You can read zillions of books, papers, and posts about the business case in project. In fact, there are four books illustrated below that have a chapter on the business case.
But, you won't find much on projects that don't have a viable economic business case, except for the universal advice: don't do the project!
Not so fast! That advice is economically sound but may not be operationally viable. Really, what do you do when there's no project outcome that's NPV positive, but you have to do something to improve or salvage operations (process, service, or product)?
"Do No Harm" becomes "Do the Least Harm". To wit, take the best of the bad as a reference case (or baseline), and then compare the other alternatives. You might wind up with a "Best (of the bad) Value" decision, to wit:
"Best (of the bad) Value" optimizes the solution operationally at the least worst cost"So, if doing nothing is not an option, or, more dramatically, failure is not an option, then go get as much funding as you can to deliver the best value you can. That'll be the best (of the bad) value for the funding.
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