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- On the computation of all supported efficient solutions in multi-objective integer network flow problemsPublication . Eusébio, Augusto; Figueira, José RuiThis paper presents a new algorithm for identifying all supported non-dominated vectors (or outcomes) in the objective space, as well as the corresponding efficient solutions in the decision space, for multi-objective integer network flow problems. Identifying the set of supported non-dominated vectors is of the utmost importance for obtaining a first approximation of the whole set of non-dominated vectors. This approximation is crucial, for example, in two-phase methods that first compute the supported non-dominated vectors and then the unsupported non-dominated ones. Our approach is based on a negative-cycle algorithm used in single objective minimum cost flow problems, applied to a sequence of parametric problems. The proposed approach uses the connectedness property of the set of supported non-dominated vectors/efficient solutions to find all integer solutions in maximal non-dominated/efficient facets.
- Funcões reais de uma variável real: derivadas e suas aplicacõesPublication . Eusébio, Augusto; Cotrim, LuísIntrodução: A formação de base em Matemática dos estudantes que ingressam no ensino superior é muito variada. Desta forma, surge, naturalmente, uma falta de uniformidade nos conhecimentos prévios dos estudantes, à qual acresce, devido a fatores vários, uma falta de conhecimentos de base fundamentais. Este trabalho tem como principal objetivo apoiar os estudantes, a fim de que possam ultrapassar o mais possível as suas lacunas. Vamos abordar, de forma sucinta, a noção de derivada de uma função real de uma variável real e algumas das suas aplicações, cujo estudo uma parte dos estudantes iniciou no ensino secundário. Foi nossa preocupação apresentar alguns exemplos resolvidos para, posteriormente, apresentar exercícios cuja resolução ficará a cargo dos estudantes. Alguns dos exercícios apresentados foram retirados de provas de avaliação das UC de Análise Matemática e Métodos Quantitativos da Escola Superior de Tecnologia e Gestão do Politécnico de Leiria.
- A primal-dual simplex algorithm for bi-objective network flow problemsPublication . Eusébio, Augusto; José Rui Figueira; Matthias EhrgottIn this paper we develop a primal–dual simplex algorithm for the bi-objective linear minimum cost network flow problem. This algorithm improves the general primal–dual simplex algorithm for multi-objective linear programs by Ehrgott et al. (J Optim Theory Appl 134:483–497, 2007). We illustrate the algorithm with an example and provide numerical results.
- Finding non-dominated solutions in bi-objective integer network flow problemsPublication . Eusébio, Augusto; Figueira, José RuiThis paper deals with an algorithm for finding all the non-dominated solutions and corresponding efficient solutions for bi-objective integer network flow problems. The algorithm solves a sequence of ε-constraint problems and computes all the non-dominated solutions by decreasing order of one of the objective functions. The optimal integer solutions for the ε-constraint problems are determined by exploring a branch-and-bound tree. The algorithm makes use of the network structure to perform the computations, i.e., the network structure of the problem is not destroyed with the inclusion of an ε-constraint. This paper presents the main features of the algorithm, the theoretical bases of the proposed approach and some computational issues. Experiments were done and the results are also reported in the paper.