Preinvexity of $n$-dimensional fuzzy number-valued functions: characterization, variational inequality and optimization problems

In this paper, the preinvexity of n-dimensional fuzzy number-valued functions are defined and discussed by means of the partial order relation in n-dimensional fuzzy number space which including preinvexity, weak preinvexity, strict preinvexity, weakly strict preinvexity, prequasiinvexity, weak prequasiinvexity, strict prequasiinvexity, weakly strict prequasiinvexity, and so on. In addition, their interrelations of the preinvexity of n-dimensional fuzzy number-valued functions are discussed, and some counterexamples are given. Furthermore, the two-parameter optimization problem, $n$-dimensional fuzzy variational-like inequality and optimality conditions related to $n$-dimensional preinvex fuzzy number-valued functions are discussed.