1. Introduction
Researchers always try to discover methods to handle imprecise and vague information, which is not possible using classical set theory. In this regard, Zadeh gave the concept of fuzzy set [
1], to cope with uncertainty. However, fuzzy sets were considered imperfect since it is not always easy to give an exact degree of membership to any element. To overcome this problem, the interval-valued fuzzy set was proposed by Turksen [
2]. Atanassov [
3] extended the notion of fuzzy sets to intuitionistic fuzzy sets by introducing the non-membership of an element with its membership in a set
X, which were proven to be a better tool than fuzzy sets. Furthermore, the intuitionistic fuzzy sets are used in many directions [
4]. Smarandache gave the notion of neutrosophic sets as a generalization of intuitionistic fuzzy sets and fuzzy sets [
5]. The idea of neutrosophic sets are further expanded to different directions [
6,
7,
8,
9] by various researchers. Jun et al. [
10] gave the idea of cubic set and it was characterized by interval-valued fuzzy set and fuzzy set, which is a more general tool to capture uncertainty and vagueness, since fuzzy set deals with single-value membership while interval-valued fuzzy set ranges the membership in the form of intervals. The hybrid platform provided by cubic set has the main advantage since it contains more information than a fuzzy set and interval-valued fuzzy set. By using this concept, different problems arising in several areas can be solved by choosing the finest choice by means of cubic sets as in the works of Abughazalah and Yaqoob [
11], Rashid et al. [
12], Gulistan et al. [
13], Ma et al. [
14], Naveed at al. [
15], Gulistan et al. [
16], Khan et al. [
17,
18], Yaqoob et al. [
19], and Aslam et al. [
20].
More recently, Jun et al. [
21] gave the idea of neutrosophic cubic set and it was subsequently used in many areas by Khan et al. [
22] and Gulistan et al. [
23,
24].
On the other hand, Molodtsov [
25] introduced the concept of soft sets that can be seen as a new mathematical theory for dealing with uncertainty. It was applied to many different fields by Maji et al. [
26] who later defined fuzzy soft set theory and some properties of fuzzy soft sets [
27]. Hybrids of soft sets were further developed [
28,
29,
30,
31,
32].
Alkhazaleh and Salleh in 2011 defined the concept of soft expert set in which the user could know the opinion of all the experts in one model and gave an application of this concept in the decision-making problem [
33]. Arokia et al. [
34] studied fuzzy parameterizations for decision-making in risk management systems via soft expert set. Arokia and Arockiarani [
35] provided a fusion of soft expert set and matrix models. Alkhazaleh and Salleh [
36] extended the concept of soft expert set in terms of fuzzy set and provided its application. Bashir and Salleh [
37] provided the concept of fuzzy parameterized soft expert set. Bashir et al. [
38] discussed possibility fuzzy soft expert set. Alhazaymeh et al. [
39] provided the application of generalized vague soft expert set in decision-making. Broumi and Smarandache [
40] extended the soft expert sets in terms of intuitionistic fuzzy sets. Abu Qamar and Hassan [
41,
42] presented the idea of Q-neutrosophic soft relation and its entropy measures of distance and similarity. Sahin et al. [
43] gave the idea of neutrosophic soft expert sets while Uluçay et al. [
44], introduced the concept of generalized neutrosophic soft expert set for multiple-criteria decision-making. Neutrosophic vague soft expert set theory was put forward by Al-Quran and Hassan [
45] and developed it further to complex neutrosophic soft expert set [
46,
47]. Qayyum et al. [
48] gave the idea of cubic soft expert sets for a more general approach. Ziemba and Becker [
49] presented analysis of the digital divide using fuzzy forecasting, which is a new approach in decision-making.
Hence it is natural to extend the concept of expert sets to neutrosophic cubic soft expert sets for a more generalized approach. The major contribution of this paper is the development of neutrosophic cubic soft expert sets(NCSESs) by using the concept of neutrosophic cubic soft sets which generalizes the concept of fuzzy soft expert sets, intuitionistic soft expert sets, and cubic soft expert sets. We define and analyze the properties of internal neutrosophic cubic soft expert sets (INCSESs) and external neutrosophic cubic soft expert sets (ENCSESs),
P-order,
P-union,
P-intersection,
P-AND,
P-OR, and
R-order,
R-union,
R-intersection,
R-AND, and
R-OR of NCSESs. The NCSESs satisfy the laws of commutativity, associativity, De Morgan, distributivity, idempotentency, and absorption. We derive some conditions for
P-union and
P-intersection of two INCSESs to be an INCSES. It is shown that
P-union and
P-intersection of ENCSESs need not be an ENCSES. The
R-union and
R-intersection of the INCSESs (resp., ENCSESs) need not be an INCSES (resp. ENCSES). Necessary conditions for the
P-union,
R-union, and
R-intersection of two ENCSESs to be an ENCSES are obtained. We also study the conditions for
R-intersection and
P-intersection of two NCSESs to be an INCSES and ENCSES. This paper is organized as follows.
Section 2 will be on preliminaries, while
Section 3 develops an approach to neutrosophic cubic soft expert set. We focus on the basic operations, namely
P-order,
R-order,
P-containment,
R-containment,
P-union,
P-intersection,
R-union,
R-intersection, complement,
P-AND,
P-OR,
R-AND, and
R-OR of NCSESs.
Section 4 will present more results on NCSESs, followed by
Section 5 on application in analyzing a cricket series. A comparison analysis will be discussed in
Section 6 and a conclusion is drawn in
Section 7.
3. Neutrosophic Cubic Soft Expert Set
In this section, we develop an approach to neutrosophic cubic soft expert set which is a more general approach for soft expert set theory. We focus on the basic operations namely, P-order, R-order, P-containment, R-containment, P-union, P-intersection, R-union, R-intersection, complement, P-AND, P-OR, R-AND, and R-OR of neutrosophic cubic soft expert sets. The symbol stands for the neutrosophic cubic soft expert set.
Definition 14. Let U be a finite set containing n alternatives, E be a set of criteria, X be a set of experts. A triplet is called neutrosophic cubic soft expert set over U, if and only if is a mapping into the set of all neutrosophic cubic set in U and defined aswheresuch that Example 1. Let be the set of countries playing a cricket series, playing conditions, historic record} be the set of factors affecting the series, be the set of experts giving their expert opinion. Let Then the neutrosophic cubic soft expert set is given by
The function of the form denotes the range of values where the experts are sure to give certain membership to a certain element, denotes the range of values where the experts are hesitant and denotes the range of values where the experts are sure to give negative points to a certain element as a non-membership. Thus, experts have a wide range of scale to make their conclusion as compared to the previous defined versions of fuzzy sets. More specific in the current example is the function of the form which gives the expert opinion for the past performance of these two countries, gives the expert opinion for running series between these two countries and gives the expert opinion for the upcoming series between these two countries which is not to be held in the near future. Definition 15. A neutrosophic cubic soft expert setover U is said to be: - (i)
Internal truth neutrosophic cubic soft experts set (briefly, ) if for all so that - (ii)
Internal indeterminacy neutrosophic cubic soft experts set (briefly, ) if for all so that - (iii)
Internal falsity neutrosophic cubic soft experts set (briefly, ) if for all so that
If a neutrosophic cubic soft expert set in satisfies (i), (ii), (iii), then it is known as internal neutrosophic cubic soft expert set in abbreviated as ().
Example 2. Consider the Example 1. Then the internal neutrosophic cubic soft expert set is given by
Remark 1. We can draw the following conclusion from Example 2;
(i) If the value of lies in the interval then it means that the respective team is going to maintain its progress in different time frames.
(ii) If the panel of experts consists of the internal panel (meaning that the experts are from the same country or same cricket board), then it is known as INCSESs.
Definition 16. A neutrosophic cubic soft expert setover U is said to be: - (i)
External truth neutrosophic cubic soft expert set (briefly, ) if for all we have - (ii)
External indeterminacy neutrosophic cubic soft expert set (briefly, ) if for all we have - (iii)
External falsity neutrosophic cubic soft expert set (briefly, ) if for all we have
If a neutrosophic cubic soft expert set over satisfies (i), (ii), (iii), then it is known as external neutrosophic cubic soft expert set in abbreviated as ().
Example 3. Let U be the set of countries playing a one-day international (ODI) triangular series provided in Example 1, then the external neutrosophic cubic soft expert set is given by;
Remark 2. We can draw the following conclusion from Example 3;
(i) If the value of does not lie in the interval then it means the respective team is not maintaining its progress in different time frames.
(ii) If the panel of experts consists of the external panel (meaning that the experts are not from the same country or same cricket board), then it is known as ENCSESs.
Our next discussion is to define some basic operations on neutrosophic cubic soft expert sets to get more insight of neutrosophic cubic soft expert sets.
Definition 17. A over U is said to be P-order contained in another over U, denoted by
if
where condition implies that Definition 18. A over U is said to be R-order contained in another over U, denoted by
if
where condition implies that Definition 19. Two and over U is said to be equal which is denoted by
if
where condition implies that Remark 3. (a) We observe from Definitions 17–19, that for any two and over U;
(i) If and
then
(ii) If and
then
(b) Using Definitions 17–19, one can easily compare the performance of two cricket teams in different time frames.
Definition 20. Let and be two in
Then we define (i) where where (ii) where where (iii) where where (iv) where where (v) The complement of a neutrosophic cubic soft expert set denoted by Proposition 1. Let , be in Then
- (i)
If and
then
- (ii)
If
then
- (iii)
If and
then
- (iv)
If and
then
- (v)
If and
then ,
Proof. The proof is straightforward. □
Theorem 1. For any two and over U the following properties hold:
- (i)
Idempotent law:
- (ii)
Commutative law:
- (iii)
Associative law:
- (iv)
Distributive and De Morgan’s laws also true.
- (v)
Involution law:
Proposition 2. For any two and over U the following properties are equivalent:
- (i)
- (ii)
- (iii)
Proof. By Definition 20, we have
as
by hypothesis. Now for any
since
using Definition 17, implies that
and
for any
where
. Since
and
Thus
and
So It is ok.
Hence
□
(ii)⇒(iii) By Definition 20, we have
as
and
, by hypothesis. Now for any
since
by Definition 20, we have
this implies that
Thus, we have
Hence
(iii)⇒(i) By hypothesis we have
as
and
and
Also
this implies that
and
for any
.
Hence
Corollary 1. If we take in the Proposition 2, then the following are equivalent:
- (i)
- (ii)
- (iii)
Proof. The proof is straightforward. □
4. More on NCSESs
In this section, we discuss different types of union and intersection of the NCSESs and their related conditions.
1. The following example shows thatR-Union of twoinUneed not bein
Example 4. Let and be two in whereandNow by Definition 20, we have As and Hence is not a in U.
The following theorem gives the condition under which R-union of two in U is also a in U.
Theorem 2. Let and be two in
where
and
such thatfor all and Then is in U. Proof. By Definition 20, we know
where
where
If
or if
then the result is trivial. If
then
□
Since
and
are
in
U. So
and
Also
for all
and
Hence
is
in
U.
2. The following example yields thatR-intersection of two need not be a.
Example 5. Let and be two in whereandNow by Definition 20, we have As and Hence is not a in U.
The following theorem gives the condition that R-intersection of two is to be a
Theorem 3. Let and be two in
where and such thatfor all and Then is a in U. Proof. Similar to the proof of the Theorem 2. □
3. The following example yields thatR-union of twoneed not be an.
Example 6. Let and be two in whereandNow by Definition 20, we have As and Hence R-union is not a in U.
The following theorem gives the condition that R-union of two to be a .
Theorem 4. Let and be two in
where and such thatfor all and Then is a in U. Proof. By Definition 20, we know
where
where
If
take
and
Then
h is one of
If we choose
or
then
and so
Thus
Hence
Now if
then
and so
Assume
then we have
So, we can write
For the case
which contradicted the fact that
and
be two
in
U. For the case
we have
Again, assume that
then we have
or
For the case
which contradict
and
be two
in
U. For the case
we have
because
If
or if
then the result is trivial. Hence
is a
in
4. The following example shows thatR-intersection of twoneed not be.
Example 7. Let and be two in whereandNow by Definition 20, we have As and Hence