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The fuzzy linear assignment problem (FLAP) is an extension of the classical linear assignment problem (LAP) to situations in which uncertainty in the cost coefficients is represented by fuzzy numbers. FLAP applications range from the assignment of workers to tasks to multiple-criteria decision analysis in fuzzy environments and many other engineering applications. Most FLAP formulations assume that all cost coefficients are fuzzy numbers of the same type (e.g. triangular, trapezoidal). The standard solution approach is the defuzzification of the cost coefficients, thus transforming the FLAP into a crisp LAP that can be solved by classical assignment algorithms such as the Hungarian method. It is known that defuzzification methods suffer from lack of discrimination when comparing fuzzy numbers which may lead to suboptimal assignments. The solution approach proposed in this paper is based on the theory of algebraic assignment problems and total orderings in the set of all fuzzy numbers, and it allows to solve FLAPs with different types of fuzzy numbers. More specifically, the FLAP is transformed into a lexicographic linear assignment problem (LLAP) which is solved in its place. We show, both theoretically and numerically, how this transformation overcomes the limitations present in existing approaches.

Singh and Yadav (2017) defined the product of unrestricted LR-type Intuitionistic Fuzzy Numbers (IFNs), and making use of the new product operation, the authors proposed a method to solve Fully Intuitionistic Fuzzy Linear Programming (FIFLP) problems. However, their method cannot be used to find the unique optimal value of FIFLP problems with inequality constraints. Recently, Perez-Canedo and Concepcion-Morales (2019) presented a method to find the unique optimal fuzzy value of Fully Fuzzy Linear Programming (FFLP) problems with equality and inequality constraints based on the optimisation of a lexicographic criterion for ranking LR fuzzy numbers. The authors suggested that their method could be extended to find the unique optimal intuitionistic fuzzy value of FIFLP problems with inequality constraints as well. In this paper, we analyse Singh and Yadav’s method and modify it to find the unique optimal intuitionistic fuzzy value of FIFLP problems with equality and inequality constraints. Thus, a new method is obtained and is demonstrated by means of a fully intuitionistic fuzzy production planning problem. Results are compared with those obtained by using Singh and Yadav’s method and show that the proposed method overcomes the shortcomings and limitations of their method.

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.

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., Spanish Ministry of Economy and Competitiveness, European Union (EU) TIN2017-86647-P

The Berth Allocation (BA) problem is an important problem in port logistics. It aims at finding optimal berthing times and positions of arriving vessels in a wharf subject to physical constraints. The optimisation criteria in the BA problem are diverse and respond to specific interests of vessels and wharves operators. Furthermore, although the BA problem has been dealt with mostly under complete certainty, it is a highly uncertain problem due to many factors that can affect vessels arrival and handling times. This paper takes fuzzy uncertainty into account and presents a fully fuzzy two-objective BA problem, by considering the minimisation of the total waiting time of vessels and the makespan of the wharf operation from the perspectives of vessels and wharves operators, respectively. A fuzzy epsilon-constraint method and a lexicographic method for fully fuzzy linear programming with inequality constraints are used jointly to solve the problem. A numerical example is given as illustration. Results demonstrate the usefulness of the proposed approach in handling fuzziness and conflicting objectives simultaneously in a BA problem.