L'incertitude est une propriété naturelle des systèmes complexes. Les paramètres de certains modèles peuvent être imprécis; la présence de perturbations aléatoires est une source majeure d'incertitude pouvant avoir un impact important sur les performances du système. Dans cette thèse, nous étudierons les problèmes d’optimisation avec contraintes en probabilités dans les cas suivants : Tout d’abord, nous passons en revue les principaux résultats relatifs aux contraintes en probabilités selon trois perspectives: les problèmes liés à la convexité, les reformulations et les approximations de ces contraintes, et le cas de l’optimisation distributionnellement robuste. Pour les problèmes d’optimisation géométriques, nous étudions les programmes avec contraintes en probabilités jointes. A l’aide d’hypothèses d’indépendance des variables aléatoires elliptiquement distribuées, nous déduisons une reformulation des programmes avec contraintes géométriques rectangulaires jointes. Comme la reformulation n’est pas convexe, nous proposons de nouvelles approximations convexes basées sur la transformation des variables ainsi que des méthodes d’approximation linéaire par morceaux. Nos résultats numériques montrent que nos approximations sont asymptotiquement serrées. Lorsque les distributions de probabilité ne sont pas connues à l’avance, le calcul des bornes peut être très utile. Par conséquent, nous développons quatre bornes supérieures pour les contraintes probabilistes individuelles, et jointes dont les vecteur-lignes de la matrice des contraintes sont indépendantes. Sur la base des inégalités de Chebyshev, Chernoff, Bernstein et de Hoeffding, nous proposons des approximations déterministes. Des conditions suffisantes de convexité. Pour réduire la complexité des calculs, nous reformulons les approximations sous forme de problèmes d'optimisation convexes solvables basés sur des approximations linéaires et tangentielles par morceaux. Enfin, des expériences numériques sont menées afin de montrer la qualité des approximations étudiées sur des données aléatoires. Dans certains systèmes complexes, la distribution des paramètres aléatoires n’est que partiellement connue. Pour traiter les incertitudes dans ces cas, nous proposons un ensemble d'incertitude basé sur des données obtenues à partir de distributions mixtes. L'ensemble d'incertitude est construit dans la perspective d'estimer simultanément des moments d'ordre supérieur. Ensuite, nous proposons une reformulation du problème robuste avec contraintes en probabilités en utilisant des données issues d’échantillonnage. Comme la reformulation n’est pas convexe, nous proposons des approximations convexes serrées basées sur la méthode d’approximation linéaire par morceaux sous certaines conditions. Pour le cas général, nous proposons une approximation DC pour dériver une borne supérieure et une approximation convexe relaxée pour dériver une borne inférieure pour la valeur de la solution optimale du problème initial. Enfin, des expériences numériques sont effectuées pour montrer que les approximations proposées sont efficaces. Nous considérons enfin un jeu stochastique à n joueurs non-coopératif. Lorsque l'ensemble de stratégies de chaque joueur contient un ensemble de contraintes linéaires stochastiques, nous modélisons ces contraintes sous la forme de contraintes en probabilité jointes. Pour chaque joueur, nous formulons les contraintes en probabilité dont les variables aléatoires sont soit normalement distribuées, soit elliptiquement distribuées, soit encore définies dans le cadre de l’optimisation distributionnellement robuste. Sous certaines conditions, nous montrons l’existence d’un équilibre de Nash pour ces jeux stochastiques. / Chance constrained optimization is a natural and widely used approaches to provide profitable and reliable decisions under uncertainty. And the topics around the theory and applications of chance constrained problems are interesting and attractive. However, there are still some important issues requiring non-trivial efforts to solve. In view of this, we will systematically investigate chance constrained problems from the following perspectives. As the basis for chance constrained problems, we first review some main research results about chance constraints in three perspectives: convexity of chance constraints, reformulations and approximations for chance constraints and distributionally robust chance constraints. For stochastic geometric programs, we formulate consider a joint rectangular geometric chance constrained program. With elliptically distributed and pairwise independent assumptions for stochastic parameters, we derive a reformulation of the joint rectangular geometric chance constrained programs. As the reformulation is not convex, we propose new convex approximations based on the variable transformation together with piecewise linear approximation methods. Our numerical results show that our approximations are asymptotically tight. When the probability distributions are not known in advance or the reformulation for chance constraints is hard to obtain, bounds on chance constraints can be very useful. Therefore, we develop four upper bounds for individual and joint chance constraints with independent matrix vector rows. Based on the one-side Chebyshev inequality, Chernoff inequality, Bernstein inequality and Hoeffding inequality, we propose deterministic approximations for chance constraints. In addition, various sufficient conditions under which the aforementioned approximations are convex and tractable are derived. To reduce further computational complexity, we reformulate the approximations as tractable convex optimization problems based on piecewise linear and tangent approximations. Finally, based on randomly generated data, numerical experiments are discussed in order to identify the tight deterministic approximations. In some complex systems, the distribution of the random parameters is only known partially. To deal with the complex uncertainties in terms of the distribution and sample data, we propose a data-driven mixture distribution based uncertainty set. The data-driven mixture distribution based uncertainty set is constructed from the perspective of simultaneously estimating higher order moments. Then, with the mixture distribution based uncertainty set, we derive a reformulation of the data-driven robust chance constrained problem. As the reformulation is not a convex program, we propose new and tight convex approximations based on the piecewise linear approximation method under certain conditions. For the general case, we propose a DC approximation to derive an upper bound and a relaxed convex approximation to derive a lower bound for the optimal value of the original problem, respectively. We also establish the theoretical foundation for these approximations. Finally, simulation experiments are carried out to show that the proposed approximations are practical and efficient. We consider a stochastic n-player non-cooperative game. When the strategy set of each player contains a set of stochastic linear constraints, we model the stochastic linear constraints of each player as a joint chance constraint. For each player, we assume that the row vectors of the matrix defining the stochastic constraints are pairwise independent. Then, we formulate the chance constraints with the viewpoints of normal distribution, elliptical distribution and distributionally robustness, respectively. Under certain conditions, we show the existence of a Nash equilibrium for the stochastic game.

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The rise of the sharing economy is forcing industries to move from traditional take-make-waste economy towards circular ones (Schuttelaar & Partners, 2019). Along with the increasing influence of e-commerce, supply-chain collaboration and globalization has increased the pressure of warehouse operations and requires more efficient operations with the use of less resources thanever before (Frazelle, 2016). Lean is a concept that has gained attention as a means of doing morewith less, through reducing wastes and continuously improving, which has been successful within the manufacturing sector. The difficulties of translating this concept into non-repetitive environments has contributed to the low level of adoption within industries like construction. Where companies within construction equipment rental have an even greater challenge, dealing with circular product flows. Thus, making their warehousing operations inherently complex along with the large amounts of different products. The thesis has focused on analysing the warehouse operations flow of collective fall protectionproducts in the equipment rental industry, from customer order to customer return. This has been enabled by performing an adapted Value Stream Mapping method in a new setting for equipmentrental industries. Where Value Stream Mapping traditionally is applied in manufacturing to visually represents the actions that are required for a product to move through the production andinformation flows. The purpose of the study is to apply VSM beyond manufacturing and tosummarize the results in order to provide suggestions for improving the handling of non-serialized products within the equipment rental industry. A case company within the equipment rental industry has been used to meet the purpose of the study by testing and evaluating the proposed frame work. The findings have been followed up by performing a return handling study, conducted by equipment rental professionals, which has been subject to a statistical analysis. The findings from the VSM are used as a foundation for the return handling study, which serves as a means of verifying or dismissing them. These are then summarized in suggestions for improvements to the case company and suggestions for future actions. Among the findings are indications of longer return times for products that are not forewarned prior to returns, and increase processing rates for sorted returns. Furthermore three product types were more frequently requiring repairs, cleaning and scrapping compared to other products, suggesting that these requiremore time to process. The company is advised to work with standardizing processes for return handling throughout the organization within invoicing specifications and product storage. Suggestions for future research within the subject area is also presented. / Den ökande utbredningen av delningsekonomi tvingar industrier att gå från traditionella linjära affärsmodeller mot cirkulära (Schuttelaar & Partners, 2019). Utöver denna förändring så ökar även trycket på lagerverksamheter som ett resultat av det ökande inflytandet av e-handel, Supply-Chainsamarbete och globalisering som kräver effektivare verksamheter med användning av mindre resurser än någonsin förut (Frazelle, 2016). Lean är ett koncept som har uppmärksammats som en filosofi för att hantera dessa förändringar, genom att åstadkomma mer med mindre resurser och därmed minska slöseri och att sträva efter ständiga förbättringar, vilket har varit framgångsrikt inom tillverkningssektorn. Svårigheterna med att översätta detta koncept till icke-repetitiva miljöer har bidragit till den begränsade appliceringen av lean inom industrier som byggnadskonstruktion. Där företag inom uthyrning av byggnadsmaskiner och utrustning har en ännu större utmaning, då deras verksamhet bygger på cirkulära produktflöden. Vilket orsakar komplex lagerhantering av många olika produkter i stora mängder. Avhandlingen har fokuserat på att analysera lagerflödet av kollektiva fallskyddsmedel i uthyrningsbranschen, från kundorder till kundreturer. Detta har möjliggjorts genom att utföra en anpassad värdeflödeskartläggning i ett nytt sammanhang, nämligen för företag inom uthyrning av byggnadsmaskiner och utrustning. Där värdeflödeskartläggning vanligtvis används inom tillverkning för att visuellt representera de aktiviteter som krävs för att en produkt ska kunna flytta genom produktions- och informationsflödena. Syftet med studien är att tillämpa värdeflödeskartläggning utanför tillverkningssektorn och att sammanfatta resultaten i form av förslag på förbättringar av hanteringen av icke-serienumrerade produkter inom uthyrningsbranschen. Ett samarbete med ett fallföretag inom uthyrningsindustrin har genomgåtts för att uppfylla syftet med studien. Tillsammans med fallföretaget har det föreslagna ramarbetet testats och utvärderats. Resultaten har följts upp genom att genomföra en returhanteringsstudie, som utförs av verksamma inom lager hanteringen för kollektiva fallskyddsmedel, studiens resultat har sedan varit föremål för en statistisk analys. Upptäckterna från värdeflödeskartläggningen användes som grund för returhanteringsstudien, som agerat medel för att verifiera eller avfärda dem. Dessa har sedan sammanfattas i förslag till förbättringar till fallföretaget och förslag till framtida åtgärder. Bland resultaten finns indikationer på längre hanteringstider för returer som inte är förvarnade innan de blir återlämnade, samt minskade hanteringstider för sorterade returer. Dessutom krävde tre produkttyper ofta reparationer, rengöring och skrotning jämfört med andra produkter, vilket tyder på att dessa produkter är mer tidskrävande vid returer. Företaget rekommenderas att arbeta med standardiseringsprocesser för returhantering i hela organisationen, mer specifikt inom bestämmelser för fakturering och produktlagring. Förslag till framtida forskning inom ämnesområdet presenteras även.

<p> The globular cluster system of a nearby giant elliptical galaxy, NGC 5128 is studied to place constraints on the formation history of the galaxy. In this thesis, we have identified 190 new globular clusters via radial velocity measurements, bringing the total known population of globular clusters to 605 within this galaxy. We have examined the colour and spatial distributions of the globular cluster system and find it is bimodal in colour, with both a red and blue globular cluster population. The blue population is more spatially extended than the red, and both populations fall off in number density with radius as a power-law. There is a clear lack of globular clusters along the isophotal minor axis of the galaxy beyond a galactocentric radius of 15' warranting further search. With this new dataset, we have measured the ages, metallicities, and formation timescales for 72 globular clusters. The spectroscopic metallicity distribution function is bimodal indicating there is a metal-rich and metal-poor globular cluster population that corresponds to the red and blue globular clusters, respectively. We find the majority of both metal-rich (56%) and metal-poor (92%) globular clusters are older than 8 Gyr, comparable to the Milky Way globular cluster system. We do find a smaller fraction, 18% of our sample, are metal-rich globular clusters with ages younger than 5 Gyr, while the remaining globular clusters have intermediate ages between 5 -8 Gyr. The formation times of these globular clusters, estimated by their alpha-to-iron abundance ratios, indicate they formed quickly, on a timescale similar to globular clusters in most spiral galaxies, but on slower timescales than those in some other giant elliptical galaxies. The kinematics of the full globular cluster system is analyzed, as well as for the metal-rich and metal-poor globular clusters separately, as a function of galactocentric radius. We find the metal-poor globular cluster system has a small rotation signature of 17±14 km s-1 around no clearly defined axis and its dynamics are dominated by dispersion. The metal-rich globular cluster system has a mild rotation of 41 ± 15 km s-1 about the galaxy's isophotal major axis, following the rotation of a representative field star population, the planetary nebulae. The motion of the metal-rich globular cluster system is also dominated by random motion. We estimate the mass of the galaxy to be (5.5 ± 1.9) x 1011 with a mass-to-light ratio of 15.35 M0 /L80 using the globular cluster M0 population out to 20'. This estimate places NGC 5128 on a mass scale similar to other giant elliptical galaxies. The evidence collected suggests that NGC 5128 formed in a hierarchical scenario, gradually building up larger structure from smaller protogalaxies at early times in the history of the Universe. The group environment of NGC 5128 may have prolonged star formation within the galaxy as well, enabling a small spread in the old ages of globular clusters and also slowing the formation timescales compared to globular clusters in other giant elliptical galaxies. Results from this thesis also support more recent accretions in the history of NGC 5128, building up the more metal-rich and young globular clusters, which have a different rotation axis than the rest of the population. </p> / Thesis / Doctor of Philosophy (PhD)

<p> We study the problem of finding approximate significance points of random variables whose exact distributions are unknown or extremely complicated . We consider the case where at least the first three moments, and possibly the lower or upper endpoint of the distribution are known. </p> <p> The methods of approximation studied include the Johnson system of transformations, Pearson curves, Pearson curves with known lower terminal, Cornish-Fisher expansions and the approximation a+bW, where W is chi-squared with p degrees of freedom . A new three-moment approximation of the form (cW)^k, with W as defined above, is also considered. These methods of approximation are discussed, with special attention to fitting procedures and computer implementation. </p> <p> The methods of approximation are compared, with respect to ease of application and accuracy of approximation, over a wide variety of exact distributions. The accuracy of each approximation is discussed and guidelines are given for determining which of several approximations should be used in a particular case. </p> / Thesis / Master of Science (MSc)

We obtain a generalization of the Sokhotskii formula for the forth degree by means of methods of theory of residues. We obtain relations of Mikusinskii and Gonzalez – Dominguez type that connect distributions and hyperdistributions. We obtain an analog of the Sokhotskii formula for an arbitrary positive integer. / Получены обобщения формулы Сохоцкого для четвертой степени методами теории вычетов. Получены соотношения типа Микусинского и Гонсалеса – Домингеса, связывающие распределения и гиперраспределения. Было получено обобщение формулы Сохоцкого для любого натурального числа.

Phylogeny, the study of evolutionary relationships among species and other taxa, plays a crucial role in understanding the history of life. Bayesian analysis using Markov chain Monte Carlo (MCMC) is a widely used approach for inferring phylogenetic trees, but it suffers from slow convergence in higher dimensions and is slow to converge. This thesis focuses on exploring variational inference (VI), a methodology that is believed to lead to improved speed and accuracy of phylogenetic models. However, VI models are known to concentrate the density of the learned approximation in high-likelihood areas. This thesis evaluates the current state of Variational Inference Bayesian Phylogenetics (VBPI) and proposes a solution using a mixture of components to improve the VBPI method's performance on complex datasets and multimodal latent spaces. Additionally, we cover the basics of phylogenetics to provide a comprehensive understanding of the field. / Fylogeni, vilket är studien av evolutionära relationer mellan arter och andra taxonomiska grupper, spelar en viktig roll för att förstå livets historia. En ofta använd metod för att dra slutsatser om fylogenetiska träd är bayesiansk analys med Markov Chain Monte Carlo (MCMC), men den lider av långsam konvergens i högre dimensioner och kräver oändligt med tid. Denna uppsats fokuserar på att undersöka hur variationsinferens (VI) kan nyttjas inom fylogenetisk inferens med hög noggranhet. Vi fokuserar specifik på en modell kallad VBPI. Men VI-modeller är allmänt kända att att koncentrera sig på höga sannolikhetsområden i posteriorfördelningar. Vi utvärderar prestandan för Variatinal Inference Baysian Phylogenetics (VBPI) och föreslår en förbättring som använder mixturer av förslagsfördelningar för att förbättra VBPI-modellens förmåga att hantera mer komplexa datamängder och multimodala posteriorfördelningar. Utöver dettta går vi igenom grunderna i fylogenetik för att ge en omfattande förståelse av området.

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