Shortest path problems arise in a variety of applications ranging from transportation planning to network routing among others. One group of these problems involves finding shortest paths in graphs where the edge weights are defined by probability distributions. While some research has addressed the problem of finding a single shortest path, no research has been done on finding multiple paths in such graphs. This thesis addresses the problem of finding paths for multiple robots through a graph in which the edge weights represent the probability that each edge will fail. The objective is to find paths for n robots that maximize the probability that at least k of them will arrive at the destination. If we make certain restrictions on the edge weights and topology of the graph, this problem can be solved in O(n log n)time. If we restrict only the topology, we can find approximate solutions which are still guaranteed to be better than the single most reliable path.

This dissertation contributes to the ongoing national project concerning the \emph{Next Generation Air Transportation System} (NextGen) that endeavors, in particular, to reshape the management of air traffic in the continental United States. Our work is part of this effort and mainly concerns modeling and algorithmic enhancements to the Airspace Planning and Collaborative Decision-Making Model (APCDM). First, we augment the APCDM to study an \emph{Airspace Flow Program} (AFP) in the context of weather-related disruptions. The proposed model selects among alternative flight plans for the affected flights while simultaneously (a) integrating slot-exchange mechanisms induced by multiple Ground Delay Programs (GDPs) to permit airlines to improve flight efficiencies through a mediated bartering of assigned slots, and (b) considering issues related to sector workloads, airspace conflicts, as well as overall equity concerns among the involved airlines in regard to accepted slot trades and flight plans. More specifically, the APCDM is enhanced to include the following: a. The revised model accommodates continuing flights, where some flight cannot depart until a prerequisite flight has arrived. Such a situation arises, for example, when the same aircraft will be used for the departing flight. b. We model a slot-exchange mechanism to accommodate flights being involved in multiple trade offers, and to permit slot trades at multiple GDP airports (whence the flight connection constraints become especially relevant). We also model flight cancelations whereby, if a flight assigned to a particular slot is canceled, the corresponding vacated slot would be made available for use in the slot-exchange process. c. Alternative equity concepts are presented, which more accurately reflect the measures used by the airlines. d. A reduced variant of the APCDM, referred to as \textbf{APCDM-Light}, is also developed. This model serves as a fast-running version of APCDM to be used for quick-turn analyses, where the level of modeling detail, as well as data requirements, are reduced to focus only on certain key elements of the problem. e. As an alternative for handling large-scale instances of APCDM more effectively, we present a \emph{sequential variable fixing heuristic} (SFH). The list of flights is first partitioned into suitable subsets. For the first subset, the corresponding decision variables are constrained to be binary-valued (which is the default for these decision variables), while the other variables are allowed to vary continuously between 0 and 1. If the resulting solution to this relaxed model is integral, the algorithm terminates. Otherwise, the binary variables are fixed to their currently prescribed values and another subset of variables is designated to be binary constrained. The process repeats until an integer solution is found or the heuristic encounters infeasibility. f. We experiment with using the APCDM model in a \emph{dynamic, rolling-horizon framework}, where we apply the model on some periodic basis (e.g., hourly), and where each sequential run of the model has certain flight plan selections that are fixed (such as flights that are already airborne), while we consider the selection among alternative flight plans for other imminent flights in a look-ahead horizon (e.g., two hours). These enhancements allow us to significantly expand the functionality of the original APCDM model. We test the revised model and its variants using realistic data derived from the \emph{Enhanced Traffic Management System} (ETMS) provided by the \emph{Federal Aviation Administration} (FAA). One of the new equity methods, which is based on average delay per passenger (or weighted average delay per flight), turns out to be a particularly robust way to model equity considerations in conjunction with sector workloads, conflict resolution, and slot-exchanges. With this equity method, we were able to solve large problem instances (1,000 flights) within 30 seconds on average using a 1\% optimality tolerance. The model also produced comparable solutions within about 20 seconds on average using the Sequential Fixing Heuristic (SFH). The actual solutions obtained for these largest problem instances were well within 1\% of the best known solution. Furthermore, our computations revealed that APCDM-Light can be readily optimized to a 0.01\% tolerance within about 5 seconds on average for the 1,000 flight problems. Thus, the augmented APCDM model offers a viable tool that can be used for tactical air traffic management purposes as an airspace flow program (particularly, APCDM-Light), as well as for strategic applications to study the impact of different types of trade restrictions, collaboration policies, equity concepts, and airspace sectorizations. The modeling of slot ownership in the APCDM motivates another problem: that of generating detoured flight plans that must arrive at a particular slot time under severe convective weather conditions. This leads to a particular class of network flow problems that seeks a shortest path, if it exists, between a source node and a destination node in a connected digraph $G(N,A)$, such that we arrive at the destination at a specified time while leaving the source no earlier than a lower bounding time, and where the availability of each network link is time-dependent in the sense that it can be traversed only during specified intervals of time. We refer to this problem as the \emph{reverse time-restricted shortest path problem} (RTSP). We show that RTSP is NP-hard in general and propose a dynamic programming algorithm for finding an optimal solution in pseudo-polynomial time. Moreover, under a special regularity condition, we prove that the problem is polynomially solvable with a complexity of order $O(|N / A|)$. Computational results using real flight generation test cases as well as random simulated problems are presented to demonstrate the efficiency of the proposed solution procedures. The current airspace configuration consists of sectors that have evolved over time based on historical traffic flow patterns. \citet{kopardekar_dyn_resect_2007} note that, given the current airspace configuration, some air traffic controller resources are likely under-utilized, and they also point out that the current configuration limits flexibility. Moreover, under the free-flight concept, which advocates a relaxation of waypoint traversals in favor of wind-optimized trajectories, the current airspace configuration will not likely be compatible with future air traffic flow patterns. Accordingly, one of the goals for the \emph{NextGen Air Transportation System} includes redesigning the airspace to increase its capacity and flexibility. With this motivation, we present several methods for defining sectors within the \emph{National Airspace System} (NAS) based on a measure of sector workload. Specifically, given a convex polygon in two-dimensions and a set of weighted grid points within the region encompassed by the polygon, we present several mixed-integer-programming-based algorithms to generate a plane (or line) bisecting the region such that the total weight distribution on either side of the plane is relatively balanced. This process generates two new polygons, which are in turn bisected until some target number of regions is reached. The motivation for these algorithms is to dynamically reconfigure airspace sectors to balance predicted air-traffic controller workload. We frame the problem in the context of airspace design, and then present and compare four algorithmic variants for solving these problems. We also discuss how to accommodate monitoring, conflict resolution, and inter-sector coordination workloads to appropriately define grid point weights and to conduct the partitioning process in this context. The proposed methodology is illustrated using a basic example to assess the overall effect of each algorithm and to provide insights into their relative computational efficiency and the quality of solutions produced. A particular competitive algorithmic variant is then used to configure a region of airspace over the U.S. using realistic flight data. The development of the APCDM is part of an ongoing \emph{NextGen} research project, which envisages the sequential use of a variety of models pertaining to three tiers. The \emph{Tier 1} models are conceived to be more strategic in scope and attempt to identify potential problematic areas, e.g., areas of congestion resulting from a severe convective weather system over a given time-frame, and provide aggregate measures of sector workloads and delays. The affected flow constrained areas (FCAs) highlighted by the results from these \emph{Tier 1} models would then be analyzed by more detailed \emph{Tier 2} models, such as APCDM, which consider more specific alternative flight plan trajectories through the different sectors along with related sector workload, aircraft conflict, and airline equity issues. Finally, \emph{Tier 3} models are being developed to dynamically examine smaller-scaled, localized fast-response readjustments in air traffic flows within the time-frame of about an hour prior to departure (e.g., to take advantage of a break in the convective weather system). The APCDM is flexible, and perhaps unique, in that it can be used effectively in all three tiers. Moreover, as a strategic tool, analysts could use the APCDM to evaluate the suitability of potential airspace sectorization strategies, for example, as well as identify potential capacity shortfalls under any given sector configuration. / Ph. D.

Lagrangian relaxation is a successful solution approach for many combinatorial optimisation problems, one of them being the train timetabling problem (TTP). We model this problem using time expanded networks for the single train schedules and coupling constraints to enforce restrictions like station capacities and headway times. Lagrangian relaxation of these coupling constraints leads to shortest path subproblems in the time expanded networks and is solved using a proximal bundle method. However, large instances of our practical partner Deutsche Bahn lead to computationally intractable models. In this thesis we develop two new algorithmic techniques to improve the solution process for this kind of optimisation problems. The first new technique, Dynamic Graph Generation (DGG), aims at improving the computation of the shortest path subproblems in large time expanded networks. Without sacrificing any accuracy, DGG allows to store only small parts of the networks and to dynamically extend them whenever the stored part proves to be too small. This is possible by exploiting the properties of the objective function in many scheduling applications to prefer early paths or due times, respectively. We prove that DGG can be implemented very efficiently and its running time and the size of nodes that have to be stored additionally does not depend on the size of the time expanded network but only on the length of the train routes. The second technique is an asynchronous and parallel bundle method (APBM). Traditional bundle methods require one solution of each subproblem in each iteration. However, many practical applications, e.g. the TTP, consist of rather loosely coupled subproblems. The APBM chooses only small subspaces corresponding to the Lagrange multipliers of strongly violated coupling constraints and optimises only these variables while keeping all other variables fixed. Several subspaces of disjoint variables may be chosen simultaneously and are optimised in parallel. The solutions of the subspace problem are incorporated into the global data as soon as it is available without any synchronisation mechanism. However, in order to guarantee convergence, the algorithm detects automatically dependencies between different subspaces and respects these dependencies in future subspace selections. We prove the convergence of the APBM under reasonable assumptions for both, the dual and associated primal aggregate data. The APBM is then further extended to problems with unknown dependencies between subproblems and constraints in the Lagrangian relaxation problem. The algorithm automatically detects these dependencies and respects them in future iterations. Again we prove the convergence of this algorithm under reasonable assumptions. Finally we test our solution approach for the TTP on some real world instances of Deutsche Bahn. Using an iterative rounding heuristic based on the approximate fractional solutions obtained by the Lagrangian relaxation we are able to compute feasible schedules for all trains in a subnetwork of about 10% of the whole German network in about 12 hours. In these timetables 99% of all passenger trains could be scheduled with no significant delay and the travel time of the freight trains could be reduced by about one hour on average.

This dissertation is concerned with the problem of estimating travel time per links in urban context using sparsely sampled GPS data. One of the challenges in this thesis is use the sparsely sampled data. A part of this research work, i developed a digital map with its new geographic information system (GIS), dealing with map-matching problem, where we come out with an enhancement tecnique, and also the shortest path problem.The thesis research work was conduct within the project PUMAS, which is an avantage for our research regarding the collection process of our data from the real world field and also in making our tests. The project PUMAS (Plate-forme Urbaine de Mobilité Avancée et Soutenable / Urban Platform for Sustainable and Advanced Mobility) is a preindustrial project that has the objective to inform about the traffic situation and also to develop an implement a platform for sustainable mobility in order to evaluate it in the region, specifically Rouen, France. The result is a framework for any traffic controller or manager and also estimation researcher to access vast stores of data about the traffic estimation, forecasting and status.

Cette thèse porte sur le problème de l'estimation des temps de parcours, de véhicules, par section de route dans un contexte urbain, en utilisant les données GPS à faible densité d’échantillon. L'un des défis de cette thèse est d'utiliser ce genre de données. Dans le cadre de ce travail de recherche, j'ai développé une carte numérique avec son nouveau système d'information géographique (SIG), qui traite la problématique du map-matching, où nous avons apporté des améliorations, ainsi que le problème du plus court chemin.La thèse s'inscrit dans le cadre du projet PUMAS (Plate-forme Urbaine de Mobilité Avancée et Soutenable), ce qui est un avantage pour nos recherches en ce qui concerne le processus de collecte de données réelles sur le terrain ainsi que pour faire nos tests. Le projet PUMAS est un projet préindustriel qui a pour objectif d'informer sur la situation du trafic mais également de développer et de mettre en œuvre une plate-forme de mobilité durable afin de l'évaluer dans la région, notamment à Rouen, France. Le résultat offre un cadre pour tout contrôleur de la situation, gestionnaire ou chercheur pour accéder à de vastes réserves de données sur l'estimation du flux du trafic, sur les prévisions et sur l'état du trafic. / This dissertation is concerned with the problem of estimating travel time per links in urban context using sparsely sampled GPS data. One of the challenges in this thesis is use the sparsely sampled data. A part of this research work, i developed a digital map with its new geographic information system (GIS), dealing with map-matching problem, where we come out with an enhancement tecnique, and also the shortest path problem.The thesis research work was conduct within the project PUMAS, which is an avantage for our research regarding the collection process of our data from the real world field and also in making our tests. The project PUMAS (Plate-forme Urbaine de Mobilité Avancée et Soutenable / Urban Platform for Sustainable and Advanced Mobility) is a preindustrial project that has the objective to inform about the traffic situation and also to develop an implement a platform for sustainable mobility in order to evaluate it in the region, specifically Rouen, France. The result is a framework for any traffic controller or manager and also estimation researcher to access vast stores of data about the traffic estimation, forecasting and status.

Les travaux présentés dans cette thèse, qui s’inscrivent dans le cadre du projet franco-allemand DSS_Evac_Logistic, visent à proposer des méthodes permettant de calculer des plans d’évacuation macroscopiques d’une ville lors d’une catastrophe majeure. Deux problèmes d’évacuations sont considérés dans cette thèse : le problème d’évacuation par bus et le problème d’évacuation par bus et voitures. Le problème d’évacuation par bus a pour objectif de définir un plan d’évacuation afin de mettre à l’abri les évacués. Dans cette thèse, nous nous sommes intéressés à l’étude de trois versions du problème d’évacuation par bus. La première version est monocritère où nous cherchons à minimiser la date de fin d’évacuation. Puis, dans le second problème et afin d’assurer la sécurité des évacués, nous avons considéré une version bicritère qui généralise le cas monocritère, en incluant le risque encouru lors de l’évacuation des personnes. Les deux critères à minimiser sont la date de fin d’évacuation et le risque. La troisième version est une version robuste bicritère qui permet d’appréhender l’incertitude sur les données. Le but est de minimiser à la fois la date de fin d’évacuation et les modifications apportées sur une solution, de sorte qu’elle soit réalisable pour n’importe quel scénario de données. Pour résoudre ces problèmes d’évacuation par bus, nous avons proposé des méthodes exactes et des méthodes heuristiques. / The work presented in this thesis, which is a part of the Franco-German project DSS_Evac_Logistic, aims at proposing methods to calculate macroscopic evacuation plans for mid-size towns after a tremendous disaster. Two evacuation problems have been tackled in this thesis : the bus evacuation problem and bus-and-vehicle evacuation problem. The bus evacuation problem aims at calculating an evacuation plan to relocate evacuees outside the endangered area. In this thesis, we consider three versions of the bus evacuation problem. The first one is a monocriterion problem, where the objective is to minimize the maximum evacuation time. In order to guarantee the safety of evacuees, we have considered a bicriteria problem, which is a generalization of the monocriterion version, in which we take into consideration the risk exposure of the evacuees. Consequently, the bicriteria problem is solved by minimizing the total evacuation time and the risk. The third version is a bicriteria robust version because most of the planning data is subject to uncertainty. The goal is to minimize both the evacuation time and the vulnerability of the schedule that is subject to different evacuation circumstances. To solve all the versions of the bus evacuation problem, we have developed exact solutions based on mathematical formulation to address small instances and heuristic solutions to deal with larger instances.

Cette thèse développe des algorithmes pour les problèmes de plus court chemin sous cont-rain-tes de ressources, et les applique à l'optimisation des rotations des avions et des équipages d'une compagnie aérienne dans le cadre d'approches par génération de colonnes.Les problèmes de plus court chemin sous contraintes de ressources sont généralement résolus grâce à une énumération intelligente de tous les chemins non dominés. Les approches récentes utilisent des bornes sur les ressources des chemins pour éliminer des solutions partielles. L'efficacité de la méthode est conditionnée par la qualité des bornes utilisées. Notre principale contribution au domaine est l'introduction d'une procédure générique pour calculer des bornes qui s'applique à la plupart des problèmes de chemins sous contraintes, et en particulier les problèmes stochastiques. A cette fin, nous introduisons une généralisation du problème de plus court chemin sous contraintes dans laquelle les ressources des chemins appartiennent à un monoïde ordonné comme un treillis. La ressource d'un chemin est la somme des ressources de ses arcs, le terme somme désignant l'opérateur du monoïde. Le problème consiste à trouver parmi les chemins qui satisfont une contrainte donnée celui dont la ressource minimise une fonction de coût croissante de la ressource des chemins. Nous généralisons les algorithmes d'énumération à ce nouveau problème. La théorie des treillis nous permet de construire une procédure polynomiale pour trouver des bornes de qualité. L'efficacité pratique de la méthode est évaluée au travers d'une étude numérique détaillée sur des problèmes de chemins déterministes et stochastiques. Les procédures de calcul des bornes peuvent être interprétées comme des généralisations aux monoïdes ordonnés comme des treillis d'algorithmes de la littérature définis pour résoudre un problème de chemin pour lequel les ressources des chemins prennent leur valeur dans un semi-anneau.Nos algorithmes de chemins ont été appliqués avec succès au problème de crew pairing. Étant donné un ensemble de vols opérés par une compagnie aérienne, les problèmes d'aircraft routing et de crew pairing construisent respectivement les séquences de vols opérées par les avions et par les équipages de manière à couvrir tous les vols à moindre coût. Comme certaines séquences de vols ne peuvent être réalisées par un équipage que s'il reste dans le même avion, les deux problèmes sont liés. La pratique actuelle dans l'industrie aéronautique est de résoudre tout d'abord le problème d'aircraft routing, puis le problème de crew pairing, ce qui aboutit à une solution non-optimale. Des méthodes de résolution pour le problème intégré ont été développées ces dix dernières années. Nous proposons une méthode de résolution pour le problème intégré reposant sur deux nouveaux ingrédients : un programme linéaire en nombre entier compact pour le problème d'aircraft routing, ainsi que de nouveaux pour le problème esclave de l'approche usuelle par génération de colonnes du problème de crew pairing. Ces algorithmes pour le problème esclave sont une application de nos algorithmes pour le problème de plus court chemin sous contraintes. Nous généralisons ensuite cette approche de manière à prendre en compte des contraintes de probabilités sur la propagation du retard. Ces algorithmes permettent de résoudre quasiment à l'optimum les instances industrielles d'Air France / This thesis develops algorithms for resource constrained shortest path problems, and uses them to solve the pricing subproblems of column generation approaches to some airline operations problems.Resource constrained shortest path problems are usually solved using a smart enumeration of the non-dominated paths. Recent improvements of these enumeration algorithms rely on the use of bounds on path resources to discard partial solutions. The quality of the bounds determines the performance of the algorithm. Our main contribution to the topic is to introduce a standard procedure to generate bounds on paths resources in a general setting which covers most resource constrained shortest path problems, among which stochastic versions. In that purpose, we introduce a generalization of the resource constrained shortest path problem where the resources are taken in a lattice ordered monoid. The resource of a path is the monoid sum of the resources of its arcs. The problem consists in finding a path whose resource minimizes a non-decreasing cost function of the path resource among the paths that satisfy a given constraint. Enumeration algorithms are generalized to this framework. We use lattice theory to provide polynomial procedures to find good quality bounds. The efficiency of the approach is proved through an extensive numerical study on deterministic and stochastic path problems. Interestingly, the bounding procedures can be seen as generalizations to lattice ordered monoids of some algebraic path problem algorithms which initially work with resources in a semiring.Given a set of flight legs operated by an airline, the aircraft routing and the crew pairing problem build respectively the sequences of flight legs operated by airplanes and crews at minimum cost. As some sequences of flight legs can be operated by crews only if they stay in the same aircraft, the two problems are linked. The current practice in the industry is to solve first the aircraft routing, and then the crew pairing problem, leading to a non-optimal solution. During the last decade, solution schemes for the integrated problem have been developed. We propose a solution scheme for the integrated problem based on two new ingredients: a compact integer program approach to the aircraft routing problem, and a new algorithm for the pricing subproblem of the usual column generation approach to the crew pairing problem, which is based on our resource constrained shortest path framework. We then generalize the algorithm to take into account delay propagation through probabilistic constraints. The algorithms enable to solve to near optimality Air France industrial instances

Lagrangian relaxation is a successful solution approach for many combinatorial optimisation problems, one of them being the train timetabling problem (TTP). We model this problem using time expanded networks for the single train schedules and coupling constraints to enforce restrictions like station capacities and headway times. Lagrangian relaxation of these coupling constraints leads to shortest path subproblems in the time expanded networks and is solved using a proximal bundle method. However, large instances of our practical partner Deutsche Bahn lead to computationally intractable models. In this thesis we develop two new algorithmic techniques to improve the solution process for this kind of optimisation problems. The first new technique, Dynamic Graph Generation (DGG), aims at improving the computation of the shortest path subproblems in large time expanded networks. Without sacrificing any accuracy, DGG allows to store only small parts of the networks and to dynamically extend them whenever the stored part proves to be too small. This is possible by exploiting the properties of the objective function in many scheduling applications to prefer early paths or due times, respectively. We prove that DGG can be implemented very efficiently and its running time and the size of nodes that have to be stored additionally does not depend on the size of the time expanded network but only on the length of the train routes. The second technique is an asynchronous and parallel bundle method (APBM). Traditional bundle methods require one solution of each subproblem in each iteration. However, many practical applications, e.g. the TTP, consist of rather loosely coupled subproblems. The APBM chooses only small subspaces corresponding to the Lagrange multipliers of strongly violated coupling constraints and optimises only these variables while keeping all other variables fixed. Several subspaces of disjoint variables may be chosen simultaneously and are optimised in parallel. The solutions of the subspace problem are incorporated into the global data as soon as it is available without any synchronisation mechanism. However, in order to guarantee convergence, the algorithm detects automatically dependencies between different subspaces and respects these dependencies in future subspace selections. We prove the convergence of the APBM under reasonable assumptions for both, the dual and associated primal aggregate data. The APBM is then further extended to problems with unknown dependencies between subproblems and constraints in the Lagrangian relaxation problem. The algorithm automatically detects these dependencies and respects them in future iterations. Again we prove the convergence of this algorithm under reasonable assumptions. Finally we test our solution approach for the TTP on some real world instances of Deutsche Bahn. Using an iterative rounding heuristic based on the approximate fractional solutions obtained by the Lagrangian relaxation we are able to compute feasible schedules for all trains in a subnetwork of about 10% of the whole German network in about 12 hours. In these timetables 99% of all passenger trains could be scheduled with no significant delay and the travel time of the freight trains could be reduced by about one hour on average.