In this paper we apply the bi-decomposition on multi-valued functions and restrict the decomposition to MIN and MAX gates. It is known from [Bi-decomposition of multi-valued relations] that the MIN and MAX bi-decomposition leads in general to small multi-level circuits, well understandable for humans. Unfortunately, there does not exist a MIN or MAX bi-decomposition for each multi-valued function. In this paper we close this gap by the MAX-MIN multi-decomposition. Experimental results show that our complete decomposition of a set of benchmarks requires approximately the same sum of gates and literals as the known incomplete approach and the number of logic levels could even be reduced in average by 20 percent.