What is the most effective way of leveling mods?

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
Uncommon
Rare

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.