Abstract: An alternative method based on the mini-max polynomial approximation (MMPA) method was studied to compute the exponential of the burnup matrix in the paper. Against Chebyshev rational approximation method (CRAM) and Quadrature-based rational approximation method (QRAM), MMPA method can solve the burnup equation with only on inverse calculation, and all computations are real arithmetic, which has the advantages of good numerical stability and high solution efficiency. Furthermore, based on MMPA method, the burnup calculation code named AMAC was developed and coupled with the Monte Carlo transport code OpenMC, which was verified by decay and fixed flux irradiation problems,a series of OECD/NEA burnup credit criticality benchmarks, including pressurized water reactor (PWR) pin-cell benchmark and boiling water reactor (BWR) assembly benchmark. The calculation results are in good agreement with the experimental values and the reference values. The correctness and effectiveness of the MMPA method in theory and numerical value are preliminarily verified.