Thesis (M.ED.) --University of Limpopo, 2011 / A deep level approach to learning leads to quality learning outcomes. Teachers should use appropriate teaching strategies to encourage learners to use deep level approaches to learning. The Conceptual Change Model (CCM) approach is one such strategy for the teaching of science concepts. Deep level approaches are a necessity when dealing with a difficult science concept like photosynthesis. The purpose of this study was to investigate Grade 10 learners’ achievements in photosynthesis using the CCM approach in order to minimize misconceptions and develop a broader and deeper understanding of the photosynthesis process in the high school context in a semi-rural South African school. The learners’ attitudes towards the CCM approach in the teaching of Life Sciences were explored. This study aimed to answer the following main question: what are the achievements of Grade 10 learners’ in photosynthesis as core knowledge? The CCM approach included worksheets based on all five steps of the CCM process: commit to an outcome, expose beliefs, confront beliefs, accommodate the concept and extend the concept. The sample consisted of 78 Grade 10 learners. The research was carried out with a quasi-experimental/control group design and lasted for six weeks. The achievement test and questionnaires were used as instruments to collect data. The analyses of results show that experimental and control group’s pre-test academic achievement scores were similar and there was no significant difference between them (p < 0.05), but when the academic achievement of the post-test results of the EG and CG were analyzed, it was clear that there is a significant difference. The results from post-tests suggest that learners from the EG, taught using the CCM approach, show significantly greater achievements in photosynthesis than learners from the CG. In addition, learners from EG show a positive attitude towards Life Sciences after CCM teaching approach, but not from the CG taught using traditional approach. These findings have implications for a science teacher and recommendations are made to improve the teaching of photosynthesis as core knowledge.

This dissertation studies optimization problems that involve probabilistic covering constraints. A probabilistic constraint evaluates and requires that the probability that a set of constraints involving random coefficients with known distributions hold satisfy a minimum requirement. A covering constraint involves a linear inequality on non-negative variables with a greater or equal to sign and non-negative coefficients. A variety of applications, such as set cover problems, node/edge cover problems, crew scheduling, production planning, facility location, and machine learning, in uncertain settings involve probabilistic covering constraints. In the first part of this dissertation we consider probabilistic covering linear programs. Using the sampling average approximation (SAA) framework, a probabilistic covering linear program can be approximated by a covering k-violation linear program (CKVLP), a deterministic covering linear program in which at most k constraints are allowed to be violated. We show that CKVLP is strongly NP-hard. Then, to improve the performance of standard mixed-integer programming (MIP) based schemes for CKVLP, we (i) introduce and analyze a coefficient strengthening scheme, (ii) adapt and analyze an existing cutting plane technique, and (iii) present a branching technique. Through computational experiments, we empirically verify that these techniques are significantly effective in improving solution times over the CPLEX MIP solver. In particular, we observe that the proposed schemes can cut down solution times from as much as six days to under four hours in some instances. We also developed valid inequalities arising from two subsets of the constraints in the original formulation. When incorporating them with a modified coefficient strengthening procedure, we are able to solve a difficult probabilistic portfolio optimization instance listed in MIPLIB 2010, which cannot be solved by existing approaches. In the second part of this dissertation we study a class of probabilistic 0-1 covering problems, namely probabilistic k-cover problems. A probabilistic k-cover problem is a stochastic version of a set k-cover problem, which is to seek a collection of subsets with a minimal cost whose union covers each element in the set at least k times. In a stochastic setting, the coefficients of the covering constraints are modeled as Bernoulli random variables, and the probabilistic constraint imposes a minimal requirement on the probability of k-coverage. To account for absence of full distributional information, we define a general ambiguous k-cover set, which is ``distributionally-robust." Using a classical linear program (called the Boolean LP) to compute the probability of events, we develop an exact deterministic reformulation to this ambiguous k-cover problem. However, since the boolean model consists of exponential number of auxiliary variables, and hence not useful in practice, we use two linear program based bounds on the probability that at least k events occur, which can be obtained by aggregating the variables and constraints of the Boolean model, to develop tractable deterministic approximations to the ambiguous k-cover set. We derive new valid inequalities that can be used to strengthen the linear programming based lower bounds. Numerical results show that these new inequalities significantly improve the probability bounds. To use standard MIP solvers, we linearize the multi-linear terms in the approximations and develop mixed-integer linear programming formulations. We conduct computational experiments to demonstrate the quality of the deterministic reformulations in terms of cost effectiveness and solution robustness. To demonstrate the usefulness of the modeling technique developed for probabilistic k-cover problems, we formulate a number of problems that have up till now only been studied under data independence assumption and we also introduce a new applications that can be modeled using the probabilistic k-cover model.

In this dissertation we study several non-convex and stochastic optimization problems. The common theme is the use of mixed-integer programming (MIP) techniques including valid inequalities and reformulation to solve these problems. We first study a strategic capacity planning model which captures the trade-off between the incentive to delay capacity installation to wait for improved technology and the need for some capacity to be installed to meet current demands. This problem is naturally formulated as a MIP with a bilinear objective. We develop several linear MIP formulations, along with classes of strong valid inequalities. We also present a specialized branch-and-cut algorithm to solve a compact concave formulation. Computational results indicate that these formulations can be used to solve large-scale instances. We next study methods for optimization with joint probabilistic constraints. These problems are challenging because evaluating solution feasibility requires multidimensional integration and the feasible region is not convex. We propose and analyze a Monte Carlo sampling scheme to simplify the probabilistic structure of such problems. Computational tests of the approach indicate that it can yield good feasible solutions and reasonable bounds on their quality. Next, we study a MIP formulation of the non-convex sample approximation problem. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. Computational results indicate that large-scale instances can be solved using the strengthened formulations. Finally, we study optimization problems with stochastic dominance constraints. A stochastic dominance constraint states that a random outcome which depends on the decision variables should stochastically dominate a given random variable. We present new formulations for both first and second order stochastic dominance which are significantly more compact than existing formulations. Computational tests illustrate the benefits of the new formulations.

In der Dissertation wurde ein innovatives Portfoliomodell entwickelt, welches den Präferenzen einer großen Gruppe von Investoren entspricht, die mit der traditionellen Portfolio Selektion auf Basis von Mittelwertrendite und Varianz nicht zufrieden sind. Vor allem bezieht sich die Unzufriedenheit auf eine sehr spezifische Definition der Risiko- und Wertmaße, die angenommene Nutzenfunktion, die Risikodiversifizierung sowie die Beschränkung des Assetuniversums. Dies erschwert vor allem die Optimierung der modernen Finanzprodukte. Das im Modell verwendete Risikomaß-Ausfallrisiko drückt die Präferenzen der Investoren im Bereich unterhalb der Renditebenchmark aus. Die Renditenabweichung von der Benchmark nach oben werden nicht, wie im Falle des Mittelwertrendite-Varianz-Portfoliomodells, minimiert oder als risikoneutral, wie bei dem Mittelwertrendite-Ausfallrisiko-Portfoliomodell, betrachtet. Stattdessen wird ein Wertmaß, das Chance-Potenzial (Upper Partial Moment), verwendet, mit welchem verschiedene Investorenwünsche in diesem Bereich darstellbar sind. Die Eliminierung der Annahme der normalverteilten Renditen in diesem Chance-Potenzial-Ausfallrisiko-Portfoliomodell erlaubt eine korrekte Asset Allokation auch im Falle der nicht normalverteilten Renditen, die z. B. Finanzderivate, Aktien, Renten und Immobilien zu finden sind. Bei diesen tendiert das traditionelle Mittelwertrendite-Varianz-Portfoliomodell zu suboptimalen Entscheidungen. Die praktische Anwendung des Chance-Potenzial-Ausfallrisiko-Portfoliomodells wurde am Assetuniversum von Covered Calls, Protective Puts und Aktien gezeigt. / This thesis presents an innovative portfolio model appropriate for a large group of investors which are not content with the asset allocation with the traditional, mean return-variance based portfolio model above all in term of its rather specific definition of the risk and value decision parameters, risk diversification, related utility function and its restrictions imposed on the asset universe. Its modifiable risk measure – shortfall risk – expresses variable risk preferences below the return benchmark. The upside return deviations from the benchmark are not minimized as in case of the mean return-variance portfolio model or considered risk neutral as in the mean return-shortfall risk portfolio model, but employs variable degrees of the chance potential (upper partial moments) in order to provide investors with broader range of utility choices and so reflect arbitrary preferences. The elimination of the assumption of normally distributed returns in the chance potential-shortfall risk model allows correct allocation of assets with non-normally distributed returns as e.g. financial derivatives, equities, real estates, fixed return assets, commodities where the mean-variance portfolio model tends to inferior asset allocation decisions. The computational issues of the optimization algorithm developed for the mean-variance, mean-shortfall risk and chance potential-shortfall risk portfolio selection are described to ease their practical application. Additionally, the application of the chance potential-shortfall risk model is shown on the asset universe containing stocks, covered calls and protective puts.

In der Dissertation wurde ein innovatives Portfoliomodell entwickelt, welches den Präferenzen einer großen Gruppe von Investoren entspricht, die mit der traditionellen Portfolio Selektion auf Basis von Mittelwertrendite und Varianz nicht zufrieden sind. Vor allem bezieht sich die Unzufriedenheit auf eine sehr spezifische Definition der Risiko- und Wertmaße, die angenommene Nutzenfunktion, die Risikodiversifizierung sowie die Beschränkung des Assetuniversums. Das im Modell verwendete Risikomaß-Ausfallrisiko drückt die Präferenzen der Investoren im Bereich unterhalb der Renditebenchmark aus. Die Renditenabweichung von der Benchmark nach oben werden nicht, wie im Falle des Mittelwertrendite-Varianz-Portfoliomodells, minimiert oder als risikoneutral, wie bei dem Mittelwertrendite-Ausfallrisiko-Portfoliomodell, betrachtet. Stattdessen wird ein Wertmaß, das Chance-Potenzial (Upper Partial Moment), verwendet, mit welchem verschiedene Investorenwünsche in diesem Bereich darstellbar sind. Die Eliminierung der Annahme der normalverteilten Renditen in diesem Chance-Potenzial-Ausfallrisiko-Portfoliomodell erlaubt eine korrekte Asset Allokation auch im Falle der nicht normalverteilten Renditen, die z. B. Finanzderivate, Aktien, Renten und Immobilien zu finden sind. Bei diesen tendiert das traditionelle Mittelwertrendite-Varianz-Portfoliomodell zu suboptimalen Entscheidungen. Die praktische Anwendung des Chance-Potenzial-Ausfallrisiko-Portfoliomodells wurde am Assetuniversum von Covered Calls, Protective Puts und Aktien gezeigt. / This thesis presents an innovative portfolio model appropriate for a large group of investors which are not content with the asset allocation with the traditional, mean return-variance based portfolio model above all in term of its rather specific definition of the risk and value decision parameters, risk diversification, related utility function and its restrictions imposed on the asset universe. Its modifiable risk measure – shortfall risk – expresses variable risk preferences below the return benchmark. The upside return deviations from the benchmark are not minimized as in case of the mean return-variance portfolio model or considered risk neutral as in the mean return-shortfall risk portfolio model, but employs variable degrees of the chance potential (upper partial moments) in order to provide investors with broader range of utility choices and so reflect arbitrary preferences. The elimination of the assumption of normally distributed returns in the chance potential-shortfall risk model allows correct allocation of assets with non-normally distributed returns as e.g. financial derivatives, equities, real estates, fixed return assets, commodities where the mean-variance portfolio model tends to inferior asset allocation decisions. The computational issues of the optimization algorithm developed for the mean-variance, mean-shortfall risk and chance potential-shortfall risk portfolio selection are described to ease their practical application. Additionally, the application of the chance potential-shortfall risk model is shown on the asset universe containing stocks, covered calls and protective puts.

In der Dissertation wurde ein innovatives Portfoliomodell entwickelt, welches den Präferenzen einer großen Gruppe von Investoren entspricht, die mit der traditionellen Portfolio Selektion auf Basis von Mittelwertrendite und Varianz nicht zufrieden sind. Vor allem bezieht sich die Unzufriedenheit auf eine sehr spezifische Definition der Risiko- und Wertmaße, die angenommene Nutzenfunktion, die Risikodiversifizierung sowie die Beschränkung des Assetuniversums. Dies erschwert vor allem die Optimierung der modernen Finanzprodukte. Das im Modell verwendete Risikomaß-Ausfallrisiko drückt die Präferenzen der Investoren im Bereich unterhalb der Renditebenchmark aus. Die Renditenabweichung von der Benchmark nach oben werden nicht, wie im Falle des Mittelwertrendite-Varianz-Portfoliomodells, minimiert oder als risikoneutral, wie bei dem Mittelwertrendite-Ausfallrisiko-Portfoliomodell, betrachtet. Stattdessen wird ein Wertmaß, das Chance-Potenzial (Upper Partial Moment), verwendet, mit welchem verschiedene Investorenwünsche in diesem Bereich darstellbar sind. Die Eliminierung der Annahme der normalverteilten Renditen in diesem Chance-Potenzial-Ausfallrisiko-Portfoliomodell erlaubt eine korrekte Asset Allokation auch im Falle der nicht normalverteilten Renditen, die z. B. Finanzderivate, Aktien, Renten und Immobilien zu finden sind. Bei diesen tendiert das traditionelle Mittelwertrendite-Varianz-Portfoliomodell zu suboptimalen Entscheidungen. Die praktische Anwendung des Chance-Potenzial-Ausfallrisiko-Portfoliomodells wurde am Assetuniversum von Covered Calls, Protective Puts und Aktien gezeigt. / This thesis presents an innovative portfolio model appropriate for a large group of investors which are not content with the asset allocation with the traditional, mean return-variance based portfolio model above all in term of its rather specific definition of the risk and value decision parameters, risk diversification, related utility function and its restrictions imposed on the asset universe. Its modifiable risk measure ─ shortfall risk ─ expresses variable risk preferences below the return benchmark. The upside return deviations from the benchmark are not minimized as in case of the mean return-variance portfolio model or considered risk neutral as in the mean return-shortfall risk portfolio model, but employs variable degrees of the chance potential (upper partial moments) in order to provide investors with broader range of utility choices and so reflect arbitrary preferences. The elimination of the assumption of normally distributed returns in the chance potential-shortfall risk model allows correct allocation of assets with non-normally distributed returns as e.g. financial derivatives, equities, real estates, fixed return assets, commodities where the mean-variance portfolio model tends to inferior asset allocation decisions. The computational issues of the optimization algorithm developed for the mean-variance, mean-shortfall risk and chance potential-shortfall risk portfolio selection are described to ease their practical application. Additionally, the application of the chance potential-shortfall risk model is shown on the asset universe containing stocks, covered calls and protective puts.

Šiame darbe atskleidžiama azartinių lošimų santykių prigimtis, jų specifinis pobūdis ir poveikis visuomenei. Pagrindžiamas specialaus azartinių lošimų reglamentavimo poreikis, formuojami tokio reglamentavimo uždaviniai bei reikalavimai šių uždavinių įgyvendinimo būdams. Azartinių lošimų srityje nurodomi svarbiausi socialiniai pokyčiai, keliantys naujus reikalavimus nagrinėjamų santykių teisinam reglamentavimui. Remiantis Lietuvos Respublikos ir pasirinktų užsienio šalių (Austrijos, Belgijos, Didžiosios Britanijos, Lenkijos, Šveicarijos ir kitų šalių) norminių teisės aktų bei kitų šaltinių analize išskiriamos specialios azartinių lošimų reguliavimo priemonės. Vertinamas šių priemonių bei jų taikymo būdų atitikimas suformuluotiems reglamentavimo uždavinių įgyvendinimo reikalavimams (suformuluotam bendrajam idealumo kriterijui). Remiantis autoriaus gautomis išvadomis darbo pabaigoje pateikiamas abstraktus, aprašomojo pobūdžio idealus azartinių lošimų reglamentavimo modelis, kuriame siūlomos principinės nuostatos, orientacinės gairės azartinių lošimų reglamentavimo modeliui šiuolaikinėje demokratinėje valstybėje. / In this graduation work the nature of gambling, specific character of it and the influence of gambling to the society is being disclosed. Also, the special regulation of gambling is being justified and the main goals of such special regulation are being formed as well as the requirements to the implements of these goals. In the area of gambling games the main social changes that brings new chalenges for gambling regulation are being pointed. The law and special sources of gambling in Lithuania and other countries (Austria, Belgium, United Kingdom, Poland, Switzerland and others) is also being analyzed. Acording to the mentioned analysis, the special instrumentalities of gambling regulation are being parted. After that, the parted instrumentalities of gambling regulation are being considered to the formed general perfect gambling regulation criterion. Finally, according to the author's conclusions, an abstract, descriptive model of gambling perfect gambling regulation is being proposed. In this model principled attitudes, orientational guidlines of gambling regulation in modern democracy are being proposed.

Η ερευνητική υπόθεση αυτής της διατριβής ήταν ότι οι διευθυντές των Σχολείων Δεύτερης Ευκαιρίας (ΣΔΕ), αντλώντας από πόρους και κανόνες των ΣΔΕ, με τις πρακτικές τους αναπαράγουν τις βασικές αρχές και χαρακτηριστικά των ΣΔΕ (σύμφωνα με την Υπουργική Απόφαση 2373/2003). Ως ερευνητική μέθοδος χρησιμοποιήθηκε η εστιασμένη συνέντευξη σε επιλεγμένο δείγμα 31 διευθυντών, ενώ παράλληλα λήφθηκαν συνεντεύξεις και από 7 στελέχη του φορέα υλοποίησης του Προγράμματος των ΣΔΕ. Η ερευνητική τεχνική ήταν η ανάλυση περιεχομένου των δεδομένων, τα οποία ερμηνεύτηκαν με τη θεωρία της δομοποίησης. Στα συμπεράσματα, σε μεγάλο βαθμό επιβεβαιώνεται η ερευνητική υπόθεση. Επιπλέον, αναδεικνύονται και άλλα σημαντικά ζητήματα εκπαιδευτικής πολιτικής: παράγοντες που επιτρέπουν αυτήν την αναπαραγωγή, κίνδυνοι και απειλές, ευκαιρίες, καθώς και παράγοντες βιωσιμότητας των ΣΔΕ. / The research hypothesis of this thesis was that the headmasters of Second Chance Schools (SCS), drawing upon resources and rules of SCS, reproduce, through their practices, the main principles and characteristics of SCS (according to Ministerial Resolution 2373/2003). The research method of focused interview was employed in a selected sample of 31 headmasters, and additionally 7 members of the SCS Implementation Institution executive stuff were interviewed. The research technique was content analysis of the research data, which were interpreted by the structuration theory. The conclusions confirm, to a great extent, the research hypothesis and, additionally, bring out other significant educational policy issues: factors that enable this reproduction, dangers and threats, opportunities, as well as factors for SCS’ sustainability.

Tendo como base a idéia de que a Modernidade configura a própria tradição da ruptura, extraída de Octavio Paz, este estudo propõe-se a examinar o poema Un coup de dés, de Stéphane Mallarmé, aproximado-o de alguns poetas e músicos. Realizado no auge da Modernidade, esse poema foi retomado, na segunda metade do século XX, pelo movimento da poesia concreta, produzido no Brasil, como referência para suas teorias, através de sua estrutura espácio-temporal. Seus criadores, Augusto e Haroldo de Campos e Décio Pignatari, também resgataram a visão que Oswald de Andrade, Mário de Andrade e Manuel Bandeira, poetas do Modernismo, tiveram sobre Mallarmé. Assim como o Concretismo demonstrou interesse especial pelo poeta francês, parte da crítica literária estruturalista e pós-estruturalista estudou a revolução empreendida por Un coup de dés como uma extensão da transparência do espaço da escritura, em textos de Jacques Derrida, Roland Barthes, Michel Foucault, Julia Kristeva e Maurice Blanchot. Nesse sentido, Un coup de dés também faz com que seja necessário compor um conjunto teórico sobre a concepção do “acaso” utilizado em sua concepção. Visto desse ângulo, Octavio Paz serve de referência para equacionar diversas questões que o poema suscita, no contexto da Modernidade, inclusive sua relação com a música. Este campo é aberto pela revelação de que Mallarmé queria realizar, em Un coup de dés, um poema ligado intrinsecamente ao universo musical em que estava mergulhado no final do século XIX, à sombra das obras de Wagner, Debussy e Stravinski. A importância que Mallarmé atribuía à música, acompanhada por seu gesto de “acaso crítico”, chamou a atenção de dois músicos do século XX, John Cage e Pierre Boulez: o primeiro trabalhando o acaso de Mallarmé não só em peças musicais, mas também em poemas, compostos a partir da década de 1960 e o segundo, tomando Mallarmé como referência em ensaios e trabalhos. Tal universo musical incide em dois criadores da poesia concreta: Augusto de Campos, tanto em sua série musical Poetamenos quanto em seus poemas visuais feitos durante a fase ortodoxa e posterior à do Concretismo, e Haroldo de Campos, para o qual Mallarmé constitui a representação mais exemplar de seu percurso teórico-crítico e criativo. Mais do que eleger Mallarmé como referência em sua poesia, Haroldo de Campos o configura como o poeta mais relevante para sua noção de escritura da Modernidade. Perseguindo a mesma tradição sincrônica, baseada na concepção de Roman Jakobson, Ezra Pound e T. S. Eliot, escritores-críticos, Haroldo, ao longo de sua trajetória poética, recupera a musicalidade do poeta francês e as formas de articulação com os limites da palavra no espaço da página, eixo nuclear do presente estudo. Pretende-se, pois, demonstrar, nessa investigação, que Un coup de dés, estabelecido seu contato com a música, é produto de um acaso racional e crítico, instigando a exploração da palavra poética. O diálogo da literatura com a música, que os concretos buscar sublinhar através do poema de Mallarmé, é o caminho que este trabalho quer traçar. / Taking as a basis the idea that Modernity configures its own tradition of rupture, extracted from Octavio Paz, this study is going to examine the poem Un coup de dés, by Stephane Mallarmé, approximating him to some poets and musicians. This poem was written in the climax of Modernity and was retaken in the second half of the twentieth century by the concrete poetry trend, which happened in Brazil, as a reference for its theories, through its spatial and temporal structure. Its creators, Augusto e Haroldo de Campos, also reaffirmed the view that Oswald de Andrade, Mário de Andrade and Manuel Bandeira, poets of Modernism, had about Mallarmé. In the same way as ‘Concretism’ showed special interest for the French poet, part of the structuralist and post-structuralist literary criticists studied the revolution presented by Un coup de dés as an extension of the transparency of the writing space in texts by Jacques Derrida, Roland Barthes, Michel Foucault, Julia Kristeva and Maurice Blanchot. In this sense, Un coup de dés also makes it necessary to compose a theoretical ensemble about the conception of “chance” used in its conception. Seen from this angle, Octavio Paz is a reference to counterbalance several questions the poem stirs up, in the context of Modernity, including its relation to music. This area is opened by the revelation that Mallarmé wanted to create, in Un coup de dés, a poem linked intrinsically to the universe of music in which he was immersed at the end of the nineteenth century, in the shadow of Wagner’s, Debussy’s and Stravinski’s works. The importance that Mallarmé attributed to music, accompanied by his gesture of “critical chance”, called the attention of two musicians of the twentieth century John Cage and Pierre Boulez: the first working Mallarmé’s “chance” not only in musical works, but also in poems, composed in the 1960’s, and the second, taking Mallarmé as a reference in essays and works. Such musical universe falls on two creators of concrete poetry: Augusto de Campos not only in his series Poetamenos but also in his visual poems composed during an orthodox phase posterior to Concretism, and Haroldo de Campos, for whom Mallarmé is the most outstanding guide of his theoretical, critical and creative course. More than just electing Mallarmé as a reference in his poetry, Haroldo de Campos, configures him as the most relevant poet for his notion of writing of Modernity. Following the same synchronical tradition, based on the conception of Roman Jakobson, Ezra Pound and T. S. Eliot, critical writers, Haroldo, along his poetic course, recovers the musicality of the French poet and the forms of articulating the nuclear axis of the present study. We intend, then, to demonstrate, in this study, that Un coup de dés establishes its contact with the music, is a product of the rational and critical ‘chance’ leading towards the exploration of the poetical word. The dialogue between literature and music, that the concretes intend to reinforce through Mallarmé’s poem, is the course that this work intends to bring forth.

[pt] Estudamos aproximações amostrais de problemas com restrições probabilísticas através da aproximação pela média amostral (SAA) e demonstramos as propriedades de convergência relacionadas. Utilizamos SAA para obter bons candidatos à solução e cotas estatísticas para o valor ótimo do problema original. Para ajustar corretamente parâmetros, aplicamos o método a dois problemas com restrições probabilísticas. O primeiro é um problema de seleção de portfolio linear com retornos seguindo uma distribuição lognormal multivariada. O segundo é uma versão com restrições probabilísticas conjuntas de um problema da mistura simplificado. Concluímos com uma aplicação mais exigente ao problema de se determinar a provisão mínima que um agente econômico deve ter de forma a satisfazer uma série de obrigações futuras com probabilidade suficientemente alta. / [en] We study sample approximations of chance constrained problems through the sample average approximation (SAA) approach and prove the related convergence properties. We discuss how to use the SAA method to obtain good candidate solutions and bounds for the optimal value of the original problem. In order to tune the parameters of SAA, we apply the method to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem. We conclude with a more demanding application of SAA methodology to the determination of the minimum provision an economic agent must have in order to meet a series of future payment obligations with sufficiently high probability.