Ricardo C. Silva

Two-phase method to solve fuzzy quadratic programming problems

Quadratic programming problems are of up most importance in a variety of relevant practical fields, as e.g., portfolio selection. This work presents and develops an original and novel fuzzy sets based method that solves a class of quadratic programming problems with vagueness in the set of constraints. The method uses two phases to solve fuzzy quadratic programming problems, which eventually can be considered in the portfolio context. In the first phase we parametrize the fuzzy problem in several classical alpha-problems with different cutting levels. In the second phase each of these alpha-problems is solved by using conventional solving techniques. The final fuzzy solution to the former problem can be obtained by integrating all of these particular alpha-solutions. Some illustrative numerical examples illustrating the solution approach are solved and analyzed to show the efficiency of this proposed method.

Extending and relating different approaches for solving fuzzy quadratic problems

Quadratic programming problems are applied in an increasing variety of practical fields. As ambiguity and vagueness are natural and ever-present in real-life situations requiring solutions, it makes perfect sense to attempt to address them using fuzzy quadratic programming problems. This work presents two methods used to solve linear problems with uncertainties in the set of constraints, which are extended in order to solve fuzzy quadratic programming problems. Also, a new quadratic parametric method is proposed and it is shown that this proposal contains all optimal solutions obtained by the extended approaches with their satisfaction levels. A few numerical examples are presented to illustrate the proposed method.

A Survey of Fuzzy Convex Programming Models

Optimization is a procedure of finding and comparing feasible solutions until no better solution can be found. It can be divided into several fields, one of which is the Convex Optimization. It is characterized by a convex objective function and convex constraint functions over a convex set which is the set of the decision variables. This can be viewed, on the one hand, as a particular case of nonlinear programming and, on the other hand, as a general case of linear programming. Convex optimization has applications in a wide range of real-world applications, whose data often cannot be formulate precisely. Hence it makes perfect sense to apply fuzzy set theory as a way to mathematically describe this vagueness. In this paper we review the theory about this topic and describe some flexible and possibilistic programming models to solve fuzzy convex programming problems. Flexible programming uses fuzzy sets to represent the vagueness of the decision maker’s aspirations and constraints, while possibilistic programming models imprecise or ambiguous data by possibility distributions.

Fuzzy costs in quadratic programming problems

Although quadratic programming problems are a special class of nonlinear programming, they can also be seen as general linear programming problems. These quadratic problems are of the utmost importance in an increasing variety of practical fields. As, in addition, ambiguity and vagueness are natural and ever-present in real-life situations requiring operative solutions, it makes perfect sense to address them using fuzzy concepts formulated as quadratic programming problems with uncertainty, i.e., as Fuzzy Quadratic Programming problems. This work proposes two novel fuzzy-sets-based methods to solve a particular class of Fuzzy Quadratic Programming problems which have vagueness coefficients in the objective function. Moreover, two other linear approaches are extended to solve the quadratic case. Finally, it is shown that the solutions reached from the extended approaches may be obtained from two proposed parametric multiobjective approaches.

The use of possibility theory in the definition of fuzzy Pareto-optimality

Pareto-optimality conditions are crucial when dealing with classic multi-objective optimization problems. Extensions of these conditions to the fuzzy domain have been discussed and addressed in recent literature. This work presents a novel approach based on the definition of a fuzzily ordered set with a view to generating the necessary conditions for the Pareto-optimality of candidate solutions in the fuzzy domain. Making use of the conditions generated, one can characterize fuzzy efficient solutions by means of carefully chosen mono-objective problems and Karush-Kuhn-Tucker conditions to fuzzy non-dominated solutions. The uncertainties are inserted into the formulation of the studied fuzzy multi-objective optimization problem by means of fuzzy coefficients in the objective function. Some numerical examples are analytically solved to illustrate the efficiency of the proposed approach.

Using the fuzzy sets theory in the multimodal transport network problem

In this work, an approach to solve multimodal transport network problems with uncertain costs in the edges is proposed. This kind of problem has been studied for several researchers who search solutions to the large numbers of problems relating on the transport systems like: traffic jam, pollution, delays, among others. In this work, the modelling of this problems is based on graph theory are presented, where each transport mode is represented by a subgraph and the whole graph is the union of all subgraphs. Besides, its mathematical formulation describes nonlinear conditions and the goal is to develop an algorithm that reaches a set of Pareto solutions, which find the best routes between origin and destination. A numerical example is used to illustrate the efficiency of the proposed approach.

Solving Real-World Fuzzy Quadratic Programming Problems by a Parametric Method

Although fuzzy quadratic programming problems are of the utmost importance in an increasing variety of practical fields, there are remaining technological areas in which has not been tested their applicability or, if tried, have been little studied possibilities. This may be the case of Renewable Energy Assessment, Service Quality, Technology Foresight, Logistics, Systems Biology, etc. With this in mind, the goal of this paper is to apply a parametric approach previously developed by authors to solve some of these problems, specifically the portfolio selection problem by using BM&FBOVESPA data of some Brazilian securities and the economic dispatch problem, which schedules a power generation in an appropriate manner in order to satisfy the load demand.

Application of an iterative method and an evolutionary algorithm in fuzzy optimization

This work develops two approaches based on the fuzzy set theory to solve a class of fuzzy mathematical optimization problems with uncertainties in the objective function and in the set of constraints. The first approach is an adaptation of an iterative method that obtains cut levels and later maximizes the membership function of fuzzy decision making using the bound search method. The second one is a meta- heuristic approach that adapts a standard genetic algorithm to use fuzzy numbers. Both approaches use a decision criterion called satisfaction level that reaches the best solution in the uncertain environment. Selected examples from the literature are presented to compare and to validate the efficiency of the methods addressed, emphasizing the fuzzy optimization problem in some import-export companies in the south of Spain.

A dual approach to solve fuzzy quadratic programming problems

Although quadratic programming can be defined as a specific class of non-linear programming, it can also be used to generalize linear programming. Therefore, one can find many applications in quadratic in real world problems. Inaccuracies are also found in a natural way in real life situations that require realistic solutions. Fuzzy logic has to deal with the uncertainties inherent in this situation. The initiative to shape the inaccuracies in real life optimization problems is applied in an increasing variety of practical fields. Knowing the importance of this problem, the purpose of this work is to present a new dual approach in fuzzy environment. It solves quadratic programming problems with uncertain order relation in the set of constraints. The approach proposal is implemented in two theoretical numerical examples, which show their efficiency.

Relação entre modelos de programação não-linear com incerteza no conjunto de restrições

In this work, we demonstrate analytical and numerically the relation between two methods, Trappey et al. (1988) and Xu (1989), found in the literature. These methods were developed to solve nonlinear programming problems with uncertainties in the set of constraints. A comparative analysis of the performance between classic and fuzzy nonlinear optimization methods are presented too in the work. For the comparison, two problems are modeled that had been shaped in terms of classic nonlinear programming, which allow the introduction of uncertainties in its formularizations. Based on the analysis of the problem proposed in Xu (1989), we verified that the two described methods provide similar results, as per some conditions.