# Facility Layout Problem with Alternative Facility Variants
## 6 Results
### Experiment 2  
In Table 1, a clear trend that the cooling schedule helps to converge to better solutions can be observed. EA_T_T significantly outperformed EA_T_F, and the same held for EA_F_T and EA_F_F.  
Using the cooling schedule led to better solutions in the multi-objective optimization case as well, but the differences were not as profound as in the single-objective case. While MOEA_T_T is not significantly better than MOEA_T_F in terms of the median hypervolume, as indicated by the \(p\)-value of 0.16, the run using the cooling schedule resulted in a better  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Median**} & \multicolumn{4}{c}{**\(p\)-Value**} \\ \cline{3-6}  & **maxFreeSpace [m\({}^{2}\)]** & **EA\_T\_T** & **EA\_T\_F** & **EA\_F\_T** & **EA\_F\_F** \\ \hline EA\_T\_T & 10,176 & - & **2.2 \(\times\) 10\({}^{-8}\)** & **3.1 \(\times\) 10\({}^{-2}\)** & **1.6 \(\times\) 10\({}^{-6}\)** \\ \hline EA\_T\_F & 9879 & - & - & **2.9 \(\times\) 10\({}^{-13}\)** & 0.37 \\ \hline EA\_F\_T & 10,501 & - & - & - & **1.6 \(\times\) 10\({}^{-11}\)** \\ \hline EA\_F\_F & 9940 & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results for the single-objective case of Experiment 1 (variants = false vs. variants = true) and Experiment 2 (cooling = false vs. cooling = true). The median of the set of 100 best-of-run values is presented for each method plus the \(p\)-values calculated pairwise with the Wilcoxon rank-sum test. The statistically significant differences are highlighted in bold.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Median**} & \multicolumn{4}{c}{**\(p\)-Value**} \\ \cline{3-6}  & **Hypervolume** & **MOEA\_T\_T** & **MOEA\_T\_F** & **MOEA\_F\_T** & **MOEA\_F\_F** \\ \hline MOEA\_T\_T & \(1.61\times 10^{7}\) & - & 0.16 & **9.76 \(\times\) 10\({}^{-4}\)** & 7.97 \(\times\) 10\({}^{-2}\)** \\ \hline MOEA\_T\_F & \(1.55\times 10^{7}\) & - & - & **1.2 \(\times\) 10\({}^{-5}\)** & **2.13 \(\times\) 10\({}^{-3}\)** \\ \hline MOEA\_F\_T & \(1.71\times 10^{7}\) & - & - & - & 5.96 \(\times\) 10\({}^{-2}\) \\ \hline MOEA\_F\_F & \(1.65\times 10^{7}\) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results for the multi-objective case of Experiment 1 (variants = false vs. variants = true) and Experiment 2 (cooling = false vs. cooling = true). The median of the set of 100 hypervolume values is presented for each method plus the \(p\)-values calculated pairwise with the Wilcoxon rank-sum test. The best values are highlighted in bold.  
set of non-dominated solutions compared to the corresponding run not using the cooling schedule, in 70 out of 100 runs. A similar ratio, 73:27, was observed for MOEA_F_T and MOEA_F_F. This indicates that the cooling schedule has a positive effect on performance, even in the multi-objective optimization scenario.  
The observed difference in the effect of the cooling schedule in the single- and multi-objective optimization may also point to the fact that in the multi-objective case, it is even more important to keep the population maximally diverse throughout the whole run. This applies particularly to the head parts of individuals' priority lists. As soon as the cooling schedule becomes effective, the head parts get 'frozen' and cannot be modified any more. Suppose a majority of head parts of the priority lists in the population determine floor plans that are good in terms of one particular objective. In that case, there is only a small chance that additional floor plans, good in terms of the other objective, will effectively evolve from then on. In the single-objective case, this is not an issue.