Eigen fuzzy sets of fuzzy relation can be used for the estimation of highest and lowest levels of involved variables when applying max-min composition on fuzzy relations. By the greatest eigen fuzzy sets (set which can be greater anymore) maximum membership degrees of any fuzzy set can be found, with the help of least eigen fuzzy set (set which can be less anymore) minimum membership degrees of any fuzzy sets can be found as well.The lowest and highest level, impact or e ffect of anything can be found by applying eigen fuzzy set theory. The implicational aspect of this research study is medical and customer satisfaction level measurement. By applying methods of eigen fuzzy set theory the e ffectiveness of medical cure and customer satisfaction can be found with high precision.

We present data structures and algorithms for sets and some generalizations thereof (fuzzy sets, multisets, and fuzzy multisets) available for R through the sets package. Fuzzy (multi-)sets are based on dynamically bound fuzzy logic families. Further extensions include user-definable iterators and matching functions. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Yes / Temporal fault trees (TFTs), an extension of classical Boolean fault trees, can model time-dependent failure behaviour of dynamic systems. The methodologies used for quantitative analysis of TFTs include algebraic solutions, Petri nets (PN), and Bayesian networks (BN). In these approaches, precise failure data of components are usually used to calculate the probability of the top event of a TFT. However, it can be problematic to obtain these precise data due to the imprecise and incomplete information about the components of a system. In this paper, we propose a framework that combines intuitionistic fuzzy set theory and expert elicitation to enable quantitative analysis of TFTs of dynamic systems with uncertain data. Experts’ opinions are taken into account to compute the failure probability of the basic events of the TFT as intuitionistic fuzzy numbers. Subsequently, for the algebraic approach, the intuitionistic fuzzy operators for the logic gates of TFT are defined to quantify the TFT. On the other hand, for the quantification of TFTs via PN and BN-based approaches, the intuitionistic fuzzy numbers are defuzzified to be used in these approaches. As a result, the framework can be used with all the currently available TFT analysis approaches. The effectiveness of the proposed framework is illustrated via application to a practical system and through a comparison of the results of each approach. / This work was supported in part by the Mobile IOT: Location Aware project (grant no. MMUE/180025) and Indoor Internet of Things (IOT) Tracking Algorithm Development based on Radio Signal Characterisation project (grant no. FRGS/1/2018/TK08/MMU/02/1). This research also received partial support from DEIS H2020 project (grant no. 732242).

Stastical pattern recognition is the most successful approach to automatic speech and speaker recognition (ASASR). Of all the statistical pattern recognition techniques, the hidden Markov model (HMM) is the most important. The Gaussian mixture model (GMM) and vector quantisation (VQ) are also effective techniques, especially for speaker recognition and in conjunction with HMMs. for speech recognition. However, the performance of these techniques degrades rapidly in the context of insufficient training data and in the presence of noise or distortion. Fuzzy approaches with their adjustable parameters can reduce such degradation. Fuzzy set theory is one of the most, successful approaches in pattern recognition, where, based on the idea of a fuzzy membership function, fuzzy C'-means (FCM) clustering and noise clustering (NC) are the most, important techniques. To establish fuzzy approaches to ASASR, the following basic problems are solved. First, a time-dependent fuzzy membership function is defined for the HMM. Second, a general distance is proposed to obtain a relationship between modelling and clustering techniques. Third, fuzzy entropy (FE) clustering is proposed to relate fuzzy models to statistical models. Finally, fuzzy membership functions are proposed as discriminant functions in decison making. The following models are proposed: 1) the FE-HMM. NC-FE-HMM. FE-GMM. NC-FEGMM. FE-VQ and NC-FE-VQ in the FE approach. 2) the FCM-HMM. NC-FCM-HMM. FCM-GMM and NC-FCM-GMM in the FCM approach, and 3) the hard HMM and GMM as the special models of both FE and FCM approaches. Finally, a fuzzy approach to speaker verification and a further extension using possibility theory are also proposed. The evaluation experiments performed on the TI46, ANDOSL and YOHO corpora showbetter results for all of the proposed techniques in comparison with the non-fuzzy baseline techniques.

Fuzzy relation equations are becoming extremely important in order to investigate the optimal solution of the inverse problem even though there is a restrictive condition for the availability of the solution of such inverse problems. We discussed the methods for finding the optimal (maximum and minimum) solution of inverse problem of fuzzy relation equation of the form $R \circ Q = T$ where for both cases R and Q are kept unknown interchangeably using different operators (e.g. alpha, sigma etc.). The aim of this study is to make an in-depth finding of best project among the host of projects, depending upon different factors (e.g. capital cost, risk management etc.) in the field of civil engineering. On the way to accomplish this aim, two linguistic variables are introduced to deal with the uncertainty factor which appears in civil engineering problems. Alpha-composition is used to compute the solution of fuzzy relation equation. Then the evaluation of the projects is orchestrated by defuzzifying the obtained results. The importance of adhering to such synopsis, in the field of civil engineering, is demonstrated by an example.

This research extends fuzzy methods to consider the fuzzy validation of fuzzy land cover data at the sub-pixel level. The study analyses the relationships between fuzzy memberships generated by field survey and those generated from the classification of remotely sensed data. In so doing it examines the variations in the relationship between observed and predicted fuzzy land cover classes. This research applies three land cover classification techniques: Fuzzy sets, Fuzzy c-means and Boolean classification, and develops three models to determine fuzzy land cover change. The first model is dependent on fuzzy object change. The second model depends on the sub-pixel change through a fuzzy change matrix, for both fuzzy sets and fuzzy c-means, to compute the fuzzy change, fuzzy loss and fuzzy gain. The third model is a Boolean change model which evaluates change on a pixel-by-pixel basis. The results show that using a fuzzy change analysis presents a subtle way of mapping a heterogeneous area with common mixed pixels. Furthermore, the results show that the fuzzy change matrix gives more detail and information about land cover change and is more appropriate than fuzzy object change because it deals with sub-pixel change. Finally the research has found that a fuzzy error matrix is more suitable than an error matrix for soft classification validation because it can compare the membership from the field with the classified image. From this research there arise some important points: • Fuzzy methodologies have the ability to define the uncertainties associated with describing the phenomenon itself and the ability to take into consideration the effect of mixed pixels. • This research compared fuzzy sets and fuzzy c-means, and found the fuzzy set is more suit-able than fuzzy c-means, because the latter suffers from some disadvantages, chiefly that the sum of membership values of a data point in all the clusters must be one, so the algorithm has difficulty in handling outlying points. • This research validates fuzzy classifications by determining the fuzzy memberships in the field and comparing them with the memberships derived from the classified image.

In order to store and process natural phenomena in Geographic Information Systems (GIS) it is necessary to model the real world to form computational representation. Since classical set theory is used in conventional GIS software systems to model uncertain real world, the natural variability in the environmental phenomena can not be modeled appropriately. Because, pervasive imprecision of the real world is unavoidably reduced to artificially precise spatial entities when the conventional crisp logic is used for modeling. An alternative approach is the fuzzy set theory, which provides a formal framework to represent and reason with uncertain information. In addition, linguistic variable concept in a fuzzy logic system is useful for communicating concepts and knowledge with human beings. In this thesis, a system to enhance commercial GIS software, namely ArcGIS, with fuzzy set theory is designed and implemented. The proposed system allows users to (a) incorporate human knowledge and experience in the form of linguistically defined variables into GIS-based spatial analyses, (b) handle impreiii cision in the decision-making processes, and (c) approximate complex ill-defined problems in decision-making processes and classification. The operation of the proposed system is presented through case studies, which demonstrate its application for classification and decision-making processes. This thesis shows how fuzzy logic approach may contribute to a better representation and reasoning with imprecise concepts, which are inherent characteristics of geographic data stored and processed in GIS.

Thesis (Ph. D.)--University of New South Wales, 1998. / Also available online.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Na tentativa de facilitar o diagnÃstico dos diversos fatores que afetam a qualidade da Ãgua e antever possÃveis impactos futuros sobre o meio ambiente , sÃo adotadas aÃÃes que racionalize m o uso da Ãgua a partir da otimizaÃÃo de processos naturais ou tecnolÃgicos. A modelagem matemÃtica à um exemplo disso e, em conjunto com a Teoria Fuzzy , que permite fazer a anÃlise dos resultados sem necessidade de significativos bancos de dados, pode - se estabelecer o risco como indicador de contaminaÃÃo das Ãguas de rios, sendo de valor prÃtico na tomada de decisÃo e concessÃo de outorga de lanÃamentos. Neste estudo, foi desenvolvido um modelo matemÃtico aplicado Ãs equaÃÃes completas de Streeter - Phelps utilizando a Teoria dos nÃmeros Fuzzy , a fim de analisar o risco de contaminaÃÃo de um curso d'Ãgua que recebe agentes poluentes de mÃltiplas fontes de lanÃamento. Pelas simulaÃÃes do modelo, foram analisados diferentes cenÃrios, verificando a influÃncia d os seus parÃmetros, bem como o lanÃamento de fontes poluidoras pontuais e difusas, nos percentuais de risco. De acordo com os resultados, observou - se que a quantidade de carga lanÃada tem influÃncia no tempo de diluiÃÃo desta massa no sistema, de forma que , para maiores valores de lanÃamento, o tempo de diluiÃÃo à menor, favorecendo os processos de decaimento e formaÃÃo da camada bentÃnica; em relaÃÃo Ãs reaÃÃes fÃsicas, quÃmicas e biolÃgicas, verifica - se que os processos de sedimentaÃÃo, fotossÃntese e res piraÃÃo, para os dados mÃdios encontrados em literatura, tem pequena influÃncia no comportamento das curvas de concentraÃÃo de OD e curvas de risco, enquanto que o processo de nitrificaÃÃo tem forte influÃncia; jà a temperatura desempenha um significativo papel no comportamento do OD, onde, para valores maiores, maior serà o dÃficit OD e, em consequÃncia, aumento dos percentuais de risco. Por fim, o modelo desenvolvido como proposta de facilitar a tomada de decisÃo no controle de lanÃamento de efluentes em rios mostrou - se uma alternativa viÃvel e de valor prÃtico de anÃlise, jà que os objetivos foram alcanÃados / In an attempt to facilitate the diagnosis of the various factors that affect water quality and predict possible future impacts on the environment, actions to rationalize the use of water from the optimization of natural and technological processes are adopted. Mathematical modeling is one example and, together with Fuzzy Theory, which allows the analysis of the results without the need for significant databases, one can establish the risk as an indicator of contamination of rivers, and of practical value in decision making and allocation of grant releases. In this study, the full Streeter-Phelps equations, using the Fuzzy set Theory, was applied, in order to analyze the risk of contamination of a watercourse that receives multiple sources release pollutants. Through the model simulations, different scenarios were analyzed, and the influence of its parameters as well as the launch point and nonpoint pollution sources, in the calculation of the risk. According to the results, it was observed that the amount of discharge released influences the time of the mass dilution in the system, so that for higher values of launch, the dilution time is less favoring the formation and decay processes of benthic layer; regarding the physical, chemical and biological reactions, it appears that sedimentation processes, photosynthesis and respiration, concerning with the average data found in literature, have little influence on the behavior of the curves of DO concentration curves and risk, while the nitrification process has a strong influence; with respect to the temperature, the results showed that it plays a significant role in the behavior of DO, where, for larger values of it, the higher the DO deficit and, consequently, increase in the risk. Finally, the model developed as a proposal to facilitate the decision making in the control of discharge of effluents into rivers proved to be a viable and practical analytical alternative way, since the goals were achieved.

Doctor of Philosophy / Department of Industrial and Manufacturing Systems Engineering / David Ben-Arieh / Group decision-making is an essential activity is many domains such as financial, engineering, and medical fields. Group decision-making basically solicits opinions from experts and combines these judgments into a coherent group decision. Experts typically express their opinion in many different formats belonging to two categories: quantitative evaluations and qualitative ones. Many times experts cannot express judgment in accurate numerical terms and use linguistic labels or fuzzy preferences. The use of linguistic labels makes expert judgment more reliable and informative for decisionmaking. In this research, a new linguistic label fusion operator has been developed. The operator helps mapping one set of linguistic labels into another. This gives decision makers more freedom to choose their own linguistic preference labels with different granularities and/or associated membership functions. Three new consensus measure methods have been developed for group decision making problem in this research. One is a Markov chain based consensus measure method, the other is order based, and the last one is a similarity based consensus measure approach. Also, in this research, the author extended the concept of Ordered Weighted Average (OWA) into a fuzzy linguistic OWA (FLOWA). This aggregation operator is more detailed and includes more information about the aggregate than existing direct methods. After measuring the current consensus, we provide a method for experts to modify their evaluations to improve the consensus level. A cost based analysis gives the least cost suggestion for this modification, and generates a least cost of group consensus. In addition, in this research I developed an optimization method to maximize two types of consensus under a budget constraint. Finally considering utilization of the consensus provides a practical recommendation to the desired level of consensus, considering its cost benefits.