In automatic decimation of polygonal models, the measure of geometric fidelity plays the key role. Among the existing measures, volume based error measure results in better quality approximations whereas the one based on normal field variation better preserves salient features. We exploit both volume and normal field to develop a more reliable and efficient two phase greedy algorithm. In the first phase, the priority of each vertex is defined using the normal field variation across its one-ring neighborhood. In the second phase, the desired number of vertices is removed according to their priorities. Once a vertex is a candidate for removal, it is eliminated by collapsing an outgoing halfedge that is selected by using the measure of geometric fidelity based on volume loss. Subjective and objective comparisons validate that the proposed algorithm not only has better speed-quality trade-off but also consumes less memory space and keeps visually important features even after drastic simplification in a better way than the similar state-of-the-art best algorithms. This method is useful for applications where computing coordinates and/or attributes other than those attached to the original vertices is not allowed by the application and the focus is on both speed and quality of LODs.