Well I have recently come back to this site and figured I'd share this tip that could help you advance in Fallout Shelter faster.

Every game that massively relies on time like this one and is offline will always succumb to the glitch where you can fast forward by resetting the game. Of course by doing this glitch the normal way you are potentially messing up many things on your saves like daily reports, rooms finishing, training, exploring the Wasteland, etc.

Works on Android. Not sure about iDevices since I don't own one but it should nevertheless.

What this glitch can be used for...
-CAPS
-Challenges
-Training
-Faster advancement
-More dwellers

-Steps-
1.) Send all of your dwellers into the Wasteland, once they're all out close the app.

2.) Go to settings and find your Time and Date settings, turn off the Automatic Date and Time option.

3.) Set the date to any day in the past. Don't go to far though, if you go to far back your phone will reset the date and time to be current. I usually go back to the first of the month.

4.) Open Fallout Shelter and call all your Dwellers back and place them where they are most suitable. Make sure you have everything set up that you want to advance. (Training/baby making/Radio Station/etc)

5.) Go back to your Time and Date settings and make it the next day. (Ex: You started on the 1st, you now change it to the 2nd)

6.) Open up Fallout Shelter again, you should have all your rooms ready to go, babies in stomachs, SPECIAL stats ready, etc.)

7.) Collect everything and set it to the next day again.

-Notes-
You don't have to set it back in time it just helps for when you set it back to the current date that everything isn't screwed.

If you want to exploit the Daily Reports you can set the date far back in the past and start a new Vault. Every day you will get a check up and get a nice amount of caps every day and every seven days you will receive a lunch box racking up more caps for you. Basically you would have a advanced start for a little bit of your spare time. You will wanna experiment with your phone to see how far it will let you go back before resetting.

Hope you guys enjoyed the tutorial.