We introduce two online models for the vertex k-center and the vertex k-median problems. Clients (i.e., graph vertices) and their corresponding links (i.e., graph edges) are revealed sequentially, determining the topology of a graph over time. Clients are revealed by an adversary to an online algorithm that selects existing graph vertices on which to open facilities; once open, a facility cannot be removed or relocated. We define two models: an online algorithm may be restricted to open a facility only at the location of the most recent client or at the location of any existing client. We examine these models on three classes of graphs under two types of adversaries. We establish lower bounds on the respective competitive ratios attainable by any online algorithm for each combination of model, adversary, and graph class. Finally, we describe algorithms whose competitive ratios provide corresponding upper bounds on the best competitive ratios achievable.

We introduce two online models for the vertex k-center and the vertex k-median problems. Clients (i.e., graph vertices) and their corresponding links (i.e., graph edges) are revealed sequentially, determining the topology of a graph over time. Clients are revealed by an adversary to an online algorithm that selects existing graph vertices on which to open facilities; once open, a facility cannot be removed or relocated. We define two models: an online algorithm may be restricted to open a facility only at the location of the most recent client or at the location of any existing client. We examine these models on three classes of graphs under two types of adversaries. We establish lower bounds on the respective competitive ratios attainable by any online algorithm for each combination of model, adversary, and graph class. Finally, we describe algorithms whose competitive ratios provide corresponding upper bounds on the best competitive ratios achievable.

<p> This dissertation examines the facility location problems in the presence of barrier regions and consists basically of four essays exploring new problems. Despite the fact that the facility location problems considering barriers to travel are more realistic than their unrestricted counterparts, research in the area is relatively limited. This is due to the computational complexity associated with them. </p> <p> The first essay analyzes the problem of locating a facility in a region in the presence of a probabilistic line barrier. The objective is to locate the facility such that the sum of the volume times distances between the facility and demand points is minimized. Some convexity results are presented and a solution algorithm is proposed. </p> <p> Another interrelated problem is locating a facility in a region where a fixed line barrier such as a borderline divides the region into two. The regions communicate with each other through a number of passage points located on the line barrier. A version of this problem with minisum objective has been studied in the literature where the locations of the passage points are known. The second essay considers a number of extensions to this problem and proposes an efficient solution methodology based on the Outer Approximation algorithm. </p> <p> The third essay discusses the problem of locating a rectangular barrier facility m an area where interactions among existing facilities are present. The problem has two objectives. The first objective is to minimize the interference of the barrier facility to the interactions among the existing facilities. The second objective is to find a center (minimax) location for the barrier facility. The problem is formulated as a bi-objective problem and a mixed integer program is proposed as a solution methodology. A Simulated Annealing algorithm is presented for an extension of the problem where expropriation of existing facilities is also possible. </p> <p> Finally, the last essay suggests a practical analog approach for facility location problems in the presence of barriers. The use of the analog for certain problems is justified through some analytical results and a number of problems that appeared in the literature are solved efficiently. </p> / Thesis / Doctor of Philosophy (PhD)

DIAS, Fábio Carlos Sousa. Algoritmos para o problema de localização simples baseados nas formulações clássica e canônica. 2008. 89 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2008. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-11T15:12:03Z No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-15T15:32:35Z (GMT) No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) / Made available in DSpace on 2016-07-15T15:32:35Z (GMT). No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) Previous issue date: 2008 / In this work, we study the Simple Plant Location Problem (SPLP). Using its classical mathematical programming formulation and another recently proposed formulation, we develop several algorithms to …nd lower and upper bounds for the problem as well as branch-and-bound algorithms. With the classical formulation, such bounds are obtained via the data correction method and dominance criteria between …xed and transportation costs. We propose a projection of this formulation that has shown to be computationally atractive. Using the new formulation, we propose and prove the correctness of several iterative procedures that attempt to …nd an optimal solution to the problem by solving a sequence of parametric sub-problems, each one obtained by removing some variables and constraints of the original formulation. At each iteration of this process, we can obtain lower and upper bounds. We also apply Lagrangean relaxation to this new formulation in order to get other bounds. We consider several possibilities of relaxing the constraints. In addition, we develop branch-and-bound algorithms based on both formulations and the obtained bounds. We evaluate the computational e¢ ciency of all proposed algorithms with hard test instances from the literature. Computational results are reported and comparisons with other algorithms from the literature are carried out. / Neste trabalho, estudamos o problema de localização simples (SPLP - Simple Plant Location Problem). Usando a formulação matemática clássica e uma outra formulação proposta recentemente, desenvolvemos vários algoritmos para encontrar limites inferiores e superiores, bem como algoritmos tipo branch-and-bound. Com a formulação clássica, tais limites são obtidos utilizando o método de correção de dados e critérios de dominância entre os custos …xos e de transporte. Propomos uma projeção dessa formulação, que se mostrou computacionalmente atrativa. Usando a nova formulação propomos e mostramos a corretude de vários procedimentos iterativos que procuram encontrar uma solução para o problema, resolvendo uma seqüência de subproblemas paramétricos obtidos com a remoção de variáveis e restrições da formulação original. Em cada iteração desse processo, podemos gerar limites inferiores e superiores. Aplicamos ainda relaxação lagrangeana a essa nova formulação para obter outros limites. Analisamos várias possibilidades de relaxação das restrições. Desenvolmento também algoritmos branch-and-bound baseados em ambas as formulações e nos limites obtidos. Avaliamos a e…ciência computacional de todos os algoritmos com instâncias de teste difíceis, disponíveis na literatura. Resultados computacionais e comparações com outros algoritmos da literatura são reportados.

DIAS, Fábio Carlos Sousa. Algoritmos para o problema de localização simples baseados nas formulações clássica e canônica. 2008. 81 f. : Dissertação (mestrado) - Universidade Federal do Ceará, Centro de Ciências, Departamento de Computação, Fortaleza-CE, 2008. / Submitted by guaracy araujo (guaraa3355@gmail.com) on 2016-06-22T17:13:52Z No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) / Approved for entry into archive by guaracy araujo (guaraa3355@gmail.com) on 2016-06-22T17:16:23Z (GMT) No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) / Made available in DSpace on 2016-06-22T17:16:23Z (GMT). No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) Previous issue date: 2008 / In this work, we study the Simple Plant Location Problem (SPLP). Using its classical mathematical programming formulation and another recently proposed formulation, we develop several algorithms to …nd lower and upper bounds for the problem as well as branch-and-bound algorithms. With the classical formulation, such bounds are obtained via the data correction method and dominance criteria between …xed and transportation costs. We propose a projection of this formulation that has shown to be computationally atractive. Using the new formulation, we propose and prove the correctness of several iterative procedures that attempt to …nd an optimal solution to the problem by solving a sequence of parametric sub-problems, each one obtained by removing some variables and constraints of the original formulation. At each iteration of this process, we can obtain lower and upper bounds. We also apply Lagrangean relaxation to this new formulation in order to get other bounds. We consider several possibilities of relaxing the constraints. In addition, we develop branch-and-bound algorithms based on both formulations and the obtained bounds. We evaluate the computational e¢ ciency of all proposed algorithms with hard test instances from the literature. Computational results are reported and comparisons with other algorithms from the literature are carried out. / Neste trabalho, estudamos o problema de localização simples (SPLP - Simple Plant Location Problem). Usando a formulação matemática clássica e uma outra formulação proposta recentemente, desenvolvemos vários algoritmos para encontrar limites inferiores e superiores, bem como algoritmos tipo branch-and-bound. Com a formulação clássica, tais limites são obtidos utilizando o método de correção de dados e critérios de dominância entre os custos …xos e de transporte. Propomos uma projeção dessa formulação, que se mostrou computacionalmente atrativa. Usando a nova formulação propomos e mostramos a corretude de vários procedimentos iterativos que procuram encontrar uma solução para o problema, resolvendo uma seqüência de subproblemas paramétricos obtidos com a remoção de variáveis e restrições da formulação original. Em cada iteração desse processo, podemos gerar limites inferiores e superiores. Aplicamos ainda relaxação lagrangeana a essa nova formulação para obter outros limites. Analisamos várias possibilidades de relaxação das restrições. Desenvolmento também algoritmos branch-and-bound baseados em ambas as formulações e nos limites obtidos. Avaliamos a e…ciência computacional de todos os algoritmos com instâncias de teste difíceis, disponíveis na literatura. Resultados computacionais e comparações com outros algoritmos da literatura são reportados.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, estudamos o problema de localizaÃÃo simples (SPLP - Simple Plant Location Problem). Usando a formulaÃÃo matemÃtica clÃssica e uma outra formulaÃÃo proposta recentemente, desenvolvemos vÃrios algoritmos para encontrar limites inferiores e superiores, bem como algoritmos tipo branch-and-bound. Com a formulaÃÃo clÃssica, tais limites sÃo obtidos utilizando o mÃtodo de correÃÃo de dados e critÃrios de dominÃncia entre os custos &#133;xos e de transporte. Propomos uma projeÃÃo dessa formulaÃÃo, que se mostrou computacionalmente atrativa. Usando a nova formulaÃÃo propomos e mostramos a corretude de vÃrios procedimentos iterativos que procuram encontrar uma soluÃÃo para o problema, resolvendo uma seqÃÃncia de subproblemas paramÃtricos obtidos com a remoÃÃo de variÃveis e restriÃÃes da formulaÃÃo original. Em cada iteraÃÃo desse processo, podemos gerar limites inferiores e superiores. Aplicamos ainda relaxaÃÃo lagrangeana a essa nova formulaÃÃo para obter outros limites. Analisamos vÃrias possibilidades de relaxaÃÃo das restriÃÃes. Desenvolmento tambÃm algoritmos branch-and-bound baseados em ambas as formulaÃÃes e nos limites obtidos. Avaliamos a e&#133;ciÃncia computacional de todos os algoritmos com instÃncias de teste difÃceis, disponÃveis na literatura. Resultados computacionais e comparaÃÃes com outros algoritmos da literatura sÃo reportados. / In this work, we study the Simple Plant Location Problem (SPLP). Using its classical mathematical programming formulation and another recently proposed formulation, we develop several algorithms to &#133;nd lower and upper bounds for the problem as well as branch-and-bound algorithms. With the classical formulation, such bounds are obtained via the data correction method and dominance criteria between &#133;xed and transportation costs. We propose a projection of this formulation that has shown to be computationally atractive. Using the new formulation, we propose and prove the correctness of several iterative procedures that attempt to &#133;nd an optimal solution to the problem by solving a sequence of parametric sub-problems, each one obtained by removing some variables and constraints of the original formulation. At each iteration of this process, we can obtain lower and upper bounds. We also apply Lagrangean relaxation to this new formulation in order to get other bounds. We consider several possibilities of relaxing the constraints. In addition, we develop branch-and-bound algorithms based on both formulations and the obtained bounds. We evaluate the computational e ciency of all proposed algorithms with hard test instances from the literature. Computational results are reported and comparisons with other algorithms from the literature are carried out.