Summary: This article is intended to solve the systems of inverse fuzzy relational equations with max-min composition, which is beneficial for solving the well-known problems of fuzzy abductive/backward reasoning. Almost all the existing methods are based on numerical algorithms, and either cannot find the globally optimal solutions or consume significant computational cost to acquire the most accurate result. Due to these drawbacks, the existing methods seem not to render the retroduction applying inverse fuzzy relation popular in many real-time expert systems such as intelligent diagnosis. It is well known that the system of inverse fuzzy relational equations is not always consistent. In this case, according to the preselected weights, we employ weighted \(L_1\) norm distances to define a variety of best approximate solutions. Then we show that there exist very straightforward algebraic formulae for finding the best approximate solutions. The proposed approach not only finds the globally optimal solutions, but also has the advantage of being computationally very simple and efficient, and hence it has a lot of potentials to perform real-time abductive reasoning in many expert systems.