4. Model Formulation

4.3. Model Constraints

The department rules for scheduling have been modelled into constraints and presented in this section.

(1)  Each employee should work at most one shift per day:

\sum\limits_{k=1}^{4} x_{i j k l} \leq 1 \quad \begin{array}{r}\forall i=1 \cdots n_{j} \\ \forall j=1 \cdots 2 \\ \forall l=1 \cdots 28\end{array}              (3)

(2)  Set the staff requirement in each shift:

\sum\limits_{i=1}^{n_{j}} x_{i j k l}=N_{j k l} \quad \begin{aligned}&\forall j=1 \cdots 2 \\&\forall k=1 \cdots 4 \\&\forall l=1 \cdots 28\end{aligned}            (4)

(3)  Each employee should not work in annual leave days:

\sum\limits_{k=1}^{4} \sum\limits_{l \in \operatorname{AL}(i, j)} x_{i j k l}=0 \quad \forall i=1 \cdots n_{j}                  (5)

(4)  The employee should not work on day shift after night shift

\begin{array}{r} \forall i=1 \cdots n_{j} \\ x_{i j 3 l}+x_{i j 1(l+1)} \leq 1 \quad \forall j=1 \cdots 2 \\ \forall l=1 \cdots 28 \end{array}             (6)

(5)  Overtimes will be considered starting from the sixth shift during each week:

\sum\limits_{k=1}^{3} \sum\limits_{l \in s_{w}} x_{i j k l}-d_{i j w} \leq 5 \quad \begin{gathered} \forall i=1 \cdots 4  \\ \forall i=1 \cdots n_{j} \\ \forall j=1 \cdots 2 . \end{gathered}      (7)

(6)  Overtimes should not exceed two shifts per week:

\begin{array}{ll} & \forall w=1 \cdots 4 \\ d_{i j w} \leq 2 & \forall i=1 \cdots n_{j} \\ & \forall j=1 \cdots 2\end{array}      (8)

(7)  Counting the number of working shifts during the preferable off days:

\sum\limits_{k=1}^{4} \sum\limits_{l \in S_{\mathrm{off}_{(i, j)}}} x_{i j k l}=R_{(i, j)} \quad \begin{gathered} \forall i=1 \cdots n_{j}\\ \forall j=1 \cdots 2\end{gathered}     (9)

(8)  The operator who leaves to work at an external branch should take two days off per week (Thursday and Friday):

\sum\limits_{L \in s_{1 w}} x_{i 24 l}=5 y_{i w} \quad \begin{aligned}&\forall i=1 \cdots n_{2} \\&w=1 \cdots 4\end{aligned}

\sum\limits_{k=1}^{4} \sum\limits_{l \in s_{2 w}} x_{i 2 k l} \leq 2\left(1-y_{i w}\right) \quad \begin{aligned} \forall i &=1 \cdots n_{2} \\ w &=1 \cdots 4 \end{aligned}      (10)