Lean, Six Sigma, and Theory of Constraints are independent applications and do not support each other.
Lean, Six sigma and Theory of constraints are dependent applications of making a process ore efficent and effective. All supports each other in some ways for a same objective to make process better.
For any non-efficient process, Lean says reduce the waste , Six sigma says redcue the variation and Theory of constraints says remove/improve bottleneck in the process. All three have diffenent way to identify the problem/issue in the process. It may be possible that due to high waste in one steps of process there is high variation in output and that step is acting as bottleneck. This can be in opposite way also, due to bottleneck step a lot of inventory/waste is getting collected and that is impacting the output and causing variation in output parameters.
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