This paper is concerned with shape and topology optimization of parts and their supports, taking into account constraints coming from the metal powder bed additive manufacturing process. Despite the high complexity of this process, it is represented by the simple inherent strain model, which has the advantage of being computationally cheap. Three optimization criteria, evaluated with this model, are proposed to minimize defects caused by additive manufacturing: vertical displacements, residual stresses and deflection of the part after baseplate separation. Combining these criteria with a constraint on the compliance for the final use of the part leads to optimization problems which deliver optimized manufacturable shapes with only a slight loss on the final use performance. The numerical results are assessed by manufacturing some optimized and reference geometries. These experimental results are also used to calibrate the inherent strain model by an inverse analysis. The same type of optimization is applied to supports in the case of a fixed non-optimizable part. For our 3-d numerical tests we rely on the level set method, the notion of shape derivatives and an augmented Lagrangian algorithm for optimization.