The Warframe Wiki has a very nice explanation of what is needed to level mods.
First the economical answer, the cost of fusing mods in credits are directly tied to the amount of benefit it can potential supply. So using the most optimal fusion of mods from a mod perspective will automaticly also be the best solution on a financial view.
To level a mod you need to add a certain amount of energy to it, this energy comes from fusing other mods with the mod you wish to level.
There are 3 factors when fusioning mods:
- Level
- Rarity
- Polarity/Type
Level
Energy required to level a mod is doubled each level and they start at level 0, except for Fusion Cores, and goes up to 10.
Energy they can supply for fusion is base + ½ * level * base.
So if at level 0 it supplies 2 energy, then at level 1 it will supply 3 energy, at level 2 it will supply 4 energy etc.
The level is the number next to the rarity, no number is equal to level 0, or the number of white stripes on the left side.
Rarity
There are 3 types of rarity:
common has a multiplier of 1, uncommon 2 and rare 3.
Calculating Energy needed to level
With these 2 factors we can calculate the amount of energy needed to get a mod to the next level, to make the math easier later when we get to Polarity/Type we make sure it is a multiplier of 4.
E = 4 * R * 2^(L - 1)
- E is energy needed to level from precious level.
- R is the rarity (Common = 1, Uncommon = 2, Rare = 3).
- L is the level.
This gives us the following chart:
Level
1 2 3 4 5 6 7 8 9 10
Common 4 8 16 32 64 128 256 512 1024 2048
Uncommon 8 16 32 64 128 256 512 1024 2048 4096
Rare 12 24 48 96 192 384 768 1536 3072 6144
Calculating Energy supplied by mod
We can also calculate how much Energy a mod supply at different levels:
E = 2 * R(2 + L)
- E is energy needed to level from precious level.
- R is the rarity (Common = 1, Uncommon = 2, Rare = 3).
- L is the level.
This gives us the following chart:
Level
0 1 2 3 4 5 6 7 8 9 10
Common 4 6 8 10 12 14 16 18 20 22 24
Uncommon 8 12 16 20 24 28 32 36 40 44 48
Rare 12 18 24 30 36 42 48 54 60 66 72
With these 2 charts you should be able to calculate how many mods are needed to upgrade a mod, but there is the 3rd factor.
Polarity/Type
Polarity/Type Is divided into 3 groups when fusing:
- Same mod (duplicates)
- Mod with same polarity or Fusion Core
- Mod with different polarity
Two of the same mods give 100% of its energy.
So fusing 2 level 0 common mods of the same kind it will transfer 4 energy from one mod to the other, resulting in it gaining g enough energy to reach level 1.
If you then want to get it to level 2 you will require 8 more energy which is equal to 2 more of the same mod
Mods that have same polarity or Fusion cores give 50% of their energy.
Looking at the chart we can see that a level 5 uncommon Fusion Core supplies 28 energy.
It will get a 50% penalty to energy supplied to it will add 14 energy to any core.
So when fusing it with a level 0 common core it will gain 2 levels (4+8 energy needed) and have 2/16 of the next level done.
Mods with different polarity gives 25% of their energy.
Having a rare level 0 mod that will provide 12 energy and fusing it to a common level 0 with different polarity will give it 3 energy, so it will be 75% of the way to level 1.
So the most efficient way of leveling mods is to use duplicates, once you get spare mods (don't need any more/higher level of a mod) you can use same polarity and fusion cores to level other mods.
My personal recommendation is not to use use spare mods/fusion cores to level common mods. Most of them drop relative often and you will be better suited waiting for some duplicates to drop and using spare mods/fusion cores on rare mods where you won't be seeing as many duplicates.
Best Answer
This answer will vary greatly by your skill level. Currently, I find that Tier 1 sabotage are actually the most efficient for me with a small group (who are skilled at finding caches). Sabotage missions get you more than once chance for a drop. If one tenno does the sabotage while the others find all the caches, they can be done very quickly and provide multiple drops. Captures are also very fast, but only provide one drop and don't necessarily provide as high an average ducat price as Tier 1 and Tier 2 sabotage. Tier 3 Defense and Survival missions with large skilled groups are also great as they tend to provide a steady stream of items with a fairly good return.