mathematics
Article
Some Operations and Properties of the Cubic Intuitionistic Set
with Application in Multi-Criteria Decision-Making
Shahzad Faizi 1 , Heorhii Svitenko 2,3 , Tabasam Rashid 4 , Sohail Zafar 4 and Wojciech Sałabun 3, *
1
2
3
4
*
Department of Mathematics, Virtual University of Pakistan, Lahore 54000, Pakistan
Department of Software Engineering, Kharkiv National University of Radio Electronics, Nauky Ave. 14,
61166 Kharkiv, Ukraine
Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied
Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of
Technology in Szczecin, ul. Żołnierska 49, 71-210 Szczecin, Poland
Department of Science and Humanities, University of Management and Technology, Lahore 54770, Pakistan
Correspondence: wojciech.salabun@zut.edu.pl; Tel.: +48-91-449-5580
Abstract: This paper proposes some operations on the cubic intuitionistic set along with useful
properties. We propose the internal cubic intuitionistic set (ICIS), the external cubic intuitionistic set
(ECIS), P-order, R-order order (P-(R-) order), P-union, R-union (P-(R-) union), P-intersection, and
R-intersection (P-(R-) intersection). We further investigate several properties of the P-(R-) union and
P-(R-) intersection of ICISs and ECISs, and present some examples in this context. Some important
theorems related to ICISs and ECISs are also presented with proof. Finally, an application example
is given to measure the effectiveness and significance of the proposed operations by solving a
multi-criteria decision-making (MCDM) problem.
Keywords: fuzzy set; interval-valued fuzzy set; intuitionistic fuzzy set; interval-valued intuitionistic
fuzzy set; cubic set; cubic intuitionistic set
MSC: 03E72;94D05
Citation: Faizi, S.; Svitenko, H.;
Rashid, T.; Zafar, S.; Sałabun, W.
Some Operations and Properties of
the Cubic Intuitionistic Set with
Application in Multi-Criteria
Decision-Making. Mathematics 2023,
11, 1190. https://doi.org/
10.3390/math11051190
Academic Editors: Jun Ye and
Yanhui Guo
Received: 9 January 2023
Revised: 15 February 2023
Accepted: 21 February 2023
Published: 28 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
Zadeh [1] proposed the idea of fuzzy sets in 1965 and further extended this idea
to an interval-valued fuzzy set (IVFS) [2]. Some complex decision-making problems in
the economy, engineering, social science, environmental science, etc., exist that cannot
be completely modeled by methods of classical mathematics because of the presence of
various types of uncertainties. Others, on the other hand, use certain data processed by
methods that are hybrid approaches, such as the INVAR method [3] or the CODAS-COMET
method [4]. However, to handle the vagueness and uncertainty occurring in such decisionmaking problems, some well-known mathematical theories have been introduced, such as
fuzzy set theory [1], intuitionistic fuzzy set (IFS) theory [5], interval-valued intuitionistic
fuzzy set (IVIFS) theory [6,7], hesitant fuzzy set theory [8], hesitant fuzzy linguistic set
theory [9], soft set theory [10], fuzzy soft set theory [11], etc. An example of this could
be the use of triangular fuzzy numbers in a fuzzy extension of a simplified best–worst
method [12].
At times, uncertainty research uses generalized approaches to better cope with the
decision-making process via approaches related to the Dempster–Shafer evidence theory
(DSET) [13], or quantum evidence theory (QET) [14]. Other ways are to use methods
based on either entropy [15] or distance measures [16]. Most of the researchers studied
IVFS [12]. For example, Zhang et al. [17] investigated the entropy of IVFSs based on
distance measures. Zeng and Guo [18] discussed the similarity measure, inclusion of the
measure, and entropy of IVFSs, while Grzegorzewski [19] proposed IVFSs based on the
Mathematics 2023, 11, 1190. https://doi.org/10.3390/math11051190
https://www.mdpi.com/journal/mathematics
Mathematics 2023, 11, 1190
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Hausdorff metric. Furthermore, IVFSs have been widely used and applied in real-life
applications. For example, Sambuc [20] and Kohout [21] used the concept of IVFSs in
medical diagnoses in thyroid pathology and medicine in a CLINAID system, respectively.
Gorzalczany [22] used the idea of IVFSs in approximate reasoning. Turksen [23,24] further
used the same idea of IVFSs in interval-valued logic in preference modeling [25].
Jun et al. [26] proposed the idea of a cubic set and presented its two important types,
called the internal cubic set and the external cubic set by using the idea of the fuzzy set and
IVFS. They further introduced some operations of union and intersection regarding the
cubic sets, such as the P-(R-) union and P-(R-) intersection, and studied important related
properties. Jun [27] further extended the idea of the cubic set, introduced the notion of the
cubic intuitionistic set, and discussed its useful applications in BCK/BCI-algebras. Recently,
studies on the cubic set theory have rapidly grown. For example, Jun et al. [28] proposed
the concept of cubic IVIFS and discussed its important applications in BCK/BCI-algebra.
With the help of using a cubic set and a neutrosophic set, Ali et al. [29] presented the
notion of a neutrosophic cubic set and studied some useful properties. Kang and Kim [30]
investigated the images and inverse images of almost-stable cubic sets and discussed
the complement, the P-union, and the P-intersection of inverse images of almost-stable
cubic sets. Chinnadurai et al. [31] investigated several properties of the P-(R-) union and
P-(R-) intersection of cubic sets and studied some properties of cubic ideals of near rings.
Jun et al. [32] proposed the ideas of cubic α-ideals and cubic p-ideals and studied several
useful properties.
Cubic sets are widely studied and are important in many areas, as discussed in
the literature by various researchers. Motivated by the advantages of cubic sets, this
paper proposes the notion of CIS based on IVFSs and intuitionistic fuzzy sets. Although
Jun [27] previously introduced the idea of CIS as cubic intuitionistic sets and discussed
their applications in BCK/BCI-algebras, this paper presents a completely different research
work under the framework of CIS. We first propose two important types of CIS, named
ICIS and ECIS. We then investigate the complement of CIS, the P-(R-) cubic intuitionistic
subsets, and the P-(R-) union and the intersection of CISs. Furthermore, we prove various
important theorems and results related to the proposed union and intersection operations.
Finally, we present an application example to demonstrate the validity of the proposed
operations by solving a MCDM problem.
The remainder of the paper can be summarized briefly as follows. Some basic concepts
related to the work are presented in Section 2. The notions of CIS, ICIS, and ECIS are
introduced in Section 3. We further investigate P-(R-)order, P-(R-)union, P-(R-)intersection,
and related important properties with proof in the same section. A MCDM approach using
CISs is presented in Section 4 along with an application example. We conclude the paper
with some concluding remarks in Section 5.
2. Preliminary
This section introduces necessary notions and presents a few auxiliary results that we
need in the rest of the paper. Throughout this paper, we let [ I ], I X , and [ I ] X stand for the
set of all closed subintervals of [0, 1], the collection of all fuzzy sets in a set X, and IVFSs in
X, respectively.
Definition 1. Let X be a non-empty set. A fuzzy set in set X is defined as function f : X → [0, 1].
the relation ≤, join (∨), meet (∧), and complement of I X for all x ∈ X can be defined, respectively,
as follows:
f 1 ≤ f 2 ⇔ f 1 ( x ) ≤ f 2 ( x ) for all f 1 , f 2 ∈ I X ,
( f 1 ∨ f 2 )( x ) = f 1 ( x ) ∨ f 2 ( x ) = max{ f 1 ( x ), f 2 ( x )},
( f 1 ∧ f 2 )( x ) = f 1 ( x ) ∧ f 2 ( x ) = min{ f 1 ( x ), f 2 ( x )},
f 1c ( x ) = 1 − f 1 ( x ),
Mathematics 2023, 11, 1190
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where f 1c represents the complement of f 1 .
Definition 2. By an interval number, we mean a closed sub-interval a = [ a− , a+ ] of Iwhere
0 ≤ a− ≤ a+ ≤ 1. The complement ac of a ∈ [ I ] is defined as follows:
a c = [1 − a + , 1 − a − ].
The refined minimum and refined maximum (briefly, rmin and rmax) and the symbols , , = of
the elements a1 = [ a1− , a1+ ] and a2 = [ a2− , a2+ ] of [ I ] is defined as follows:
rmin{ a1 , a2 } = [min{ a1− , a2− }, min{ a1+ , a2+ }],
rmax{ a1 , a2 } = [max{ a1− , a2− }, max{ a1+ , a2+ }],
a1 a2 if and only if a1− ≥ a2− and a1+ ≥ a2+ .
Similarly, we can define a1 a2 and a1 = a2 .
Definition 3. For a non-empty set X, a function A : X → [ I ] is called an IVFS in X. The element
A = [ A− ( x ), A+ ( x )] for every A ∈ [ I ] X and x ∈ X, is called the membership degree of an element
x to the set A. The IVFS is simply denoted as A = [ A− , A+ ]. The complement Ac of A can be
defined as Ac = [1 − A+ , 1 − A− ].
For every A1 , A2 ∈ [ I ] X , the following are true:
A1 ⊆ A2 if and only ifA1 A2 ,
A1 = A2 if and only ifA1 = A2 .
Definition 4 ([5]). Let E be a crisp set. An IFS Ã can be defined as
à = {h x, µ à ( x ), νà ( x )i : x ∈ E}.
where µ à : E → [0, 1] and νà : E → [0, 1] indicate, respectively, the membership and nonmembership degrees of x ∈ E with the condition 0 ≤ µ à ( x ) + νà ( x ) ≤ 1 for every x ∈ E.
Definition 5 ([6]). An expression of the form given by
B = {h x, MB ( x ), NB ( x )i : x ∈ X }
is called the IVIFS in X, where MB : X → [ I ] and NB : X → [ I ] are IVFSs with the condition that
0 ≤ MB+ ( x ) + NB+ ( x ) ≤ 1 for all x ∈ X.
The intervals MB and NB denote, respectively, the membership and non-membership degrees of
x ∈ X.
Definition 6 ([26]). A mathematical structure of the form
A = {h x, A( x ), λ( x )i : x ∈ X },
is called the cubic set in X, where A and λ are, respectively, the IVFS and a fuzzy set in X. Jun [27]
introduced the notion of the cubic intuitionistic set as follows:
Definition 7 ([27]). A mathematical structure of the form
A = {h x, A( x ), λ( x )i : x ∈ X },
is called the cubic intuitionistic set where A is an IVIFS in X and λ is an IFS in X.
Mathematics 2023, 11, 1190
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3. Some Operations on the Cubic Intuitionistic Set
This section introduces the concept of CIS with some modifications as proposed by
Jun in [27] as follows:
Definition 8. By CIS in a non-empty set X, we mean a mathematical structure of the form
A = {h x, M A ( x )/α A ( x ), NA ( x )/β A ( x )i| x ∈ X }
where M A : X → [ I ] and NA : X → [ I ] are IVFSs of the form M A ( x ) = [ M− ( x ), M+ ( x )],
NA ( x ) = [ N − ( x ), N + ( x )] with the conditions that
+
0 ≤ M+
A ( x ) + NA ( x ) ≤ 1 and 0 ≤ α A ( x ) + β A ( x ) ≤ 1 f or all x ∈ X.
M A ( x ) and NA ( x ) denote, respectively, the membership and non-membership degrees of x and
α A : X → [0, 1], β A : X → [0, 1] are fuzzy sets in X. For simplicity, we denote CIS( X ) as the
collection of all CISs A = h M A /α A , NA /β A i in X. In the rest of the paper, we will use the same
notations with symbols for CIS as presented in the above definition.
Remark 1. For any non-empty set X, let 1( x ) = 1 and 0( x ) = 0 for all x ∈ X. Then, A =
h M A /1, NA /0( x )i, B = h MB /0, NB /1i and C = h MC /
in X.
−
MC
+ MC+
NC− + NC+
i
,
N
/
C
2
2
are all CISs
Definition 9. For A = h M A /α A , NA /β A i ∈ CIS( X ), the score value of A is defined as
Sc(A) =
1
M−
+ M+
+ α A − N − (x) + N + (x) + β A
A
A
3
where Sc(A) ∈ [−1, 1].
Definition 10. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i ∈ CIS( X ), then
(i) A = B ⇔ M A = MB , α A = α B ; NA = NB , β A = β B (Equality)
(ii) A ⊆ P B ⇔ M A ⊆ MB , α A ≤ α B ; NA ⊇ NB , β A ≥ β B (P-order)
(iii) A ⊆ R B ⇔ M A ⊆ MB , α A ≥ α B ; NA ⊇ NB , β A ≤ β B (R-order)
Definition 11. Let 0 = [0, 0] and 1 = [1, 1]. Then, a CIS A = h M A /α A , NA /β A i in which
M A = 0, α A = 1, NA = 1 and β A = 0 (respectively, M A = 1, α A = 0, NA = 0 and β A = 1) is
denoted by 0̈ (respectively 1̈).
A CIS B = h MB /α B , NB /β B i in which MB = 0, α B = 0, NB = 1, β B = 1 (respectively
MB = 1, α B = 1, NB = 0 and β A = 0) is denoted by 0̂ (respectively, 1̂).
We can see that the score values of 0̈, 1̈, 0̂ and 1̂ can be computed, respectively, as
Sc(0̈) = −0.33, Sc(1̈) = 0.33, Sc(0̂) = −1 and Sc(1̂) = 1.
Definition 12. Consider the family of CISs Ai = h Mi /αi , Ni /β i i, i ∈ ℧ in X, we define
(a) P-union
∪ P Ai
i ∈℧
= h ∪ Mi / ∨ αi , ∩ Ni / ∧ β i i
∩ P Ai
i ∈℧
= h ∩ Mi / ∧ αi , ∪ Ni / ∨ β i i
∪ R Ai
i ∈℧
= h ∪ Mi /( ∧ αi , ∩ Ni / ∨ β i i
i ∈℧
i ∈℧
i ∈℧
i ∈℧
(b) P-intersection
i ∈℧
i ∈℧
i ∈℧
i ∈℧
(c) R-union
i ∈℧
i ∈℧
i ∈℧
i ∈℧
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(d) R-intersection
∩ R Ai
i ∈℧
= h ∩ Mi / ∨ αi ), ∪ Ni /( ∧ β i i
i ∈℧
i ∈℧
i ∈℧
i ∈℧
Remark 2. The complement of A = h M A /α A , NA /β A i is defined as
c
Ac = h McA /1 − α A , NA
/1 − β A i.
Obviously, (Ac )c = A, 0̈c = 1̈, 1̈c = 0̈, 0̂c = 1̂, 1̂c = 0̂.
Remark 3. For the family of CISs Ai = h Mi /αi , Ni /β i i, i ∈ ℧ in X, we have
(∪ P Ai )c = ∩ P (Ai )c , (∩ P Ai )c = ∪ P (Ai )c , (∪ R Ai )c = ∩ R (Ai )c and (∩ R Ai )c = ∪ R (Ai )c .
i ∈℧
i ∈℧
i ∈℧
i ∈℧
i ∈℧
i ∈℧
i ∈℧
i ∈℧
Definition 13. Let X be a non-empty set.
1
2
+
A CIS A = h M A /α A , NA /β A i is said to be ICIS if M−
A ≤ α A ≤ M A and
−
+
NA ≤ β A ≤ NA .
A CIS B = h MB /α B , NB /β B i in X is said to be ECIS if α B ∈
/ ( MB− , MB+ )
−
+
and β B ∈
/ ( NB , NB ).
Example 1. For a non-empty set X,
1
2
Let A = h M A /α A , NA /β A i be a CIS with M A = [0.1, 0.3], α A = 0.2, NA = [0.4, 0.6] and
β A = 0.5, then A is ICIS.
Let B = h MB /α B , NB /β B i be a CIS with MB = [0.2, 0.4], α B = 0.1, NB = [0.5, 0.6] and
β B = 0.7, then B is ECIS.
Remark 4. Every CIS in X can be considered a Zadeh fuzzy set, IFS, IVFS, IVIFS, and cubic
set according to ( M = N = 0, β = 0), ( M = N = 0), ( N = 0, β = 0), ( β = α = 0) and
( N = 0, β = 0), respectively.
Theorem 1. Let A = h M A /α A , NA /β A i be A CIS which is not an ECIS in X. Then there exist
−
+
+
x ∈ X such that α A ( x ) ∈ ( M−
A ( x ), M A ( x )) and β A ( x ) ∈ ( NA ( x ), NA ( x )).
Proof. Straightforward.
Theorem 2. Let A = h M A /α A , NA /β A i be A CIS in X. If A is both ICIS and ECIS, then
α( x ) ∈ U ( M ) ∪ L( M ) and β( x ) ∈ U ( N ) ∪ L( N ) f or all x ∈ X where U ( M ) = { M+ ( x )| x ∈
X }, L( M ) = { M− ( x )| x ∈ X }, U ( N ) = { N + ( x )| x ∈ X } and L( N ) = { N − ( x )| x ∈ X }.
Proof. Assume that A is both ICIS and ECIS. Then, using Definition 13, we have M− ( x ) ≤
α( x ) ≤ M+ ( x ), N − ( x ) ≤ β( x ) ≤ N + ( x ) and α( x ) ∈
/ ( M− ( x ), M+ ( x )), β( x ) ∈
/ ( N − ( x ),
+
−
+
−
N ( x )) for all x ∈ X. Thus α( x ) = M ( x ) or α( x ) = M ( x ) and β( x ) = N ( x ) or β( x ) =
N + ( x ). Hence α( x ) ∈ U ( M ) ∪ L( M) and β( x ) ∈ U ( N ) ∪ L( N ) for all x ∈ X.
Theorem 3. Let A = h M A /α A , NA /β A i be A CIS in X. If A is ICIS (respectively, ECIS), then
Ac is ICIS (respectively ECIS).
Proof. Since A = h M A /α A , NA /β A i is ICIS in X, we have
+
−
+
M−
A ≤ α A ≤ M A andNA ≤ β A ≤ NA
+
−
+
respectively,α A ∈
/ ( M−
/ ( NA
, NA
) .
A , M A ) and β A ∈
This implies that
−
+
−
1 − M+
A ≤ 1 − α A ≤ 1 − M A and 1 − NA ≤ 1 − β A ≤ 1 − NA
Mathematics 2023, 11, 1190
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−
+
−
, 1 − NA
) .
respectively,1 − α A ∈
/ (1 − M +
/ (1 − NA
A , 1 − M A ) and 1 − β A ∈
c /1 − β i is ICIS (respectively, ECIS)
Hence Ac = h McA /1 − α A , NA
A
We will show (through the following example) that the P-union and P-intersections of
ECISs are not necessarily ECISs.
Example 2. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X.
Let M A = [0.1, 0.3], α A = 0.5, NA = [0.4, 0.6], β A = 0.2, MB = [0.4, 0.6], α B = 0.2,
NB = [0.1, 0.3] and β B = 0.5 for all x ∈ X. Then A ∪ p B = h MB /α A , NB /β A i and A ∩ p B =
h M A /α B , NA /β B i. Hence, A ∪ p B and A ∩ p B are not ECISs.
From the following example, it can be easily seen that the R-union and R-intersection
of ICIS need not be ICISs.
Example 3. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ICISs in X. Let
M A = [0.1, 0.3], α A = 0.2, NA = [0.5, 0.7], β A = 0.6, MB = [0.5, 0.7], α B = 0.6, NB =
[0.1, 0.3] and β B = 0.2 for all x ∈ X. Then A ∪ R B = h MB /α A , NB /β A i and A ∩ R B =
h M A /α B , NA /β B i. Hence, A ∪ R B and A ∩ p B are not ICISs.
In the following examples, we will show that the R-union and R-intersection of ECIS
may not be ECIS.
Example 4.
1
2
Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X. Let M A =
[0.1, 0.3], α A = 0.5, NA = [0.35, 0.4], β A = 0.2, MB = [0.4, 0.6], α B = 0.7, NB =
[0.2, 0.3] and β B = 0.1 for all x ∈ X. Then A ∪ R B = h MB /α A , NB /β A i and note that
α A ∈ ( MB− , MB+ ); therefore, A ∪ R B is not ECIS.
Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X. Let M A =
[0.2, 0.4], α A = 0.1, NA = [0.4, 0.6], β A = 0.5, MB = [0.5, 0.7], α B = 0.3, NB = [0.1, 0.3]
and β B = 0.6 for all x ∈ X. Then A ∩ R B = h M A /α B , NA /β A i and, hence, A ∩ R B is not
ECIS.
Theorem 4. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ICISs in X, such
−
+
+
that max{ M−
A , MB } ≤ ( α A ∧ α B ) and min{ NA , NB } ≥ ( β A ∨ β B ). Then the R-union and
R-intersection of A and B are ICISs.
Proof. A and B are ICISs; therefore,
+
−
+
M−
A ≤ α A ≤ M A , NA ≤ β A ≤ NA
MB− ≤ α B ≤ MB+ and NB− ≤ β B ≤ NB+
which implies that
(α A ∧ α B ) ≤ ( M A ∪ MB )+ and ( β A ∨ β B ) ≥ ( NA ∩ NB )− .
It follows that
−
+
( M A ∪ MB )− = max{ M−
A , MB } ≤ (α A ∧ α B ) ≤ ( M A ∪ MB )
and
( NA ∩ NB )− ≤ ( β A ∨ β B ) ≤ min{ NA+ , NB+ } = ( NA ∩ NB )+ .
Hence, A ∪ R B is ICIS. Similar arguments work in the case of A ∩ R B.
Mathematics 2023, 11, 1190
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Given two CISs A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i in X. If we
exchange α A for α B and β A for β B , we denote these CISs by A∗ = h M A /α B , NA /β B i and
B∗ = h MB /α A , NB /β A i, respectively.
The next example shows that, for any two ECISs in X, A∗ and B∗ need not be ICISs in X.
Example 5.
Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be ICIS in X. Let M A = [0.1, 0.3],
α A = 0.7, NA = [0.5, 0.7], β A = 0.15, MB = [0.4, 0.6], α B = 0.35, NB = [0.2, 0.3]
and β B = 0.1 for all x ∈ X. Then it is easy to see that A∗ = h M A /α B , NA /β B i and
B∗ = h MB /α A , NB /β A i are not ICISs in X.
Let X = { a, b}. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in
X defined in Table 1. Moreover, A∗ = h M A /α B , NA /β B i and B∗ = h MB /α A , NB /β A i are
not ICISs in X because α B ( a) = 0.35 ∈
/ [0.4, 0.6] = M A ( a), β B ( a) = 0.25 ∈
/ [0.1, 0.2] =
/ [0.2, 0.4] = M A (b) and β B (b) = 0.35 ∈
/ [0.4, 0.6] =
NA ( a). Moreover, α B (b) = 0.15 ∈
NA ( b ).
1
2
Table 1. CISs A and B.
X
M A /α A
NA /β A
MB /α B
NB /β B
a
b
[0.4, 0.6]/0.65
[0.2, 0.4]/0.1
[0.1, 0.2]/0.35
[0.4, 0.6]/0.7
[0.1, 0.3]/0.35
[0.4, 0.5]/0.15
[0.4, 0.5]/0.25
[0.1, 0.3]/0.35
We will show through the following example that the P-union of two ECISs in X may
not be an ICIS in X.
Example 6. Consider again two ECISs, A and B, as shown in Table 1. In this case, A ∪ P B is
not ICIS in X because (α A ∨ α B )( a) = 0.65 ∈
/ [0.4, 0.6] = M A ∪ MB , ( β A ∧ β B )( a) = 0.25 ∈
/
[0.1, 0.2] = NA ∩ NB .
In the following result, we will find a condition for the P-union of two ECISs to be
an ICIS.
Theorem 5. For two ECISs A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i in X. If
A∗ = h M A /α B , NA /β B i and B∗ = h MB /α A , NB /β A i are ICISs in X. Then A ∪ P B and
A ∩ P B are ICISs in X.
Proof. Since A and B are ECISs in X, then
+
−
+
αA ∈
/ ( M−
/ ( NA
, NA
),
A , M A ), β A ∈
αB ∈
/ ( MB− , MB+ ) and β B ∈
/ ( NB− , NB+ ).
For all x ∈ X. Since A∗ and B∗ are ICISs in X, then
+
−
+
M−
A ≤ α B ≤ M A , NA ≤ β B ≤ NA
MB− ≤ α A ≤ MB+ and NB− ≤ β A ≤ NB+
for all x ∈ X. Thus, we can consider the following cases for any x ∈ X.
Case 1
+
−
+
α A ≤ M−
A ≤ α B ≤ M A , β A ≤ NA ≤ β B ≤ NA ,
α B ≤ MB− ≤ α A ≤ MB+ and β B ≤ NB− ≤ β A ≤ NB+ .
Case 2
+
−
+
M−
A ≤ α B ≤ M A ≤ α A , NA ≤ β B ≤ NA ≤ β A ,
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MB− ≤ α A ≤ MB+ ≤ α B and NB− ≤ β A ≤ NB+ ≤ β B .
Case 3
+
−
+
α A ≤ M−
A ≤ α B ≤ M A , β A ≤ NA ≤ β B ≤ NA ,
MB− ≤ α A ≤ MB+ ≤ α B and NB− ≤ β A ≤ NB+ ≤ β B .
Case 4
+
−
+
M−
A ≤ α B ≤ M A ≤ α A , NA ≤ β B ≤ NA ≤ β A ,
α B ≤ MB− ≤ α A ≤ MB+ and β B ≤ NB− ≤ β A ≤ NB+ .
The arguments in all cases are similar; therefore, we consider the first case.
−
−
−
We have α A = M−
A = MB = α B and β A = NA = NB = β B .
∗
∗
Since A and B are ICISs in X, then
+
+
+
α B ≤ M+
A , α A ≤ MB , β B ≤ NA and β A ≤ NB .
It follows that
−
( M A ∪ MB )− = max{ M−
A , MB } = (α A ∨ α B )
+
+
≤ max{ M+
A , MB } = ( M A ∪ MB ) and
( NA ∩ NB )− = min{ NA− , NB− } = ( β A ∧ β B )
≤ min{ NA+ , NB+ } = ( NA ∪ NB )+ .
Hence, A ∪ P B is ICIS. Similar steps can be used for A ∩ P B.
From Example 2, it can be easily seen that the P-union and P-intersections of ECISs
are not necessarily the ECISs in X. In the next result, we will show when the P-union and
P-intersection of two ECISs are ECISs in X.
Theorem 6. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X,
such that
−
−
+
min{max{ M+
A , MB }, max{ M A , MB }} ≥ ( α A ∧ α B )
−
−
+
> max{min{ M+
A , MB }, min{ M A , MB }} and
+
−
min{max{ NA
, NB− }, max{ NA
, NB+ }} > ( β A ∨ β B )
≥ max{min{ NA+ , NB− }, min{ NA− , NB+ }},
then A ∩ P B is ECIS in X.
Proof. Take
−
−
+
α x = min{max{ M+
A , MB }, max{ M A , MB }},
−
−
+
β x = max{min{ M+
A , MB }, min{ M A , MB }},
+
−
α∗x = min{max{ NA
, NB− }, max{ NA
, NB+ }} and
+
−
β∗x = max{min{ NA
, NB− }, min{ NA
, NB+ }}
−
+
+
−
−
+
+
∗
then α x is one of M−
A , MB , M A , MB and α x is one of NA , NB , NA , NB . We will consider the
−
+
−
+
∗
∗
case when α x = M A and α x = NA or α x = M A and α x = NA . Similar arguments will work
for all remaining cases.
−
∗
If α x = M−
A and α x = NA , then
+
MB− ≤ MB+ ≤ M−
A ≤ MA
−
NB− ≤ NB+ ≤ NA
≤ NA+
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and so β x = MB+ and β∗x = NB+ . Thus,
MB− = ( M A ∩ MB )− ≤ ( M A ∩ MB )+ = MB+ = β x < (α A ∧ α B ),
−
NA
= ( NA ∪ NB )− = α x > ( β A ∨ β B )
and, hence,
(α A ∧ α B ) ∈
/ (( M A ∩ MB )− , ( M A ∩ MB )+ ) and
(β A ∨ βB ) ∈
/ (( NA ∪ NB )− , ( NA ∪ NB )+ ).
+
∗
If α x = M+
A and α x = NA , then
+
−
+
+
MB− ≤ M+
A ≤ MB andNB ≤ NA ≤ NB
so
−
∗
−
−
β x = max{ M−
A , MB } andβ x = max{ NA , NB }.
−
∗
Assume that β x = M−
A and β x = NA , then
+
+
MB− ≤ M−
A < ( α A ∧ α B ) ≤ M A ≤ MB and
−
NB− ≤ NA
≤ ( β A ∨ β B ) < NA+ ≤ NB+ .
From the above inequality, we have the following cases
Case-1
+
+
MB− ≤ M−
A < ( α A ∧ α B ) < M A ≤ MB and
−
< ( β A ∨ β B ) < NA+ ≤ NB+
NB− ≤ NA
Case-2
+
+
MB− ≤ M−
A < ( α A ∧ α B ) = M A ≤ MB and
−
NB− ≤ NA
= ( β A ∨ β B ) ≤ NA+ ≤ NB+ .
Case-1 contradicts the fact that CISs A and B are ECISs. From Case-2, it implies that
(α A ∧ α B ) ∈
/ (( M A ∩ MB )− , ( M A ∩ MB )+ ) and
(β A ∨ βB ) ∈
/ (( NA ∪ NB )− , ( NA ∪ NB )+ )
since
+
(α A ∧ α B ) = M+
A = ( M A ∩ MB ) and
( β A ∨ β B ) = NA− = ( NA ∪ NB )− .
Assume that β x = MB− and β∗x = NB− , then
−
+
+
M−
A ≤ MB < ( α A ∧ α B ) ≤ M A ≤ MB and
−
NA
≤ NB− ≤ ( β A ∨ β B ) ≤ NA+ ≤ NB+ .
We now have two cases.
Case-1
−
+
+
M−
A ≤ MB < ( α A ∧ α B ) < M A ≤ MB and
−
NA
≤ NB− < ( β A ∨ β B ) < NA+ ≤ NB+ .
Case-2
−
+
+
M−
A ≤ MB < ( α A ∧ α B ) = M A ≤ MB and
−
NA
≤ NB− = ( β A ∨ β B ) < NA+ ≤ NB+ .
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Case-1 contradicts that A and B are ECISs. From Case-2, it implies that
(α A ∧ α B ) ∈
/ (( M A ∩ MB )− , ( M A ∩ MB )+ ) and
(β A ∨ βB ) ∈
/ (( NA ∪ NB )− , ( NA ∪ NB )+ )
since
+
(α A ∧ α B ) = M+
A = ( M A ∩ MB ) and
( β A ∨ β B ) = NB− = ( NA ∪ NB )− .
Similar results can be obtained if we assume
−
∗
−
β x = MB− andβ∗x = NA
orβ x = M−
A and β x = NB
Hence, the P-intersection of A and B is ECIS in X.
Theorem 7. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X,
such that
−
−
+
min{max{ M+
A , MB }, max{ M A , MB }} > ( α A ∨ α B )
−
−
+
≥ max{min{ M+
A , MB }, min{ M A , MB }} and
+
−
min{max{ NA
, NB− }, max{ NA
, NB+ }} ≥ ( β A ∧ β B )
> max{min{ NA+ , NB− }, min{ NA− , NB+ }},
then A ∪ P B is ECIS in X.
Proof. The proof is similar to Theorem 6; therefore, we omit the details.
Example 7. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X =
{ a, b, c} as shown in Table 2. Then, A and B always satisfy the following conditions.
−
−
+
min{max{ M+
A , MB }, max{ M A , MB }} = ( α A ∨ α B )
−
−
+
> max{min{ M+
A , MB }, min{ M A , MB }} and
+
−
min{max{ NA
, NB− }, max{ NA
, NB+ }} > ( β A ∧ β B )
= max{min{ NA+ , NB− }, min{ NA− , NB+ }}.
However, the P-union of A and B is not ECIS because (α A ∨ α B )( a) = 0.2 ∈ [0.1, 0.3] = [( M A ∪
MB )− ( a), ( M A ∪ MB )+ ( a)] and ( β A ∧ β B )( a) = 0.45 ∈ [0.4, 0.5] = [( NA ∩ NB )− ( a), ( NA ∩
NB )+ ( a)].
Table 2. CISs A and B.
X
M A /α A
NA /β A
MB /α B
NB /β B
a
b
c
[0.1, 0.2]/0.2
[0.1, 0.4]/0.05
[0.6, 0.7]/0.7
[0.45, 0.6]/0.45
[0.5, 0.6]/0.7
[0.1, 0.15]/0.1
[0.05, 0.3]/0.03
[0.2, 0.3]/0.3
[0.5, 0.8]/0.4
[0.4, 0.5]/0.6
[0.55, 0.65]/0.55
[0.05, 0.2]/0.3
From Example 4, it can be easily observed that the R-union and R-intersection of ECISs
may not be ECISs in X. In the next result, we will show that the R-union and R-intersection
of two ECISs are ECISs in X.
Theorem 8. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X,
such that
−
−
+
min{max{ M+
A , MB }, max{ M A , MB }} > ( α A ∧ α B )
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−
−
+
≥ max{min{ M+
A , MB }, min{ M A , MB }} and
+
−
min{max{ NA
, NB− }, max{ NA
, NB+ }} ≥ ( β A ∨ β B )
> max{min{ NA+ , NB− }, min{ NA− , NB+ }},
then A ∪ R B is ECIS in X.
Proof. Take
−
−
+
α x = min{max{ M+
A , MB }, max{ M A , MB }},
−
−
+
β x = max{min{ M+
A , MB }, min{ M A , MB }},
+
−
α∗x = min{max{ NA
, NB− }, max{ NA
, NB+ }} and
+
−
β∗x = max{min{ NA
, NB− }, min{ NA
, NB+ }}
−
+
+
−
−
+
+
∗
then α x is one of M−
A , MB , M A , MB and α x is one of NA , NB , NA , NB . We will consider the
−
+
−
+
∗
∗
case when α x = MB and α x = NB or α x = MB and α x = NB . Similar arguments will work
for all remaining cases.
If α x = MB− and α∗x = NB− , then
+
−
+
M−
A ≤ M A ≤ MB ≤ MB and
−
NA
≤ NA+ ≤ NB− ≤ NB+
+
∗
so β x = M+
A and β x = NA . Thus,
MB− = ( M A ∪ MB )− = α x > (α A ∧ α B ) and
+
NA
= ( NA ∩ NB )+ = β∗x < ( β A ∧ β B )
and, hence,
(α A ∧ α B ) ∈
/ (( M A ∪ MB )− , ( M A ∪ MB )+ ) and
(β A ∨ βB ) ∈
/ (( NA ∩ NB )− , ( NA ∩ NB )+ ).
If α x = MB+ and α∗x = NB+ , then
+
+
−
+
+
M−
A ≤ MB ≤ M A andNA ≤ NB ≤ NA
and so
−
∗
−
−
β x = max{ M−
A , MB } andβ x = max{ NA , NB }.
−
∗
Assume that β x = M−
A and β x = NA , then
+
+
MB− ≤ M+
A < ( α A ∧ α B ) < MB ≤ M A and
−
NB− ≤ NA
< ( β A ∨ β B ) ≤ NB+ ≤ NA+ .
We have two cases
Case-1
+
+
MB− ≤ M−
A < ( α A ∧ α B ) < MB ≤ M A and
−
NB− ≤ NA
< ( β A ∨ β B ) < NB+ ≤ NA+ .
Case-2
+
+
MB− ≤ M−
A = ( α A ∧ α B ) ≤ MB ≤ M A and
−
NB− ≤ NA
< ( β A ∨ β B ) = NB+ ≤ NA+ .
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Case-1 contradicts the fact that CISs A and B are ECISs. From Case-2, it implies that
(α A ∧ α B ) ∈
/ (( M A ∪ MB )− , ( M A ∪ MB )+ ) and
(β A ∨ βB ) ∈
/ (( NA ∩ NB )− , ( NA ∩ NB )+ )
since
+
(α A ∧ α B ) = M−
A = ( M A ∪ MB ) and
( β A ∨ β B ) = NB+ = ( NA ∩ NB )+ .
Assume that β x = MB− and β∗x = NB− , then
−
+
+
M−
A ≤ MB ≤ ( α A ∧ α B ) ≤ MB ≤ M A and
−
NA
≤ NB− < ( β A ∨ β B ) ≤ NB+ ≤ NA+ .
We have two cases
Case-1
−
+
+
M−
A ≤ MB < ( α A ∧ α B ) < MB ≤ M A and
−
NA
≤ NB− < ( β A ∨ β B ) < NB+ ≤ NA+
Case-2
−
+
+
M−
A ≤ MB = ( α A ∧ α B ) < MB ≤ M A and
−
NA
≤ NB− < ( β A ∨ β B ) = NB+ ≤ NA+ .
Case-1 contradicts the fact that CISs A and B are ECISs. From Case-2, it implies that
(α A ∧ α B ) ∈
/ (( M A ∪ MB )− , ( M A ∪ MB )+ ) and
(β A ∨ βB ) ∈
/ (( NA ∩ NB )− , ( NA ∩ NB )+ )
since
(α A ∧ α B ) = MB− = ( M A ∪ MB )− and
( β A ∨ β B ) = NB+ = ( NA ∩ NB )+ .
Similar results can be obtained if we assume
−
∗
−
orβ x = M−
β x = MB− andβ∗x = NA
A andβ x = NB
Hence A ∪ R B is ECIS in X.
Example 8. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in a set
X = { a, b, c} as shown in Table 3. Then it is easy to see that A and B satisfy the conditions
−
−
+
min{max{ M+
A , MB }, max{ M A , MB }} = ( α A ∧ α B )
−
−
+
> max{min{ M+
A , MB }, min{ M A , MB }} and
+
−
min{max{ NA
, NB− }, max{ NA
, NB+ }} > ( β A ∨ β B )
= max{min{ NA+ , NB− }, min{ NA− , NB+ }}.
However, A ∪ R B is not ECIS because
(α A ∧ α B )( a) = 0.7 ∈ [0.6, 0.8] = [( M A ∪ MB )− ( a), ( M A ∪ MB )+ ( a)] and ( β A ∨
β B )( a) = 0.1 ∈ [0.05, 0.15] = [( NA ∩ NB )− ( a), ( NA ∩ NB )+ ( a)].
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Table 3. CISs A and B.
X
M A /α A
NA /β A
MB /α B
NB /β B
a
b
c
[0.6, 0.7]/0.7
[0.1, 0.4]/0.5
[0.1, 0.2]/0.2
[0.1, 0.15]/0.1
[0.5, 0.6]/0.5
[0.45, 0.6]/0.45
[0.5, 0.8]/0.9
[0.2, 0.3]/0.3
[0.05, 0.3]/0.4
[0.05, 0.2]/0.03
[0.55, 0.65]/0.55
[0.4, 0.5]/0.3
The following theorems can be easily verified and proved; therefore, we omit the details.
Theorem 9. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ECISs in X, such that
−
−
+
min{max{ M+
A , MB }, max{ M A , MB }} ≥ ( α A ∨ α B )
−
−
+
> max{min{ M+
A , MB }, min{ M A , MB }} and
+
−
, NB+ }} > ( β A ∧ β B )
min{max{ NA
, NB− }, max{ NA
≥ max{min{ NA+ , NB− }, min{ NA− , NB+ }},
then A ∩ R B is also an ECIS in X.
Theorem 10. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ICISs in X. If
−
(α A ∧ α B ) ≤ max{ M−
A , MB }
( β A ∨ β B ) ≥ min{ NA+ , NB+ },
then A ∪ R B is an ECIS in X.
Theorem 11. Let A = h M A /α A , NA /β A i and B = h MB /α B , NB /β B i be two ICISs in X. If
+
(α A ∨ α B ) ≥ min{ M+
A , MB }
( β A ∧ β B ) ≤ max{ NA− , NB− },
then A ∩ R B is ECIS in X.
4. MCDM Method Based on Cubic Intuitionistic Sets
In this section, we will apply the proposed operations to deal with the MCDM problems using CISs.
Let A = { A1 , A2 , . . . , Am } be a set of alternatives, C = {C1 , C2 , . . . , Cn } be a set of
criteria, and E = {e1 , e2 , . . . , eK } be a set of experts. Suppose each alternative Ai (i =
1, 2, . . . , m) is assessed by the expert ek (k = 1, 2, . . . , K ) with respect to the criteria Cj ( j =
1, 2, . . . , n) using CISs. The proposed MCDM method is based on the following steps.
Step 1 Construct the decision matrices Rk = (rijk )m×n based on the assessed values of
expert ek (k = 1, 2, . . . , K ) in the form of CISs rijk .
Step 2 Calculate the aggregated decision matrix R = (rij )m×n by using the proposed
operations as discussed in Definition 12 where rij = ∪ P rijk or rij = ∪ R rijk .
k =1,2,...K
k =1,2,...K
Step 3 Calculate the score value of each rij of the aggregated decision matrix R by using
Definition 9.
Step 4 Calculate the preference values of each alternative Ai (i = 1, 2, . . . , m) where P( Ai ) =
∑im=1 ∑nj=1 rij .
Step 5 Generate the ranking order of alternatives according to the non-increasing order of
the preference values.
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An Application Example
Let us suppose that a technical committee composed of three technicians/experts
E = {e1 , e2 , e3 } wishes to select the best available washing machine on the market. Suppose,
there are four types of washing machines A = { A1 , A2 , A3 , A4 } available in the market and
the experts are requested to select the best one amongst the four with respect to the criteria
set C = {C1 = eco-friendly, C2 = capacity, C3 = price}. Suppose the expert ek (k = 1, 2, 3)
assessed each alternative Ai (i = 1, 2, . . . , 4) under the criteria Cj ( j = 1, 2, 3) by using the
CISs. We will now proceed with the following steps.
Step 1 According to the expert’s opinion, the individual decision matrices R1 , R2 , R3 are
constructed, which can be seen in Tables 4–6.
Table 4. Decision matrix R1 provided by expert e1 .
Alt.
C1
C2
C3
A1
A2
A3
A4
h[0.7, 0.8]/0.35, [0.1, 0.2]/0.6i
h[0.2, 0.3]/0.25, [0.3, 0.4]/0.7i
h[0.8, 0.9]/0.7, [0.05, 0.1]/0.3i
h[0.5, 0.6]/0.5, [0.1, 0.2]/0.3i
h[0.5, 0.6]/0.5, [0.2, 0.3]/0.2i
h[0.3, 0.4]/0.65, [0.5, 0.6]/0.2i
h[0.7, 0.8]/0.2, [0.1, 0.2]/0.4i
h[0.6, 0.7]/0.3, [0.2, 0.3]/0.4i
h[0.6, 0.7]/0.4, [0.1, 0.2]/0.7i
h[0.2, 0.3]/0.7, [0.4, 0.5]/0.2i
h[0.6, 0.7]/0.3, [0.2, 0.3]/0.6i
h[0.4, 0.5]/0.6, [0.2, 0.3]/0.2i
Table 5. Decision matrix R2 provided by expert e2 .
Alt.
C1
C2
C3
A1
A2
A3
A4
h[0.6, 0.7]/0.3, [0.1, 0.2]/0.5i
h[0.25, 0.4]/0.5, [0.4, 0.5]/0.4i
h[0.7, 0.8]/0.8, [0.1, 0.2]/0.1i
h[0.4, 0.5]/0.4, [0.1, 0.2]/0.5i
h[0.45, 0.5]/0.6, [0.25, 0.35]/0.3i
h[0.4, 0.5]/0.6, [0.3, 0.4]/0.3i
h[0.8, 0.9]/0.7, [0, 0.1]/0.3i
h[0.5, 0.6]/0.4, [0.2, 0.3]/0.6i
h[0.5, 0.6]/0.7, [0.2, 0.3]/0.2i
h[0.3, 0.4]/0.4, [0.5, 0.6]/0.6i
h[0.5, 0.6]/0.8, [0.1, 0.2]/0.2i
h[0.5, 0.6]/0.5, [0.2, 0.3]/0.4i
Table 6. Decision matrix R3 provided by expert e3 .
Alt.
C1
C2
C3
A1
A2
A3
A4
h[0.6, 0.7]/0.7, [0.2, 0.3]/0.2i
h[0.2, 0.3]/0.5, [0.4, 0.5]/0.4i
h[0.7, 0.85]/0.6, [0.1, 0.15]/0.2i
h[0.5, 0.6]/0.7, [0.2, 0.3]/0.2i
h[0.55, 0.6]/0.8, [0.2, 0.3]/0.1i
h[0.3, 0.4]/0.6, [0.4, 0.5]/0.1i
h[0.75, 0.8]/0.6, [0.1, 0.2]/0.3i
h[0.5, 0.6]/0.5, [0.2, 0.3]/0.4i
h[0.65, 0.7]/0.6, [0.2, 0.3]/0.3i
h[0.25, 0.3]/0.4, [0.5, 0.6]/0.5i
h[0.6, 0.7]/0.8, [0.1, 0.2]/0.2i
h[0.4, 0.5]/0.7, [0.1, 0.2]/0.1i
Step 2 The aggregated decision matrix R = (rij )4×3 is calculated with the help of the
proposed operation (P-union) as introduced in Definition 12 where rij = ∪ P rijk .
k =1,2,3
The aggregated decision matrix R is shown in Table 7.
Table 7. Aggregated decision matrix R by applying the P-union operation.
Alt.
C1
C2
C3
A1
A2
A3
A4
h[0.7, 0.8]/0.7, [0.1, 0.2]/0.2i
h[0.25, 0.4]/0.5, [0.3, 0.4]/0.4i
h[0.8, 0.9]/0.8, [0.05, 0.1]/0.1i
h[0.5, 0.6]/0.7, [0.1, 0.2]/0.2i
h[0.55, 0.6]/0.8, [0.2, 0.3]/0.1i
h[0.4, 0.5]/0.65, [0.3, 0.4]/0.1i
h[0.8, 0.9]/0.7, [0, 0.1]/0.3i
h[0.6, 0.7]/0.5, [0.2, 0.3]/0.4i
h[0.65, 0.7]/0.7, [0.1, 0.2]/0.2i
h[0.3, 0.4]/0.7, [0.4, 0.5]/0.2i
h[0.6, 0.7]/0.8, [0.1, 0.2]/0.2i
h[0.5, 0.6]/0.7, [0.1, 0.2]/0.1i
Step 3 By using Definition 9, we will calculate the score value of each rij of the aggregated
decision matrix R. The matrix of the score values of the elements of R is shown in
Table 8.
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Table 8. Score values of the aggregated decision matrix.
Alt.
C1
C2
C3
A1
A2
A3
A4
0.5667
0.0167
0.7500
0.4333
0.4500
0.2500
0.6667
0.3000
0.5167
0.1000
0.5333
0.4667
Step 4,5 Finally, the preference value P( Ai ), i = 1, 2, ..., 4 of each alternative is calculated
where P( Ai ) = ∑4i=1 ∑3j=1 rij . The preference values of alternatives by using the
P-union operation are given below:
P( A1 ) = 0.5111, P( A2 ) = 0.1222, P( A3 ) = 0.6500, P( A4 ) = 0.4000.
We can see that the ranking order of alternatives according to the non-increasing
order of their preference values is A3 A1 A4 A2 . Similarly, the preference
value of each alternative by using the R-union operation is calculated and given as
follows:
P( A1 ) = 0.2778, P( A2 ) = −0.0444, P( A3 ) = 0.4389, P( A4 ) = 0.2333
In this case, the ranking order of alternatives is A3 A1 A4 A2 .
We can observe that the ranking order of alternatives by using the R-union operation is
exactly the same as that obtained with the help of the P-union operation, which shows the
robustness of the proposed approach. We can easily see that by using the P-intersection and
R-intersection operations as discussed in Definition 12, the ranking order of alternatives
will lead to the reverse order of the raking orders obtained in the P-union and R-union
operations, respectively.
5. Conclusions
In this research work, we introduced a new modified form of CIS and discussed some
of its related properties. We further introduced two types of CISs, i.e., ICIS and ECIS.
The P-(R-) order, P-(R-) union, P-(R-) intersection of CISs, and some useful properties
were also discussed with necessary examples. As a supplement, we proved that the Punion and P-intersection of ICISs are also ICISs. Some conditions for the P-(R-) union
and P-(R-) intersection of two ECISs to be ICISs were also provided in this paper. We also
provided a few conditions for the P-(R-) union and P-(R-) intersection of two ECISs to be
ECISs. To check the effectiveness and validity of the proposed operations, we provided an
application example at the end by solving a MCDM problem.
In future work, more research can be conducted regarding the intuitionistic cubic soft
set and its application in information science and knowledge systems. We intend to apply
the intuitionistic cubic soft sets to algebraic structures.
Author Contributions: Conceptualization, S.F., T.R., S.Z., H.S. and W.S.; methodology, S.F., T.R., S.Z.,
H.S. and W.S.; software, S.F., T.R., S.Z., H.S. and W.S.; validation, S.F., T.R., S.Z., H.S. and W.S.; formal
analysis, S.F., T.R., S.Z., H.S. and W.S.; investigation, S.F., T.R., S.Z., H.S. and W.S.; resources, S.F., T.R.,
S.Z., H.S. and W.S.; data curation, S.F., T.R., S.Z., H.S. and W.S.; writing—original draft preparation,
S.F., T.R., S.Z., H.S. and W.S.; writing—review and editing, S.F., T.R., S.Z., H.S. and W.S.; visualization,
S.F., T.R., S.Z., H.S. and W.S.; supervision, S.F., T.R., S.Z., H.S. and W.S.; project administration, S.F.,
T.R., S.Z., H.S. and W.S.; funding acquisition, S.F., T.R., S.Z., H.S. and W.S. All authors have read and
agreed to the published version of the manuscript.
Funding: The work was supported by the National Science Centre 2021/41/B/HS4/01296 (W.S.).
and 2022/01/4/ST6/00028 (G.S.)
Institutional Review Board Statement: Not applicable.
Mathematics 2023, 11, 1190
16 of 17
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: The authors would like to thank the editor and the anonymous reviewers, whose
insightful comments and constructive suggestions helped us to significantly improve the quality of
this paper.
Conflicts of Interest: The authors declare no conflict of interest.
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