J. M. Herrero

A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.

Multiobjective evolutionary algorithms for multivariable PI controller design

A multiobjective optimisation engineering design (MOED) methodology for PI controller tuning in multivariable processes is presented. The MOED procedure is a natural approach for facing multiobjective problems where several requirements and specifications need to be fulfilled. An algorithm based on the differential evolution technique and spherical pruning is used for this purpose. To evaluate the methodology, a multivariable control benchmark is used. The obtained results validate the MOED procedure as a practical and useful technique for parametric controller tuning in multivariable processes.

Comparison of design concepts in multi-criteria decision-making using level diagrams

In this work, we address the evaluation of design concepts and the analysis of multiple Pareto fronts in multi-criteria decision-making using level diagrams. Such analysis is relevant when two (or more) design concepts with different design alternatives lie in the same objective space, but describe different Pareto fronts. Therefore, the problem can be stated as a Pareto front comparison between two (or more) design concepts that only differ in their relative complexity, implementation issues, or the theory applied to solve the problem at hand. Such analysis will help the decision maker obtain a better insight of a conceptual solution and be able to decide if the use of a complex concept is justified instead of a simple concept. The approach is validated in a set of multi-criteria decision making benchmark problems.

Hybrid DE algorithm with adaptive crossover operator for solving real-world numerical optimization problems

In this paper, the results for the CEC 2011 Competition on testing evolutionary algorithms on real world optimization problems using a hybrid differential evolution algorithm are presented. The proposal uses a local search routine to improve convergence and an adaptive crossover operator. According to the obtained results, this algorithm shows to be able to find competitive solutions with reported results.

Applied Pareto multi-objective optimization by stochastic solvers

It is well known that many engineering design problems with different objectives, some of which can be opposed to one another, can be formulated as multi-objective functions and resolved with the construction of a Pareto front that helps to select the desired solution. Obtaining a correct Pareto front is not a trivial question, because it depends on the complexity of the objective functions to be optimized, the constraints to keep within and, in particular, the optimizer type selected to carry out the calculations. This paper presents new methods for Pareto front construction based on stochastic search algorithms (genetic algorithms, GAs and multi-objective genetic algorithms, MOGAs) that enable a very good determination of the Pareto front and fulfill some interesting specifications. The advantages of these applied methods will be proven by the optimization of well-known benchmarks for metallic supported I-beam and gearbox design.

Hole distribution in phononic crystals: Design and optimization

An exhaustive study has been made into the potential improvement in attenuation and focusing of phononic crystal arrays resulting from the deliberate creation of vacancies. Use is made of a stochastic search algorithm based on evolutionary algorithms called the epsilon variable multi-objective genetic algorithm which, in conjunction with the application of multiple scattering theory, enables the design of devices for effectively controlling sound waves. Several parameters are analyzed, including the symmetries used in the distribution of holes and the optimum number of holes. The validity and utility of the general rules obtained have been confirmed experimentally.

Non-linear robust identification using evolutionary algorithms

This work describes a new methodology for robust identification (RI), meaning the identification of the parameters of a model and the characterization of uncertainties. The alternative proposed handles non-linear models and can take into account the different properties demanded by the model. The indicator that leads the identification process is the identification error (IE), that is, the difference between experimental data and model response. In particular, the methodology obtains the feasible parameter set (FPS, set of parameter values which satisfy a bounded IE) and a nominal model in a non-linear identification problem. To impose different properties on the model, several norms of the IE are used and bounded simultaneously. This improves the model quality, but increases the problem complexity. The methodology proposes that the RI problem is transformed into a multimodal optimization problem with an infinite number of global minima which constitute the FPS. For the optimization task, a special genetic algorithm (@e-GA), inspired by Multiobjective Evolutionary Algorithms, is presented. This algorithm characterizes the FPS by means of a discrete set of models well distributed along the FPS. Finally, an application for a biomedical model which shows the blockage that a given drug produces on the ionic currents of a cardiac cell is presented to illustrate the methodology.

Well-Distributed Pareto Front by Using the \epsilon \hskip-0.9em \nearrow \hskip-0.4em-MOGA Evolutionary Algorithm

In the field of multiobjective optimization, important efforts have been made in recent years to generate global Pareto fronts uniformly distributed. A new multiobjective evolutionary algorithm, called \(\epsilon \hskip-0.9em \nearrow \hskip-0.4em-MOGA\), has been designed to converge towards \(\mathbf{\Theta}_P^*\), a reduced but well distributed representation of the Pareto set Θ P . The algorithm achieves good convergence and distribution of the Pareto front J(Θ P ) with bounded memory requirements which are established with one of its parameters. Finally, a optimization problem of a three-bar truss is presented to illustrate the algorithm performance.In the field of multiobjective optimization, important efforts have been made in recent years to generate global Pareto fronts uniformly distributed. A new multiobjective evolutionary algorithm, called ∉-MOGA, has been designed to converge towards ΘP*, a reduced but well distributed representation of the Pareto set ΘP. The algorithm achieves good convergence and distribution of the Pareto front J(ΘP) with bounded memory requirements which are established with one of its parameters. Finally, a optimization problem of a three-bar truss is presented to illustrate the algorithm performance.

A Smart-Distributed Pareto Front Using the ev-MOGA Evolutionary Algorithm

Obtaining multi-objective optimization solutions with a small number of points smartly distributed along the Pareto front is a challenge. Optimization methods, such as the normalized normal constraint (NNC), propose the use of a filter to achieve a smart Pareto front distribution. The NCC optimization method presents several disadvantages related with the procedure itself, initial condition dependency, and computational burden. In this article, the epsilon-variable multi-objective genetic algorithm (ev-MOGA) is presented. This algorithm characterizes the Pareto front in a smart way and removes the disadvantages of the NNC method. Finally, examples of a three-bar truss design and controller tuning optimizations are presented for comparison purposes.

Decision Making Graphical Tool for Multiobjective Optimization Problems

Multiobjective optimization problems have become an important issue at many engineering problems. A tradeoff between several design criteria is required and important efforts are made for the development of Multiobjective Optimization Techniques and, in particular, Evolutionary Multiobjective Optimization. Usually these algorithms produce a set of optimum solutions in Pareto sense, there is not a unique solution. The designer (Decision Maker) has to finally select one solution for each particular problem, then he has to select from a set of Pareto solutions, the most adequate solution according to his preferences. It is widely accepted that visualization tools are valuable tools to provide the Decision Maker (DM) with a meaningful way to analyze Pareto set and then to help to select an adequate solution. This work describes a new graphical way to represent high dimensional and large sets of Pareto solutions, allowing an easier analysis, and helping the DM to select an adequate solution.