Open Access. Powered by Scholars. Published by Universities.®

Physical Sciences and Mathematics Commons

Open Access. Powered by Scholars. Published by Universities.®

Fuzzy logic

Discipline
Institution
Publication Year
Publication
Publication Type
File Type

Articles 31 - 60 of 105

Full-Text Articles in Physical Sciences and Mathematics

Design And Implementation Of A New Speed Planner For Semiautonomous Systems, Mustafa Demi̇r, Volkan Sezer Jan 2018

Design And Implementation Of A New Speed Planner For Semiautonomous Systems, Mustafa Demi̇r, Volkan Sezer

Turkish Journal of Electrical Engineering and Computer Sciences

In this paper, a new speed planning method is developed for semiautonomous systems by improving our previous fuzzy logic-based approach. In this proposed method, an extended risk factor is calculated on top of the classical risk factor, using environmental factors and the user's speed reference. We obtain safer speed values in critical scenarios with this new extended risk factor definition. Another improvement comes from the design of the semiautonomous architecture. Instead of the fully autonomous solution of the previous work, we calculate the final speed reference by combining the risk factor-based speed value and user's speed reference, which provides a …


Why Modeling Complex Dynamic Systems Using Fuzzy Cognitive Maps?, Peter P. Groumpos Oct 2016

Why Modeling Complex Dynamic Systems Using Fuzzy Cognitive Maps?, Peter P. Groumpos

UBT International Conference

The difficult problem of modeling Complex Dynamic Systems (CDS) is carefully reviewed. Main characteristics of CDS are considered and analyzed. Today’s mathematical models and approaches cannot provide satisfactory answers to the challenging problems of the society. The key problem of complex dynamic systems and control theory consists in the development of methods of qualitative analysis of the dynamics and behavior of such systems and in the construction of efficient control algorithms for their efficient operation. The purpose of control to bring the system to a point of its phase space which corresponds to maximal or minimal value of the chosen …


Cognitive Big Data Analytics And Persuasive Social Influence Diffusion, Eman Ahmed Ghanim Abukhousa May 2016

Cognitive Big Data Analytics And Persuasive Social Influence Diffusion, Eman Ahmed Ghanim Abukhousa

Dissertations

Current demands in local and global economies and the pursuit of competitiveness are calling for data-driven strategies. Data-driven solutions analyze trends, make predictions about future events, and prescribe what to do next in an actionable manner. However, cognitive and behavioral data are distinguished by their multiplicity and rapid changes to meet evolving and dynamic goals of individuals. This research work is concerned with the utility of analytical solutions to synthesize and influence cognitive and behavioral adoption. We propose a multidimensional data model to identify and extract cognitive indicators for analysis and persuasive interventions. The process starts by discovering behavioral features …


Implementation Of An Air Supply Unit Control Scheme For The Uc2av (Unmanned Circulation Control Aerial Vehicle), Cameron Rosen Jan 2016

Implementation Of An Air Supply Unit Control Scheme For The Uc2av (Unmanned Circulation Control Aerial Vehicle), Cameron Rosen

Electronic Theses and Dissertations

The expanded prevalence of Unmanned Aerial Vehicles (UAVs) in recent years has created many opportunities to research novel applications for their use, enabled by the reduced cost, mission flexibility, and reduced risk that small-scale unmanned platforms provide in comparison to larger aircraft. Despite the versatility of unmanned aviation, limitations on payload size and weight, fuel and power capacity, and takeoff and landing infrastructure can restrict UAV applications, and have created a need for lift augmenting technologies that can reduce the impact of these limitations. Circulation Control (CC) is an active flow technique that has been proven as a method for …


Pure Fuzzy Hall Effect Sensors For Permanent Magnet Synchronous Motor, İbrahi̇m Alişkan, Rüstem Yilmazel Jan 2016

Pure Fuzzy Hall Effect Sensors For Permanent Magnet Synchronous Motor, İbrahi̇m Alişkan, Rüstem Yilmazel

Turkish Journal of Electrical Engineering and Computer Sciences

An investigation about Hall effect sensors' efficiency is confirmed in permanent magnet synchronous motor (PMSM) drive systems. A fuzzy control algorithm is used as an artificial intelligence controller. Large scale and low slopes are used for creating membership functions and a sensitive controller is obtained. Speed is wanted to be taken under control and a minimum error value is aimed. PMSM drive systems are established using MATLAB-Simulink/SimPower. Simulations are realized with real-time parameters in discrete mode. A fuzzy logic controller is designed by using the MATLAB/Fuzzy Logic Toolbox. A normalization technique and high resolution output of the fuzzy logic controller …


Fuzzy Speed Estimation In The Case Of Sensorless Induction Machine Vector Control, Mohamed Bahloul, Mansour Souissi, Mohamed Chaabane, Larbi Chrifi Alaoui, Said Drid Jan 2016

Fuzzy Speed Estimation In The Case Of Sensorless Induction Machine Vector Control, Mohamed Bahloul, Mansour Souissi, Mohamed Chaabane, Larbi Chrifi Alaoui, Said Drid

Turkish Journal of Electrical Engineering and Computer Sciences

This paper deals with the problem of sensorless indirect field oriented control of an induction machine. Based on the fourth-order nonlinear model, we propose a novel structure of a full-order Takagi-Sugeno (TS) adaptive observer, through which we ensure the estimation of IM states and the rotor speed, which is considered as an immeasurable premise variable. This approach satisfies global stability for all operating conditions and operates under unknown applied load torque. In this work, a TS fuzzy model is presented to approximate a nonlinear IM model in the stationary rotating reference frame. A TS adaptive observer is then described. We …


A Comprehensive Control Strategy Of Railway Power Quality Compensator For Ac Traction Power Supply Systems, Hamed Jafari Kaleybar, Siamak Farshad Jan 2016

A Comprehensive Control Strategy Of Railway Power Quality Compensator For Ac Traction Power Supply Systems, Hamed Jafari Kaleybar, Siamak Farshad

Turkish Journal of Electrical Engineering and Computer Sciences

The development of electrical railway systems leads to critical power quality problems in the power grid. This paper discusses a kind of half-bridge-based railway power quality compensator system (HBRPQC) that can compensate negative sequence currents, harmonics, and reactive power simultaneously. In order to keep the HBRPQC performance efficient for the different kinds of transformers used in traction power supply substations, a new multifunctional control strategy that performs better than previous methods is proposed. Due to the fast dynamicity of traction loads, a recessive self-tuning PI controller based on fuzzy logic is adopted in the current control system. The output control …


Modeling And Control Of A 6-Control-Area Interconnected Power System To Protect The Network Frequency Applying Different Controllers, Ngockhoat Nguyen, Qi Huang, Thi-Mai-Phuong Dao Jan 2016

Modeling And Control Of A 6-Control-Area Interconnected Power System To Protect The Network Frequency Applying Different Controllers, Ngockhoat Nguyen, Qi Huang, Thi-Mai-Phuong Dao

Turkish Journal of Electrical Engineering and Computer Sciences

In this paper, we first analyze and build a model of the 6-control-area interconnected power system as the typical case study. Subsequently, different frequency controllers based on tie-line bias control strategy, namely integral, proportional--integral, proportional--integral--derivative, and PI-based fuzzy logic, will be investigated and applied to the power system model. The most important control performances, such as overshoots and settling times, are considered in this study to evaluate the stability of the power system and choose the most suitable controller for maintaining the network frequency. The simulation results have been achieved by using MATLAB/Simulink package version 2013 in this work. According …


Fuzzy Decision Tree-Based Inference System For Liver Disease Diagnosis, Himaja Sivaraju Jan 2016

Fuzzy Decision Tree-Based Inference System For Liver Disease Diagnosis, Himaja Sivaraju

Theses and Dissertations

Medical diagnosis can be challenging because of a number of factors. Uncertainty in the diagnosis process arises from inaccuracy in the measurement of patient attributes, missing attribute data and limitation in the medical expert’s ability to define cause and effect relationships when there are multiple interrelated variables. Given this situation, a decision support system, which can help doctors come up with a more reliable diagnosis, can have a lot of potential. Decision trees are used in data mining for classification and regression. They are simple to understand and interpret as they can be visualized. But, one of the disadvantages of …


Optimization Of Automatic Steering Control On A Vehiclewith A Steer-By-Wire System Using Particle Swarm Optimization, Fachrudin Hunaini, Imam Robandi, Nyoman Sutantra Jan 2016

Optimization Of Automatic Steering Control On A Vehiclewith A Steer-By-Wire System Using Particle Swarm Optimization, Fachrudin Hunaini, Imam Robandi, Nyoman Sutantra

Turkish Journal of Electrical Engineering and Computer Sciences

This paper presents a simulation of the automatic steering control system on a vehicle model using particle swarm optimization (PSO) to optimize the parameters of the control system. The control system involves fuzzy logic control (FLC) and proportional-integral-derivative (PID) control working in a cascade; the main control (FLC) is used to control lateral motion, and the secondary control (PID) is an enhancement to control the yaw motion in vehicle models representing 10 degrees of freedom of the vehicle dynamics system. Optimization by PSO is carried out simultaneously on both control systems. On FLC it is done by setting the width …


Fuzzy Logic-Based Guaranteed Lifetime Protocol For Real-Time Wireless Sensor Networks, Babar Shah, Farkhund Iqbal, Ali Abbas, Ki Il Kim Aug 2015

Fuzzy Logic-Based Guaranteed Lifetime Protocol For Real-Time Wireless Sensor Networks, Babar Shah, Farkhund Iqbal, Ali Abbas, Ki Il Kim

All Works

© 2015 by the authors; licensee MDPI, Basel, Switzerland. Few techniques for guaranteeing a network lifetime have been proposed despite its great impact on network management. Moreover, since the existing schemes are mostly dependent on the combination of disparate parameters, they do not provide additional services, such as real-time communications and balanced energy consumption among sensor nodes; thus, the adaptability problems remain unresolved among nodes in wireless sensor networks (WSNs). To solve these problems, we propose a novel fuzzy logic model to provide real-time communication in a guaranteed WSN lifetime. The proposed fuzzy logic controller accepts the input descriptors energy, …


Image Segmentation Using Fuzzy-Spatial Taxon Cut, Lauren Barghout May 2015

Image Segmentation Using Fuzzy-Spatial Taxon Cut, Lauren Barghout

MODVIS Workshop

Images convey multiple meanings that depend on the context in which the viewer perceptually organizes the scene. This presents a problem for automated image segmentation, because it adds uncertainty to the process of selecting which objects to include or not include within a segment. I’ll discuss the implementation of a fuzzy-logic-natural-vision-processing engine that solves this problem by assuming the scene architecture prior to processing. The scene architecture, a standardized natural-scene-perception-taxonomy comprised of a hierarchy of nested spatial-taxons. Spatial-taxons are regions (pixel-sets) that are figure-like, in that they are perceived as having a contour, are either `thing-like', or a `group of …


How Success In A Task Depends On The Skills Level: Two Uncertainty-Based Justifications Of A Semi-Heuristic Rasch Model, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich Mar 2015

How Success In A Task Depends On The Skills Level: Two Uncertainty-Based Justifications Of A Semi-Heuristic Rasch Model, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The more skills a student acquires, the more successful this student is with the corresponding tasks. Empirical data shows that the success in a task grows as a logistic function of skills; this dependence is known as the Rasch model. In this paper, we provide two uncertainty-based justifications for this model: the first justification provides a simple fuzzy-based intuitive explanation for this model, while the second -- more complex one -- explains the exact quantitative behavior of the corresponding dependence.


Development Of An Automated Method For Identification Of Wet And Dry Channel Segments Using Lidar Data And Fuzzy Logic Cluster Analysis, Chris Rowney Jan 2015

Development Of An Automated Method For Identification Of Wet And Dry Channel Segments Using Lidar Data And Fuzzy Logic Cluster Analysis, Chris Rowney

Electronic Theses and Dissertations

Research into the use of LiDAR data for purposes other than simple topographic elevation determination, such as urban land cover classification and the identification of forest biomass, has become prominent in recent years. In many cases, alternative analysis methodologies conducted using airborne LiDAR data are possible because the raw data collected during a survey can include information other than the classically used elevation and coordinate points, the X, Y, and Z of the plane. In particular, intensity return values for each point in a LiDAR grid have been found to provide a useful data set for wet and dry channel …


A New High-Performance Current-Mode Fuzzy Membership Function Circuit And Its Application, Turgay Temel Jan 2015

A New High-Performance Current-Mode Fuzzy Membership Function Circuit And Its Application, Turgay Temel

Turkish Journal of Electrical Engineering and Computer Sciences

A new current-mode fuzzy-membership function circuit is proposed. The circuit has a very compact and simple architecture based on the simultaneous use of winner- and loser-take-all topologies. The circuit has also the advantage of easily adjusting generated membership function characteristics such as center or mean and width with a straightforward application of respective input currents. It is shown that the new circuit outperforms previous counterparts in terms of speed, power consumption, layout area, and robustness to variations in design parameters and errors. As an application, a fuzzy-classifier is designed with a new membership circuit as a seven fuzzy level controller.


Energy Economy In Regulated And Market-Based Power System: Case Study In Serbia, Mileta Zarkovic, Ivan Skokljev Jan 2015

Energy Economy In Regulated And Market-Based Power System: Case Study In Serbia, Mileta Zarkovic, Ivan Skokljev

Turkish Journal of Electrical Engineering and Computer Sciences

This study presents a possible process of simulating power plant generation planning. The process combines expected overall industry costs, associated cost uncertainty, and expected CO$_{2}$ emissions for different generations, variations of future fossil fuel costs, carbon prices, plant investment costs, and demand, including price elasticity impacts. Uncertainty in the decision stems from the elasticity of prices of fuel and electricity. The aim of this paper is to apply fuzzy numbers to power generation planning and to use a Monte Carlo simulation to check. Simulations are demonstrated through a case study of an electricity industry with coal and lignite, combined cycle …


Computational Intelligence Based Complex Adaptive System-Of-Systems Architecture Evolution Strategy, Siddharth Agarwal Jan 2015

Computational Intelligence Based Complex Adaptive System-Of-Systems Architecture Evolution Strategy, Siddharth Agarwal

Doctoral Dissertations

The dynamic planning for a system-of-systems (SoS) is a challenging endeavor. Large scale organizations and operations constantly face challenges to incorporate new systems and upgrade existing systems over a period of time under threats, constrained budget and uncertainty. It is therefore necessary for the program managers to be able to look at the future scenarios and critically assess the impact of technology and stakeholder changes. Managers and engineers are always looking for options that signify affordable acquisition selections and lessen the cycle time for early acquisition and new technology addition. This research helps in analyzing sequential decisions in an evolving …


A Framework And System For A Multi-Model Decision Aid For Sustainable Farming Practices, Kasi Bharath Vegesana Jan 2015

A Framework And System For A Multi-Model Decision Aid For Sustainable Farming Practices, Kasi Bharath Vegesana

Computational Modeling & Simulation Engineering Theses & Dissertations

Decision support systems (DSS) for farmers address the need for modeling multiple processes and scenarios that affect farmer decision making. Existing DSS have various drawbacks that stop them from being deployed as decision support tools. This research proposes a multi-model simulation framework that can be used to analyze farm management practices at the crop level, individual farm level and at the community level to show the impact and alternatives for smallholder farming practices. A generic crop growth model is proposed, based on existing equations. We run sensitivity analysis on the model to identify important variables. The outputs from the crop …


Postmodern Fuzzy System Theory: A Deconstruction Approach Based On Kabbalah, Gabriel Burstein, Constantin Virgil Negoita, Menachem Kranz Nov 2014

Postmodern Fuzzy System Theory: A Deconstruction Approach Based On Kabbalah, Gabriel Burstein, Constantin Virgil Negoita, Menachem Kranz

Publications and Research

Modern general system theory proposed a holistic integrative approach based on input-state-output dynamics as opposed to the traditional reductionist detail based approach. Information complexity and uncertainty required a fuzzy system theory, based on fuzzy sets and fuzzy logic. While successful in dealing with analysis, synthesis and control of technical engineering systems, general system theory and fuzzy system theory could not fully deal with humanistic and human-like intelligent systems which combine technical engineering components with human or human-like components characterized by their cognitive, emotional/motivational and behavioral/action levels of operation. Such humanistic systems are essential in artificial intelligence, cognitive and behavioral science …


Community Detection And Influence Maximization In Online Social Networks, Kanna Gharib Al-Falahi May 2014

Community Detection And Influence Maximization In Online Social Networks, Kanna Gharib Al-Falahi

Dissertations

The detecting and clustering of data and users into communities on the social web are important and complex issues in order to develop smart marketing models in changing and evolving social ecosystems. These marketing models are created by individual decision to purchase a product and are influenced by friends and acquaintances. This leads to novel marketing models, which view users as members of online social network communities, rather than the traditional view of marketing to individuals. This thesis starts by examining models that detect communities in online social networks. Then an enhanced approach to detect community which clusters similar nodes …


An Information Security Control Assessment Methodology For Organizations, Angel Rafael Otero Jan 2014

An Information Security Control Assessment Methodology For Organizations, Angel Rafael Otero

CCE Theses and Dissertations

In an era where use and dependence of information systems is significantly high, the threat of incidents related to information security that could jeopardize the information held by organizations is more and more serious. Alarming facts within the literature point to inadequacies in information security practices, particularly the evaluation of information security controls in organizations. Research efforts have resulted in various methodologies developed to deal with the information security controls assessment problem. A closer look at these traditional methodologies highlights various weaknesses that can prevent an effective information security controls assessment in organizations. This dissertation develops a methodology that addresses …


New Operations Over Interval Valued Intuitionistic Hesitant Fuzzy Set, Florentin Smarandache, Said Broumi Jan 2014

New Operations Over Interval Valued Intuitionistic Hesitant Fuzzy Set, Florentin Smarandache, Said Broumi

Branch Mathematics and Statistics Faculty and Staff Publications

Hesitancy is the most common problem in decision making, for which hesitant fuzzy set can be considered as a useful tool allowing several possible degrees of membership of an element to a set. Recently, another suitable means were defined by Zhiming Zhang [1], called interval valued intuitionistic hesitant fuzzy sets, dealing with uncertainty and vagueness, and which is more powerful than the hesitant fuzzy sets. In this paper, four new operations are introduced on interval-valued intuitionistic hesitant fuzzy sets and several important properties are also studied.


A Control Scheme Employing An Adaptive Hysteresis Current Controller And An Uncomplicated Reference Current Generator For A Single-Phase Shunt Active Power Filter, Hüseyi̇n Doğan, Ramazan Akkaya Jan 2014

A Control Scheme Employing An Adaptive Hysteresis Current Controller And An Uncomplicated Reference Current Generator For A Single-Phase Shunt Active Power Filter, Hüseyi̇n Doğan, Ramazan Akkaya

Turkish Journal of Electrical Engineering and Computer Sciences

In recent years, active power filters (APFs) have become popular because of their excellent current harmonics and reactive power mitigation ability. The performance of the APF is directly related to the selected control strategy, which is mainly divided into 2, the reference current generation and current control. Most of the reference current generation methods are very complicated and require precise mathematical expressions. Moreover, being the most preferred current control method, conventional hysteresis controllers have a fixed bandwidth, which causes different total demand distortion values under different load conditions. In this study, a novel control scheme for a single-phase shunt APF …


A New Approach In The Evaluation Of Hospital Information Systems, Güney Gürsel, Neşe Zayi̇m, Kemal Hakan Gülkesen, Ali̇ Ari̇foğlu, Osman Saka Jan 2014

A New Approach In The Evaluation Of Hospital Information Systems, Güney Gürsel, Neşe Zayi̇m, Kemal Hakan Gülkesen, Ali̇ Ari̇foğlu, Osman Saka

Turkish Journal of Electrical Engineering and Computer Sciences

Hospital information system (HIS) evaluation frameworks have largely been discussed in the literature. However, existing frameworks lack one important aspect: to what extent user expectations of HISs are met. In this study, user expectation data are collected by means of the 'expectation questionnaire'. The internal consistencies of the answers to the questionnaire are measured by Cronbach's alpha coefficient. Fuzzy logic methodologies are used to evaluate the expectation variables in the proposed evaluation framework. The evaluation variables are not represented in the results equally; they are reflected by the weights assigned by the users. Our proposed framework provides the overall degree …


Fuzzy Logic Approach To Henry Factor For Distributed Feedback Laser Case, Abdullah Çavuşoğlu Jan 2014

Fuzzy Logic Approach To Henry Factor For Distributed Feedback Laser Case, Abdullah Çavuşoğlu

Turkish Journal of Electrical Engineering and Computer Sciences

In this study, a simple approach for intelligent modeling of the Henry factor (\alpha-alpha parameter, antiguiding factor, phase-amplitude coupling factor) or the so-called linewidth enhancement factor, which is an actual analysis and design parameter for semiconductor laser diodes and optical communication systems, is proposed based on the fuzzy logic (FL) phenomenon. The proposed FL-based model easily computes the Henry factor in terms of different wavelengths and injection current levels (i.e. the inputs of the model). The experimental data belong to a distributed feedback laser, obtained from amplified spontaneous emission spectra, which is among the techniques required for the characterization of …


Algebraic Structures On Fuzzy Unit Square And Neutrosophic Unit Square, Florentin Smarandache, W.B. Vasantha Kandasamy Jan 2014

Algebraic Structures On Fuzzy Unit Square And Neutrosophic Unit Square, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors build algebraic structures on fuzzy unit semi open square UF = {(a, b) | a, b  [0, 1)} and on the fuzzy neutrosophic unit semi open square UN = {a + bI | a, b  [0, 1)}. This study is new and we define, develop and describe several interesting and innovative theories about them. We cannot build ring on UN or UF. We have only pseudo rings of infinite order. We also build pseudo semirings using these semi open unit squares. We construct vector spaces, S-vector spaces and strong pseudo special vector space using …


Fuzzy Impedance And Force Control Of A Stewart Platform, Selçuk Ki̇zi̇r, Zafer Bi̇ngül Jan 2014

Fuzzy Impedance And Force Control Of A Stewart Platform, Selçuk Ki̇zi̇r, Zafer Bi̇ngül

Turkish Journal of Electrical Engineering and Computer Sciences

A new, simple, and effective control method is proposed for the position-based impedance and force control of a Stewart platform (SP). The control approach can be divided into 3 parts, namely position control in free space, impedance control in contact, and force control. An impedance filter is developed to achieve the desired behavior between the position and force. The gain of the filter is modified by a fuzzy logic proportional-integral-derivative controller. Kinematic and dynamic models of the SP are simulated in a MATLAB/Simulink environment. Several real-time experiments are conducted with the SP developed in our laboratory. The results of the …


Complete Description Of Idempotent Hedges In Fuzzy Logic, Jaime Nava Aug 2013

Complete Description Of Idempotent Hedges In Fuzzy Logic, Jaime Nava

Departmental Technical Reports (CS)

In describing expert knowledge, it is often important to properly take into account hedges} like "very", "somewhat", etc. In particular, fuzzy logic provides a consistent way of describing hedges. For some of the hedges, a repetition changes the meaning: e.g., "very very small" is smaller than "very small". However, other hedges -- like "somewhat" -- are idempotent, in the sense that repeating this hedge twice does not change the meaning. In this paper, we provide a complete description of such idempotent hedges.


Why Trapezoidal And Triangular Membership Functions Work So Well: Towards A Theoretical Explanation, Aditi Barua, Lalitha Snigdha Mudunuri, Olga Kosheleva Aug 2013

Why Trapezoidal And Triangular Membership Functions Work So Well: Towards A Theoretical Explanation, Aditi Barua, Lalitha Snigdha Mudunuri, Olga Kosheleva

Departmental Technical Reports (CS)

In fuzzy logic, an imprecise ("fuzzy") property is described by its membership function μ(x), i.e., by a function which describes, for each real number x, to what degree this real number satisfies the desired property. In principle, membership functions can be of different shape, but in practice, trapezoidal and triangular membership functions are most frequently used. In this paper, we provide an interval-based theoretical explanation for this empirical fact.


Computing With Words: Towards A New Tuple-Based Formalization, Olga Kosheleva, Vladik Kreinovich, Ariel Garcia, Felipe Jovel, Luis A. Torres Escobedo, Thavatchai Ngamsantivong Jul 2013

Computing With Words: Towards A New Tuple-Based Formalization, Olga Kosheleva, Vladik Kreinovich, Ariel Garcia, Felipe Jovel, Luis A. Torres Escobedo, Thavatchai Ngamsantivong

Departmental Technical Reports (CS)

An expert opinion describes his or her opinion about a quantity by using imprecise ("fuzzy") words from a natural language, such as "small", "medium", "large", etc. Each of these words provides a rather crude description of the corresponding quantity. A natural way to refine this description is to assign degrees to which the observed quantity fits each of the selected words. For example, an expert can say that the value is reasonable small, but to some extent it is medium. In this refined description, we represent each quantity by a tuple of the corresponding degrees.

Once we have such a …