在本篇論文裡, 我們提出一個"油輪排程規畫"的方法。處理在計畫期間中, 有關原油供應商的供給, 油輪的運輸與需求地儲油庫存貨管理的問題。藉由一些性質的探討來分析運輸排程與存貨管理。由供應油商契約量中供應量的情形, 提出最大累積可提運量表, 並藉由該表來輔助我們做排程; 由分油的概念與訂貨週期的制定, 分析運輸工具的排程狀況與儲油庫油量存貨的關係; 由運輸與儲油庫油料之間的互動關係, 提出管理的方案; 最後再藉由理想存貨量的關念來確立租用臨時運輸工具的可行性。並藉由所提出的性質來處理租用運輸工具的排程。在各個階段的處理過程中, 包括分析限制條件是否滿足的情形等, 總共提出了四個演算法, 找出滿足整個問題限制條件的可行排程。最後, 引用實務上的例子, 來說明油輪排程規畫的方法。 / In this paper, we consider an inventory problem of which the lead time depends on vehicles chosen from a restricted set. The inventory itself hasother constraints that must satisfy the demand and capacity limitation. Inspecific, a problem that considers scheduling oil tankers for transportingcrude oils from supplies to a refinery is represented. With the help of some properties discovered while analyzing the problem of oil tanker scheduling with inventory management, such as largest accumulated supplies from oil suppliers, demand intervals and reorder intervals, and an ideal inventory level, we develope an oil tanker scheduling model to determine each vehicle's schedule and satisfy all constraints. The model consists of four heuristic algorithms which are described step by step asthe solution procedure. A realworld example from an oil company is used to illustrate the four algorithms and suggest a feasible schedule for transporting crude oils. In short, we not only give 1 overall viewpoint of inventory and transportation problems but also provide a heuristic procedurefor solving it.

MSc (Statistics) / Department of Statistics / Haulage trucks and cross border traders cross through Beitbridge border post from landlocked countries such as Zimbabwe and Zambia for the sake of trading. Because of global warming, South Africa has lately been experiencing extreme weather patterns in the form of very high temperatures and heavy rainfall. Evidently, in 2013 tra c could not cross the Limpopo River because water was owing above the bridge. For planning, its important to predict the likelihood of such events occurring in future. Extreme value models o er one way in which this can be achieved. This study identi es suitable distributions to model the annual maximum heights of Limpopo river at Beitbridge border post. Maximum likelihood method and the Bayesian approach are used for parameter estimation. The r -largest order statistics was also used in this dissertation. For goodness of t, the probability and quantile- quantile plots are used. Finally return levels are calculated from these distributions. The dissertation has revealed that the 100 year return level is 6.759 metres using the maximum likelihood and Bayesian approaches to estimate parameters. Empirical results show that the Fr echet class of distributions ts well the ood heights data at Beitbridge border post. The dissertation contributes positively by informing stakeholders about the socio- economic impacts that are brought by extreme flood heights for Limpopo river at Beitbridge border post

In this thesis we present results about some eigenvalue statistics of the beta-Hermite ensembles, both in the classical cases corresponding to beta = 1, 2, 4, that is the Gaussian orthogonal ensemble (consisting of real symmetric matrices), the Gaussian unitary ensemble (consisting of complex Hermitian matrices) and the Gaussian symplectic ensembles (consisting of quaternionic self-dual matrices) respectively. We also look at the less explored general beta-Hermite ensembles (consisting of real tridiagonal symmetric matrices). Specifically we look at the empirical distribution function and two different scalings of the largest eigenvalue. The results we present relating to these statistics are the convergence of the empirical distribution function to the semicircle law, the convergence of the scaled largest eigenvalue to the Tracy-Widom distributions, and with a different scaling, the convergence of the largest eigenvalue to 1. We also use simulations to illustrate these results. For the Gaussian unitary ensemble, we present an expression for its level density. To aid in understanding the Gaussian symplectic ensemble we present properties of the eigenvalues of quaternionic matrices. Finally, we prove a theorem about the symmetry of the order statistic of the eigenvalues of the beta-Hermite ensembles. / I denna kandidatuppsats presenterar vi resultat om några olika egenvärdens-statistikor från beta-Hermite ensemblerna, först i de klassiska fallen då beta = 1, 2, 4, det vill säga den gaussiska ortogonala ensemblen (bestående av reella symmetriska matriser), den gaussiska unitära ensemblen (bestående av komplexa hermitiska matriser) och den gaussiska symplektiska ensemblen (bestående av kvaternioniska själv-duala matriser). Vi tittar även på de mindre undersökta generella beta-Hermite ensemblerna (bestående av reella symmetriska tridiagonala matriser). Specifikt tittar vi på den empiriska fördelningsfunktionen och två olika normeringar av det största egenvärdet. De resultat vi presenterar för dessa statistikor är den empiriska fördelningsfunktionens konvergens mot halvcirkel-fördelningen, det normerade största egenvärdets konvergens mot Tracy-Widom fördelningen, och, med en annan normering, största egenvärdets konvergens mot 1. Vi illustrerar även dessa resultat med hjälp av simuleringar. För den gaussiska unitära ensemblen presenterar vi ett uttryck för dess nivåtäthet. För att underlätta förståelsen av den gaussiska symplektiska ensemblen presenterar vi egenskaper hos egenvärdena av kvaternioniska matriser. Slutligen bevisar vi en sats om symmetrin hos ordningsstatistikan av egenvärdena av beta-Hermite ensemblerna.

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Dans cette thèse, on s'intéresse aux plus grands facteur premiers d'entiers consécutifs. Désignons par $P^+(n)$ (resp. $P^-(n)$) le plus grand (resp. plus petit) facteur premier d'un entier générique $n\geq 1$ avec la convention que $P^+(1)=1$ (resp. $P^-(1)=\infty$). Dans le premier chapitre, nous étudions les plus grands facteurs premiers d'entiers consécutifs dans les petits intervalles. Nous démontrons qu'il existe une proportion positive d'entiers $n$ tels que $P^+(n)<P^+(n+1)$ pour $n\in\, ]x,\, x+y]$ avec $y=x^{\theta}, \tfrac{7}{12}<\theta\leq 1$. Nous obtenons un résultat similaire pour la condition $P^+(n)>P^+(n+1)$. Dans le deuxième chapitre, nous nous intéressons à la fonction $P_y^+(n)$, où $P_y^+(n)=\max\{p|n:\, p\leq y\}$ et $2\leq y\leq x.$ Nous montrons qu'il existe une proportion positive d'entiers $n$ tels que $P_y^+(n)<P_y^+(n+1)$. En particulier, la proportion d'entiers $n$ avec $P^+(n)<P^+(n+1)$ est plus grande que $0,1356$ en prenant $y=x.$ Les outils principaux sont le crible et un système de poids bien adapté. Dans le troisième chapitre, nous démontrons que les deux configurations $P^+(n-1)>P^+(n)<P^+(n+1)$ et $P^+(n-1)<P^+(n)>P^+(n+1)$ ont lieu pour une proportion positive d'entiers $n$, en utilisant le système de poids bien adapté que l'on a introduit dans le Chapitre 2. De façon similaire, on peut obtenir un résultat plus général pour $k$ entiers consécutifs, $k\in \mathbb{Z}, k\geq3$. Dans le quatrième chapitre, on étudie les plus grands facteurs premiers d'entiers consécutifs voisins d'un entier criblé. Sous la conjecture d'Elliott-Halberstam, nous montrons d'abord que la proportion de la configuration $P^+(p-1)<P^+(p+1)$ est plus grande que $0,1779$. Puis, nous démontrons qu'il existe une proportion positive d'entiers $n$ tels que $P^+(n)<P^+(n+2), P^-(n)>x^{\beta}$ avec $0<\beta<\frac{1}{3}$ / In this thesis, we study the largest prime factors of consecutive integers. Denote by $P^+(n)$ (resp. $P^-(n)$) the largest (resp. the smallest) prime factors of the integer $n\geq 1$ with the convention $P^+(1)=1$ (resp. $P^-(1)=\infty$). In the first chapter, we consider the largest prime factors of consecutive integers in short intervals. We prove that there exists a positive proportion of integers $n$ for $n\in\, (x,\, x+y]$ with $y=x^{\theta}, \tfrac{7}{12}<\theta\leq 1$ such that $P^+(n)<P^+(n+1)$. A similar result holds for the condition $P^+(n)>P^+(n+1)$. In the second chapter, we consider the function $P_y^+(n)$, where $P_y^+(n)=\max\{p|n:\, p\leq y\}$ and $2\leq y\leq x$. We prove that there exists a positive proportion of integers $n$ such that $P_y^+(n)<P_y^+(n+1)$. In particular, the proportion of the pattern $P^+(n)<P^+(n+1)$ is larger than $0.1356$ by taking $y=x.$ The main tools are sieve methods and a well adapted system of weights. In the third chapter, we prove that the two patterns $P^+(n-1)>P^+(n)<P^+(n+1)$ and $P^+(n-1)<P^+(n)>P^+(n+1)$ occur for a positive proportion of integers $n$ respectively, by the well adapted system of weights that we have developed in the second chapter. With the same method, we derive a more general result for $k$ consecutive integers, $k\in \mathbb{Z}, k\geq 3$. In the fourth chapter, we study the largest prime factors of consecutive integers with one of which without small prime factor. Firstly we show that under the Elliott-Halberstam conjecture, the proportion of the pattern $P^+(p-1)<P^+(p+1)$ is larger than $0.1779$. Then, we prove that there exists a positive proportion of integers $n$ such that $P^+(n)<P^+(n+2), P^-(n)>x^{\beta}$ with $0<\beta<\frac{1}{3}$