I am studying for PMP exam.
Is there a process to decide on an approach to a project (i.e, traditional, hybrid or Agile)? It seems this decision is an output of a process. ITTOs? And where would it be documented; in the project Charter?
The selection of the project management methods and techniques would be part of what the Guide to the PMBOK 6th Edition refers to as "Develop Project Management Plan".
An input to this process is, as Bogdan mentioned, the organizational process assets. These include policies and procedures that may influence the decision on what methods and techniques may or should be used. If the organization doesn't have policies or guidance, then there may be judgments made by the project manager(s) or data gathering efforts to determine what options are best in the circumstance.
The output of the project management plan and various subsidiary plans would capture the approach. The project life cycle and development approach are components of the project management plan. Other processes, such as those related to quality management and monitoring and controlling project work may influence the project management plan throughout the project.
PMP is for waterfall management. Agile is another track for PMI; PMI-ACP
To answer the question out of the PMP scope, I can say that's a management decision for a department. Some projects need to be managed as waterfall, and some projects need to be agile.
I can't imagine applying agile to a construction project. 😁
This decision is usually made as part of the company's practices and policies, sort of like "meta data" on the approach to run projects. In the context of the PMP this is called the Organizational Process Assets.
When you are building the project charter, as per your example, the decision of what methodology or practice you will use to run the project was already taken.
I think that the essential idea here is that you need to plan ... at the earliest point in a project ... what the "ground rules" are going to be. "Traditional, Agile, and Hybrid" are merely terms – fairly meaningless by themselves, but loosely descriptive. The project charter should anticipate how the project is to proceed, how inevitable changes are going to be handled, and so on, and strive to settle these issues in some specific detail. So that, when the project is underway, everyone in the various management roles is, so to speak, "singing from the same songbook." All of this should be contained in the Charter, and revised when and if necessary. This becomes especially important in big, long-lasting, and complex projects – or when any sort of third-party contractor involvement is anticipated.