Systems that utilise type-2 fuzzy sets to handle uncertainty have not been implemented in real world applications unlike the astonishing number of applications involving standard fuzzy sets. The main reason behind this is the complex mathematical nature of type-2 fuzzy sets which is the source of two major problems. On one hand, it is difficult to mathematically manipulate type-2 fuzzy sets, and on the other, the computational cost of processing and performing operations using these sets is very high. Most of the current research carried out on type-2 fuzzy logic concentrates on finding mathematical means to overcome these obstacles. One way of accomplishing the first task is to develop a meaningful mathematical representation of type-2 fuzzy sets that allows functions and operations to be extended from well known mathematical forms to type-2 fuzzy sets. To this end, this thesis presents a novel alpha-cut representation theorem to be this meaningful mathematical representation. It is the decomposition of a type-2 fuzzy set in to a number of classical sets. The alpha-cut representation theorem is the main contribution of this thesis. This dissertation also presents a methodology to allow functions and operations to be extended directly from classical sets to type-2 fuzzy sets. A novel alpha-cut extension principle is presented in this thesis and used to define uncertainty measures and arithmetic operations for type-2 fuzzy sets. Throughout this investigation, a plethora of concepts and definitions have been developed for the first time in order to make the manipulation of type-2 fuzzy sets a simple and straight forward task. Worked examples are used to demonstrate the usefulness of these theorems and methods. Finally, the crisp alpha-cuts of this fundamental decomposition theorem are by definition independent of each other. This dissertation shows that operations on type-2 fuzzy sets using the alpha-cut extension principle can be processed in parallel. This feature is found to be extremely powerful, especially if performing computation on the massively parallel graphical processing units. This thesis explores this capability and shows through different experiments the achievement of significant reduction in processing time.

<p>Predmet istraživanja ovog rada su mere neodređenosti, posebno mera<br />kredibliteta, kao i mogućnost njihove primene u aktuarstvu. U cilju<br />generalizacije teorije kredibiliteta, uvedena je nova mera, nazvana mera c-<br />kredibiliteta. Mera c-kredibiliteta na X je skupovna funkcija takva da su<br />zadovoljene osobine normalnosti, monotonosti, samodualnosti i maksimalnosti.<br />Za nju su dokazane neke osobine kao što su npr. subaditivnost i<br />poluneprekidnost. Nadalje, definisan je integral zasnovan na meri c-<br />kredibiliteta, a navedena su i dokazana određena svojstva. Nova mera je uvedena<br />i u fazi okruženju kao agregirana vrednost mera mogućnosti i neophodnosti.</p> / <p>This thesis studies uncertainty measures, especially credibility measure, as well as<br />the possibility of their application in actuaries. In order to generalize credibility theory,<br />a new fuzzy measure is proposed, called c &minus; credibility measure. C &minus; credibility<br />measure on X is a set function that satisfies normality, monotonicity, self-duality and<br />maximality. Certain properties of the c&minus;credibility measure are proved, such as, for<br />example, subadditivity and semicontinuity. Furthermore, an integral based on this<br />measure is defined, in analogy to the existing integrals, and its properties are proved.<br />Then, the credibility measure in a fuzzy environment is introduced as the aggregate<br />value of the possibility and necessity measures.</p>

Continuous development has been occurring in the area of decision support systems. Modern systems focus on applying decision models that can provide intelligent support to the decision maker. These systems focus on modelling the human reasoning process in situations requiring decision. This task may be achieved by using an appropriate decision model. Multicriteria decision making (MCDM) is a common decision making approach. This research investigates and seeks a way to resolve various issues associated with the application of this model. MCDM is a formal and systematic decision making approach that evaluates a given set of alternatives against a given set of criteria. The global evaluation of alternatives is determined through the process of aggregation. It is well established that the aggregation process should consider the importance of criteria while determining the overall worth of an alternative. The importance of individual criteria and of sub-sets of the criteria affects the global evaluation. The aggregation also needs to consider the importance of the sub-set of criteria. Most decision problems involve dependent criteria and the interaction between the criteria needs to be modelled. Traditional aggregation approaches, such as weighted average, do not model the interaction between the criteria. Non-additive measures such as fuzzy measures model the interaction between the criteria. However, determination of non-additive measures in a practical application is problematic. Various approaches have been proposed to resolve the difficulty in acquisition of fuzzy measures. These approaches mainly propose use of past precedents. This research extends this notion and proposes an approach based on similarity-based reasoning. Solutions to the past problems can be used to solve the new decision problems. This is the central idea behind the proposed methodology. The methodology itself applies the theory of reasoning by analogy for solving MCDM problems. This methodology uses a repository of cases of past decision problems. This case base is used to determine the fuzzy measures for the new decision problem. This work also analyses various similarity measures. The illustration of the proposed methodology in a case-based decision support system shows that interactive models are suitable tools for determining fuzzy measures in a given decision problem. This research makes an important contribution by proposing a similarity-based approach for acquisition of fuzzy measures.

With growing demands for reliability, availability, safety and cost efficiency in modern machinery, accurate fault diagnosis is becoming of paramount importance so that potential failures can be better managed. Although various methods have been applied to machinery condition monitoring and fault diagnosis, the diagnostic accuracy that can be attained is far from satisfactory. As most machinery faults lead to increases in vibration levels, vibration monitoring has become one of the most basic and widely used methods to detect machinery faults. However, current vibration monitoring methods largely depend on signal processing techniques. This study is based on the recognition that a multi-parameter data fusion approach to diagnostics can produce more accurate results. Fuzzy measures and fuzzy integral data fusion theory can represent the importance of each criterion and express certain interactions among them. This research developed a novel, systematic and effective fuzzy measure and fuzzy integral data fusion approach for machinery fault diagnosis, which comprises feature set selection schema, feature level data fusion schema and decision level data fusion schema for machinery fault diagnosis. Different feature selection and fault diagnostic models were derived from these schemas. Two fuzzy measures and two fuzzy integrals were employed: the 2-additive fuzzy measure, the fuzzy measure, the Choquet fuzzy integral and the Sugeno fuzzy integral respectively. The models were validated using rolling element bearing and electrical motor experiments. Different features extracted from vibration signals were used to validate the rolling element bearing feature set selection and fault diagnostic models, while features obtained from both vibration and current signals were employed to assess electrical motor fault diagnostic models. The results show that the proposed schemas and models perform very well in selecting feature set and can improve accuracy in diagnosing both the rolling element bearing and electrical motor faults.

En ésta tesis se estudian 2 aspectos de la toma decisiones. Primero, se investiga la forma en que las personas categorizan atributos numéricos. Se presenta y se prueba experimentalmente un modelo del proceso mental que usan las personas para trasformar una cantidad en una categoría verbal. Bajo ciertas condiciones situacionales, el modelo es capaz de predecir conceptualizaciones verbales. Segundo, se exploran las interconexiones entre la información cognitiva y emocional durante la decisión. Se elaboran y se prueban experimentalmente cuatro modelos de la forma en que se combina la información cognitiva y emocional durante el proceso de elección, para determinar el valor de una alternativa. Los modelos muestran una alta capacidad de predicción. Esta varía en función de (1) la interacción de la información verbal y numérica con la capacidad cognitiva situacional del individuo y (2) la correlación entre los juicios cognitivos y las reacciones emocionales. / The present dissertation investigates two aspects of decision making: First, I study the way in which people understand and categorize numerical attributes of products. I develop and experimentally test a model of the mental process people use to transform a quantitative attribute into a verbal category. Under certain environmental conditions, the model is able to predict the verbal conceptualization of people. Second, I explore the interconnections of cognitive and emotional information during the process of decision making. I propose and experimentally test four different models of the way cognitive and affective information is combined during the decision making process in order to determine the value of an alternative. The models display a high predictive power. Their performance is influenced by (1) the interaction of verbal and numerical information with the situational cognitive capacities of the individual and (2) by the correlation of cognitive judgments and affective reactions.

The design of hydropower projects requires a comprehensive planning process in order to achieve the objective to maximise exploitation of the existing hydropower potential as well as future revenues of the plant. For this purpose and to satisfy approval requirements for a complex hydropower development, it is imperative at planning stage, that the conceptual development contemplates a wide range of influencing design factors and ensures appropriate consideration of all related aspects. Since the majority of technical and economical parameters that are required for detailed and final design cannot be precisely determined at early planning stages, crucial design parameters such as design discharge and hydraulic head have to be examined through an extensive optimisation process. One disadvantage inherent to commonly used deterministic analysis is the lack of objectivity for the selection of input parameters. Moreover, it cannot be ensured that the entire existing parameter ranges and all possible parameter combinations are covered. Probabilistic methods utilise discrete probability distributions or parameter input ranges to cover the entire range of uncertainties resulting from an information deficit during the planning phase and integrate them into the optimisation by means of an alternative calculation method. The investigated method assists with the mathematical assessment and integration of uncertainties into the rational economic appraisal of complex infrastructure projects. The assessment includes an exemplary verification to what extent the Random Set Theory can be utilised for the determination of input parameters that are relevant for the optimisation of hydropower projects and evaluates possible improvements with respect to accuracy and suitability of the calculated results. / Die Auslegung von Wasserkraftanlagen stellt einen komplexen Planungsablauf dar, mit dem Ziel das vorhandene Wasserkraftpotential möglichst vollständig zu nutzen und künftige, wirtschaftliche Erträge der Kraftanlage zu maximieren. Um dies zu erreichen und gleichzeitig die Genehmigungsfähigkeit eines komplexen Wasserkraftprojektes zu gewährleisten, besteht hierbei die zwingende Notwendigkeit eine Vielzahl für die Konzepterstellung relevanter Einflussfaktoren zu erfassen und in der Projektplanungsphase hinreichend zu berücksichtigen. In frühen Planungsstadien kann ein Großteil der für die Detailplanung entscheidenden, technischen und wirtschaftlichen Parameter meist nicht exakt bestimmt werden, wodurch maßgebende Designparameter der Wasserkraftanlage, wie Durchfluss und Fallhöhe, einen umfangreichen Optimierungsprozess durchlaufen müssen. Ein Nachteil gebräuchlicher, deterministischer Berechnungsansätze besteht in der zumeist unzureichenden Objektivität bei der Bestimmung der Eingangsparameter, sowie der Tatsache, dass die Erfassung der Parameter in ihrer gesamten Streubreite und sämtlichen, maßgeblichen Parameterkombinationen nicht sichergestellt werden kann. Probabilistische Verfahren verwenden Eingangsparameter in ihrer statistischen Verteilung bzw. in Form von Bandbreiten, mit dem Ziel, Unsicherheiten, die sich aus dem in der Planungsphase unausweichlichen Informationsdefizit ergeben, durch Anwendung einer alternativen Berechnungsmethode mathematisch zu erfassen und in die Berechnung einzubeziehen. Die untersuchte Vorgehensweise trägt dazu bei, aus einem Informationsdefizit resultierende Unschärfen bei der wirtschaftlichen Beurteilung komplexer Infrastrukturprojekte objektiv bzw. mathematisch zu erfassen und in den Planungsprozess einzubeziehen. Es erfolgt eine Beurteilung und beispielhafte Überprüfung, inwiefern die Random Set Methode bei Bestimmung der für den Optimierungsprozess von Wasserkraftanlagen relevanten Eingangsgrößen Anwendung finden kann und in wieweit sich hieraus Verbesserungen hinsichtlich Genauigkeit und Aussagekraft der Berechnungsergebnisse ergeben.

The design of hydropower projects requires a comprehensive planning process in order to achieve the objective to maximise exploitation of the existing hydropower potential as well as future revenues of the plant. For this purpose and to satisfy approval requirements for a complex hydropower development, it is imperative at planning stage, that the conceptual development contemplates a wide range of influencing design factors and ensures appropriate consideration of all related aspects. Since the majority of technical and economical parameters that are required for detailed and final design cannot be precisely determined at early planning stages, crucial design parameters such as design discharge and hydraulic head have to be examined through an extensive optimisation process. One disadvantage inherent to commonly used deterministic analysis is the lack of objectivity for the selection of input parameters. Moreover, it cannot be ensured that the entire existing parameter ranges and all possible parameter combinations are covered. Probabilistic methods utilise discrete probability distributions or parameter input ranges to cover the entire range of uncertainties resulting from an information deficit during the planning phase and integrate them into the optimisation by means of an alternative calculation method. The investigated method assists with the mathematical assessment and integration of uncertainties into the rational economic appraisal of complex infrastructure projects. The assessment includes an exemplary verification to what extent the Random Set Theory can be utilised for the determination of input parameters that are relevant for the optimisation of hydropower projects and evaluates possible improvements with respect to accuracy and suitability of the calculated results. / Die Auslegung von Wasserkraftanlagen stellt einen komplexen Planungsablauf dar, mit dem Ziel das vorhandene Wasserkraftpotential möglichst vollständig zu nutzen und künftige, wirtschaftliche Erträge der Kraftanlage zu maximieren. Um dies zu erreichen und gleichzeitig die Genehmigungsfähigkeit eines komplexen Wasserkraftprojektes zu gewährleisten, besteht hierbei die zwingende Notwendigkeit eine Vielzahl für die Konzepterstellung relevanter Einflussfaktoren zu erfassen und in der Projektplanungsphase hinreichend zu berücksichtigen. In frühen Planungsstadien kann ein Großteil der für die Detailplanung entscheidenden, technischen und wirtschaftlichen Parameter meist nicht exakt bestimmt werden, wodurch maßgebende Designparameter der Wasserkraftanlage, wie Durchfluss und Fallhöhe, einen umfangreichen Optimierungsprozess durchlaufen müssen. Ein Nachteil gebräuchlicher, deterministischer Berechnungsansätze besteht in der zumeist unzureichenden Objektivität bei der Bestimmung der Eingangsparameter, sowie der Tatsache, dass die Erfassung der Parameter in ihrer gesamten Streubreite und sämtlichen, maßgeblichen Parameterkombinationen nicht sichergestellt werden kann. Probabilistische Verfahren verwenden Eingangsparameter in ihrer statistischen Verteilung bzw. in Form von Bandbreiten, mit dem Ziel, Unsicherheiten, die sich aus dem in der Planungsphase unausweichlichen Informationsdefizit ergeben, durch Anwendung einer alternativen Berechnungsmethode mathematisch zu erfassen und in die Berechnung einzubeziehen. Die untersuchte Vorgehensweise trägt dazu bei, aus einem Informationsdefizit resultierende Unschärfen bei der wirtschaftlichen Beurteilung komplexer Infrastrukturprojekte objektiv bzw. mathematisch zu erfassen und in den Planungsprozess einzubeziehen. Es erfolgt eine Beurteilung und beispielhafte Überprüfung, inwiefern die Random Set Methode bei Bestimmung der für den Optimierungsprozess von Wasserkraftanlagen relevanten Eingangsgrößen Anwendung finden kann und in wieweit sich hieraus Verbesserungen hinsichtlich Genauigkeit und Aussagekraft der Berechnungsergebnisse ergeben.