A Study in function optimization with the breeder genetic algorithm

Optimization is concerned with the finding of global optima (hence the name) of problems that can be cast in the form of a function of several variables and constraints thereof. Among the searching methods, {em Evolutionary Algorithms} have been shown to be adaptable and general tools that have often outperformed traditional {em ad hoc} methods. The {em Breeder Genetic Algorithm} (BGA) combines a direct representation with a nice conceptual simplicity. This work contains a general description of the algorithm and a detailed study on a collection of function optimization tasks. The results show that the BGA is a powerful and reliable searching algorithm. The main discussion concerns the choice of genetic operators and their parameters, among which the family of Extended Intermediate Recombination (EIR) is shown to stand out. In addition, a simple method to dynamically adjust the operator is outlined and found to greatly improve on the already excellent overall performance of the algorithm.Postprint (published version

Measuring efficiency of a hierarchical organization with fuzzy DEA method

The paper analyses how the data envelopment analysis (DEA) and fuzzy set theory can be used to measure and evaluate the efficiency of a hierarchical system with n decision making units and a coordinating unit. It is presented a model for determining the of activity levels of decision making units so as to achieve both fuzzy objectives of achieving global target levels of coordination unit on the inputs and outputs and individual target levels of decision making units, and then some methods to resolve fuzzy models are proposed.fuzzy DEA, policy making in multi-level organisations, efficiency analysis

Lexicographic Methods for Fuzzy Linear Programming

Fuzzy Linear Programming (FLP) has addressed the increasing complexity of real-world decision-making problems that arise in uncertain and ever-changing environments since its introduction in the 1970s. Built upon the Fuzzy Sets theory and classical Linear Programming (LP) theory, FLP encompasses an extensive area of theoretical research and algorithmic development. Unlike classical LP, there is not a unique model for the FLP problem, since fuzziness can appear in the model components in different ways. Hence, despite fifty years of research, new formulations of FLP problems and solution methods are still being proposed. Among the existing formulations, those using fuzzy numbers (FNs) as parameters and/or decision variables for handling inexactness and vagueness in data have experienced a remarkable development in recent years. Here, a long-standing issue has been how to deal with FN-valued objective functions and with constraints whose left- and right-hand sides are FNs. The main objective of this paper is to present an updated review of advances in this particular area. Consequently, the paper briefly examines well-known models and methods for FLP, and expands on methods for fuzzy single- and multi-objective LP that use lexicographic criteria for ranking FNs. A lexicographic approach to the fuzzy linear assignment (FLA) problem is discussed in detail due to the theoretical and practical relevance. For this case, computer codes are provided that can be used to reproduce results presented in the paper and for practical applications. The paper demonstrates that FLP that is focused on lexicographic methods is an active area with promising research lines and practical implications.Spanish Ministry of Economy and CompetitivenessEuropean Union (EU) TIN2017-86647-

FLIP - Multiobjective Fuzzy Linear Programming Package

FLIP (Fuzzy LInear Programming) is a package designed to help in analysis of multiobjective linear programming (MOLP) problems in an uncertain environment. The uncertainty of data is modeled by L-R type fuzzy numbers. They can appear in the objective functions as well as on the both sides of the constraints. The input data to the FLIP package include the characteristics of the analyzed fuzzy MOLP problem, i.e., the number of criteria, constraints and decision variables, fuzzy cost coefficients for every objective and fuzzy coefficients of LHS and RHS for all constraints. The data loading is supported by a graphical presentation of fuzzy coefficients. The calculation is preceded by a transformation of the fuzzy MOLP problem into a multiobjective linear fractional program. It is then solved with an interactive method using a linear programming procedure as the only optimiser. In every iteration, one gets a series of solutions that are presented very clearly in a graphical and numerical form. In FLIP, interaction with the user takes place at two levels: first, when safety parameters have to be defined in the transformation phase, and second, when the associate deterministic problem is solved. The package is written in TURBO-Pascal and can be used on microcomputers compatible with IBM-PC XT/AT with hard disc and a graphic card

Possibility/Necessity-Based Probabilistic Expectation Models for Linear Programming Problems with Discrete Fuzzy Random Variables

This paper considers linear programming problems (LPPs) where the objective functions involve discrete fuzzy random variables (fuzzy set-valued discrete random variables). New decision making models, which are useful in fuzzy stochastic environments, are proposed based on both possibility theory and probability theory. In multi-objective cases, Pareto optimal solutions of the proposed models are newly defined. Computational algorithms for obtaining the Pareto optimal solutions of the proposed models are provided. It is shown that problems involving discrete fuzzy random variables can be transformed into deterministic nonlinear mathematical programming problems which can be solved through a conventional mathematical programming solver under practically reasonable assumptions. A numerical example of agriculture production problems is given to demonstrate the applicability of the proposed models to real-world problems in fuzzy stochastic environments

An ε-Constraint Method for Multiobjective Linear Programming in Intuitionistic Fuzzy Environment

Effective decision-making requires well-founded optimization models and algorithms tolerant of real-world uncertainties. In the mid-1980s, intuitionistic fuzzy set theory emerged as another mathematical framework to deal with the uncertainty of subjective judgments and made it possible to represent hesitancy in a decision-making problem. Nowadays, intuitionistic fuzzy multiobjective linear programming (IFMOLP) problems are a topic of extensive research, for which a considerable number of solution approaches are being developed. Among the available solution approaches, ranking function-based approaches stand out for their simplicity to transform these problems into conventional ones. However, these approaches do not always guarantee Pareto optimal solutions. In this study, the concepts of dominance and Pareto optimality are extended to the intuitionistic fuzzy case by using lexicographic criteria for ranking triangular intuitionistic fuzzy numbers (TIFNs). Furthermore, an intuitionistic fuzzy epsilon-constraint method is proposed to solve IFMOLP problems with TIFNs. The proposed method is illustrated by solving two intuitionistic fuzzy transportation problems addressed in two studies (S. Mahajan and S. K. Gupta's, "On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions," Ann Oper Res, vol. 296, no. 1, pp. 211-241, 2021, and Ghosh et al.'s, "Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem," Complex Intell Syst, vol. 7, no. 2, pp. 1009-1023, 2021). Results show that, in contrast with Mahajan and Gupta's and Ghosh et al.'s methods, the proposed method guarantees Pareto optimality and also makes it possible to obtain multiple solutions to the problems.MCIN/AEI PID2020-112754GB-I00FEDER/Junta de Andalucia-Consejeria de Transformacion Economica, Industria, Conocimiento y Universidades/Proyecto B-TIC-640-UGR2