TITLE: How to solve the Few Scientists Problem (big word problem) in its general form?
QUESTION [6 upvotes]: I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form.
Let me start with the concrete form, however:
Concrete Form:
You start with 5 scientists. A scientist can train 50 students for 5 years, after which each student becomes a scientist. (Assume a perfect graduation rate always, and assume you have an infinite population from which to draw students). Or a scientist can work on a project.
The problem is you have 30 Type-A projects, 50 Type-B projects, and 75 Type-C Projects, and they all need to be completed in minimal time. Each Type-A Project requires at least 10 scientists and takes 200/x years to complete, where x is the number of scientists assigned to them. Type-B's require at least 18 scientists and take 150/x years to complete. Type-C's require at least 25 scientist and take 120/x to complete.
What is the minimum time necessary to complete all projects, and what is the "event-order" of such an optimal solution?
I could solve this numerically by doing simulations in a computer program (although that will still be a pain in the neck), but what I really need is how to solve this in its general form.
General Form:
Just assign constants to everything. You start with s scientists, who can train t students for y years. There are A type-a projects, B type-b's, and C type-c's. Respectively, they require a minimum of d, e, and f scientists, and take g/x, h/x, and i/x years to complete.
How do you go about solving this? Is that even possible? Solving this requires finding an optimal solution (completing all projects in minimal time), and proving that no other solution exists that has a smaller finish time.
EDIT: Thanks to @Paul for this clarification. For projects, scientists can join or leave at any time. This is indiscrete time. For training, however, only 1 scientist can train a group of 50. (Two scientists training 50 does not make it go 2x faster.) The training has to be "atomic", which I think is the right word.

REPLY [2 votes]: There are some features to this problem that you might not think are present in the general form.  All the projects require a fixed number of scientist-years of research and all the scientists are interchangeable.  We just need to assign $22500$ scientist-years to research.  The minimum number per project does not matter because we will have lots of scientists and can assign them all to the same project, finishing it quickly, then have them move to another.  The only option is how many scientists to assign to training and how many to research.  Because the training ratio is so high, we make back the cost of training in $0.1$ years.  We should spend all our efforts training until we can complete the projects in less than $5$ years.  After that, training is in vain, because the trainees will not be ready until the research is done.  So we start with all five scientists training, then all $255$ scientists training, so at the end of $10$ years we have $13005$ scientists.  We will be done in $\frac {16500}{13005}\approx 1.730$ years for a total of $11.730$ years.