https://orcid.org/0000-0002-3923-2277
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Abstract
This paper investigates N-bipolar hypersoft topology (N-BHST), a novel extension of both the well-established N-hypersoft topology (N-HST) and hypersoft topology (HST). Deviating significantly from its precursor, the N-bipolar hypersoft (N-BHS) set, N-BHST introduces a multi-opinion approach to decision-making, augmenting robustness and adaptability. This innovative framework addresses identified limitations in N-bipolar soft topology (N-BST), especially in managing multi-argument approximate functions. The study analyzes various operators (closure, interior, exterior, and boundary) within the N-BHST framework, elucidating their interrelationships. Additionally, an examination is carried out on the enhancement of multi-criteria group decision-making (MCGDM) using N-BHST, setting it apart from existing models. A numerical example is presented to illustrate its application in real-world decision scenarios.
Citation: Musa SY, Asaad BA (2024) A progressive approach to multi-criteria group decision-making: N-bipolar hypersoft topology perspective. PLoS ONE 19(5): e0304016. https://doi.org/10.1371/journal.pone.0304016
Editor: Naeem Jan, Korea National University of Transportation, KOREA, REPUBLIC OF
Received: January 24, 2024; Accepted: May 3, 2024; Published: May 21, 2024
Copyright: © 2024 Musa, Asaad. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: The author(s) received no specific funding for this work;.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
In the information age, the challenges associated with decision-making have become increasingly complex due to intricate decision-making environments. Expressing attribute values for alternatives has consequently become more difficult. To address this, Zadeh introduced the concept of fuzzy sets [1], offering a convenient way to represent fuzzy information in decision-making scenarios. Alongside this, probability theory has emerged as a crucial tool for handling intricate information in practical issues, extensively explored by numerous scholars [2, 3]. This heightened interest in fuzzy sets has led to various extensions, intuitionistic fuzzy sets [4], and interval-valued intuitionistic fuzzy sets [5].
The rapid growth in the volume and complexity of gathered information in our contemporary society has led to various forms of ambiguity, particularly in addressing complex issues across diverse fields such as engineering, economics, social science, environmental science, and biology. To effectively describe and extract valuable insights from data enveloped in uncertainty, researchers in computer science, mathematics, and related disciplines have explored several theories, including fuzzy set theory [1], vague set theory [6], probability theory, rough set theory [7–11], and interval mathematics [12]. Additionally, the theory of soft set, introduced by Molodtsov [13] in 1999, has emerged as a novel mathematical tool designed to handle vagueness independently of existing methods’ limitations. Soft set has found wide-ranging applications in various fields, including game theory, probability theory, smoothness of functions, Riemann integration, operations research, and Perron integration. Recently, researchers have shown significant interest in soft set and its various extensions [14–22].
2 Related works
In the realm of social judgment frameworks, Alcantud and Laruelle [23] introduced a ternary voting framework within the soft set context. Non-binary assessments are also crucial in ranking and rating scenarios, prompting Fatimah et al. [24] to propose an extended soft set model named N-soft (N-S) set, emphasizing the importance of ordered grades in real-world problems. Building on this, Alcantud et al. [25] extended N-S set to the realm of rough set theory. Akram et al. [26] introduced hesitant N-S set, while their subsequent work [27] delved into parameter reduction in N-S set. The semantics of N-S set were explored by Alcantud [28], and Akram et al. [29] further modified N-S set to devise fuzzy N-S set. Fatimah and Alcantud [30] contributed to the field with the invention of multi-fuzzy N-S set. Shabir and Fatima [31] presented N-bipolar soft (N-BS) set and its application to decision making. Another extension, intuitionistic fuzzy N-S set, was introduced by Akram et al. [32], while the notion of bipolar fuzzy N-S set was proposed by Akram et al. [33]. Expanding the scope, Mahmood et al. [34] studied complex fuzzy N-S set, and Rehman and Mahmood [35] investigated complex intuitionistic fuzzy N-S set. Riaz et al. [36] introduced N-soft topology (N-ST) as an extension of soft topology, exploring its applications in multi-criteria group decision-making. In a related context, Mustafa [37] proposed N-BST as an extension of both N-ST and soft topology, applying it to multi-criteria group decision-making. These extensions demonstrate the diverse applications and adaptability of soft set in addressing complex scenarios in recent research efforts.
In 2018, Smarandache [38] introduced the concept of hypersoft (HS) sets, offering an improved framework for handling indistinct and uncertain data within models similar to soft sets. HS sets employ a multi-argument approximation function strategy, enhancing adaptability and reliability compared to traditional soft sets. Building on this, Musa and Asaad [39] introduced bipolar hypersoft (BHS) sets demonstrating their applicability in decision-making problems [40]. Their subsequent work delved into the topological structures associated with BHS sets [41, 42]. The exploration of core principles, properties, and operations of HS sets and their extensions has been the focus of extensive investigations [43–47].
Recent investigations into hybrid HS set models underscore a rising interest among researchers in innovative methodologies. These models incorporate standard HS sets with binary evaluations and fuzzy HS sets utilizing real numbers between 0 and 1. Real-world challenges often entail non-binary and discrete data structures. Addressing this, Musa et al. [48] introduced N-HS sets, presenting a more comprehensive framework than conventional HS sets. N-HS sets integrate a parametrized object characterization based on ordered grades, providing a versatile representation for intricate real-world scenarios. Furthermore, they [49] introduced the notion of N-HST, extending the concept from HST [50].
In addition to Musa’s significant contributions to HS sets, their groundbreaking work on the N-BHS set [51] is noteworthy. This innovative hybridization of N-HS sets and bipolarity settings enhances HS model capabilities, providing nuanced data representation solutions for non-binary and discrete data structures. Building upon N-BHS sets, our article extends the exploration into BHS set theory by introducing N-BHST. This extension aims to enhance our understanding of structural properties and relationships within the N-BHS framework. We further explore the practical implications of N-BHST by examining its application in MCGDM. Integrating topological considerations into N-BHS set models enriches the toolkit for researchers and practitioners, offering insights and methodologies for tackling decision-making challenges in real-world scenarios.
2.1 Motivation
Inspired by the complexities inherent in decision-making scenarios, this manuscript explores the conceptual realm of the N-BHST, marking a substantial departure from its forerunner, the N-BHS set. Notably, the N-BHST introduces a paradigm shift by embracing a multi-opinion approach to decision-making, in contrast to the single-opinion framework utilized by the N-BHS set.
In addition to its unique decision-making approach, the N-BHST pioneers a fresh line of inquiry aimed at addressing limitations observed in the N-BST regarding the handling of multi-argument approximate functions. The manuscript presents a systematic partitioning of attributes into discrete subattribute values through the application of disjoint sets.
2.2 Main contributions
The ensuing key contributions underscore our inventive approaches and methodologies:
- Traced the evolution of soft set models from conventional to advanced concepts like HS sets.
- Introduced N-BHST as a resilient multi-opinion approach to decision-making.
- Innovatively partitioned attributes into discrete subattribute values, tackling challenges in multi-argument approximations.
- Proposed algorithms within the N-BHST framework for accommodating diverse opinions in decision-making.
- Conducted a comparative analysis, establishing N-BHST as a versatile approach for critical evaluation factors.
- Explored fuzzy integration in N-BHST and outlined plans to extend the study to some novel models, aiming to derive structures for solving real-world problems with uncertainties.
2.3 Paper organization
The organization of this paper is outlined as follow: In Section 1, we provide an introduction to the context and motivation behind our research. In Section 2, we review related works, discussing the state-of-the-art literature, identifying gaps, and highlighting the methodologies proposed by other scholars in addressing similar problems. In Section 3, we review HS sets, N-HS sets, and N-BHS sets covering associated definitions and results. In Section 4, we delve into the extended N-HST, labeled as N-BHST, and introduce the corresponding operators. In Section 5, we provide details on MCGDM approaches suitable for N-BHST. In Section 6, we conduct a comparative analysis between N-BHST and relevant existing models. Finally, the conclusion of our study is presented in Section 7.
3 Preliminaries
In this section, we delve into the concepts of HS sets, N-HS sets, and N-BHS sets, providing an overview and presenting relevant findings associated with each.
3.1 Hypersoft sets
Definition 1. [38] Let ¥ denote a collection of alternatives, the power set of ¥. Consider a1, a2, …, an as n distinct attributes, each associated with the sets ξ1, ξ2, …, ξn, where ξi ∩ ξj = ∅ for i ≠ j, and i, j ∈ {1, 2, …, n}. An HS set is denoted as
, where
.
Let ξ1 × ξ2 × … × ξn = ξ and . It’s crucial to highlight that every element in ξ, ζ, and
is an n-tuple element. The representation of the HS set
is as follows:
Definition 2. [43] Consider a set of attributes ξ = {ϵ1, ϵ2, …, ϵn}. The NOT set of ξ, denoted as ¬ξ, is defined as ¬ξ = {¬ϵ1, ¬ϵ2, …, ¬ϵn}, where ¬ϵi = not ϵi for i = 1, 2, …, n.
3.2 N-hypersoft sets
Definition 3. [48] Let ¥ denote a collection of alternatives, ξ be a set representing attributes, and . Let R be a set of ordered grades, specifically R = {0, 1, …, N − 1}, where N ∈ {2, 3, …}. An N-HS set is a pair
, where
and satisfies the following condition: for each ϵ ∈ ξ, there exists a unique
such that
or, equivalently,
, where η ∈ ¥ and rϵ ∈ R.
The N-HS set can be expressed as follows:
The N-HS set can be visualized in tabular form, where rij denotes
or
. This tabular representation is depicted in Table 1.
Definition 4. [48] Suppose and
are two N-HS sets. Then,
-
is an N-HS subset of
, denoted by
, if ξ ⊆ ζ and
, ∀ ϵ ∈ ξ and η ∈ ¥.
-
and
are N-HS equal if
and ξ = ζ.
-
is a null N-HS set, denoted by Λ0, if ∀ ϵ ∈ ξ and η ∈ ¥,
.
-
is a whole N-HS set, denoted by ΛN−1, if ∀ ϵ ∈ ξ and η ∈ ¥,
.
- The N-HS complement of
is denoted as
where
.
- The N-HS extended union of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∪ ζ and η ∈ ¥:
- The N-HS extended intersection of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∪ ζ and η ∈ ¥:
- The N-HS union of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∩ ζ and η ∈ ¥:
.
- The N-HS intersection of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∩ ζ and η ∈ ¥:
.
Definition 5. [49] Suppose is a collection of N-HS sets, then
is considered an N-HST if:
- Λ0, ΛN−1 ∈
.
-
,
.
-
.
We define
as an N-hypersoft topological space (N-HSTS). The elements of
are called N-HS open sets, and their complements in the N-HS context are referred to as N-HS closed sets.
3.3 N-bipolar hypersoft sets
Definition 6. [51] Let ¥ denote a collection of alternatives, ξ be a set representing attributes, and . Let R be a set of ordered grades, specifically R = {0, 1, …, N − 1}, where N ∈ {2, 3, …}. An N-BHS set is a triple
, where
and
and satisfies the following condition: for each ϵ ∈ ξ, there exists a unique (η, rϵ), (η, r¬ϵ)
such that
(or
) and
(or
), with the condition that rϵ + r¬ϵ ≤ N − 1, where η ∈ ¥ and rϵ, r¬ϵ ∈ R.
The N-BHS set can be written in this simplified form:
.
The N-BHS set can be represented in tabular form, where
denotes
(or
), and
(or
). This representation is illustrated in Table 2.
Definition 7. [51] Suppose and
are two N-BHS sets. Then,
-
is an N-BHS subset of
, denoted by
, if ξ ⊆ ζ and
and
, ∀ ϵ ∈ ξ and η ∈ ¥.
-
and
are N-BHS equal if
,
, and ξ = ζ.
-
is a null N-BHS set, denoted by
, if ∀ ϵ ∈ ξ and η ∈ ¥,
and
.
-
is a whole N-BHS set, denoted by
, if ∀ ϵ ∈ ξ and η ∈ ¥,
and
.
- The N-BHS complement of
is
, where, ∀ ϵ ∈ ξ and η ∈ ¥,
and
.
- The N-BHS extended union of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∪ ζ and η ∈ ¥:
- The N-BHS extended intersection of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∪ ζ and η ∈ ¥:
- The N-BHS union of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∩ ζ and η ∈ ¥:
and
.
- The N-BHS intersection of
and
is denoted and defined as
, where ∀ ϵ ∈ ξ ∩ ζ and η ∈ ¥:
and
.
Proposition 1. [51] Suppose and
are two N-BHS sets. Then,
-
.
-
.
-
.
-
.
-
-
.
-
.
-
.
4 N-bipolar hypersoft topology and its operators
In this section, we present N-BHST as an extension of the N-HST and HST. Additionally, we explore various operators, including closure, interior, exterior, and boundary, within the framework of N-BHST, examining their interrelationships.
Definition 8. Suppose is a collection of N-BHS sets, then
is considered an N-BHST if:
-
,
.
-
,
.
-
.
We define
as an N-bipolar hypersoft topological space (N-BHSTS). The elements of
are called N-BHS open sets, and their complements in the N-BHS context are referred to as N-BHS closed sets.
Proposition 2. Consider the N-BHSTS . Then, the following statements hold:
- The sets
and
are N-BHS closed.
- The union of any two N-BHS closed sets remains a closed set in the N-BHS context.
- The intersection of any number of N-BHS closed sets remains a closed set in the N-BHS context.
Proof. Straightforward.
Proposition 3. Let be an N-BHSTS. Then, the following collections define N-HSTs. In particular,
and
are N-HSTSs.
- i.
.
- ii.
(provided that ¥ and ξ are finite sets).
Proof. i. Suppose that is an N-BHSTS. Then,
-
,
implies that Λ0, ΛN−1 ∈
.
- Let
,
. Since
,
, then
. This implies that
.
- Let
. Since
for all i ∈ I, so that
, thus
.
Hence, defines an N-HST, and
is an N-HSTS.
ii. The proof is similar to part (i).
Corollary 1 Let be an N-BHSTS. Then, the following collections define N-HST.
- i.
.
- ii.
(provided that ¥ and
are finite sets).
The following proposition demonstrates when the converse of proposition 3 holds true.
Proposition 4. Let be an N-HSTS. Then, the collection
consisting of N-BHS sets
such that
and
for all ¬ϵ ∈ ¬ξ and η ∈ ¥, defines an N-BHST. In particular,
is an N-BHSTS.
Proof. Straightforward.
Corollary 2. Let be an N-HSTS. Then, the collection
consisting of N-BHS sets
such that
and
for all ¬ϵ ∈ ¬ξ and η ∈ ¥, defines an N-BHST.
Remark 1. If the condition described in proposition 4 is not met, then the following example demonstrates that the converse of proposition 3 is generally not true. This is the case even if the collections and
define an N-HST.
Example 1. Let . Let ξ1 = {e1, e2}, ξ2 = {e3}, and ξ3 = {e4}, then ξ = ξ1 × ξ2 × ξ3 = {ϵ1 = (e1, e3, e4), ϵ2 = (e2, e3, e4)}. Let
= {Λ0, Λ5,
,
and
= {Λ0, Λ5,
,
be two 6-HSTs, where
= {(ϵ1, {η1, 3}, {η2, 0}), (ϵ2, {η1, 2}, {η2, 4})}.
= {(ϵ1, {η1, 3}, {η2, 5}), (ϵ2, {η1, 2}, {η2, 4})}.
And,
= {(ϵ1, {η1, 1}, {η2, 0}), (ϵ2, {η1, 1}, {η2, 0})}.
= {(ϵ1, {η1, 2}, {η2, 0}), (ϵ2, {η1, 2}, {η2, 1})}.
Then,
where
= {(ϵ1, {η1, 3, 1}, {η2, 0, 0}), (ϵ2, {η1, 2, 1}, {η2, 4, 0})}.
= {(ϵ1, {η1, 3, 2}, {η2, 5, 0}), (ϵ2, {η1, 2, 2}, {η2, 4, 1})}.
If we take
,
then,
= {(ϵ1, {η1, 3, 2}, {η2, 0, 0}), (ϵ2, {η1, 2, 2}, {η2, 4, 1})}.
But, . Hence,
is not a 6-BHST.
Definition 9. Let ¥ denote a set of alternatives, ξ be a set representing attributes, and R = {0, 1, …, N − 1} be a set representing ordered grades, where N ∈ {2, 3, …}. The collection is referred to as an N-BHS indiscrete topology.
Definition 10. Let ¥ represent a set of alternatives, ξ be a set representing attributes, R = {0, 1, …, N − 1} be a set representing ordered grades, where N ∈ {2, 3, …}, and be a collection of all N-BHS sets that can be defined over ¥. Then,
is termed an N-BHS discrete topology.
Proposition 5. Let and
be two N-BHSTSs, then
ξ, N) is an N-BHSTS.
Proof.
-
.
- Let
. Then,
and
. Since
and
. Then,
.
- Let
. Then,
and
, for all i ∈ I, and so
and
. Therefore,
.
Thus, defines an N-BHST and
ξ, N) is an N-BHSTS.
Remark 2. The union of two N-BHSTs may not be an N-BHST.
Example 2. Let . Let ξ1 = {e1, e2}, ξ2 = {e3}, and ξ3 = {e4}, then ξ = ξ1 × ξ2 × ξ3 = {ϵ1 = (e1, e3, e4), ϵ2 = (e2, e3, e4)}. Let
and
be two 6-BHSTs where
= {(ϵ1, {η1, 5, 1}, {η2, 4, 1}), (ϵ2, {η1, 0, 3}, {η2, 2, 1})}.
= {(ϵ1, {η1, 4, 2}, {η2, 4, 0}), (ϵ2, {η1, 0, 0}, {η2, 0, 5})}.
= {(ϵ1, {η1, 4, 2}, {η2, 4, 1}), (ϵ2, {η1, 0, 3}, {η2, 0, 5})}.
= {(ϵ1, {η1, 5, 1}, {η2, 4, 0}), (ϵ2, {η1, 0, 0}, {η2, 2, 1})}.
And
= {(ϵ1, {η1, 2, 2}, {η2, 1, 4}), (ϵ2, {η1, 3, 1}, {η2, 4, 1})}.
= {(ϵ1, {η1, 1, 1}, {η2, 0, 5}), (ϵ2, {η1, 0, 4}, {η2, 0, 3})}.
= {(ϵ1, {η1, 1, 2}, {η2, 0, 5}), (ϵ2, {η1, 0, 4}, {η2, 0, 3})}.
= {(ϵ1, {η1, 2, 1}, {η2, 1, 4}), (ϵ2, {η1, 3, 1}, {η2, 4, 1})}.
Then, ,
.
If we take
.
Then,
= {(ϵ1, {η1, 5, 1}, {η2, 4, 1}), (ϵ2, {η1, 3, 1}, {η2, 4, 1})}.
But, . Hence,
is not a 6-BHST. Therefore, in general, the union of two N-BHSTs may not be an N-BHST.
Definition 11. Let be an N-BHSTS and
be an N-BHS set. The intersection of all N-BHS closed supersets of
is called an N-BHS closure of
and is denoted by
.
In other words, .
Remark 3. Let be an N-BHSTS and
be an N-BHS set. Then,
-
is the smallest N-BHS closed set containing
.
-
is an N-BHS closed set if and only if
.
Example 3. Let . Let ξ1 = {e1, e2}, ξ2 = {e3}, and ξ3 = {e4}, then ξ = ξ1 × ξ2 × ξ3 = {ϵ1 = (e1, e3, e4), ϵ2 = (e2, e3, e4)}. Let
be a 6-BHST, where
= {(ϵ1, {η1, 4, 1}, {η2, 2, 3}), (ϵ2, {η1, 1, 2}, {η2, 5, 0})}.
= {(ϵ1, {η1, 2, 2}, {η2, 3, 1}), (ϵ2, {η1, 1, 4}, {η2, 0, 3})}.
= {(ϵ1, {η1, 2, 2}, {η2, 2, 3}), (ϵ2, {η1, 1, 4}, {η2, 0, 3})}.
= {(ϵ1, {η1, 4, 1}, {η2, 3, 1}), (ϵ2, {η1, 1, 2}, {η2, 5, 0})}.
Let be any 6-BHS set defined as follows:
= {(ϵ1, {η1, 0, 5}, {η2, 2, 2}), (ϵ2, {η1, 1, 4}, {η2, 0, 5})}.
Then,
= {(ϵ1, {η1, 1, 4}, {η2, 3, 2}), (ϵ2, {η1, 2, 1}, {η2, 0, 5})}.
Proposition 6. Let be an N-BHSTS and let
and
be two N-BHS sets. Then,
-
and
.
-
.
-
implies
.
-
.
-
.
-
.
Proof. We prove only (3), (4), and (5), and the remaining are straightforward.
- 3. By (2.),
. Since
, we have
. But
is an N-BHS closed set. Thus,
is an N-BHS closed set containing
. Since
is the smallest N-BHS closed set containing
, so we have
.
- 4. Since
and
. By (3.), we have
and
. Hence,
. Now, since
and
are N-BHS closed sets,
is also an N-BHS closed. Also,
and
implies that
. Thus,
is an N-BHS closed containing
. Since
is the smallest N-BHS closed set containing
, we have
. Hence,
.
- 5. Since
and
, then
and
. Therefore,
.
Remark 4. In general, the equality in Proposition 6 (5) does not hold.
Example 4. Consider the 6-BHSTS in Example 3 and let
and
be any 6-BHS sets defined as follows:
= {(ϵ1, {η1, 0, 5}, {η2, 1, 2}), (ϵ2, {η1, 0, 5}, {η2, 0, 5})}.
= {(ϵ1, {η1, 1, 4}, {η2, 0, 5}), (ϵ2, {η1, 1, 1}, {η2, 0, 5})}.
Then,
= {(ϵ1, {η1, 1, 4}, {η2, 3, 2}), (ϵ2, {η1, 2, 1}, {η2, 0, 5})}.
And,
= {(ϵ1, {η1, 1, 4}, {η2, 1, 3}), (ϵ2, {η1, 2, 1}, {η2, 0, 5})}.
Hence,
= {(ϵ1, {η1, 1, 4}, {η2, 1, 3}), (ϵ2, {η1, 2, 1}, {η2, 0, 5})}.
On the other hand,
= {(ϵ1, {η1, 0, 5}, {η2, 0, 5}), (ϵ2, {η1, 0, 5}, {η2, 0, 5})} =
.
Then,
.
It follows that
. Therefore, in general,
.
Definition 12. Let be an N-BHSTS. Then, the N-BHS interior of
is denoted by
and is defined as the N-BHS union of all N-BHS open set contained in
.
In other words, .
Remark 5. Let be an N-BHSTS and
be an N-be an BHS set. Then,
-
be an is the largest N-be an BHS open set contained in
.
-
be an is an N-BHS open set if and only if
.
Example 5. Consider the 6-BHSTS and the 5-BHS set
in Example 3. Then,
= {(ϵ1, {η1, 0, 5}, {η2, 0, 5}), (ϵ2, {η1, 0, 5}, {η2, 0, 5})} =
.
Proposition 7. Let be an N-BHSTS and let
and
be two N-BHS sets. Then,
-
and
.
-
.
-
implies
.
-
.
-
.
-
.
Proof. We prove only (3), (4), and (5), and the remaining are straightforward.
- 3. Since
and
, then we have
. But,
is the largest N-BHS open set contained in
. Thus,
.
- 4. Since
and
, then
and
. This implies
. Now, since
and
, then
. This implies
. Hence,
. Thus,
.
- 5. Since
, then
and
, then
. This implies
.
Remark 6. The next example illustrates that, in general, the equality in Proposition 7 (5) does not hold.
Example 6. Consider the 6-BHSTS in Example 3 and let
and
be any 6-BHS sets defined as follows:
= {(ϵ1, {η1, 5, 0}, {η2, 4, 0}), (ϵ2, {η1, 1, 2}, {η2, 5, 0})}.
= {(ϵ1, {η1, 2, 2}, {η2, 5, 0}), (ϵ2, {η1, 5, 0}, {η2, 4, 1})}.
Then,
= {(ϵ1, {η1, 4, 1}, {η2, 3, 1}), (ϵ2, {η1, 1, 2}, {η2, 5, 0})} And,
= {(ϵ1, {η1, 2, 2}, {η2, 3, 1}), (ϵ2, {η1, 1, 4}, {η2, 0, 3})}.
Hence,
{(ϵ1, {η1, 4, 1}, {η2, 3, 1}), (ϵ2, {η1, 1, 2}, {η2, 5, 0})}.
On the other hand,
= {(ϵ1, {η1, 5, 0}, {η2, 5, 0}), (ϵ2, {η1, 5, 0}, {η2, 5, 0})} =
.
Then,
.
It follows that
. Therefore, in general,
.
Remark 7. Let be an N-BHSTS and
be an N-BHS set. Then,
.
Proposition 8. Let be an N-BHSTS and let
be an N-BHS set. Then,
-
.
-
.
Proof.
Definition 13. Let be an N-BHSTS and
be an N-BHS set. Then, the N-BHS exterior of
is denoted and defined as follows:
.
Example 7. Consider the 6-BHSTS and the 6-BHS set
in Example 3. Then,
= {(ϵ1, {η1, 4, 1}, {η2, 2, 3}), (ϵ2, {η1, 1, 2}, {η2, 5, 0})}.
Proposition 9. Let be an N-BHSTS and let
and
be two N-BHS sets. Then,
-
and
.
-
implies
.
-
.
-
.
Proof.
-
.
.
-
then
. This implies
. Hence,
.
-
.
-
.
Definition 14. Let be an N-BHSTS. Then, N-BHS boundary of
is denoted by
and is defined as
.
Remark 8. According to Definition 14, it follows that the N-BHS sets and
share the same N-BHS boundary.
Example 8. Consider the 6-BHSTS and the 6-BHS set
in Example 3. Then,
= {(ϵ1, {η1, 1, 4}, {η2, 3, 2}), (ϵ2, {η1, 2, 1}, {η2, 0, 5})}.
Proposition 10. Let be an N-BHSTS and let
be an N-BHS set. Then,
-
.
-
.
-
.
-
.
Proof.
-
. Hence,
.
-
.
-
.
-
.
Proposition 11. Let be an N-BHSTS and let
be an N-BHS set. Then,
-
.
-
.
Proof.
-
.
- By Proposition 10 (2),
, then
.
5 Advancements in MCGDM through N-bipolar hypersoft topology
In this section, we delve into the advancements made in MCGDM using N-BHST. Our methodology sets itself apart from existing literature by harnessing to incorporate a spectrum of opinions, encompassing complete disagreement
, complete agreement
, and expertise in partial agreement levels denoted as
for k = 1, 2, …, p.
Two distinct algorithms, namely Algorithm 1 and Algorithm 2, are introduced to augment the MCGDM process through N-BHST. Notably, despite their unique methodologies, both algorithms converge towards the same optimal choice. The rankings generated by these algorithms exhibit remarkable similarity, underscoring the robust nature of our approach and its ability to provide a consistent and reliable decision-making outcome. Further sections will unravel the complexities of these algorithms, complemented by a numerical example that illustrates their practical application in real-world decision scenarios.
Algorithm 1. N-BHST-Aggregate Rank.
- Input:
- (a) ¥ = {ηi, i = 1, 2, …, m} as a set of alternatives.
- (b) ξ = {ϵj, j = 1, 2, …, n} as a set of attributes.
- (c)
as an N-BHST.
- Calculations:
- Output: Choose the optimal choice η as the alternative with the maximum aggregate value:
The flowchart of Algorithm 1 is given in Fig 1.
Algorithm 2. N-BHST-Cardinal Rank.
- Input:
- (a) ¥ = {ηi, i = 1, 2, …, m} as a set of alternatives.
- (b) ξ = {ϵj, j = 1, 2, …, n} as a set of attributes.
- (c)
as an N-BHST.
- Calculations:
- (a) For each N-BHS open set
, compute the cardinal N-HS set:
- (b) Find the aggregate N-HS set by using the formula:
where
, and
are matrices corresponding to
, and
, respectively. The matrix
represents the transpose of the matrix
.
- (c) Find the aggregate N-BHS set for each N-BHS open set by using the formula:
- (d) For each alternative ηi, compute the aggregate value as the sum of the aggregate values across all N-BHS open sets:
- (a) For each N-BHS open set
- Output: Choose the optimal choice η as the alternative with the maximum aggregate value:
The flowchart of Algorithm 2 is given in Fig 2.
Numerical example: Selecting a project manager
Suppose a project team is tasked with selecting a project manager from a pool of candidates, denoted as ¥ = {η1,η2, η3, η4, η5}. The selection process involves evaluating candidates based on three critical attributes: Leadership Skills, Technical Expertise, and Communication Skills. These attributes are further broken down into sub-attributes, forming the set of all possible skills ξ = ξ1 × ξ2 × ξ3, where:
- ξ1 = Leadership Skills with sub-attributes e1 = Team Management, e2 = Decision- Making, e3 = Conflict Resolution,
- ξ2 = Technical Expertise with sub-attributes e4 = Project Planning,
- ξ3 = Communication Skills with sub-attributes e5 = Written Communication.
The ordered grades R = {0, 1, 2} are used to assess candidates, representing Novice to Master levels. A 3-BHST is employed to capture different levels of agreement and disagreement. The expertise assessments for complete disagreement
, complete agreement
, and two partial agreement scenarios
and
are provided in Tables 3 and 4.
We employ both Algorithm 1 and Algorithm 1 to determine the most suitable candidate for the project manager position. It’s important to highlight that, in the computations, we intentionally exclude the N-BHS open sets and
as their inclusion does not influence the ranking.
5.1 Approach 1: Utilizing Algorithm 1 for project manager selection.
In this methodology, Algorithm 1 is utilized to enhance the MCGDM process with the integration of N-BHST. The specific steps involved are outlined below, applying them to a practical example of selecting a project manager.
- Compute the aggregate 3-BHS set for each 3-BHS open set:
- Compute the aggregate value for each alternative:
- Choose the optimal choice based on the maximum aggregate value:
Therefore, η2 emerges as the most fitting choice for the project manager role, exhibiting the highest overall value among all candidates. The order of preference is η2 ≻ η7 ≻ η6 ≻ η1 ≻ η3 ≻ η5 ≻ η4.
5.2 Approach 2: Utilizing Algorithm 2 for project manager selection.
Continuing with our example focused on selecting a project manager, we proceed with Approach 2, leveraging Algorithm 2 to further enhance the MCGDM approach within the context of N-BHST. The following steps elucidate the decision-making process for identifying the best candidate.
- Compute the cardinal 3-HS set:
- Compute the aggregate 3-HS set:
This means:
Similarly,
This means: - Compute the aggregate 3-BHS set for each 3-BHS open set:
- Compute the aggregate value for each alternative:
- Choose the optimal choice based on the maximum aggregate value:
Hence, η2 emerges as the optimal candidate for the project manager position, exhibiting the highest aggregate value compared to all other candidates. The ranking order is η2 ≻ η7 ≻ η1 ≻ η6 ≻ η3 = η5 ≻ η4.
6 Comparative analysis
This section presents a comparative analysis of the proposed N-BHST in conjunction with relevant existing models. Key evaluation factors, including SF (single-argument approximate function), MF (multi-argument approximate function), SO (single opinion), MO (multi-opinion), and BS (bipolarity setting), are taken into account. The objective is to emphasize the versatility and effectiveness of the N-BHST concerning these crucial features in comparison to other models in the field.
Table 5 provides a comprehensive comparative overview, illustrating the applicability or limitations of each model with respect to the identified evaluation factors. A check-mark (✓) signifies the successful incorporation of the respective feature, while the symbol (×) denotes its absence or limited implementation.
The analysis reveals that the N-BHST adeptly deals with all the evaluated features, positioning itself as a comprehensive and versatile approach. Conversely, comparative models have limitations in one or more of these aspects, underscoring the distinct strengths and advancements of the N-BHST.
This comparative assessment highlights the superiority of the proposed N-BHST, demonstrating its capability to encompass SF, MF, SO, MO, and BS. This differentiation distinguishes it from existing models, emphasizing the significance and relevance of our research and offering valuable insights for the further advancement and adoption of the N-BHST in practical applications.
7 Concluding remarks
In conclusion, this research has explored the realm of N-BHST, an innovative extension of the well-established N-HST and HST. Departing significantly from its predecessor, the N-BHS set, N-BHST introduces a multi-opinion approach to decision-making, enhancing robustness and adaptability. Our analysis has scrutinized various operators within the N-BHST framework, shedding light on their interrelationships.
Moreover, we conducted a thorough examination of the application of N-BHST in enhancing MCGDM, distinguishing it from existing models. The inclusion of a sensitivity analysis further contributes to the reliability of the proposed N-BHST method, ensuring its robustness in various applications. This analysis provides practitioners with valuable insights into how variations in attributes impact decision outcomes, thereby facilitating informed decision-making.
As a compelling direction for future research, we propose extending the envisioned N-BHST by integrating fuzzy N-BHST. The incorporation of fuzzy logic principles into the current N-BHST framework augments the model’s capability to manage uncertainty and imprecision inherent in the evaluation process. This expansion allows for the representation of evaluations with degrees of membership, fostering a more comprehensive and nuanced approach.
Furthermore, our future research endeavors will explore novel hybrid models such as hesitant N-BHS set, neutrosophic N-BHS set, fermatean N-BHS set, and others. By deducing algebraic and topological structures for these models, we aim to provide solutions to various real-world problems characterized by uncertainties. These efforts will contribute to the advancement of decision-making methodologies, enabling practitioners to tackle complex challenges more effectively.
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