Bogdana Stanojević’s research while affiliated with Mathematical Institute of the Serbian Academy of Sciences and Arts and other places

We approach the full fuzzy data envelopment analysis (DEA) strictly relying on the extension principle. So far in the literature, fuzzy DEA (that uses fuzzy inputs and outputs but crisp weights) and full fuzzy DEA (that uses fuzzy inputs, outputs and weights) were treated distinctly. However, the crisp weights from fuzzy models only act as crisp weights within optimization, but in fact their feasible values finally describe fuzzy weights. As a consequence, the distinction between full fuzzy models and fuzzy models is due to the distinction between establishing or not an a priori shape for the fuzzy weights. In this paper we advance the idea that the methodologies introduced for fuzzy DEA are valuable for full fuzzy DEA as well; and propose a Monte Carlo simulation algorithm to offer an empirical visualization of the shapes of the fuzzy efficiencies of DMUs in full fuzzy DEA. Such visualization firstly can certify whether a solution approach to a full fuzzy DEA derives solutions complying to the extension principle or not; and secondly discloses the fuzzy shapes of the weights obtained by applying a methodology from fuzzy DEA to solving full fuzzy DEA. The complexity of the proposed algorithm is the same as the solution approach to the crisp DEA model that corresponds to the observed full fuzzy DEA model. We report the numerical results of our experiments, compare them to results found in the recent literature, and discuss the misleading consequences of ignoring the extension principle in the context of full fuzzy DEA. Keywords: data envelopment analysis, efficiency, fuzzy programming.

Optimization involving fuzzy numbers generally, and intuitionistic fuzzy numbers particularly is more and more an essential part of any computational intelligence method; and might be of interest in modeling fuzzy control systems, or carrying out a fuzzy sensitivity analysis. The recent literature includes many research papers related to both theoretical modelling, and practical implementation of fuzzy decision support systems. Fuzzy optimization is one of the main part in such decisionmaking tools. A realistic solution to a fuzzy optimization problem is always desired, and similarly, a Pareto optimal solution to a multiple objective optimization problem is always a need. In this paper we analyze the shortcomings; and eliminate the weaknesses of a solution approach from the literature proposed by Moges et al. in 2023. Firstly, Moges et al. used an accuracy function to de-fuzzify the original intuitionistic fuzzy optimization problem; then linearized the obtained crisp problem; and finally involved fuzzy goals to solve the derived multiple objective linear problem. We present two faulty key points of their approach, namely the de-fuzzification and linearization steps; prove the inappropriateness of their results; and propose some improvements. The final aim in addressing the first key point is to clarify the border between defuzzifications made in accordance to true theoretical statements, and those made for modeling reasons. The methodological improvement we propose is related to the second key point and it assures that Pareto optimal solutions - highly required in multiple objective optimization - are obtained. To illustrate our point of view, we use a numerical example from the literature, and report better numerical results derived by the improved methodology.

Solving optimization problems in a fuzzy environment is an area widely addressed in the recent literature. De-fuzzification of data, construction of crisp more or less equivalent problems, unification of multiple objectives, and solving a single crisp optimization problem are the general descriptions of many procedures that approach fuzzy optimization problems. Such procedures are misleading (since relevant information is lost through de-fuzzyfication and aggregation of more objectives into a single one), but they are still dominant in the literature due to their simplicity. In this paper, we address the full fuzzy linear programming problem, and provide solutions in full accordance with the extension principle. The main contribution of this paper is in modeling the conjunction of the fuzzy sets using the “product” operator instead of “min” within the definition of the solution concept. Our theoretical findings show that using a generalized “min” operator within the extension principle assures thinner shapes to the derived fuzzy solutions compared to those available in the literature. Thinner shapes are always desirable, since such solutions provide the decision maker with more significant information.

Business Analytics – which unites Descriptive, Predictive and Prescriptive Analytics – represents an important component in the framework of Big Data. It aims to transform data into information, enabling improvements in making decisions. Within Big Data, optimization is mostly related to the prescriptive analysis, but in this paper, we present one of its applications to a predictive analysis based on regression in fuzzy environment. The tools offered by a regression analysis can be used either to identify the correlation of a dependency between the observed inputs and outputs; or to provide a convenient approximation to the output data set, thus enabling its simplified manipulation. In this paper we introduce a new approach to predict the outputs of a fuzzy in – fuzzy out system through a fuzzy regression analysis developed in full accordance to the extension principle. Within our approach, a couple of mathematical optimization problems are solve for each desired α−level. The optimization models derive the left and right endpoints of the α−cut of the predicted fuzzy output, as minimum and maximum of all crisp values that can be obtained as predicted outputs to at least one regression problem with observed crisp data in the α−cut ranges of the corresponding fuzzy observed data. Relevant examples from the literature are recalled and used to illustrate the theoretical findings.

The role of the regression analysis is crucial in many disciplines. Addressing the fuzzy quadratic least square regression for observed data modeled by fuzzy numbers, we aim to emphasize how a methodology that does not fully comply to the extension principle may fail to predict fuzzy valued numbers. We also propose a solution approach that functions in full accordance to the extension principle, thus overcoming the shortcomings arisen from the practice of splitting the optimization of a fuzzy number in independent optimizations of its components.

We address the full fuzzy linear fractional programming problem with LR fuzzy numbers. Our goal is to revitalize a strict use of extension principle by employing it in all stages of our solution approach, thus deriving results that fully comply to it. Using the $\alpha$-cuts of the coefficients we present the linear optimization models that empirically derive the $\alpha$-cuts of the optimal objective fuzzy value, and discuss the optimization models able to derive the exact endpoints of the optimal objective values intervals. For initial maximization (minimization) problems the main issue is related to how to solve two stage min-max (max-min) problems to obtain the left (right) most endpoints. Our goals are as it follows: to obtain exact solutions to small-size problems; to obtain relevant information about solutions to large-scale problems that are in accordance to the extension principle; and to provide a procedure able to measure to which extent the solutions obtained by an approach to full fuzzy linear fractional programming comply to the extension principle. We illustrate the theoretical findings reporting numerical results, and including a relevant comparison to the results from the literature.KeywordsLinear fractional programmingLR fuzzy numbersExtension principle

The regression method is widely used in predictive analysis. Its role is to derive an analytic estimation of the outputs expected for given inputs based on observed input-output data. The objective function that is optimized within the regression model is generally the representation of the approximation error comparing to the observed data. Nowadays, the uncertainty is commonly taken into consideration when modeling real systems, and vectorizing information is an important aspect of addressing big data in computer science. Consequently, finding pertinent fuzzy regression models is of great importance within mathematical modeling. In this paper we report our findings related to the full use of the extension principle in solving the optimization model comprised in a least square fuzzy linear regression methodology. We propose a solution approach based on mathematical programming to estimate the fuzzy outputs of the observed fuzzy data; and group our experiments in two categories with respect to the crispness of the observed input data. The first category uses crisp input data and is considered to better explain the advantage of using the extension principle within the solution approach; while the second category, having fuzzy both input and output observed data, is included to prove the relevance of the new approach compared to methodologies from the recent literature.KeywordsFuzzy linear regressionExtension principleLeast square