New exact approaches to row layout problems

Abstract

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.

Appendix

In the following we repeat the DRFLP model of Secchin and Amaral presented in [68, 69] and show how these ideas can be used to improve the SF-DRFLP model of Amaral [9] as well. For an DRFLP instance given by n departments with lengths $\ell _i, i\in [n],$ pairwise transport weights $w_{ij},i,j\in [n], i<j,$ and two rows we use the position variables ${\tilde{x}}_i, i\in [n],$ distance variables ${\tilde{d}}_{ij}, i,j\in [n],i<j,$ and binary ordering variables ${\tilde{\alpha }}_{ij}, i,j\in [n], i\ne j,$ that are one if and only if department i lies left to department j in the same row, otherwise they are zero. In comparison to [10] we additionally use the binary betweenness-type variables ${\tilde{y}}_{ijk},i,j,k\in [n], |\{i,j,k\}|=3,i<k,$ with the interpretation

$$\begin{aligned} {\tilde{y}}_{ijk} = {\left\{ \begin{array}{ll} 1, &{} j\text { lies between departments }i,k\text { in the same row},\\ 0, &{} \text { otherwise}. \end{array}\right. } \end{aligned}$$

We set $L:=\sum _{i\in [n]} \ell _i$. Then the model reads:

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

Additionally, Secchin and Amaral introduced the following constraints:

$$\begin{aligned}&\left. \begin{array}{l}{\tilde{d}}_{ij}\le {\tilde{d}}_{ik}+{\tilde{d}}_{jk}, \\ {\tilde{d}}_{ik}\le {\tilde{d}}_{ij}+{\tilde{d}}_{jk}, \\ {\tilde{d}}_{jk}\le {\tilde{d}}_{ij}+{\tilde{d}}_{ik},\end{array}\right\}&i,j,k\in [n],i<j<k, \end{aligned}$$

(48)

$$\begin{aligned}&\left. \begin{array}{l}{\tilde{y}}_{ikj} \le {\tilde{\alpha }}_{ik}+{\tilde{\alpha }}_{ki},\\ {\tilde{y}}_{ikj} \le {\tilde{\alpha }}_{ij}+{\tilde{\alpha }}_{ji},\\ {\tilde{y}}_{ikj} \le {\tilde{\alpha }}_{jk}+{\tilde{\alpha }}_{kj},\end{array}\right\}&i,j,k\in [n], |\{i,j,k\}|=3, i<j. \end{aligned}$$

(49)

After an analysis of the running times of different separation variants, it was suggested in [69] to use the triangle constraints (48) from the beginning and to not use (49). In our tests in Sect. 6, where we compare our new models with approaches from the literature, we use the same variant. Additionally, for the DRFLP we strengthened the model above by reducing the value of L according to Lemma 10.

Finally, we show how the SF-DRFLP model in [9] can be strengthened using the ideas above. In this model position variables ${\tilde{x}}_i, i\in [n],$ are not needed because the left border is fixed and spaces between neighboring departments are not allowed. So one can get rid of them. Using the variables ${\tilde{\alpha }}_{ij}, {\tilde{d}}_{ij}, i,j\in [n], i\ne j,$ as well as the betweenness variables ${\tilde{y}}_{ijk}, i,j,k\in [n], |\{i,j,k\}|=3, i<k,$ from above the extended model reads:

$$\begin{aligned} \min \;&\sum _{\begin{array}{c} i,j\in [n]\\ i<j \end{array}} w_{ij} {\tilde{d}}_{ij} \\ \text {s.t.}\;&(40)-(47)\\&{\tilde{d}}_{ij} \ge \tfrac{\ell _i-\ell _j}{2} + \sum _{k\in [n]{{\setminus }} \{i\}} \ell _k {\tilde{\alpha }}_{ki} - \sum _{k\in [n]{{\setminus }} \{j\}} \ell _k {\tilde{\alpha }}_{kj},&i,j\in [n], i<j,\\&{\tilde{d}}_{ij} \ge \tfrac{\ell _j-\ell _i}{2} + \sum _{k\in [n]{{\setminus }} \{j\}} \ell _k {\tilde{\alpha }}_{kj} - \sum _{k\in [n]{{\setminus }} \{i\}} \ell _k {\tilde{\alpha }}_{ki},&i,j\in [n], i<j. \end{aligned}$$

This model can be improved by using (48) and (49). Additionally, one could fix ${\tilde{\alpha }}_{{\hat{\imath }} {\hat{\jmath }}}$ to zero for one pair ${\hat{\imath }}, {\hat{\jmath }} \in [n], {\hat{\imath }} < {\hat{\jmath }}$. In our computational tests we added (48) from the beginning and did not use constraints (49) as suggested for the DRFLP in [69].