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In this article, various properties of triangular membership functions are formally proven, including the relationship between a triangular membership function composed of two straight lines and a MAX function, as well as a triangular membership function defined by the horizontal coordinates of the triangle’s vertices. Furthermore, we formalize defuzzified value of a triangular membership function and the integration of two connected functions.

The application of neural network model in intelligent diagnosis usually encounters challenges such as continuous adjustment of network parameters and significant cost in training the network facing numerous complex physiological data. To address this challenge, this paper introduces a fuzzy SZGWO-ELM neural network model for medical disease aid diagnosis with fuzzy membership function and ELM network to refine the improved Gray Wolf optimization algorithm. Firstly, the Z-type membership function is introduced as the inertia weight to get a balance for the grey wolf in seeking the optimal solution globally and locally and ensuring fast convergence. Secondly, the S-type membership function is utilized as the adaptive weight to flexibly adjust the grey wolf search step size to facilitate a quick approximation of the optimal solution. Finally, the improved Gray Wolf optimization algorithm is used to optimize the parameters of the ELM neural network model, termed as SZGWO-ELM. This method can eliminate the need for extensive network parameter adjustments and quickly locate the optimal solution to the problem using a lightweight neural network. The performance of the SZGWO is assessed by using metrics like convergence, mean, and standard deviation. Multiple experiments reveal that this method shows superior performance. Furthermore, five publicly accessible medical disease datasets from UCI were conducted to evaluate the performance of SZGWO-ELM network model comparing with different classify model, and the results in terms of precision, sensitivity, specificity and accuracy can achieve 99.52%, 94.14%, 99.26% and 96.08%, respectively, which illustrate that the proposed SZGWO-ELM neural network significantly enhance the model’s accuracy, providing better support for doctors in disease diagnosis.

The main aim of the presented study is estimating the parameters of Weibull distribution by utilizing simulation to generated the samples size when n=10, 50,100. Considering in the current study the parameters estimator of Weibull membership function, then using the nonlinear membership function for Gaussian function to find the fuzzy number for these parameters estimator. After that utilizing the ranking function to transform the fuzzy number to crisp number.

According to the literature, we can use several linearisation processes to develop a novel technique for solving the multi-objective linear fractional programming problem (MOLFPP) under fuzzy environment. Nevertheless, there hasn’t been any discussion of the comparative studies of solutions in the literature. This article discusses various approaches for applying linearisation techniques to convert multi-objective linear fractional programming problem (MOLFPP) into multi-objective linear programming problem (MOLPP). A comparative analysis is then conducted. In order to showcase our work, we employ several strategies for linearisation, and we explore membership functions such as linear membership function, hyperbolic membership function and exponential membership function.

【Background and Objective】Improving water use efficiency in agriculture is a critical to alleviating global water stress. As agriculture consumes approximately 62% of global water, optimizing agricultural water use is essential for sustainable resource management. In the soil-plant-atmosphere continuum (SPAC), water transport in soil-root system is a key water cycle component and has attracted increasing research attention. Recent studies have shown that rational an approximate root pruning can improve plant water use efficiency, offering promising strategies for water-efficient agriculture. However, study on root pruning in rice remains limited, with most studies focusing on either timing or intensity, while ignoring their combined effects. This study aims to address this gap.【Method】A field experiment was conducted using two rice varieties, Jingzhan 1 and Luhan 639. Root pruning was performed on the 7 th, 9 th, and 11 th day after germination by cutting the root at 1.5 cm from the root tip, removing half of the root, and retaining 2 cm root segment. An unpruned root treatment was the control. Overall, there were eight treatments, which were arranged in a randomized design with three replicates each. Key indicators such as seedling vigor index, root-to-shoot ratio, photosynthetic traits, and water use efficiency were measured during the. Using the membership function method, seven variables were used to comprehensively assess the impact of different pruning treatments.【Result】For the Jingzhan 1 variety, pruning the root at 1.5 cm from the root tip on the 9 th day after germination resulted in the greatest improvement in seedling growth and water use efficiency. For the Luhan 639 variety, removing half of the root on the 9 th day after germination produced the best results.【Conclusion】Root pruning, when applied at the optimal time and intensity, can significantly enhance rice seedling growth and water use efficiency. This study provides practical guidelines for improving water-efficient rice cultivation and contributes to the broader goal of sustainable agricultural water management.

Poverty is a multidimensional one, which means that the poor can suffer multiple disadvantages at the same time. This paper aims to further develop and refine the multidimensional poverty measure using Fuzzy Sets Theory (FST). The application of FST starts with properly conveying the realities of attributes inequality and poverty proposing appropriate and justified membership functions for both variables. And then, applying fuzzy rules to integrate attributes inequality in multidimensional poverty measures. We obtain a class of fuzzy multidimensional poverty integrated indices. An application based on individual well-being data from Tunisian households in 2015 is presented to illustrate the use of proposed concepts.

An interval type-2 (IT2) mutual subsethood Cauchy fuzzy neural inference system has been proposed in this paper. The network architecture consists of 3-layers with all connection weights being IT2 Cauchy fuzzy membership functions (CMFs). The crisp inputs to the system are fuzzified into IT2CMFs with fixed centers and uncertain spreads. The hidden layer represents the rule-based knowledge. The firing degree of the antecedent part of each rule at the hidden layer is computed by aggregating the product of the mutual subsethood similarity measures between the inputs and the connection weights. A volume defuzzification is used to compute the numeric output. A gradient descent back-propagation algorithm is used to train the model. The novelty of the proposed model is threefold. First, is enriching the theory of the mutual subsethood fuzzy neural models by adopting the Cauchy membership function (CMF) as another powerful fuzzy basis function (FBF) rather than the classical choice of Gaussian fuzzy membership functions (GMFs). Second, is the success of computing the mutual subsethood similarity measure between the IT2CMFs and all the model parameters’ updating equations in analytic closed-form formulas, not numerically or approximately. Third, is the ability to extract the type-1 (T1) mutual subsethood Cauchy fuzzy neural inference system (T1MSCFuNIS) with all its analytic closed-form formulas directly as a special case from the general formulas of IT2MSCFuNIS model. Such a novelty makes the proposed model a concrete and effective development of the theory of mutual subsethood fuzzy neural models. Both IT2MSCFuNIS and T1MSCFuNIS models have been tested using different examples from the domains of function approximation, classification, and prediction. The results ensure the efficacy of both models compared with other models reported in the literature.

Fuzzy approaches that are proposed to describe uncertain, impressive or vague concepts, are based on the construction of membership function (MF), which reflects what is known about the linguistic variables in the application domain. However, a non-trivial problem exists in how to construct the most appropriate MF that has the best-fit representation of the analysed problem. Therefore, many authors propose their own ways to construct MF using a certain technique in a particular application domain. Consequently, the need for a general approach for constructing MF led us to systematise and to generalise the analysed approaches into a general methodological framework (GMF) of constructing MF. The novelty of this paper is that the proposed GMF is general, domain independent and free of a chosen understanding of fuzziness (i.e., similarity (imprecision), preference (vagueness), and uncertainty). To verify the proposed GMF, it was applied for the enterprise business service quality (QoS EBS ) planning problem. The obtained results showed that a semi-automatic MF construction for QoS EBS planning was more sensitive, less subjective and more precise than a manual construction. Moreover, illustrative examples showed that our proposed GMF is applicable and implementable. The reliability of the results was assessed using experts and users’ experience, which is based on general guidelines of the “acceptable” response time limits for various activities.

In this research paper, a generator of fuzzy methods based on theorems and axioms of fuzzy logic is derived, analyzed and applied. The family presented generates fuzzy implications according to the value of a selected parameter. The obtained fuzzy implications should satisfy a number of axioms, and the conditions of satisfying the maximum number of axioms are denoted. New theorems are stated and proven based on the rule that the fuzzy function of fuzzy implication, which is strong, leads to fuzzy negation. In this work, the data taken were fuzzified for the application of the new formulae. The fuzzification of the data was undertaken using four kinds of membership degree functions. The new fuzzy functions were compared based on the results obtained after a number of repetitions. The new proposed methodology presents a new family of fuzzy implications, and also an algorithm is shown that produces fuzzy implications so as to be able to select the optimal method of the generator according to the value of a free parameter.

Predicting the behaviour of a manufacturing operation is always challenging. Predictive analytics plays a major role in tackling errors present in the data acquired during the manufacturing process. Data uncertainties are unavoidable; however, they need to be mapped appropriately for the effective implementation of suitable control schemes. In this work, an attempt is made to predict the machinability of α–β titanium alloy during turning operation using three cooling agents such as dry, liquid nitrogen, and carbon dioxide. Interval type-2 fuzzy logic system (IT2FLS) along with centre of sets type reduction is considered to handle uncertainties present during the turning operation. The computational complexity of IT2FLS is overcome by reducing it to type 1 fuzzy logic system using Mendel's first results. Simulation results of both IT2FLS and T1FLS are compared with semi-elliptic membership function and trapezoidal membership function. The results obtained validate the Mendel's statement by reflecting similar behaviour in both the fuzzy logic systems. The results also confirm that the predictions of machinability parameters in turning operation using SEMF are a preferred option.