Uma introdução à partição de inteiros Boletim da Sociedade Portuguesa de Matemática, 56 (2007): 51-61.
Abstract - An introduction to integer partitions is given and some relevant identities are analyzed applying generating
functions, the bijection principle and Ferrers boards. Furthermore, throughout the text, the use of simple problems as didactic
tools for the study and understanding of increasingly more dificult results and concepts is exemplified. |
A multi-attribute ranking solutions confirmation procedure Annals of Operations Research 138 (2005): 127-141.
Abstract - Ranking problems arise from the knowledge of several binary relations defined on the same set of alternatives,
which we intend to rank. Despite the fact that many approaches of multiple criteria decision analysis applied to this type of problems
have been developed, in general, it is not possible to decide which is the most appropriate one. On the other hand, inconsistencies
can emerge during the intermediate steps of many methodologies, with negative consequences on the quality of the obtained ranking
solutions. In a previous work, the authors introduced some numerical tools to confirm the solutions of multi-attribute ranking problems
as linear extensions of a weighted sum of preference relations. An extension of this technique allows the recognition of critical
preference pairs of alternatives, which are often caused by the aforementioned inconsistencies. Herein, a multi-attribute ranking
solutions confirmation procedure is introduced and applied to confirm the results obtained by a multi-attribute decision methodology
on a tender for the supply of buses to the Porto Public Transport Operator (STCP). |
The Generalised Simplex Method - a non-vertex-simplex-like approach to linear programming PhD dissertation (in Portuguese), Universidade de Aveiro (1992).
Abstract - A new approach for linear programming problems based on a generalization of
the simplex method is introduced. This approach is aimed at getting feasible movements between
faces of arbitrary dimension of a polytope, converging to the optimal face. The method starts
from a face of the polytope of high dimension, and moves systematically from one face to another,
by overtaking many faces of lower dimension (in particular many vertices, which are faces of zero
dimension). The method is particularly suited for a parallel processing implementation and allows
a straightforward post-optimal analysis. |
Equitable bipartitions of graphs and related results Journal of Mathematical Sciences, Vol 20 (Issue 1: Aveiro Seminar on Control, Optimization and Graph Theory) (2004): 869-880.
Abstract - A set of vertices S is (k,p)-regular if induces a
k-regular subgraph of G such that any vertex out of S has p
neighbors in S. A bipartition of the set of vertices (V1,V2) is equitable
if V1 is (m11,m21)-regular and V2 is (m22,m12)-regular. For
different values of k and p the properties of (k,p)-regular sets
are analyzed as well as the recognition of graphs which includes such regular sets.
Connections between (k,p)-regular sets, equitable bipartitions and switching
operations are exploited with applications on the determination of maximum matchings,
maximum induced matchings, maximum stable sets and maximum cliques of graphs with particular
properties. |
Euclidean Jordan algebras with strongly regular graphs Journal of Mathematical Sciences, Vol 20 (Issue 1: Aveiro Seminar on Control, Optimization and Graph Theory) (2004): 881-894.
Abstract - An Euclidean Jordan algebra Vn of dimension
m = Int(n/2)+1,
for arbitrary n, is produced. The algebraic and combinatorial properties of a basis
of Vn allows the determination of strongly regular graphs of order
n. The character table of this Euclidean Jordan algebra is deduced and
applied on the spectra recognition of the above strongly regular graphs
included in Vn. |
A numerical tool for multi-attribute ranking problems NETWORKS, Vol. 41, n. 4 (2003): 229-234 .
Abstract - A large variety of techniques have been developed to solve or approximate the
solution of multi-attribute ranking problems. From such techniques, several implicit or explicit
partial orders, defined on the same set of alternatives, are obtained (in many cases by pairwise
comparisons) with the goal of determining a linear order. Often, this goal is attained by
assigning positive weights to each partial order relation. However, the imprecise judgements of
the pairwise comparisons as well as other factors lead to inconsistencies which have been
analyzed in an extensive literature devoted to this type of problem. In this paper, numerical
results about linear extensions of weighted sum relations are applied to the recognition of
pairwise imprecise judgements between alternatives as well as to the confirmation of a ranking
solution as a linear extension of a quasi-order defined by a weighted sum of binary preference
relations. |
On graphs with stability number equal to the optimal value of a convex quadratic program Matemática Contemporânea (Sociedade Brasileira de Matemática), 25 (2003): 9-24.
Abstract - Since the Motzkin-Straus result on the clique number of graphs, published in 1965, where they show that
the size of the largest clique in a graph can be obtained by solving a quadratic programming problem,
several results on the continuous approach to the determination of the clique number of a graph or,
equivalently, to the determination of the stability number of its complement, have been published. In
this paper, a Motzkin-Straus-like approach to the stability number of graphs is presented and extended to
the study of graphs for which the stability number is equal to the optimal value of a convex quadratic
programming problem (called graphs with convex-QP stability number) as well as the determination of
convex quadratic lower and upper bounds on the stability number of arbitrary graphs. In the presence of
adverse conditions, it is proved that the recognition of graphs with convex-QP stability number is
equivalent to the recognition of graphs with a particular combinatorial structure (called regular-stable
graphs). Additionally, for particular types, as is the case of line graphs of forests or threshold graphs,
the polynomial-time recognition of graphs with convex-QP stability number is introduced. |
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On regular-stable graphs Ars Combinatoria (to appear).
Abstract - We introduce graphs G, with at least a maximum
independent set of vertices, I, such that for all v in V(G)\I,
the number of vertices in the intersection of NG(v) with I
is constant. When this number of vertices is equal to l
we say that I has the l-property and that G
is l-regular-stable. Furthermore we extend the
study of this property to the well-covered graphs (that is, graphs where all
maximal independent sets of vertices have the same cardinality). In a such
study we consider well-covered graphs for which all maximal independent set
of vertices have the l-property, herein called
well-covered l-regular-stable graphs. |
|
Analytical properties of a stochastic system with MMPP input and an acces function Journal of Telecommunications and Information Technology 3 (2002):17-23.
Abstract - Stochastic modeling of teletraffic systems with restricted availability and
correlated input arrival rates is of great interest in GoS (grade of service) analysis and
design of certain telecommunication networks. This paper presents some analytical properties
of a recursive nature, associated with the infinitesimal generator of the Markov process
which describes the state of a teletraffic system with MMPP (Markov modulated Poisson process)
input traffic, negative exponentially distributed service times, finite queue and restricted
availability defined through a loss function. Also the possible application of the derived
properties to a direct method of resolution of the linear system, which gives the stationary
probability distribution of the system, will be discussed. |
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Confirmation results on multiattribute ranking problems Proceeding of XI Latin-Iberian American Congress of Operations Research (2002): CDROM.
Abstract - Ranking problems arise from the knowledge of several binary relations defined
on the same set of objects, which we intend to rank. However, in this process, some
inconsistencies can arise. Herein, a confirmation technique for ranking solutions of
multi-attribute problems is proposed by confirming them as linear extensions of a weighted sum
of preference relations. A technique to recognize critical preference pairs of alternatives and
a threshold level for the confirmation of ranking solutions obtained by arbitrary methodologies
are presented. A partial order on the set of different solutions obtained by arbitrary
methodologies applied to the same multi-attribute ranking problem is also introduced. Finally, a
numerical example and a case study highlight the application of the proposed techniques. |
|
Influência da reciclagem e compostagem na receita duma instalação de incineração Aguas & Resíduos - Ano V, Março/Julho (2001): 31-43.
Resumo - Estudou-se o efeito do aumento da taxa de reciclagem de resíduos de embalagem
(por exigência da directiva comunitária 94/62) na receita da instalação de incineração da
LIPOR II, usando optimização linear com análise de sensibilidade, tendo-se determinado as taxas
que optimizam o sistema integrado: [reciclagem + incineração]. |
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Self-concordance and damped Newton method Investigação Operacional 20 (2000): 147-166 (in Portuguese).
Abstract - In 1994 Nesterov and Nemirovskii introduced a unified theory of interior
point methods in convex programming based on self-concordant functions. This concept shows
the good behaviour of certain functions related to the local metric defined by themselves.
Due to the Lipschitz properties of this functions, Nestorov and Nemirovskii proposed a damped
version of the Newton method and proved its convergence. In this paper we describe the damped
Newton method and present the most relevant properties of self-concordance to prove its
convergence. Finally some computational experiments are referred. |
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On the starting feasible solution in linear programming Investigação Operacional 14 (1994): 61-75 (in Portuguese).
Abstract - In most approachs, the resolution of a linear programming problem needs a
starting feasible solution. Among the techniques commonly used to overcome this dificulty, the
construction of an artificial linear program having a known feasible solution and then the
application of the big-M method is certainly the most popular. In this paper we study some
criteria that provide an estimation of M, in order to get an optimal solution of the original
linear program from the feasible solution of the constructed artificial linear program. We also
provide some techniques for updating M, which allow to avoid the undesirable consequences that
arise from an initial unsafe estimate M. Although the fact that the main discussed results are
oriented to the simplex method at the end of the paper we extend some of the results to interior
point methods. |
|
Duality in linear fractional programming Investigação Operacional 10 (1990): 65-84 (in Portuguese).
Abstract - Firstly we present an overview of the main developments on duality in linear
fractional programming and we analyse some of the difficulties raised by the classical approach.
Taking as basis a method directed to the parametric analysis of linear fractional programs, we
put forward a dual formulation for this type of problems, the resolution of which does not
present great difficulty. The proposed formulations verifies the more common dual-primal type
of relationships and allows for an easy sensitivity analysis. Finally, geometric interpretations
in a space determined by the images of the admissible region (obtained from the primal and dual
problems, respectively) are presented. |
|
Optimal flow and equipment assignement problem Casos de aplicação da investigação operacional (coordenação de Carlos H. Antunes e Luís V. Tavares), McGraw-Hill, Lisboa (2000): 52-63 (in Portuguese).
Abstract - The storage and assignment of containers to storage locations in warehouses
with high level of input and output flow are current problems in automatic storage systems,
flexible manufacturing systems (systems with a large variety of products), wide distribution
centres, among others. Such problems imply a good choice of the storage locations, minimising
the scattering of the products of the same order and its transportation distances runs by the
stack-trucks. This paper is about a case study related with the storage of equipment in a
warehouse during a certain time that it is up to two days, between the production and the
delivery operations. The problems related with the flow of the equipment on the loading area and
their loads into the storage locations are described. Furthermore the corresponding optimisation
mathematical models are analysed. Finally, several computational results are presented, comparing
the values obtained from the resolution of the proposed models with the ones obtained from the
load decisions registered on the computer files. |
|
The overflow traffic from the Erlang-B system - what convexity properties? Matemática em Telecomunicações Que Problemas?. Edited by Joaquim J. Júdice and Maria C. Gouveia, Instituto de Telecomunicações, Coimbra (1998): 6-11.
Abstract - This contribution focuses on the study of some analytical properties of an
high transcendental function well known by teletraffic theorists, designated as Erlang-B
function, which gives the blocking probability of an arriving call in the M/M/x loss system.
This and other related functions play an important role in many models of teletraffic
theory
and has many applications in multiple teletraffic engineering problems. Denoting by a the
offered traffic and x the number of circuits, this work focus on the properties of the
overflow traffic function Â(a,x) from an Erlang-B system, when (a,x) belongs to
R+xR. |
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A reduced recursion for computing Erlang-B function derivatives Teletraffic Contributions for the Information Age. Edited by V. Ramaswami & P. E. Wirth, Elsevier Science, B. V., Amsterdam, Netherlands, vol 2B (1997): 1315-1326.
Abstract - A generic algorithm approach for calculating the
derivatives of order n of the Erlang-B function in the number of servers x,
which may be envisaged as a generalization of the classical recursion used
for calculating the Erlang-B formula, was proposed by the authors in a
previous work. This method compares favourably, in terms of efficiency with
a method prposed by D. L. Jagerman based on cardinal series quadrature,
excepting for very high values of x where the situation is inverse.
The paper presents an algorithm designated as Reduced Recursion and
shows that if the arguments of the function are sufficiently high then it is
possible to reduce significantly the number of steps of the recursion given
in the above referred previous work., without jeopardizing the required
precision of the results. Computational results and comparison with the
Jagerman algorithm are presented and discussed. These results show a very
good performance of the method even for very high values of the arguments. |
|
Analysis and resolution of fractional programming problems - an approach distinct from the
ones based on Dinkelbach's algorithm Proceedings of AIRO'90 Congress - Models and Methods for Decision Support (1990): 789-806.
Abstract - The modeling of many pratical problems lead to the necessity of optimizing a
ratio of two real-valued continuous functions over a nonempy subset of Rn
called feasible region. These problems are denoted by fractional programs,
for which, in general terms, we only impose that the denominator function
must not change the signal in the above-mentioned region, that also be
considered a compatcs set. In this work, we present a analysis of the
fractional programming problem, emphasizing the geometric relationship
between the image of the feasible region (obtained by the numerator and
denominator functions of the objective function) and the graphic
representation of some connected functions. Several results are developed,
in order to obtain a new algorithmic approach for convex-concave fractional
programs. In this algorithmic approach we integrate a succession of
mathematical programs, more simple than the original one. The objective
function of these problems are the numerator function of the original
fractional program. We also propose a new dual formulation for
convex-concave fractional programming problems. This enables the improvement
of the propose algorithm. |
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Análise de uma aplicação da programação em lógica ao ensino assistido por computador Actas do Segundo Encontro de Inteligência Artificial - Epia86, Lisboa (1986): 219-225.
Resumo - Implementou-se, em Prolog, um sistema de apoio ao ensino de Matemática, mais
particularmente de estruturas algébricas. Descreve-se o referido sistema, quer do ponto de vista
de utilização, quer de implementação. Discutem-se opções que se poderiam tomar quanto a eventuais
extensões do sistema implementado. Finalmente, tiram-se algumas conclusões quanto às vantagens e
desvantagens do recurso ao Prolog como linguagem para o desenvolvimento de sistemas de apoio ao
ensino. |