We study the scheduling situation in which a set of jobs subjected to release dates and deadlines are to be performed on a single machine. The objective is to minimize a piecewise linear objective function $\sum_j F_j$ where $F_j(C_j)$ corresponds to the cost of the completion of job $j$ at time $C_j$. This class of function is very large and thus interesting both from a theoretical and practical point of view: It can be used to model total (weighted) completion time, total (weighted) tardiness, earliness and tardiness, etc. We introduce a new Mixed Integer Program (MIP) based on time interval decomposition. Our MIP is closely related to the well-known time-indexed MIP formulation but uses much less variables and constraints. Experiments on academic benchmarks as well as on real-life industrial problems show that our generic MIP formulation is efficient.