Fuzzy numbers are commonly represented with fuzzy sets. Their objective is to better represent imprecise data. However, operations on fuzzy numbers are not as straightforward as maths on crisp numbers. Commonly, the Zadeh's extension rule is applied to elaborate a result. This can produce two problems: (1) high computational complexity and (2) for some fuzzy sets and some operations the results is not a fuzzy set with the same features (eg. multiplication of two triangular fuzzy sets does not produce a triangular fuzzy set). One more problem is the fuzzy spread -- fuzziness of the result increases with the number of operations. These facts can severely limit the application field of fuzzy numbers. In this paper we would like to revisite this problem with a different kind of fuzzy numbers -- extensional fuzzy numbers. The paper defines operations on extensional fuzzy numbers and relational operators (=, >, >=, <, <=) for them. The proposed approach is illustrated with several applicational examples. The C++ implementation is available from a public GitHub repository.

Since its inception, Fuzzy Set has been widely used to handle uncertainty and imprecision in decision-making. However, conventional fuzzy sets, often referred to as type-1 fuzzy sets (T1FSs) have limitations in capturing higher levels of uncertainty, particularly when decision-makers (DMs) express hesitation or ambiguity in membership degree. To address this, Interval Type-2 Fuzzy Sets (IT2FSs) have been introduced by incorporating uncertainty in membership degree allocation, which enhanced flexibility in modelling subjective judgments. Despite their advantages, existing IT2FS construction methods often lack active involvement from DMs and that limits the interpretability and effectiveness of decision models. This study proposes a socio-technical co-constructive approach for developing IT2FS models of linguistic terms by facilitating the active involvement of DMs in preference elicitation and its application in multicriteria decision-making (MCDM) problems. Our methodology is structured in two phases. The first phase involves an interactive process between the DM and the decision analyst, in which a modified version of Deck-of-Cards (DoC) method is proposed to construct T1FS membership functions on a ratio scale. We then extend this method to incorporate ambiguity in subjective judgment and that resulted in an IT2FS model that better captures uncertainty in DM's linguistic assessments. The second phase formalizes the constructed IT2FS model for application in MCDM by defining an appropriate mathematical representation of such information, aggregation rules, and an admissible ordering principle. The proposed framework enhances the reliability and effectiveness of fuzzy decision-making not only by accurately representing DM's personalized semantics of linguistic information.

Random fuzzy variables join the modeling of the impreciseness (due to their ``fuzzy part'') and randomness. Statistical samples of such objects are widely used, and their direct, numerically effective generation is therefore necessary. Usually, these samples consist of triangular or trapezoidal fuzzy numbers. In this paper, we describe theoretical results and simulation algorithms for another family of fuzzy numbers -- LR fuzzy numbers with interval-valued cores. Starting from a simulation perspective on the piecewise linear LR fuzzy numbers with the interval-valued cores, their limiting behavior is then considered. This leads us to the numerically efficient algorithm for simulating a sample consisting of such fuzzy values.

The precise and large dataset concerning catastrophic events is very important for insurers. To improve the quality of such data three methods based on the bootstrap, bootknife, and GAN algorithms are proposed. Using numerical experiments and real-life data, simulated outputs for these approaches are compared based on the mean squared (MSE) and mean absolute errors (MAE). Then, a direct algorithm to construct a fuzzy expert's opinion concerning such outputs is also considered.

Numerous techniques of multi-criteria decision-making (MCDM) have been proposed in a variety of business domains. One of the well-known methods is the Analytical Hierarchical Process (AHP). Various uncertain numbers are commonly used to represent preference values in AHP problems. In the case of multi-granularity linguistic information, several methods have been proposed to address this type of AHP problem. This paper introduces a novel method to solve this problem using shadowed fuzzy numbers (SFNs). These numbers are characterized by approximating different types of fuzzy numbers and preserving their uncertainty properties. The new Shadowed AHP method is proposed to handle preference values which are represented by multi-types of uncertain numbers. The new approach converts multi-granular preference values into unified model of shadowed fuzzy numbers and utilizes their properties. A new ranking approach is introduced to order the results of aggregation preferences. The new approach is applied to solve a supplier selection problem in which multi-granular information are used. The features of the new approach are significant for decision-making applications.

The credibility theory was introduced by B. Liu as a new way to describe the fuzzy uncertainty. The credibility measure is the fundamental notion of the credibility theory. Recently, L.Yang and K. Iwamura extended the credibility measure by defining the parametric measure $m_{\lambda}$ ($\lambda$ is a real parameter in the interval $[0,1]$ and for $\lambda= 1/2$ we obtain as a particular case the notion of credibility measure). By using the $m_{\lambda}$-measure, we studied in this paper a risk neutral multi-item inventory problem. Our construction generalizes the credibilistic inventory model developed by Y. Li and Y. Liu in 2019. In our model, the components of demand vector are fuzzy variables and the maximization problem is formulated by using the notion of $m_{\lambda}$-expected value. We shall prove a general formula for the solution of optimization problem, from which we obtained effective formulas for computing the optimal solutions in the particular cases where the demands are trapezoidal and triangular fuzzy numbers. For $\lambda=1/2$ we obtain as a particular case the computation formulas of the optimal solutions of the credibilistic inventory problem of Li and Liu. These computation formulas are applied for some $m_{\lambda}$-models obtained from numerical data.

Factors which attract customers and persuade them to buy new car are various regarding different consumer tastes. There are some methods to extract pattern form mass data. In this case we firstly asked passenger car marketing experts to rank more important factors which affect customer decision making behavior using fuzzy Delphi technique, then we provided a sample set from questionnaires and tried to apply a useful artificial neural network method called selforganizing map (SOM) to find out which factors have more effect on Iranian customer's buying decision making. Fuzzy tools were applied to adjust the study to be more real. MATLAB software was used for developing and training network. Results report four factors are more important rather than the others. Results are rather different from marketing expert rankings. Such results would help manufacturers to focus on more important factors and increase company sales level.

We introduce an evidential model for time-to-event prediction with censored data. In this model, uncertainty on event time is quantified by Gaussian random fuzzy numbers, a newly introduced family of random fuzzy subsets of the real line with associated belief functions, generalizing both Gaussian random variables and Gaussian possibility distributions. Our approach makes minimal assumptions about the underlying time-to-event distribution. The model is fit by minimizing a generalized negative log-likelihood function that accounts for both normal and censored data. Comparative experiments on two real-world datasets demonstrate the very good performance of our model as compared to the state-of-the-art.

Optimal information and knowledge management is crucial for organizations to achieve their objectives efficiently. As a rare and valuable resource, effective knowledge management provides a strategic advantage and has become a key determinant of organizational success. The study aims to identify critical success and failure factors for implementing knowledge management systems in government organizations. This research employs a descriptive survey methodology, collecting data through random interviews and questionnaires. The study highlights the critical success factors for knowledge management systems in government organizations, including cooperation, an open atmosphere, staff training, creativity and innovation, removal of organizational constraints, reward policies, role modeling, and focus. Conversely, failure to consider formality, staff participation, collaboration technologies, network and hardware infrastructure, complexity, IT staff, and trust can pose significant obstacles to successful implementation.

Fault tree analysis is a vital method of assessing safety risks. It helps to identify potential causes of accidents, assess their likelihood and severity, and suggest preventive measures. Quantitative analysis of fault trees is often done via the dependability metrics that compute the system's failure behaviour over time. However, the lack of precise data is a major obstacle to quantitative analysis, and so to reliability analysis. Fuzzy logic is a popular framework for dealing with ambiguous values and has applications in many domains. A number of fuzzy approaches have been proposed to fault tree analysis, but -- to the best of our knowledge -- none of them provide rigorous definitions or algorithms for computing fuzzy unreliability values. In this paper, we define a rigorous framework for fuzzy unreliability values. In addition, we provide a bottom-up algorithm to efficiently calculate fuzzy reliability for a system. The algorithm incorporates the concept of $\alpha$-cuts method. That is, performing binary algebraic operations on intervals on horizontally discretised $\alpha$-cut representations of fuzzy numbers. The method preserves the nonlinearity of fuzzy unreliability. Finally, we illustrate the results obtained from two case studies.