Biobjective Optimization over the Efficient Set Methodology for Pareto Set Reduction in Multiobjective Decision Making: Theory and Application

A large number of available solutions to choose from poses a significant challenge for multiple criteria decision making. This research develops a methodology that reduces the set of efficient solutions under consideration. This dissertation is composed of three major parts: (i) the formalization of a theoretical framework; (ii) the development of a solution approach; and (iii) a case study application of the methodology. In the first part, the problem is posed as a multiobjective optimization over the efficient set and considers secondary robustness criteria when the exact values of decision variables are subjected to uncertainties during implementation. The contributions are centered at the modeling of uncertainty directly affecting decision variables, the use of robustness to provide additional trade-off analysis, the study of theoretical bounds on the measures of robustness, and properties to ensure that fewer solutions are identified. In the second part, the problem is reformulated as a biobjective mixed binary program and the secondary criteria are generalized to any convenient linear functions. A solution approach is devised in which an auxiliary mixed binary program searches for unsupported Pareto outcomes and a novel linear programming filtering excludes any dominated solutions in the space of the secondary criteria. Experiments show that the algorithm tends to run faster than existing approaches for mixed binary programs. The algorithm enables dealing with continuous Pareto sets, avoiding discretization procedures common to the related literature. In the last part, the methodology is applied in a case study regarding the electricity generation capacity expansion problem in Texas. While water and energy are interconnected issues, to the best of our knowledge, this is the first study to consider both water and cost objectives. Experiments illustrate how the methodology can facilitate decision making and be used to answer strategic questions pertaining to the trade-off among different generation technologies, power plant locations, and the effect of uncertainty. A simulation shows that robust solutions tend to maintain feasibility and stability of objective values when power plant design capacity values are perturbed

Techniques for Multiobjective Optimization with Discrete Variables: Boxed Line Method and Tchebychev Weight Set Decomposition

Many real-world applications involve multiple competing objectives, but due to conflict between the objectives, it is generally impossible to find a feasible solution that optimizes all, simultaneously. In contrast to single objective optimization, the goal in multiobjective optimization is to generate a set of solutions that induces the nondominated (ND) frontier. This thesis presents two techniques for multiobjective optimization problems with discrete decision variables. First, the Boxed Line Method is an exact, criterion space search algorithm for biobjective mixed integer programs (Chapter 2). A basic version of the algorithm is presented with a recursive variant and other enhancements. The basic and recursive variants permit complexity analysis, which yields the first complexity results for this class of algorithms. Additionally, a new instance generation method is presented, and a rigorous computational study is conducted. Second, a novel weight space decomposition method for integer programs with three (or more) objectives is presented with unique geometric properties (Chapter 3). The weighted Tchebychev scalarization used for this weight space decomposition provides the benefit of including unsupported ND images but at the cost of convexity of weight set components. This work proves convexity-related properties of the weight space components, including star-shapedness. Further, a polytopal decomposition is used to properly define dimension for these nonconvex components. The weighted Tchebychev weight set decomposition is then applied as a “dual” perspective on the class of multiobjective “primal” algorithms (Chapter 4). It is shown that existing algorithms do not yield enough information for a complete decomposition, and the necessary modifications required to yield the missing information is proven. Modifications for primal algorithms to compute inner and outer approximations of the weight space components are presented. Lastly, a primal algorithm is restricted to solving for a subset of the ND frontier, where this subset represents the compromise between multiple decision makers’ weight vectors.Ph.D

New extesions of the scalarizations techiques in the multiobjective one-dimensional cutting stock problem

Orientador: Antonio Carlos MorettiTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O presente trabalho trata do Problema de Corte Unidimensional Inteiro Multiobjetivo (PCUIM). Este problema possui uma importância prática enorme e a sua abordagem multiobjetiva foi pouco reportada na literatura. O modelo biobjetivo considerado visa minimizar a soma das frequências dos padrões de corte para atender à mínima demanda e ao número de diferentes padrões a serem usados (\textit{setup}), sendo estas metas conflitantes entre si. Neste caso, o PCUIM possui um conjunto não unitário de soluções, ditas de \textit{soluções eficientes}, todas igualmente importantes para o problema. A geração de cada solução eficiente necessita a otimização de um Problema de Programação Linear Inteiro e a obtenção de todas estas soluções pode ser uma tarefa relativamente cara, principalmente quando os padrões de corte não são fornecidos pelo usuário a priori. Nesta tese, foram utilizados sete métodos distintos que transformam o PCUIM em problemas de otimização escalares, que por sua vez, geram as soluções eficientes. Seis métodos foram adaptados da literatura e um foi originalmente desenvolvido. A fim de acelerar a obtenção do conjunto de soluções eficientes, no caso com os padrões fornecidos pelo usuário, foi adotada uma estratégia que relaxa as condições de integralidade das variáveis do problema e, posteriormente, cada solução eficiente produzida é integralizada por meio de uma heurística ineditamente desenvolvida. Os extensos testes computacionais presentes no Capítulo 8, comprovaram que esta ideia foi adequada e eficaz. Além disso, a nova técnica de escalarização se mostrou muito competitiva com as demais consagradas na literatura, possibilitando um crescimento e um avanço na área de Problemas de Corte bem como na Otimização Combinatória MultiobjetivoAbstract: The present work deals with the Multiobjective One-Dimensional Cutting Stock Problem (MODCSP). This problem has an enormous practical importance, and the multiobjective approach has been little reported in the literature. The bi-objective model considered aims to minimize the sum of the frequency of cutting patterns to meet minimal demand and the number of different cutting patterns to be used (setup), being these objectives conflicting. In this case, the MODCSP has a non-unitary set of solutions, said \textit{efficient solutions}, equally important for the problem. The generation of each efficient solution requires the optimization of an Integer Linear Problem. So, the complete enumeration of these solutions can be an expensive task, especially when cutting patterns are not provided by the user. In this thesis, we applied seven different methods that transform the MODCSP on scalar optimization problems, where each problem provide an efficient solution. Six scalarization methods were adapted from literature and one was unprecedentedly developed. In the case of the cutting patterns be provided a priori, we used a relaxation strategy (heuristic) to accelerate obtaining of the set efficient solutions. In this approach, we relaxed the integrality condition of the variables and each efficient solution was rounded by a specially developed heuristic. The extensive results in Chapter 8 validated that this idea was adequate and effective. Furthermore, the new scalarization technique proved to be very competitive with other established in the literature, enabling growth and advancement in the area of the Cutting Problems and in Multiobjective Combinatorial OptimizationDoutoradoMatematica AplicadaDoutor em Matemática Aplicada2013/06035-0FAPESPCAPE

Actas de las XXXIV Jornadas de Automática

Postprint (published version

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Aeronautical enginnering: A cumulative index to a continuing bibliography (supplement 312)

This is a cumulative index to the abstracts contained in NASA SP-7037 (301) through NASA SP-7073 (311) of Aeronautical Engineering: A Continuing Bibliography. NASA SP-7037 and its supplements have been compiled by the Center for AeroSpace Information of the National Aeronautics and Space Administration (NASA). This cumulative index includes subject, personal author, corporate source, foreign technology, contract number, report number, and accession number indexes

Modified interactive Chebyshev algorithm (MICA) for convex multiobjective programming

In this paper, we describe an interactive procedural algorithm for convex multiobjective programming based upon the Tchebycheff method, Wierzbicki's reference point approach, and the procedure of Michalowski and Szapiro. At each iteration, the decision maker (DM) has the option of expressing his or her objective-function aspirations in the form of a reference criterion vector. Also, the DM has the option of expressing minimally acceptable values for each of the objectives in the form of a reservation vector. Based upon this information, a certain region is defined for examination. In addition, a special set of weights is constructed. Then with the weights, the algorithm of this paper is able to generate a group of efficient solutions that provides for an overall view of the current iteration's certain region. By modification of the reference and reservation vectors, one can "steer" the algorithm at each iteration. From a theoretical point of view, we prove that none of the efficient solutions obtained using this scheme impair any reservation value for convex problems. The behavior of the algorithm is illustrated by means of graphical representations and an illustrative numerical example.Multiobjective programming Interactive procedures Tchebycheff method Reference point methods Aspiration criterion vectors Reservation levels

Aeronautical engineering: A continuing bibliography with indexes (supplement 289)

This bibliography lists 792 reports, articles, and other documents introduced into the NASA scientific and technical information system in Mar. 1993. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics