A Revised Version of a Lexicographical-based Method for Solving Fully Fuzzy Linear Programming Problems with Inequality Constraints

Ezzati et al. (A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl Math Model. 2015;39(12):3183-3193) introduced a lexicographic criterion for ranking triangular fuzzy numbers (TFNs), and proposed a method to solve fully fuzzy linear programming (FFLP) problems based on the lexicographic method of multi-objective optimisation; the authors assumed that fuzzy inequality constraints can be transformed into fuzzy equality constraints by introducing non-negative fuzzy slack and surplus variables. They illustrated the proposed method by means of a fully fuzzy investment problem. Bhardwaj and Kumar (A note on “A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem”. Appl Math Model. 2015;39(19):5982-5985) demonstrated that introducing fuzzy slack and surplus variables is mathematically incorrect, and showed that the solution of the fuzzy investment problem is unfeasible. Towards the end of their paper, they claimed that there is no feasible solution to the fuzzy investment problem when considering Ezzati et al.'s ranking criterion. In this paper, we propose a revised version of Ezzati et al.'s method whereby the optimal solution of FFLP problems with equality and inequality constraints can be obtained. Furthermore, by using the revised method, we show that feasible solutions of the fuzzy investment problem actually exist, and therefore Bhardwaj and Kumar's claim is false. To show the applicability of the revised method, we also consider a fully fuzzy project scheduling problem with budget constraint.