In this paper, a novel hexahedral refinement method is introduced which finds the optimal tradeoff between the number of inserted elements and inserted singularities according to user-prescribed weighting. The input of our algorithm is the hexahedral mesh with quads tagged for refinement. The quad sets to be inserted by sheet inflation are determined by solving a integer optimization problem to minimize the number of inserted elements and inserted singularities while maintaining mesh consistency. Finally, we design the optimized sheet structures by selecting the optimal local shrink set at the cross section of intersecting quad sets to ensure the quality of mesh refinement. The refinement scheme can be applied iteratively until the mesh density meets the requirements. Experimental results for some mechanical parts verified the effectiveness of the proposed method.