This paper deals with the switching law design problem for discrete-time switched affine systems. While the existing literature addresses the case of globally stabilizing switching laws, in discrete time, many switched affine systems can only be stabilized locally. In this work, methods are proposed for the design of switching control laws ensuring the local uniform ultimate boundedness of the equilibrium. The approach is based on the existence of a stabilizing continuous state feedback for a bilinear system. Estimates of the domain of attraction and of the invariant attractive neighborhood of the equilibrium are given. Numerical methods that allows a simple implementation of the proposed results are given.