# Assignment problem

• Multi-dimensional assignment. More than 2 indexes, the variables are of the form, $LaTeX: x_{i,j,k, ...}$. The constraints sum all but one index -- for example, the 3-index assignment problem is
$LaTeX: \textstyle \begin{array}{rrcl} \min & \sum_{i,j,k} c_{i,j,k} x_{i,j,k} & & \\ \\ \mbox{s.t.} & \sum_{j,k} x_{i,j,k} & = & 1 \mbox{ for all } i, \\ & \sum_{i,k} x_{i,j,k} & = & 1 \mbox{ for all } j, \\ & \sum_{i,j} x_{i,j,k} & = & 1 \mbox{ for all } k. \end{array}$