# Maximizing the time we reach to a threshold in a series of numbers

I have a problem and I really don't know what kind of mathematical method should I apply to solve or model my problem. I would be thankful If anyone can give me some answer or help.

Suppose we have more than one option to select a set of numbers. like A={a1,a2,a3,..an} and B={b1,b2,b3,..,bn} (number of members in each set is equal to n) each ai or bi is an integer value. We also have a threshold value lets say "TH". If we choose Option A, the new value of set A becomes A={a1+1,a2+1,....an+1} and also same case for B or any other options. Each round we have options, my question is that which option every time we must select to increase the number of rounds until we have a number (ai or bi ..) reaches to the Threshold? (Assume we have a global set which contains sets A,B,.. I mean the numbers are finite) Im trying to find a criteria by which every time I choose a set which help me to increase the number of rounds(because once in a set we have a number which is greater than or equal the threshold, we should stop). Also considering that we have a periodic time "t" by which we decrease values of each sets(the set that is not currently in use) by "1".

-Example : A={7,4,5,5}, B={5,6,2,4} and TH = 8 If we choose A then after choosing we have A={8,5,6,6} and if we choose B we get={6,7,3,5}. So is better to select B, because after selecting A we have a value equal to our threshold.

-Another Example: A={5,5,3,4}, B={5,4,3,3} In this case we get A={6,6,4,5} and B={6,5,4,4} so maybe is better choose B again, because we have two 6 in A but one in B and most probably A goes to threshold sooner, so we choose B ?

-Third Example A={6,4,3,3}, B={5,4,3,3} : considering that we have a periodic time "t" by which we decrease values of each sets by 1 .meaning that each t time, t time, we decrease the values by 1. So If in third example we choose B then we are giving a time to A to have that "6" value becomes to 5.

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