To begin this solution, sorry if it is a little long winded. I have quite a few hours invested into Fallout Shelter and not only am I trying to help anyone get the achievement for leveling a dweller to 50, but I am also providing tips that will ultimately allow you to get other achievements more easily.
Undoubtedly the fastest way reach level 50 is to send a dweller into the Wasteland. Dwellers continue to level up in the Wasteland when you are not playing where they only level up while you are playing in the Vault. The down side is the Wasteland is not kind to your dweller.
To prepare your dweller for the leveling grind, and to just stay alive in the Wasteland period, spend some time, caps, and praying to the RNG gods with the following.
S.P.E.C.I.A.L StatsTo best maximize your dwellers survivability in the wastes, max out all of their S.P.E.C.I.A.L stats. The dweller becomes almost completely self-sufficient with maxed out S.P.E.C.I.A.L stats allowing you to leave them out for days with almost no supervision. I have done my own personal testing; with max stats a dweller at level 1 should reach level 50 between 3 - 5 days in the Wasteland depending on RNG events. They also earn between 1000 and 5000 caps a day.
All stats max out at 10. This is important to remember for Endurance.
If you are trying to get the dweller out the door fast, max out at least Strength, Endurance, and Agility.
Strength is upgraded via the Weight room. This determines how much damage the dweller takes. The higher the strength the less damage they take.
Endurance (END so I don’t have to type Endurance every time) is upgraded via the Fitness room. This determines how many hit points (health) the dweller gains per level. A dweller with END 1 will gain 3 hit points per level where a dweller with END 10 will gain 7.5 hit points per level. Endurance can be increased beyond 10 with outfits that have an END number resulting in more hit points per level.
Here is the equation to calculate your dwellers hit points.
2.5 + (END Number * 0.5)
Hit points are not retroactive.
Agility is unlocked via the Athletics room. This determines how many shots your dweller can fire before the enemy attacks, or how much damage they can avoid if fleeing. Obviously, the more damage they can inflict before the enemy attacks the faster the battle is completed.
Stimpacks and RadawayYou will need a Medbay and Science Lab to produce Stimpacks and Radaway. I recommend building 3 double room Medbays and 1 double room Science Lab. This way you can load your dweller up on 25 Stimpacks and 5 Radaway when sending them into the wastes, but still have enough to deal with any Vault emergencies that may happen immediately after your dweller leaves. Too many times I have had a pack of 3 Deathclaws attack within 30 seconds of a dweller leaving. Clever girl.
Just in case you are wondering why I recommend double instead of triple room sizes for the Medbay and Science Lab, it is more efficient and has a higher yield. Three double Medbays has the same storage of 65 as two triples, but the three double Medbays produce 9 stimpacks (3 each) where the two triples produce 8 (4 each). Production exponentially increases when fully upgrading the double Medbays. Same exact thing applied to Science Labs. See, I saved you some time and made your Vault more efficient.
Weapons, Armor, and PetsAnd this is where RNG comes into play. Basically, give your dweller the best gun, best armor, (and if you have some animals with damage resistance, extra XP gain, or increased damage) the best animal that will help them stay alive or level up faster in the Wasteland.
I recommend any gun of 6 attack or higher for early treks into the wastes and weapons with only double-digit attack values at level 15 or higher.
Obviously whatever outfit you give them should have the highest Endurance you can provide them so they can earn those hit points.
OverseeingOverseeing the dweller all depends on what S.P.E.C.I.A.L. stats you maxed out.
If you maxed out all S.P.E.C.I.A.L. stats on a dweller then you literally can go for days without playing if you want. I have several Dwellers I have power leveled to 50 where I send them out and 4 days later I return to the game with a batch of new level 50 dwellers.
If you maxed out only Strength, Endurance, and Agility and you know you are going to be away for 10 or more hours before you can play Fallout Shelter again make sure they have 25 Stimpacks and 10 Radaway in the dwellers inventory. If the Wasteland is kind, in some cases you can go 24 hours with little or no supervision and find your dweller with half his supply of Stimpacks and Radaway remaining.
Hitting Level 50Once your dweller reaches level 50 call them back to the Vault.
Alternatively, the dweller will auto return to the vault after their inventory is full. So, you can send out a dweller at level 1, have them level to 50, and have them return without you having to give the command if you so desire. This, from my experience, takes about 5 to 7 days. It all depends on how quickly they find stuff to fill their inventory. Don’t worry, the dweller always hits level 50 before finding enough stuff to auto-return.
Either way, once they return the achievement should pop.
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References used: Fallout Shelter portion of the Fallout Wiki.