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Prioritization and throughput would be key challenges, so I'd recommend using the principles of ToC to identify, elevate and meter work based on the constraint in your system & process.
If you have 3 distinct types of projects, I would caution against too much standardization as it tends to force square pegs into round holes.
Personally, I would us a systems architecting approach. Note: This is not IT systems, but rather general systems theory. In short, I would examine your 3 pillars and identify what qualities are most important to a good solution. They may very well be different. The types of solution approaches would be selected based on how they lead to the qualities you desire. Develop a general solution architecture for each of the different problem types, and see whether they can use similar solutions or not.
If the outcomes are very predictable with low variance, you might create a tool kit for more classic "waterfall" PM. For high complexity/high change environments, you might look to more agile approaches. Product development processes and continuous improvement will use some of the same elements but applied different ways.
If all you have is a hammer, all problems look like a nail so equip your team with a tool kit broad enough to address the scope of your problems.
I ado agree with Kiron and Keith.
understand that you are trying to formalize project portfolio management (also PPM). The benefits are, as you say yourself, transparency (for management), consistency (or standardization) and centralization. There are 2 aspects to PPM, the work (projects , programs) and the resources (people, but also assets). I would start with the work and shy away from resources as this includes often disputes.
Organizations may create a PMO to deal with this task, and there is a vast amount of concepts and best practices about PPM and PMO and Portfolio Management.
Methodology is good to consider, but perhaps there are low hanging fruit worthwhile focusing on first? Have you been able to diagnose the issues and corresponding impacts to the organization? What are the "limitations"?
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