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.

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.

Most existing fuzzy set methods use points as their input, which is the finest granularity from the perspective of granular computing. Consequently, these methods are neither efficient nor robust to label noise. Therefore, we propose a frame-work called granular-ball fuzzy set by introducing granular-ball computing into fuzzy set. The computational framework is based on the granular-balls input rather than points; therefore, it is more efficient and robust than traditional fuzzy methods, and can be used in various fields of fuzzy data processing according to its extensibility. Furthermore, the framework is extended to the classifier fuzzy support vector machine (FSVM), to derive the granular ball fuzzy SVM (GBFSVM). The experimental results demonstrate the effectiveness and efficiency of GBFSVM.

Field canals improvement projects (FCIPs) are one of the ambitious projects constructed to save fresh water. To finance this project, Conceptual cost models are important to accurately predict preliminary costs at the early stages of the project. The first step is to develop a conceptual cost model to identify key cost drivers affecting the project. Therefore, input variables selection remains an important part of model development, as the poor variables selection can decrease model precision. The study discovered the most important drivers of FCIPs based on a qualitative approach and a quantitative approach. Subsequently, the study has developed a parametric cost model based on machine learning methods such as regression methods, artificial neural networks, fuzzy model and case-based reasoning.

Supplier selection problem has gained extensive attention in the prior studies. However, research based on Fuzzy Multi-Attribute Decision Making (F-MADM) approach in ranking resilient suppliers in logistic 4.0 is still in its infancy. Traditional MADM approach fails to address the resilient supplier selection problem in logistic 4.0 primarily because of the large amount of data concerning some attributes that are quantitative, yet difficult to process while making decisions. Besides, some qualitative attributes prevalent in logistic 4.0 entail imprecise perceptual or judgmental decision relevant information, and are substantially different than those considered in traditional suppler selection problems. This study, for the first time, develops a Decision Support System (DSS) that will help the decision maker to incorporate and process such imprecise heterogeneous data in a unified framework to rank a set of resilient suppliers in the logistic 4.0 environment. The proposed framework induces a triangular fuzzy number from large-scale temporal data using probability-possibility consistency principle. Large number of non-temporal data presented graphically are computed by extracting granular information that are imprecise in nature. Fuzzy linguistic variables are used to map the qualitative attributes. Finally, fuzzy based TOPSIS method is adopted to generate the ranking score of alternative suppliers. These ranking scores are used as input in a Multi-Choice Goal Programming (MCGP) model to determine optimal order allocation for respective suppliers. Finally, a sensitivity analysis assesses how the Cost versus Resilience Index (SCRI) changes when differential priorities are set for respective cost and resilience attributes.

We apply the Ordered Weighted Averaging (OWA) operator in multi-criteria decision-making. To satisfy different kinds of uncertainty, measure based dominance has been presented to gain the order of different criterion. However, this idea has not been applied in fuzzy system until now. In this paper, we focus on the situation where the linguistic satisfactions are fuzzy measures instead of the exact values. We review the concept of OWA operator and discuss the order mechanism of fuzzy number. Then we combine with measure-based dominance to give an overall score of each alternatives. An example is illustrated to show the whole procedure.

In fuzzy decision-making processes based on linguistic information, operations on discrete fuzzy numbers are commonly performed. Aggregation and defuzzification operations are some of these often used operations. Many aggregation and defuzzification operators produce results independent to the decision makers strategy. On the other hand, the Weighted Average Based on Levels (WABL) approach can take into account the level weights and the decision makers optimism strategy. This gives flexibility to the WABL operator and, through machine learning, can be trained in the direction of the decision makers strategy, producing more satisfactory results for the decision maker. However, in order to determine the WABL value, it is necessary to calculate some integrals. In this study, the concept of WABL for discrete trapezoidal fuzzy numbers is investigated, and analytical formulas have been proven to facilitate the calculation of WABL value for these fuzzy numbers. Trapezoidal and their special form, triangular fuzzy numbers, are the most commonly used fuzzy number types in fuzzy modeling, so in this study, such numbers have been studied. Computational examples explaining the theoretical results have been performed.

In an earlier work we have used the Triangular Fuzzy Numbers (TFNs)as an assessment tool of student skills.This approach led to an approximate linguistic characterization of the students' overall performance, but it was not proved to be sufficient in all cases for comparing the performance of two different student groups, since tywo TFNs are not always comparable. In the present paper we complete the above fuzzy assessment approach by presenting a defuzzification method of TFNS based on the Center of Gravity (COG) technique, which enables the required comparison. In addition we extend our results by using the Trapezoidal Fuzzy Numbers (TpFNs) too, which are a generalization of the TFNs, for student assessment and we present suitable examples illustrating our new results in practice.