\begin{subequations}\label{eq:ctr_shale}
\begin{align}
\min \sum_{j, k} r_{k} n_{k} \frac{V_{j}}{\theta_{j}}\left(x_{k j} -\theta_{j}\right)^{2} +\sum_{j} p_{j} u_{j} - \lambda \sum_{j, k} r_{k} n_{k} x_{k j} c_{k j} \ \ \tag{\ref{eq:ctr_shale}}
\end{align}
\begin{alignat}{2}
\text{s.t.} & \sum_{k \in \Gamma(j)} r_{k} n_{k} x_{k j}+u_{j} \geq d_{j}, \forall j &\text{ (demand constraints)} \\
&\sum_{j \in \Gamma(k)} x_{k j} \leq 1, \ \forall k & \text{(supply constraints)} \\
&r_{k} x_{k j} \leq f_{j}, \ \forall k,j &\text{ (frequency constraints)} \\
&n_{k} x_{k j} \leq 1, \ \forall k, j &\text{ (slot constraints)} \\
& x_{k j}, u_j\ge 0, \ \forall k, j &\text{ (non-negativity constraint)}
\end{alignat}
\end{subequations}