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