Fallout Shelter is a mobile game where you need to create and manage your dwellers and the vaults where they live. To expand your vault you need more dwellers and increase the population. Here is the Guide on how to increase the population in Fallout Shelter.
As we all know we can increase the population through sex(a man and a woman in a vault). But is this enough to grow the population fast? The dweller(Man) can have sex once with one woman. But he can spread his seeds to multiple women outside his vault. This means we need more female dwellers(This is not Game of thrones so don't try with your Family members :p). You should always have a couple in each quarter to accelerate the process.
At a time, 2 babies is easy than to manage 4 or more. So do not try to use Baby room when not required otherwise the resources will get affected indirectly. Unfortunately, Couple with higher stats will not produce a baby with high. But sometimes it might work as you get a baby with one stat as 3 from the start(randomly picks the average of both).
When you have a pregnant dweller it's always good to keep an extra space for the baby. There is no harm if you make her work when she is pregnant but make sure that when the vault is under radiation or the roach attack she runs to another room. If you are unsure about which room will be vacant then take time as it takes 3-4 hours(Real time) for a baby to born in the game. Till then press the gear button on the upper left of the screen to see the list of everyone and show you the space remaining of the rooms. If you don't have enough space left than she won't deliver be a baby.
Note: Space is needed to deliver a baby same as for a baby to grow adult.
You can also try the alternative method by placing dwellers with Higher C in Radio. Chances will increase that you will get another person from the wasteland at the door.
Hint: Try with high L dweller to increase the Luck chances.
If you have enough space but not many dwellers to produce and as mentioned above that you should not try with family members, but you can try with generation gaps.
For eg:
A + B = C
D + E = F
C + F = Z
So, A + Z = Eureka. It works.
Hope this helps you. Enjoy :)