Gracia Sánchez

An evolutionary algorithm for constrained multi-objective optimization

The paper follows the line of the design and evaluation of new evolutionary algorithms for constrained multi-objective optimization. The evolutionary algorithm proposed (ENORA) incorporates the Pareto concept of multi-objective optimization with a constraint handling technique and with a powerful diversity mechanism to obtain multiple nondominated solutions through the simple run of the algorithm. Constraint handling is carried out in an evolutionary way and using the min-max formulation, while the diversity technique is based on the partitioning of search space in a set of radial slots along which are positioned the successive populations generated by the algorithm. A set of test problems recently proposed for the evaluation of this kind of algorithm has been used in the evaluation of the algorithm presented. The results obtained with ENORA were very good and considerably better than those obtained with algorithms recently proposed by other authors.

Multi-objective evolutionary computation and fuzzy optimization

Abstract In fuzzy optimization it is desirable that all fuzzy solutions under consideration be attainable, so that the decision maker will be able to make “a posteriori” decisions according to current decision environments. No additional optimization runs will be needed when the decision environment changes or when the decision maker needs to evaluate several decisions to establish the most appropriate ones. In this sense, multi-objective optimization is similar to fuzzy optimization, since it is also desirable to capture the Pareto front composing the solution. The Pareto front in a multi-objective problem can be interpreted as the fuzzy solution for a fuzzy problem. Multi-objective evolutionary algorithms have been shown in the last few years to be powerful techniques in solving multi-objective optimization problems because they can search for multiple Pareto solutions in a single run of the algorithm. In this contribution, we first introduce a multi-objective approach for nonlinear constrained optimization problems with fuzzy costs and constraints, and then an “ad hoc” multi-objective evolutionary algorithm to solve the former problem. A case study of a fuzzy optimization problem arising in some import–export companies in the south of Spain is analyzed and the proposed solutions from the evolutionary algorithm considered here are given.

A multi-objective evolutionary approach for fuzzy optimization in production planning

This paper outlines, first, a real-world industrial problem for product-mix selection involving 8 variables and 21 constraints with fuzzy coefficients and thereafter, a multi-objective optimization approach to solve the problem. This problem occurs in production planning in which a decisionmaker plays a pivotal role in making decision under fuzzy environment. Decision-maker should be aware of his/her level-of-satisfaction as well as degree of fuzziness while making the product-mix decision. Thus, the authors have analyzed using a modified S-curve membership function the fuzziness patterns and fuzzy sensitivity of the solution found from the multi-objective optimization methodology. An ad hoc Pareto-based multi-objective evolutionary algorithm is proposed to capture multiple non dominated solutions in a single run of the algorithm. Results obtained have been compared with the well-known multi-objective evolutionary algorithm NSGA-II.

Multi-objective evolutionary algorithms for fuzzy classification in survival prediction

OBJECTIVE This paper presents a novel rule-based fuzzy classification methodology for survival/mortality prediction in severe burnt patients. Due to the ethical aspects involved in this medical scenario, physicians tend not to accept a computer-based evaluation unless they understand why and how such a recommendation is given. Therefore, any fuzzy classifier model must be both accurate and interpretable. METHODS AND MATERIALS The proposed methodology is a three-step process: (1) multi-objective constrained optimization of a patients data set, using Pareto-based elitist multi-objective evolutionary algorithms to maximize accuracy and minimize the complexity (number of rules) of classifiers, subject to interpretability constraints; this step produces a set of alternative (Pareto) classifiers; (2) linguistic labeling, which assigns a linguistic label to each fuzzy set of the classifiers; this step is essential to the interpretability of the classifiers; (3) decision making, whereby a classifier is chosen, if it is satisfactory, according to the preferences of the decision maker. If no classifier is satisfactory for the decision maker, the process starts again in step (1) with a different input parameter set. RESULTS The performance of three multi-objective evolutionary algorithms, niched pre-selection multi-objective algorithm, elitist Pareto-based multi-objective evolutionary algorithm for diversity reinforcement (ENORA) and the non-dominated sorting genetic algorithm (NSGA-II), was tested using a patients data set from an intensive care burn unit and a standard machine learning data set from an standard machine learning repository. The results are compared using the hypervolume multi-objective metric. Besides, the results have been compared with other non-evolutionary techniques and validated with a multi-objective cross-validation technique. Our proposal improves the classification rate obtained by other non-evolutionary techniques (decision trees, artificial neural networks, Naive Bayes, and case-based reasoning) obtaining with ENORA a classification rate of 0.9298, specificity of 0.9385, and sensitivity of 0.9364, with 14.2 interpretable fuzzy rules on average. CONCLUSIONS Our proposal improves the accuracy and interpretability of the classifiers, compared with other non-evolutionary techniques. We also conclude that ENORA outperforms niched pre-selection and NSGA-II algorithms. Moreover, given that our multi-objective evolutionary methodology is non-combinational based on real parameter optimization, the time cost is significantly reduced compared with other evolutionary approaches existing in literature based on combinational optimization.

Solving fuzzy optimization problems by evolutionary algorithms

In this paper mathematical programming problems with fuzzy constraints are dealt with. Fuzzy solutions are obtained by means of a parametric approach in conjunction with evolutionary techniques. Some relevant characteristics of the evolutionary algorithm are for instance a real-coded representation of solutions and the preselection scheme as niche formation and elitist technique. Three test problems with fuzzy constraints and different structures are used in order to check and compare the proposed technique. The results obtained are very good in comparison with those from another methods.

Fuzzy Optimization with Multi-Objective Evolutionary Algorithms: a Case Study

This paper outlines a real-world industrial problem for product-mix selection involving 8 decision variables and 21 constraints with fuzzy coefficients. On one hand, a multi-objective optimization approach to solve the fuzzy problem is proposed. Modified S-curve membership functions are considered. On the other hand, an ad hoc Pareto-based multi-objective evolutionary algorithm to capture multiple non dominated solutions in a single run of the algorithm is described. Solutions in the Pareto front corresponds with the fuzzy solution of the former fuzzy problem expressed in terms of the group of three (xrarr, mu, alpha), i.e., optimal solution - level of satisfaction - vagueness factor. Decision-maker could choose, in a posteriori decision environment, the most convenient optimal solution according to his level of satisfaction and vagueness factor. The proposed algorithm has been evaluated with the existing methodologies in the field and the results have been compared with the well-known multi-objective evolutionary algorithm NSGA-II

Improving interpretability in approximative fuzzy models via multiobjective evolutionary algorithms

Current research lines in fuzzy modeling mostly tackle improving the accuracy in descriptive models and improving of the interpretability in approximative models. This article deals with the second issue, approaching the problem by means of multiobjective optimization in which accuracy and interpretability criteria are simultaneously considered. Evolutionary algorithms are especially appropriated for multiobjective optimization because they can capture multiple Pareto solutions in a single run of the algorithm. We propose a multiobjective evolutionary algorithm to find multiple Pareto solutions (fuzzy models) showing a trade‐off between accuracy and interpretability. Additionally, neural‐network‐based techniques in combination with ad hoc techniques for improving interpretability are incorporated into the multiobjective evolutionary algorithm to improve the efficiency of the algorithm.

Nonlinear Optimization with Fuzzy Constraints by Multi-Objective Evolutionary Algorithms

Fuzzy constrained optimization problems have been extensively studied since the seventies. In the linear case, the first approaches to solve the so-called fuzzy linear programming problem were made in [12] and [15]. Since then, important contributions solving different linear models have been done and these models have been recipients of a great dealt of work. In the nonlinear case the situation is quite different, as there is a wide variety of specific and both practical and theoretically relevant nonlinear problems, each having a different solution method. In the following we consider a Nonlinear Programming problem with fuzzy constraints. From a mathematical point of view the problem can be addressed as:

Fuzzy modeling with multi-objective neuro-evolutionary algorithms

\begin{gathered} Min f(x) \hfill \\ s.t.:g_j (x) \lesssim b_j , j = 1, \ldots ,m \hfill \\ x_i \in [l_i ,u_i ], i = 1, \ldots ,n, l_i \geqslant 0 \hfill \\ \end{gathered}