Adaptive non-additive generalized fuzzy systems

An adaptive non-additive generalized fuzzy model (GFM) is presented in this paper using the framework of Gaussian mixture model (GMM) which provides the membership functions for the input fuzzy sets. By replacing the consequent part of the additive GFM rule by a non-additive function, we obtain the non-additive GFM. The coefficients of the non-additive function then become the fuzzy measures. The defuzzified output constructed from both the premise and consequent parts of the modified GFM rules in the wake of non-additiveness takes the form of Choquet fuzzy integral. The parameters of the premise and the consequent parts of the non-additive fuzzy rules are updated based on the estimation error on the arrival of each online data to make the system adaptive. The resulting adaptive non-additive fuzzy model is applied on two benchmark applications and the results demonstrate the advantage of the adaptive feature.     

Hierarchical MADM with fuzzy integral for evaluating enterprise intranet web sites

In the real world, most criteria have inter-dependent or interactive characteristics so they cannot be evaluated by conventional additive measures. Thus, to approximate the human subjective evaluation process, it would be more suitable to apply a fuzzy integral model, in which it is not necessary to assume additivity and independence. This research proposes an effective algorithm to determine the λ-value using the input data of fuzzy densities and the fuzzy integral based on λ-fuzzy measure to determine the overall evaluation. This research also gives an example of evaluating enterprise intranet web sites with illustrations of the hierarchical structure of λ-fuzzy measure for a Choquet integral model. The results show that the fuzzy integral is more suitable than a traditional multi-criteria evaluation method for human subjective evaluation, or when criteria are not mutually independent.     

Multiple-attribute group decision-making based on power Bonferroni operators of linguistic q-rung orthopair fuzzy numbers

In this paper, a new conception of linguistic q-rung orthopair fuzzy number (Lq-ROFN) is proposed where the membership and nonmembership of the q-rung orthopair fuzzy numbers ( q-ROFNs) are represented as linguistic variables. Compared with linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers, the Lq-ROFNs can more fully describe the linguistic assessment information by considering the parameter q to adjust the range of fuzzy information. To deal with the multiple-attribute group decision-making (MAGDM) problems with Lq-ROFNs, we proposed the linguistic score and accuracy functions of the Lq-ROFNs. Further, we introduce and prove the operational rules and the related properties characters of Lq-ROFNs. For aggregating the Lq-ROFN assessment information, some aggregation operators are developed, involving the linguistic q-rung orthopair fuzzy power Bonferroni mean (BM) operator, linguistic q-rung orthopair fuzzy weighted power BM operator, linguistic q-rung orthopair fuzzy power geometric BM (GBM) operator, and linguistic q-rung orthopair fuzzy weighted power GBM operator, and then presents their rational properties and particular cases, which cannot only reduce the influences of some unreasonable data caused by the biased decision-makers, but also can take the interrelationship between any two different attributes into account. Finally, we propose a method to handle the MAGDM under the environment of Lq-ROFNs by using the new proposed operators. Further, several examples are given to show the validity and superiority of the proposed method by comparing with other existing MAGDM methods.     

Multiple attribute group decision-making based on order-α divergence and entropy measures under q-rung orthopair fuzzy environment

The q-rung orthopair fuzzy set (^qROPFS), proposed by Yager, is a more effective and proficient tool to represent uncertain or vague information in real-life situations. Divergence and entropy are two important measures, which have been extensively studied in different information environments, including fuzzy, intuitionistic fuzzy, interval-valued fuzzy, and Pythagorean fuzzy. In the present communication, we study the divergence and entropy measures under the q-rung orthopair fuzzy environment. First, the work defines two new order-α divergence measures for ^qROPFSs to quantify the information of discrimination between two ^qROPFSs. We also examine several mathematical properties associated with order-α ^qROPF divergence measures in detail. Second, the paper introduces two new parametric entropy functions called “order-α ^qROPF entropy measures” to measure the degree of fuzziness associated with a ^qROPFS. We show that the proposed order-α divergence and entropy measures include several existing divergence and entropy measures as their particular cases. Further, the paper develops a new decision-making approach to solve multiple attribute group decision-making problems under the ^qROPF environment where the information about the attribute weights is completely unknown or partially known. Finally, an example of selecting the best enterprise resource planning system is provided to illustrate the decision-making steps and effectiveness of the proposed approach.     

Single variable differential calculus under q-rung orthopair fuzzy environment: Limit,derivative, chain rules,and its application

The q-rung orthopair fuzzy set ( q-ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q-rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q-rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q-rung orthopair single variable fuzzy function ( q-ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q-ROSVFFs, and offer the relationship between the limit of q-ROSVFF and that of q-ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q-ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q-ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q-ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q-ROSVFFs.     

Rule-based Mamdani-type fuzzy modeling of skin permeability

Two Mamdani type fuzzy models (three inputs–one output and two inputs–one output) were developed to predict the permeability of compounds through human skin. The models were derived from multiple data sources including laboratory data, published data bases, published statistical models, and expert opinion. The inputs to the model include information about the compound (molecular weight and octonal–H₂O partition coefficient) and the application temperature. One model included all three parameters as inputs and the other model only included information about the compound. The values for mole molecular weight ranged from 30 to 600 Da. The values for the log of the octonal–H₂O partition coefficient ranged from −3.1 to 4.34. The values for the application temperature ranged from 22 to 39 °C. The predicted values of the log of permeability coefficient ranged from −5.5 to −0.08.Each model was a collection of rules that express the relationship of each input to the permeability of the compound through human skin. The quality of the model was determined by comparing predicted and actual fuzzy classification and defuzzification of the predicted outputs to get crisp values for correlating estimates with published values. A modified form of the Hamming distance measure is proposed to compare predicted and actual fuzzy classification. An entropy measure is used to describe the ambiguity associated with the predicted fuzzy outputs.The three input model predicted over 70% of the test data within one-half of a fuzzy class of the published data. The two input model predicted over 40% of the test data within one-half of a fuzzy class of the published data. Comparison of the models show that the three input model exhibited less entropy than the two input model.     

Design and analysis of a class of self-organizing and trainable fuzzy controllers

As the applications of fuzzy-controllers become more complicated, the attributes of self-organization and trainability become increasingly important. Indeed, the specification of fuzzy rules and membership functions for systems with a large number of state variables is extremely difficult. This paper introduces a new class of self-organizing and trainable fuzzy-controllers that can be designed without specific information regarding either the membership functions or the fuzzy rules. The proposed controller derives the fuzzy rules from clusters formed in the input space, through a self-organizing process. The clustering is performed through a simple method which can adaptively allocate new clusters as more date are available to the controller. Then, the membership values of crisp inputs are determined by K-nearest-neighbor (KNN) distance measures applied to the centers of the input clusters. Finally, a KNN defuzzification processes directly estimates of the crisp output of unknown input data. An adaptation procedure for the center vector of each cluster and the corresponding output value is developed. The overall design is analyzed in terms of the existence and the uniqueness of the solution of the proposed model. The performance of the proposed controller is considered through the modeling of the Mackey—Glass time-series.     

A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector

Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs and multiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with proportional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial models directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radial model in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure requires crisp data which may not always be available in real world applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) a concept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using α – cut approach, (ii) a fuzzy correlation coefficient method using expected value approach which calculates the expected intervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs, and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposed approaches are then applied to the State Bank of Patiala in the Punjab state of India with districts as the DMUs.     

Some q-rung orthopair uncertain linguistic aggregation operators and their application to multiple attribute group decision making

q-Rung orthopair fuzzy sets (q-ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the $q\mathrm{t}\mathrm{h}$ power of the membership and the $q\mathrm{t}\mathrm{h}$ power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q-rung orthopair uncertain linguistic sets (q-ROULSs) based on the q-ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q-ROULSs, several q-rung orthopair uncertain linguistic aggregation operators are developed, including the q-rung orthopair uncertain linguistic weighted arithmetic average operator, the q-rung orthopair uncertain linguistic ordered weighted average operator, the q-rung orthopair uncertain linguistic hybrid weighted average operator, the q-rung orthopair uncertain linguistic weighted geometric average operator, the q-rung orthopair uncertain linguistic ordered weighted geometric operator, and the q-rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q-rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision-making steps and explain the validity and feasibility of two methods by comparing with other methods.     

Kingfisher: an efficient algorithm for searching for both positive and negative dependency rules with statistical significance measures

Statistical dependency analysis is the basis of all empirical science. A commonly occurring problem is to find the most significant dependency rules, which describe either positive or negative dependencies between categorical attributes. In medical science, for example, one is interested in genetic factors, which can either predispose or prevent diseases. The requirement of statistical significance is essential, because the discoveries should hold also in future data. Typically, the significance is estimated either by Fisher??s exact test or the ?? ²-measure. The problem is computationally very difficult, because the number of all possible dependency rules increases exponentially with the number of attributes. As a solution, different kinds of restrictions and heuristics have been applied, but a general, scalable search method has been missing. In this paper, we introduce an efficient algorithm, called Kingfisher, for searching for the best non-redundant dependency rules with statistical significance measures. The rules can express either positive or negative dependencies between a set of positive attributes and a single consequent attribute. The algorithm itself is independent from the used goodness measure, but we concentrate on Fisher??s exact test and the ?? ²-measure. The algorithm is based on an application of the branch-and-bound search strategy, supplemented by several pruning properties. Especially, we prove a new lower bound for Fisher??s p and introduce a new effective pruning principle. According to our experiments on classical benchmark data, the algorithm is well scalable and can efficiently handle even dense and high-dimensional data sets. An interesting observation was that Fisher??s exact test did not only produce more reliable rules than the ?? ²-measure, but it also performed the search much faster.     

Some q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their applications to multiple-attribute group decision making

Multiple-attribute group decision making (MAGDM) under linguistic environment is an important part of modern decision sciences, and information aggregation operator plays an import role in solving MAGDM problems. In this paper, an approach for solving MAGDM problem with q-rung orthopair fuzzy 2-tuple linguistic information is developed. First, the q-rung orthopair fuzzy 2-tuple linguistic weighted averaging (q-ROFTLWA) operator and the q-rung orthopair fuzzy 2-tuple linguistic weighted geometric (q-ROFTLWG) operator are presented. Furthermore, the q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean (q-ROFTLMM) operator and the q-rung orthopair fuzzy 2-tuple linguistic dual Muirhead mean (q-ROFTLDMM) operator are proposed on the basis of Muirhead mean (MM) operator and dual Muirhead mean (DMM) operator. Then, an approach is developed to deal with MAGDM problem under q-rung orthopair fuzzy 2-tuple linguistic environment based on the proposed operators. Finally, a numerical example for selecting desirable emergency alternative(s) in the process of designing emergency preplan is given to illustrate the application of the developed method and demonstrate its effectiveness.     

采用非线性规则和不均匀分布的输入输出隶属函数时典型模糊控制器的结构分析

针对两输入 (e,Δe)一输出 (Δu)的典型模糊控制器, 其输入变量采用三角形、全交迭、对称、不均匀分布的隶属函数, 输出变量采用对称、不均匀分布的单点隶属函数, 当采用非线性控制规则和Sum Product推理方法时, 推导了输出的解析表达式, 分析了其结构特性和极限特性, 证明了此类模糊控制器具有通用逼近性, 并讨论了典型模糊控制系统的局部稳定性.     

A novel multiple-attribute group decision-making method based on q-rung orthopair fuzzy generalized power weighted aggregation operators

In this paper, a novel approach is developed to deal with multiple-attribute group decision-making (MAGDM) problem under q-rung orthopair fuzzy environment. Firstly, some operators have been proposed to aggregate q-rung orthopair fuzzy information, such as the q-rung orthopair fuzzy generalized power averaging (q-ROFGPA) operator, the q-rung orthopair fuzzy generalized power weighted averaging (q-ROFGPWA) operator, the q-rung orthopair fuzzy generalized power geometric (q-ROFGPG) operator, and the q-rung orthopair fuzzy generalized power weighted geometric (q-ROFGPWG) operator. In addition, some desirable properties and special cases of these operators are discussed. Second, a novel approach is developed to solve MAGDM problem under the q-rung orthopair fuzzy environment based on the proposed q-ROFGPWA and q-ROFGPWG operators. Finally, a practical example is given to illustrate the application of the proposed method, and further the sensitivity analysis and comparative analysis are carried out.     

Validation of overlapping clustering: A random clustering perspective

As a widely used clustering validation measure, the F-measure has received increased attention in the field of information retrieval. In this paper, we reveal that the F-measure can lead to biased views as to results of overlapped clusters when it is used for validating the data with different cluster numbers (incremental effect) or different prior probabilities of relevant documents (prior-probability effect). We propose a new “IMplication Intensity” (IMI) measure which is based on the F-measure and is developed from a random clustering perspective. In addition, we carefully investigate the properties of IMI. Finally, experimental results on real-world data sets show that IMI significantly alleviates biased incremental and prior-probability effects which are inherent to the F-measure.     

q-Rung orthopair fuzzy sets-based decision-theoretic rough sets for three-way decisions under group decision making

As an extension of Pythagorean fuzzy sets, the q-rung orthopair fuzzy sets (q-ROFSs) can easily solve uncertain information in a broader perspective. Considering the fine property of q-ROFSs, we introduce q-ROFSs into decision-theoretic rough sets (DTRSs) and use it to portray the loss function. According to the Bayesian decision procedure, we further construct a basic model of q-rung orthopair fuzzy decision-theoretic rough sets (q-ROFDTRSs) under the q-rung orthopair fuzzy environment. At the same time, we design the corresponding method for the deduction of three-way decisions by utilizing projection-based distance measures and TOPSIS. Then, we extend q-ROFDTRSs to adapt the group decision-making (GDM) scenario. To fuse different experts’ evaluation results, we propose some new aggregation operators of q-ROFSs by utilizing power average (PA) and power geometric (PG) operators, that is, q-rung orthopair fuzzy power average, q-rung orthopair fuzzy power weighted average (q-ROFPWA), q-rung orthopair fuzzy power geometric, and q-rung orthopair fuzzy power weighted geometric (q-ROFPWG). In addition, with the aid of q-ROFPWA and q-ROFPWG, we investigate three-way decisions with q-ROFDTRSs under the GDM situation. Finally, we give the example of a rural e-commence GDM problem to illustrate the application of our proposed method and verify our results by conducting two comparative experiments.     

Some q‐Rung Orthopai Fuzzy Bonferroni Mean Operators and Their Application to Multi‐Attribute Group Decision Making

In the real multi‐attribute group decision making (MAGDM), there will be a mutual relationship between different attributes. As we all know, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between parameters. In addition, in describing uncertain information, the eminent characteristic of q‐rung orthopair fuzzy sets (q‐ROFs) is that the sum of the qth power of the membership degree and the qth power of the degrees of non‐membership is equal to or less than 1, so the space of uncertain information they can describe is broader. In this paper, we combine the BM operator with q‐rung orthopair fuzzy numbers (q‐ROFNs) to propose the q‐rung orthopair fuzzy BM (q‐ROFBM) operator, the q‐rung orthopair fuzzy weighted BM (q‐ROFWBM) operator, the q‐rung orthopair fuzzy geometric BM (q‐ROFGBM) operator, and the q‐rung orthopair fuzzy weighted geometric BM (q‐ROFWGBM) operator, then the MAGDM methods are developed based on these operators. Finally, we use an example to illustrate the MAGDM process of the proposed methods. The proposed methods based on q‐ROFWBM and q‐ROFWGBM operators are very useful to deal with MAGDM problems.     

A multi-objective genetic optimization of interpretability-oriented fuzzy rule-based classifiers

The paper presents a multi-objective genetic approach to design interpretability-oriented fuzzy rule-based classifiers from data. The proposed approach allows us to obtain systems with various levels of compromise between their accuracy and interpretability. During the learning process, parameters of the membership functions, as well as the structure of the classifier's fuzzy rule base (i.e., the number of rules, the number of rule antecedents, etc.) evolve simultaneously using a Pittsburgh-type genetic approach. Since there is no particular coding of fuzzy rule structures in a chromosome (it reduces computational complexity of the algorithm), original crossover and mutation operators, as well as chromosome-repairing technique to directly transform the rules are also proposed. To evaluate both the accuracy and interpretability of the system, two measures are used. The first one – an accuracy measure – is based on the root mean square error of the system's response. The second one – an interpretability measure – is based on the arithmetic mean of three components: (a) the average length of rules (the average number of antecedents used in the rules), (b) the number of active fuzzy sets and (c) the number of active inputs of the system (an active fuzzy set or input means a set or input used by at least one fuzzy rule). Both measures are used as objectives in multi-objective (2-objective in our case) genetic optimization approaches such as well-known SPEA2 and NSGA-II algorithms. Moreover, for the purpose of comparison with several alternative approaches, the experiments are carried out both considering the so-called strong fuzzy partitions (SFPs) of attribute domains and without them. SFPs provide more semantically meaningful solutions, usually at the expense of their accuracy. The operation of the proposed technique in various classification problems is tested with the use of 20 benchmark data sets and compared to 11 alternative classification techniques. The experiments show that the proposed approach generates classifiers of significantly improved interpretability, while still characterized by competitive accuracy.     

Complexity classes of provable recursive functions

Complexity measures and provable recursive functions (p-functions) are combined to define a p-measure as a measure for which Blum's axioms can be proved in a given axiomatic system. For p-measures, it is shown that the complexity class of a p-function contains only p-functions and that all p-functions form a single complexity class. Various other classes and a variation of a complexity measure, all suggested by the notion of provability, are also investigated. Classical results in complexity theory remain true when relativized to p-functions.     

Processing time estimations by variable structure TSK rules learned through genetic programming

Accuracy in processing time estimation of different manufacturing operations is fundamental to get more competitive prices and higher profits in an industry. The manufacturing times of a machine depend on several input variables and, for each class or type of product, a regression function for that machine can be defined. Time estimations are used for implementing production plans. These plans are usually supervised and modified by an expert, so information about the dependencies of processing time with the input variables is also very important. Taking into account both premises (accuracy and simplicity in information extraction), a model based on TSK (Takagi–Sugeno–Kang) fuzzy rules has been used. TSK rules fulfill both requisites: the system has a high accuracy, and the knowledge structure makes explicit the dependencies between time estimations and the input variables. We propose a TSK fuzzy rule model in which the rules have a variable structure in the consequent, as the regression functions can be completely distinct for different machines or, even, for different classes of inputs to the same machine. The methodology to learn the TSK knowledge base is based on genetic programming together with a context-free grammar to restrict the valid structures of the regression functions. The system has been tested with real data coming from five different machines of a wood furniture industry.

Orthogonal Fuzzy Rule-Based Systems: Selection of Optimum Rules

In this paper, the concept of orthogonal fuzzy rule-based systems is introduced. Orthogonal rules are an extension to the definition of orthogonal vectors when the vectors are vectors of membership functions in the antecedent part of rules. The number and combination of rules in a fuzzy rule-based system will be optimised by applying orthogonal rules. The number of rules, and subsequently the complexity of the fuzzy rule-based systems, are directly associated with the number of input variables and distinguishable membership functions for each individual input variable. A subset of rules can be used if it is known which subset provides closer behaviour to the case when all rules are used. Orthogonal fuzzy rule-based systems are proposed as a judgment as to whether the optimal rules are selected. The application of orthogonal fuzzy rules becomes essential when fuzzy rule-based systems containing many inputs are used. An illustrative example is presented to create a model for the solder paste printing stage of surface mount tech-nology.