Thesis (M.Sc. (Mathematics)) - Rhodes University, 2009.

Thesis (Ph. D.)--University of Hong Kong, 2005. / Title proper from title frame. Also available in printed format.

An important trend in fuzzy group theory in recent years has been the notion of classification of fuzzy subgroups using a suitable equivalence relation. In this dissertation, we have successfully used the natural equivalence relation defined by Murali and Makamba in [81] and a natural fuzzy isomorphism to classify fuzzy subgroups of some finite abelian p-groups of rank three of the form Zpn + Zp + Zp for any fixed prime integer p and any positive integer n. This was achieved through the usage of a suitable technique of enumerating distinct fuzzy subgroups and non-isomorphic fuzzy subgroups of G. We commence by giving a brief discussion on the theory of fuzzy sets and fuzzy subgroups from the perspective of group theory through to the theory of sets, leading us to establish a linkage among these theories. We have also shown in this dissertation that the converse of theorem 3.1 proposed by Das in [24] is incorrect by giving a counter example and restate the theorem. We have then reviewed and enriched the study conducted by Ngcibi in [94] by characterising the non-isomorphic fuzzy subgroups in that study. We have also developed a formula to compute the crisp subgroups of the under-studied group and provide its proof. Furthermore, we have compared the equivalence relation under which the classification problem is based with various versions of equivalence studied in the literature. We managed to use this counting technique to obtain explicit formulae for the number of maximal chains, distinct fuzzy subgroups, non-isomorphic maximal chains and non-isomorphic fuzzy subgroups of these groups and their proofs are provided.

Made available in DSpace on 2016-01-13T13:27:45Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-07-08. Added 1 bitstream(s) on 2016-01-13T13:31:37Z : No. of bitstreams: 1 000855574.pdf: 806510 bytes, checksum: 5165b6086fbf8e4db7aeff948269d0e3 (MD5) / Neste trabalho são apresentadas algumas de nições básicas da Teoria de Conjuntos Fuzzy e alguns exemplos teóricos que, na maioria dos textos, são deixados como exercí- cio para o leitor. Dessa forma, pretende-se que o texto que mais didático e completo, podendo ser aproveitado para cursos introdutórios da teoria. Algumas aplicações, voltadas para a área de Geologia, também são apresentadas. Destacamos a proposta de modelagem realizada utilizando-se informações geofísicas [1] e um sistema baseado em regras fuzzy para o estudo de locais na região de Rio Claro (São Paulo - Brasil) que são mais propícios para se perfurar poços com boas vazões / This work presents some basic de nitions of fuzzy set theory and some theoretical examples that in most of the texts are left as an exercise for the reader. Thus, it is intended that the text is more didactic, complete and can be used to introductory courses theory . Some applications, focused on the Geology eld, are also presented. Highlight the proposed modeling performed using geophysical [1] and a system based on fuzzy rules for study sites in Rio Claro region (São Paulo - Brazil) that are more conducive to drill wells with good flow

Orientador: Renata Zotin Gomes de Oliveira / Banca: Magda da Silva Peixoto / Banca: Laécio Carvalho de Barros / Resumo: Neste trabalho são apresentadas algumas de nições básicas da Teoria de Conjuntos Fuzzy e alguns exemplos teóricos que, na maioria dos textos, são deixados como exercí- cio para o leitor. Dessa forma, pretende-se que o texto que mais didático e completo, podendo ser aproveitado para cursos introdutórios da teoria. Algumas aplicações, voltadas para a área de Geologia, também são apresentadas. Destacamos a proposta de modelagem realizada utilizando-se informações geofísicas [1] e um sistema baseado em regras fuzzy para o estudo de locais na região de Rio Claro (São Paulo - Brasil) que são mais propícios para se perfurar poços com boas vazões / Abstract: This work presents some basic de nitions of fuzzy set theory and some theoretical examples that in most of the texts are left as an exercise for the reader. Thus, it is intended that the text is more didactic, complete and can be used to introductory courses theory . Some applications, focused on the Geology eld, are also presented. Highlight the proposed modeling performed using geophysical [1] and a system based on fuzzy rules for study sites in Rio Claro region (São Paulo - Brazil) that are more conducive to drill wells with good flow / Mestre

Magister Commercii - MCom / With the transition to a democratic society, one of the challenges to emerge was to provide economic freedoms for all – that is, freedom from isolation, freedom from powerlessness, freedom from hunger, and freedom from poverty. Fighting poverty has always been high on the agenda of the democratic government since the transition. Numerous measures and policies were instituted aimed at reducing poverty. Researchers initiated serious efforts to try and understand the nature and extent of poverty, through various studies being conducted. Many of these studies focused only on money-metric measures, despite the fact that poverty is a multidimensional concept. What has resulted are various poverty-measurement methods producing different results on the extent of poverty. In addition, certain aspects of these money-metric approaches have serious shortcomings. The fuzzy sets approach addresses many of these shortcomings, as it is a multidimensional approach. Few studies have used this approach to measure poverty in South Africa. This thesis plans to use this method to analyse poverty levels and trends in South Africa, focusing on multidimensional, non-money-metric poverty. Data from Census 1996, Census 2001 and Community Survey 2007 will be used for the study. From the results of the analysis it was established that there is some divergence in the findings of money-metric approaches and those of fuzzy sets. A key result to emerge is the difference in poverty trends over the period 1996 – 2007. Most studies reviewed in Chapter Three that used the money-metric approach showed that poverty trends were upward in the 1990s, before a downward trend took place in the 2000s. This took place irrespective of the survey data used. The non-money-metric poverty trends derived in this chapter, however, show a continuous downward trend over the period. The overall mean deprivation in South Africa has declined since 1996. For people residing in provinces such as Limpopo, the Eastern Cape and KwaZulu-Natal, they are more likely to be poor under the fuzzy sets approach. This may be an indication of inadequate service delivery and the extent to which recent government measures to address poverty have been successful or not. In terms of race, blacks still have the highest mean deprivation, but enjoyed the biggest decline of mean deprivation between 1996 and 2007. Finally, mean deprivation for female-headed households in South Africa was also higher than for male-headed households over the period.

For a fuzzy uniform space, the notion of a Cauchy prefilter, a precompact fuzzy set, a complete fuzzy set and a bounded fuzzy set are defined in such a way that these notions are good extensions of the corresponding notions for a uniform space. A theory of fuzzy uniform spaces is developed which generalises the theory of uniform spaces.

We start by discussing fuzzy sets and the algebra of fuzzy sets. We consider some properties of fuzzy modeling tools. This is followed by considering the Mamdani and Sugeno models for designing fuzzy controllers. Various methods for using sets of data for desining controllers are discussed. This is followed by a chapter illustrating the use of genetic algorithms in designing and optimizing fuzzy controllers.Finally we look at some previous applications of fuzzy control in telecommunication networks, and illustrate a simple application that was developed as part of the present work.

Many real-world problems could be formulated in a way to fit the necessary form for discrete optimization. Discrete optimization problems can be solved by numerous different techniques that have developed over time. Some of the techniques provide optimal solution(s) to the problem and some of them give "good enough" solution(s). The fundamental reason for developing techniques capable of producing solutions that are not necessarily optimal is the fact that many discrete optimization problems are NP-complete. Metaheuristic algorithms are a common name for a set of general-purpose techniques developed to provide solution(s) to the problems associated with discrete optimization. Mostly the techniques are based on natural metaphors. Discrete optimization could be applied to countless problems in transportation engineering. Recently, researchers started studying the behavior of social insects (ants) in an attempt to use the swarm intelligence concept to develop artificial systems with the ability to search a problem's solution space in a way that is similar to the foraging search by a colony of social insects. The development of artificial systems does not entail the complete imitation of natural systems, but explores them in search of ideas for modeling. This research is partially devoted to the development of a new system based on the foraging behavior of bee colonies — Bee System. The Bee System was tested through many instances of the Traveling Salesman Problem. Many transportation-engineering problems, besides being of combinatorial nature, are characterized by uncertainty. In order to address these problems, the second part of the research is devoted to development of the algorithms that combine the existing results in the area of swarm intelligence (The Ant System) and approximate reasoning. The proposed approach — Fuzzy Ant System is tested on the following two examples: Stochastic Vehicle Routing Problem and Schedule Synchronization in Public Transit. / Ph. D.

published_or_final_version / Economics and Finance / Master / Master of Economics