1. Introduction
Decision-making is a common task associated with intelligent and complicated actions. Here, humans face situations in which they must select between many options using logic and mental processes. Depending on the nature of the circumstance, many sorts of uncertainty may be present. Different strategies and techniques are used to cope with uncertainty in decision-making difficulties. Researchers have introduced many theories and tools—for example, fuzzy set theory [
1], interval-valued fuzzy set theory [
2], intuitionistic fuzzy set theory [
3], rough set theory [
4], as well as soft set theory [
5]. These theories were created to address the problem of the lack of parameterization tools in classic uncertainty theories. Soft set theory, in addition, is not an extension of earlier mathematical ideas. When it comes to dealing with uncertainty, soft set theory differs drastically from traditional models. Soft set theory has been claimed to have practical and prospective applications in a variety of disciplines, including game theory, measurement theory, decision-making, medical diagnostics, and others.
Recently, soft set theory and its extension to other mathematical approaches were vigorously investigated by many authors. Soft set theory combined with a fuzzy set theory introduced a new concept, namely fuzzy soft set theory [
6], which has been applied in decision-making [
7,
8,
9,
10,
11,
12]. Soft set theory can be combined with an intuitionistic fuzzy set theory [
13,
14] and applied in decision-making [
15,
16]. Furthermore, Yang et al. [
17] developed a hybrid model known as interval-valued fuzzy soft sets and presented several fundamental characteristics. The authors then utilized interval-valued fuzzy choice values to address decision-making issues that constitute the sum of lower and upper objects’ membership with respect to each parameter. The number of parameters fulfilled by the object may not be explained as the concept of interval-valued choice values. To address this restriction, Feng et al. [
18] utilized reduced fuzzy soft sets with a level soft set of interval-valued fuzzy soft sets to gain a better understanding of the decision-making processes as described by Yang et al. [
17]. Then, depending on (weighted) interval-valued fuzzy soft sets, they introduced flexible methods for decision-making procedures. In addition, the concept of interval-valued fuzzy topology was presented in [
19] and was extended later by [
20] based on the interval-valued fuzzy topology.
The decision-making methods based on interval-valued fuzzy soft set were first used by Yang et al. [
17]. Moreover, Row and Maji [
7] proposed the fuzzy soft sets concept, which was then implemented to solve decision-making processes. In addition, Kong et al. [
17] modified the method of Row and Maji [
7] by proposing a new fuzzy soft set based on multi-criteria decision-making utilizing a level soft set. However, Basu et al. [
9] discussed that the procedure of selecting the level soft set is not unique. Moreover, Ma et al. [
21] gave four distinct types of parameter reduction for interval-valued fuzzy soft sets, which were then contrasted concerning the computation complexity, the exact applicability, and reduction findings level. Furthermore, Ref. [
22] presented a new decision-making algorithm based on two types of tables, namely the average table and the antitheses for interval-valued fuzzy soft sets, while Me et al. [
23] discussed two different methods. Here, the first method was suggested by Yang et al. [
17] and the other proposed by [
24,
25]. In particular, Khameneh et al. [
26,
27] demonstrated the preference relationship of both intuitionistic fuzzy soft sets and fuzzy soft sets, which were subsequently used to address group decision-making issues. Moreover, Ali et al. [
28] expanded Khameneh et al. [
26]’s work on the interval-valued fuzzy soft set preference relationship. This work concentrates on using interval-valued fuzzy soft topology to generalize the equivalence and preorder of interval-valued fuzzy soft sets. Depending on the preference relationship, this generalized technique provides a deeper understanding of the decision-making process. This paper is outlined as follows. In
Section 2, we provide several definitions and theorems acquired for this paper. In
Section 3, we study interval-valued fuzzy preordered and interval-valued fuzzy soft equivalences. Then, by using
cut, two different crisp preorders and equivalences are defined. In
Section 4, the interval-valued fuzzy soft data rank is formulated depending on a new score function to solve the decision-making problem.
2. Preliminaries
This section reviews several fundamental properties and definitions acquired. Note that, in this study, X denotes the set of objects, E denotes the set of parameters, denotes the set of all fuzzy subsets, and in which and denotes the set of all interval-valued fuzzy subsets of X. Then, a fuzzy subset f over X is the mapping where the value of denotes the membership degree of
Definition 1 ([
2])
. An interval-valued fuzzy set set of pair is a mapping expressed by provided that for any represent a closed subinterval of in which and denote the upper and lower degrees of membership x to f with . Molodtsov [
5] introduced the soft sets (SS) concept for the first time in 1999 as a pair of
or
in which
E denotes a parameter set and
f denotes the mapping
in which for any
denotes a subset of
X. A novel hybrid tool is defined as follows by merging the soft sets concept with interval-valued fuzzy sets.
Definition 2 ([
17])
. An interval-valued fuzzy soft set set, as a pair of , is the mapping f given by in which for any and , . Assume two sets over the common universe X. Then, the union of and , expressed by , is the set in which for any and , we obtain . The intersection of and , expressed by , denotes the set in which and , and we obtain . The complement of is denoted by and is expressed by in which and any . The null set, expressed by , is denoted as an set over X in which ∀ and any . Moreover, the absolute set, expressed by is denoted as an set over , for any and ∀.
By employing the matrix form of interval-valued fuzzy relations, researchers in [
29,
30] assembled a finite
set given by the following
matrix:
in which
,
and
for
and
As a result, the properties of complement, intersection, union, and others may be expressed in the finite case’s matrix format.
Definition 3 ([
20])
. The collection τ of an subset of which is closed under arbitrary union with finite intersection and containing absolute and null sets, is known as the interval-valued fuzzy soft topology. Definition 4 ([
28])
. The α-upper and β-lower crisp concepts of all parameters e of f, in which and are defined aswhich is formulated into the two matrices given belowandin which and are the given threshold vectors. Theorem 1 ([
28])
. The following collection form α-upper topology and β-lower topology over in which and is given by Theorem 2 ([
28])
. The following binary relations are two preorder relations, in which and such that Definition 5 ([
28])
. Let the binary relations be and and threshold intervals . We then expressas well as 3. Generating Preorder and Equivalence Relations from Interval-Valued Fuzzy Soft Data
In this section, the interval-valued fuzzy soft preorder and the interval-valued fuzzy soft equivalence are presented. We then provide upper crisp preorder and lower crisp preorder by using -cut.
Theorem 3. Letbe antopological space and letandbe two-points with distinct support x and y with e-lower and e-upper values ofandaccordingly.
- 1.
Thebinary relation “” on X expressed byis anpreorder onwhile the pairis known as anpreordered set. - 2.
Thebinary relation “” on X expressed byis anequivalence relation overIfthenandareequivalence.
Proof. - 1.
Firstly, if is a - open set containing then for all , where Thus, “ ” is reflexive. Now, assume and where and are any -points. Then, if is a - open set containing , then and also Thus, Therefore, “ ” is transitive.
Hence, in general, for any two -points and with distinct support x and y with e-lower and e-upper values of and accordingly. Here, we say that if and only if for each -open set we have Then, implies that ∀ and and we obtain and
- 2.
It is straightforward.
□
Definition 6. Let be an topological space and let be an set induced by an preorder on The concepts of α-upper crisp “” and β-lower crisp “” relations on X, in which and and for all , are given as follows: It is obvious that for the
open set
induced by an
preorder on
the
-upper crisp relation
and
-lower crisp relation
on
in which
are given by
or
as well as
or
are considered as
-upper preorder and
-lower preorder relations, respectively.
Definition 7. Let denote an topological space and let be an set induced by an equivalence on The α-upper crisp “ ” and β-lower crisp “ ” relation concepts on X, in which and ∀, are given as follows: Similarly, the
-upper crisp relation
and
-lower crisp relation
on
X given by
and
are defined as
-upper equivalence and
-lower equivalence relations, respectively, ∀
,
.
Proposition 1. Let denote an topological space and let and denote two sets induced by an preorder on Then, for all the threshold intervals and where , , and the following hold.
- 1.
If then and Similarly, if then and
- 2.
If then and Similarly, if then and
- 3.
If then and Similarly, if then and
- 4.
and
- 5.
If then and Similarly, if then and
Proof. The proof follows immediately thereafter. □
Proposition 2. Let denote an topological space and let be an set induced by an preorder on Then, for all the threshold interval and in which as well as for and we have
- 1.
- 2.
Proof. Let and Thus, we have
- 1.
Therefore,
- 2.
For
and
we have
Hence, □
3.1. Comparison between Preorder Matrices
Let the finite set denote the set of objects and resemble the set of parameters. The matrix forms of the upper preorderings “” and the lower preorderings “” on X are utilized to express two comparison matrices, and These are two square matrices having columns and rows labeled by objects of the universe X given below.
Definition 8. Consider the upper binary relations and the lower binary relation on while is an set induced by an preordered set and threshold intervals and We then expressandwhere Proposition 3. Assume that is an topological space and , are two matrices defined in Equations (5) and (6), where the threshold intervals Then, the following hold. - 1.
For and
- 2.
If then If then
- 3.
and resemble symmetric matrices.
in which
Proof. We only prove part 2. The other parts are derived similarly.
Assume that then, and Thus, Since is a transitive relation. Then, Similarly, assume that then and Thus, □
Proposition 4. Let denote an topological space and the threshold intervals as well as are given, where Suppose that is an set induced by an preorder on Then, the following hold:
- 1.
= if and only if ∀ and
- 2.
= if and only if ∀ and
- 3.
if and only if , ∀ and
- 4.
if and only if , ∀ and
where are an identity and a unit matrix, respectively.
Proof. We prove parts 1 and 4. The other parts are derived similarly.
For part 1, assume that
=
Then, ∀
and we have
and
if
Hence, by Equation (
5), we obtain
, while
if
Assume that ∀ such that and we have Thus, However, by Proposition (1), we have for Then, =
For part 4, assume that
Then,
for all
Then, by Equation (
6), we have
for all
Assume that ∀
and
Hence, by Equation (
6), we obtain
and
□
Proposition 5. Let denote an topological space and denote an set induced by an preorder on with in which are the threshold intervals. Then,
- 1.
if and only if
- 2.
if and only if
- 3.
if and only if
- 4.
if and only if
where resemble the upper and lower triangular matrices, accordingly.
Proof. We prove parts 1 and 2. The other parts are derived similarly.
- 1.
Assume that
Then, ∀
and we have
if
and
if
By (Equation (
5)) and ∀
, we have
, while for
we obtain
Thus,
∀
∀
but
and finally
but
for all
Then,
on
Assume that
on
Then, ∀
and we have
if
and
if
By (Equation (
5)), we obtain
if
and
if
Therefore,
- 2.
Assume that
Then, for all
, we have
; we have
if
and
if
By Equation (
6), we have
Thus, for all for all but and finally but for all
Thus, in
Assume that
For all
if
then
and if
then
By Equation (
6), we have
if
and
if
Therefore,
□
Proposition 6. Let denote an topological space and the threshold intervals as well as are given, where Suppose that is an set induced by an preorder on Then, the following hold:
- 1.
then
- 2.
then
- 3.
If is the maximal set, then
- 4.
If is the minimal set, then
Proof. We prove part 1. The other parts are derived similarly.
For part 1, assume that
Then, by Equation (
3), we have
if
and
if
Thus, we have also by Equation (
5)
if
and
if
Therefore,
□
3.2. Equivalence Matrices
Similarly, we can apply the upper equivalence relations and the lower equivalence relations on X to compute two square matrices given by
and accordingly, in which and
Definition 9. Consider the two binary relations and on X and is an set induced by an equivalence on X with threshold intervals and Then, we expressandwhere Proposition 7. Let denote an topological space with given threshold intervals and where Then, the following hold:
- 1.
and for all
- 2.
If then . If then ,
- 3.
If then . If then
- 4.
and resemble symmetric matrices,
in which
Proof. The proof follows immediately thereafter. □
Proposition 8. Let denote an topological space with given threshold intervals as well as , where Suppose that is an set induced by an equivalence on Then, the following hold:
- 1.
For any : then
- 2.
For any : then
- 3.
If is the maximal set, then
- 4.
If is the minimal set, then
Proof. The proof follows immediately thereafter. □
4. Application in Decision-Making
Decision-making is a common term in daily life and is associated with intelligent and complicated procedures that humans might face. However, decision-making is also a fundamental part of organization and management. In particular, correct and efficient decision-making is the primary objective and goal for management. In fact, in any management structure, decision-making sub-consciously or consciously becomes an important parameter in the role of organization. Thus, the decision-making will follow certain sequential steps, such as defining the problem; collection of information and data, and determination of weighing options; selection of the best possible option; and performing the execution and applications. Now, in these stages, if any uncertainties occur, then the decision-making process will involve taking a decision in an uncertain environment, where information can be handled by fuzzy sets and systems.
In real-world problems and applications, the sequential stages may be more complicated due to complexities and uncertainties; thus, the decision-makers will adopt an alternative method, and it is also possible to prefer fuzzy methods rather than the crisp ones. In this section, we present a new formula to compute the score function of each object based on the preference relationship between two different upper
(see Equations (
1) and (
5)) and lower preorderings
(see Equations (
2) and (
6)), respectively.
Definition 10. Let X denote the universal set of objects, E denote the set of parameters, and let the threshold intervals be given, in which and The mapping is expressed byin which is the score function of object . According to this flowchart (see
Figure 1), the following algorithm is proposed.
Figure 1.
The flowchart for Algorithm 1.
Figure 1.
The flowchart for Algorithm 1.
Algorithm 1 Ranking Assessments by Interval-Valued Fuzzy Soft Preorder Relation |
|
Example 1. Let denote a set of five-star hotels for one customer and denote a set of parameters. Suppose that customers wish to choose the parameters given by “chromatic exterior and interior design”, “cleanliness”, “facilities”, and “excellent service”, respectively. We can assess the hotels as three matrices given in the following Table 1, Table 2 and Table 3. Assume that and
Step
The upper and lower crisp matrices are given as:
Step
The upper and lower topology are expressed in
Table 4 and
Table 5, accordingly.
Step
Compute matrices
and
by using Equations (
5) and (
6). Moreover, matrices
and
are computed using Equations (
3) and (
4) over
in which
and
given by:
Step 4. By using Equation (
9), in which
as well as
we have
Step 5. Then, the ranking of the overall assessment is obtained as below:
Steps 6 and 7. Therefore, is the best object, while cannot be selected.
5. Conclusions
Fuzzy ordered structures on a universal set are an important research tool to model uncertainty or fuzziness in the real world, which is closely related to fuzzy topology. This paper introduced interval-valued fuzzy soft preorderings, and subsequently an interval-valued fuzzy soft equivalence based on interval-valued fuzzy soft topology. We then presented two different crisp preorderings and equivalence relations over the X-associated interval-valued fuzzy soft topology. Employing a new method for ranking data, a score function was defined to solve multi-group decision-making problems. Finally, a numerical example was given. For future research, interval-valued fuzzy soft ordering is the most powerful concept in system analysis. It can be implemented from the decision-making methods in conflict handling, recommender systems, and practical evaluation systems.