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Several new resampling methods for generating bootstrap samples of fuzzy numbers are proposed. To avoid undesired repetitions in the secondary samples we do not draw randomly directly observations from the primary samples but construct them allowing for some modifications in their membership functions, however only such which do not disturb the canonical representation of the initial fuzzy numbers. We consider both two-parameter and three-parameter canonical representations, as well as the triangular and trapezoidal outputs in the secondary samples. Numerical experiments concerning some statistical tests based on fuzzy samples show that the suggested methods may appear helpful in statistical reasoning with imprecise data.

In recent years, many researchers have utilized metaheuristic optimization algorithms along with fuzzy logic theory in their studies for various purposes. The harmony search (HS) algorithm is one of the metaheuristic optimization algorithms that is widely employed in different studies along with fuzzy logic (FL) theory. FL theory is a mathematical approach to expressing uncertainty by applying the conceptualization of fuzziness in a system. This review paper presents an extensive review of published papers based on the combination of HS and FL systems. In this regard, the functional characteristics of models obtained from integration of FL and HS have been reported in various articles, and the performance of each study is investigated. The basic concept of the FL approach and its derived models are introduced to familiarize readers with the principal mechanisms of FL models. Moreover, appropriate descriptions of the primary classifications acquired from the coexistence of FL and HS methods for specific purposes are reviewed. The results show that the high efficiency of HS to improve the exploration of FL in achieving the optimal solution on the one hand, and the capability of fuzzy inference systems to provide more flexible and dynamic adaptation of the HS parameters based on human perception on the other hand, can be a powerful combination for solving optimization problems. This review paper is believed to be a useful resource for students, engineers, and professionals.

A fuzzy least squares estimator in the multiple with fuzzy-input–fuzzy-output linear regression model is considered. The paper provides a formula for the L2 estimator of the fuzzy regression model. This paper proposes several operations for fuzzy numbers and fuzzy matrices with fuzzy components and discussed some algebraic properties that are needed to use for proving theorems. Using the proposed operations, the formula for the variance, provided and this paper, proves that the estimators have several important optimal properties and asymptotic properties: they are Best Linear Unbiased Estimator (BLUE), asymptotic normality and strong consistency. The confidence regions of the coefficient parameters and the asymptotic relative efficiency (ARE) are also discussed. In addition, several examples are provided including a Monte Carlo simulation study showing the validity of the proposed theorems.