Scheduling IT Staff at a Bank: A Mathematical Programming Approach

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)