The optimization of sawing processes in the wood industry is critical for maximizing efficiency and profitability. The introduction of computerized tomography scanners provides sawmill operators with three-dimensional internal models of logs, which can be used to assess value and yield more accurately. We present a methodology for solving the sawing optimization problem employing a flexible sawing scheme that allows greater flexibility in cutting logs into products while considering product quality classes influenced by wane defects. The methodology has two phases: preprocessing and optimization. In the preprocessing phase, two alternative algorithms are given that generate and evaluate the potential sawing positions of products by considering the 3D surface of the log, product size requirements, and product quality classes. In the optimization phase, a maximum set-packing problem is solved for the preprocessed data using mixed-integer programming (MIP), aiming to obtain a feasible cut pattern that maximizes value yield. This is implemented in a system named FlexSaw, which takes advantage of parallel computation during the preprocessing phase and utilizes a MIP solver during the optimization phase. The proposed sawing methods are evaluated on the Swedish Pine Stem Bank. Additionally, FlexSaw is compared with an existing tool that utilizes cant sawing. Results demonstrate the superiority of flexible sawing. While the practical feasibility of implementing a flexible way of sawing logs is constrained by the limitations of current sawmill machinery, the potential increase in yield promotes the exploration of alternative machinery in the wood industry.