Fuzzy linear programming problems : models and solutions

We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, alpha-cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately

Duality in Fuzzy Linear Programming with Symmetric Trapezoidal Numbers

Linear programming problems with trapezoidal fuzzy numbers have recently attracted much interest. Various methods have been developed for solving these types of problems. Here, following the work of Ganesan and Veeramani and using the recent approach of Mahdavi-Amiri and Nasseri, we introduce the dual of the linear programming problem with symmetric trapezoidal fuzzy numbers and establish some duality results. The results will be useful for post optimality analysis

A Constructive Proof of Fundamental Theory for Fuzzy Variable Linear Programming Problems

Two existing methods for solving fuzzy variable linear programming problems based on ranking functions are the fuzzy primal simplex method proposed by Mahdavi-Amiri et al. (2009) and the fuzzy dual simplex method proposed by Mahdavi-Amiri and Nasseri (2007). In this paper, we prove that in the absence of degeneracy these fuzzy methods stop in a finite number of iterations. Moreover, we generalize the fundamental theorem of linear programming in a crisp environment to a fuzzy one. Finally, we illustrate our proof using a numerical example

A Novel Technique for Solving Multiobjective Fuzzy Linear Programming Problems

This study considers multiobjective fuzzy linear programming (MFLP) problems in which the coefficients in the objective functions are triangular fuzzy numbers. The study proposing a new technique to transform MFLP problems into the equivalent single fuzzy linear programming problem and then solving it via linear ranking function using the simplex method, supported by numerical example

A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach

The first author would like to appreciate from the research grant of University of Mazandaran. The research of Jose Luis Verdegay is supported in part by the project TIN2017-86647-P (Spanish Ministry of Economy and Competitiveness) which includes FEDER funds from the European Union.Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decisionmaking problems is generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper,weconcentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, numerical example is included to illustrate the mentioned solving process.University of MazandaranSpanish Ministry of Economy and Competitiveness TIN2017-86647-PEuropean Commissio

Robust optimization in data envelopment analysis: extended theory and applications.

Performance evaluation of decision-making units (DMUs) via the data envelopment analysis (DEA) is confronted with multi-conflicting objectives, complex alternatives and significant uncertainties. Visualizing the risk of uncertainties in the data used in the evaluation process is crucial to understanding the need for cutting edge solution techniques to organizational decisions. A greater management concern is to have techniques and practical models that can evaluate their operations and make decisions that are not only optimal but also consistent with the changing environment. Motivated by the myriad need to mitigate the risk of uncertainties in performance evaluations, this thesis focuses on finding robust and flexible evaluation strategies to the ranking and classification of DMUs. It studies performance measurement with the DEA tool and addresses the uncertainties in data via the robust optimization technique. The thesis develops new models in robust data envelopment analysis with applications to management science, which are pursued in four research thrust. In the first thrust, a robust counterpart optimization with nonnegative decision variables is proposed which is then used to formulate new budget of uncertainty-based robust DEA models. The proposed model is shown to save the computational cost for robust optimization solutions to operations research problems involving only positive decision variables. The second research thrust studies the duality relations of models within the worst-case and best-case approach in the input \u2013 output orientation framework. A key contribution is the design of a classification scheme that utilizes the conservativeness and the risk preference of the decision maker. In the third thrust, a new robust DEA model based on ellipsoidal uncertainty sets is proposed which is further extended to the additive model and compared with imprecise additive models. The final thrust study the modelling techniques including goal programming, robust optimization and data envelopment to a transportation problem where the concern is on the efficiency of the transport network, uncertainties in the demand and supply of goods and a compromising solution to multiple conflicting objectives of the decision maker. Several numerical examples and real-world applications are made to explore and demonstrate the applicability of the developed models and their essence to management decisions. Applications such as the robust evaluation of banking efficiency in Europe and in particular Germany and Italy are made. Considering the proposed models and their applications, efficiency analysis explored in this research will correspond to the practical framework of industrial and organizational decision making and will further advance the course of robust management decisions

Decision support model for the selection of asphalt wearing courses in highly trafficked roads

The suitable choice of the materials forming the wearing course of highly trafficked roads is a delicate task because of their direct interaction with vehicles. Furthermore, modern roads must be planned according to sustainable development goals, which is complex because some of these might be in conflict. Under this premise, this paper develops a multi-criteria decision support model based on the analytic hierarchy process and the technique for order of preference by similarity to ideal solution to facilitate the selection of wearing courses in European countries. Variables were modelled using either fuzzy logic or Monte Carlo methods, depending on their nature. The views of a panel of experts on the problem were collected and processed using the generalized reduced gradient algorithm and a distance-based aggregation approach. The results showed a clear preponderance by stone mastic asphalt over the remaining alternatives in different scenarios evaluated through sensitivity analysis. The research leading to these results was framed in the European FP7 Project DURABROADS (No. 605404).The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under Grant Agreement No. 605404

Robust optimization in data envelopment analysis: extended theory and applications.

Performance evaluation of decision-making units (DMUs) via the data envelopment analysis (DEA) is confronted with multi-conflicting objectives, complex alternatives and significant uncertainties. Visualizing the risk of uncertainties in the data used in the evaluation process is crucial to understanding the need for cutting edge solution techniques to organizational decisions. A greater management concern is to have techniques and practical models that can evaluate their operations and make decisions that are not only optimal but also consistent with the changing environment. Motivated by the myriad need to mitigate the risk of uncertainties in performance evaluations, this thesis focuses on finding robust and flexible evaluation strategies to the ranking and classification of DMUs. It studies performance measurement with the DEA tool and addresses the uncertainties in data via the robust optimization technique. The thesis develops new models in robust data envelopment analysis with applications to management science, which are pursued in four research thrust. In the first thrust, a robust counterpart optimization with nonnegative decision variables is proposed which is then used to formulate new budget of uncertainty-based robust DEA models. The proposed model is shown to save the computational cost for robust optimization solutions to operations research problems involving only positive decision variables. The second research thrust studies the duality relations of models within the worst-case and best-case approach in the input – output orientation framework. A key contribution is the design of a classification scheme that utilizes the conservativeness and the risk preference of the decision maker. In the third thrust, a new robust DEA model based on ellipsoidal uncertainty sets is proposed which is further extended to the additive model and compared with imprecise additive models. The final thrust study the modelling techniques including goal programming, robust optimization and data envelopment to a transportation problem where the concern is on the efficiency of the transport network, uncertainties in the demand and supply of goods and a compromising solution to multiple conflicting objectives of the decision maker. Several numerical examples and real-world applications are made to explore and demonstrate the applicability of the developed models and their essence to management decisions. Applications such as the robust evaluation of banking efficiency in Europe and in particular Germany and Italy are made. Considering the proposed models and their applications, efficiency analysis explored in this research will correspond to the practical framework of industrial and organizational decision making and will further advance the course of robust management decisions