Modernization

Ship modernization is used to increase Firepower, Torpedo, Anti-Air, and Armor stats. In addition, Maruyu can be used to increase Luck, while DE can increase HP, Anti-Submarine, and Luck stats.

To modernize a ship, first pick the ship you want to upgrade, then choose 1 to 5 ships to "merge". You will lose them in the process, make sure you didn't pick the ones you want to keep. A consumed ship's equipment is wasted as well, you will not retain any equipment still equipped on the consumed ship. Remember to remove any valuable equipment the consumable ship might possibly have. Also stock equipment can be broken down for small amounts of material though most likely not worth the effort.

General Modernization (Firepower, Torpedo, Anti-Air, Armor)

You will lose those modernization upgrades after remodel. Don't let it discourage you from modernizing your girls though, unless they are close to remodel level.
There is a maximum cap for those upgrades (refer to infoboxes on ship pages or to tables listed here).
Modernization can fail with +1 only, see the table below for details (0 will give you the penguin failure screen, all other results will display as the success message).
A formula to calculate the amount achieved in a successful modernization is (i is summary input):

i + \left \lfloor \frac{i + 1}{5} \right \rfloor

^[1]

A formula to calculate the amount achieved in a failed modernization is:

\left \lfloor \frac{1}{2} \left ( i + \frac{i + 2}{5} \right ) \right \rfloor

^[1]

Input	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Success	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20	21	23	24
Fail	0	1	2	2	3	3	4	5	5	6	6	7	8	8	9	9	10	11	11	12
Efficiency	50	75	83.3	87.5	90	83.3	85.7	87.5	88.9	90	86.4	87.5	88.5	89.3	90	87.5	88.2	88.9	89.5	90
Input	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
Success	25	26	27	29	30	31	32	33	35	36	37	38	39	41	42	43	44	45	47	48
Fail	12	13	14	14	15	15	16	17	17	18	18	19	20	20	21	21	22	23	23	24
Efficiency	88.1	88.6	89.1	89.6	90	88.5	88.9	89.3	89.7	90	88.7	89.1	89.4	89.7	90	88.9	89.2	89.5	89.7	90

The chance of success is 50% for each stat you're trying to boost with modernization, i.e. you can attempt to modernize for +15 firepower and +15 armor, but end up with +15 firepower and +8 armor as the result.
Values that are most efficient in the local cluster are bolded. Players should attempt to modernize at these values for maximum points per ship consumed. Efficiency here is the average number of points you get factoring the 50% chance for modernization success, relative to the number of points input.
For instance, if attempting to maximize Anti-Air efficiency, you can use 5 Isuzu Kai for +5, +5, +20 and +10. More commonly for low-level admirals, combining an Isuzu Kai from 1-5 with a Fusou from 2-2 leads to +5, +1, +5 and +5 which can be used as is, or supplemented with destroyers/submarines to add torpedo stat more efficiently, though torpedoes are rarely the limiting stat.
The calculations are modified as a ship reaches its cap. When the cap is approached, success is taken as the max addable points, while failure is still the failure points. For instance, if there are 5 points remaining, using a 5 point input yields 5-3 or an average efficiency of 80%, which is significantly inferior to either the 4 point input or 3 point input.

Luck Modernization (Maruyu)

Following the December 2013 update, Maruyu can be used as Modernization material for +1.2 luck (+1.6 luck if she is remodeled to Maruyu Kai), result will be rounded down after adding random number below 1. You can still increase a maxed-out ship's luck through modernization. Luck gained through modernization is retained upon remodeling.

Maruyu	1	2	3	4	5
Expected	1.2	2.4	3.6	4.8	6.0
Success	2	3	4	5	6
Fail	1	2	3	4	6
Chance	20%	40%	60%	80%	N/A

Maruyu Kai	1	2	3	4	5
Expected	1.6	3.2	4.8	6.4	8.0
Success	2	4	5	7	8
Fail	1	3	4	6	8
Chance	60%	20%	80%	40%	N/A

Note:

Although the indicator will display +6 luck for 5 Maruyu Kai before modernization, it is guaranteed to award +8 luck once carried out.
DE modernization can also be used to increase luck.

DE Modernization (HP, ASW, Luck)

After the 29/9/17 update it's possible to use DE to increase HP, ASW and Luck^[2]^[3]^[4].

Using at least one DE will have a chance to increase ASW and/or Luck.
Using two different DE of the same class (i.e., Shimushu + Kunashiri or Etorofu + Matsuwa, referred as pair below) will have a chance to increase HP.
For unmarried ships HP can always be increased by maximum 2 points overall. For married ships it can be increased by maximum 2 points with the following exceptions^[5]:
- Maximum 1 point: I-13, I-14, I-19, I-26, I-58, U-511 Kai, Hayasui.
- Can't be increased: I-168 Kai, I-401, I-8, Maruyu, Hayasui Kai, Bismarck (all forms), Italia/Littorio, Musashi, Mutsu Kai, Nagato Kai/Nagato Kai Ni, Richelieu, Roma/Roma Kai, Yamato.
ASW can be increased by maximum 9 points overall.
- Ship classes that have 0 base ASW stat will not receive the modernization.
Luck can not be increased past its maximal value.
Bonus values are preserved after remodel.
The following are some preliminary results/speculations about success rates (note that or assumes non-maxed stats, that is, for example, +1 HP or +1 Luck with maxed Luck can give 0 HP and Luck, ~50% for +1 HP and 0% for Luck since it's maxed).^[6]

Combination	Forms	Result	Samples
1 DE	Base and Kai	+1 ASW or +1 Luck, 60/40	~50
1 DE pair	Kai	+1 HP or +1 Luck, 50/50? Independent 20-30% for +1 ASW?	50-60+
1+ DE pair + 0+ DE	Base and Kai	Still +1 HP or +1 Luck, 50/50?	10+
2 different DE pairs + DE	Kai	Can give +2 ASW	1

Leveling (Evasion, LOS, ASW)

Generally, those stats are unaffected by modernization and follow the following formula instead (ASW can also be raised using DE modernization):

f(lv) = f(1) + \left \lfloor (f(99) - f(1)) \times \frac{lv}{99} \right \rfloor

The current value is determined only by ship's current level lv and remodel form (Base, Kai, Kai Ni, etc.) with its associated constants f(1) and f(99) (refer to infoboxes on ship pages or to tables listed here).
f(1) is the base or starting value for this remodel form: value at level 1 for ships in base form, arbitrary value otherwise (that is, not the value at remodel level).
f(99) is the value at level 99 (that is, not the maximal value).
For example, for Shoukaku Kai Ni A evasion(1) = 36 and evasion(99) = 81, so evasion at remodel level (88) is 36 + floor((81 - 36) * 88 / 99) = 76, while the maximal value at level 165 is 36 + floor((81 - 36) * 165 / 99) = 111.
After marriage the value continues to grow as lv > 99. For example, Shimakaze Kai has less evasion than Yukikaze Kai before marriage (67 + floor((99 - 67) * lv / 99) > 55 + floor((99 - 55) * lv / 99) when lv < 97) and more after (67 + floor((99 - 67) * lv / 99) < 55 + floor((99 - 55) * lv / 99) when lv > 101).
With the DE modernization bonus the ASW formula is

asw(lv) = asw(1) + \left \lfloor (asw(99) - asw(1)) \times \frac{lv}{99} \right \rfloor + bonus