2) Compact Formulation
\[
\begin{align}
\min\;\;
& \sum_{f\in F}\sum_{k\in K} \mathrm{CAPEX}_{f,k}\, y_{f,k}
+ \sum_{t\in T}\Big(\mathrm{OPEX}^{\mathrm{trans}}_t + \mathrm{OPEX}^{\mathrm{ops}}_t\Big)
\tag{1}
\end{align}
\]
\[
\begin{align}
\mathrm{OPEX}^{\mathrm{trans}}_t
&= \frac{\mathrm{days}_t}{10^6}\Big[
\alpha \Big(\sum_{h,f} x^{HF}_{t,h,f} + \sum_{f,d} x^{FD}_{t,f,d}\Big)
+ r \Big(\sum_{h,f} d^{HF}_{h,f}\, x^{HF}_{t,h,f} + \sum_{f,d} d^{FD}_{f,d}\, x^{FD}_{t,f,d}\Big)
\Big] \\
\mathrm{OPEX}^{\mathrm{ops}}_t
&= \sum_{f\in F} \mathrm{OpsYear}_{k(f)} \cdot \frac{\mathrm{days}_t}{365}\, z_{f,t}
\tag{2}
\end{align}
\]
\[
\begin{align}
q_{t,d} &= \mathrm{Topline}_t \cdot s_d, \quad s_d=\sum_{h\in H} s_{h,d}, \quad \sum_{d\in D} s_d \approx 1 \\
500\sum_{f} x^{FD}_{t,f,d} &\ge q_{t,d} && \forall t,d \\
500\sum_{h} x^{HF}_{t,h,f} &\ge 500\sum_{d} x^{FD}_{t,f,d} && \forall t,f \\
500\sum_{d} x^{FD}_{t,f,d} &\le \mathrm{cap}_{k(f)}\, z_{f,t} && \forall t,f \\
z_{f,t} &= 1 \ \text{iff}\ t\ge \text{start}(f), \qquad y_{f,k}\in\{0,1\},\ \sum_{k} y_{f,k}\le 1
\tag{3}
\end{align}
\]
\(x^{HF}\) and \(x^{FD}\) are trucks (not orders, capacity = 500 orders/truck); \(\alpha=50\) €/truck and \(r=1.5\) €/km.
The allocator uses nearest-FC assignment under capacity; costs follow (1)–(3) across all periods.
