A secret sharing scheme is a way of protecting a secret by distributing partial information to a set of participants P in such a way that only authorized subsets of P can recover the secret. The family of authorized subsets is called the access structure of the scheme. In 1979, threshold schemes were proposed to realize threshold access structures, and in 1987, multiple assignment schemes were proposed to realize monotone access structures. In this paper, we propose a new method for constructing multiple assignment schemes. Basically, our construction method is a combination of the threshold scheme and the cumulative scheme. We also show that the new method yields better results for some special access structures.