Search Results (23)

A neutrosophic set is an important tool for handling vagueness and impreciseness in real-world problems, and distance measures are also exhibited in neutrosophic set theory. The (m,a,n)-fuzzy neutrosophic set is more flexible and efficient than the existing extensions of neutrosophic sets when discussing the distance between multiple objects. This paper invents ten distance measures for comparing (m,a,n)-fuzzy neutrosophic sets. Moreover, the created distance measures are applied in pattern classifications and multi-criteria decisions. Additionally, numerical examples demonstrate these distance measures in practical and scientific applications involving classifying materials and investment problems. The comparative analysis, along with graphical interpretations, further illustrates the effectiveness and superiority of the proposed measures. Full article

In the present paper, we first introduced the notion of k-fuzzy $\gamma$-open (k-$\mathcal{F}$-$\gamma$-open) sets as a generalized novel class of fuzzy open ($\mathcal{F}$-open) sets on fuzzy topological spaces ($\mathcal{FTS}s$) in the sense of Šostak. The class of k-$\mathcal{F}$-$\gamma$-open sets is contained in the class of k-$\mathcal{F}$-$\beta$-open sets and contains all k-$\mathcal{F}$-semi-open and k-$\mathcal{F}$-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-$\mathcal{F}$-$\gamma$-closed and k-$\mathcal{F}$-$\gamma$-open sets and discussed some of their properties. After that, we defined and studied the notions of $\mathcal{F}$-$\gamma$-continuous (resp. $\mathcal{F}$-$\gamma$-irresolute) functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,\mathsf{Ϝ})$. However, we displayed and investigated the notions of $\mathcal{F}$-almost (resp. $\mathcal{F}$-weakly) $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions. Next, we presented and characterized some new $\mathcal{F}$-functions via k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-closed sets, called $\mathcal{F}$-$\gamma$-open (resp. $\mathcal{F}$-$\gamma$-irresolute open, $\mathcal{F}$-$\gamma$-closed, $\mathcal{F}$-$\gamma$-irresolute closed, and $\mathcal{F}$-$\gamma$-irresolute homeomorphism) functions. The relationships between these classes of functions were investigated with the help of some examples. We also introduced some new types of $\mathcal{F}$-separation axioms called k-$\mathcal{F}$-$\gamma$-regular (resp. k-$\mathcal{F}$-$\gamma$-normal) spaces via k-$\mathcal{F}$-$\gamma$-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of $\mathcal{F}$-compactness called k-$\mathcal{F}$-almost (resp. k-$\mathcal{F}$-nearly) $\gamma$-compact sets. Full article

The conventional point-estimate human health risk assessment (HHRA) primarily uses average concentrations of a limited number of samples due to the high monitoring costs of heavy metals in groundwater. The results can be erroneous when concentrations significantly deviate from the average across the collected samples in an investigation region. The present research developed a hierarchical fuzzy-based HHRA (F-HHRA) framework to handle variations in limited data sets and subjectively established a broader range of risks for various exposure groups. Groundwater samples from 80 to 120 m deep in shallow wells were collected from agricultural farms along Wadi Rumah in the Qassim Region of Saudi Arabia. Laboratory testing found total dissolved solids much higher than the promulgated drinking water quality standards. As the aftertaste issue eliminated the raw water potability, the study considered dermal exposure for HHRA. The collected samples were tested for thirteen potential heavy metals (HMs), including barium (Ba), boron (B), cadmium (Cd), chromium (Cr), copper (Cu), iron (Fe), lead (Pb), lithium (Li), manganese (Mn), silver (Ag), strontium (Sr), thallium (TI), and zinc (Zn). Cu, Fe, Pb, Ag, and TI were lower than the detectable limit of the inductively coupled plasma mass spectrometry device. Concentrations of the remaining HMs in wastewater outfalls that were much less than the groundwater eradicated the impact of anthropogenic activities and affirmed natural contamination. Apart from 10% of the samples for Mn and 90% of the samples for Sr, all the other HMs remained within the desired maximum allowable concentrations. Point-estimate and fuzzy-based approaches yielded ‘low’ dermal non-cancer risk and cancer risk for all groups other than adults, where dermal cancer risk of Cr remained in the ‘acceptable’ (1 × 10⁻⁶ and 1 × 10⁻⁵) risk zone. Although dermal risk does not require controls, scenario analysis established the rationality of F-HHRA for more contaminated samples. The proposed hierarchical F-HHRA framework will facilitate the decision-makers in concerned agencies to plan risk mitigation strategies (household level and decentralized systems) for shallow well consumers in Saudi Arabia and other arid regions. Full article

The Neutrosophic Set $\left(N_{set}\right)$ represents the uncertainty in data with fuzzy attributes beyond true and false values independently. The problem arises when the summation of true $\left(\mathcal{T}\mathit{r}\right)$, false $\left(\mathcal{F}\mathit{a}\right)$, and indeterminacy $\left(\mathcal{I}\mathit{n}\right)$ values crosses the membership value of one, that is, $\mathcal{T}\mathit{r}+\mathcal{I}\mathit{n}+\mathcal{F}\mathit{a}<1$. It becomes more crucial during decision-making processes like medical diagnoses or any data sets where $\mathcal{T}\mathit{r}+\mathcal{I}\mathit{n}+\mathcal{F}\mathit{a}<1$. To achieve this goal, the $F{N}_{set}$ is recently introduced. This study employs the Interval-Valued Fermatean Neutrosophic Set ($IVF{N}_{set}$) 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 $IVF{N}_{set}$, which deals with partial ignorance in true, false, or uncertain regions independently for multi-decision processes. The $IVF{N}_{set}$ contains the interval-valued ${\mathcal{T}\mathit{r}}_{\mathit{m}\mathit{e}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{s}\mathit{h}\mathit{i}\mathit{p}}$ value, ${\mathcal{I}\mathit{n}}_{\mathit{m}\mathit{e}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{s}\mathit{h}\mathit{i}\mathit{p}}$ value, and ${\mathcal{F}\mathit{a}}_{\mathit{m}\mathit{e}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{s}\mathit{h}\mathit{i}\mathit{p}}$ for knowledge representation. The algebraic properties and set theory between the interval-valued $F{N}_{set}$ have also been presented with an illustrative example. Full article

A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the n^th power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a n^th power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPA${}_m^n$) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPA${}_m^n$ operator with the square-root fuzzy weighted power average (SR-FWPA), the n^th power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators. Full article

The article provides an analysis of the metrological properties of a 5 MN·m torque transducer. The relative electrical signal (given in electrical units, mV/V) as a function of torque measured in both the clockwise and anticlockwise directions was monitored. To fit the data, the weighted method of least squares with both a straight-line and a cubic spline curve was used. The results of the analysis indicated that the straight-line fitting method produced smaller values of expanded uncertainty than the cubic spline fitting method. Additionally, the study confirmed the assumptions that the Monte Carlo method for propagating uncorrelated distributions was more accurate than the uncertainty propagation method, regardless of the type of curve fitting used. From the estimated uncertainty coverage corridor at selected measurement points, confidence intervals–expanded uncertainties were determined. Additionally, the fuzzy sets approach to the evaluation of uncertainty was presented, and the approximate value of the expanded uncertainty was calculated. Full article

Based upon the Pascal distribution series ${\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +{\displaystyle \sum _{j=m+1}^{\infty }}\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{a}_j{\zeta }^j$, we can obtain a set of fuzzy differential subordinations in this investigation. We also newly obtain class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ of univalent analytic functions defined by the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$, give certain properties for the class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ and also obtain some applications connected with a special case for the operator. New research directions can be taken on fuzzy differential subordinations associated with symmetry operators. Full article

This work is concerned with the branch of complex analysis known as geometric function theory, which has been modified for use in the study of fuzzy sets. We develop a novel operator $L_{\alpha ,\lambda }^m:{\mathrm{A}}_n\to {\mathrm{A}}_n$ in the open unit disc $\Delta$ using the Noor integral operator and the generalized Sălăgean differential operator. First, we develop fuzzy differential subordination for the operator $L_{\alpha ,\lambda }^m$ and then, taking into account this operator, we define a particular fuzzy class of analytic functions in the open unit disc $\Delta$, represented by $R_{ϝ}^{\lambda }(m,\alpha ,\delta )$. Using the idea of fuzzy differential subordination, several new results are discovered that are relevant to this class. The fundamental theorems and corollaries are presented, and then examples are provided to illustrate their practical use. Full article

A fuzzy set is a powerful tool to handle uncertainty and ambiguity, and generally, the notions of symmetry and similarity are also exhibited in the fuzzy set theory. The class of (m, n)-rung orthopair fuzzy sets through two universes are more flexible and efficient than the q-rung orthopair fuzzy sets when discussing the symmetry and similarity between multiple objects. This research article comprehensively investigates ten similarity measures that employ cosine and cotangent functions for comparing (m, n)-rung orthopair fuzzy sets, which are a superclass of q-rung orthopair fuzzy sets. Moreover, the proposed weighted similarity measures are applied to real-world problems in building material analysis. A comparative analysis is conducted between the proposed measures and the existing cosine and cotangent measures of q-rung orthopair fuzzy sets, showing that the proposed measures are more efficient than existing ones. Additionally, a numerical example demonstrates the practical and scientific applications of these similarity measures in classifying plant leaf diseases. The sensitivity analysis shows that the existing measures cannot be applied to (m, n)-fuzzy data for distinct values of m and n. The results are supported by graphical interpretations, further illustrating the efficacy of the proposed measures. Full article

Fuzzy Topological Topographic Mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, denoted as $FTT{M}_n$, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. Later, the $FTT{M}_n$ can also be viewed as a graph. Previously, a group of researchers defined an assembly graph and utilized it to model a DNA recombination process. Some researchers then used this to introduce the concept of tangled cords for assembly graphs. In this paper, the tangled cord for $FTT{M}_4$ is used to calculate the Eulerian paths. Furthermore, it is utilized to determine the least upper bound of the Hamiltonian paths of its assembly graph. Hence, this study verifies the conjecture made by Burns et al. Full article

Let H_eff (V_eff, E_eff) be a finite simple connected graph of order m with vertex set V_eff and edge set E_eff. A dominating set $S_{ds}\subseteq V_{eff}$ is called an efficiently dominating set if, for every vertex $u_a\in V_G,\hspace{0.33em}\left|N_G\left[u_a\right]\cap S_{ds}\right|=1$—where NG [u_a] denotes the closed neighborhood of the vertex u_a. Using efficient domination techniques and labelling, we constructed the fuzzy network. An algorithm has been framed to encrypt and decrypt the secret information present in the network, and furthermore, the algorithm has been given in pseudocode. The mathematical modelling of a strong fuzzy network is defined and constructed to elude the burgeoning intruder. Using the study of the efficient domination of fuzzy graphs, this domination parameter plays a nuanced role in encrypting and decrypting the framed network. One of the main purposes of fuzzy networks is encryption, so one of our contributions to this research is to build a novel combinatorial technique to encrypt and decode the built-in fuzzy network with a secret number utilizing effective domination. An illustration with an appropriate secret message is provided along with the encryption and decryption algorithms. Furthermore, we continued this study in intuitionistic fuzzy networks. Full article

In this paper, we provide different variants of the Hermite–Hadamard ($H\cdot H$) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically $s$-convex fuzzy-number-valued functions ($U\mathcal{D}\mathcal{H}$ $s$-convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the $\frac{2\rho \sigma }{\rho +\sigma }$ type and are innovative in the setting of up and down harmonically $s$-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research. Full article

Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “$(m,n)$-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank $(m,n)$-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with $(m,n)$-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of $(m,n)$-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example. Full article

Fuzzy topological topographic mapping ($FTTM$) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. The key to the model is its topological structure that can accommodate electrical or magnetic recorded brain signal. A sequence of FTTM, $FTT{M}_n$, is an extension of FTTM whereby its form can be arranged in a symmetrical form, i.e., polygon. The special characteristic of $FTTM$, namely, the homeomorphisms between its components, allows the generation of new $FTTM$. The generated $FTTM$s can be represented as pseudo graphs. A pseudo-graph consists of vertices that signify the generated $FTTM$ and edges that connect their incidence components. A graph of pseudo degree zero, $G_0\left(FTT{M}_n^k\right),$ however, is a special type of graph where each of the $FTTM$ components differs from its adjacent. A researcher posted a conjecture on $G_0^3\left(FTT{M}_n^3\right)$ in 2014, and it was finally proven in 2021 by researchers who used their novel grid-based method. In this paper, the extended $G_0^3\left(FTT{M}_n^3\right)$, namely, the conjecture on $G_0^4\left(FTT{M}_n^4\right)$ that was posed in 2018, is narrated and proven using simple mathematical induction. Full article

The smart energy consumption of any household, maintaining the thermal comfort level of the occupant, is of great interest. Sensors and Internet-of-Things (IoT)-based intelligent hardware setups control the home appliances intelligently and ensure smart energy consumption, considering environment parameters. However, the effects of environment-driven consumer body dynamics on energy consumption, considering consumer comfort level, need to be addressed. Therefore, an Energy Management System (EMS) is modeled, designed, and analyzed with hybrid inputs, namely environmental perturbations, and consumer body biological shifts, such as blood flows in skin, fat, muscle, and core layers (affecting consumer comfort through blood-driven-sensations). In this regard, our work incorporates 69 Multi-Node (MN) Stolwijik’s consumer body interfaced with an indoor (room) electrical system capable of mutual interactions exchange from room environmental parameters and consumer body dynamics. The mutual energy transactions are controlled with classical PID and Adaptive Neuro-Fuzzy-Type II (NF-II) systems inside the room dimensions. Further, consumer comfort, room environment, and energy consumption relations with bidirectional control are demonstrated, analyzed, and tested in MATLAB/Simulink to reduce energy consumption and energy cost. Finally, six different cases are considered in simulation settings and for performance validation, one case is validated as real-time hardware experimentation. Full article