Teresa León

A fuzzy mathematical programming approach to the assessment of efficiency with DEA models

In many real applications, the data of production processes cannot be precisely measured. This is particularly worrying when assessing efficiency with frontier-type models, such as data envelopment analysis (DEA) models, since they are very sensitive to possible data errors. For this reason, the possibility of having available a methodology that allows the analyst to deal with imprecise data becomes an issue of great interest in these contexts. To that end, we develop some fuzzy versions of the classical DEA models (in particular, the BCC model) by using some ranking methods based on the comparison of α-cuts. The resulting auxiliary crisp problems can be solved by the usual DEA software. We show, in a numerical example, how our models become specially useful for detecting sensitive decision-making units. Our approaches can be seen as an extension of the DEA methodology that provides users and practitioners with models which represent some real life processes more appropriately.

Viability of infeasible portfolio selection problems: A fuzzy approach☆

Abstract This paper deals with fuzzy optimization schemes for managing a portfolio in the framework of risk–return trade-off. Different models coexist to select the best portfolio according to their respective objective functions and many of them are linearly constrained. We are concerned with the infeasible instances of such models. This infeasibility, usually provoked by the conflict between the desired return and the diversification requirements proposed by the investor, can be satisfactorily avoided by using fuzzy linear programming techniques. We propose an algorithm to repair infeasibility and we illustrate its performance on a numerical example.

Applying logistic regression to relevance feedback in image retrieval systems

This paper deals with the problem of image retrieval from large image databases. A particularly interesting problem is the retrieval of all images which are similar to one in the users mind, taking into account his/her feedback which is expressed as positive or negative preferences for the images that the system progressively shows during the search. Here we present a novel algorithm for the incorporation of user preferences in an image retrieval system based exclusively on the visual content of the image, which is stored as a vector of low-level features. The algorithm considers the probability of an image belonging to the set of those sought by the user, and models the logit of this probability as the output of a generalized linear model whose inputs are the low-level image features. The image database is ranked by the output of the model and shown to the user, who selects a few positive and negative samples, repeating the process in an iterative way until he/she is satisfied. The problem of the small sample size with respect to the number of features is solved by adjusting several partial generalized linear models and combining their relevance probabilities by means of an ordered averaged weighted operator. Experiments were made with 40 users and they exhibited good performance in finding a target image (4 iterations on average) in a database of about 4700 images. The mean number of positive and negative examples is of 4 and 6 per iteration. A clustering of users into sets also shows consistent patterns of behavior.

Solving a class of fuzzy linear programs by using semi-infinite programming techniques☆

This paper deals with a class of Fuzzy Linear Programming problems characterized by the fact that the coefficients in the constraints are modeled as LR-fuzzy numbers with different shapes. Solving such problems is usually more complicated than finding a solution when all the fuzzy coefficients have the same shape. We propose a primal semi-infinite algorithm as a valuable tool for solving this class of Fuzzy Linear programs and, we illustrate it by means of several examples.

On the numerical treatment of linearly constrained semi-infinite optimization problems

Abstract We consider the application of two primal algorithms to solve linear semi-infinite programming problems depending on a real parameter. Combining a simplex-type strategy with a feasible-direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently. The second algorithm runs a feasible-direction method first and then switches to the purification procedure. The linear programming subproblems that yield the search direction involve only a small subset of the constraints. These subsets are updated at each iteration using a multi-local optimization algorithm. Numerical test examples, taken from the literature in order to compare the numerical effort with other methods, show the efficiency of the proposed algorithms.

A multi-local optimization algorithm

The development of efficient algorithms that provide all the local minima of a function is crucial to solve certain subproblems in many optimization methods. A “multi-local” optimization procedure using inexact line searches is presented, and numerical experiments are also reported. An application of the method to a semi-infinite programming procedure is included.

A fuzzy method to repair infeasibility in linearly constrained problems

Abstract In this paper we introduce a fuzzy method to deal with infeasibility in linearly constrained programs. Given an infeasible instance, we determine how much we should perturb the right-hand side coefficients in order to attain feasibility and propose a ‘feasible reformulation’ of the problem. Although we prove that our algorithm always finds such a reformulation the convenience of using it can be decided by the analyst. By this, we mean that the method also provides a simple way to compute lower bounds on the changes on every right-hand side coefficient, and if the decision maker considers that some of the magnitudes are unacceptable, he or she simply stops at this step. We think that it will be specially useful for those situations in which the cause of the infeasibility is in the requirement of specifying exact values for the parameters in the mathematical programs formulation.

A purification algorithm for semi-infinite programming

Abstract In this paper we present a purification algorithm for semi-infinite linear programming. Starting with a feasible point, the algorithm either finds an improved extreme point or concludes with the unboundedness of the problem. The method is based on the solution of a sequence of linear programming problems. The study of some recession conditions has allowed us to establish a weak assumption for the finite convergence of this algorithm. Numerical results illustrating the method are given.

New descent rules for solving the linear semi-infinite programming problem

The algorithm described in this paper approaches the optimal solution of a continuous semi-infinite linear programming problem through a sequence of basic feasible solutions. The descent rules that we present for the improvement step are quite different when one deals with non-degenerate or degenerate extreme points. For the non-degenerate case we use a simplex-type approach, and for the other case a search direction scheme is applied. Some numerical examples illustrating the method are given.

Optimization under Uncertainty and Linear Semi-Infinite Programming: A Survey

This paper deals with the relationship between semi-infinite linear programming and decision making under uncertainty in imprecise environments. Actually, we have reviewed several set-inclusive constrained models and some fuzzy programming problems in order to see if they can be solved by means of a linear semi-infinite program. Finally, we present some numerical examples obtained by using a primal semi-infinite programming method.