The solution of linear programming with LR-fuzzy numbers in objective function

This paper considers a class of fuzzy linear programming (FLP) problems where the coefficients of the objective function are characterized by LR-fuzzy intervals or LR-fuzzy numbers with the same shapes. The existing methods mainly focus on the problems whose fuzzy coefficients are linear shape. In this paper, we are interested in the solution of FLP problems whose fuzzy coefficients are nonlinear shapes. Specifically, we show that the FLP problem can be transformed into a multi-objective linear programming problem with four objectives if LR-fuzzy numbers have the same shape, and therefore the optimal solutions can be found by the relationships between FLP problems and the resulting parameter linear programming problems. An example is also presented to demonstrate that the method is invalid if fuzzy coefficients have different shapes. Then, we discuss the relationships among FLP problems, possibility and necessity maximization problems, and obtain a conclusion that all Pareto optimal solutions of the possibility-necessity approaches are the subset of the weak dominated solutions of the FLP problem. Finally, one numerical example is given to illustrate the solution procedure.