Fuzzy Reasoning Method Research Articles

Application of the Sugeno integral in Fuzzy Rule-Based Classification

Fuzzy Rule-Based Classification System (FRBCS) is a well-known technique to deal with classification problems. Recent studies have considered the usage of the Choquet integral and its generalizations (e.g.: CT-integral, CF-Integral and CC-integral) to enhance the performance of such systems. Such fuzzy integrals were applied to the Fuzzy Reasoning Method (FRM) to aggregate the fired fuzzy rules when classifying new data. However, the Sugeno integral, another well-known aggregation operator, obtained good results in other applications, such as brain–computer interfaces. These facts led to the present study, in which we consider the Sugeno integral in classification problems. That is, the Sugeno integral is applied in the FRM of a widely used FRBCS, and its performance is analyzed over 33 different datasets from the literature, also considering different fuzzy measures. To show the efficiency of this new approach, the results obtained are also compared with previous studies that involved the application of different aggregation functions. Finally, we perform a statistical analysis of the application.

Fuzzy reasoning method based on picture similarity measure and its application in medical diagnosis

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule’s antecedent and input. Both algorithms did not consider the similarity of rule’s antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods.

Application of Fuzzy Case-Based Reasoning and Fuzzy Analytic Hierarchy Process for Machining Cutter Planning and Control

Cutter planning and control are the crucial problems in machining processes. The current literature indicates that the issue of cutter planning and control problem was not adequately researched in the past in a metal-cutting process. Usually, cutter planning and control problems were addressed using different optimization, simulation, and computer-aided planning (CAP) methods. To bridge this knowledge gap, this study proposed a decision support system (DSS) that can integrate fuzzy case-based reasoning (F-CBR) and fuzzy analytic hierarchy process (F-AHP) methods. This integration was applied to determine hybrid similarity measures between new and prior cases. The study provides new insights into the integration of fuzzy set theory (FST), CBR, and AHP for solving machining cutter planning and control problems. Our proposed system retrieves the best similar prior cases to reuse and adapt them to new order arrivals. A numerical example was illustrated to validate the soundness of the researched DSS.

A novel failure mode and effect analysis method using a flexible knowledge acquisition framework based on picture fuzzy sets

Failure mode and effect analysis (FMEA) is an effective reliability management tool for identifying potential failures in a system/component that has been widely utilized in various fields by combining fuzzy sets theory. However, the difficulty in assessment expression and acquisition, imprecision in assessment aggregation, and missing relationships among risk factors are prominent challenges in the existing fuzzy FMEA methods. Thus, this work employs the picture fuzzy sets (PFSs) theory to meet these challenges, allowing experts to express assessments more efficiently and accurately. Meanwhile, we propose a novel FMEA method to improve the three essential processes of FMEA, involving the following steps. Firstly, a flexible knowledge acquisition framework (FKAF) is established to simplify the expert evaluation process, allowing experts to express fuzzy information with various forms of non-fuzzy assessments. Then, a strategy-based picture fuzzy conversion (PFC) method is developed to standardize the non-fuzzy values to picture fuzzy numbers (PFNs). Secondly, to improve the assessment aggregation accuracy in uncertain environments, we suggest the picture fuzzy evidential reasoning (PFER) method that extend the existing fuzzy evidential reasoning (FER) methods. Thirdly, to completely describe the parallel and causal relationships among risk factors, four alternative models are established using picture fuzzy Petri nets (PFPNs) and risk priority rankings are determined through inference. Finally, the effectiveness and superiority of the proposed FMEA method are verified through two case studies and one extended experiment, demonstrating that it overcomes the shortcomings of the existing FMEA methods and reduces application costs while ensuring the rationality of rankings.

Fuzzy Reasoning for Mixture of Fuzzy/Intuitionistic Fuzzy Information Based on Triple I Method

Generalized modus ponens (GMP) is a basic model of approximate reasoning. GMP in fuzzy propositions is called fuzzy modus ponen (FMP) and it is called intuitionistic fuzzy modus ponens (IFMP) when fuzzy propositions are generalized to intuitionistic fuzzy propositions. In this paper, we aim to investigate fuzzy reasoning methods for GMP with mixture of fuzzy/intuitionistic fuzzy information. For mixed types of GMP, we present two basic methods to solve GMP problems based on the triple I method (TIM). One is to transform the GMP problem into two FMP problems by decomposition and aggregate the solutions of TIM for FMP problems to obtain the solution for GMP problem. The other is to transform the GMP problem into the IFMP problem by expansion and aggregate the solutions of TIM for the IFMP problem to obtain the solution for the GMP problem. These two methods are called the decomposition triple I method (DTIM) and expansion triple I method (ETIM), respectively. We analyze the relationship between DTIM and ETIM. Furthermore, we discuss the reversibility of DTIM and ETIM.

The Relationship between Fuzzy Reasoning Methods Based on Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets

Two important basic inference models of fuzzy reasoning are Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT). In order to solve FMP and FMT problems, the full implication triple I algorithm, the reverse triple I algorithm and the Subsethood Inference Subsethood (SIS for short) algorithm are proposed, respectively. Furthermore, the existing reasoning algorithms are extended to intuitionistic fuzzy sets and interval-valued fuzzy sets according to different needs. The purpose of this paper is to study the relationship between intuitionistic fuzzy reasoning algorithms and interval-valued fuzzy reasoning algorithms. It is proven that there is a bijection between the solutions of intuitionistic fuzzy triple I algorithm and the interval-valued fuzzy triple I algorithm. Then, there is a bijection between the solutions of intuitionistic fuzzy reverse triple I algorithm and the interval-valued fuzzy reverse triple I algorithm. At the same time, it is shown that there is also a bijection between the solutions of intuitionistic fuzzy SIS algorithm and interval-valued fuzzy SIS algorithm.

Intuitionistic fuzzy rule-base evidential reasoning with application to the currency trading system on the Forex market

In this paper, the application of the intuitionistic fuzzy rule-base evidential reasoning (IFRBER) to the development of a new optimized automated trading system (ATS) for the Forex market is presented. The used IFRBER approach represents the intuitionistic fuzzy sets in the framework of the evidence theory that allows us to avoid the revealed drawbacks of the IFS operational laws and enhance the overall performance of the IFRBER approach. It is shown that the IFRBER approach extracts from an analyzed system considerably more of useful for the decision making information than the usual fuzzy rule-base evidential reasoning (FRBER). Then based on the IFRBER, a new approach to make the justified transaction buying and selling decisions was proposed. This approach was used to develop a new optimized ATS for the Forex market. It is shown that due to the ability of a new approach to use more of useful information that present implicitly in the problem formulation than the proposed earlier usual fuzzy rule-base evidential reasoning method, the developed ATS provides a considerably more profitable and comfortable (with a higher percent of winning trades and with low risks) trading than the earlier developed ATS.

The use of Fuzzy rule-based systems in the design process of the metallic products on example of microstructure evolution prediction

The challenge on the contemporary market of consumer goods is a quick response to customer needs. It entails time restrictions, which a semi-finished products’ (including metal products) manufacturer must meet. This issue must be addressed during a design phase, which for the most of semi-finished products suppliers, takes part during a quotation preparation process. Our research is aimed at investigating possibility of application of Fuzzy Reasoning methods for shortening of a design process, being a part of this process. We present a study on application of simplified models for solving technological tasks, allowing obtaining expected properties of designed products. The core of our concept is replacing numerical models and classical metamodels with a rule-based reasoning. A quotation preparation process can be supported by solving a technological problem without numerical experiments. Our goal was to validate the thesis basing not only on the presentation of some potential solutions but also on the results of simulation studies. The problem is illustrated with an example of thermal treatment of aluminum alloys, aimed at evaluation of a summary fraction of precipitations as a function of time and technological parameters. We assumed that it is possible to use both unstructured and point numerical experiments for knowledge acquisition. Implementation of this concept required the use of hybrid knowledge acquisition methods that combine the results of point experiments with expert knowledge. A comparison of obtained results to the ones obtained with metamodels shows a similar efficiency of both approaches, while our method is less time and laborious.

Identification of artificial groundwater recharge zones in semi-arid region of southern India using geospatial and integrated decision-making approaches

The present study integrates various decision-making methods viz. Fuzzy Logic (FL), Frequency Ratio (FR), Analytical Hierarchy Process (AHP) to delineate the artificial groundwater recharge zones (AGRZs) in semi-arid region of south India. Eight criterions used in this process, including lithology, soil types, lineament density (LD), drainage density (DD), vadose zone (VZ), rainfall, land use/ land cover (LULC), and slope. All of these criteria and their sub-criteria used to derive weights for different thematic layers and combined using a geospatial method to classify AGRZs. The resultant map was categorised into five classes: very poor, poor, moderate, good and very good. The FL, AHP, FR, and integrated approaches were validated against groundwater fluctuation data using receiver operating characteristic (ROC) and field AUC [Area Under the Curve] approaches, showed an accuracy of 72%, 76% and 78% and 82%. As a result, the output of the integrated models outperforms that of the FR, FL and AHP models. Fuzzy reasoning and AHP methods rely on assigning a relative score based on the significance of the variables, which necessitates extensive knowledge of the criterions. FR, on the other hand, is simpler and more reliable since it does not rely on user-based scoring for the various criterions. The results of this study can be useful for decision makers responsible for the sustainable management of groundwater resources.

On the symmetric quintuple implicational method of fuzzy reasoning

A novel fuzzy reasoning method called the SQI (symmetric quintuple implicational) methodis put forward, which is a generalization of the QIP (quintuple implication principle) method. First of all, the symmetric quintuple implicational principles are presented, which are distinct from the ones of the QIP method. Then unified optimal solutions of the SQI method are obtained for FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens), meanwhile corresponding reversible properties are verified. Furthermore, focusing on the case of multiple rules, optimal solutions of the SQI method are achieved, which involves two general approaches, i.e., FITA (first-infer-then-aggregate) and FATI (first-aggregate-then-infer). Equivalence relation of continuity and interpolation is analyzed for both FITA and FATI under the environment of the SQI method. Finally, one computing example arising in the field of affective computing is given for the SQI method with FATI. It is found that the SQI method preserves the same properties as the QIP method.

Integrating fuzzy case-based reasoning, parametric and feature-based cost estimation methods for machining process

PurposeThis paper aims to propose an intelligent system that serves as a cost estimator when new part orders are received from customers.Design/methodology/approachThe methodologies applied in this study were case-based reasoning (CBR), analytic hierarchy process, rule-based reasoning and fuzzy set theory for case retrieval. The retrieved cases were revised using parametric and feature-based cost estimation techniques. Cases were represented using an object-oriented (OO) approach to characterize them in n-dimensional Euclidean vector space.FindingsThe proposed cost estimator retrieves historical cases that have the most similar cost estimates to the current new orders. Further, it revises the retrieved cost estimates based on attribute differences between new and retrieved cases using parametric and feature-based cost estimation techniques.Research limitations/implicationsThe proposed system was illustrated using a numerical example by considering different lathe machine operations in a computer-based laboratory environment; however, its applicability was not validated in industrial situations.Originality/valueDifferent intelligent methods were proposed in the past; however, the combination of fuzzy CBR, parametric and feature-oriented methods was not addressed in product cost estimation problems.

Characterizations of Fuzzy Implications Generated by Continuous Multiplicative Generators of T-Norms

In this article, we define the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications by continuous multiplicative generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms, along with the research lines of Yager's <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$f$</tex-math></inline-formula> -generated implications by continuous additive generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$f$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g$</tex-math></inline-formula> -generated implications by continuous additive generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -conorms and of Balasubramaniam's <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> -generated implications by continuous multiplicative generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -conorms. This article is mainly motivated by many desirable properties of such generator generating implications for their flexible ordering property, unique determinations by induced natural negations, close relations to nilpotent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> -conditionality, the law of importation with more choices of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> other than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T_{P}$</tex-math></inline-formula> , and good distributivity over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -conorms. The relationships of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications to other well-known classes of fuzzy implications are clarified: they will unify <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g$</tex-math></inline-formula> -generated implications with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g(1)&lt; \infty$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> -generated implications with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h(1)&gt;0$</tex-math></inline-formula> and hence, have intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(S,N)$</tex-math></inline-formula> -implications; the only intersection with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R$</tex-math></inline-formula> -implications is the Goguen implication and they have no intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$f$</tex-math></inline-formula> -generated implications. The main results of this article are several characterizing theorems for (sometimes subclasses of) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications by means of aforementioned desirable properties from the respective perspectives, of which some partially solved or enriched related open problems in the literature. Finally, the superiorities of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications for modeling strict fuzzy preference relations and for constructing novel fuzzy reasoning methods are analyzed.

Fuzzy Reasoning Method Based on Distance Measure and Its Reductive Property

This paper shows a basic and original fuzzy reasoning method that can draw a novel study direction of the approximate inference in fuzzy systems with uncertainty. Firstly we propose a criterion function for checking of the reductive property about fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). Secondly unlike fuzzy reasoning methods based on the similarity measure, we propose a principle of new fuzzy reasoning method based on distance measure and then present two theorems for FMP and FMT. Thirdly through the several computational experiments, we show that proposed method is simple and effective, and in accordance with human thinking. Finally we pointed out conclusion that proposed method does satisfy the convergence of the fuzzy control and has not information loss.

Failure mode and effects analysis (FMEA) for risk assessment based on interval type-2 fuzzy evidential reasoning method

Failure mode and effect analysis (FMEA) has been widely adopted to define, identity, and remove potential and recognized hazards. As an indicator in traditional FMEA, the risk priority number (RPN) is an effective tool for measuring risk and the calculation of RPN is also very simple. Nevertheless, there are many drawbacks in the conventional FMEA method. It is necessary to seek approaches that can make up for the deficiency of traditional FMEA method and strengthen assessment capability of ranking failure modes according to three relevant risk factors. This paper presents a way to combine interval type-2 fuzzy sets (IT2FSs) with evidential reasoning (ER) method, which is able to overcome some disadvantages of the conventional FMEA approach and deal with uncertainties more efficiently. First, we give a more precise expression of the risk factors in the form of IT2FSs and gain the relative weight of three risk factors. Second, one can judge the failure modes in relation to each risk factors with belief structures. Finally, the ER method is used to combine the belief structures under the weight of the three risk factors. To verify the feasibility of the method, an application for steam valve system is performed and the obtained results show the effectiveness of the method.

Application of fuzzy intelligent reasoning method in art color measurement

Under the influence of the COVID-19 epidemic situation, many countries have taken many measures to control the flow of people. The inability of people to gather makes art color measurement a problem. Color matching and coordination and color space conversion have always been the research focus of art color measurement. This paper studies a method of fuzzy intelligent reasoning in art color measurement. Based on case-based reasoning and rule-based reasoning, this method is an important self-learning and self-maintenance method in the field of artificial intelligence and expert system. On the basis of expounding the basic characteristics of color and color space, this paper designs the process principle of case-based reasoning and the process of rule-based reasoning. In this paper, case and rule knowledge representation method, case retrieval technology and rule conflict resolution strategy are established. Based on the above strategy, the color case base is established. In addition, the rule table is established by combining color-brewer with rule reasoning and referring to color-brewer. The rule table has a certain reference value for the application of intelligent reasoning method in the field of art color measurement under the influence of COVID-19 epidemic.

Logical foundation of symmetric implicational methods for fuzzy reasoning

Logical foundation of symmetric implicational methods for fuzzy reasoning

A fuzzy evidential reasoning based approach for submarine power cable routing selection for offshore wind farms

Offshore wind farms have developed fast as an environmentally friendly source of energy. The submarine power cable of the offshore wind farm, used for connecting power generation devices to onshore equipment, may have a significant impact on navigation safety and is prone to being damaged (e.g. caused by fishing or emergency anchoring by ships) when adopting unfavorable routing. This paper proposes a fuzzy evidential reasoning method for submarine power cable routing selection of the offshore wind farm by comprehensively considering the conditions for cable laying and its influence on maritime safety. The kernel of this approach is to establish a three-layer decision-making framework after fuzzification of the input variables, to derive the belief rule base, and to obtain the optimal routing from the submarine power cable candidates using evidential reasoning and index value. The proposed approach is applied to a real routing selection problem of a submarine power cable for an offshore wind farm in Zhejiang Province of China. The resulting choice corresponds to the discussions in a workshop unanimously.

The state-of-art of the generalizations of the Choquet integral: From aggregation and pre-aggregation to ordered directionally monotone functions

In 2013, Barrenechea et al. used the Choquet integral as an aggregation function in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems. After that, starting from 2016, new aggregation-like functions generalizing the Choquet integral have appeared in the literature, in particular in the works by Lucca et al. Those generalizations of the Choquet integral, namely CT-integrals (by t-norm T), CF-integrals (by a fusion function F satisfying some specific properties), CC-integrals (by a copula C), CF1F2-integrals (by a pair of fusion functions (F1, F2) under some specific constraints) and their generalization gCF1F2-integrals, achieved excellent results in classification problems. The works by Lucca et al. showed that the aggregation task in a FRM may be performed by either aggregation, pre-aggregation or just ordered directional monotonic functions satisfying some boundary conditions, that is, it is not necessary to have an aggregation function to obtain competitive results in classification. The aim of this paper is to present and discuss such generalizations of the Choquet integral, offering a general panorama of the state of the art, showing the relations and intersections among such five classes of generalizations. First, we present them from a theoretical point of view. Then, we also summarize some applications found in the literature.

New Method of Fuzzy Conditional Inference and Reasoning

ABSTRACT We consider fuzzy inference of the form ‘if ··· then ··· ’. ‘if ··· then ··· else ···’ and ‘if ··· and/or ··· and/or ··· then ···’. Muzomoto proposed logical constructs for ‘if. then.’ by applying Godelien and Standard Sequence methods. These method do not satisfy all the intuitions In this paper, We propose method of fuzzy inference and applied on logical constructs. We have shown that our fuzzy inference method satisfy all the intuitions under several criterions.

Some notes on optimal fuzzy reasoning methods

The optimal fuzzy reasoning (OFR) method proposed by Zhang and Cai (2004) is treated as a control problem and it is dramatically different from other existing fuzzy reasoning methods. The aim of this paper is to make some notes on OFR methods. We give a modification of an OFR method presented by Zhang and Cai. Some advantages and limitations of OFR methods are examined. We introduce several new OFR methods based on the Łukasiewicz implication, the Kleene-Dienes implication and the Zadeh implication. The reasoning consistency of the proposed methods is studied. The relationship between the OFR method and other fuzzy reasoning methods (e.g., the compositional rule of inference method and the pointwise optimizing fuzzy inference method) is examined.