## 3 Custom Interior-Point Method for Solving FLPRC
### Starting-Point Using Flexible Bay Structure
Therefore, the main computational load is consistent with solving this dense matrix. Table 5 shows the calculation effort based on the calculation shown in Table 1. The required flops to compute the Newton steps are reduced a lot by the proposed block eliminations. This reduction leads to the faster CPU Time for computing Newton steps as \begin{table}
\begin{tabular}{l r r r} \hline \hline
**Problem** & **Proposal** & **CPLEX** & **NEOS** \\ \hline O7 & 14 & 14 & 15 \\ O8 & 14 & 13 & 22 \\ O9 & 14 & 22 & 22 \\ VC10 & 15 & 34 & 59 \\ Ba10 & 15 & 37 & 89 \\ M15a & 15 & 47 & 94 \\ M15s & 15 & 66 & 136 \\ AB20 & 15 & 77 & 162 \\ Tam30 & 16 & 87 & 207 \\ SC30 & 16 & 102 & 225 \\ SC35 & 16 & 117 & 320 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average CPU TIME for computing an optimal arrangement under relative-constraints (\(\upmu\) sec). \begin{table}
\begin{tabular}{l r r} \hline \hline
**Problem** & _with_ **Eliminations** & _without_ **Eliminations** \\ \hline O7 & 0.1 & 4.2 \\ O8 & 0.1 & 4.5 \\ O9 & 0.1 & 4.7 \\ VC10 & 0.2 & 4.8 \\ Ba10 & 0.2 & 5.3 \\ M15a & 0.2 & 7.8 \\ M15s & 0.2 & 8.7 \\ AB20 & 0.2 & 8.9 \\ Tam30 & 0.2 & 10.1 \\ SC30 & 0.2 & 12.6 \\ SC35 & 0.2 & 13.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average CPU TIME for computing Newton Steps (\(\upmu\) sec). ### Discussion

The method proposed in this study is the optimization of department position and shape given a relative position \(\mathcal{H}_{ij},\mathcal{V}_{ij}\). This technique can be used as a subroutine for a solution method using MIP. This approach also includes [PERSON] (2012) and [PERSON] and

Figure 4: Sparsity pattern of the coefficient matrix after the 2 nd elimination for SC35. Figure 3: Sparsity pattern of the coefficient matrix after the 1 st elimination for SC35. [PERSON] [42], which combines MIP and MH. Among these algorithms, FLPRC is solved many times. Therefore, by using the proposed method, the calculation time can be shortened and more time can be devoted to the search of the relative positions. The proposed method can also be used as a subroutine of the layout technique using FBS, such as [30] because FBS is used to derive the starting point in the proposed method. FBS has an encoding that divides the building into bays. Although this approach is simple, it does not necessarily guarantee optimality due to the layout that cannot be expressed by encoding. On the other hand, in the process of searching, if the proposed method is applied based on the relative position obtained by FBS, the layout of FBS can be _improved_, and there is a possibility that a better layout can be searched. In recent years, many MH approaches have used FBS and have achieved very good results. Further improvement can be considered by applying this technique. ## 5 Conclusions

We have considered the problem of computing an optimal arrangement of department within a facility, as measured by the minimum material handling cost, subject to the constraints for departments with respect to within-boundary, aspect-ratio, non-overlapping, and relative positioning. We have developed a primal-dual interior-point method that exploits the special structure of a problem, and is much faster than a standard method. We have proposed an efficient computation of the Newton step, using three types of block eliminations. To illustrate the effectiveness of our algorithm, we have given numerical results, using problem instances from several benchmark problems for FLP with specified relative-positioning given from randomized based rule. For a typical problem with 35 departments, the proposed algorithm is roughly 10 times faster. Using the proposed algorithm, we can accelerate a MIP (Mixed-Integer Programming) based algorithm for solving FLP, which is one of the most efficient methods for a continuous representation based problem. Future research includes applying the proposed method to the MIP-based approach and MH-based approach.