engineering)
Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco.
Department of Mathematics and Science
A. A. Salama, Faculty of Science, Port Said University, Egypt.
University of New Mexico
Yanhui Guo, School of Science, St. Thomas University, Miami, USA.
705 Gurley Avenue
Francisco Gallego Lupiaňez, Universidad Complutense, Madrid, Spain.
Gallup, NM 87301, USA
Peide Liu, Shandong Universituy of Finance and Economics, China.
E-mail: [email protected]
Pabitra Kumar Maji, Math Department, K. N. University, W. B, India.
Home page: http://fs.gallup.unm.edu/NSS
S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia.
Jun Ye, Shaoxing University, China.
Ştefan Vlăduţescu, University of Craiova, Romania.
Valeri Kroumov, Okayama University of Science, Japan.
Volume 7 Contents 2015
H. E. Khalid. An Origional Notion to Find Maximal Neutrosophic Soft Sets………………………...………. 53
Solution in the Fuzzy Neutosophic Relation Equations 3
K. Mondal, and S. Pramanik. Neutrosophic Decision
(FNRE) with Geometric Programming (GP)……….. …
Making Model of School Choice …………………...…. 62
K. Mondal, and S. Pramanik. Rough Neutrosophic
S. Broumi, and F. Smarandache. Soft Interval-Valued
Multi-Attribute Decision-Making Based on Grey Rela- 8
Neutrosophic Rough Sets…………...…………………. 69
tional Analysis………………………………………….
M. Ali, M. Shabir, F. Smarandache, and L. Vladareanu
A. A. Salama. Basic Structure of Some Classes of Neu-
Neutrosophic LA-semigroup Rings……………………. 81
trosophic Crisp Nearly Open Sets and Possile Applica- 18
tion to GIS Topology..………………………………….
S. Broumi, and F. Smarandache. Interval Neutrosophic
Rough Set…………... …………………………………. 23
F. Yuhua. Examples of Neutrosophic Probability in
Physics……………....…………………………………. 32
M. Ali, F. Smarandache, S. Broumi, and M. Shabir. A
New Approach to Multi-spaces Through the Applica- 34 The Educational Publisher Inc.
tion of Soft Sets...…...………………………………….
1313 Chesapeake Ave.
V. Patrascu. The Neutrosophic Entropy and its Five
Components……………………………………………. 40 Columbus, Ohio 43212, USA.
S. Ye, J. Fu, and J. Ye. Medical Diagnosis Using Dis-
tance-Based Similarity Measures of Single Valued 47
Neutrosophic Multisets ...…………………………….
A. Hussain, and M. Shabir. Algebraic Structures Of
Copyright © Neutrosophic Sets and Systems
Neutrosophic Sets and Systems, Book Series, Vol. 7, 2015
Neutrosophic Sets and Systems
An International Book Series in Information Science and Engineering
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Copyright © Neutrosophic Sets and Systems
Neutrosophic Sets and Systems, Vol. 7, 2015 3
An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic
Relation Equations (FNRE) with Geometric Programming (GP)
Dr. Huda E. Khalid
University of Telafer, Head ; Mathematics Department, College of Basic Education, Telafer- Mosul –
Iraq. E-mail: [email protected]
Abstract. In this paper, finding - a maximal attempt to establish the structure of solution set
solution is introduced to (𝑉,𝛬) fuzzy on model. The NRE have a wide applications
neutrosophic relation equation. the notion of in various real world problems like flow rate in
fuzzy relation equation was first investigated chemical plants, transportation problem, study
by Sanchez in 1976, while Florentin of bounded labor problem, study of
Smarandache put forward a fuzzy interrelations among HIV/AIDS affected
neutrosophic relation equations in 2004 with patients and use of genetic algorithms in
innovative investigation. This paper is first chemical problems .
Keyword Neutrosophic Logic, Neutrosophic Relation Equations (NRE),Integral Neutrosophic
lattices, Fuzzy Integral Neutrosophic Matrices, Maximal Solution, Fuzzy Geometric Programming
(FGP).
Introduction
The analysis of most of the real world 𝑇, i varies in 𝐼,𝑓 varies in 𝐹. Statically 𝑇,𝐼,𝐹
problems involves the concept of are subsets, but dynamically 𝑇,𝐼,𝐹 are
indeterminacy. One cannot establish or cannot functions operators depending on many known
rule out the possibility of some relation but or unknown parameters.
says that cannot determine the relation or link;
this happens in legal field, medical diagnosis 1.2 Physics Example for Neutrosophic
even in the construction of chemical flow in Logic [4]
industries and more chiefly in socio economic
problems prevailing in various countries.[4], as For example the Schrodinger's Theory says
well as the importance of geometric that the quantum state of a photon can
programming and the fuzzy neutrosophic basically be in more than one place in the same
relation equations in theory and application, I time , which translated to the neutrosophic set
have proposed a new structure for maximum means that an element (quantum state) belongs
solution in FNRE with geometric and does not belong to a set (one place) in the
programming. same time; or an element (quantum state)
belongs to two different sets (two different
1.1 Definition [4] places) in the same time. It is a question of
“alternative worlds” theory very well
Let 𝑇,𝐼,𝐹 be real standard or non- represented by the neutrosophic set theory. In
standard subsets 𝑜𝑓 , with Schroedinger’s Equation on the behavior of
electromagnetic waves and “matter waves” in
- +
𝑠𝑢𝑝 𝑇 = 𝑡_𝑠𝑢𝑝,𝑖𝑛𝑓] 0𝑇,1 =[ 𝑡_𝑖𝑛𝑓,𝑠𝑢𝑝 𝐼 = quantum theory, the wave function Psi which
𝑖_𝑠𝑢𝑝,𝑖𝑛𝑓 𝐼 = 𝑖_𝑖𝑛𝑓,𝑠𝑢𝑝 𝐹 = describes the superposition of possible states
𝑓_𝑠𝑢𝑝,𝑖𝑛𝑓 𝐹 = 𝑓_𝑖𝑛𝑓,𝑎𝑛𝑑 𝑛_𝑠𝑢𝑝 = may be simulated by a neutrosophic function,
𝑡_𝑠𝑢𝑝+𝑖_𝑠𝑢𝑝+𝑓_𝑠𝑢𝑝,𝑛_𝑖𝑛𝑓 = 𝑡_𝑖𝑛𝑓+ i.e. a function whose values are not unique for
𝑖_𝑖𝑛𝑓+𝑓_𝑖𝑛𝑓.𝐿𝑒𝑡 𝑈 each argument from the domain of definition
(the vertical line test fails, intersecting the
be a universe of discourse, and 𝑀 a set graph in more points). Don’t we better
included in 𝑈. An element 𝑥 from U is noted describe, using the attribute “neutrosophic”
with respect to the set 𝑀 𝑎𝑠 𝑥(𝑇,𝐼,𝐹) and than “fuzzy” or any others, a quantum particle
belongs to 𝑀 in the following way: It is 𝑡% that neither exists nor non-exists?
true in the set, 𝑖% indeterminate (unknown if it
is) in the set, and 𝑓% false, where 𝑡 varies in
Dr. Huda E. Khalid, An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equations
(FNRE) with Geometric Programming (GP)
Neutrosophic Sets and Systems, Vol. 7, 2015 4
1.3 Application for Neutrosophic to intuitionistic fuzzy set, of "indeterminacy" -
due to unexpected parameters hidden in some
Logic [4]
sets, and let the superior limits of the
components to even boil over 1 (over flooded)
A cloud is a neutrosophic set, because its
and the inferior limits of the components to
borders are ambiguous, and each element
even freeze under 0 (under dried). For
(water drop) belongs with a neutrosophic
example: an element in some tautological sets
probability to the set. (e.g. there are a kind of
may have 𝑡 > 1, called "over included".
separated water props, around a compact mass
Similarly, an element in a set may be "over
of water drops, that we don't know how to
indeterminate" (for 𝑖 > 1, in some paradoxist
consider them: in or out of the cloud). Also, we
sets), "over excluded" (for 𝑓 > 1, in some
are not sure where the cloud ends nor where it
unconditionally false appurtenances); or
begins, neither if some elements are or are not
in the set. That's why the percent of "under true" (for 𝑡 < 0, in some
indeterminacy is required and the neutrosophic unconditionally false appurtenances), "under
probability (using subsets - not numbers - as indeterminate" (for 𝑖 < 0, in some
components) should be used for better unconditionally true or false appurtenances),
modeling: it is a more organic, smooth, and "under fals some unconditionally true
especially accurate estimation. Indeterminacy appurtenances). This is because we should
is the zone of ignorance of a proposition’s make a distinction between unconditionally
value, between truth and falsehood. From the true (𝑡 > 1,𝑎𝑛𝑑 𝑓 < 0 𝑜𝑟 𝑖 < 0) and
intuitionistic logic, paraconsistent logic, conditionally true appurtenances
dialetheism, fallibilism, paradoxes, (𝑡 [ 1,𝑎𝑛𝑑 𝑓 [ 1 𝑜𝑟 𝑖 [ 1). In a rough set RS, an
pseudoparadoxes, and tautologies we transfer element on its boundary-line cannot be
the "adjectives" to the sets, i.e. to intuitionistic classif ied neither as a member of RS nor of its
set (set incompletely known), paraconsistent complement with certainty. In the neutrosophic
set, dialetheist set, faillibilist set (each element set a such element may be characterized by
has a percentage of indeterminacy), paradoxist 𝑥(𝑇,𝐼,𝐹), with corresponding set-values for 𝑇,
set (an element may belong and may not 𝐼,𝐹 . One first presents the evolution of
belong in the same time to the set), sets fro m fuzzy set to neutrosophic set. Then
pseudoparadoxist set, and tautological set on e ]in-0tr,o1d+u[ces the neutrosophic components 𝑇,
respectively. hence, the neutrosophic set 𝐼, 𝐹 which represent the membership,
generalizes: indeterminacy, and non-membership values
re spectively, where is the non-standard
• the intuitionistic set, which supports unit interval, and thus one defines the
incomplete set theories (𝑓𝑜𝑟 0 < 𝑛 < neutrosophic set.[4] ]-0,1+[
1,0 [ 𝑡,𝑖,𝑓 [ 1) and incomplete known
elements belonging to a set; 2 Basic Concepts for NREs
• the fuzzy set (for 𝑛 = 1 𝑎𝑛𝑑 𝑖 = 2.1 Definition [3].
0,𝑎𝑛𝑑 0 [ 𝑡,𝑖,𝑓 [ 1);
A Brouwerian lattice L in which, for any
•the classical set (for 𝑛 = 1 𝑎𝑛𝑑 𝑖 = 0, with
given elements 𝑎 & 𝑏 the set of all 𝑥 ∈𝐿 such
𝑡,𝑓 𝑒𝑖𝑡ℎ𝑒𝑟 0 𝑜𝑟 1);
that 𝑎Ʌ𝑥 ≤𝑏 contains a greatest element,
• the paraconsistent set (for 𝑛 >
denoted 𝑎∝𝑏, the relative pseudocomplement
1,𝑤𝑖𝑡ℎ 𝑎𝑙𝑙 𝑡,𝑖 ,𝑓 < 1 );
of 𝑎 𝑖𝑛 𝑏 [san].
•the faillibilist set (𝑖 +> 0);
• the dialethe ist set, a set 𝑀 whose at least one 2.2 Remark [4]
of its elements also belongs to its complement
If 𝐿 = [0,1], then it is easy to see that for
𝐶(𝑀); thus, the intersection of some disjoint
any given 𝑎,𝑏 ∈𝐿,
sets is not empty
1 𝑎≤𝑏
•the paradoxist set (𝑡 = 𝑓 = 1); 𝑎∝𝑏={
𝑏 𝑎>𝑏
• the pseudoparadoxist set (0 < 𝑖 < 1,𝑡 =
1 𝑎𝑛𝑑 𝑓 > 0 𝑜𝑟 𝑡 > 0 𝑎𝑛𝑑 𝑓 = 1); 2.3 Definition [4]
•the tautological set (𝑖 ,𝑓 < 0). Let 𝑁 = 𝐿 ∪{𝐼} where L is any lattice and I
Compared with all other types of sets, in the an indeterminate.
neutrosophic set each element has three Define the max, min operation on N as
components which are subsets (not numbers as follows
in fuzzy set) and considers a subset, similarly 𝑀𝑎𝑥 {𝑥,𝐼} = 𝐼 for 𝑎𝑙𝑙 𝑥 ∈𝐿\ {1}
Dr. Huda E. Khalid, An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equations (FNRE) with
Geometric Programming (GP)
5 Neutrosophic Sets and Systems, Vol. 7, 2015
class of 𝑚 × 𝑛 matrices is contained in the
𝑀𝑎𝑥 {1,𝐼} = 1 class of fuzzy integral neutrosophic matrices.
𝑀𝑖𝑛 {𝑥,𝐼} = 𝐼 for 𝑎𝑙𝑙 𝑥 ∈ 𝐿 \ {0}
𝑀𝑖𝑛 {0,𝐼} = 0 2.7 Example [4]
We know if 𝑥,𝑦 ∈ 𝐿 then max and min are
well defined in L. N is called the integral 𝐼 0.1 0
Let 𝐴 =( )
neutrosophic lattice. 0.9 1 𝐼
2.3 Example [4]
A is a 2×3 integral fuzzy neutrosophic
Let 𝑁 = 𝐿 ∪{𝐼} given by the following
matrix. We define operation on these matrices.
diagram:
An integral fuzzy neutrosophic row vector is a
1×𝑛 integral fuzzy neutrosophic matrix.
Similarly an integral fuzzy neutrosophic
column vector is a 𝑚 × 1 integral fuzzy
I
neutrosophic matrix.
Clearly [4]
2.8 Example [4]
𝑀𝑖𝑛 {𝑥,𝐼 } = 𝐼 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝐿\ {0}
𝑀𝑖𝑛 {0,𝐼 } = 0
𝐴 = (0.1,0.3,1,0,0,0.7,𝐼,0.002,0.01
𝑀𝑎𝑥 {𝑥,𝐼 } = 𝐼 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝐿 \ {1}
𝑀𝑎𝑥 {1,𝐼 } = 1 ,𝐼,1,0.12) is a integral row vector or a

We see N is an integral neutrosophic lattice
1 × 1 , integral fuzzy neutrosophic matrix.
and clearly the order of N is 6.
2.9 Ex2ample [4]
2.4 Remark [4]
𝐵 = (1,0.2,0.111,𝐼,0.32,0.001,𝐼,0,1) 𝑇 is
1- If L is a lattice of order n and N = L ∪ {I }
an integral neutrosophic column vector or B is
be an integral neutrosophic lattice then order of
a 9 × 1 integral fuzzy neutrosophic matrix.
N is n + 1.
We would be using the concept of fuzzy
2. For an integral neutrosophic lattice
neutrosophic column or row vector in our
𝑁 also {0} is the minimal element and {1} is
study.
the maximal element of N.
2.10 Definition [4]
2.5 Conventions About Neutrosophic
Sets [4] Let P = (p ) be a m × n integral fuzzy
ij
neutrosophic matrix and Q = (q ) be a n ×
ij
Let 𝐴,𝐵 ∈𝑁(𝑋) 𝑖.𝑒., p integral fuzzy neutrosophic matrix. The
𝐴∶ 𝑋 → [0,1]∪ 𝐼 ,𝐵 ∶𝑋 →[0,1] ∪ 𝐼 composition map P Q is defined by R
(𝐴 ∩𝐵 ) (𝑥 ) = 𝑚𝑖 𝑛 {𝐴 (𝑥),𝐵 (𝑥)},if (r ) which is a m ×= p matrix where
ij
𝐴 (𝑥) = 𝐼 or 𝐵 ( 𝑥) = 𝐼 th en (𝐴 ∩ 𝐵) (𝑥) is o
r =maxmin(p q )
define d to be 𝐼 i.e., 𝑚𝑖𝑛 {𝐴 (𝑥 ),𝐵 (𝑥))} = ( ij ik kj ) with the
k
𝐼 (𝐴 ∪𝐵) (𝑥) = 𝑚𝑎𝑥 {𝐴 (𝑥),𝐵 (𝑥)} if one
assumption max(p ,I)= I and min(p ,I) =
of 𝐴 (𝑥) = 𝐼 𝑜𝑟 𝐵 (𝑥) = 𝐼 𝑡ℎ𝑒 𝑛 (𝐴 ∪ I9 ij ij
I where pij ϵ(0,1). min (0,I)= 0 and
𝐵) (𝑥) = 𝐼 i.e., 𝑚𝑎𝑥 {𝐴 (𝑥),𝐵 (𝑥)} = 𝐼.
max(1,I) = 1.
Thus it is pertinent to mention here that if one
of 𝐴 (𝑥)= 𝐼 𝑜𝑟 𝐵(𝑥)= 𝐼 then (𝐴∪𝐵)(𝑥)=
2.11 Example [4]
(𝐴 ∩ 𝐵)( 𝑥). i.e., is the existence of
indeterminacy 𝑚𝑎𝑥 {𝐴 (𝑥),𝐵 (𝑥)}=
0.3 𝐼 1
𝑚𝑖𝑛 {𝐴 (𝑥),𝐵(𝑥)} = 𝐼 Let 𝑝=[ 0 0.9 0.2] ,𝑄 =(0.1,𝐼,0)𝑇
𝐴̅ (𝑥)= 1− 𝐴 (𝑥); 𝑖𝑓 𝐴 (𝑥)= 𝐼
0.7 0 0.4
𝑡ℎ𝑒𝑛 𝐴̅ (𝑥) = 𝐴 (𝑥) = 𝐼. be two integral fuzzy neutrosophic matrices.
0.3 𝐼 1 0.1
2.6 Definition [4] P Q=[ 0 0.9 0.2] [ 𝐼 ]=
0.7 0 0.4 0
Let 𝑁 = [0,1] ∪ 𝐼 where I is the (𝐼o ,𝐼 ,0 .1) o
indeterminacy. The 𝑚 × 𝑛 matrices 𝑀 = T
𝑚×𝑛

{(𝑎 ) / 𝑎 ∈ [0,1] ∪ 𝐼} is called the fuzzy
𝑖𝑗 𝑖𝑗
integral neutrosophic matrices. Clearly the
Dr. Huda E. Khalid, An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equations (FNRE)
with Geometric Programming (GP)
Neutrosophic Sets and Systems, Vol. 7, 2015 6
3 Structure of the Maximal Solution [0,1] or equal to 𝐼 , so we have that following
status:
Set
1- If 𝑎∈[0,1] & 𝑏∈𝐼 , 𝑎𝛼𝑏=𝑥 =𝐼 where
B.Y.Cao proposed the structure set of maximal
𝑎≠0 therefore 𝑎∈(0,1] , here we must
and minimal solution for FRGP (fuzzy relation
remember that min(𝐼,𝑥)=𝐼 ∀ 𝑥 ∈(0,1]∪
geometric programming) with (V,Λ) operator
𝐼.
[optimal models & meth with fuzzy quantities
2- If 𝑎 =𝐼 & 𝑏∈[0,1], then 𝑎𝛼𝑏=𝑥 =0
2010] [1-2]. Its useful and neces sary to call
,here min(𝐼,𝑥)=𝐼 ∀ 𝑥 ∈(0,1]∪𝐼 also
back the following ideas
min(𝐼,0)=0.
3- At 𝑎&𝑏∈[0,1], the solution will back to
3.1 Definition
the same case that stated by Sanchez ,i.e.
1 𝑎≤𝑏
If 𝑋(𝐴,𝑏)≠∅ it can be completely 𝑎𝛼𝑏={
𝑏 𝑎>𝑏
determined by a unique maximum solution and
4- At 𝑎=𝑏=𝐼 ,this implies that 𝑎𝛼𝑏=1.
a finite number of minimal solution. The
maximum solution can be obtained by Note that , 𝑎˄𝑥 ≤𝑏 →min(𝑎,𝑥)≤𝑏 →
applying the following operation:-
min(𝐼,𝑥)≤𝐼 →𝑥 =1
𝑥̂𝑗=˄{𝑏𝑖|𝑏𝑖<𝑎𝑖𝑗} (1≤𝑖≤𝑚,1≤𝑗≤𝑛) Consequently :
Stipulate that set {˄∅=1}. If 𝑥̂ =
(𝑥 ̂ ,𝑥̂ , … ,𝑥̂) is a so lut io n t o 𝐴 𝜊𝑋 = 1 𝑎 ≤𝑏 𝑜𝑟 𝑎 =𝑏 =𝐼
𝑏 . th1en 𝑥̂ 2m𝑛ust be thT e g reatest sjo lution. 𝑏 𝑎𝑖𝑗 >𝑏𝑖 𝑖𝑗 𝑖𝑗
. 𝑖 𝑖𝑗 𝑖
0 𝑎 =𝐼 𝑎𝑛𝑑 𝑏 =[0,1]
3.2 An original notion to find 𝑖𝑗 𝑖𝑗
maximal solution 𝑥̂𝑗 =𝑎𝛼𝑏= 𝐼 𝑏𝑖 =𝐼 𝑎𝑛𝑑 𝑎𝑖𝑗 =(0,1]
𝑛𝑜𝑡 𝑐𝑜𝑚𝑝.𝑎 =0 𝑎𝑛𝑑 𝑏 =𝐼
𝑖𝑗 𝑖𝑗
The most important question:
{
What is the structure of the maximum element
for any fuzzy neutrosophic relation equations 3.3 Lemma
in the interval [0,1]∪𝐼 ??
If 𝑎 =0 𝑎𝑛𝑑 𝑏 =𝐼 then 𝐴𝜊𝑋 =𝑏 is not
𝑖𝑗 𝑖
We know that compatible.
max{0,𝐼}=𝐼 & max{𝑥,𝐼}=𝐼 ∀ 𝑥 ∈ Proof
[0,1)∪𝐼.
Let 𝑎 =0 , 𝑏 =𝐼
𝑖𝑗 𝑖
Depending upon the definition ( 3.1), the
stipulation that {˄∅=1} will be fixed. What is the value of 𝑥 ∈[0,1]∪𝐼 satisfying
𝑗
Also by Sanchez (1976) we have ˅𝑛 (𝑎 ˄𝑥 )=𝑏 ∀ 1≤𝑖 ≤𝑚 ?
𝑖𝑗 𝑖𝑗 𝑗 𝑖
1 𝑎≤𝑏
𝑎𝛼𝑏 ={ We have
𝑏 𝑎>𝑏
1- 𝑎𝛼𝑏=𝑎 ˄𝑥 ≤𝑏
Don’t forget that 𝛼 is relative psedo 𝑖𝑗 𝑗 𝑖
2- min(0,𝑥 )=0 ∀ 𝑥 ∈[0,1]∪𝐼
complement of 𝑎 𝑖𝑛 𝑏 . On the other hand, 𝑗 𝑗
Florentin was deffind the neutrosophic lattice
see ref. [4] p.235 So 𝑎𝛼𝑏=𝑎𝑖𝑗˄𝑥𝑗 =min(0,𝑥𝑗)=0 not equal
nor less than to 𝐼
So if we want to establish the maximum
solution for any FNRE in the interval [0,1]∪𝐼 We know that the incomparability occurs only
, we must redefine 𝑎𝛼𝑏 where 𝑎𝛼𝑏=𝑎˄𝑥 ≤ when 𝑥 ∈𝐹𝑁 𝑎𝑛𝑑 𝑦∈[0,1] see ref. [4] p.233
𝑏 ;𝑎,𝑥,𝑏 ∈[0,1]∪𝐼 .
∴𝐴𝜊𝑋=𝑏 is not compatible
Note that, all matrices in this work are an
integral fuzzy neutrosophic matrices. It is
obvious that 𝑎 or 𝑏 are either belonging to
Dr. Huda E. Khalid, An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equations (FNRE)
with Geometric Programming (GP)
Neutrosophic Sets and Systems, Vol. 7, 2015 7
Without loss of generality , suppose 𝑏 ≥ s.t. 𝐴𝜊𝑥 =𝑏 where
1
𝑏 ≥𝑏 ≥⋯≥𝑏 when we rearranged the
2 3 𝑛
components of 𝑏 in decreasing order we also 𝐼 .2 .85 .9
adjusted 𝐴,𝑥 and 𝑓(𝑥) accordingly 𝑏. .8 .2 𝐼 .1
𝐴=[ ]
Now, in fuzzy neutrosophic numbers, how can .9 .1 𝐼 .6
we classify numbers to rearrange them? For 𝐼 .8 .1 𝐼
more details see ref. [5] page 245.
𝑏=(𝐼,.6,.5,𝐼) , 0≤𝑥 ≤1 ,1≤𝑖 ≤4
𝑖
3.4 Example
𝑥̂ =˄(1,1,.5,1 )=0.5
1
𝑥̂ =˄(𝐼,1,1,𝐼 )=𝐼
Rearranged the following matrices in 2
decreasing order 𝑥̂ =˄(𝐼,0,0,𝐼 )=0
3
0.85 𝐼 𝐼 𝑥̂ =˄(𝐼,1,.5,1 )=𝐼
4
𝐼 𝐼 0.6
1)𝑏=[ ] 2) 𝑏=[ ] 3) 𝑏=[ ]
0.5 0.5 𝐼
∴ 𝑥̂ =(0.5,𝐼,0,𝐼 )𝑇
𝐼 0.1 0.1
Solution.
𝐼 𝐼 𝐼 Conclusion
𝐼 𝐼 𝐼
1)𝑏=[ ] 2) 𝑏=[ ] 3) 𝑏=[ ]
0.85 0.5 0.6
0.5 0.1 0.1
In this article, the basic notion for finding
maximal solution in a geometric programming
subject to a system of fuzzy neotrosophic
4 Numerical Examples:-
relational equation with max-min composition
was introduced. In 1976,Sanchez gave the
Find the maximum solution for the following
formula of the maximal solution for fuzzy
FNREGP problems:-
relation equation concept and describing in
details its structure. Some numerical examples
1)
have shown that the proposed method is
𝑚𝑖𝑛𝑓(𝑥)= 5𝑥−.5𝑥−1.5𝑥2𝑥−2𝑥−1
1 2 3 4 5 betimes step to enter in this kind of problems
to search for minimal solutions which remains
𝐼 0.8 0.6 𝐼 0 𝐼
𝑥1 as unfathomable issue.
𝑠.𝑡. 𝐼000...386 0.0002...379 000 𝐼...588 000.16..29 0000....1852 𝜊 𝑥𝑥𝑥234 = ..𝐼𝐼55 AIh ewclpko unladon wdli kleemd ytog etdmheaenepkn tmtsh ya nhku sbtoa ndS ifro r Phriosf .fDurll.
[𝑥 ]
[ 𝐼 0.1 0.2 0.3 𝐼 ] 5 [.4] Florentin Smarandache for his advice and kind
assistance to publish this paper.
Solution :- The greatest solution is
𝑥̂ =˄{1,𝐼,𝐼,0,1,0}=0 Reference
1
𝑥̂2 =˄{𝐼,𝐼,𝐼,1,1,1}=𝐼 [1] B. Y. Cao "Optimal models and methods
𝑥̂3 =˄{𝐼,𝐼,𝐼,0,1,1}=0 with fuzzy quantities". Springer-Verlag, Berlin
𝑥̂4 =˄{1,𝐼,𝐼,.5,1,1}=𝐼 (2010).
𝑥̂5 =˄{𝑛𝑜𝑡 𝑐𝑜𝑚𝑝𝑎𝑟𝑎𝑏𝑙𝑒,…….} [2] B.Y. Cao and J. Yang "Monomial
geometric programming with fuzzy relation
Therefore by lemma (3.3) the system 𝐴𝜊𝑥 =𝑏 equation constraints". Fuzzy Optimization and
is not comparable. Decision Making 6 (4): 337-349(2007).
[3] E. Sanchez, Resolution of Composite
2) Fuzzy Relation Equations,Information and
min𝑓(𝑥)= Control 30,38-48 (1976)
(1.5𝐼𝛬𝑥1.5)𝑉(2𝐼𝛬𝑥2)𝑉(.8𝛬𝑥3−.5)𝑉(4𝛬𝑥4−1) [4] V. Kandasamy & F. Smarandache, Fuzzy
Relational Maps and Neutrosophic Relational
Maps. American Research Press, Rehoboth,2004.
[5] V. Kandasamy & F. Smarandache, Fuzzy
Iinterval Matrices Neutrosophic Interval
Received: October 12, 2014. Accepted: December 20, 2014.
Dr. Huda E. Khalid, An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equations (FNRE) with
Geometric Programming (GP)
Neutrosophic Sets and Systems, Vol. 7, 2015 8
Rough Neutrosophic Multi-Attribute Decision-Making
Based on Grey Relational Analysis
Kalyan Mondal1, and Surapati Pramanik2*
¹Birnagar High School (HS), Birnagar, Ranaghat, District: Nadia, Pin Code: 741127, West Bengal, India. E mail:[email protected]
²*Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, PO-Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West
Bengal, India.
*Correosponding Address: Email: [email protected]
Abstract. This paper presents rough netrosophic multi- grey relational analysis method and apply it to multi-
attribute decision making based on grey relational attribute decision making problem. Information entropy
analysis. While the concept of neutrosophic sets is a method is used to obtain the partially known attribute
powerful logic to deal with indeterminate and weights. Accumulated geometric operator is defined to
inconsistent data, the theory of rough neutrosophic sets transform rough neutrosophic number (neutrosophic pair)
is also a powerful mathematical tool to deal with to single valued neutrosophic number. Neutrosophic grey
incompleteness. The rating of all alternatives is relational coefficient is determined by using Hamming
expressed with the upper and lower approximation distance between each alternative to ideal rough
operator and the pair of neutrosophic sets which are neutrosophic estimates reliability solution and the ideal
characterized by truth-membership degree, rough neutrosophic estimates un-reliability solution.
indeterminacy-membership degree, and falsity- Then rough neutrosophic relational degree is defined to
membership degree. Weight of each attribute is partially determine the ranking order of all alternatives. Finally, a
known to decision maker. We extend the neutrosophic numerical example is provided to illustrate the
grey relational analysis method to rough neutrosophic applicability and efficiency of the proposed approach.
Keywords: Neutrosophic set, Rough Neutrosophic set, Single-valued neutrosophic set, Grey relational analysis, Information
Entropy, Multi-attribute decision making.
Introduction the concept of rough neutrosophic sets [39, 40] is recently
proposed and very interesting. Literature review reveals
The notion of rough set theory was originally proposed by
that only two studies on rough neutrosophic sets [39, 40]
Pawlak [1, 2]. The concept of rough set theory [1, 2, 3, 4]
are done.
is an extension of the crisp set theory for the study of
Neutrosophic sets and rough sets are two different
intelligent systems characterized by inexact, uncertain or
concepts. Literature review reflects that both are capable of
insufficient information. It is a useful tool for dealing with
handing uncertainty and incomplete information. New
uncertainty or imprecision information. It has been
hybrid intelligent structure called “rough neutrosophic
successfully applied in the different fields such as artificial
sets” seems to be very interesting and applicable in
intelligence [5], pattern recognition [6, 7], medical
realistic problems. It seems that the computational
diagnosis [8, 9, 10, 11], data mining [12, 13, 14], image
techniques based on any one of these structures alone will
processing [15], conflict analysis [16], decision support
not always provide the best results but a fusion of two or
systems [17,18], intelligent control [19], etc. In recent
more of them can often offer better results [40].
years, the rough set theory has caught a great deal of
Decision making process evolves through crisp
attention and interest among the researchers. Various
environment to the fuzzy and uncertain and hybrid
notions that combine the concept of rough sets [1], fuzzy
environment. Its dynamics, adaptability, and flexibility
sets [20], vague set [21], grey set [22, 23] intuitionistic
continue to exist and reflect a high degree of survival value.
fuzzy sets [24], neutrosophic sets [25] are developed such
Approximate reasoning, fuzziness, greyness, neutrosophics
as rough fuzzy sets [26], fuzzy rough sets [27, 28, 29],
and dynamic readjustment characterize this process. The
generalized fuzzy rough sets [30, 31], vague rough set [32],
decision making paradigm evolved in modern society must
rough grey set [33, 34, 35, 36] rough intuitionistic fuzzy
be strategic, powerful and pragmatic rather than retarded.
sets [37], intuitionistic fuzzy rough sets [38], rough
Realistic model cannot be constructed without genuine
neutrosophic sets [ 39, 40]. However neutrosophic set [41,
understanding of the most advanced decision making
42] is the generalization of fuzzy set, intuitionistic fuzzy
model evolved so far i.e. the human decision making
set, grey set, and vague set. Among the hybrid concepts,
Kalyan Mondal, and Surapati Pramanik, Rough Neutrosophic Multi-Attribute Decision-Making Based on Grey
Relational Analysis
Neutrosophic Sets and Systems, Vol. 7, 2015 9
process. In order to perform this, very new hybrid concept [25] gains very popularity because of its capability to deal
such as rough neutrosophic set must be introduced in with the origin, nature, and scope of neutralities, as well as
decision making model. their interactions with different conceptional spectra.
Decision making that includes more than one measure of Definition2.1.1: Let E be a space of points (objects) with
performance in the evaluation process is termed as multi- generic element in E denoted by y. Then a neutrosophic set
attribute decision making (MADM). Different methods of N1 in E is characterized by a truth membership function
MADM are available in the literature. Several methods of T , an indeterminacy membership function I and a
N1 N1
MADM have been studied for crisp, fuzzy, intuitionistic falsity membership function F . The functions T and F
N1 N1 N1
fuzzy, grey and neutrosophic environment. Among these, are real standard or non-standard subsets of ]−0,1+[that is
the most popular MADM methods are Technique for Order ] [ ] [ ] [
Preference by Similarity to Ideal Solution (TOPSIS) TN1:E→ −0,1+ ; IN1:E→ −0,1+ ; FN1: E→ −0,1+ .
proposed by Hwang & Yoon [43], Preference Ranking It should be noted that there is no restriction on the
Organization Method for Enrichment Evaluations (PRO- sum of TN1(y),IN1(y),FN1(y) i.e.
MVIEšeTkHriEteEri)j umpskroop oKseOdm prboym isnBo raRnsa ngireatn je al(.V IK[O44R]), -0≤TN1(y)+IN1(y)+FN1(y)≤3+
Definition2.1.2: (complement) The complement of a
developed by Opricovic & Tzeng [45], ELimination Et
neutrosophic set A is denoted by N1c and is defined by
Choix Traduisant la REalité (ELECTRE) studied by Roy
[46], ELECTRE II proposed by Roy and Bertier [47], TN1c(y)={1+}−TN1(y); IN1c(y)={1+}−IN1(y)
ELECTREE III proposed by( Roy [48], ELECTRE IV FN1c(y)={1+}−FN1(y)
proposed by Roy and Hugonnard [49), Analytical
Definition2.1.3: (Containment) A neutrosophic set
Hierarchy Process(AHP) developed by Satty [50], fuzzy
N1 is contained in the other neutrosophic set N2,
AHP developed by Buckley [51], Analytic Network
N1⊆N2if and only if the following result holds.
Process (ANP) studied by Mikhailov [52], Fuzzy TOPSIS ( ) ( ) ( ) ( )
proposed by Chen [53], single valued neutrosophic multi infTN1 y ≤infTN2 y, supTN1 y ≤supTN2 y
( ) ( ) ( ) ( )
criteria decision making studied by Ye [54, 55, 56], infI y ≥infI y, supI y ≥supI y
N1 N2 N1 N2
neutrosophic MADM studied by Biswas et al. [57], ( ) ( ) ( ) ( )
infF y ≥infF y, supF y ≥supF y
N1 N2 N1 N2
Entropy based grey relational analysis method for MADM
for all y in E.
studied by Biswas et al. [58]. A small number of
Definition2.1.4: (Single-valued neutrosophic set).
applications of neutrosophic MADM are available in the
Let E be a universal space of points (objects) with a
literature. Mondal and Pramanik [59] used neutrosophic
generic element of E denoted by y.
multicriteria decision making for teacher selection in
A single valued neutrosophic set [61] S is characterized by
higher education. Mondal and Pramanik [60] also
developed model of school choice using neutrosophic a truth membership function TN(y), a falsity membership
MADM based on grey relational analysis. However, function FN(y) and indeterminacy membership function
[ ]
MADM in rough neutrosophic environment is yet to IN(y)with TN(y),FN(y),IN(y)∈ 0,1 for all y in E.
appear in the literature. In this paper, an attempt has been
When E is continuous, a SNVS S can be written as
made to develop rough neutrosophic MADM based on
follows:
grey relational analysis.
Rest of the paper is organized in the following way. S=∫ TS(y),FS(y),IS(y) y,∀y∈E
y
Section 2 presents preliminaries of neutrosophic sets and
and when E is discrete, a SVNS S can be written as
rough neutrosophic sets. Section 3 is devoted to present
follows:
rough neutrosophic multi-attribute decision-making based
on grey relational analysis. Section 4 presents a numerical S=∑ TS(y),FS(y),IS(y) y,∀y∈E
example of the proposed method. Finally, section 5 It should be observed that for a SVNS S,
( ) ( ) ( )
presents concluding remarks and direction of future 0≤supTS y +supFS y +supIS y ≤3,∀y∈E
research. Definition2.1.5: The complement of a single v alued
neutrosophic set S is denoted bySc and is defined by
2 Mathematical Preliminaries
TSc(y)=FS(y); ISc(y)=1−IS(y); FSc(y)=TS(y)
2.1 Definitions on neutrosophic Set Definition2.1.6: A SVNS S is contained in the other
N1
The concept of neutrosophy set is originated from the new SVNS SN2 , denoted as SN1⊆SN2 iff,TSN1(y)≤TSN2(y);
branch of philosophy, namely, neutrosophy. Neutrosophy I (y)≥I (y); F (y)≥F (y), ∀y∈E.
SN1 SN2 SN1 SN2
Kalyan Mondal, and Surapati Pramanik, Rough Neutrosophic Multi-Attribute Decision-Making Based on Grey
Relational Analysis