How do Lean Six Sigma principles and methodologies enhance operational efficiency in manufacturing and service industries? Discuss the core concepts of Lean Six Sigma, including waste reduction, process optimization, and continuous improvement. Explore how tools such as DMAIC $($Define$,$ Measure, Analyze, Improve, Control$)$ and value stream mapping are utilized to identify and eliminate inefficiencies in operations. Analyze case studies or examples of organizations that have successfully implemented Lean Six Sigma to streamline processes, reduce costs, and enhance quality. Furthermore, discuss the challenges and barriers to implementing Lean Six Sigma and propose strategies for overcoming them to achieve sustainable improvements in operations management.