José Manuel Cadenas

Using fuzzy numbers in linear programming

Managers, decision makers, and experts dealing with optimization problems often have a lack of information on the exact values of some parameters used in their problems. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve these problems. This paper studies a linear programming (LP) problem in which all its elements are defined as fuzzy sets. Special cases of this general model are found and reproduced, and it is shown that they coincide with the particular problems proposed in the literature by different authors and distinct approaches. Solution methods are also provided. They show how it is possible to address and solve linear programming problems with data given in a qualitative form, instead of the usual quantitative and precise way.

Using ranking functions in multiobjective fuzzy linear programming

Multiobjective mathematical programming problems, in particular the vector optimization problems, define a well known and studied area because of its relevance to numerous practical applications. In this paper vector optimization problems with a fuzzy nature are considered. In these problems usually it is assumed that all the objective functions involved come from the same decision maker. The problem considered here assumes, however, that the objective functions can be defined by different decision makers, and that the coefficients in each of these objective functions are fuzzy numbers. Hence, solution methodologies for these multiobjective fuzzy mathematical programming problems, using different ordering methods ranking fuzzy numbers, are proposed. As an illustration a bi-objective model for land use is presented.

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.

Application of fuzzy optimization to diet problems in Argentinean farms

Abstract The problem of designing diets for cattle in an Argentinean farm is addressed in this paper. Usually the livestock is not confined, so it is impossible to assess the amount of food each animal will eat. Therefore it makes no sense to design diets verifying the nutritional requirements exactly. It is more valuable to allow for constraint violations, i.e., fuzzy constraints which in turn may lead to cheaper diets. Under this scenario the problem is modelled as a fuzzy linear programming (FLP) one and then solved by using a decision support systems (DSS) named SACRA (a Spanish acronym for support system for the construction of cattle diets) that the authors have specifically developed for this problem. SACRA is based on PROBO, an already experimented DSS also developed by the authors, and it is highly friendship, interactive and does not require any knowledge about FLP. The tests carried out with SACRA have shown a high level of satisfaction from the side of the decision makers.

Membership functions in the fuzzy C-means algorithm

A membership function and an objective function deduced from the geometrical properties associated to the metric defined by the covariance matrix of a sample have been proposed recently. We use these functions to determine the membership probabilities and the criterion function in the fuzzy C-means algorithm.

Using machine learning in a cooperative hybrid parallel strategy of metaheuristics

This paper proposes the construction of a centralized hybrid metaheuristic cooperative strategy to solve optimization problems. Knowledge (intelligence) is incorporated into the coordinator to improve performance. This knowledge is incorporated through a set of rules and models obtained from a knowledge extraction process applied to the records of the results returned by individual metaheuristics. The effectiveness of the approach is tested in several computational experiments in which we compare the results obtained by the individual metaheuristics, by several non-cooperative and cooperative strategies and by the strategy proposed in this paper.

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.

Towards a new strategy for solving fuzzy optimization problems

Fuzzy Optimization models and methods has been one of the most and well studied topics inside the broad area of Soft Computing. Particularly relevant is the field of fuzzy linear programming (FLP). Its applications as well as practical realizations can be found in all the real world areas. As FLP problems constitute the basis for solving fuzzy optimization problems, in this paper a basic introduction to the main models and methods in FLP is presented and, as a whole, Linear Programming problems with fuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrix are analyzed. But fuzzy sets and systems based optimization methods do not end with FLP, and hence in order to solve more complex optimization problems, fuzzy sets based Meta-heuristics are considered, and two main operative approaches described. Provided that these techniques obtain efficient and/or effective solutions, we present a fuzzy rule based methodology for coordinating Meta-heuristics and in addition, to provide intelligence, we propose a process of extraction of the knowledge to conduct the coordination of the system.

Partition validity and defuzzification

The fuzzy partition validity is a necessary measure to fix the number of groups in the fuzzy algorithms which need to know this data. We propose a measure based on the membership probabilities. Moreover, we propose a new concept of the hard partition based on the metric of the groups. The hard partition can be obtained from the fuzzy partition, from which it is possible to find the hard membership probabilities and the hard partition validity. A similarity relation is proposed over the set of groups belonging to the hard partition.

A local geometrical properties application to fuzzy clustering

Possibilistic clustering is seen increasingly as a suitable means to resolve the limitations resulting from the constraints imposed in the fuzzy C-means algorithm. Studying the metric derived from the covariance matrix we obtain a membership function and an objective function whether the Mahalanobis distance or the Euclidean distance is used. Applying the theoretical results using the Euclidean distance we obtain a new algorithm called fuzzy-minimals, which detects the possible prototypes of the groups of a sample. We illustrate the new algorithm with several examples.