DEMiCs is a software package written in C++ for computing the mixed volume of the Newton polytopes of a general semi-mixed polynomial system through dynamic enumeration of all mixed cells. The underlying mixed cells play an essential role for computing all isolated zeros of a polynomial system by polyhedral homotopy continuation method. A notable feature of DEMiCs is in the construction of a dynamic enumeration tree for finding all mixed cells. The dynamic enumeration method, proposed by Mizutani, Takeda and Kojima for fully mixed polynomial systems, is extended to semi-mixed systems and incorporated in the package. Numerical results show that DEMiCs is significantly faster than existing software packages for semi-mixed polynomial systems with many distinct supports. The software package DEMiCs is available at \url{ mizutan8/DEMiCs/}.