The Mayans may have had the first timeboxed project. They had a strict 2012 timebox cutoff with little room for extension since the world would no longer exist. Although agile methods have been preaching the benefits of fixed timeboxed schedules since their creation, it still raises concerns with many stakeholders.
The triangles diagram from DSDM created in 1994 shows the shift from fixed functionality (vary resources and time) to fixed time and cost (vary functionality):
So instead of fixing functionality (scope) via the signoff of a specification document and completing all of this functionality (hopefully within the time and budget specified), the resources and time are fixed and as much of the functionality as can be completed is done before the time and money runs out.
This sounds a bit like time and materials, but there is an understanding that the core functionality--the Must Haves, the Priority 1s or whatever you want to call them--will be delivered. In fact, 80 percent of the outlined functionality should be delivered--and it is the last 20 percent that is up for replacement with late-breaking changes that could add even more value.
So is it the best of both worlds? It provides all the important features--and an opportunity to swap out low-priority elements with things that might crop up as we go. However, this is not how many stakeholders view
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