Modelo de otimização binível e biobjetivo para a escolha e localização de sensores

Tese de mestrado em Investigação Operacional, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2012Um grande número de estudos tem sido desenvolvido para a resolução de problemas com vários objetivos (problemas multiobjetivo) ou vários níveis de decisão (problemas multinível). No entanto, estudos sobre problemas que combinem estas duas características ainda são escassos. Este trabalho tem como objetivo propor uma resolução do problema, binível e biobjetivo, da escolha da configuração de uma rede de sensores e a sua respetiva localização num espaço bi-dimensional (nomeadamente num quadrado), de forma a minimizar os custos totais (aquisição e instalação) e maximizar a percentagem de área coberta (e consequentemente, reduzir o risco de não-deteção). Para resolver o problema foi desenvolvido um algoritmo evolutivo, que percorre o conjunto de soluções de Nível Superior (com o número de cada tipo de sensor a usar), explorando para cada uma dessas soluções, no Nível Inferior, as melhores localizações para os sensores. O algoritmo apresenta como output uma lista de soluções potencialmente não-dominadas. De forma a compreender para que conjuntos de parâmetros o algoritmo apresenta melhores resultados, foram executados diversos testes, com as várias fronteiras potencialmente de Pareto resultantes a serem comparadas através do indicador de qualidade, hipervolume. Este indicador permite comparar diferentes fronteiras potencialmente de Pareto, avaliando a dominância das soluções e a sua distribuição ao longo do Espaço dos Objetivos. Os resultados dos testes permitiram identificar oportunidades para tornar o algoritmo mais eficiente. Assim será conveniente integrar um método que permita uma pesquisa não-exaustiva e inteligente no Nível Superior, bem como um método que seja capaz de distribuir melhor os sensores no espaço admissível. Adicionalmente, este algoritmo deverá permitir resolver problemas com regiões admissíveis e avaliações de custo e risco mais realistas.A large number of studies have been developed to solve problems with multiple objectives (multi-objective optimization) or multiple decision levels (multi-level optimization). Nevertheless, studies about problems that combine these two characteristics are still rare. The objective of this dissertation is to offer a possible resolution to the bi-level and bi-objective problem, that is choosing a sensor network configuration and its distribution on a bidimensional space (namely, a square), in order to minimize the total costs (procurement, installation) and to maximize the percentage of covered area (and therefore, to reduce the risk of non-detection). In order to solve this problem, an evolutionary algorithm was developed that analyzes the solution set at the Upper Level (with the number of each type of sensor to be used), searching, in the Lower Level, the best locations for any of those sensors. The algorithm produces as output a list of potentially non-dominated solutions. In order to better understand the set of parameters to which the algorithm provides betters results, various tests were executed, with the potential Pareto frontiers given as an output, being compared by the quality indicator, hypervolume. This indicator allows us to compare different potential Pareto frontiers, by evaluating the dominance of their solutions and their distribution in the Objective Space

Models for the piecewise linear unsplittable multicommodity flow problems

International audienceIn this paper, we consider multicommodity flow problems, with unsplit-table flows and piecewise linear routing costs. We first focus on the case where the piecewise linear routing costs are convex. We show that this problem is N P-hard for the general case, but polynomially solvable when there is only one commodity. We then propose a strengthened mixed-integer programming formulation for the problem. We show that the linear relaxation of this formulation always gives the optimal solution of the problem for the single commodity case. We present a wide array of computational experiments, showing this formulation also produces very tight linear programming bounds for the multi-commodity case. Finally, we also adapt our formulation for the non-convex case. Our experimental results imply that the linear programming bounds for this case, are only slightly than the ones of state-of-the-art models for the splittable flow version of the problem

Optimal design of switched Ethernet networks implementing the Multiple Spanning Tree Protocol

International audienceSwitched Ethernet networks rely on the Spanning Tree Protocol (STP) to ensure a cycle-free connectivity between nodes, by reducing the topology of the network to a spanning tree. The Multiple Spanning Tree Protocol (MSTP) allows for the providers to partition the traffic in the network and assign it to different virtual local area networks, each satisfying the STP. In this manner, it is possible to make a more efficient use of the physical resources in the network. In this paper we consider the traffic engineering problem of finding optimal designs of switched Ethernet networks implementing the MSTP, such that the worst-case link utilization is minimized. We show that this problem is N P-hard. We propose three mixed-integer linear programming formulations for this problem. Through a large set of computational experiments, we compare the performance of these formulations. Until now, the problem was almost exclusively solved with heuristics. Our objective here is provide a first comparison of different models that can be used in exact methods

Modèles et méthodes pour les problèmes d'ingéniérie de trafic mono-routage.

Traffic Engineering (TE) uses methods and models from a variety of mathematical fields, such as statistics and optimization, to improve the performance of telecommunication networks. In this thesis, we study TE problems dealing with networks that impose single-path routing. As the name infers, in this type of routing, the traffic flow of each "commodity" cannot be split in its path between its origin and destination. Given its cheap cost, single-path routing is widely used in today's data centers, where thousands of stored servers perform computations or host Internet services. One common case of single-path routing is the one enforced by the Spanning Tree Protocol (STP) in switched Ethernet networks. The STP requires the network to keep its activated links loop-free, while maintaining the other redundant links ready for back-up, in case of link failure. The Multiple Spanning Tree Protocol (MSTP) extends the STP by installing multiple virtual networks compliant with the STP, over a single physical topology. Therefore, the MSTP is greatly beneficial for network service providers, as it allows for a more efficient use of the existing resources.Network design problems dealing with the MSTP are generally highly combinatorial and very hard to solve. As such, TE literature mainly suggests heuristic methods, which can quickly produce reasonable designs. Notwithstanding, due to a scarce existence of lower bounds to the optimum values of such problems, there is little knowledge about the quality of the solutions provided by these heuristics.In this sense, we propose mathematical programming models and methods that can provide optimal designs for these networks, or at the very least, obtain valid lower bounds. Taking into mind the goal of avoiding congestion in the network, we focus on two problems that deal with the following load-balancing objectives: the minimization of the worst-case link utilization, and the minimization of flow costs given by piecewise linear functions that penalize heavily-loaded links. The study of both these problems yielded relevant by-products: the first is the study of a MSTP network design problem, where we minimize the total load, and the second is the study of a fundamental unsplittable multicommodity flow problem with piecewise linear costs.For all the considered problems, we provide studies of complexity, extensive polyhedral studies to compare the proposed formulations, and a wide array of computational experiments to evaluate the performance of the proposed models and methods.

Optimal design of switched Ethernet networks implementing the Multiple Spanning Tree Protocol

International audience; Switched Ethernet networks rely on the Spanning Tree Protocol (STP) to ensure a cycle-free connectivity between nodes, by reducing the topology of the network to a spanning tree. The Multiple Spanning Tree Protocol (MSTP) allows for the providers to partition the traffic in the network and assign it to different virtual local area networks, each satisfying the STP. In this manner, it is possible to make a more efficient use of the physical resources in the network. In this paper we consider the traffic engineering problem of finding optimal designs of switched Ethernet networks implementing the MSTP, such that the worst-case link utilization is minimized. We show that this problem is N P-hard. We propose three mixed-integer linear programming formulations for this problem. Through a large set of computational experiments, we compare the performance of these formulations. Until now, the problem was almost exclusively solved with heuristics. Our objective here is provide a first comparison of different models that can be used in exact methods

Models and methods for Traffic Engineering problems with single-path routing

Traffic Engineering (TE) uses methods and models from a variety of mathematical fields, such as statistics and optimization, to improve the performance of telecommunication networks. In this thesis, we study TE problems dealing with networks that impose single-path routing. As the name infers, in this type of routing, the traffic flow of each "commodity" cannot be split in its path between its origin and destination. Given its cheap cost, single-path routing is widely used in today's data centers, where thousands of stored servers perform computations or host Internet services. One common case of single-path routing is the one enforced by the Spanning Tree Protocol (STP) in switched Ethernet networks. The STP requires the network to keep its activated links loop-free, while maintaining the other redundant links ready for back-up, in case of link failure. The Multiple Spanning Tree Protocol (MSTP) extends the STP by installing multiple virtual networks compliant with the STP, over a single physical topology. Therefore, the MSTP is greatly beneficial for network service providers, as it allows for a more efficient use of the existing resources.Network design problems dealing with the MSTP are generally highly combinatorial and very hard to solve. As such, TE literature mainly suggests heuristic methods, which can quickly produce reasonable designs. Notwithstanding, due to a scarce existence of lower bounds to the optimum values of such problems, there is little knowledge about the quality of the solutions provided by these heuristics.In this sense, we propose mathematical programming models and methods that can provide optimal designs for these networks, or at the very least, obtain valid lower bounds. Taking into mind the goal of avoiding congestion in the network, we focus on two problems that deal with the following load-balancing objectives: the minimization of the worst-case link utilization, and the minimization of flow costs given by piecewise linear functions that penalize heavily-loaded links. The study of both these problems yielded relevant by-products: the first is the study of a MSTP network design problem, where we minimize the total load, and the second is the study of a fundamental unsplittable multicommodity flow problem with piecewise linear costs.For all the considered problems, we provide studies of complexity, extensive polyhedral studies to compare the proposed formulations, and a wide array of computational experiments to evaluate the performance of the proposed models and methods.Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe

Optimal design of switched Ethernet networks implementing the multiple spanning tree protocol

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On the convex piecewise linear unsplittable multicommodity flow problem

International audienceWe consider the problem of finding the cheapest routing for a set of commodities over a directed graph, such that: i) each commodity flows through a single path, ii) the routing cost of each arc is given by a convex piecewise linear function of the load (i.e. the total flow) traversing it. We propose a new mixed-integer programming formulation for this problem. This formulation gives a complete description of the associated polyhedron for the single commodity case, and produces very tight linear programming bounds for the multi-commodity case

Branch-and-cut methods for the network design problem with vulnerability constraints

The aim of Network Design Problem with Vulnerability Constraints (NDPVC), is to design survivable telecommunications networks that impose length bounds on the communication paths of each commodity pair, before and after the failure of any k links. This problem was proposed as an alternative to the Hop-Constrained Survivable Network Design Problem (kHSNDP), which addresses similar issues, but imposes very conservative constraints, possibly leading to unnecessarily expensive solution or even rendering instances infeasible. In fact, it is known that the cost of the optimal solutions of the NDPVC never exceeds that of the related kHSNDP. However, previous results using the standard methods of a general-purpose integer linear (ILP) solver, combined with several ILP formulations, show that such methods fail to solve most instances in the benchmarking test set, within a time limit of two hours. In this paper, we propose three branch-and-cut algorithms, which are significantly more efficient in solving the NDPVC. The first algorithm is a cutting-plane method devised in the context of a new layered graph ILP formulation, whereas the other two are based on Benders decomposition methods of previously known formulations. With the proposed new methods, we are able to solve substantially more instances of the NDPVC and therefore able to provide a more complete comparison of its solutions to those of the kHSNDP