# Space-Constrained Scheduling Optimization Method for Minimizing the Effects of Stacking of Trades
## 4 Method Verification  
### Verifying the Effectiveness of SSO  
The activity-on-node network consisting of 13 activities shown in Figure 6 was used to illustrate the procedure described in the preceding section and allows us to validate each step of the SSO method in detail. The case study is a network for electrical and mechanical work on a site with the floor plan as shown in Figure 7. The schedule information is stored in Table 1 with the activity index (i), activity duration (\(D_{i}\)), the number of workers per day (\(r_{i}\)) assigned to an activity, the number of equipment per day (\(e_{i}\)), required amount of material (\(m_{i}\)), required temporary facilities (\(f_{i}\)), and predecessor (\(PS_{i}\)), etc. The case site was divided into three specific work areas (\(h\)), and their area size were 51.3 m\({}^{2}\), 42.1 m\({}^{2}\), and 98.6 m\({}^{2}\). The occupation density function (\(f\)s) for each activity was found by considering the resources needed to execute the activity and the characteristics (e.g., location and area size) of specific work areas (Table 2). The number of alternatives for occupation density function (\(f\)s) obtained by changing the execution pattern of activity (\(i\)) were [3, 2, 2, 1, 4, 2, 1, 2, 2, 1, 2, 2, 3]. The CPM computes the critical path and total project completion time as A- > D- > L and 66 days, respectively. The non-critical activities (\(N_{CP}\)) are B, C, E, F, G, H, I, J, K, and M. The total floats (\(f_{i}\)) of activities were [0, 8, 17, 0, 9, 8, 9, 8, 17, 27, 8, 0, 27]. Therefore, based on Equation (9), the size of the optimal solution search range in this test case was  
\(7.68\times 10^{14}\) = (3 \(\times\) 1) \(\times\) (2 \(\times\) 9) \(\times\) (2 \(\times\) 18) \(\times\) (1 \(\times\) (4 \(\times\) 10) \(\times\) (2 \(\times\) 9) \(\times\) (1 \(\times\) 10) \(\times\) (2 \(\times\) 9) \(\times\) (2 \(\times\) 18) \(\times\) (1 \(\times\) 28) \(\times\) (2 \(\times\) 9) \(\times\) (2 \(\times\) 1) \(\times\) (3 \(\times\) 28). Figure 6: Network for the case study. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Activity ID** & **Activity** & **Predecessors** & **Duration** & **Workers Num.** & **Equip. Num.** & **Material** & **Temp.** & **Total Float** & **Alt. Num.** \\
**ID** & **Index (\(\hat{u}\))** & **(PS.)** & **(D.)** & **(Unit Size)** & **(Unit Size)** & **Qty.** & **Facility Qty.** & **(\(\hat{f}\))** & **(\(\hat{f}\))** & **(\(\hat{k}\))** \\ \hline A & 1 & - & 14 & 5 (4 m\({}^{2}\)) & 1 (10 m\({}^{2}\)) & - & 2 (8 m\({}^{2}\)) & 0 & 3 \\ B & 2 & - & 11 & 3 (4 m\({}^{2}\)) & 1 (10 m\({}^{2}\)) & 23 (1 m\({}^{2}\)) & - & 8 & 2 \\ C & 3 & A & 15 & 2 (4 m\({}^{2}\)) & 1 (15 m\({}^{2}\)) & 15 (2 m\({}^{2}\)) & 1 (10 m\({}^{2}\)) & 17 & 2 \\ D & 4 & A & 21 & 2 (4 m\({}^{2}\)) & - & - & 2 (10 m\({}^{2}\)) & 0 & 1 \\ E & 5 & B & 15 & 5 (4 m\({}^{2}\)) & 1 (15 m\({}^{2}\)) & - & - & 9 & 4 \\ F & 6 & B & 16 & 3 (4 m\({}^{2}\)) & - & 15 (2m\({}^{2}\)) & - & 8 & 2 \\ G & 7 & D,E & 13 & 3 (4 m\({}^{2}\)) & - & - & - & 9 & 1 \\ H & 8 & D,E,F & 14 & 5 (4 m\({}^{2}\)) & - & 5 (1 m\({}^{2}\)) & 1 (10 m\({}^{2}\)) & 8 & 2 \\ I & 9 & C & 11 & 3 (4 m\({}^{2}\)) & 8 (1 m\({}^{2}\)) & 2 (7 m\({}^{2}\)) & 17 & 2 \\ J & 10 & C & 1 & 1 (4 m\({}^{2}\)) & 2 (15 m\({}^{2}\)) & 20 (1 m\({}^{2}\)) & 1 (5 m\({}^{2}\)) & 27 & 1 \\ K & 11 & G,HJ & 9 & 3 (4 m\({}^{2}\)) & - & 5 (4 m\({}^{2}\)) & 1 (8 m\({}^{2}\)) & 8 & 2 \\ L & 12 & D,E,F & 31 & 3 (4 m\({}^{2}\)) & 1 (15 m\({}^{2}\)) & 15 (2 m\({}^{2}\)) & - & 0 & 2 \\ M & 13 & J & 9 & 2 (4 m\({}^{2}\)) & 3 (5 m\({}^{2}\)) & 8 (1 m\({}^{2}\)) & - & 27 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Schedule information of the case study. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Activity ID** & **Index of** & \multicolumn{3}{c}{**Occupation Density Function (\(f_{\text{s},\text{k},\text{k}}\)(\(p\)), \(p\): Progress Rate)**} \\ \cline{3-6}
**(Index)** & **Alt.