Conformance checking is a branch of process mining that aims to assess to what degree event data originating from the execution of a (business) process and a corresponding reference model conform to each other. Alignments have been recently introduced as a solution for conformance checking and have since rapidly developed into becoming the de facto standard. The state-of-the-art method to compute alignments is based on solving a shortest path problem derived from the reference model and the event data. Within such a shortest path problem, a cost function is used to guide the search to an optimal solution. The standard cost-function treats mismatches in the model and log as equal. In this paper, we consider a variant of this standard cost fu...