Multidimensional Topological Measure Spaces and Their Applications in Decision‐Making Problems

In this paper, Multi-dimensional fuzzy set calculation model is introduced, and the method is called Konohen fuzzy. For some actual purposes, formal expression functional model for multi-dimensional solution space can not be obtained so easily. We propose the SPF : sample points based fuzzy set definition method to descrive multi-dimensional fuzzy set which is used to model problems on high-dimensional solution spaces. Konohen fuzzy handles more free styled fuzzy set shapes in multi-dimensional spaces than previous methods. The SPF is consist of a set of random finite sample points and a set of their membership values. Membership values of arbitrary points are obtained using interpolation, and this method makes it possible to handle complex and complicated systems in simple way that is an original goal of fuzzy logic. A numerical example results show that calculation error between SPF and overall interpolation is very small.

This study introduces the concepts of complements, t-norms, t-conorms, and cut sets in the context of multi-dimensional fuzzy sets. Some fundamental results including DeMorgan-type identities, concerning these, are obtained. The advantage of using multidimensional fuzzy sets for analyzing data is that each element in the universe can be given individual attention as per the requirement. By utilizing this scope, as an application of the introduced concepts, stage identification of certain diseases based on Multidimensional fuzzy sets and intuitionistic fuzzy sets is provided together with a comparative study on typical biomedical data.

Food quality is a fuzzy category, which could be evaluated using fuzzy logic. Our approach to food quality evaluation is based on mapping food quality attributes into a fuzzy domain as a multi-dimensional fuzzy sets. First, the data representing quality attributes are mapped into orthogonal coordinates using PCA to reduce dimensionality. Second, subtractive clustering (SC) is applied to determine a representative number of clusters. Each point in the dataset is associated with each cluster by credibilistic fuzzy C-means clustering (CFCM). After data organized in fuzzy clusters, an artificial neural network (ANN) is trained to associate each point in the dataset with its membership degree in each cluster. Trained ANN serves as a predictive model to convert real-time data stream into the multi-dimensional fuzzy domain. The application of this methodology is illustrated for real-time quality evaluation in shrimp batch drying. In this study 27 quality attributes have been merged into 9 orthonormal vectors, which have been clustered into 10 fuzzy sets. This structuring of the experimental fuzzy domain allowed the development of a multi-dimensional kinetic model, which improved the quality of shrimp drying. The computational time for quality identification in the fuzzy domain is below 1 sec, which is satisfactory for most real-time applications. This data-driven algorithm is completely automated and has unlimited potential for real-time fuzzy control and optimization. Highlights Multi-dimensional fuzzy sets are unique identifiers of food quality Extracting principal information in orthonormal coordinates Using the artificial neural network for predicting membership functions A multi-dimensional fuzzy kinetics model was developed Structuring of fuzzy domain decreased computational time for fuzzy control

Problems can arise in decision making when a different number of evaluations for the different criteria or alternatives may be available. This is the case, for instance, when for some reason, one or more experts refrain from evaluating certain criteria for an alternative. In these situations, approaches using n-dimensional fuzzy sets and hesitant fuzzy sets are not appropriate. This is because the first approach works only with a fixed dimension, whereas the second modifies the information, fusing equal evaluations given by different experts for the same pair of alternatives/criteria or adjusting the elements so that an order defined on hesitant fuzzy sets can be used. As such, in addition to being a natural extension of n-dimensional fuzzy sets, multidimensional fuzzy sets contemplate the aforementioned situation without modifying their elements. In this article, we introduce the concept of multidimensional fuzzy sets as a generalization of the n-dimensional fuzzy sets in which the elements can have different dimensions. We begin by presenting a way to generate a family of partial orders and conditions under which these sets have a lattice structure. Moreover, the concepts of admissible orders on multidimensional fuzzy sets and m-ary multidimensional aggregation functions with respect to an admissible order are defined and studied. Finally, we provide two multicriteria group decision-making methods based on these m-ary multidimensional aggregation functions.

This article examines distance and similarity measures in multidimensional fuzzy sets, which are essential in decision-making and aggregation across various fields. It defines the axioms for multidimensional distance measures and introduces a framework for normalized distance and similarity measures within a suitable fuzzy space. The concept of complement-invariant proximity measures is also discussed. The paper further explores the relationship between distance and similarity, linking them with multidimensional entropy. It presents sigma-distance, sigma-similarity, and sigma-entropy measures that balance values between fuzzy sets and their complements. Finally, two decision-making problems are analyzed, with a comparative study showing the proposed model’s advantage over existing approaches.

For the purpose of improving the resultant performance processes of microsurface precise grinding in different machining conditions, the obtained grinded surface should be quality-evaluated and influence-assessed under the instructions of topography modeling. Since microsurface fitting algorithms exerts a considerable influence on those constructed topography mathematical features in geometric domain, which results to different grinding micro-topography processes caused by one specific influence mechanism, and makes the performance investigation of fitting algorithms in surface micro-topography grinding processes indispensable. Through extracting the coordinate positions of those selected physical control points in one objective surface topography sample to be grinded by using a series of precise spatial coordinate measurement apparatuses, several typical algorithms of surface fitting in geometric domain were used for constructing the micro-topography models of those sample section surface with complicated point cloud. On the base of computing the newly proposed mathematical features caused from those micro-topography models, fuzzy evaluation data sequences were established and one new multi-dimensional mathematical quantitative evaluation method derived from fuzzy relation set was proposed and employed to deal with their inherent mutual-relationships. As the performance comparisons of resultant grinded topography between using surface modeling operations and not using surface modeling operations were clearly quantified, it is obvious that the multi-dimensional fuzzy influence relation set between surface fitting algorithms, topography geometric features, and practical grinding processes in different experimental conditions will be quantitatively analyzed in detail, which contributes to the acquirement of final conclusions concerned with the inherent influence mechanism and mutual mathematical relation set emerged from the performance results of different surface fitting algorithms, together with the topography spatial features and practical surface grinding process characteristics in one specific experimental conditions as well. Through realizing grinding processes evaluation based on multi-dimensional fuzzy relation set and making performance comparisons with several other typical statistical evaluation methods in index analysis, an in-depth performance inspection of surface precise grinding with objective micro-topography will be facilitated and optimized; simultaneously, a new research idea for improving microsurface modeling and its subsequent grinding processes in a practical experimental operation can also be demonstrated in the long run.

In this paper we introduce a new way to determine a threshold for making a multidimensional fuzzy set from sample data set using persistent homology on these data. A multi dimensional fuzzy set is a fuzzy set that has arbitrary shape on the support parameter space. In previous paper, network distance of data is used to make a multi dimensional fuzzy set from sampling data keeping its topological properties. There are some control parameters to make sample data network and they are ad-hoc adjusted parameters with trial and errors. Persistent homology is a new mathematical methodology to make a feasible network structure from sample data set. We employ persistent homology and introduce a new algorithm and shown a feasibility of proposed method through numerical examples.

Multidimensional fuzzy sets: Negations and an algorithm for multi-attribute group decision making

A new approach is presented to enhance fuzzy modeling using multidimensional fuzzy sets directly in the fuzzy model in place of decomposed fuzzy sets by projection. The antecedent fuzzy sets in the form of multivariable functions are represented in continuous form. This is accomplished by using a multi input and multi output neural network, which is in particular radial basis functions (RBF) network providing the required multivariable function approximation properties in a fuzzy model. The inputs of the network are fuzzy model inputs, outputs are the membership function values as to the multivariable fuzzy sets involved in the model. Thus, each output is restricted to one multivariable fuzzy set. The RBF network is trained according to the multidimensional fuzzy sets which are identified after fuzzy clustering of the data. Based on this, the linear model parameters are determined by the method of least squares in a straightforward manner. The transparency issues can be handled by means of projection process to have information about the shape and locations of the membership functions pertinent to each variable at the input. The accurate fuzzy modeling having been thus guaranteed the information on the linear model parameters are used to establish the multivariable fuzzy rules with their respective validity regions. The redundancy of the number of fuzzy sets can be circumvented by classical cluster merging algorithms available. In this approach, the fuzzy rules are determined from the local linear models, however in contrast with the conventional fuzzy modeling, the model outputs are determined by the multivariable fuzzy sets. To obtain regular sets, RBF network is trained by means of orthogonal least squares (OLS) method so that at the same time the convergence issues of general neural network training is eliminated.

In this study, we propose fuzzy modeling algorithm to improve Takagi-Sugeno fuzzy model. This algorithm initially finds desirable number of rules at once, in advance, and then identifies the premise and consequent parameters separately by fixing number determined. The proposed algorithm consists of three stages: determination of the optimal number of fuzzy rules, coarse tuning of parameters and fine tuning of these parameters. To find the optimal number of rules, the new cluster validity algorithm that is based on the validity criterion V sv adapted to the usage of FCRM-like clustering, is proposed. In coarse tuning, by using the mentioned clustering algorithm for input-output data and the projection scheme, the consequent and premise parameters are coarsely defined. In fine tuning, the gradient descent (GD) method is used to precisely adjust parameters of fuzzy model but unlike other similar modeling algorithms, the premise parameters are adjusted with respect to multidimensional membership function in premise part of rule. Finally, two examples are given to demonstrate the validity of suggested modeling algorithm and show its excellent predictive performance.

Interpretability of fuzzy rule-based models has always been of significant interest to the research community and the research in this area led to a number of far-reaching results. In this study, we briefly revisit the methodology and concepts of interpretability of Takagi–Sugeno (T-S) rule-based models and develop a conceptual framework involving several levels at which rules are interpreted. The layers at which interpretability is positioned are structured hierarchically by starting with the initial fuzzy set level (originating from the design of the rules), moving to information granules of finite support (where interval calculus is engaged) and finally ending up with symbols built at the higher level. As T-S rule-based models are endowed with local functions forming the conclusion parts of the rules, with the use of the principle of justifiable granularity, we develop a way of forming an interpretable conclusion in the form of information granule. To facilitate interpretability of conditions of the rules, multidimensional fuzzy sets (coming as a result of clustering) are decomposed into a Cartesian product of 1-D fuzzy sets and the quality of the resulting decomposition is evaluated. The quality of granular rules is assessed by analyzing the relationship between specificity of condition and conclusion information granules. The rules emerging at the level of symbols are further interpreted by engaging linguistic approximation, which helps approximate a collection of linguistic terms of subconditions producing a linguistic summarization in the form <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</i> (inputs are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> ) consisting of a certain linguistic quantifier <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</i> . The performance of summarization is provided in the form of ranking of the relevance of the rules. Experimental studies using publicly available data are completed and analyzed.

Fuzzy clustering of fuzzy ecological data

The analysis of the perceptual properties of texture plays a fundamental role in tasks where some interaction with subjects is needed. In order to face the imprecision related to these properties, several fuzzy approaches can be found in the literature, but they do not properly take into account the subjectivity of users. In this paper, a generic technique is proposed in order to adapt any multidimensional fuzzy set to the different perceptions of texture properties that a particular user can have. This way, we combine the improvement in the texture characterization given by the use of several computational measures as reference set with the adaptation to the subjectivity of the human perception. In the proposed adaptation method, the membership functions are automatically transformed on the basic of the information given by the user.

In this paper, an approach for the tuning of a model-based non-linear predictive control (NMPC) is presented. The proposed control uses the pattern search optimization algorithm (PSM), which is applied to the pH non-linear control in the alkalinization process of sugar juice. First, the model identification is made using the Takagi Sugeno T-S fuzzy inference systems with multidimensional fuzzy sets; the next step is the controller parameters tuning. The PSM algorithm is used in both cases. The proposed approach allows the minimization of model uncertainty and decreases, in the response, the error in a steady state when compared with other authors who perform the same procedure but apply other optimization algorithms. The results show an improvement in the steady-state error in the plant response.

On Data-driven Takagi-Sugeno Modeling of Heterogeneous Systems with Multidimensional Membership Functions