Mathematics

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14 pages, 284 KiB

Open AccessArticle

On Soft ω_δ-Open Sets and Some Decomposition Theorems

by Dina Abuzaid, Samer Al-Ghour and Monia Naghi

Mathematics 2024, 12(6), 924; https://doi.org/10.3390/math12060924 - 21 Mar 2024

Viewed by 508

Abstract

In this paper, we present a novel family of soft sets named “soft

ω_{δ}

-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft

δ

-open sets and soft

ω^{0}

-open [...] Read more.

In this paper, we present a novel family of soft sets named “soft

ω_{δ}

-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft

δ

-open sets and soft

ω^{0}

-open sets. Also, we introduce certain sufficient conditions for the equivalence between this new soft topology and several existing soft topologies. Moreover, we verify several relationships that contain soft covering properties, such as soft compactness and soft Lindelofness, which are related to this new soft topology. Furthermore, in terms of the soft interior operator in certain soft topologies, we define four classes of soft sets. Via them, we obtain new decomposition theorems for soft

δ

-openness and soft

θ

-openness, and we characterize the soft topological spaces that have the soft “semi-regularization property”. In addition, via soft

ω_{δ}

-open sets, we introduce and investigate a new class of soft functions named “soft

ω_{δ}

-continuous functions”. Finally, we look into the connections between the newly proposed soft concepts and their counterparts in classical topological spaces. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

14 pages, 482 KiB

Open AccessArticle

Dominations in Intutionistic Fuzzy Directed Graphs with Applications towards Influential Graphs

by Hao Guan, Waheed Ahmad Khan, Amna Fida, Khadija Ali, Jana Shafi and Aysha Khan

Mathematics 2024, 12(6), 872; https://doi.org/10.3390/math12060872 - 16 Mar 2024

Viewed by 509

Abstract

In this manuscript, we introduce a few new types of dominations in intuitionistic fuzzy directed graphs (IFDGs) based on different types of strong arcs (SAs). Our work is not only a direct extension of domination in directed fuzzy graphs (DFGs) but also fills [...] Read more.

In this manuscript, we introduce a few new types of dominations in intuitionistic fuzzy directed graphs (IFDGs) based on different types of strong arcs (SAs). Our work is not only a direct extension of domination in directed fuzzy graphs (DFGs) but also fills the gap that exists in the literature regarding the dominations in different extended forms of fuzzy graphs (FGs). In the beginning, we introduce several types of strong arcs in IFDGs, like semi-

β

strong arcs, semi-

δ

strong arcs, etc. Then, we introduce the concepts of domination in IFDGs based on these strong arcs and discuss its various useful characteristics. Moreover, the dominating set (DS), minimal dominating set (MDS), etc., are described with some fascinating results. We also introduce the concept of an independent set in IFDGs and investigate its relations with the DS, minimal independent set (MIS) and MDS. We also provide numerous important characterizations of domination in IFDGs based on minimal and maximal dominating sets. In this context, we discuss the lower and upper dominations of some IFDGs. In addition, we introduce the terms status and structurally equivalent and examine a few relationships with the dominations in IFDGs. Finally, we investigate the most expert (influential) person in the organization by utilizing the concepts of domination in IFGs. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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24 pages, 759 KiB

Open AccessArticle

Calculating Insurance Claim Reserves with an Intuitionistic Fuzzy Chain-Ladder Method

by Jorge De Andrés-Sánchez

Mathematics 2024, 12(6), 845; https://doi.org/10.3390/math12060845 - 13 Mar 2024

Viewed by 592

Abstract

Estimating loss reserves is a crucial activity for non-life insurance companies. It involves adjusting the expected evolution of claims over different periods of active policies and their fluctuations. The chain-ladder (CL) technique is recognized as one of the most effective methods for calculating [...] Read more.

Estimating loss reserves is a crucial activity for non-life insurance companies. It involves adjusting the expected evolution of claims over different periods of active policies and their fluctuations. The chain-ladder (CL) technique is recognized as one of the most effective methods for calculating claim reserves in this context. It has become a benchmark within the insurance sector for predicting loss reserves and has been adapted to estimate variability margins. This variability has been addressed through both stochastic and possibilistic analyses. This study adopts the latter approach, proposing the use of the CL framework combined with intuitionistic fuzzy numbers (IFNs). While modeling with fuzzy numbers (FNs) introduces only epistemic uncertainty, employing IFNs allows for the representation of bipolar data regarding the feasible and infeasible values of loss reserves. In short, this paper presents an extension of the chain-ladder technique that estimates the parameters governing claim development through intuitionistic fuzzy regression, such as symmetric triangular IFNs. Additionally, it compares the results obtained with this method with those derived from the stochastic chain ladder by England and Verrall. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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10 pages, 278 KiB

Open AccessArticle

Hyperconnectedness and Resolvability of Soft Ideal Topological Spaces

by Ahmad Al-Omari and Wafa Alqurashi

Mathematics 2023, 11(22), 4697; https://doi.org/10.3390/math11224697 - 19 Nov 2023

Cited by 2 | Viewed by 894

Abstract

This paper introduces and explores the concept of soft ideal dense sets, utilizing soft open sets and soft local functions, to examine their fundamental characteristics under some conditions for the following notions: soft ideal hyperconnectedness, soft ideal resolvability, soft ideal irresolvability, and soft [...] Read more.

This paper introduces and explores the concept of soft ideal dense sets, utilizing soft open sets and soft local functions, to examine their fundamental characteristics under some conditions for the following notions: soft ideal hyperconnectedness, soft ideal resolvability, soft ideal irresolvability, and soft ideal semi-irresolvability in soft ideal topological spaces. Moreover, it explores the relationship between these notions if

τ ⊓ \bar{I} = ϕ_{E}

is obtained in the soft set environment. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

15 pages, 328 KiB

Open AccessArticle

Some Classes of Soft Functions Defined by Soft Open Sets Modulo Soft Sets of the First Category

by Zanyar A. Ameen and Mesfer H. Alqahtani

Mathematics 2023, 11(20), 4368; https://doi.org/10.3390/math11204368 - 20 Oct 2023

Cited by 6 | Viewed by 948

Abstract

Soft continuity can contribute to the development of digital images and computational topological applications other than the field of soft topology. In this work, we study a new class of generalized soft continuous functions defined on the class of soft open sets modulo [...] Read more.

Soft continuity can contribute to the development of digital images and computational topological applications other than the field of soft topology. In this work, we study a new class of generalized soft continuous functions defined on the class of soft open sets modulo soft sets of the first category, which is called soft functions with the Baire property. This class includes all soft continuous functions. More precisely, it contains various classes of weak soft continuous functions. The essential properties and operations of the soft functions with the Baire property are established. It is shown that a soft continuous with values in a soft second countable space is identical to a soft function with the Baire property, apart from a topologically negligible soft set. Then we introduce two more subclasses of soft functions with the Baire property and examine their basic properties. Furthermore, we characterize these subclasses in terms of soft continuous functions. At last, we present a diagram that shows the relationships between the classes of soft functions defined in this work and those that exist in the literature. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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18 pages, 663 KiB

Open AccessArticle

Utilizing m-Polar Fuzzy Saturation Graphs for Optimized Allocation Problem Solutions

by Abdulaziz M. Alanazi, Ghulam Muhiuddin, Bashair M. Alenazi, Tanmoy Mahapatra and Madhumangal Pal

Mathematics 2023, 11(19), 4136; https://doi.org/10.3390/math11194136 - 30 Sep 2023

Viewed by 607

Abstract

It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m-polar fuzzy graph. Since [...] Read more.

It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m-polar fuzzy graph. Since there is only one component for this idea, we are unable to manage this kind of circumstance using the fuzzy model since we take into account m components for each node as well as edges. Again, since each edge or node only has two components, we are unable to apply a bipolar or intuitionistic fuzzy graph model. In contrast to other fuzzy models, mPFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of mPFGs using examples and related theorems. Considering all those things together, we define saturation for a m-polar fuzzy graph (mPFG) with multiple membership values for both vertices and edges; thus, a novel approach is required. In this context, we present a novel method for defining saturation in mPFG involving m saturations for each element in the membership value array of a vertex. This explains

α

-saturation and

β

-saturation. We investigate intriguing properties such as

α

-vertex count and

β

-vertex count and establish upper bounds for particular instances of mPFGs. Using the concept of

α

-saturation and

α

-saturation, block and bridge of mPFG are characterized. To identify the

α

-saturation and

β

-saturation mPFGs, two algorithms are designed and, using these algorithms, the saturated mPFG is determined. The time complexity of these algorithms is

O (| V |^{3})

, where

| V |

is the number of vertices of the given graph. In addition, we demonstrate a practical application where the concept of saturation in mPFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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20 pages, 933 KiB

Open AccessArticle

Modeling and Verification of Uncertain Cyber-Physical System Based on Decision Processes

by Na Chen, Shengling Geng and Yongming Li

Mathematics 2023, 11(19), 4122; https://doi.org/10.3390/math11194122 - 29 Sep 2023

Cited by 2 | Viewed by 635

Abstract

Currently, there is uncertainty in the modeling techniques of cyber-physical systems (CPS) when faced with the multiple possibilities and distributions of complex system behavior. This uncertainty leads to the system’s inability to handle uncertain data correctly, resulting in lower reliability of the system [...] Read more.

Currently, there is uncertainty in the modeling techniques of cyber-physical systems (CPS) when faced with the multiple possibilities and distributions of complex system behavior. This uncertainty leads to the system’s inability to handle uncertain data correctly, resulting in lower reliability of the system model. Additionally, existing technologies struggle to verify the activity and safety of CPS after modeling, lacking a dynamic verification and analysis approach for uncertain CPS properties.This paper introduces a generalized possibility decision process as a system model. Firstly, the syntax and semantics of generalized possibility temporal logic with decision processes are defined. Uncertain CPS is extended by modeling it based on time-based differential equations and uncertainty hybrid time automaton. After that, model checking is performed on the properties of activity and safety using fuzzy linear time properties. Finally, a cold–hot hybrid constant-temperature system model is used for simulation experiments. By combining theory and experiments, this paper provides a new approach to the verification of uncertain CPS, effectively addressing the state explosion problem. It plays a crucial role in the design of uncertain CPS and offers a key solution for model checking in the presence of uncertainty. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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21 pages, 1585 KiB

Open AccessArticle

Faculty Performance Evaluation through Multi-Criteria Decision Analysis Using Interval-Valued Fermatean Neutrosophic Sets

by Said Broumi, Raman Sundareswaran, Marayanagaraj Shanmugapriya, Prem Kumar Singh, Michael Voskoglou and Mohamed Talea

Mathematics 2023, 11(18), 3817; https://doi.org/10.3390/math11183817 - 5 Sep 2023

Cited by 7 | Viewed by 935

Abstract

The Neutrosophic Set

(N_{s e t})

represents the uncertainty in data with fuzzy attributes beyond true and false values independently. The problem arises when the summation of true

(T r)

, false

(F a)

, and [...] Read more.

The Neutrosophic Set

(N_{s e t})

represents the uncertainty in data with fuzzy attributes beyond true and false values independently. The problem arises when the summation of true

(T r)

, false

(F a)

, and indeterminacy

(I n)

values crosses the membership value of one, that is,

T r + I n + F a < 1

. It becomes more crucial during decision-making processes like medical diagnoses or any data sets where

T r + I n + F a < 1

. To achieve this goal, the

F N_{s e t}

is recently introduced. This study employs the Interval-Valued Fermatean Neutrosophic Set (

I V F N_{s e t}

) as its chosen framework to address instances of partial ignorance within the domains of truth, falsehood, or uncertainty. This selection stands out due to its unique approach to managing such complexities within multi-decision processes when compared to alternative methodologies. Furthermore, the proposed method reduces the propensity for information loss often encountered in other techniques. IVFNS excels at preserving intricate relationships between variables even when dealing with incomplete or vague information. In the present work, we introduce the

I V F N_{s e t}

, which deals with partial ignorance in true, false, or uncertain regions independently for multi-decision processes. The

I V F N_{s e t}

contains the interval-valued

{T r}_{m e m b e r s h i p}

value,

{I n}_{m e m b e r s h i p}

value, and

{F a}_{m e m b e r s h i p}

for knowledge representation. The algebraic properties and set theory between the interval-valued

F N_{s e t}

have also been presented with an illustrative example. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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19 pages, 689 KiB

Open AccessArticle

A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System

by Anushree Bhattacharya and Madhumangal Pal

Mathematics 2023, 11(13), 2992; https://doi.org/10.3390/math11132992 - 4 Jul 2023

Cited by 5 | Viewed by 1785

Abstract

A fuzzy graph G is stated to have a set of trees as its tree cover if all the vertices of G are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree [...] Read more.

A fuzzy graph G is stated to have a set of trees as its tree cover if all the vertices of G are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer

β > 0

, finding a set of trees to cover all the vertices of a graph with minimum cost and at most

β

number of spanning trees is known as the

β

-tree cover problem. Combining the tree-covering concept and facility location problem in a fuzzy environment for solving critical real-life problems in the recent era is a more fruitful approach. This issue strongly inspires us to develop a model with a practical algorithm. This paper provides an algorithm and complexity analysis to determine the number of rooted trees s covering the given fuzzy graph. In addition, a model is constructed with three optimization programming problems in the facility location problem and a tree covering fuzzy graphs. The model includes two types of the facility location problem, simultaneously addressing a variable covering radius and a fixed covering radius. A numerical example is provided to further describe the model, then, in the application part of the paper, the proposed model is applied to solve the real-life problem of maximizing demand saturation by minimizing the number of small denominations in the Indian banking system. This problem involves the data input of different indicators in the banking system along with details of the denominations of banknotes. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

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