A method to find the unique optimal fuzzy value of fully fuzzy linear programming problems with inequality constraints having unrestricted L-R fuzzy parameters and decision variables

• Published in: Expert systems with applications 2019-06, Vol.123, p.256-269
• Main Authors: Pérez-Cañedo, Boris, Concepción-Morales, Eduardo René
• Format: Article
• Language: English
• Subjects: Fully fuzzy linear programming; Fuzzy sets; Inequality; L-R Fuzzy number; Lexicographic ranking criterion; Linear functions; Linear programming; Nonlinear programming; Parameters; Slack variables
• ISSN: 0957-4174; 1873-6793
• DOI: 10.1016/j.eswa.2019.01.041

•A new lexicographic method to find unique optimal values of FFLP problems is proposed.•Fuzzy inequality relations in the constraint set are defined lexicographically.•Existing lexicographic criteria for ranking L-R fuzzy numbers are easily incorporated.•An application example is given to illustrate the proposed method.•Obtained solutions have better objective values as compared to existing methods. In the literature, a wide variety of methods exist for solving fully fuzzy linear programming (FFLP) problems. However, there is still no method to find the unique optimal fuzzy value of FFLP problems with inequality constraints having unrestricted L-R fuzzy parameters and decision variables. Alternatively, some researchers introduce non-negative fuzzy slack and surplus variables to transform the FFLP problems with inequality constraints into FFLP problems having only equality constraints. Others, replace each fuzzy inequality constraint with a set of crisp linear inequalities. However, these two approaches cannot guarantee solutions with unique optimal fuzzy values and may lead to unfeasible problems. In this paper, based on the total order properties of a lexicographic criterion for ranking L-R fuzzy numbers, a method to find the unique optimal fuzzy value of FFLP problems with equality and inequality constraints having unrestricted L-R fuzzy parameters and decision variables is proposed. By following the steps of the proposed method, the FFLP problem is transformed into an equivalent mixed 0–1 lexicographic non-linear programming (MLNLP) problem. A numerical example is provided to illustrate the proposed method, and the results are compared with those obtained by other alternative methods, showing that the proposed method overcomes the shortcomings and limitations of the existing ones.