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.

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.

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 ε‐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.