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Decision Making: Applications in Management and Engineering
ISSN: 2560-6018
eISSN: 2620-0104
DOI:_ https://doi.org/10.31181/dmame04012023b
* Corresponding author.
Email addresses: nadiabatool1512@gmail.com (N. Batool), sadaqatku@gmail.com (S.
Hussain), kausar.nasreen57@gmail.com (N. Kausar), dr.mohammadmunir@gmail.com (M.
Munir), ritarec1@yahoo.com.hk (Y.M.L.Rita), salmakhan359@gmail.com (S. Khan)
DECISION MAKING UNDER INCOMPLETE DATA:
INTUITIONISTIC MULTI FUZZY IDEALS OF NEAR-RING
APPROACH
Nadia Batool1*, Sadaqat Hussain1, Nasreen Kausar2, Mohammed Munir3,
Rita Yi Man Li4* and Salma Khan5
1Department of Mathematics, University of Baltistan Skardu, Gilgit Baltistan 16100,
Pakistan,
2Department of Mathematics, Faculty of Arts and Sciences, Yildiz Technical University,
Esenler 34220, Istanbul, Turkey,
3Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan,
4Sustainable Real Estate Research Center, Department of Economics and Finance, Hong
Kong Shue Yan University, Hong Kong 999077, China
5Department of Mathematics and Statistics, Hazara University Mansehra 21120,
Khyber Pakhtunkhwa, Pakistan.
Received: 10 September 2022;
Accepted: 30 November 2022;
Available online: 4 January 2023.
Original Scientific Paper
Abstract. Real-world data is often partial, uncertain, or incomplete. Decision
making based on data as such can be addressed by fuzzy sets and related
systems. This article studies the intuitionistic multi-fuzzy sub-near rings and
Intuitionistic multi-fuzzy ideals of near rings. It presents some of the
elementary operations and relations defined on these structures. The concept
of level subsets and support of the Intuitionistic multi-fuzzy sub-near ring is
also presented. It looks into and demonstrated a few characteristics of
intuitionistic multi-fuzzy near-rings and ideals. This research advances fuzzy
set theory, which is often applied to problems involving pattern recognition
and multiple criterion decision-making. Thus, the results may be beneficial to
artificial intelligence related research. Alternatively, the intuitionistic multi-
fuzzy approach may be applied to vector spaces and modules or extended to
inter-valued fuzzy systems.
Keywords: Intuitionistic Fuzzy Set, Near-ring, Fuzzy Multi Near-ring,
Intuitionistic multi fuzzy Near-ring, Ideals, fuzzy multi ring
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
2
1. Introduction
The concept of "fuzzy sets" was initially proposed by Zadeh (Zadeh, 1965) (Zadeh,
1965), which opened the door for set theory researchers. Many versions and
generalizations of fuzzy sets have appeared to solve problems such as multi-criteria
decision-making, recognition of patterns and diagnosis of diseases (Broumi, Ajay, et
al., 2022) (Ashraf et al., 2022; Broumi, Sundareswaran, et al., 2022) waste
management (Zhumadillayeva et al., 2020) and voltage balancing (Taghieh et al.,
2022). Applications of the fuzzy systems to some other areas can be studied (Gulzar,
Alghazzawi, et al., 2020; Gulzar, Mateen, et al., 2020; Kausar, 2019; Kausar et al.,
2020; Riaz et al., 2022b). Due to the appearance of fuzzy sets and their associated
systems as problem-solving tools in MCDM (Abbas et al., n.d.; Abdullah, 2013;
Kahraman, 2008) and other disciplines of our daily life, more people were attracted
towards this area of research. Theoretical mathematicians use this idea to generalize
well-known mathematical structures. For example, Rosenfeld used it to develop a
fuzzy group structure and generalize the classical group. This was the initial
development of fuzzy group theory. Liu introduced fuzzy rings and studied various
properties of rings and ideals in a fuzzy context. Within a few years, the fuzzification
of algebraic structures became a hot topic within the research community.
Researchers developed more fuzzy algebraic structures like fuzzy modules, fuzzy
algebras, fuzzy sub near rings etc. were developed [see (Al-Husban, 2021; Fathi &
Salleh, 2009; Hur et al., 2005; Rahman & Saikia, 2012; Zhan & Ma, 2005)]. Salah
Abuzaid was the first who introduced the notion of fuzzy sub near-rings (Abou-Zaid,
1991) and later, its variants were studied(Asif et al., 2020; Hussain et al., 2022).
The concept of multisets is initiated by Yager (Yager, 1986). The multi-fuzzy
groups were proposed by T. K. Shinoj et al. [see (Dresher & Ore, 1938; Shinoj et al.,
2015)]. They studied the basic properties of multi-fuzzy groups and presented a few
preliminary results. The fuzzy versions of multi-subrings and their ideals were
established by L. Sujatha (Sujatha, 2014) in 2014. She also proved that the finite
multi-fuzzy subrings (ideals) intersection is a multi-fuzzy subring (ideal). Fuzzy
multi-near-rings and their associated multi-ideals were introduced by Tahan et al.
(Al Tahan et al., 2021) in 2021. They defined various operations on multi-ideals of
fuzzy near-rings. They presented foundational results related to fuzzy multi sub-
near-rings, fuzzy multi-ideals and the operations defined on multi-ideals. The anti-
fuzzy multi-ideals of near-rings were considered by Hoskova (Hoskova-Mayerova &
Al Tahan, 2021).
One of the well-known generalizations of fuzzy sets is the intuitionistic fuzzy set
proposed by Atanassov(Atanassov, 1986). Renowned mathematicians also use this
set to fuzzify algebraic structures. Fathi was the first to describe the notion
intuitionistic fuzzy group (Fathi & Salleh, 2009) Consequently, the intuitionistic
versions of groups, rings, ideals, modules, near-rings etc., have been established (Hur
et al., 2005). Many researchers (Kausar & Waqar, 2019; Kousar et al., 2021, 2022;
Riaz et al., 2022a) used intuitionistic fuzzy sets and developed different structures.
These included triangular intuitionistic fuzzy linear programming, lattice-valued
intuitionistic fuzzy subgroup type-3, algebraic codes over lattice-valued intuitionistic
fuzzy type-3 submodules, codes over lattice-valued intuitionistic fuzzy set type-3,
non-associative ordered semi-groups by intuitionistic fuzzy bi ideals. Others further
developed different versions of intuitionistic fuzzy sets, such as complex
intuitionistic fuzzy sets in group theory and t-intuitionistic fuzzy subgroups(Gulzar,
Alghazzawi, et al., 2020; Gulzar, Mateen, et al., 2020). It also encompasses the finite
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
3
intuitionistic anti-fuzzy normal subrings’ direct product (Kausar, 2019) intuitionistic
fuzzy normal subrings (Shah et al., 2012).
The primary purpose of this study is to find the connections between multi-fuzzy
near-rings and intuitionistic fuzzy sets. For this, we first define intuitionistic fuzzy
multi-near-rings (IFM𝑁𝑅s) and ideals associated with this structure. We also define
the basic operations and study the critical properties of this structure. Moreover, we
study the support and level subsets of IFM𝑁𝑅s and produce a few related
results. This work contributes to the fuzzy set theory and fuzzy algebra, which are
extensively used to solve multi-criteria decision-making and pattern recognition
problems (Kaharman et. al., 2008).
2. Motivation and Scope
Fuzzy sets and associated systems deal with theoretical and practical problems
equipped with incomplete, uncertain or ambiguous data. Although this is a very
young discipline started with the work of L. A. Zadeh but gained the attention of
many researchers quickly. The reason is that it can be directly applied to theoretical
and daily life problems, including Multi-criteria decision-making, Pattern
recognition, disease diagnosis and Management. So it is worth studying the fuzzy
systems to produce more sufficient and adequate theoretical bases used to develop
better problem-solving tools. This article also proposes a fuzzy algebraic system
intuitionistic fuzzy multi-near-rings, a generalization of well-known fuzzy multi-
near-rings.
3. Preliminaries
This study first recalls some basic definitions of fuzzy and multi-fuzzy sets.
Definition 2.1: Let 𝒩 be a non-empty set, then a fuzzy set is given by an ordered
pair (Zadeh, 1965)
is the degree of membership, and is the membership function.
Example 1. If then be fuzzy set of.
Definition 2.2: Let 𝒩 be a set that is not empty. An intuitionistic fuzzy set will be
(Atanassov, 1986)
Whereand are the degrees of membership and non-membership function,
respectively, they are defined as and.
For each .
Remark 1. Every fuzzy set is an intuitionistic fuzzy set.
Definition 2.3: Let be a non-empty set, then be a multiset drawn from,
characterized by a count function , where represent the number of
repetition of an element in and is set of positive integers (Yager, 1987, Hoskova
et. al., 2020).
Let be a set; then a multiset will be represented as
where represents the
number of repetition of an element in.
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
4
Example 2. Let be a set, then is a multiset, and it can
also be represented as
.
Definition 2.4: Per Tahan et al. (2021), let be a non-empty set then a fuzzy
multiset constructed from can be represented as;
Where represents the count membership function, here is a set of all
crisp multisets which is constructed from unit interval and for each,
is a decreasingly ordered sequence that is
.
Definition 2.5: Let be a non-empty set. An intuitionistic version of a fuzzy multiset
can be represented (Onasanya et. al., 2019);
represents the count membership function, and
shows the count non-membership function. Here is a set of all crisp multisets
which is constructed from unit interval , and for each, is a
decreasingly ordered sequence that is
and
is denoted by
. For each
.
Remark 2. is ordered decreasingly, but the corresponding may
need to be in decreasing and increasing order.
Remark 3. An intuitionistic fuzzy set on set can be treated as a particular case of
an intuitionistic fuzzy multiset, if
.
Example 3. Let. Then is an intuitionistic fuzzy multiset over with
count functions:
Definition 2.6: According to Tahan et. al (2021), Shinoj et. al., (2015), Hoskova et. al.
(2021), let be a set, and be two intuitionistic fuzzy multisets overwith
fuzzy count functionsand respectively, then:
if and
if and
is defined as;
} and
is defined as;
} and
The complement of an intuitionistic fuzzy multiset is defined as;
Example 4. Let and be two intuitionistic fuzzy multisets over
that is;
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
5
,
Then;
And;
.
Definition 2.7: Let be a non-empty set, then is said to be left (right) if
(Tahan et. al., 2021, Asif, et. al., 2020, Hussain et. al., 2022);
is a group;
is a semi-group;
Satisfies left (right) distributive law, that is;
(Left distributive)
(Right distributive)
Example 5. Let be a set of real numbers, then form under standard
and multiplication is defined as.
Note: Throughout this text, we write instead of near-ring and s for near-rings.
Definition 2.8: Let be a , and be a sub- of then is said to be an
ideal of (Tahan et. al., 2021, Asif, et. al., 2020, Hussain et. al., 2022), if;
i.
ii.
iii.
Remark 4 If fulfils conditions (i) and (ii), then is said to be the left ideal, and if
satisfies (i) and (iii), then is said to be the ideal of . If is the left and right ideal
of , then is said to be an ideal of
Example 6. Let be the set of all possible matrices with entries from
and form under standard addition and multiplication is defined as
for some then is right ideal of
.
Definition 2.9: (Abou-Zaid, 1991) Let be a 𝑁𝑅 and be fuzzy set over 𝒩
then is said to be fuzzy sub-NR of 𝒩 if the following condition satisfies;
I.
II.
Definition 2.10: (Abou-Zaid, 1991) Let be a 𝑁𝑅 and be fuzzy set over 𝒩
then is said to be the fuzzy ideal of 𝑁𝑅 if the following condition
satisfies;
I.
II.
III.
IV.
V.
Definition 2.11: (Tahan et. al., 2021) Let be a 𝑁𝑅 and be fuzzy multiset
over 𝒩 then is said to be fuzzy multi sub-NR of 𝒩 if the following
condition satisfies;
I.
II.
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
6
Definition 2.12: (Tahan et. al., 2021) Let be a 𝑁𝑅 and be a fuzzy multiset
over 𝒩 then is said to be a fuzzy multi ideal of 𝑁𝑅 𝒩 if the following
condition satisfies;
III.
IV.
V.
VI.
VII.
4. Main Result
Definition 3.1: Let be a . An intuitionistic multi-fuzzy set is an
intuitionistic multi-fuzzy sub-NR over. If the following conditions are
satisfied;
I. and
II. and
Where and count membership and count non-membership functions,
respectively.
Example 7. Let be a under standard addition and multiplication. Then, A
is an intuitionistic multi-fuzzy sub-NR overwith fuzzy count functions given by;
Definition 3.2: Let be a . An intuitionistic multi-fuzzy set is considered
an intuitionistic fuzzy multi-ideal of. If the following conditions are
satisfied;
and
and
and
and
and
Example 8. Let be a under standard addition and multiplication. Then, A is
an intuitionistic multi-fuzzy ideal ofwith fuzzy count functions given by;
Remark 5. Every intuitionistic multi-fuzzy ideal of a is an intuitionistic multi-
fuzzy sub-NR of
This study then demonstrates some exciting results for intuitionistic multi-fuzzy
near-rings, which are proven for other algebraic structures (Al-Tahan et. al., 2021).
Proposition 3.3: Let be a andbe two intuitionistic multi-fuzzy sub-
NRs of, then is also an intuitionistic multi-fuzzy sub of.
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
7
Proof. Let be two intuitionistic multi-fuzzy sub-NRs ofthen we are
to show conditions of definition 3.1.
And,
Also,
And,
Corollary 1. Let be a , andbe an intuitionistic multi-fuzzy sub-NRs
of for then
is also an intuitionistic multi-fuzzy sub-NR
of.
Proposition 3.4: (Al-Tahan et. al., 2021 & Shinoj et. al., 2015) Let be a ,
andbe intuitionistic multi-fuzzy ideals of . Then, is also an
intuitionistic multi-fuzzy ideal of.
Proof. Let be two intuitionistic multi-fuzzy ideals ofthen we are
to show conditions of definition 3.2. The first two have been done in proposition 3.3.
Since
And,
Moreover,
Also,
Let then
Because
And,
Corollary 2. Let be a , andbe the intuitionistic multi-fuzzy ideal of
s of for then
is also an intuitionistic multi-fuzzy ideal
of.
Example 9. Let be a andbe two intuitionistic multi-fuzzy ideals
ofgiven by;
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
8
And
It also satisfies the conditions of definition 3.2 and forms an intuitionistic multi-fuzzy
ideal of.
Remark 6. Let be a andbe two intuitionistic multi-fuzzy ideals
of. Then, may or may not be an intuitionistic multi-fuzzy ideal of.
Example 10. Let be a andbe two intuitionistic multi-fuzzy ideals
ofgiven by;
And,
Then,
So, Vble number of definition
3.2 and form intuitionistic fuzzy multi ideal of
It does not satisfy the conditions of definition 3.2 and does not form the intuitionistic
multi-fuzzy ideal of.
Proposition 3.5: Let be a , andbe an intuitionistic multi-fuzzy ideal
ofthen;
and
and
and
Proof 1). Let, then we have
Also,
Proof 2). Let, then we have
Also,
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
9
Proof 3). Is straight forward
Proposition 3.6: Let be a , and be an intuitionistic multi-fuzzy ideal
ofif;
Proof. Let be an intuitionistic multi-fuzzy ideal ofand
then,
Using proposition 3.5
Also, suppose that . Then,
Definition 3.7: Let be a , and be an intuitionistic multi-fuzzy set
ofthen is defined as
.
Example 11. Let be a andbe two intuitionistic multi-fuzzy ideals
ofdefined in example 10
Example 12. Let be a andbe an intuitionistic multi-fuzzy ideal of
defined in example 9
Lemma3.8: Let be a , and be an intuitionistic multi-fuzzy ideal
ofthen;
Proof. The proof is done in proposition 3.5.
Proposition 3.9: Let be a and be an intuitionistic multi-fuzzy sub-NR
ofthen is also sub-NR of
Proof. Let then,
Also, that
Moreover,
Also, that
Proposition 3.10: (Al-Tahan et. al. 2021) Let be a , and be an
intuitionistic multi-fuzzy ideal ofthen is also an ideal of
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
10
Proof. Let then,
Also,
Moreover,
Also, that
Now,
Hence
Definition 3.11: Let be a non-empty set, and be an intuitionistic multi-fuzzy set
of ; then the support of can be defined as
.
Example 13. Let be a under standard addition and multiplication,
andbe an intuitionistic multi-fuzzy sub-NR over defined in example 7, then,
Proposition 3.12: Let be a , and be an intuitionistic multi-fuzzy sub
ofthen is also sub of
Proof. Let then, also
Also,
Moreover,
Also,
Proposition 3.13: Let be a , and be an intuitionistic multi-fuzzy ideal
ofthen is also an ideal of
Proof. Let then
Also,
Moreover,
Also,
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
11
Now,
Hence
Definition 3.14: Let be two non-empty sets, A and be two intuitionistic
fuzzy multisets of , respectively, then can be defined as,
Proposition 3.15: Let be two s and be intuitionistic multi-fuzzy sub-
NR of respectively, then is also an intuitionistic multi-fuzzy sub-NR
of.
Proof. Let be two intuitionistic multi-fuzzy sub-NRs and
. Then we are to show the conditions of Definition 3.1.
(I)
And,
(II)
And,
Corollary 3. Let be s and be intuitionistic multi-fuzzy sub-NR of for
then
is an intuitionistic multi-fuzzy sub-NR of
where
and
.
Proposition 3.16: Let be two s, and be the intuitionistic multi-
fuzzy ideal of , respectively. Then is also an intuitionistic multi-fuzzy
ideal of (Al-Tahan et. al., 2021 & Sujatha, 2104).
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
12
Proof. Let be two intuitionistic multi-fuzzy sub-NRs and
then we are to show conditions of Definition 3.2. (I) and
(II) has been done in proposition 3.15 then;
Also,
And,
Let then,
And
Corollary 4. Let be s and be intuitionistic multi-fuzzy ideals of
for then
is an intuitionistic multi-fuzzy ideal of
where
and
.
Example 14. Let be a andbe two intuitionistic multi-fuzzy ideals
ofdefined in example 10. Then is an intuitionistic multi-fuzzy ideal of
with count functions given by;
Definition 3.17: Let be a non-empty set andbe an intuitionistic multi-fuzzy set
of and where
then level subset of an intuitionistic multi fuzzy set can be defined as;
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
13
Proposition 3.18: Let be a non-empty set, and A andbe intuitionistic multi-
fuzzy sets ofthen the following results hold;
, if
Remark 7. Equality hold for Vii if.
Theorem 3.19: (Sujatha, 2104) Let be a and be an intuitionistic multi-
fuzzy sub-NR of then is sub-NR of
.
Proof. Given that
Hence
Also,
Hence
Theorem 3.20: Let be a and be an intuitionistic multi-fuzzy ideal of
then is an ideal of
.
Proof. Given that
Then we are to show the
conditions of Definition 3.2.
Hence
Also,
Hence
Now,
Hence
Definition 3.21: If is the Cartesian product of two intuitionistic fuzzy multi-
sets and where
then level subset ofcan be defined as;
Proposition 3.22: Let be a non-empty set andbe two intuitionistic fuzzy
multisets of, then
Proof. Let
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
14
Hence
Theorem 3.23: Let be a andbe two intuitionistic multi-fuzzy sub-
NR ofandis an intuitionistic multi-fuzzy sub-NR ofthen
is sub-NR of.
Proof. ;
Theorem 3.24: Let be a andbe two intuitionistic multi-fuzzy ideals
ofandis an intuitionistic multi-fuzzy ideal ofis
an ideal of (Sujatha, 2014).
Proof.
;
And,
Also,
Also,
And
Lemma 3.25: Let be a , and be an intuitionistic multi-fuzzy ideal
ofthen;
and
Theorem 3.26: Let be a under standard addition and multiplication is
defined as , andbe an intuitionistic multi-fuzzy ideal of .
Then, is an intuitionistic multi-fuzzy ideal of (matrix near ring) where
can be defined as;
Proof. Let then we are to show conditions of Definition 3.2.
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
15
And
(II) and (IV) of definition 3.2; Since
And
Also,
Let
And
Example 15. Let be a andbe an intuitionistic multi fuzzy ideal
ofdefined in example 10 then
is an intutionistic multi fuzzy ideal of
Theorem 3.27: According to Al-Tahan et. al. (2021), let be a under
standard addition and multiplication is defined as andbe an
intuitionistic multi-fuzzy ideal of. Then, is an intuitionistic multi-fuzzy ideal of
where can be defined as;
Proof. Let
then we are to show conditions of Definition 3.2.
(I)
And
(II) and (IV)
And
Nadia et al./Decis. Mak. Appl. Manag. Eng. (2023)
16
(III),
(V), Let
And,
Example 16. Let be a andbe an intuitionistic multi-fuzzy ideal
ofdefined in example 10 then;
It is an intuitionistic multi-fuzzy ideal of
Proposition 3.28. Let be a , andbe an intuitionistic multi-fuzzy ideal
ofthen is an intuitionistic multi-fuzzy ideal of where
can be defined as;
Proof. Same as theorem 3.27.
5. Conclusion
This study constructed the concept of intuitionistic multi-fuzzy near-rings and
intuitionistic multi-fuzzy ideals. It explored and illustrated some properties related
to intuitionistic multi-fuzzy near-rings and intuitionistic multi-fuzzy ideals.
Moreover, it investigated the support, level subsets and Cartesian product of
intuitionistic multi-fuzzy near-rings and ideals. It established results associated with
all these new constructions. This work contributes to fuzzy set theory, widely used in
multi-criteria decision-making and pattern recognition problems. In the future, one
may extend these notions to AI-related decision-making and pattern recognition
research. Alternatively, extending to inter-valued fuzzy systems or applying the
intuitionistic multi-fuzzy idea to vector spaces and modules is possible.
Author Contributions: B.N. and H.S. developed the theoretical formalism,
performed the analytic calculations, and performed the numerical simulations; K.N.
contributed to the final version of the manuscript; M.M. and K.S. contributed to the
design and implementation of the research, to the analysis of the results and to the
writing of the manuscript; L.R.Y.M. wrote and revised the manuscript. All authors
discussed the results and commented on the manuscript.
Data Availability Statement: No data were used to support this study.
Conflicts of Interest: The authors declare that there are no conflicts of interest
regarding the publication of this article.
Decision making under incomplete data: Intuitionistic multi fuzzy ideals of near-ring approach
17
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