We propose a variant of the Mumford-Shah model for the segmentation of a pair of overlapping objects with additive intensity value. Unlike standard segmentation models, it does not only determine distinct objects in the image, but also recover the possibly multiple membership of the pixels. To accomplish this, some a priori knowledge about the smoothness of the object boundary is integrated into the model. Additivity is imposed through a soft constraint which allows the user to control the degree of additivity and is more robust than the hard constraint. We also show analytically that the additivity parameter can be chosen to achieve some stability conditions. To solve the optimization problem involving geometric quantities efficiently, we apply a multiphase level set method. Segmentation results on synthetic and real images validate the good performance of our model, and demonstrate the model's applicability to images with multiple channels and multiple objects.