問題:頻度を最小限に抑えながら多様なチームをスケジュールする

私は解決するように求められている次の問題があります。私は数年前、私はそれを自分で行うことができるだろうと思っていますが、最近は助けがよかったと認めなければなりません:)

Given $n$ employees $E = {e_1,ldots,e_n}$, an organisation
wants to construct a plan for the next $w$ weeks: each week, $k

For each employee $e_i$, we would like to maximize the time
between any two consecutive weeks that $e_i$ is on duty. This goes
for all $i$, so it should be fair in the sense that no employee is
favored.

Further, we would like to have as diverse teams as possible, so
for any $1 leq i,j leq n$, we would like to minimize the number
of times $e_i$ and $e_j$ are on the same team.

As an additional hiccup, $m

Any ideas and tips on how to model this, e.g. as a MIP would be
great!

As a concrete example, $n=17$, $w=17$, $k=3$ and $m=4$ where
$q_1=q_2=q_3=5$ and $q_4=9$.

Edit: I want to add that this is not a homework
question, but a real life problem. There is no mathematically
stringent way to pose my constraints; this is up to interpretation,
as is how the constraints should combine and weigh against each
other in an objective function. In this lies the heart of my
question. As some approaches may be a lot simpler than others, I am
willing to sacrifice some stringency in return of a simpler
problem/model.

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