\(\large
(\text{ULSP}_{\mathit{id}})~~\left\{~~
\begin{align*}
& \begin{aligned}
& \rlap{\text{Min.}}\phantom{\text{u. d. N.}} && C(\boldsymbol{\Delta q},y)=\sum_{t=1}^T \sum_{i\in S}\bigg\{\sum_{\ell\in L_i}\Big[\pi_{i\ell t}\cdot \Delta q_{i\ell t} + I\cdot \pi_{i\ell t}\sum_{t'=t+1}^T (t'-t)\cdot \Delta q_{i\ell tt'}\Big]+k_i\cdot y_{i1 t}\bigg\}
\end{aligned} \\
& \begin{aligned}
& \text{u. d. N.} && \Delta q_{i\ell t}=\sum_{t'=t}^T \Delta q_{i\ell tt'} && (i\in S;~\ell\in L_i;~t=1, \ldots, T) \\
& && \sum_{i\in S}\sum_{\ell\in L_i}\sum_{t=1}^{t'} \Delta q_{i\ell tt'} \ge d_{t'} && (t'=1, \ldots, T) \\
& && \smash{\underline{q}}_{i1}{}\cdot y_{i1t} \le \Delta q_{i1t} \le{} \overline{q}_{i1}{}\cdot y_{i1t} && (i\in S;~t=1, \ldots, T) \\
& && \Delta q_{i1t}\ge{} \overline{q}_{i1}{}\cdot y_{i2t} && (i\in S;~t=1, \ldots, T) \\
& && (\overline{q}_{i\ell}{}-\smash{\underline{q}}_{i\ell}{})\cdot y_{i(\ell+1) t} \le \Delta q_{i\ell t} && (i\in S;~\ell\in L_i\setminus\{1,\,\overline{\ell}_i\};~t=1, \ldots, T) \\
& && \Delta q_{i\ell t} \le (\overline{q}_{i\ell}{}-\smash{\underline{q}}_{i\ell}){}\cdot y_{i\ell t} && (i\in S;~\ell\in L_i\setminus\{1\};~t=1, \ldots, T)\\
& && \sum_{\ell\in L_i}\Delta q_{i\ell t} \le R_{it} && (i\in S;~t=1, \ldots, T)\\
& && \Delta q_{i\ell tt'}\ge 0 && (i\in S;~\ell\in L_i;~t,t'=1, \ldots, T:t'\ge t)\\
& && y_{i\ell t}\in\{0,1\} && (i\in S;~\ell\in L_i;~t=1, \ldots, T)
\end{aligned}
\end{align*}\right.
\)
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