Lean and Kanban are geared to performance improvement, and their principles are applied regardless of the development process in-play, and in concert with the existing process. As opposed to something like Scrum, Lean doesn't require the wholesale, instantaneous replacement of existing processes with a prescriptive process template like Scrum (or others).
Using Lean principles and Kanban will show you the hotspots in your process where you might need to focus your improvement efforts. The software process mechanics that you end up with could very likely end up looking a whole lot like a prescriptive Agile development process, but the adoption and use of specific practices and mechanics are driven by tangible need, based on the repercussions of several instances of pulling harder and deploying and observing the countermeasures that clear out the chaotic oscillations that result immediately from pulling harder.
Pulling harder means increasing the expectations for the capacity of the system. Lean organizations aren't sitting on their laurels. Successions of pulling harder, solving the resulting organizational, process, and mechanical problems, and understanding and working with resulting performance improvements is Lean's continuous improvement. Continuous improvement means an inculcated value system that doesn't allow its past achievements to distract from the next level of goals, and the inevitable ardors of reaching them.
The organization is there to serve the goal. A Lean organization isn't a goal in and of itself. The organization bends to serve the nexus of value creation at the center of the vaue-add work. An obstruction to the ability to successfully pull harder might be found in a part of the organization outside of the immediate product development team. If that's the case, it's understood that the system as a whole must be optimized in order to continue to improve performance.
Once you've got your bicycle up to speed, pedal harder and see what might get in the way of being able to. And then devise and manage a well-considered plan to adapt the process and the organization to get those obstructions out of the way.
Pulling harder creates process improvement simply by setting expectations that are designed to clearly and brightly illuminate the ugly obstructions the lurk in organizations and processes and even in the tools being used. Simply: improve by improving. Set new expectations for the well-understood process and team and take a principled approach to understanding and working with the team's capacity.
Merely overlaying new process templates on a team isn't really the most reasonable and effective approach to improving performance. A change in process should be goal-driven and executed methodically. You might make some sweeping changes in the process, but the wholesale adoption of something like Scrum's agile project management prescriptions isn't likely necessary, and in some cases such a course might exasperate the effort for performance improvement.
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