Linear ranking functions are often used to transform fuzzy multiobjective linear programming (MOLP) problems into crisp ones. The crisp MOLP problems are then solved by using classical methods (eg, weighted sum, epsilon‐constraint, etc), or fuzzy ones based on Bellman and Zadeh's decision‐making model. In this paper, we show that this transformation does not guarantee Pareto optimal fuzzy solutions for the original fuzzy problems. By using lexicographic ranking (LR) criteria, we propose a fuzzy epsilon‐constraint method that yields Pareto optimal fuzzy solutions of fuzzy variable and fully fuzzy MOLP problems, in which all parameters and decision variables take on LR fuzzy numbers. The proposed method is illustrated by means of three numerical examples, including a fully fuzzy multiobjective project crashing problem.

The tourist trip design problem (TTDP) is a well-known extension of the orienteering problem, where the objective is to obtain an itinerary of points of interest for a tourist that maximizes his/her level of interest. In several situations, the interest of a point depends on when the point is visited, and the tourist may delay the arrival to a point in order to get a higher interest. In this paper, we present and discuss two variants of the TTDP with time-dependent recommendation factors (TTDP-TDRF), which may or may not take into account waiting times in order to have a better recommendation value. Using a mixed-integer linear programming solver, we provide solutions to 27 real-world instances. Although reasonable at first sight, we observed that including waiting times is not justified: in both cases (allowing or not waiting times) the quality of the solutions is almost the same, and the use of waiting times led to a model with higher solving times. This fact highlights the need to properly evaluate the benefits of making the problem model more complex than is actually needed.

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.