Santa Claus needs help. While the demand for presents is at an all-time high - 2 billion children - he is struggling with Fachkräftemangel (lack of skilled labor), especially a lack of drone engineers.
He has a fleet of 1 million delivery drones. To keep his operation running, he absolutely cannot afford a crash rate higher than 5% of his fleet this year, as he won't be able to have more drones repaired in time for Christmas 2026.
The drones are fully charged, but due to variable battery and weather conditions, their maximum flight time follows a triangular distribution between 15 and 45 minutes, with the most likely time (mode) being 40 minutes.
Each drone can deliver 100 presents per minute of flight time.
Assuming that drones cannot be recharged during the night of nights, how must Santa set the global timer for the drones to automatically return to the sleigh so that:
- At most 5% of his drones crash.
- He can still deliver all 2 billion presents.
Can you help?
Santa has two considerations: the number of drones that can be expected to crash and the number of presents delivered. Which one do you choose to constrain and which one to optimize? Or should you consider both objectives simultaneously?
If you constrain the presents and optimize for surviving drones, then you will find that Santa should set the timer to 21 minutes and 2 seconds. The triangular distribution parameters are: a=15, c=40, b=45. Since T = 21.0333 < 40, we use the lower segment of the cumulative distribution function (CDF):
This gives us a crash rate of
This means, we can expect 951,465 drones to deliver presents, which would give us
This is cutting it awfully close when it comes to the total number of presents delivered, though. Little is won when we have enough drones for next year, but already this year some children do not find a present under the tree.
Which is why Santa called InsideOpt and asked that we multi-objectively optimize crash rate and number of delivered presents with Seeker. He wants a good buffer for both concerns.
The result: Santa now sets the timer to 21 minutes and 3 seconds. We expect a crash rate of 4.88% and the ability to deliver 2.2 million more presents than needed.
Merry Christmas!