Este trabalho apresenta um estudo teórico da Técnica de Geração de Colunas (GC) aplicada em alguns Problemas de Roteamento de Veículo (PRV). Essa técnica foi inicialmente utilizada para tratar problemas de otimização de grande porte com estruturas especiais[Dantzig & Wolfe, 1960]. Dentre as diversas classes de problemas de roteamento; revisamos a aplicação dessa técnica a dois casos particulares: O problema de roteamento de helicópteros em plataformas marítimas, cujo objetivo minimizar o custo total do transporte; O problema de roteamento com janela de tempo, onde a função objetivo é descrita pelo tamanho da frota e o custo do percurso. Revisamos e implementamos um algoritmo de caminho mínimo com janela de tempo (CMJT). Esse algoritmo surge como um sub-problema do algoritmo Primai Simplex para resolver o problema de partição de conjunto, utilizado para modelar o problema de roteamento com janela de tempo. / This work presents a study about the Column Generation Technique (CG) applied to some Vehicle Ftouting Problems. The technique was first used to deal with optimization problems having special structures. Among the vaa-ious classes of routing problems, we review the use of the technique in two specific cases: Ftouting helicopters for crew exchanges on off-shore locations, where the objective is to minimize the total transportation cost; Ftouting with time windows, where the objective function is composed by the size of the fleet and the cost of route. We review and implement a shortest path algorithm with time windows. This algorithm aa-ises as a sub-problem in the Primai Simplex algorithm to solve the linear relaxation of the set partitioning problem used to model the routing problem with time windows.

Este trabalho apresenta um estudo teórico da Técnica de Geração de Colunas (GC) aplicada em alguns Problemas de Roteamento de Veículo (PRV). Essa técnica foi inicialmente utilizada para tratar problemas de otimização de grande porte com estruturas especiais[Dantzig & Wolfe, 1960]. Dentre as diversas classes de problemas de roteamento; revisamos a aplicação dessa técnica a dois casos particulares: O problema de roteamento de helicópteros em plataformas marítimas, cujo objetivo minimizar o custo total do transporte; O problema de roteamento com janela de tempo, onde a função objetivo é descrita pelo tamanho da frota e o custo do percurso. Revisamos e implementamos um algoritmo de caminho mínimo com janela de tempo (CMJT). Esse algoritmo surge como um sub-problema do algoritmo Primai Simplex para resolver o problema de partição de conjunto, utilizado para modelar o problema de roteamento com janela de tempo. / This work presents a study about the Column Generation Technique (CG) applied to some Vehicle Ftouting Problems. The technique was first used to deal with optimization problems having special structures. Among the vaa-ious classes of routing problems, we review the use of the technique in two specific cases: Ftouting helicopters for crew exchanges on off-shore locations, where the objective is to minimize the total transportation cost; Ftouting with time windows, where the objective function is composed by the size of the fleet and the cost of route. We review and implement a shortest path algorithm with time windows. This algorithm aa-ises as a sub-problem in the Primai Simplex algorithm to solve the linear relaxation of the set partitioning problem used to model the routing problem with time windows.

Avec les problèmes présents dans le secteur de transport, qu'ils soient financiers ou environnementaux, la mobilité avancée peut y remédier avec la mise à profit de la complémentarité entre les différents modes de transport. Dans ce contexte, nous nous focalisons dans cette thèse à la mise en œuvre d'un système d’information de transport avec la recherche et la composition des itinéraires comodaux pour les clients. L'enjeu est d'être capable de répondre aux attentes des usagers avec des solutions satisfaisantes permettant de proposer des itinéraires optimaux pour gérer efficacement l’intermodalité. Dans un souci pratique, nous fournirons des itinéraires attractifs respectant les contraintes imposées même pour les requêtes simultanées. Nous utilisons des techniques d'accélération permettant de réduire l'espace de recherche pour la planification d’itinéraire. Les itinéraires attractifs sont décomposés en sections de route sur lesquelles les différentes demandes et les offres disponibles sont mises en relation. Les combinaisons des sections de route permettent d'aboutir à un ensemble de solutions intéressantes. L’aspect distribué et dynamique du problème nous a permis d'employer une modélisation basée sur le paradigme agent. Ainsi, l’alliance entre les systèmes multi-agents et les algorithmes génétiques que nous avons mis en place s'avère très utile pour gérer l’articulation de l’intermodalité entre ces différents modes de transport. Les résultats de simulation présentés montrent l’efficacité des méthodes proposées. / Nowadays, the environment impact of transport is significant. In an attempt to address these problems, in this work, we are interested in the implementation of a transport information system, which integrates the existing means of transport to respond users' requests, including public transport and the shared transport like carpooling and car-sharing. In this context of application, we elaborate algorithms to provide attractive paths with respect to the imposed constraints, even for simultaneous requests. Different acceleration techniques for path planning are used to reduce the search space for a better performance. The attractive paths are divided into route sections on which the available offers are allocated to different requests, which is treated as one resource allocation problem using metaheuristics algorithms. With consideration of the distributed and dynamic aspects of the problem, the solving strategy makes use of several concepts like multi-agents system and different optimization methods. The proposed methods are tested with realistic scenarios with instances extracted from real world transport networks. The obtained results indicate that our proposed approaches can efficiently solve the itinerary planning problems by providing good and complete solutions.

<p>Traffic planning in a modern congested society is an important and time consuming procedure. Finding fast algorithms for solving traffic problems is therefore of great interest for traffic planners allover the world. </p><p>This thesis concerns solving the fixed demand traffic assignment problem (TAP) on a number of different transportation test networks. TAP is solved using the Frank-Wolfe algorithm and the shortest path problems that arise as subproblems to the Frank-Wolfe algorithm are solved using the network simplex algorithm. We evaluate how a number of existing pricing strategies to the network simplex algorithm performs with TAP. We also construct a new efficient pricing strategy, the Bucket Pricing Strategy, inspired by the heap implementation of Dijkstra's method for shortest path problems. This pricing strategy is, together with the actual use of the network simplex algorithm, the main result of the thesis and the pricing strategy is designed to take advantage of the special structure of TAP. In addition to performing tests on the conventional Frank-Wolfe algorithm, we also test how the different pricing strategies perform on Frank-Wolfe algorithms using conjugate and bi-conjugate search directions. </p><p>These test results show that the updating shortest path calculations obtained by using the network simplex outperforms the non-updating Frank-Wolfe algorithms. Comparisons with Bar-Gera's OBA show that our implementation, especially together with the bucket pricing strategy, also outperforms this algorithm for relative gaps down to 10E-6.</p>

Empirical studies show that similar patterns emerge from a large number of different biological systems. For example, the group size distributions of several fish species and house sparrows all follow power law distributions with an exponential truncation. Networks built by ant colonies, slime mold and those are designed by engineers resemble each other in terms of structure and transportation efficiency. Based on the investigation of experimental data, we propose a variety of simple stochastic models to unravel the underlying mechanisms which lead to the collective phenomena in different systems. All the mechanisms employed in these models are rooted in the concept of selective reinforcement. In some systems the reinforcement can build optimal solutions for biological problem solving. This thesis consists of five papers. In the first three papers, I collaborate with biologists to look into group formation in house sparrows  and the movement decisions of damsel fish.  In the last two articles, I look at how shortest paths and networks are  constructed by slime molds and pheromone laying ants, as well as studying  speed-accuracy tradeoffs in slime molds' decision making. The general goal of the study is to better understand how macro level patterns and behaviors emerges from micro level interactions in both spatial and non-spatial biological systems. With the combination of mathematical modeling and experimentation, we are able to reproduce the macro level patterns in the studied biological systems and predict behaviors of the systems using minimum number of parameters.

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We study energy-efficient power allocation among relays for lifetime maximization in a dual-hop relay network operated by amplify-and-forward relays with battery limitations. Power allocation algorithms are proposed for three different scenarios. First, we study the relay cooperation case where all the relays jointly support transmissions for a targeted data rate. By exploring the correlation of time-varying relay channels, we develop a prediction-based relay cooperation method for optimal power allocation strategy to improve the relay network lifetime over existing methods that do not predict the future channel state, or assume the current channel state remains static in the future. Next, we consider energy-efficient relay selection for the single source-destination case. Assuming finite transmission power levels, we propose a stochastic shortest path approach which gives the optimal relay selection decision to maximize the network lifetime. Due to the high computational complexity, a suboptimal prediction-based relay selection algorithm, directly coming from previous problem, is created. Finally, we extend our study to multiple source-destination case, where relay selection needs to be determined for each source-destination pair simultaneously. The network lifetime in the presence of multiple source-destination pairs is defined as the longest time when all source-destination pairs can maintain the target transmission rate. We design relay-to-destination mapping algorithms to prolong the network lifeii time. They all aim at maximizing the perceived network lifetime at the current time slot. The optimal max-min approach and suboptimal user-priority based approach are proposed with different levels of computational complexity. / UOIT

Traffic planning in a modern congested society is an important and time consuming procedure. Finding fast algorithms for solving traffic problems is therefore of great interest for traffic planners allover the world. This thesis concerns solving the fixed demand traffic assignment problem (TAP) on a number of different transportation test networks. TAP is solved using the Frank-Wolfe algorithm and the shortest path problems that arise as subproblems to the Frank-Wolfe algorithm are solved using the network simplex algorithm. We evaluate how a number of existing pricing strategies to the network simplex algorithm performs with TAP. We also construct a new efficient pricing strategy, the Bucket Pricing Strategy, inspired by the heap implementation of Dijkstra's method for shortest path problems. This pricing strategy is, together with the actual use of the network simplex algorithm, the main result of the thesis and the pricing strategy is designed to take advantage of the special structure of TAP. In addition to performing tests on the conventional Frank-Wolfe algorithm, we also test how the different pricing strategies perform on Frank-Wolfe algorithms using conjugate and bi-conjugate search directions. These test results show that the updating shortest path calculations obtained by using the network simplex outperforms the non-updating Frank-Wolfe algorithms. Comparisons with Bar-Gera's OBA show that our implementation, especially together with the bucket pricing strategy, also outperforms this algorithm for relative gaps down to 10E-6.

En aquesta tesi es solucionen problemes de visibilitat i proximitat sobre superfícies triangulades considerant elements generalitzats. Com a elements generalitzats considerem:punts, segments, poligonals i polígons. Les estrategies que proposem utilitzen algoritmesde geometria computacional i hardware gràfic. Comencem tractant els problemes de visibilitat sobre models de terrenys triangulats considerant un conjunt d'elements de visió generalitzats. Es presenten dos mètodes per obtenir, de forma aproximada, mapes de multi-visibilitat. Un mapa de multi-visibilitat és la subdivisió del domini del terreny que codifica la visibilitat d'acord amb diferents criteris. El primer mètode, de difícil implementació, utilitza informació de visibilitat exacte per reconstruir de forma aproximada el mapa de multi-visibilitat. El segon, que va acompanyat de resultats d'implementació, obté informació de visibilitat aproximada percalcular i visualitzar mapes de multi-visibilitat discrets mitjançant hardware gràfic. Coma aplicacions es resolen problemes de multi-visibilitat entre regions i es responen preguntessobre la multi-visibilitat d'un punt o d'una regió. A continuació tractem els problemes de proximitat sobre superfícies polièdriques triangulades considerant seus generalitzades. Es presenten dos mètodes, amb resultats d'implementació, per calcular distàncies des de seus generalitzades sobre superfícies polièdriques on hi poden haver obstacles generalitzats. El primer mètode calcula, de forma exacte, les distàncies definides pels camins més curts des de les seus als punts del poliedre. El segon mètode calcula, de forma aproximada, distàncies considerant els camins més curts sobre superfícies polièdriques amb pesos. Com a aplicacions, es calculen diagrames de Voronoi d'ordre k, i es resolen, de forma aproximada, alguns problemes de localització de serveis. També es proporciona un estudi teòric sobre la complexitat dels diagrames de Voronoi d'ordre k d'un conjunt de seus generalitzades en un poliedre sense pesos. / In this thesis, we solve visibility and proximity problems on triangulated surfaces concerning generalized elements. As generalized elements, we consider: points, segments, polygonal chains and polygonal regions. The proposed strategies use algorithms of Computational Geometry and Graphics Hardware. We start by studying multi-visibility problems on triangulated terrain models concerning a set of generalized view elements. We present two methods to obtain approximate multi-visibility maps. A multi-visibility map is a subdivision of the terrain domain encoding visibility according to different criteria. The first method, of complex implementation, uses exactly computed visibility information to approximately reconstruct the unknown multi-visibility map. The second, from which implementation results are provided, uses approximate visibility information to compute and visualize discrete multi-visibility maps by exploiting graphics hardware capabilities. As applications, we compute multi-visibility maps, solve inter-region multi-visibility problems and approximately answer point and polygonal region multi-visibility queries. Next, we tackle proximity problems on triangulated polyhedral surfaces, where generalized obstacles are allowed, considering generalized sources. We present two methods, with implementation results, to compute distances on polyhedral surfaces from a generalized source. The first method computes exact shortest path distances from generalized sources. The second provides approximate weighted shortest path distances from generalized sites on weighted polyhedral surfaces. Both methods are posteriorly extended to handle the multiple-site problem where the corresponding distance field is obtained. As applications, we compute discrete order-k Voronoi diagrams and approximately solve some facility location problems. We also provide a theoretical study on the order-k Voronoi diagram complexity of a set of generalized sources for the non-weighted case.

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.