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In this paper, deriving the type-m fuzzy sets, intuitionistic fuzzy sets, Φ-fuzzy sets, rough sets, fuzzy rough sets and rough fuzzy sets as particular genuine sets, and establishing their connections with genuine sets, it is demonstrated that the theory of genuine sets provides a powerful tool to model various different kinds of uncertainty in a mathematical way. Furthermore, it is also shown that the genuine set theoretic descriptions of type-m fuzzy sets, intuitionistic fuzzy sets and fuzzy rough sets point out new features of these set notations, originated from the peculiar characteristics of genuine sets.

In this paper, we give similarity measures between type-2 fuzzy sets and provide the axiom definition and properties of these measures. For practical use, we show how to compute the similarities between Gaussian type-2 fuzzy sets. Yang and Shih's [22] algorithm, a clustering method based on fuzzy relations by beginning with a similarity matrix, is applied to these Gaussian type-2 fuzzy sets by beginning with these similarities. The clustering results are reasonable consisting of a hierarchical tree according to different levels.

In this paper, new similarity, inclusion measure and entropy between type-2 fuzzy sets corresponding to grades of memberships are proposed. We also create the relationships among these measures between type-2 fuzzy sets. Several examples are used to present the calculation of these similarity, inclusion measure and entropy between type-2 fuzzy sets. The comparison results show that the proposed similarity measure presents better than those of Hung and Yang (2004) and Yang and Lin (2009). Moreover, measuring the similarity between type-2 fuzzy sets is important in clustering. We also use the proposed similarity measure as a clustering method for type-2 fuzzy sets.

The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se.

Since Zadeh introduced fuzzy sets, a lot of extensions of this concept have been proposed, such as type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models, to represent higher levels of uncertainty. This paper provides a comparative investigation of type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models. Type-2 fuzzy sets study the fuzziness of the membership function (MF) using primary MF and secondary MF based on analytic mathematical methods; nonstationary fuzzy sets study the randomness of the MF using primary MF and variation function based on type-1 fuzzy sets theory; cloud models study the randomness of the distribution of samples in the universe and generate random membership grades (MGs) using two random variables based on probability and statistic mathematical methods. They all concentrate on dealing with the uncertainty of the MF or the MG which type-1 fuzzy sets do not consider, and thus have many similarities. Moreover, we find out that, the same qualitative concept “moderate amount” can be represented by an interval type-2 fuzzy set, a nonstationary fuzzy set or a normal cloud model, respectively. Then, we propose a unified mathematical expression for the interval type-2 fuzzy set, nonstationary fuzzy set and normal cloud model. On the other hand, we also find out that, the theory fundament and underlying motivations of these models are quite different. Therefore, We summarize detailed comparisons of distinctive properties of type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models. Further, we study their diverse characteristics of distributions of MGs across vertical slices. The comparative investigation shows that these models are complementary to describe the uncertainty from different points of view. Thus, this paper provides a fundamental contribution and makes a basic reference for knowledge representation and other applications with uncertainty.

Better performance at a country level will provide benefits to the whole population. This issue has been studied from various perspectives using empirical methods. However, little effort has as yet been made to address the issue of endogeneity in the interrelationships between productive performance and its determinants. We address this issue by proposing a Two-Dimensional Fuzzy-Monte Carlo Analysis (2DFMC) approach. The joint use of stochastic and fuzzy approaches – within the ambit of 2DFMCA – offers methodological tools to mitigate epistemic uncertainty while increasing research validity and reproducibility: (i) preliminary performance assessment by fuzzy ideal solutions; and (ii) robust stochastic regression of the performance scores into the epistemic sources of uncertainty related to the levels of physical and human capitals measured in distinct countries at different epochs. By applying the proposed method to a sample of 23 countries for 1890–2018, our results show that the best and worst-performing countries were Norway and Portugal, respectively. We further found that the intensity of human capital and the age of equipment (capital stock) have different impacts on productive performance – it has been established that capital intensity and total factor productivity are influenced by productivity performance, which, in turn, has a negative impact on labor productivity and GDP per capita. Our analysis provides insights to enable government policies to coordinate productive performance and other macroeconomic indicators.

In this paper, an axiomatic approach to Pareto set reduction problem is considered. The reduction is based on accounting for preferences of a decision maker which are modeled with the use of a type-2 fuzzy binary relation. This relation is only partially known through a set of so-called information quanta. Since these quanta are provided by the decision maker, it must be verified that they are consistent with the requirements of the axiomatic approach — the axioms of rational choice. Several theorems giving necessary and sufficient conditions of quanta consistency are proved. Ideas for dealing with inconsistencies are discussed with a few examples.

In this paper, the texture property "coarseness" is modeled by means of type-2 fuzzy sets, relating representative coarseness measures (our reference set) with the human perception of this texture property. The type-2 approach allows to face both the imprecision in the interpretation of the measure value and the uncertainty about the coarseness degree associated with a measure value. In our study, a wide variety of measures is analyzed, and assessments about coarseness perception are collected from pools. This information is used to obtain type-2 fuzzy sets where the secondary fuzzy sets are modeled by means of triangular membership functions fitted to the collected data.

The boil-turbine system is a multivariable and strong coupling system with the characteristics of nonlinearity, time-varying parameters, and large delay. The accurate model can effectively improve the performance of turbine–boiler coordinated control system. In this paper, the boil-turbine model is established by interval type-2 (IT2) T-S fuzzy model. The premise parameters of IT2 T-S fuzzy model are identified by robust IT2 fuzzy c-regression model (RIT2-FCRM) clustering algorithm. The RIT2-FCRM is based on interval type-2 fuzzy sets (IT2FS) and applies a robust objective function, this clustering algorithm can reduce the impacts of outliers and noise points. The effectiveness and practicability of RIT2-FCRM are demonstrated by the identification results of the boiler–turbine system.

The United Nation’s Sustainable Development Goals (SDGs) encourage countries to solve many social problems. One of these problems is homelessness. We rank states in the United States with respect to youth homelessness and the achievement of these goals. We use fuzzy similarity measures to determine the degree of similarity between these rankings. Given a similarity measure between two categories, we introduce a method using type-2 fuzzy sets that determines a similarity of a subcategory of one category with the other category. In particular, given a similarity measure between youth homelessness and the SDGs, we determine a similarity measure between youth vulnerable to toxic stress and the SDGs and also youth vulnerable to human trafficking and the SDGs. We also use a relatively new similarity measure, an $n$-dimensional similarity measure.