A note on discrete claims problems

In this note, we consider claims problems with indivisible goods. Specifically, by applying recursively the P-rights lower bound (Jiménez-Gómez and Marco-Gil (2008)), we ensure the fulfillment of Weak Order Preservation, considered by many authors as a minimal requirement of fairness. Moreover, we retrieve the Discrete Constrained Equal Losses and the Discrete Constrained Equal Awards rules (Herrero and Martíınez (2008)). Finally, by the recursive double imposition of a lower and an upper bound, we obtain the average between them. Keywords: Claims problems, Indivisibilities, Order Preservation, Constrained Egalitarian rules, Midpoint. JEL classification: C71, D63, D71.

A Method for Solving a Class of Multiple-Criteria Analysis Problems

AMS subject classification: 90C29.

Stability Analysis in Multicriteria Discrete Portfolio Optimization.

Almost every problem of design, planning and management in the technical and organizational systems has several conflicting goals or interests. Nowadays, multicriteria decision models represent a rapidly developing area of operation research. While solving practical optimization problems, it is necessary to take into account various kinds of uncertainty due to lack of data, inadequacy of mathematical models to real-time processes, calculation errors, etc. In practice, this uncertainty usually leads to undesirable outcomes where the solutions are very sensitive to any changes in the input parameters. An example is the investment managing. Stability analysis of multicriteria discrete optimization problems investigates how the found solutions behave in response to changes in the initial data (input parameters). This thesis is devoted to the stability analysis in the problem of selecting investment project portfolios, which are optimized by considering different types of risk and efficiency of the investment projects. The stability analysis is carried out in two approaches: qualitative and quantitative. The qualitative approach describes the behavior of solutions in conditions with small perturbations in the initial data. The stability of solutions is defined in terms of existence a neighborhood in the initial data space. Any perturbed problem from this neighborhood has stability with respect to the set of efficient solutions of the initial problem. The other approach in the stability analysis studies quantitative measures such as stability radius. This approach gives information about the limits of perturbations in the input parameters, which do not lead to changes in the set of efficient solutions. In present thesis several results were obtained including attainable bounds for the stability radii of Pareto optimal and lexicographically optimal portfolios of the investment problem with Savage's, Wald's criteria and criteria of extreme optimism. In addition, special classes of the problem when the stability radii are expressed by the formulae were indicated. Investigations were completed using different combinations of Chebyshev's, Manhattan and Hölder's metrics, which allowed monitoring input parameters perturbations differently.

Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability

The purpose of this thesis is twofold. The first and major part is devoted to sensitivity analysis of various discrete optimization problems while the second part addresses methods applied for calculating measures of solution stability and solving multicriteria discrete optimization problems. Despite numerous approaches to stability analysis of discrete optimization problems two major directions can be single out: quantitative and qualitative. Qualitative sensitivity analysis is conducted for multicriteria discrete optimization problems with minisum, minimax and minimin partial criteria. The main results obtained here are necessary and sufficient conditions for different stability types of optimal solutions (or a set of optimal solutions) of the considered problems. Within the framework of quantitative direction various measures of solution stability are investigated. A formula for a quantitative characteristic called stability radius is obtained for the generalized equilibrium situation invariant to changes of game parameters in the case of the H¨older metric. Quality of the problem solution can also be described in terms of robustness analysis. In this work the concepts of accuracy and robustness tolerances are presented for a strategic game with a finite number of players where initial coefficients (costs) of linear payoff functions are subject to perturbations. Investigation of stability radius also aims to devise methods for its calculation. A new metaheuristic approach is derived for calculation of stability radius of an optimal solution to the shortest path problem. The main advantage of the developed method is that it can be potentially applicable for calculating stability radii of NP-hard problems. The last chapter of the thesis focuses on deriving innovative methods based on interactive optimization approach for solving multicriteria combinatorial optimization problems. The key idea of the proposed approach is to utilize a parameterized achievement scalarizing function for solution calculation and to direct interactive procedure by changing weighting coefficients of this function. In order to illustrate the introduced ideas a decision making process is simulated for three objective median location problem. The concepts, models, and ideas collected and analyzed in this thesis create a good and relevant grounds for developing more complicated and integrated models of postoptimal analysis and solving the most computationally challenging problems related to it.

Systematic Approach for the selection of monitoring technologies in CO2 geological storage projects. Application of multicriteria decision making

Geologic storage of carbon dioxide (CO2) has been proposed as a viable means for reducing anthropogenic CO2 emissions. Once injection begins, a program for measurement, monitoring, and verification (MMV) of CO2 distribution is required in order to: a) research key features, effects and processes needed for risk assessment; b) manage the injection process; c) delineate and identify leakage risk and surface escape; d) provide early warnings of failure near the reservoir; and f) verify storage for accounting and crediting. The selection of the methodology of monitoring (characterization of site and control and verification in the post-injection phase) is influenced by economic and technological variables. Multiple Criteria Decision Making (MCDM) refers to a methodology developed for making decisions in the presence of multiple criteria. MCDM as a discipline has only a relatively short history of 40 years, and it has been closely related to advancements on computer technology. Evaluation methods and multicriteria decisions include the selection of a set of feasible alternatives, the simultaneous optimization of several objective functions, and a decision-making process and evaluation procedures that must be rational and consistent. The application of a mathematical model of decision-making will help to find the best solution, establishing the mechanisms to facilitate the management of information generated by number of disciplines of knowledge. Those problems in which decision alternatives are finite are called Discrete Multicriteria Decision problems. Such problems are most common in reality and this case scenario will be applied in solving the problem of site selection for storing CO2. Discrete MCDM is used to assess and decide on issues that by nature or design support a finite number of alternative solutions. Recently, Multicriteria Decision Analysis has been applied to hierarchy policy incentives for CCS, to assess the role of CCS, and to select potential areas which could be suitable to store. For those reasons, MCDM have been considered in the monitoring phase of CO2 storage, in order to select suitable technologies which could be techno-economical viable. In this paper, we identify techniques of gas measurements in subsurface which are currently applying in the phase of characterization (pre-injection); MCDM will help decision-makers to hierarchy the most suitable technique which fit the purpose to monitor the specific physic-chemical parameter.

Location of bioelectricity plants in the Madrid Community based on triticale crop: a multicriteria methodology

This paper presents a work whose objective is, first, to quantify the potential of the triticale biomass existing in each of the agricultural regions in the Madrid Community through a crop simulation model based on regression techniques and multiple correlation. Second, a methodology for defining which area has the best conditions for the installation of electricity plants from biomass has been described and applied. The study used a methodology based on compromise programming in a discrete multicriteria decision method (MDM) context. To make a ranking, the following criteria were taken into account: biomass potential, electric power infrastructure, road networks, protected spaces, and urban nuclei surfaces. The results indicate that, in the case of the Madrid Community, the Campiña region is the most suitable for setting up plants powered by biomass. A minimum of 17,339.9 tons of triticale will be needed to satisfy the requirements of a 2.2 MW power plant. The minimum range of action for obtaining the biomass necessary in Campiña region would be 6.6 km around the municipality of Algete, based on Geographic Information Systems. The total biomass which could be made available in considering this range in this region would be 18,430.68 t.

Multicriteria analysis in decision making under information uncertainty

This paper presents results of research related to multicriteria decision making under information uncertainty. The Bell-man-Zadeh approach to decision making in a fuzzy environment is utilized for analyzing multicriteria optimization models (< X, M > models) under deterministic information. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. This circumstance permits one to generalize the classic approach to considering the uncertainty of quantitative information (based on constructing and analyzing payoff matrices reflecting effects which can be obtained for different combinations of solution alternatives and the so-called states of nature) in monocriteria decision making to multicriteria problems. Considering that the uncertainty of information can produce considerable decision uncertainty regions, the resolving capacity of this generalization does not always permit one to obtain unique solutions. Taking this into account, a proposed general scheme of multicriteria decision making under information uncertainty also includes the construction and analysis of the so-called < X, R > models (which contain fuzzy preference relations as criteria of optimality) as a means for the subsequent contraction of the decision uncertainty regions. The paper results are of a universal character and are illustrated by a simple example. (c) 2007 Elsevier Inc. All rights reserved.

Group preference structures in AHP–group decision making

This paper presents a procedure that allows us to determine the preference structures (PS) associated to each of the different groups of actors that can be identified in a group decision making problem with a large number of individuals. To that end, it makes use of the Analytic Hierarchy Process (AHP) (Saaty, 1980) as the technique to solve discrete multicriteria decision making problems. This technique permits the resolution of multicriteria, multienvironment and multiactor problems in which subjective aspects and uncertainty have been incorporated into the model, constructing ratio scales corresponding to the priorities relative to the elements being compared, normalised in a distributive manner (wi = 1). On the basis of the individuals’ priorities we identify different clusters for the decision makers and, for each of these, the associated preference structure using, to that end, tools analogous to those of Multidimensional Scaling. The resulting PS will be employed to extract knowledge for the subsequent negotiation processes and, should it be necessary, to determine the relative importance of the alternatives being compared using anyone of the existing procedures

Multicriteria Evaluation and Optimization of Hierarchical Systems

It is shown that any multicriteria problem can be represented by a hierarchical system. Separate properties of the object are evaluated at the lower level of the system, using a criteria vector, and a composition mechanism is used to evaluate the object as a whole at the upper level. The paper proposes a method to solve complex multicriteria problems of evaluation and optimization. It is based on nested scalar convolutions of vector- valued criteria and allows simple structural and parametrical synthesis of multicriteria hierarchical systems.

Particle swarm optimization for two-connected networks with bounded rings

The Two-Connected Network with Bounded Ring (2CNBR) problem is a network design problem addressing the connection of servers to create a survivable network with limited redirections in the event of failures. Particle Swarm Optimization (PSO) is a stochastic population-based optimization technique modeled on the social behaviour of flocking birds or schooling fish. This thesis applies PSO to the 2CNBR problem. As PSO is originally designed to handle a continuous solution space, modification of the algorithm was necessary in order to adapt it for such a highly constrained discrete combinatorial optimization problem. Presented are an indirect transcription scheme for applying PSO to such discrete optimization problems and an oscillating mechanism for averting stagnation.

« Resolution Search » et problèmes d’optimisation discrète

Thèse réalisée en cotutelle avec l'Université d'Avignon.

Um estudo algorítmico para otimização do plano de tratamento da radioterapia conformal

This work performs an algorithmic study of optimization of a conformal radiotherapy plan treatment. Initially we show: an overview about cancer, radiotherapy and the physics of interaction of ionizing radiation with matery. A proposal for optimization of a plan of treatment in radiotherapy is developed in a systematic way. We show the paradigm of multicriteria problem, the concept of Pareto optimum and Pareto dominance. A generic optimization model for radioterapic treatment is proposed. We construct the input of the model, estimate the dose given by the radiation using the dose matrix, and show the objective function for the model. The complexity of optimization models in radiotherapy treatment is typically NP which justifyis the use of heuristic methods. We propose three distinct methods: MOGA, MOSA e MOTS. The project of these three metaheuristic procedures is shown. For each procedures follows: a brief motivation, the algorithm itself and the method for tuning its parameters. The three method are applied to a concrete case and we confront their performances. Finally it is analyzed for each method: the quality of the Pareto sets, some solutions and the respective Pareto curves

Networks, Uncertainty, Applications and a Crusade for Optimality

In this thesis we address a collection of Network Design problems which are strongly motivated by applications from Telecommunications, Logistics and Bioinformatics. In most cases we justify the need of taking into account uncertainty in some of the problem parameters, and different Robust optimization models are used to hedge against it. Mixed integer linear programming formulations along with sophisticated algorithmic frameworks are designed, implemented and rigorously assessed for the majority of the studied problems. The obtained results yield the following observations: (i) relevant real problems can be effectively represented as (discrete) optimization problems within the framework of network design; (ii) uncertainty can be appropriately incorporated into the decision process if a suitable robust optimization model is considered; (iii) optimal, or nearly optimal, solutions can be obtained for large instances if a tailored algorithm, that exploits the structure of the problem, is designed; (iv) a systematic and rigorous experimental analysis allows to understand both, the characteristics of the obtained (robust) solutions and the behavior of the proposed algorithm.

Algoritmos de decisión para priorizar las acciones de restauración en el mantenimiento de redes

En la actualidad, cualquier compañía de telecomunicaciones que posea su propia red debe afrontar el problema del mantenimiento de la misma. Ofrecer un mínimo de calidad de servicio a sus clientes debe ser uno de sus objetivos principales. Esta calidad debe mantenerse aunque ocurran incidencias en la red. El presente trabajo pretende resolver el problema de priorizar el orden en que se restauran los cables, caminos y circuitos, dañados por una incidencia, dentro de una red troncal de transporte perteneciente a una operadora de telecomunicaciones. Tras un planteamiento detallado del problema y de todos los factores que influirán en la toma de decisión, en primer lugar se acomete una solución basada en Métodos Multicriterio Discretos, concretamente con el uso de ELECTRE I y AHP. A continuación se realiza una propuesta de solución basada en Redes Neuronales (con dos aproximaciones diferentes al problema). Por último se utiliza un método basado en la Optimización por Enjambre de Partículas (PSO), adaptado a un problema de permutación de enteros (ordenación), y con una forma particular de evaluar la mejor posición global del enjambre. Complementariamente se realiza una exposición de lo que es una empresa Operadora de telecomunicaciones, de sus departamentos y procesos internos, de los servicios que ofrece, de las redes sobre las que se soportan, y de los puntos clave a tener en cuenta en la implementación de sus sistemas informáticos de gestión integral. ABSTRACT: Nowadays, any telecommunications company that owns its own network must face the problem of maintaining it (service assurance). Provide a minimum quality of service to its customers must be one of its main objectives. This quality should be maintained although any incidents happen to occur in the network. This thesis aims to solve the problem of prioritizing the order in which the damaged cables, trails, path and circuits, within a backbone transport network, should be restored. This work begins with a detailed explanation about network maintenance issues and all the factors that influence decision-making problem. First of all, one solution based on Discrete Multicriteria methods is tried (ELECTRE I and AHP algorithms are used). Also, a solution based on neural networks (with two different approaches to the problem) is analyzed. Finally, this thesis proposes an algorithm based on Particle Swarm Optimization (PSO), adapted to a problem of integers permutation, and a particular way of evaluating the best overall position of the swarm method. In addition, there is included an exhibition about telecommunications companies, its departments, internal processes, services offered to clients, physical networks, and key points to consider when implementing its integrated management systems.

Hybrid sampling on mutual information entropy-based clustering ensembles for optimizations

In this paper, we focus on the design of bivariate EDAs for discrete optimization problems and propose a new approach named HSMIEC. While the current EDAs require much time in the statistical learning process as the relationships among the variables are too complicated, we employ the Selfish gene theory (SG) in this approach, as well as a Mutual Information and Entropy based Cluster (MIEC) model is also set to optimize the probability distribution of the virtual population. This model uses a hybrid sampling method by considering both the clustering accuracy and clustering diversity and an incremental learning and resample scheme is also set to optimize the parameters of the correlations of the variables. Compared with several benchmark problems, our experimental results demonstrate that HSMIEC often performs better than some other EDAs, such as BMDA, COMIT, MIMIC and ECGA. © 2009 Elsevier B.V. All rights reserved.