Upper and Lower Bounds of the α-Universal Triple I Method for Unified Interval Implications

Theα-universal triple I(α-UTI)method is a recognized scheme in the field of fuzzy reasoning,whichwas proposed by our research group previously.The robustness of fuzzy reasoning determines the quality of reasoning algorithms to a large extent,which is quantified by calculating the disparity between the output of fuzzy reasoning with interference and the output without interference.Therefore,in this study,the interval robustness(embodied as the interval stability)of theα-UTI method is explored in the interval-valued fuzzy environment.To begin with,the stability of theα-UTI method is explored for the case of an individual rule,and the upper and lower bounds of its results are estimated,using four kinds of unified interval implications(including the R-interval implication,the S-interval implication,the QL-interval implication and the interval t-norm implication).Through analysis,it is found that theα-UTI method exhibits good interval stability for an individual rule.Moreover,the stability of theα-UTI method is revealed in the case of multiple rules,and the upper and lower bounds of its outcomes are estimated.The results show that theα-UTI method is stable for multiple rules when four kinds of unified interval implications are used,respectively.Lastly,theα-UTI reasoning chain method is presented,which contains a chain structure with multiple layers.The corresponding solutions and their interval perturbations are investigated.It is found that theα-UTI reasoning chain method is stable in the case of chain reasoning.Two application examples in affective computing are given to verify the stability of theα-UTImethod.In summary,through theoretical proof and example verification,it is found that theα-UTImethod has good interval robustness with four kinds of unified interval implications aiming at the situations of an individual rule,multi-rule and reasoning chain.

Differently implicational α-universal triple I restriction method of (1, 2, 2) type

From the viewpoints of both fuzzy system and fuzzy reasoning, a new fuzzy reasoning method which contains the α- triple I restriction method as its particular case is proposed. The previous α-triple I restriction principles are improved, and then the optimal restriction solutions of this new method are achieved, especially for seven familiar implications. As its special case, the corresponding results of α-triple I restriction method are obtained and improved. Lastly, it is found by examples that this new method is more reasonable than the α-triple I restriction method.

General Expressions of Triple Ⅰ Restriction Methods for Fuzzy Soft Reasoning

The aim of this paper is to discuss the Triple Ⅰ restriction reasoning methods for fuzzy soft sets. Triple Ⅰ restriction principles for fuzzy soft modus ponens(FSMP) and fuzzy soft modus tollens(FSMT) are proposed, and then, the general expressions of the Triple Ⅰ restriction reasoning method for FSMP and FSMT with respect to residual pairs are presented respectively. Finally, the optimal restriction solutions for Lukasiewicz and Godel implication operators are examined.

Universal triple I fuzzy reasoning algorithm of function model based on quotient space

Aiming at the deficiencies of analysis capacity from different levels and fuzzy treating method in product function modeling of conceptual design, the theory of quotient space and universal triple I fuzzy reasoning method are introduced, and then the function modeling algorithm based on the universal triple I fuzzy reasoning method is proposed. Firstly, the product function granular model based on the quotient space theory is built, with its function granular representation and computing rules defined at the same time. Secondly, in order to quickly achieve function granular model from function requirement, the function modeling method based on universal triple I fuzzy reasoning is put forward. Within the fuzzy reasoning of universal triple I method, the small-distance-activating method is proposed as the kernel of fuzzy reasoning; how to change function requirements to fuzzy ones, fuzzy computing methods, and strategy of fuzzy reasoning are respectively investigated as well; the function modeling algorithm based on the universal triple I fuzzy reasoning method is achieved. Lastly, the validity of the function granular model and function modeling algorithm is validated. Through our method, the reasonable function granular model can be quickly achieved from function requirements, and the fuzzy character of conceptual design can be well handled, which greatly improves conceptual design.

Reverse triple Ⅰ method of fuzzy reasoning

A theory of reverse triple I method with sustention degree is presented by using the implication operator R0 in every step of the fuzzy reasoning. Its computation formulas of supremum for fuzzy modus ponens and infimum for fuzzy modus tollens are given respectively. Moreover, through the generalization of this problem, the corresponding formulas of α-reverse triple I method with sustention degree are also obtained. In addition, the theory of reverse triple I method with restriction degree is proposed as well by using the operator R0, and the computation formulas of infimum for fuzzy modus ponens and supremum for fuzzy modus tollens are shown.

Assessing green management performance of industrial enterprise:A case study of BH Petrochemical Co.,Ltd at the high-efficiency ecological economic zone of the Yellow River Delta

This paper puts forward the connotation of enterprise green management and green management performance evaluation,analyzes the characteristics of enterprise green management performance evaluation and decomposes its content.Using the triple performance method,the paper constructs the performance evaluation index system of enterprise green management.Taking BH Petrochemical Co.,Ltd.at the high-efficiency ecological economic zone of the Yellow River Delta as an example,by using analytic hierarchy process(AHP)and Matlab programming,this paper determines the weight of each index in the enterprise green management performance evaluation index system,the fuzzy comprehensive evaluation method is used to evaluate the green management performance evaluation of industrial enterprises.The results show that the performance evaluation index system of enterprise green management constructed by"triple performance method"has important theoretical significance and application value for evaluating and guiding enterprises to strengthen their green management.

Jellyfish-inspired alginate composite hydrogel filter prepared by macro-micro double bionic strategy for efficient water purification

Recently,research on hydrogel materials with a porous structure and superior water absorption capabilities significantly grown.However,the hydrogel under gravity-driven separation conditions often exhibit an unstable pore structure,poor mechanical properties,and limited functionality.To this end,this work presents a novel approach that combines a macro-micro double bionic strategy with a triple crosslinking method to develop a multifunctional alginate composite hydrogel filter(2%-SA-κCG-PVA-Ca^(2+),2%-SKP-Ca^(2+)for short)with a stable pore structure and superior mechanical properties,which possessed an umbrella-shaped structure resembling that of jellyfish.The 2%-SKPCa^(2+)filter was synthesized using polyvinyl alcohol(PVA)as a stable structure-directing agent,and sodium alginate(SA)andκ-carrageenan(κ-CG)as polymer hydrogels.The distinctive umbrellashaped hydrogel of 2%-SKP-Ca^(2+)filter,formed through the triple crosslinking method,overcomes the limitations of unstable pore structure and poor durability seen in hydrogels prepared by traditional crosslinking methods.Furthermore,the utilization of the 2%-SKP-Ca^(2+)filter in water treatment demonstrates its good selective permeability,excellent resistance to fouling,and extended longevity,which enables it to simultaneously achieve the multifunctional water purification and the coating of multi-substrate anti-fouling coatings.Therefore,not only does this research provide an efficient,multifunctional,highly pollution-resistant preparation method for designing a new filter,but it also confirms the application prospect of the macro-micro dual bionic strategy developed in this study in complex water treatment.

Probability representations of fuzzy systems

In this paper, the probability significance of fuzzy systems is revealed. It is pointed out that COG method, a defuzzification technique used commonly in fuzzy systems, is reasonable and is the optimal method in the sense of mean square. Based on different fuzzy implication operators, several typical probability distributions such as Zadeh distribution, Mamdani distribution, Lukasiewicz distribution, etc, are given. Those distributions act as ＂inner kernels＂ of fuzzy systems. Furthermore, by some properties of probability distributions of fuzzy systems, it is also demonstrated that CRI method, proposed by Zadeh, for constructing fuzzy systems is basically reasonable and effective. Besides, the special action of uniform probability distributions in fuzzy systems is characterized. Finally, the relationship between CRI method and triple I method is discussed. In the sense of construction of fuzzy systems, when restricting three fuzzy implication operators in triple I method to the same operator, CRI method and triple I method may be related in the following three basic ways： 1） Two methods are equivalent; 2） the latter is a degeneration of the former; 3） the latter is trivial whereas the former is not. When three fuzzy implication operators in triple I method are not restricted to the same operator, CRI method is a special case of triple I method; that is, triple I method is a more comprehensive algorithm. Since triple I method has a good logical foundation and comprises an idea of optimization of reasoning, triple I method will possess a beautiful vista of application.

The completeness and applications of the formal system B

Since the formal deductive system (?) was built up in 1997, it has played important roles in the theoretical and applied research of fuzzy logic and fuzzy reasoning. But, up to now, the completeness problem of the system (?) is still an open problem. In this paper, the properties and structure of R0 algebras are further studied, and it is shown that every tautology on the R0 interval [0,1] is also a tautology on any R0 algebra. Furthermore, based on the particular structure of (?) -Lindenbaum algebra, the completeness and strong completeness of the system (?) are proved. Some applications of the system (?) in fuzzy reasoning are also discussed, and the obtained results and examples show that the system (?) is suprior to some other important fuzzy logic systems.

Truth-value transmittal fuzzy reasoning inter-polator

In this paper, we firstly associate fuzzy reasoning algorithm with the interpolation algorithm and discuss the limitation of defuzzification methods used commonly in the fuzzy reasoning algorithm. Secondly, we give a new fuzzy reasoning algorithm in case of single input, called the truth-value transmittal method, and discuss its properties. Finally, we analyze the rationality to adopy the truth-value transmittal method as the defuzzification method of full implication triple I method, and show that although CRI and triple I fuzzy reasoning method are different from fuzzy output set, they are uniform finally under the truth-value transmittal defuzzification method.