[DirectorTsaarx]

Warning: you asked for a numerical analysis, so here it is...

First, the assumptions:
2000/2000/2000 Min/Org/Rad build rate
No population bonuses
No happiness bonuses
Upgrade costs 50% of new facility

Next, the simple part of the analysis: building the Mineral Miner Facility III's (MMF3's) directly, without mucking around in building lower-tech facilities first (and faster) and upgrading later.

We build N MMF3's, at two turns per facility (cost=2500 minerals, just a bit higher than the build rate), we get:

1000*[(2N-1)+(2N-3)+(2N-5)+...+5 + 3 + 1] minerals produced, from the time building starts until the turn the Last facility starts producing. This simplifies to:

1000*N*N

(standard mathematical formula: the sum of a series of numbers is equal to (n/2)(a+m), where n is the number of elements in the series, a is the value of the first element and m is the value of the Last element).

We then subtract the cost of the facilities (2500*N) to get the net gain. Pretty straightforward; net gain from beginning of build cycle through the end of the first turn all facilities can produce is:

N*N*1000 - N*2500.

Now, if we instead build N Mineral Miner Facility II's (MMF2's), at one turn per facility (cost=2000 minerals, exactly the build rate), we get (N-1)*900 + (N-2)*900 +... + 1*900 minerals produced, from the time building starts until the Last building turn; note that at this point, the Last facility has not started producing. I'll explain why in a moment. This simplifies to:

900*N*(N-1)/2

We then subtract the cost of the facilities (2000*N) to get the net gain so far. Now, if we upgrade those facilities (to MMF3's), it costs (1250*N) for the upgrade. Everyone still with me?

Obviously, we still produce minerals during the upgrade cycle; this amounts to:

1250*N/2000 [cost divided by build rate]

Technically, that number should be rounded up to the nearest integer to get actual number of turns, but we'll ignore that for a moment. In addition, this figure includes that first turn of production for the Last facility; that's why I didn't include it in the previous formula.

Now that the upgrade is finished, we can produce at MMF3 rates. In order to compare the "upgrade" strategy to the "build once" strategy, we include enough turns of production to equal the amount of time it takes to finish building the MMF3s from scratch. This amounts to:

[(2*N)+1] - N - (1250/2000)*N

Obviously, (2*N)+1 is the number of turns required to build the MMF3's; N is the number of turns required to build the MMF2's; and (1250/2000)*N is the number of turns required to upgrade MMF2's to MMF3's. Again, that Last number should be rounded up; however, in the interest of simplifying the algebra, I've avoiding the rounding. Which really gives a slight overestimate in the amount of minerals produced in the "upgrade" strategy, since we're now calculating mineral production as 900/turn for part of a turn, and 1000/turn for the rest of that turn. After combining the above formulas and doing some algebra, we come up with the following calculation for the upgrade strategy:

450*N*N - 2762.5*N + 1000

for net gain. In the same amount of time, the "build once" strategy gets:

N*N*1000 - N*2500

for net gain.

If we set the two formulas equal to each other, N comes out to slightly more than 1. Solving both equations for values of N ranging from 1 to 25 shows that the upgrade strategy is better if we build 1 facility and upgrade it. The "build once" strategy is better if we build more than 1 facility. The attached spreadsheet shows the calculations (if anyone's interested).

Comments are welcome - especially if someone finds an error in the math...