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In this paper, multi-criteria decision-making (MCDM) methods with probabilistic hesitant fuzzy information are proposed based on the dominance degree of probabilistic hesitant fuzzy elements (PHFEs) and best worst method (BWM). First, we discuss the probabilistic distribution function of PHFE and the dominance degree matrix between two PHFEs. The dominance degree matrix is constructed based on the probabilistic distribution function of PHFE, which can be characterized as a fuzzy complementary judgment matrix. Second, BWM is extended to fuzzy preference relations based on the constructed dominance degree matrix. Subsequently, an algorithm is designed for selecting the best and worst weight vectors, and then two models are developed based on additive consistency and multiplicative consistency of fuzzy preference relations to derive the criteria weights. In addition, an algorithm is presented to improve the consistency of the dominance degree matrix when a desired consistency level is not achieved. Finally, the selection of best investment company is provided as an example to demonstrate the feasibility and effectiveness of the proposed methods.

Over the past decades, the mathematical formulation of wind turbines (WTs) has been handled using different methodologies to model the probabilistic nature via different distribution functions. Many recently published articles have applied either the wind speed or the obtained active power from the WT on various probabilistic curves, such as Weibull, log-normal, and Gamma. In this work, the wind speed was modeled at five different locations in Egypt via a novel mixture probability distribution function (MPDF) that included four well-known distribution functions used to imitate the probabilistic nature of wind speed. Moreover, a decision-making multiple objective formulation was developed to optimally fit the MPDF with a minimum root mean square error (RMSE) and ensure reliable fitting by two other effective indices. Two methodologies, namely, equal and variable class widths, were investigated to model the density of wind speed and obtain a more realistic model for the tested wind speed profiles. The results showed the effectiveness of the proposed MPDF model as the RMSE was effectively minimized using multiobjective particle swarm optimization (MOPSO), showing nearly 10% improvement compared to the nondominated sorting genetic algorithm (NSGA-II).

The present note is devoted to a generalization of the notion of shift invariant operators that we call it$\\lambda $ -invariant operators$(\\lambda \\ge 0)$ . Some properties of this new class are presented. By using probabilistic methods, three examples are delivered.

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain We also study in detail the mother body problem associated to To construct the mother body measure, we define a quadratic differential Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the For each Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

Fréchet mean and variance provide a way of obtaining a mean and variance for metric space-valued random variables, and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and operations. Examples of such data objects include covariance matrices, graph Laplacians of networks and univariate probability distribution functions. We derive a central limit theorem for the Fréchet variance under mild regularity conditions, using empirical process theory, and also provide a consistent estimator of the asymptotic variance. These results lead to a test for comparing k populations of metric space-valued data objects in terms of Fréchet means and variances. We examine the finite-sample performance of this novel inference procedure through simulation studies on several special cases that include probability distributions and graph Laplacians, leading to a test for comparing populations of networks. The proposed approach has good finite-sample performance in simulations for different kinds of random objects. We illustrate the proposed methods by analysing data on mortality profiles of various countries and resting-state functional magnetic resonance imaging data.

This article systematically investigates the inverse Gaussian (IG) process as an effective degradation model. The IG process is shown to be a limiting compound Poisson process, which gives it a meaningful physical interpretation for modeling degradation of products deteriorating in random environments. Treated as the first passage process of a Wiener process, the IG process is flexible in incorporating random effects and explanatory variables that account for heterogeneities commonly observed in degradation problems. This flexibility makes the class of IG process models much more attractive compared with the Gamma process, which has been thoroughly investigated in the literature of degradation modeling. The article also discusses statistical inference for three random effects models and model selection. It concludes with a real world example to demonstrate the applicability of the IG process in degradation analysis. Supplementary materials for this article are available online.

Recently, the probabilistic fuzzy set has been applied by the researchers in various domains to model the uncertainties in the system due to both fuzziness and randomness. In this research paper, we propose a novel high-order probabilistic fuzzy set-based forecasting method in the environment of both non-probabilistic and probabilistic uncertainties. We have also proposed a novel probability-based discretization approach to model probabilistic uncertainty during partitioning of time series data. Gaussian probability distribution function is used in this research paper to associate probabilities to membership grades and probabilistic fuzzy elements are aggregated to a fuzzy row vector using an aggregation operator. Major advantages of the proposed method are that it includes both types of uncertainties in a single framework and enhances accuracy in forecast as well. To show its suitability and outperformance over other existing forecasting methods, the proposed method is implemented in University of Alabama enrolments and TAIFEX time series datasets. Various statistical parameters, e.g., coefficient of correlation, coefficient of determination, performance parameter, evaluation parameter and tracking signal are used to verify the validity of proposed PFS-based high-order time series forecasting method.

In the present study, a total of 360 FE analyses were carried out on tubular X-joints strengthened with collar plates under brace tension under laboratory testing conditions (20 °C) and various fire conditions. The generated FE models were validated based on 31 tests. The FE analyses produced a comprehensive dataset that encapsulated resistance metrics, with detailed simulations of welds, contacts, and the incorporation of non-linear geometrical and material attributes. Twelve theoretical probability density functions (PDFs) were matched to the constructed histograms, with the maximum likelihood (ML) technique utilized to assess the parameters of these fitted PDFs. The theoretical PDFs, rigorously evaluated against the Anderson–Darling, Kolmogorov–Smirnov, and Chi-squared tests, identified the Generalized Petrov distribution as the optimal model for capturing the resistance behaviors of X-joints under tensile load and varying fire conditions. The findings have led to the proposition of five detailed theoretical PDFs and cumulative distribution functions (CDFs), introducing a novel perspective for assessing and reinforcing the structural resilience of strengthened CHS X-joints in engineering practices.

Appropriate classification of medical insurance fraud events can not only be effective in preventing and combating fraud, but also greatly improve the utilization of medical resources. Due to the uncertainty inherent in medical insurance fraud, identifying and classifying the fraud are non-trivial tasks. In addition, the selection of classification radius by traditional methods is often highly subjective. To this end, a case-based reasoning (CBR) approach in probabilistic hesitant fuzzy environment and its application to classifying the severity of medical insurance fraud events are investigated in this article. At first, the probabilistic hesitant fuzzy element (PHFE) is regarded as a discrete probability distribution, and its distribution function is defined. On this basis, a distribution discrepancy degree is proposed to make up for the shortage of existing measures between PHFEs. Then, a probabilistic hesitant fuzzy decision-making method based on CBR is proposed, which considers both decision data and the expert’s own knowledge and experience. Finally, the proposed method is used to classify the severity of medical insurance fraud events, and the rationality and superiority of the method are verified by comparative analysis. First published online 15 July 2025