La 1ere partie de ce travail traite l'optimisation des tournées sous forme d'un problème d'optimisation nommé Le problème de Voyageur de Commerce. Dans cette partie nous nous intéressons à faire une riche présentation du problème de Voyageur de Commerce, ses variantes, puis nous proposons une stratégie de génération de contrainte pour la résolution du TSP. Ensuite on traite sa version stochastique : le problème de Voyageur de commerce Probabiliste. Nous proposons une formulation mathématique du PTSP et nous présentons des résultats numériques obtenus par résolution exacte pour une série d'instances de petite taille. Dans la seconde partie, nous proposons une méthode d'approximation générale permettant d'approcher différents type de données, d'abord nous traitons l'approximation d'un signal de vent (cas simple, ID), ensuite l'approximation d'un champ de vecteurs avec prise en compte de la topographie qui constitue la principale contribution de cette partie. / The first part of this work deals with route optimization in the form of an optimization problem named The Traveler's Business Problem. In this part we are interested to make a rich presentation of the problem of Traveler Commerce, its variants, then we propose a strategy of constraint generation for the resolution of the TSP. Then we treat its stochastic version : the probabilistic business traveler problem. We propose a mathematical formulation of the PTSP and we present numerical results obtained by exact resolution for a series of small instances. In the second part, we propose a method of general approximation to approximate different type of data, first we treat the approximation of a wind signal (simple case, 1D), then the approximation of a vector field taking into account the topography which is the main contribution of this part.

Orientador: Fábio Bertequini Leão / Resumo: Este trabalho apresenta modelos de programação não linear e linear inteira binária como novos métodos para resolver o problema de diagnóstico de faltas em sistemas de transmissão de energia elétrica. Os modelos de otimização são desenvolvidos com base no conjunto de coberturas mínimas e possui como restrições as equações que descrevem a lógica e a filosofia de proteção empregadas por empresas de energia elétrica. As equações de restrições modelam a associação dos alarmes dos relés de proteção informados pelo sistema de supervisão e aquisição de dados (SCADA) com os estados esperados das funções dos relés de proteção. Os modelos de programação matemática realizam o diagnóstico de falta em uma única etapa, identificando a seção em falta através da análise dos estados dos disjuntores e das funções de proteção associadas a cada equipamento do sistema elétrico. O modelo proposto é um problema muito complexo de programação não linear inteira binária, portanto é reformulado como outro problema, em que algumas expressões são linearizadas, o que resulta em um modelo matemático de programação linear inteiro binário. A solução ótima obtida pelo modelo proposto é encontrada utilizando solvers comerciais de programação matemática. Os resultados obtidos mostram eficiência e robustez do modelo matemático. Na literatura, o problema de diagnóstico de falta é resolvido principalmente por técnicas heurísticas, portanto, o método proposto é inovador. / Doutor

Dans un graphe non orienté, le Problème de Coloration de Graphe (PCG) consiste à assigner à chaque sommet du graphe une couleur de telle sorte qu'aucune paire de sommets adjacents n'aient la même couleur et le nombre total de couleurs est minimisé. DSATUR est un algorithme exact efficace pour résoudre le PCG. Un de ses défauts est qu'une borne inférieure est calculée une seule fois au noeud racine de l'algorithme de branchement, et n'est jamais mise à jour. Notre nouvelle version de DSATUR surpasse l'état de l'art pour un ensemble d'instances aléatoires à haute densité, augmentant significativement la taille des instances résolues. Nous étudions trois formulations PLNE pour le Problème de la Somme Chromatique Minimale (PSCM). Chaque couleur est représentée par un entier naturel. Le PSCM cherche à minimiser la somme des cardinalités des sous-ensembles des sommets recevant la même couleur, pondérés par l'entier correspondant à la couleur, de telle sorte que toute paire de sommets adjacents reçoive des couleurs différentes. Nous nous concentrons sur l'étude d'une formulation étendue et proposons un algorithme de Branch-and-Price. / Given an undirected graph, the Vertex Coloring Problem (VCP) consists of assigning a color to each vertex of the graph such that two adjacent vertices do not share the same color and the total number of colors is minimized. DSATUR is an effective exact algorithm for the VCP. We introduce new lower bounding techniques enabling the computing of a lower bound at each node of the branching scheme. Our new DSATUR outperforms the state of the art for random VCP instances with high density, significantly increasing the size of solvable instances. Similar results can be achieved for a subset of high density DIMACS instances. We study three ILP formulations for the Minimum Sum Coloring Problem (MSCP). The problem is an extension of the classical Vertex Coloring Problem in which each color is represented by a positive natural number. The MSCP asks to minimize the sum of the cardinality of subsets of vertices receiving the same color, weighted by the index of the color, while ensuring that vertices linked by an edge receive different colors. We focus on studying an extended formulation and devise a complete Branch-and-Price algorithm.

Die 2^(k – 1) Typen der ungeordneten Zahlpartitionen mit k Parts (k-Partitionen) werden hier mit Hilfe der geordneten Partitionen von k definiert. Für jeden Typ gibt es eine erzeugende Funktion der geschlossenen Form mit eindeutiger Nummerierung. Die bekannte erzeugende Funktion der k-Partitionen ist die Summe dieser 2^(k – 1) typspezifischen erzeugenden Funktionen. Die Expansion dieser typspezifischen erzeugenden Funktionen in (unendlich lange) Potenzreihen ist rekursiv möglich. Untersucht werden Zerlegungen von erzeugenden Funktionen der einfachen Typen in erzeugende Funktionen anderer Typen. Damit lassen sich Bijektionen zwischen den Partitionen verschiedener Typen aufspüren. Die typspezifischen Betrachtungen werden auf die geordneten Partitionen und auf ihre erzeugenden Funktionen ausgeweitet.:1. Kurze Vorbetrachtung 2. Die Typen der ungeordneten k-Partitionen 3. Konstruktion einer typspezifischen GF (generating function) 4. Nummerierung und Symbolik für typspezifische GF’s 5. Die Summe aller typspezifischen GF’s 6. Multiplizieren elementarer Potenzreihen, Erzeugungsformeln 7. Rekursives Expandieren typspezifischer GF’s 8. Zahlen, die in k-Partitionen aller 2^(k – 1) Typen zerlegbar sind 9. Die Konjugierten der typspezifischen k-Partitionen 10. GF-Zerlegungen 10.1 Zerlegung der GF des Typs r = 2 10.2 Zerlegung der GF des Typs r = 3 11. Die typspezifischen GF’s der geordneten Partitionen 12. Literaturverzeichnis 13. Nachwort

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The development of sustainable business models is a challenging task since various factors might influence the results of an assessment. Given the complexity at the municipal level, system interdependencies between different alternatives need to be considered. One possibility to support decision makers is to apply energy system optimization models. Existing optimization models, however, ignore the roles different actors play and the resulting impact they have. To address this research issue, this thesis presents an integrated techno-economic optimization framework called IRPopt (Integrated Resource Planning and Optimization). A proven graph-based energy system approach allows the accurate modeling of deployment systems by considering different energy carriers and technical processes. In addition, a graph-based commercial association approach enables the integration of actor-oriented coordination. This is achieved by the explicit modeling of market actors on one layer and technology processes on another layer as well as resource flow interrelations and commercial agreements mechanism among and between the different layers. Using the optimization framework, various optimization problems are solvable on the basis of a generic objective function. For demonstration purposes, this thesis assesses the business models demand response and community storage. The applied examples demonstrate the modeling capabilities of the developed optimization framework. Further, the dispatch results show the usefulness of the described optimization approach.

Jede ungeordnete Zahlpartition mit k Parts (k-Partiton) hat einen Typ, der mittels einer geordneten Partition von k definiert werden kann. Es können somit 2^(k - 1) Typen definiert werden. Pro Typ gibt es eine eindeutig nummerierbare erzeugende Funktion der geschlossenen Form. Mit Rekursionen können diese Funktionen in (unendlich lange) Potenzreihen expandiert werden. Mit diesen erzeugenden Funktionen lassen sich Bijektionen zwischen den Partitionsmengen verschiedener Typen aufspüren.:1. Kurze Vorbetrachtung 2. Typen der ungeordneten k-Partitionen 3. Konstruktion der GF (generating function) des allgemeinen Typs 4. Nummerierung der konstruierten GF 5. Weitere Analysen zur konstruierten GF 6. Die konjugierten der typspezifischen k-Partitionen 7. Vereinfachte GF-Symbolik 8. Eine programmierbare Basis-GF 9. Dekomposition von Q(x, k) in typspezifische GF''s 10. Rekursives Expandieren typspezifischer GF''s 11. GF-Zerlegungen und Bijektionen 12. Zahlen, die in k-Partitionen aller Typen zerlegbar sind 13. Referenzen

Le traitement médical de choix pour la maladie rénale chronique est la transplantation d'organe. Cependant, plusieurs patients ne sont en mesure que de trouver un donneur direct avec lequel ils ne sont pas compatibles. Les Programmes de Don Croisé de Reins peuvent aider plusieurs paires donneur-patient incompatibles à échanger leur donneur entre elles. Typiquement, l'objectif principal d'un tel programme est de maximiser le nombre total de transplantations qui seront effectuées grâce à un plan d'échange. Plusieurs solutions optimales peuvent co-exister et comme la plupart correspondent à différents ensembles de patients obtenant un donneur compatible, il devient important de considérer quels individus seront sélectionnés. Fréquemment, ce problème n'est pas abordé et la première solution fournie par un solveur est choisie comme plan d'échange. Ceci peut mener à des parti-pris en faveur ou défaveur de certains patients, ce qui n'est pas considéré une approche juste. De plus, il est de la responsabilité des informaticiens de s'assurer du contrôle des résultats fournis par leurs algorithmes. Pour répondre à ce besoin, nous explorons l'emploi de multiples solutions optimales ainsi que la manière dont il est possible de sélectionner un plan d'échange parmi celles-ci. Nous proposons l'emploi de politiques aléatoires pour la sélection de solutions optimales suite à leur enumération. Cette tâche est accomplie grâce à la programmation en nombres entiers et à la programmation par contraintes. Nous introduisons aussi un nouveau concept intitulé équité individuelle. Ceci a pour but de trouver une politique juste pouvant être utilisée en collaboration avec les solutions énumerées. La mise à disposition de plusieurs métriques fait partie intégrante de la méthode. En faisant usage de la génération de colonnes en combinaison au métrique $L_1$, nous parvenons à applique la méthode à de plus larges graphes. Lors de l'évaluation de l'équité individuelle, nous analysons de façon systématique d'autres schémas d'équité tels que le principle d'Aristote, la justice Rawlsienne, le principe d'équité de Nash et les valeurs de Shapley. Nous étudions leur description mathématiques ainsi que leurs avantages et désavantages. Finalement, nous soulignons le besoin de considérer de multiples solutions, incluant des solutions non optimales en ce qui concerne le nombre de transplantations d'un plan d'échange. Pour la sélection d'une politique équitable ayant comme domaine un tel ensemble de solutions, nous notons l'importance de trouver un équilibre entre les mesures d'utilité et d'équité d'une solution. Nous utilisons le Programme de Bien-être Social de Nash afin de satisfaire à un tel objectif. Nous proposons aussi une méthodologie de décomposition qui permet d'étendre le système sous-jacent et de faciliter l'énumeration de solutions. / The preferred treatment for chronic kidney disease is transplantation. However, many patients can only find direct donors that are not fully compatible with them. Kidney Exchange Programs (KEPs) can help these patients by swapping the donors of multiple patient-donor pairs in order to accommodate them. Usually, the objective is to maximize the total number of transplants that can be realized as part of an exchange plan. Many optimal solutions can co-exist and since a large part of them features different subsets of patients that obtain a compatible donor, the question of who is selected becomes relevant. Often, this problem is not even addressed and the first solution returned by a solver is chosen as the exchange plan to be performed. This can lead to bias against some patients and thus is not considered a fair approach. Moreover, it is of the responsibility of computer scientists to have control of the output of the algorithms they design. To resolve this issue, we explore the use of multiple optimal solutions and how to pick an exchange plan among them. We propose the use of randomized policies for selecting an optimal solution, first by enumerating them. This task is achieved through both integer programming and constraint programming methods. We also introduce a new concept called individual fairness in a bid to find a fair policy over the enumerated solutions by making use of multiple metrics. We scale the method to larger instances by adding column generation as part of the enumeration with the $L_1$ metric. When evaluating individual fairness, we systematically review other fairness schemes such as Aristotle's principle, Rawlsian justice, Nash's principle of fairness, and Shapley values. We analyze their mathematical descriptions and their pros and cons. Finally, we motivate the need to consider solutions that are not optimal in the number of transplants. For the selection of a good policy over this larger set of solutions, we motivate the need to balance utility and our individual fairness measure. We use the Nash Social Welfare Program in order to achieve this, and we also propose a decomposition methodology to extend the machinery for an efficient enumeration of solutions.

This research is concerned with the modeling and development of algorithmic approaches for solving airport operational problems that arise in Air Traffic Control (ATC) systems within the terminal area at hub airports. Specifically, the problems addressed include the Aircraft Sequencing Problem (ASP) for runway operations, the Network Assignment Problem (NAP) for taxiway operations, and a simulation model for the evaluation of current or proposed ATC system in detail. For the ASP, we develop a mathematical model and apply the Reformulation-Linearization-Technique (RLT) of Sherali and Adams to construct an enhanced tightened version of the proposed model. Since ASP is NP-Hard and in fact, it is a variation of the well-known Traveling Salesman Problem with time-windows, sub-optimal solutions are usually derived to accommodate the real-time constraints of ATC systems. Nevertheless, we exhibit a significant advancement in this challenging class of problem. Also for the purpose of solving relatively large sized problems in practice, we develop and test suitable heuristic procedures. For the NAP, we propose a quasi-dynamic assignment scheme which is based on the incremental assignment technique. This quasi-dynamic assignment method assumes that the current aircraft route is influenced only by the previous aircraft assigned to the network. This simplified assumption obviates the need for iterative rerouting procedures to reach a pure equilibrium state which might not be achievable in practical taxiway operations. To evaluate the overall system, we develop a microscopic simulation model. The simulation model is designed to have the capability for reproducing not only the dynamic behavior of aircraft, but also incorporates communication activities between controllers and pilots. These activities are critical in ATC operations, and in some instances, might limit the capacity of the facility. Finally, using the developed simulation model named Virginia Tech Airport Simulation Model (VTASM) in concert with ASP and NAP, we compare the overall efficiencies of several control strategies, including that of the existing control system as well as of the proposed advanced control system. / Ph. D.

A novel approach for a robust computation of positional relations of two-dimensional geometric features is presented which guarantees reliable results, provided that the initial data is valid. The method is based on the use of integer coordinates and a method to generate a complete, gap-less and non-overlapping spatial decomposition. The spatial relationships of two geometric features are then represented using DE-9IM matrices. These allow the spatial relationships to be represented compactly. The DE-9IM matrices are based on the spatial decomposition using explicit neighborhood relations. No further geometric calculations are required for their computation. Based on comparative tests, it could be proven that this approach, up to a predictable limit, provides correct results and thus offers advantages over classical methods for the calculation of spatial relationships. This novel method can be used in all fields, especially where guaranteed reliable results are required.:Introduction Related Research Materials and Methods Results Discussion Outlook Author Contributions Funding Institutional Review Board Statement Informed Consent Statement Data Availability Statement Conflicts of Interest Abbreviations References