In this example, we have a group of students that each needs to be paired with a teacher. Each student has a different number of questions to ask the teacher, and each student-teacher pairing has a different difficulty for their interactions.
The workload for a teacher for a student-teacher pairing is given by the number of questions that the student has $Q_s$ multiplied by the interaction difficulty $\beta_{s,t}$
$$ Q_s(\beta_{s,t}) $$
How can we define the LP model for evenly distributing the workload across the teachers?
Tried an objective function that minimizes
$$ \sum_{s=1}^{S} \sum_{t=1}^{T} Q_s(\beta_{s,t}) X_{s,t} \\ $$
where $Q_s$ is the number of questions a student $s$ has,
$\beta_{s,t}$ is the difficulty of interaction between student $s$ and teacher $t$,
$X_{s,t}$ is a binary variable that shows whether student $s$ is assigned to teacher $t$.
However, the results does not suggest that the workload for each teacher $\sum_{s=1}^{S} Q_s(\beta_{s,t}) X_{s,t}$ is evenly distributed.
Is there a better objective function to achieve this?
Note: I'm doing this in Python with PuLP and Scipy