Interval Membership Functions Research Articles

ПРОБЛЕМЫ АНАЛИЗА ОТРАСЛЕВОЙ ЭКОНОМИЧЕСКОЙ УСТОЙЧИВОСТИ СРЕДСТВАМИ НЕЧЁТКОЙ ЛОГИКИ

During the analysis of economic resilience of enterprises and industries within existing scientific approaches based on fuzzy sets and soft computing methodology, a number of problems have arisen that require solutions, namely: a) making adjustments to existing linguistic norms of resilience factors; b) incorporating mechanisms of intelligent filtering when forming industry indices; c) improving the accuracy of decision models embedded in the corresponding strategic map. This article describes a series of problematic cases that can be improved within the proposed approaches, including by: a) transitioning from interval membership functions in the classifier to trapezoidal ones; b) excluding noise-contaminated context from the sample that has been rejected based on formal criteria; c) phasifying decision parameters on the strategic map as part of a 4х6 matrix. In the latter case, the technology of pairwise comparison of fuzzy numbers is applied to assess the risk of inefficiency in the decision-making process. It can be formulated that in solving problems related to fuzzy descriptions, we have proposed a series of solutions using the same fuzzy descriptions, but in a more advanced version. In fact, we suggest upgrading a number of previously proven approaches for analyzing economic resilience within the fuzzy scientific paradigm.

Interval type-2 generalizing fuzzy model for monitoring the states of complex systems using expert knowledge

A type-2 interval generalizing fuzzy model is proposed for monitoring complex systems’ states. A set-theoretic approach is proposed to generalize the results of type-2 fuzzy models with interval membership functions. The study of the correspondence of expert assessments to the output value of the generalizing fuzzy model over the observation interval is presented. Examples of the use of generalizing fuzzy model type-2 in the task of monitoring the conditions of an artesian well are given. It is shown that in order to improve the quality of decisions made, the expert needs to pay attention to the value of the interval output of the generalizing type-2 fuzzy model. Recommendations are presented to experts to improve decision-making regarding the estimation of the output interval of the generalizing model.

Multi-Objective Reliability Optimization using Fuzzy Nonlinear Programming with Interval Membership Functions and Bias Functions: A Comparison of Particle Swarm Optimization and Genetic Algorithm

Multi-objective reliability optimization is a complex problem that involves simultaneously optimizing multiple objectives while ensuring that the system meets certain reliability requirements. In this paper, we present a methodology for solving multi-objective reliability optimization problems using fuzzy nonlinear programming. The methodology involves representing the reliability of each component as a triangular interval number and each objective function as an interval membership function. Conflicts between objectives are resolved using linear and nonlinear membership functions, and exponential and quadratic membership functions are used to obtain definite biases towards the objective. The proposed methodology employs Particle Swarm Optimization (PSO) or Genetic Algorithm (GA) to solve the problem, and the approach is compared with GA for linear and nonlinear membership functions. The results indicate the effectiveness of the methodology in addressing multi-objective reliability optimization problems

Fuzzy Hidden Markov Model Using Aggregation Operators

The Fuzzy Markov Model (FMM) is a fascinating domain for dealing with ambiguity in real-world scenarios. In type 2 fuzzy set (T2F), it has an uncertainty footprint, and the region circumscribed by the lower and upper interval membership functions is uncertain. In fuzzy sets, Triangular norms (t-norms) are a valuable tool for understanding the conjunction in fuzzy logic and, as a result, determining where fuzzy sets intersect. Norms and conforms in Triangular operations that generalize logical conjunction and disjunction. They also provide a natural explanation for the conjunction and disjunction in mathematical fuzzy logic semantics. Fuzzy Frank Triangular Norms have been used to verify this t-norm as there are many of them the aggregation qualities of Trapezoidal Interval Type-2 numbers (TpIT2FNs) because triangular norm meets the compatibility with Frank norms. Frank triangular norms provide more flexibility and robustness this requires more justification in the information fusion process than other triangular norms. Previous works on not concentrate on Frank’s norms. Other Aggregation works on norms that are not flexible to get the solution. Because of that, the Frank norms are used for the hidden Markov model. We've also used them in the Viterbi method with TpIT2FNs for Fuzzy Hidden Markov Model in the staff selection process.

НЕЧІТКІ МНОЖИНИ ТИПУ-2 В ЗАДАЧАХ МОДЕЛЮВАННЯ ТА ОЦІНЮВАННЯ СТАНІВ КРИТИЧНИХ СИСТЕМ З НЕДОВИЗНАЧЕНИМИ ВХІДНИМИ ДАНИМИ ТА ВИКОРИСТАННЯМ ЕКСПЕРТІВ

A method of type-2 fuzzy sets implementation for critical systems’ modeling and state assessment tasks with uncertain input data is proposed. It is shown that the basis for solving the modeling task is designing of a fuzzy logic system with interval membership functions of type-2. The paper presents the task of further developing the process of estimating the interval output of a fuzzy system with experts involvement. An approach based on fuzzy sets is proposed for solving the task of critical systems’ modeling and states assessment. Using the example of energy grid systems, where a high degree of uncertainty is present, it is shown that the main factors that influence the appearance of uncertainty in the initial data set of such systems are caused by the lack of sufficient information in the open print and the high variability of threats under the influence of the growing pace of digitalization of business processes. An analysis of expert evaluations of the interval output of fuzzy systems based on examples of modeling complex objects in various fields of application is given. The first example demonstrates the results of modeling in the field of natural sciences with uncertain initial data for assessing the prospects of an artesian well, where the final assessment is made by an expert. The second example demonstrates the implementation of the interval fuzzy model in the task of social orientation, where the problem of recruiting personnel in social groups from the point of view of professional suitability is modeled. The third task refers to modeling in the field of medical diagnosis of diseases of the endocrine system. Evaluation by experts of the results of interval fuzzy modeling in this field makes it possible to determine the state of a person's disease for endocrine pathology and prescribe timely treatment. The given examples of evaluating the interval output of a fuzzy system, taking into account the opinion of experts, confirm the possibilities for making decisions that are adequate for the subject area in the conditions of uncertain input data. Prospects for the application of the proposed models for the problems of cyber security of critical systems are given.

Three-Dimensional Fuzzy Control of Ultrasonic Cleaning

Abstract Consideration of ultrasonic cleaning as a process with distributed parameters enables reduction of power consumption. This approach is based on establishment of control over the process depending on fixed values of ultrasonic responses in set points. The initial intensity of radiators is determined using a three-dimensional (3D) interval type-2 fuzzy logic controller essentially created for processes with distributed parameters, as well as complex expert evaluation of the input data. The interval membership functions for the input and output data consider the space heterogeneity of ultrasonic cleaning. A rule base is formed, which is 2D and not dependent upon the number of input and output parameters. A model illustrating ultrasonic cleaning with a 3D interval type-2 fuzzy logic controller is designed. Comparative analysis of the output parameters of the proposed model and the traditional method indicates an increase in the energy efficiency by 41.17% due to application of only those ultrasonic radiators that are located next to the contamination.

Energy-water nexus planning of regional electric power system within an inexact optimization model in Tangshan City, China

Under the increasingly serious energy and water resources crisis, the correlation between two resources has become more and more close. Promoting the sustainable development of energy and water resources has been an inevitable choice for regional energy system planning. In this study, a hybrid interval two-stage fuzzy credibility constrained programming (ITFCP) model is developed for supporting energy-water nexus system management. ITFCP can explicitly address system uncertainties presented as probability distributions, fuzzy sets, and intervals, as well as dual dynamics through the concept of solutions with interval membership function. The method could provide effective linkages between predefined policies and economic penalties as well as system economy and risk. Then, ITFCP is applied to electric power system management under considering energy-water nexus in Tangshan City, which is a typical resource-based, high pollution, and water-deficient region in China. Multiple scenarios of different credibility levels about water availability are designed. Results disclosed that coal-fired power and imported electricity would play primary role in meeting tremendous electricity demand under the current situation. Results also revealed that the total water consumption control would be an effective way to accelerate electric power structural adjustment, protect energy and water resources as well as improve atmospheric environment. These findings are valuable for decision makers to gain in-depth analysis for more reasonable and applicable development patterns within a complex energy-water system.

FUZZY INFERENCE SYSTEMS BASE ON POLYNOMIALCONSEQUENTS OF FUZZY RULES

Various fuzzy inference systems that operate on the basis of polynomial consequents of fuzzy rules. As well as inference methods for such systems, in particular, Takagi-Sugeno fuzzy inference systems, their differences from other popular fuzzy systems, such as Mamdani systems, etc., are considered. The attention is focused on the features of the functioning of such systems both in the construction of elementary fuzzy systems. The Systems for which the calculation of the general logical conclusion involves intermediate levels of logical inference with many hierarchically interconnected blocks of fuzzy rules. Fuzzy sets of type 2 are considered, the membership index of which is a fuzzy term of the first type. This allows you to take into account the secondary fuzziness of linguistic concepts in the design of intelligent systems based on fuzzy inference. Fuzzy systems of the second type based on Takagi-Sugeno systems and the iterative Karnik-Mendel algorithm are considered to obtain a logical conclusion for fuzzy systems with the interval membership functions of the second type in the antecedents of fuzzy rules. The proposed procedure for lowering the order of fuzzy rules for higher-order Takagi-Sugeno fuzzy systems is described and justified. A fuzzy inference method for higher-order fuzzy systems based on the partition of a set of input variables is proposed. It is proposed to build a separate block of fuzzy rules for each of the input subspaces in the presence of a common polynomial. Which is a higher-order consequent, that reduces the total number of fuzzy rules in blocks.

Commercial Vehicle-Based Robust Control of Seated Whole-Body Vibration Using Adaptive Indirect Type-2 Fuzzy Neural Network

Drivers of commercial vehicles are invariably subject to chronic diseases such as back pain as a result of exposure to severe cabin excitations. In this paper, a six degrees of freedom (6-DOF) coupled human-body and seat suspension system is modeled in order to reduce the vibrations transmitted to the head, seat, and the relative seat and cabin floor displacement. The contributions of the present paper are: 1) two significant but inherently conflicting control objectives are employed, namely the seat acceleration and the relative displacement between seat and cabin floor to account for the effect of seat endstops, 2) A novel learning rate gradient descent based neural network approximator algorithm coupled to an adaptive indirect type-2 fuzzy neural network (T2FNN) controller to converge the controller to the ideal parameters of the uncertain model. 3) The controller model takes into account the seat suspension nonlinearities due to the nonlinear asymmetric piecewise damper and the cubic hardening of the suspension spring. The proposed controller employs the principle of type-2 fuzzy systems with interval membership function and unknown specifications. The effectiveness of the closed-loop system is validated regarding the uncertainties compared to observer-based sliding mode controller (SMC) and a high-fidelity virtual lab MSC.ADAMS-Simulink platform to validate the results in practical scenarios.

Non-PDC Interval Type-2 Fuzzy Model Predictive Microclimate Control of a Greenhouse

Greenhouse climate control is known as a challenging task since it is characterized by the strong interaction among the involved variables, the time-varying uncertainties, and the external disturbances that severely affect its internal environment. In this paper, a discrete-time fuzzy model predictive control approach of a greenhouse is investigated. An interval type-2 (IT2) Takagi–Sugeno fuzzy model is employed to represent the nonlinear dynamics of the plant subject to parameter uncertainties, which are effectively captured by interval membership functions. To design the fuzzy model predictive controller, an optimization problem which minimizes a quadratic cost function respecting the input and state constraints is formulated and solved at each sampling instant in the prediction time horizon. By introducing a non-parallel distributed compensation (non-PDC) design concept and based on the non-quadratic Lyapunov function, less conservative conditions are developed in terms of linear matrix inequality to guarantee the stability analysis. Simulation results illustrate the effectiveness of the proposed control approach in leading to promote a comfortable greenhouse microclimate for the growth of the crops.

An interval risk assessment method and management of water inflow and inrush in course of karst tunnel excavation

An interval risk assessment method of water inflow and inrush in karst tunnel is proposed. Based on the method, both a conceptual and a computational form of the risk assessment model is introduced, and risk surroundings, construction factors and feedback information are analyzed. The conceptual model consists of three hierarchies and three stages. Thirteen factors were chosen as assessment indices and can be grouped into three categories: karst hydrogeological conditions and geological engineering conditions, construction factors and feedback information for management. An analytic hierarchy process and fuzzy mathematical theory were used to develop a computational model based on the interval number. The interval membership function and the weight of each factor were quantified and a relative superiority analysis of the interval matrix was performed. Moreover, a construction permission mechanism of water inflow and inrush in karst tunnels was established. In the present study, the three-stage risk assessment model and construction permission mechanism were successfully applied to the Qiyueshan tunnel. Thus, the interval risk assessment method may provide a practical reference for the risk assessment and management of karst tunnel construction and is worth exploring to control risk in other engineering construction projects.

Extended MABAC method based on divergence measures for multi-criteria assessment of programming language with interval-valued intuitionistic fuzzy sets

Programming languages (PLs) are symbolic instruction for writing programs, which have qualifications of evaluation of algorithm. The basic idea of programming languages (PLs) is to make it easier to write computer programs, now while that is true, what is often overlooked by language designers is that the other purpose, and maintaining its primary purpose, is to make it easier for people to read and understand programs. A programming language selection intrinsically is a multi-criteria decision-making (MCDM) problem. To handle the uncertainty of information in a MCDM problem, the theory of fuzzy sets is an effective tool. Interval-valued intuitionistic fuzzy sets (IVIFSs), which are characterized by the interval membership, non-membership and hesitancy functions have more flexibility to model the uncertainty of the MCDM problems than fuzzy sets (FSs). In this paper, a new integrated method based on multi-attributive border approximation area comparison (MABAC) method is proposed to handle MCDM problems with IVIFSs. This method is based on the IVIFS operators, some modifications in the classical MABAC method and a new procedure for calculation of the criteria weight. For calculation of criteria weight, we aggregate the subjective weights expressed by decision experts with the objective weights obtained from the proposed entropy and divergence measures method to obtain more realistic weights. Since the uncertainty is an inevitable characteristic of MCDM problems, the developed method can be a useful tool for decision making in an uncertain environment. To demonstrate the applicability of the developed method in the real-world MCDM problems, a programming language selection problem is taken. We perform a sensitivity analysis with different weights of criteria to show the stability of the proposed approach. This analysis shows that combining the subjective and objective weights can help to increase the stability of the proposed method with different weights of criteria. A comparison is also discussed between the results of the proposed and some existing methods for validating the proposed approach. This analysis shows that the proposed approach is efficient and well-consistent with the other methods.

A New Interval-Valued Intuitionistic Fuzzy Numbers: Ranking Methodology and Application

Ranking of interval-valued intuitionistic fuzzy (IVIF) numbers is a most popular and elegant work in the area of decision-making of several real-world problems. Some limited methods have been presented concerning the ranking of IVIF sets in literature. In the present paper, we generalize the intuitionistic fuzzy (IF) number to interval-valued intuitionistic fuzzy number by defining interval membership and nonmembership functions instead of fixed-valued function and hence it will present uncertain situation better than IF numbers. It may also be applied in data analysis, industrial management, artificial intelligence, forecasting, time series and so on. In this paper, ranking methodology of IVIF numbers is presented, for this first we define the value and ambiguity of IVIF numbers. Proposed ranking method also is compared with existing ranking methods. Further, IVIF numbers are used to capture fuzziness and hesitation in transportation problem (TP), and we propose a new method to find optimal solutions of TP with IVIF number parameters and finally, a numerical example is given to demonstrate the proposed method.

Type-1 and Type-2 Fuzzy Logic Control Algorithms for Semi-Active Seismic Vibration Control of the College Urban Bridge Using MR Dampers

In this study, the application of type-1 and type-2 fuzzy inference system (FIS) in semi-active seismic vibration control of the College Bridge using magnetorheological (MR) dampers is investigated. For this purpose, a detailed 3D finite element model of the bridge fitted with MR dampers is created in OpenSees. The command voltage of MR dampers is determined by employing both types of FISs in Matlab environment and making connection between both software. The results show the higher performance of the type-2 FIS for reducing the undesirable vibrations than that of type-1. This is because of the fact that the type-2 FIS considers interval membership functions for inputs in order to obtain the command voltage of MR dampers. Moreover, type-2 FIS effectively includes the effect of uncertainties and time delay. The results demonstrate that type-2 fuzzy controller is capable of reducing further the maximum displacement, base shear, and moment of the bridge by 24.6, 22.8, and 39.25%, respectively, compared to the type-1 fuzzy controller.

NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data

An approach to Nonlinear Output Error (NOE) modelling using Takagi–Sugeno (TS) fuzzy model for a class of nonlinear dynamic systems having variability in their outputs is presented. Furthermore, the approach is compared and graphically illustrated with other alternate approaches on the basis of interval data and interval membership functions. Assuming the identification method can be repeated offline a number of times under similar conditions, multiple input–output time series can be obtained from the underlying system. These time series are pre-processed using the techniques of statistics and probability theory to generate the envelopes of response (curves outlining the upper and lower extremes of response) at each time instant. Two types of envelopes are described in this research: the max–min envelopes and the envelopes based on the confidence intervals provided by extended Chebyshev's inequality. By incorporating interval data in fuzzy modelling and using the theory of symbolic interval-valued data, a TS fuzzy model with interval antecedent and consequent parameters is obtained. This algorithm provides a model for predicting the expected response as well as envelopes. In order to validate the presented model, a simulation case study is devised in this paper. Moreover, it is demonstrated on the real data obtained from an electro-mechanical throttle valve.

An optimization model design for energy systems planning and management under considering air pollution control in Tangshan City, China

In this study, an interval-parameter fuzzy programming mixed integer programming method (IFMIP) is designed for supporting the planning of energy systems management (ESM) and air pollution mitigation control under multiple uncertainties. The IFMIP-ESM model is based on an integration of interval-parameter programming (IPP), fuzzy programming (FP), and mixed-integer programming (MIP), which can reflect multiple uncertainties presented as both interval values and fuzzy distributions numbers. Moreover, it can successfully identify dynamics of capacity expansion schemes, reflect dual dynamics in terms of interval membership function, and analyze various emission-mitigation scenarios through incorporating energy and environmental policies. The designed model is applied to a case of energy systems management in Tangshan City, China, and the results indicate that reasonable solutions obtained from the model would be helpful for decision makers to effectively (a) adjust the allocation patterns of energy resources and transform the patterns of energy consumption and economic development, (b) facilitate the implement of air pollution control action plan, and (c) analysis dynamic interactions among system cost, energy-supply security, and environmental requirement.

Multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy sets

The main goal of green supply chain management (GSCM) is to reduce the negative environmental impacts in all activities and stages of a supply chain. Evaluation of suppliers in a supply chain according to environmental criteria can help us to achieve this goal of GSCM. Since this evaluation usually comprises some alternatives and some criteria, green supplier selection (GSS) could be considered as a multi-criteria decision-making (MCDM) problem. To handle the uncertainty of information in an MCDM problem, the theory of fuzzy sets is an effective tool. Interval type-2 fuzzy sets (IT2FSs), which are characterized by an interval membership function, is very flexible to model the uncertainty of the MCDM problems. In this study, a new integrated approach based on Weighted Aggregated Sum Product Assessment (WASPAS) method, is proposed to deal with multi-criteria group decision-making problems with IT2FSs. This approach is based on the operators of IT2FSs, some modifications in the classical WASPAS method and a new procedure for calculation of criteria weights. In the procedure of calculation of criteria weights, we combine the subjective weights expressed by decision-makers with objective weights resulted from an entropy method to obtain more realistic weights. To show the applicability of the proposed approach in the real-world MCDM problems, a green supplier selection problem is used. We perform a sensitivity analysis with different weights of criteria and different values of method's parameters to show the stability of the proposed approach. This analysis shows that combining the subjective and objective weights can help to increase the stability of the proposed approach with different weights of criteria. A comparison is also made between the results of the proposed approach and some existing methods for validating the proposed approach. This analysis shows that the proposed approach is efficient and well consistent with the other methods.

Application of Type-2 Membership Functions in Fuzzy Logic Systems

Background. Developing decision making models for problems, that are not easily formalized, and which operate the expert information, became possible by utilizing the capabilities of fuzzy sets theory and building fuzzy logic systems. Use of fuzzy sets theory techniques for knowledge formalization automatically results in the researcher having to select the type of fuzzy sets used for constructing membership functions, as well as the fuzzy model, that would fit the selected fuzzy set type. Therefore, the task of investigating the capabilities of type-2 membership functions in fuzzy logic systems is one of great interest. Objective. Expanding the capabilities of fuzzy logic systems by using type-2 membership functions. Methods. Research methods are directed towards utilizing interval and three-dimensional type-2 fuzzy sets in forecasting problems and medical diagnostics. The matter of investigating the capabilities of interval membership functions built on experimental data will be considered in two aspects: the first one – of how well interval membership functions reflect the uncertainties present in the source data, and the second one – of the advantages and disadvantages of an interval output of a fuzzy model with interval membership functions. Results. A research of interval type-2 membership functions in fuzzy logic systems was conducted in the areas of forecasting problem and medical diagnostics. Applicability of three-dimensional type-2 membership functions built from experimental data was shown. Conclusions. This paper shows the advantages of using interval membership functions in fuzzy logic systems, when developing fuzzy models using the multiple models principle. A research of three-dimensional type-2 membership functions’ applicability when modeling existing uncertainties is given. A technique for generating three-dimensional membership functions in fuzzy logic systems generated from experimental data is proposed.

Developing an MCDM method for robot selection with interval type-2 fuzzy sets

Industrial robots have been increasingly used by many manufacturing firms in different industries. The selection of robots for a particular application and manufacturing environment is a difficult task for decision makers. It has become more and more complicated due to increase in complexity, advanced features and facilities that are continuously being incorporated into the robots by different manufacturers. The decision maker needs to identify and select the best-suited robot in order to achieve the desired output with respect to many criteria. In the decision-making process, we usually confront with ambiguity and uncertainty for evaluating the criteria weights and alternatives of the problem. Interval type-2 fuzzy sets which are characterized by an interval membership function can provide us with more degrees of freedom to represent the uncertainty of the real-world problems. This paper presents a multi-criteria group decision-making approach for robot selection in the context of type-2 fuzzy sets. We propose a method based on Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method with interval type-2 fuzzy numbers to handle this problem. The best alternative (robot) is selected in the developed method according to both the ideal and the nadir solutions without defuzzification. An example with eight alternatives (robots) and seven commonly used criteria is used to illustrate the proposed method. We compare the results with some existing methods to show the validity of the extended method. Seven sets of criteria weights and the Spearman correlation coefficient are also utilized for analyzing the stability of the proposed method. It is observed that the obtained rankings of the proposed method are relatively consistent with the other methods and have good stability in different criteria weights. We focus on fuzzy multi-criteria group decision-making for robot selection.Interval type-2 fuzzy sets are considered in this research.An extended version of VIKOR method is developed for group decision-making.An example is utilized for showing the application of the proposed method in robot selection.We compare and analyze the results of the proposed method to represent the validity of it.

Granular Neural Networks With Evolutionary Interval Learning

To deal with different membership functions of the same linguistic term, a new interval reasoning method using new granular sets is proposed based on Yin Yang methodology. To make interval-valued granular reasoning efficiently and optimize interval membership functions based on training data effectively, a granular neural network (GNN) with a new high-speed evolutionary interval learning is designed. Simulation results in nonlinear function approximation and bioinformatics have shown that the GNN with the evolutionary interval learning is able to extract interval-valued granular rules effectively and efficiently from training data by using the new evolutionary interval learning algorithm.