The classical process capability indices are still the most prominently used by practitioners for asymmetrical tolerances even while not accurately reflecting on process capability. It appears that an adequate measure of capability for asymmetrical tolerances is yet to be discovered. This report formulates a list of five desirable PCI properties and explores some proposed indices developed for asymmetrical tolerances by comparing them to the properties. As none of the discussed indices satisfy all properties, four new proposals are made that improve upon existing indices. The new indices are related to process yield and centering, and compared to the existing indices. Further research is required to determine whether the new proposals are to be used in practice, but for now they serve as a source of inspiration in the development of PCIs for asymmetric tolerances.

In the divisible sandpile model, we consider a collection of i.i.d. Gaussian heights on a finite graph. It was shown by Levine et al. (2015) that the odometer function in this case equals a discrete, bi-Laplacian field. Subsequently, Cipriani et al. (2016) proved that the scaling limit of the odometer is a continuum bi-Laplacian field, this time on the unit torus. In this thesis, we will determine the scaling limit of a divisible sandpile with an initial configuration of correlated Gaussians, where the covariance is given by a stationary covariance function K(x-y). We show that after appropriate scaling, the odometer still converges to a bi-Laplacian field on the unit torus.