Multidimensional Fuzzy Sets Research Articles

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

Multidimensional fuzzy sets (MFS) is a new extension of fuzzy sets on which the membership values of an element in the universe of discourse are increasingly ordered vectors on the set of real numbers in the interval [0,1]. This paper aims to investigate fuzzy negations on the set of increasingly ordered vectors on [0,1], i.e. on L∞([0,1]), MFN in short, with respect to some partial order. In this paper we study partial orders, giving special attention to admissible orders on Ln([0,1]) and L∞([0,1]). In addition, we study the possibility of existence of strong multidimensional fuzzy negations and some properties and methods to construct such operators. In particular, we define the ordinal sums of n-dimensional negations and ordinal sums of multidimensional fuzzy negations on a multidimensional product order. A multi-attribute group decision making algorithm is presented.

Integrating Rough Sets and Multidimensional Fuzzy Sets for Approximation Techniques: A Novel Approach

Integrating Rough Sets and Multidimensional Fuzzy Sets for Approximation Techniques: A Novel Approach

Multidimensional fuzzy norms and cut sets in the context of medical decision making

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.

A Hierarchical Approach to Interpretability of TS Rule-Based Models

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.

Multidimensional fuzzy implications

Multidimensional fuzzy sets (MFS) is a new extension of fuzzy sets on which the membership values of an element in the discourse universe are increasingly ordered vectors on the set of real numbers in the interval [0,1]. This paper aims to investigate fuzzy negations and, mainly, fuzzy implications on the set of increasingly ordered vectors on [0,1], i.e. on L∞([0,1]), MFN and MFI in short, with respect to some partial order. In this paper we study partial orders, giving special attention to admissible orders on L∞([0,1]). In addition, some properties and methods to construct such operators from fuzzy negations and fuzzy implications, respectively, are provided and we demonstrate that the action of the group of automorphisms on fuzzy implications on L∞([0,1]) preserves several original properties of the implication. Finally, through a specific type of representable MFI, we are able to generate a class of MFN called natural m-negations.

Optimization Based on Pattern Search Algorithm Applied to pH Non-Linear Control: Application to Alkalinization Process of Sugar Juice

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.

Multidimensional interval type 2 epistemic fuzzy arithmetic

The article presents a new, multidimensional arithmetic of type 2 fuzzy numbers (M-IT2-F arithmetic) in which the result is a multidimensional fuzzy set. This arithmetic increases the accuracy of calculations and the scope of problems solved in relation to the currently used interval type 2 standard fuzzy arithmetic (IT2-SF arithmetic). The proposed M-IT2-F arithmetic has mathematical properties that IT2-SF arithmetic does not have. Thanks to these properties, it provides accurate calculation results that are not over- or under-estimated in terms of uncertainty. The paper contains comparisons of both types of arithmetic in application to two problems. Fuzzy arithmetic is not a finished work and is in a phase of continuous improvement and development. M-IT2-F arithmetic is a higher form of M-IT2 (non-fuzzy) arithmetic.

Food quality evaluation in drying: Structuring of measurable food attributes into multi-dimensional fuzzy sets

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

(2008-6074) Multidimensional interval type 2 epistemic fuzzy arithmetic

The article presents a new, multidimensional arithmetic of type 2 fuzzy numbers (M-IT2-F arithmetic) in which the result is a multidimensional fuzzy set. This arithmetic increases the accuracy of calculations and the scope of problems solved in relation to the currently used interval type 2 standard fuzzy arithmetic (IT2-SF arithmetic). The proposed M-IT2-F arithmetic has mathematical properties that IT2-SF arithmetic does not have. Thanks to these properties, it provides accurate calculation results that are not over- or under-estimated in terms of uncertainty. The paper contains comparisons of both types of arithmetic in application to two problems. Fuzzy arithmetic is not a finished work and is in a phase of continuous improvement and development. M-IT2-F arithmetic is a higher form of M-IT2 (non-fuzzy) arithmetic.

Multidimensional Fuzzy Sets

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.

Multidimensional fuzzy finite tree automata

This paper introduces the notion of multidimensional fuzzy finite tree automata (MFFTA) and investigates its closure properties from the area of automata and language theory. MFFTA are a superclass of fuzzy tree automata whose behavior is generalized to adapt to multidimensional fuzzy sets. An MFFTA recognizes a multidimensional fuzzy tree language which is a regular tree language so that for each dimension, a fuzzy membership grade is assigned to each tree. We study MFFTA by extending some classical problems and properties of automata and regular languages such as determinization, reduction, duality and operations on languages. Furthermore, we provided the method of converting every complete fuzzy tree automata to an MFFTA as well as an example to show the efficiency of MFFTA in comparison to FFTA.}

Approximation of real data by fuzzy sets for the classification problem

The article deals with the method of classification of real data using the apparatus of fuzzy sets and fuzzy logic as a flexible tool for learning and recognition of natural objects on the example of oil and gas prospecting sections of the Dnieper-Donetsk basin. The real data in this approach are the values for the membership function that are obtained not through subjective expert judgment but from objective measurements. It is suggested to approximate the fuzzy set membership functions by using training data to use the approximation results obtained during the learning phase at the stage of identifying unknown objects. In the first step of learning, each traditional future of a learning data is matched by a primary traditional one-dimensional set whose membership function can only take values from a binary set — 0 if the learning object does not belong to the set, and 1 if the learning object belongs to the set. In the second step, the primary set is mapped to a fuzzy set, and the parameters of the membership function of this fuzzy set are determined by approximating this function of the traditional set membership. In the third step, the set of one-dimensional fuzzy sets that correspond to a single feature of the object is mapped to a fuzzy set that corresponds to all the features of the object in the training data set. Such a set is the intersection of fuzzy sets of individual features, to which the blurring and concentration operations of fuzzy set theory are applied in the last step. Thus, the function of belonging to a fuzzy set of a class is the operation of choosing a minimum value from the functions of fuzzy sets of individual features of objects, which are reduced to a certain degree corresponding to the operation of blurring or concentration. The task of assigning the object under study to a particular class is to compare the values of the membership functions of a multidimensional fuzzy set and to select the class in which the membership function takes the highest value. Additionally, after the training stage, it is possible to determine the degree of significance of an object future, which is an indistinctness index, to remove non-essential data (object futures) from the analysis.

Adaptive multidimensional fuzzy sets for texture modeling

The modeling 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, fuzzy sets defined on the domain of computational measures of the corresponding property are usually employed. In this sense, the most interesting approaches show that the combination of different measures as reference sets improve the texture characterization. However, the main drawback of these proposals is that they do not take into account the subjectivity associated with human perception. For example, the perception of a texture property may change depending on the user, and in addition, the image context may influence the global perception of a given property. In this paper, we propose to solve these problems by combining the use of several computational measures in a reference set with adaptation to the subjectivity of human perception. To do this, we propose a generic methodology that automatically transforms any multidimensional fuzzy set modeling a texture property to the particular perception of a new user or to the image context. For this purpose, the information given by the user, or extracted from the textures present in the image, are employed.

Information Processing in HFMFs Based Risk Evaluation for Virtual Enterprise

The method of HFMFs (Hypertrapezoidal Fuzzy Membership Functions) is a new mechanism for designing multidimensional fuzzy sets. HFMFs models the correlation that exists between the variables of the state space. Therefore, it is able to describe the problems of fuzzy situation more exactly. As we known, it needs a great many information to evaluate the risk level for VE using fuzzy synthetic evaluation method. However, there are lots of difficulties to get the needed information for VE. Accordingly, the HFMFs method is introduced to estimate the risk level of VE. Considering not only the relations between different fuzzy sets but also the method of HFMFs is able to describe the problems of fuzzy situation more exactly with less information. Consequently, the HFMFs method is more suitable to the virtual enterprise with little historical experiences than the fuzzy synthetic evaluation method. Simulation has shown that this method is a simple and effective risk evaluation method for VE.

Performance investigation of fitting algorithms in surface micro-topography grinding processes based on multi-dimensional fuzzy relation set

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.

Modelling ellipsoidal uncertainty by multidimensional fuzzy sets

The paper addresses the problem of an aggregation of knowledge obtained from many sets of experiments each with different confidence level during identification of a linear dynamical system with uncertain parameters. The proposed approach describes the parameter space by a multidimensional fuzzy set specified by non-symmetric ellipsoidal α-cuts. An algorithm for determination of minimum confidence level guaranteeing stability of such a system is presented.

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

AbstractIn case of a structural mismatch between a Takagi-Sugeno model and a nonlinear system that is to be modeled, identification algorithms tend to compensate this by a finer granular partitioning. This is in conflict with the objective of parsimonious models. A novel TS model type with multi-dimensional fuzzy sets and locally adjustable/heterogeneous fuzziness is presented with a method for the adjustment. This new approach enhances the structural flexibility of the model while the number of parameters increases negligibly. Two case studies, including compressor modeling, illustrate the performance of the proposed method.

An improved Takagi-Sugeno fuzzy model with multidimensional fuzzy sets

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.

Comparison of multidimensional fuzzy set ordination with CCA and DB‐RDA

Multidimensional fuzzy set ordination (MFSO) is an ordination algorithm based on the principles of fuzzy set theory that employs a dissimilarity matrix and environmental or experimental variables directly in the calculation of ordination coordinates. The objective is similar to constrained ordinations such as canonical correspondence analysis (CCA) and distance-based redundancy analysis (DB-RDA), although MFSO employs a different conceptual and algorithmic basis. Results from MFSO, CCA, and DB-RDA are compared on four data sets to determine the relative ability of the methods to identify and quantify environmental variable effect sizes on community variability. Models were fit by best all-subsets solutions and tested for significance by permutation tests. Methods were compared on the basis of variables selected and dimensionality of results and were evaluated on the basis of the correlation of the pair-wise distances of the derived ordination solutions to a Bray-Curtis dissimilarity matrix of the taxon data, and by cross-comparisons of the inertia explained by the fitted or derived values substituted into DB-RDA and CCA analyses. MFSO and DB-RDA achieved the highest correlation with the Bray-Curtis dissimilarity matrix on one data set each, and they tied for highest correlation on the two other data sets (differences <1%). CCA tied for best on one data set, tied for second on another, and achieved the lowest correlation on two data sets. When substituting MFSO- and DB-RDA-derived values into CCA, MFSO results achieved higher inertia explained on two of the four data sets; MFSO and DB-RDA achieved equivalent results on the other two. Substituting MFSO- or CCA-derived or fitted values into DB-RDA, MFSO results achieved higher inertia explained on all four data sets. While goodness-of-fit statistics were often similar across methods, the methods sometimes chose solutions of different dimensionality or employed different variables. In general, all algorithms performed well, finding relatively low-dimensional solutions that achieved high correlation with the Bray-Curtis dissimilarity matrix and explained significant amounts of inertia in the eigenanalyses. Considering both the correlation and substitution tests, MFSO achieved the best results of the three methods. DB-RDA results were nearly as good as MFSO, however, and clearly superior to CCA.

An Agent-based Multi-attribute Sealed-bid Design for Bilateral Contract

This paper presents an agent-based sealed-bid design for multi-issue negotiation in an incomplete information setting. The information of both agents is considered completely private. In our agent-based model, each agent builds a multi-dimensional fuzzy satisfaction set for the attributes of a commodity using fuzzy linguistic terms. The issues relating to an offer are divided into two categories (i.e., quantitative issues and qualitative issues). At each time-step of the negotiation, both agents submit a combined offer of quantitative issues for each discrete value of qualitative issue to the mediate agent simultaneously. This method offers flexibility for the agents represent their satisfaction for the combinations of different attributes using a multi-dimensional membership functions, taking into account the interdependencies between the attributes. In addition, using the proposed method of calculating agreed-price, agents can be stimulated to reach an agreement as early as possible. Therefore, this method may shorten negotiation process, which is desired for both parties. And the design can discourage counter-speculation and effectively control fraud and misrepresentation in a certain extent. Through a case study, the capabilities of the proposed method are illustrated.