So I have a problem with school and I am not sure if it is a employee schedueling problem, here is the situation:
We have to minimize the amount of employees at a non profit organization(Sanquin) and we have to optimize it for 3 locations. Here is the given info: $x_{ij}$ = 1 if employee has started shift i on day j; = 0 else.
And I have the amount of employees needed noted as a parameter p --> $p_i$ = amount of employees needed.
Monday: open from 12:30 - 20:00, amount of employees needed is here each half hour: 12:30 - 13:00 --> 2 13:00 - 13:30 --> 2 13:30 - 14:00 --> 2 14:00 - 14:30 --> 4 14:30 - 15:00 --> 2 15:00 - 15:30 --> 3 15:30 - 16:00 --> 5 16:00 - 16:30 --> 6 16:30 - 17:00 --> 5 17:00 - 17:30 --> 7 17:30 - 18:00 --> 5 18:00 - 18:30 --> 7 18:30 - 19:00 --> 8 19:00 - 19:30 --> 8 19:30 - 20:00 --> 7
Then I have the goal function: $$Z = \sum_{i=0}^8 x_{i1} \cdot p_i$$ $s.t x_{ij} \ge p_i $
With $x_{ij} = {0,1 } $ and $\mathbb{N}$
Now my question: how can I apply this to different locations and different days, because different days have different $p_i$. One of my thoughts were $\sum_{i=0}^8 \sum_{j=1}^5 x_{ij}$
Oh and the shifts can sometimes vary from day to day, so I have written that all out per day, but I want it all written out in 1 single formula in 1 location if possible, the main goal is to simplify it.
You need to assign employees to each of the shifts by mixed integer (binary) otherwise the solver will just assign the largest number of employees $x$ in your case to satisfy the constraints -shift requiring the most number of employees.
If you already have a list of employees $E$, hen your variable will be binary $ x_{s,d}^{e,l} = 1$ if employee $e$ is assigned o shift $s$ on day $d$ at location $l$ else $0$.
If you're not sure how many employees, take it as $N$ where $N$ is the maximum of employees needed by any shift at any location.
You may need pre-determined combination of sets like employee-location ($ E,L)$ and parameters like maximum number of shifts per day for an employee $S_e$
Obj $\min \sum_{e,l}\sum_{s,d}x_{s,d}^{e,l}$
s.t.
$ \sum_e x_{s,d}^{e,l} \ge p_{s,d}^l \quad \forall s,d,l$
$\sum_s x_{s,d}^{e,l} \le S_e \quad \forall e,l,d$