Given a set of departments, a number of rows and pairwise connectivities between these departments, the multi-row facility layout problem (MRFLP) looks for a non-overlapping arrangement of these departments in the rows such that the weighted sum of the center-to-center distances is minimized. As even small instances of the MRFLP are rather challenging, several special cases have been considered in the literature. In this paper we present new mixed-integer linear programming formulations for the (space-free) multi-row facility layout problem with given assignment of the departments to the rows that combine distance and betweenness variables. Using these formulations instances with up to 25 departments can be solved to optimality (within at most 6 h) for the first time. Furthermore, we are able to reduce the running times for instances with up to 23 departments significantly in comparison to the literature. Later on we use these formulations in an enumeration scheme for solving the (space-free) multi-row facility layout problem. In particular, we test all possible row assignments, where some assignments are excluded due to our new combinatorial investigations. For the first time this approach enables us to solve instances with two rows with up to 16 departments, with three rows with up to 15 departments and with four and five rows with up to 13 departments exactly in reasonable time.