\(\large
(\text{ULSP}_{\mathit{aud}})~~\left\{~~
\begin{align*}
& \begin{aligned}
& \rlap{\text{Min.}}\phantom{\text{u. d. N.}} && C(\boldsymbol{q}, y)=\sum_{t=1}^T \sum_{i\in S}\sum_{\ell\in L_i}\Big[\pi_{i\ell t}\cdot q_{i\ell t} + I\cdot \pi_{i\ell t}\sum_{t'=t+1}^T (t'-t)\cdot q_{i\ell tt'}+k_i\cdot y_{i\ell t}\Big]
\end{aligned} \\
& \begin{aligned}
& \text{u. d. N.} && \sum_{\ell\in L_i} y_{i\ell t} \le 1 && (i\in S;~t=1, \ldots, T) \\
& && q_{i\ell t}=\sum_{t'=t}^T q_{i\ell tt'} && (i\in S;~\ell\in L_i;~t=1, \ldots, T) \\
& && \sum_{i\in S}\sum_{\ell\in L_i}\sum_{t=1}^{t'} q_{i\ell tt'} \ge d_{t'} && (t'=1, \ldots, T) \\
& && \smash{\underline{q}}_{i\ell}{}\cdot y_{i\ell t}\le q_{i\ell t} \le \overline{q}_{i\ell}{}\cdot y_{i\ell t} && (i\in S;~\ell\in L_i;~t=1, \ldots, T) \\
& && \sum_{\ell\in L_i}q_{i\ell t} \le R_{it} && (i\in S;~t=1, \ldots, T) \\
& && q_{i\ell tt'}\ge 0 && (i\in S;~\ell\in L_i;~t,t'=1, \ldots, T:t'\ge t)\\
& && y_{i\ell t}\in\{0,1\} && (i\in S;~\ell\in L_i;~t=1, \ldots, T)
\end{aligned}
\end{align*}\right.
\)
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