Definition Of Membership Functions Research Articles

Fermatean fuzzy Z-analytic hierarchy process: An application to third party logistics providers

To decrease the impact of ambiguity, fuzzy sets are frequently used to decide if an element belongs to a set. Various fuzzy set extensions differ from others in how uncertainty is expressed in the definition of membership functions. This paper presents a novel extension named Fermatean fuzzy Z-numbers, introducing fuzzy restriction and fuzzy reliability functions with Fermatean fuzzy membership functions. By utilizing fuzzy Z-numbers to delineate membership degrees, considering both restriction and its reliability, in tandem with Fermatean membership functions offering an expanded value spectrum, the assessment of linguistic evaluations using Fermatean fuzzy Z-numbers can yield more pragmatic outcomes in decision evaluations amidst high uncertainty. Moreover, the interval-valued Fermatean fuzzy-Z-analytic hierarchy process is developed to explore the impact of proposed interval-valued Fermatean fuzzy-Z numbers on the multi-criteria decision-making process, demonstrated in the context of third-party logistics provider selection. Adding fuzzy reliability functions to the decision-making process improves how uncertainty is represented by considering variations in confidence levels associated with each decision, thus aiding in the clearer expression of decisions. Additionally, the proposed method's outcomes are compared with the outcomes of the current approaches.

Spherical fuzzy sets based integrated DEMATEL, ANP, VIKOR approach and its application for renewable energy selection in Turkey

Renewable energy continues occupying the global agenda for the low-carbon transition. Identifying suitable sources for renewable energy involves different criteria and alternatives, which can be defined as a Multi-Criteria Decision-Making (MCDM) problem. MCDM methods can be enriched with Group Decision Making (GDM) to merge individual judgments into group opinions, as well as with fuzzy sets to better simulate human opinions. To handle uncertainties of DMs’ preferences, a recently developed fuzzy set extension, the Spherical Fuzzy Sets (SFSs), is combined in this study with different MCDM tools to establish a decision support tool for the energy source selection problem. This article proposes a novel MCDM approach that integrates SF-DEMATEL (Spherical Fuzzy Decision-Making Trial and Evaluation Laboratory), SF-ANP (Spherical Fuzzy Analytic Network Process), and SF-VIKOR (Spherical Fuzzy Vlse Kriterijumska Optimizacija Kompromisno Resenje) algorithms in a GDM environment and presents its application in a case study for selecting renewable energy sources in Turkey. Comparing wind, geothermal, solar, hydropower, and biogas, the proposed method highlighted both wind and solar energy as a compromise solution as the most suitable options for Turkey. It has been found that compared to similar fuzzy sets, SFS can better reflect the hesitation degree to address the lack of knowledge and errors in the membership function definitions. The robustness and plausibility of these findings are demonstrated with sensitivity and comparative analyses, suggesting this MCDM method can also be applied to similar problems in other settings.

Intuitionistic Fuzzy Sets with Ordered Pairs and Their Usage in Multi-Attribute Decision Making: A Novel Intuitionistic Fuzzy TOPSIS Method with Ordered Pairs

Intuitionistic Fuzzy Sets with Ordered Pairs (IFSOP) are the recent extension of intuitionistic fuzzy sets by incorporating functional and dysfunctional points of view into the definition of membership functions. This paper extends the Technique of Order Preference Similarity to the Ideal Solution (TOPSIS) method to the Intuitionistic Fuzzy TOPSIS (IF TOPSIS) with ordered pairs method and applies it to a multi-criteria risk-based supplier selection problem under fuzziness. IF TOPSIS with ordered pairs involves finding a positive ideal solution and a negative ideal solution, and measuring the distance between each alternative and these solutions. The final ranking of the alternatives is obtained based on the proportion of distances between the positive and negative ideal solutions. By asking functional and dysfunctional questions in this ranking process, the developed IF TOPSIS with ordered pairs method incorporates the accuracy and consistency of expert judgments, enhancing the decision-making process. A sensitivity analysis is also presented in order to show the robustness of the rankings obtained by IF TOPSIS with ordered pairs.

A Novel Membership Function Definition for Fuzzy Classification

In this paper, a novel membership function is defined for fuzzy sets using a supervised learning approach. Firstly, the training dataset is separated using the previously defined polyhedral conic functions in a supervised learning approach. Then obtained polyhedral conic functions are used for defining a new membership function. After that, a new fuzzy classification algorithm is formed to classify fuzzy sets with a similar structure. The algorithm with all suggested methods is implemented on real-world datasets, and the performance values are compared with the state of art classification algorithms.

Fuzzy Logic-Based System for the Estimation of the Usability Level in User Tests

Starting from the challenge of obtaining the usability level in numerical and linguistic terms within a user test based on the three attributes that define usability, the development of a system based on fuzzy logic is proposed for estimating the level of output usability in a test with users based on the ISO 9241-11 standard. The attributes are effectiveness, efficiency, and satisfaction, which have different metrics and numerical scales. For the development of the system, five methodological stages were defined: characterization of the structure of a usability test, definition of membership functions for the inputs and outputs of the system, design of the inference rules that relate the inputs and outputs, design and implementation of the fuzzy system, and development of the case study. The proposed system was implemented using the FCL (Fuzzy Control Language) and the jFuzzyLogic API that takes as inputs the values calculated for the attributes of effectiveness, efficiency, and satisfaction, and obtains the usability level as output considering the membership functions of the inputs and outputs, as well as a set of inference rules defined by a set of experts. As a case study, the proposed fuzzy system was validated from the results obtained in a usability test with 5 users which was developed on the Sigma Electrónica website. From the results obtained in the case study, it could be concluded that the implemented system is adequate in terms of obtaining a level of usability in numerical and linguistic terms in conventional usability tests developed in a usability laboratory considering the attributes of ISO 9241-11.

Design and Implementation of a Fuzzy Logic Controller for Power Plant Temperature Monitoring and Control using Fuzzylite

Design and implementation of a fuzzy logic controller for power plant temperature monitoring and control used fuzzy lite software to simulate using triangular method and compared with bell shape membership function. Fuzzy logic technology was deployed; the motor temperature and RPM being used as crisp inputs to the fuzzy logic controller with appropriate membership function definitions. The fuzzy logic was designed and simulated using the Fuzzy Lite and Proteus software. It was implemented as firmware written in C++ programming language being executed on the PIC16F877A microcontroller. The results gotten from the simulations and implementation were in concordance as variations in motor temperature influenced motor speed. Triangular method used in this work was compared with Bell shape method used in previous work to ascertain its contribution to knowledge and also discovered that triangular method is more precise in its result.

Application of Fuzzy Logic in the Selection of a Suitable Sanitary Landfill Site for Solid Waste Disposal and Management in Coastal Area of Nigeria

Solid waste management has been identified as one of the most challenging environmental problems facing cities in most developing countries because of the risks to human health and the general environment. Different methods are available for solid waste disposal and they include; ocean dumping open burning and sanitary landfill. Of these available methods, sanitary landfill remains the most effective and hygienic method of solid waste disposal. Inspite of its effectiveness, the method of waste disposal using sanitary landfill is challenged by the availability of suitable land. To select a suitable site for landfill application numerous factors must be considered, ranging from; flood extent, stream, rivers, swampy areas, ground water, built-up areas, roads, slopes, airport, palaces and point of interests (POIs). In this study, fuzzy logic technique of data analysis was employed to analyze the various factors in order to select the most suitable site for landfill application in the study area. The basic steps involved in the application of fuzzy logic in the modeling and prediction of suitable site for landfill application are as follows; definition of input and output variables, conversion of crisp variables into fuzzy sets, definition of membership functions for each inputs and output variables, creation of fuzzy rules and simulation. Result of the fuzzy simulation revealed that; for river buffer distance of 0.9909km, ground water elevation of 7.13m, road buffer distance of 0.7727km, land slope of 2.673 0 and built-up area buffer distance of 1.37km, suitability ranking is 2 which mean “least suitable. For river buffer distance of 1.355km, ground water elevation of 9.537m, road buffer distance of 1.645km, land slope of 3.655 0 and built-up area buffer distance of 1.778km, suitability ranking is 3 which means “moderately suitable” and for river buffer distance of 2.5km, ground water elevation of 15m, road buffer distance of 2.5km, land slope of 6 0 and built-up area buffer distance of 2.5km, suitability ranking is 4 which means “highly suitable” Keywords: Sanitary landfill, Fuzzy logic, Membership functions, Fuzzy rules and Suitability rating DOI : 10.7176/JEES/10-1-07 Publication date: January 31 st 2020

A new interactive approach for solving fully fuzzy mixed integer linear programming

In this paper, a novel method to solve Fully Fuzzy Mixed Integer Linear Programming (FFMILP) problems is presented. Our method is based on the definition of membership function and a fuzzy interactive technique for solving the classical multiobjective programming. It is worthwhile to note that this is the first time that the fully fuzzy mixed integer linear programming problem is discussed and a solving method is presented. To illustrate the steps of the proposed method, some numerical examples are solved and the results are compared with other methods in the literature. Computational results present the application of the method.

Fuzzy Space Partitioning Based on Hyperplanes Defined by Eigenvectors for Takagi-Sugeno Fuzzy Model Identification

This article presents a novel method for fuzzy space partitioning and the identification of Takagi-Sugeno fuzzy models. The novelty is in its region-splitting mechanism and membership function definition, which is based on hyperplanes. The proposed algorithm introduces a concept of principal component analysis to define the hyperplanes that split the problem space and uses the distances to these hyperplanes as metrics instead of center-oriented clusters. In contrast with many other methods, the presented method delivers reproducible results and has an easy tuning procedure. The performance is illustrated with analytical examples, benchmark problems from the literature, and real-process data. The obtained results are very promising; however, as with most learning methods, the results depend on the data distribution and input variable selection.

INTERVAL-VALUED NEUTROSOPHIC AHP WITH POSSIBILITY DEGREE METHOD

The Analytic Hierarchy Process (AHP) is one of the most widely used methods in multi criteria decision making (MCDM) in many areas. The method has been extended with hesitant fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, and type -2 fuzzy sets etc. These extended methods can consider the vagueness in decision making problems through different definitions of membership functions. Each of them tries to increase the effectiveness of AHP under uncertainty. Decision makers can fully express their judgments through neutrosophic sets (NS) since NS are based on three independent parameters, truthiness (T), indeterminancy (I) and falsity (F), providing a distinction between a ‘relative truth’ and an ‘absolute truth’. In this paper, we employ the possibility degree method for ranking interval numbers in our neutrosophic AHP approach by utilizing NS’ representation power. Besides, we employ interval-valued NS since a larger domain for the definition of T, I, and F is provided. Pairwise comparison matrices can be filled in by using linguistic terms such as weakly more important, moderately more important or extremely important. Then, we obtain the relative importance degrees of criteria by using the possibility degree method. In order to show the effectiveness of our method, a MCDM application is given in energy planning. Comparative and sensitivity analyses are also presented in the paper.

A novel pythagorean fuzzy AHP and its application to landfill site selection problem

Multi-criteria decision-making (MCDM) methods are susceptible to the subjectivity of experts when especially they use linguistic terms for assessment. This subjectivity and vagueness in the evaluation process have been handled by the recent extensions of ordinary fuzzy sets such as type-2 fuzzy sets, hesitant fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets and neutrosophic sets. Pythagorean fuzzy sets are superior to the other extensions with a more flexible definition of membership function. A novel Pythagorean fuzzy AHP method has been developed for MCDM. The developed method has been applied to a landfill site selection problem for the city of Istanbul in Turkey. The proposed method has successfully evaluated the landfill location alternatives with respect to the considered criteria. The results are compared with ordinary fuzzy AHP, and it is revealed that the proposed method produces consistent and informative outcomes better representing the uncertainty of decision-making environment. Robustness of the decision given by the proposed method is ensured by conducting one-at-a-time sensitivity analysis.

A Combined Fuzzy Logic and Analytical Hierarchy Process Method for Optimal Selection and Locating of Pedestrian Crosswalks

One of the main challenges for transportation engineers is the consideration of pedestrian safety as the most vulnerable aspect of the transport system. In many countries around the world, a large number of accidents recorded by the police are composed of accidents involving pedestrians and vehicles, for example when pedestrians may be struck by passing vehicles when crossing the street. Careful consideration of the parameters that are involved in selecting the type and optimum location of pedestrian crosswalks results in a higher pedestrian safety coefficient and a reduced accident rate at these facilities. At the start of this study, these parameters that are important in specifying the optimum type and location of pedestrian crosswalks were determined. Then the data layers of these identified parameters were defined using the ArcGIS software. These layers can subsequently be used for determination of the optimal positioning of pedestrian crosswalks. To specify the boundary changes for each parameter, fuzzy membership functions were defined for each parameter using fuzzy logic. The Analytical Hierarchy Process method (AHP) was used in order to combine these layers of information after the fuzzy membership functions were defined. Expert Choice software was used to determine the final weight resultant of the professionals' poll that was conducted. A field study sample has been carried out to determine the optimal location of pedestrian crosswalks in the city of Tehran. The final output from the ArcGIS software shows the ideal locations and the appropriate type of pedestrian crosswalks in the field study sample. The results indicate that the use of fuzzy logic in definition of membership functions of location parameters, along with using AHP for determination of the weight of data layers built in ArcGIS, is a satisfactory combined method for specifying the location of pedestrian crosswalks.

Exact analytical inversion of interval type-2 TSK fuzzy logic systems with closed form inference methods

In this study, an exact analytical inversion procedure is presented for interval type-2 Takagi–Sugeno–Kang fuzzy logic systems (IT2 TSK FLSs) with closed form inference methods. In this inversion procedure, the inversion variable is needed to be defined by piecewise linear interval type-2 fuzzy sets (PWL IT2 FSs). This provides a way to define an IT2 TSK FLS in a form that is composed of several fuzzy subsystems with respect to the inversion variable. In the proposed procedure, the output definitions of fuzzy subsystems are derived in a unique parametric form in terms of the inversion variable by using parameters of linear definition of membership functions. By rearranging this definition with respect to the inversion variable; either a linear, quadratic or cubic equation is obtained depending on the types of the inference method and rule consequents of the IT2 TSK FLS. Since explicit analytical solutions are available for these types of equations, the exact inverse solution(s) of each fuzzy subsystem can easily be calculated. The inverse solution set of overall IT2 TSK FLS is obtained by composing the inverse solutions of all fuzzy subsystems. The exactness and feasibility of the proposed inversion procedure has been illustrated through simulation examples.

АЛГОРИТМ ФУНКЦИОНИРОВАНИЯ ИНФОРМАЦИОННО- КОММУНИКАЦИОННОГО МЕХАНИЗМА ПРЕДПРИЯТИЯ

The article deals with the organizational and economic mechanism of management of information and communications in the enterprise, which is a set of functions, tools, methods and management decisions aimed at achieving the set goals of the enterprise. The model of fuzzy logic which takes into account the uncertainty level of parameters during their evaluation forms the basis of this mechanism. The proposed algorithm of functioning of the information and communications management mechanism is a sequence of commands that enable one to make grounded management decisions. The initial stage is assessment of the level of development of the communications system of the enterprise, which consists of information, technical, managerial, organizational and social components. For this purpose, there are three coefficients for each component, which help to assess their current level of development. Each coefficient according to the fuzzy model undergoes a fuzzification stage, bringing to fuzziness, which includes a definition of membership functions for both incoming and outgoing variables. Due to the fact that the model is a two-stage, outgoing variables of the first stage (sufficiency of technical, managerial, organizational, informational and social components) become incoming at the second stage of simulation. The next step of the mechanism of information and communications management is development of a criterion base, which is a set of production rules, being the basics of the knowledge base, enabling to justify the acceptance of one or another managerial decision. For producing a value of the output variable the author uses the Mamdani rule. The realization of the presented algorithm is performed with the help of a MATLAB tool in the Fuzzy Logic Toolbox package. The sufficiency of development of each component of the communications system, which is the basis for determination of a general state of the entire communications system development, is checked at the first stage of the algorithm. The final step is defuzzification of the acquired value, which is the basis for formation of tools for improvement of the communications system. The functional algorithm of the organizational and economic mechanism of information and communications management has been used to identify the level of development of the communications system components of PJSC CB “Privat Bank”. The analysis helped to reveal the insufficient level of the communications system development of the aforesaid commercial bank, due to a low level of development of the social component. The management decisions aimed at improving the labour system, namely, improvement of the motivation system and implementation of feedback channels, have been made.

GIS-fuzzy logic approach for building indices: regional feasibility and natural potential of ranching in tropical wetland

The regional feasibility of ranching (RFR) index was obtained in order to evaluate the productive potential of farms in the Pantanal. Five indicators were selected by expert and employed for the developing of the index. One of the five indicators corresponded to the natural potential for livestock ranching (NPLR) index which was generated by GIS-fuzzy logic. Fuzzy inference process, involving definitions of membership functions, fuzzy set operations and inference rules was implemented and validated with the participation of primary stakeholders. Different scenarios were simulated in a batch, next validated and adjusted with the participation of stakeholders. Both procedures were performed by the use of the Webfuzzy software. The NPLR and RFR index values, calculated for the pilot ranch, corresponded to the expectations of both expert and stakeholders. Fuzzy logic combined with landscape metric seems to be suitable for the definition of the productive natural potential of ranches to produce livestock in the Pantanal. The indices can assess the regional feasibility of ranching, contributing to decision-making of stakeholders.

A fuzzy approach for drawing S-N curves in the finite life region

In this work a fuzzy approach has been developed in order to estimate the probability of fatigue failure. In particular, with the proposed method the S–N curves of a material in the finite life region can be drawn. The experimental data are represented in terms of fuzzy sets and are fitted using a fuzzy linear regression. Data scattering and uncertainty in the empirical failure model are reflected in the definition of membership functions. Several examples are shown to illustrate the procedure. Failure probabilities and fatigue curves obtained by the fuzzy method are similar to those obtained by traditional statistical analysis, based on normal distributions of strength with standard deviation that remains constant with different load levels. In particular, the results obtained indicate that the possibilities offered by fuzzy systems are also applicable for estimating the Wohler curve of a material under fatigue stresses. To evaluate its reliability, the proposed method is compared with the traditional one, with particular attention to the case in which a small amount of experimental data is available. The new fuzzy method is slightly less accurate than traditional statistical analysis to outline S–N curves in the finite life region. This is mainly due to the fact that the method is influenced by the nonuniformity of data dispersion at each level of stress.

Inferring descriptive and approximate fuzzy rules for credit scoring using evolutionary algorithms

Generating both accurate as well as explanatory classification rules is becoming increasingly important in a knowledge discovery context. In this paper, we investigate the power and usefulness of fuzzy classification rules for data mining purposes. We propose two evolutionary fuzzy rule learners: an evolution strategy that generates approximate fuzzy rules, whereby each rule has its own specific definition of membership functions, and a genetic algorithm that extracts descriptive fuzzy rules, where all fuzzy rules share a common, linguistically interpretable definition of membership functions in disjunctive normal form. The performance of the evolutionary fuzzy rule learners is compared with that of Nefclass, a neurofuzzy classifier, and a selection of other well-known classification algorithms on a number of publicly available data sets and two real life Benelux financial credit scoring data sets. It is shown that the genetic fuzzy classifiers compare favourably with the other classifiers in terms of classification accuracy. Furthermore, the approximate and descriptive fuzzy rules yield about the same classification accuracy across the different data sets.

A robust fuzzy logic approach to modelling algae biomass

Modelling of algal bloom is an ambitious and difficult topic due to the complexity of aquatic ecosystem, insufficient knowledge of the detailed processes and mechanism involved and shortage of high quality data. Owing to the ability to deal with imprecise, uncertain data or ambiguous relationships among data, fuzzy logic (FL) has proved to be a useful and practical method in algal bloom modelling. However, common to any FL modelling approach, the definition of membership functions and inference rules remains difficult. Although, generating rules directly from measurements and observations (rather than consulting the expert) has recently been explored, the procedures turn out to be quite cumbersome. In this paper, a robust FL approach is explored to derive a model directly from measurements, using expert knowledge as a reference only. Its strength lies in the capability to combine partial knowledge on processes with partially available data from observations. The approach was successfully tested in a case for the North Sea to model chlorophyll a (Chl-a) concentration. It had also been introduced to the Dutch pilot study of the European Commission project Harmful Algal Bloom Expert System, which involves 13 institutes and universities from nine EU countries. The objectives of the paper are to illustrate how the robust fuzzy model is developed.

A robust fuzzy logic approach to modelling algae biomass / Une approche robuste de logique floue pour modéliser la biomasse d'algues

Modelling of algal bloom is an ambitious and difficult topic due to the complexity of aquatic ecosystem, insufficient knowledge of the detailed processes and mechanism involved and shortage of high quality data. Owing to the ability to deal with imprecise, uncertain data or ambiguous relationships among data, fuzzy logic (FL) has proved to be a useful and practical method in algal bloom modelling. However, common to any FL modelling approach, the definition of membership functions and inference rules remains difficult. Although, generating rules directly from measurements and observations (rather than consulting the expert) has recently been explored, the procedures turn out to be quite cumbersome. In this paper, a robust FL approach is explored to derive a model directly from measurements, using expert knowledge as a reference only. Its strength lies in the capability to combine partial knowledge on processes with partially available data from observations. The approach was successfully tested in a case ...

Integration of data mining techniques and heuristic knowledge in fuzzy logic modelling of eutrophication in Taihu Lake

A fuzzy logic (FL) model was developed in this study to predict algal biomass concentration in the eutrophic Taihu Lake, China. Common to any FL model, definition of membership functions and induction of inference rules remain difficult. They conventionally rely on the use of “heuristic knowledge”, which usually seems to be not enough to fulfil practical requirement or even unavailable. In this fuzzy model, a method combining data mining techniques with heuristic knowledge is developed. It used principal component analysis (PCA) to identify the major abiotic driving factors and to reduce dimensionality. Self-organising feature map (SOFM) technique and empirical knowledge were applied jointly to define the membership functions and to induce inference rules. As indicated by the results, the fuzzy model successfully demonstrated the potentials of exploring “embedded information” by combining data mining techniques with heuristic knowledge. The developed method had also been introduced to the European Commission project Harmful Algal Bloom Expert System (HABES) which involves 13 institutes and universities from 9 EU countries. The objectives of this paper are to illustrate how these techniques are integrated and how the developed fuzzy model is applied to predict the algal biomass concentration in the eutrophic lake.