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\(\large
(\text{HLP1})~~\left\{~
\begin{align*}
& \begin{aligned}
& \rlap{\text{Min.}}\phantom{\text{u. d. N.}} && f(x, y, z)=\sum_{j=1}^n\sum_{i=1}^m\Big[\big(c_{ji}\cdot\sum_{j'=1}^n b_{jj'} + c_{ij}\cdot\sum_{j'=1}^n b_{j'j}\big)\cdot z_{ij} + \sum_{i'=1:i'\neq i}^m c_{ii'}\cdot x_{ii'}^j\Big]+\sum_{i=1}^m c_i^f\cdot y_i+\sum_{i=1}^m \sum_{i'=2:i'>i}^m c_{ii'}^f\cdot y_{ii'}
\end{aligned} \\
& \begin{aligned}
& \text{u. d. N.} && \sum_{i=1}^m z_{ij}=1 && (j=1, \ldots, n) \\
& && \big(\sum_{j'=1}^n b_{jj'}\big)\cdot z_{ij}+\sum_{i'=1:i'\neq i}^m x_{i'i}^j = \sum_{i'=1:i'\neq i}^m x_{ii'}^j + \sum_{j'=1}^n b_{jj'}\cdot z_{ij'} && (i=1, \ldots, m;~j=1, \ldots, n) \\
& && \sum_{j=1}^n \Big[\big(\sum_{j'=1}^n b_{jj'}\big)\cdot z_{ij} + \sum_{i'=1:i'\neq i}^m x_{i'i}^j\Big]\le a_i \cdot y_i && (i=1, \ldots, m) \\
& && \sum_{j=1}^n (x_{ii'}^j+x_{i'i}^j) \le \big(\sum_{j=1}^n\sum_{j'=1}^n b_{jj'}\big)\cdot y_{ii'} && (i, i'=1, \ldots, m: i < i')\\
& && x_{ii'}^j \ge 0 && (i, i'=1, \ldots, m:i\neq i';~j=1, \ldots, n) \\
& && y_i,~y_{ii'},~z_{ij}\in\{0, 1\} && (i, i'=1, \ldots, m: i < i';~j=1, \ldots, n)
\end{aligned}
\end{align*}\right.
\)
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