I have a MIP model that allocates 24 rides between 6 drivers (with many constraints irrelevant here), and an important part of my objective is splitting the rides fairly.
The easiest implementation I could think of is to punish for any driver that gets less (or more?) than the average 24/6=4 rides.
However, I saw that in many works and publications the concept of leximin fairness is used and considered intuitive and justified. Given an allocation y, we sort it and then compare it to other allocations lexicographically, and if it's larger it's considered fairer.
Question: why doesn't this criteria always agree with my easy implementation? e.g.
$y^1$=<2,3,3,3,3,10> is considered leximin fairer than $y^2$=<2,2,5,5,5,5>. However:
- If the punishment would be $\sum_i abs(avg-y_i)$, than $y^1$ would be punished by 12 while $y^2$ which would be punished only by 8
- If the punishment would be $\sum_i max(0, avg-y_i)$, than $y^1$ would be punished by 6 while $y^2$ which would be punished only by 4
is this just a matter of subjective preferences in the definition of fairness? or is there a bug in my implementation? Is there a known way of incorporating leximin fairness in a MIP model?