Intro

So looking through the reddit pages during the downtime, I found this. While my ^brainless gut says that raids are efficient for specific TMR farming, it got me wondering. And with more raids in the future, this question will keep haunting me for a while til it is answered.

Is raid-refilling more efficient than Earth Shrine for TMR farming from a Lapis cost-analysis? And if so, by how much?

Drop Rates

Before we can start figuring out if one is better than the other, we must first figure out the raid drop rates. Luckily, there is this detailed google doc with just that information. I'll look at the last 4 non-reruns to get an general idea of our drop rates.

Special thanks to /u/Jack_Mikeson for the many Raid Data Analysis posts. I appreciate your hard work.

Essentially, everytime you do a single pull, you average out to a 0.0033% TMR increase.

As for the chasing the raid specific TMR (ignoring ALL moogles)...

5 Star Units

4 Star Units

3 Stau Units

1^{targetted at a specific 5☆ unit. If multiple on pull, average them}

Calculations

The makers of Abstraction are proud to introduce: Math. New and improved Math gives you exactly what you need, exactly when you need it. Math isn't right for everyone and may cause headaches, nausea, fits of rage, and spending sprees. Children under three should not be exposed to Math. If you experience trouble calculating, stop doing Math and consult your local Gumi medical support ^{He's dead Jim}.

^{MATH IS MATH, why would they change math?}

ALL Moogle

For Raid > ES (for any TMR)
  (Moogle Value/Pull) * ($R/Orb) * (pull/cost) * (Raid Orbs/Refill) * (Refill/Refill Cost) > (TMR gain/run) * (run/cost) * (Energy/Refill) * (Refill/Refill Cost)

  (0.0033625/1 Pull) * ([Raid Currency Gain/Orb] Raid-$/1 Orb) * (1 Pull/100 Raid-$) * (5 Orbs/Refill) * (1 Refill/100 Lapis) > (0.05/1 run) * (1 run / energy) * ([Max Energy] energy / 1 Refill) * (1 Refill/100 Lapis)
  (1.68125 * 10^-6) * [Raid Currency Gain/Orb] / Lapis > (5 * 10^-4) * [Max Energy] / Lapis

  [Raid Currency Gain/Orb] >~ 297.378 * [Max Energy]

 * This calculation ignores the TMR % gained for the units that are fighting the raid
 * Unit cancellation is fun for some.

Just use the last equation. If it is true, then raid is better. Otherwise, stick to ES-ing. I've also included the first formula just in case the moogle rate changes in the later future.

As for Raid-TMR...

5 Star Units

For Raid > ES (for Raid TMR's only)...
 (Moogle Value/Pull) * ($R/Orb) * (pull/cost) * (Raid Orbs/Refill) * (Refill/Refill Cost) > (TMR gain/run) * (run/cost) * (Energy/Refill) * (Refill/Refill Cost)
 (1.15625 * 10^-5) * [Raid Currency Gain/Orb] / Lapis > [Number of Raid Units] * (10^-4) * [Max Energy] / Lapis
 [Raid Currency Gain/Orb] >~ 8.6486 * [Max Energy] * [Number of Raid Units ES-ing]

4 Star Units

For Raid > ES...
 (Moogle Value/Pull) * ($R/Orb) * (pull/cost) * (Raid Orbs/Refill) * (Refill/Refill Cost) > (TMR gain/run) * (run/cost) * (Energy/Refill) * (Refill/Refill Cost)
 (1.6 * 10^-5) * [Raid Currency Gain/Orb] / Lapis > [Number of Raid Units] * (10^-4) * [Max Energy] / Lapis
 [Raid Currency Gain/Orb] >~ 6.25 * [Max Energy ] * [Number of Raid Units ES-ing]

3 Star Unit

For Raid > ES...
 (Moogle Value/Pull) * ($R/Orb) * (pull/cost) * (Raid Orbs/Refill) * (Refill/Refill Cost) > (TMR gain/run) * (run/cost) * (Energy/Refill) * (Refill/Refill Cost)
 (2.525 * 10^-5) * [Raid Currency Gain/Orb] / Lapis > [Number of Raid Units] * (10^-4) * [Max Energy] / Lapis
 [Raid Currency Gain/Orb] >~ 3.9604 * [Max Energy ] * [Number of Raid Units ES-ing]

Applied

ALL Moogle

[Raid Currency Gain/Orb] >~ 297.378 * [Max Energy] iff Raid-refreshing is better

Elt Rate: 700/Orb (Before bonus)
Raid Level: 30
   Elt Rate : 2324/Orb

120-Energy (bundle refill, daily)
    2324 !> 35,685.36
      6% effective
215-Energy (Rank 250)
    2324 !> 63,936.27
      3% efficient

5 Star Raid Unit

[Raid Currency Gain/Orb] >~ 8.6486 * [Max Energy] * [Number of Raid Units ES-ing] iff Raid-refreshing is better

120-Energy (bundle refill, daily)
    2324 ?> 1037.832 * [# of Raid Units]
      True only if 2 or less 5* raid units are being farmed
          1-Unit: 224% efficient
          2-Unit: 112% efficient
          3-Unit:  75% efficient
          4-Unit:  56% efficient
          5-Unit:  45% efficient
215-Energy (Rank 250)
    2324 ?> 1859.45 * [# of Raid Units]
      True only if one 5* raid unit is being farmed
          1-Unit: 124% efficient
          2-Unit:  62% efficient
          3-Unit:  42% efficient
          4-Unit:  31% efficient
          5-Unit:  25% efficient

4 Star Raid Unit

[Raid Currency Gain/Orb] >~ 6.25 * [Max Energy ] * [Number of Raid Units ES-ing] iff Raid-refreshing is better

120-Energy (bundle refill, daily)
    2324 ?> 750 * [Number of Raid Units ES-ing]
      True only if three or less 4* raid units are being farmed
          1-Unit: 310% efficient
          2-Unit: 155% efficient
          3-Unit: 103% efficient
          4-Unit:  77% efficient
          5-Unit:  62% efficient

215-Energy (Rank 250)
    2324 ?> 1343.75 * [Number of Raid Units ES-ing]
      True only if one 4* raid units are being farmed
          1-Unit: 173% efficient
          2-Unit:  86% efficient
          3-Unit:  58% efficient
          4-Unit:  43% efficient
          5-Unit:  35% efficient

3 Star Raid Unit

[Raid Currency Gain/Orb] >~ 3.9604 * [Max Energy ] * [Number of Raid Units ES-ing] iff Raid-refreshing is better

120-Energy (bundle refill, daily)
    2324 ?> 475.248 * [Number of Raid Units ES-ing]
      True only if four or less 4* raid units are being farmed
          1-Unit: 489% efficient
          2-Unit: 245% efficient
          3-Unit: 163% efficient
          4-Unit: 122% efficient
          5-Unit:  98% efficient

215-Energy (Rank 250)
    2324 ?> 851.486 * [Number of Raid Units ES-ing]
      True only if two or less 4* raid units are being farmed
          1-Unit: 273% efficient
          2-Unit: 136% efficient
          3-Unit:  91% efficient
          4-Unit:  68% efficient
          5-Unit:  55% efficient

Conclusion

Don't raid refresh if you only care about TMR progress for non-raid units.

If you are looking to farm for a raid 5* TMR, then raid is only better if you are farming 1 or 2 raid TMR only (exactly where depends on your rank. Use formula). That said, if you only have 1 raid unit, you might not need to refresh as you would probably have the 100% moogle from regular effort if you diligently run Elt raids.

As for 3* and 4* units, the numbers varies. But as a general rule of thumb, 2 or less raid units, then raid refreshing is fine. And once again, chance are, you already can get that many moogles from diligently not refreshing.

^{If you refill and can do Elt without issues, then Raid-refilling will be better for TMR farming than ES-refilling, even without the raid-level bonus. Depending on the raid level and the currency gain before bonus, Pro might also be better for refreshing at the peak raid level in case you have trouble with Elt difficulty.
Is it harder to macro? Probably unless your rocking a Olive 7 star or similar.
If you do plan on doing raid refreshing instead, hope you have your raid summon macro for this...}

^{Edit #1: Math fix. Percentile vs decimal math meant I was off by 2 orders of magnitude. Thank you /u/dangderr for math-checking me.
Edit #2: Added Raid TMR into math, as I owe it to the community after my initial screw up.
Edit #3: Abstracted the Raid TMR math more. It will now tell you if it is efficient depending on the number of 5 star raid units you are farming.
Edit #4: Added 3 and 4 star units}