egin{subequations}label{eq:ctr_shale}
egin{align}
min sum_{j, k} r_{k} n_{k} frac{V_{j}}{ heta_{j}}left(x_{k j} - heta_{j}
ight)^{2} +sum_{j} p_{j} u_{j} - lambda sum_{j, k} r_{k} n_{k} x_{k j} c_{k j} ag{
ef{eq:ctr_shale}}
end{align}
egin{alignat}{2}
ext{s.t.} & sum_{k in Gamma(j)} r_{k} n_{k} x_{k j}+u_{j} geq d_{j}, forall j & ext{ (demand constraints)} \
&sum_{j in Gamma(k)} x_{k j} leq 1, forall k & ext{(supply constraints)} \
&r_{k} x_{k j} leq f_{j}, forall k,j & ext{ (frequency constraints)} \
&n_{k} x_{k j} leq 1, forall k, j & ext{ (slot constraints)} \
& x_{k j}, u_jge 0, forall k, j & ext{ (non-negativity constraint)}
end{alignat}
end{subequations}