Network Function Virtualization (NFV) orchestration and management have attracted a lot of attention in recent years as it provides new opportunities regarding performance and deployment. In particular, several models have attempted to capture the behavior of such systems under various restricted assumptions. However, previously proposed mathematical models can only handle problems of fairly small size. This paper proposes a Mixed Integer Linear Programming (MILP) model for the resource utilization problem in an NFV dynamic context with several enhancements regarding the state of the art. We include the utilization of flow constraints to ensure the order of functions in a service chain. By systematic generation of Flow Cover inequalities, significant improvements in processing time are obtained with a standard MILP solver to compute exact optimal solutions. We also propose three efficient heuristics (two MILP-based heuristics) to find high-quality feasible solutions for large-scale systems within reduced execution time. We also carry out a set of experiments to evaluate the proposed algorithms and provide valuable guidelines for the efficient design of such systems. The results show that our approach is capable of handling large size instances of the NFV deployment problem involving up to 200 nodes and 100 demands.