This dissertation consists of two self-contained studies. The first study, in the domain of stochastic inventory theory, addresses the structure of optimal ordering policies in a periodic review setting. We take multiple sources of a single product to imply an ordering cost function that is nondecreasing, piecewise linear, and convex. Our main contribution is a proof of the optimality of a finite generalized base stock policy under an average cost criterion. Our inventory model is formulated as a Markov decision process with complete observations. Orders are delivered immediately. Excess demand is fully backlogged, and the function describing holding and backlogging costs is convex. All parameters are stationary, and the random demands are independent and identically distributed across periods. The (known) distribution function is subject to mild assumptions along with the holding and backlogging cost function. Our proof uses a vanishing discount approach. We extend our results from a continuous environment to the case where demands and order quantities are integral. The second study is in the area of capacity planning. Our overarching contribution is a relatively simple and fast solution approach for the fleet composition problem faced by a retail distribution firm, focusing on the context of a major beverage distributor. Vehicles to be included in the fleet may be of multiple sizes; we assume that spot transportation capacity will be available to supplement the fleet as needed. We aim to balance the fixed costs of the fleet against exposure to high variable costs due to reliance on spot capacity. We propose a two-stage stochastic linear programming model with fixed recourse. The demand on a particular day in the planning horizon is described by the total quantity to be delivered and the total number of customers to visit. Thus, daily demand throughout the entire planning period is captured by a bivariate probability distribution. We present an algorithm that efficiently generates a "definitive" collection of bases of the recourse program, facilitating rapid computation of the expected cost of a prospective fleet and its gradient. The equivalent convex program may then be solved by a standard gradient projection algorithm.

Dans cette thèse, nous nous intéressons à différents opérateurs d'inf-convolutions et à leurs applications à une classe d'inégalités de transport générales, plus spécifiquement sur les graphes. Notre objet de recherche s'inscrit donc dans les théories du transport de mesure et de l'analyse fonctionnelle. En introduisant une notion de gradient adapté au cadre discret (et plus généralement à tout espace métrique dont les boules sont compactes), nous prouvons que certains opérateurs d'inf-convolution sont solutions d'une inéquation d'Hamilton Jacobi sur les graphes. Ce résultat nous permet d'étendre au cadre discret un théorème classique de Bobkov, Gentil et Ledoux. Plus précisément nous montrons que des inégalités de transport faible (adaptées au cadre discret) sont équivalentes, sur un graphe, à l'hypercontractivité des opérateurs d'inf-convolutions. On en déduit plusieurs résultats concernant différentes inégalités fonctionnelles, dont celle de Sobolev logarithmique et de transport faible. Nous étudions par ailleurs les propriétés générales de différents opérateurs d'inf-convolutions, incluant le précédent, mais aussi un opérateur relié à un modèle issu de la physique (et au phénomène de grande déviation), toujours sur les graphes (dérivabilités, convexité, points extremum etc.). Dans un deuxième temps, nous nous intéressons aux liens entre différentes notions de courbure de Ricci sur les graphes -- proposées récemment par plusieurs auteurs -- et les inégalités fonctionnelles de type transport-entropie, ou transport-information associées à une chaîne de Markov. Nous obtenons également une borne supérieure sur le diamètre d'un graphe dont la courbure, en un certain sens, est minorée, un résultat à la Bonnet-Myers. Enfin, en nous restreignant au cas de la dimension 1, sur la droite réelle, nous obtenons une caractérisation d'une inégalité de transport faible et de l'inégalité de Sobolev logarithmique restreinte aux fonctions convexes. Ces résultats utilisent des propriétés géométriques liés à l'ordre convexe. / In this thesis, we interest in different inf-convolution operators and their applications to a class of general transportation inequalities, more specifically in the graphs. Therefore, our research topic fits in the theories of transportation and functional analysis. By introducing a gradient notion adapting to a discrete space (more generally to all space in which all closed balls are compact), we prove that some inf-convolution operators are solutions of a Hamilton-Jacobi's inequation. This result allows us to extend a classical theorem from Bobkov, Gentil and Ledoux. More precisely, we prove that, in a graph, some weak transport inequalities are equivalent to the hypercontractivity of inf-convolution operators. Thanks to this result, we deduce some properties concerning different functional inequalities, including Log-Sobolev inequalities and weak-transport inequalities. Besides, we study some general properties (differentiability, convexity, extreme points etc.) of different inf-convolution operators, including the one before, but also an operator related to a physical model (and to a large deviation phenomenon). We stay always in a graph. Secondly, we interest in connections between different notions of discrete Ricci curvature on the graphs which are proposed by several authors in the recent years, and functional inequalities of type transport-entropy, or transport-information related to a Markov chain. We also obtain an extension of Bonnet-Myers' result: an upper bound on the diameter of a graph of which the curvature is floored in some ways. Finally, restricting in the real line, we obtains a characterisation of a weak transport inequality and a log-Sobolev inequality restricted to convex functions. These results are from the geometrical properties related to the convex ordering.