# Space-Constrained Scheduling Optimization Method for Minimizing the Effects of Stacking of Trades
## 4 Method Verification
### Verifying the Effectiveness of SSO
(\(k_{\text{s}}\))** & **Work Area A (\(h\) = 1)** & **Work Area B (\(h\) = 2)** & **Work Area C (\(h\) = 3)** \\ \hline \multirow{3}{*}{A (\(i=1\))} & 1 & \(f_{\text{s}1,\text{1}}\)(\(p\)) = 0.4, 0 \(<p\leq 0.7\) & \(f_{\text{s}1,\text{2}}\)(\(p\)) = 0.33p - 0.03, 0.7 \(<p\leq 1\) & \(f_{\text{s}1,\text{3}}\)(\(p\)) = 0.3, 0 \(<p\leq 0.7\) \\  & 2 & \(f_{\text{s}1,\text{2}}\The space interference level (\(f_{\text{SC}}\)) of the original schedule, which is computed by commencing each activity at its original EST (\(\varepsilon_{i}\)), was 47.48. The space interference level of each specific work area was 8.85, 16.35, and 22.28. In the case of exceeding the occupation allowance of the specific work areas (\(\sum f\text{s}>1\)) was found to occur three times (at 23, 24, and 30 days). The space interference level (\(f_{\text{SC}}\)) of the LST (\(l_{\text{s}i}\)) schedule was calculated to be 66.58 (h\({}_{1}\) = 29.36, h\({}_{2}\) = 32.88, h\({}_{3}\) = 4.34). The allowed occupation density of specific work areas was exceeded four times (at 46, 54, 56, and 57 days). Because early start time (EST) and late start time (LST) schedules both exceeded the allowed value for occupation density, it was predicted that there would be inevitable construction delays and cost increases. Therefore, it is necessary to establish a schedule that minimizes space interference by performing optimization. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Activity ID** **(Index)**} & \multirow{2}{*}{**Index of Alt. (\(k_{i}\))**} & \multicolumn{3}{c}{**Occupation Density Function (\(f_{\text{s}_{\text{UL  
The GA parameters were initialized with [_PS_ = 400, _CR_ = 0.4, _MR_ = 0.05, StallGenLimit = 200 times, TimeLimit = Inf, StallTimeLimit = Inf] to execute the GA optimization operation. The GA operation stopped as the allowable cumulative change of the objective function reached 1 \(\times\) 10\({}^{-6}\) after undergoing 124 generations. SSO found [(3,0), (1,0), (2,0), (1,0), (1,4), (2,8), (1,1) (2,0), (1,13), (1,0), (1,2), (1,0), (3,1)] as the near-global solution (i.e., the set of pairs of execution pattern alternatives and deferred start times of activities) that minimizes space interference without exceeding the occupation allowance of each specific work area (Table 3). These results were expressed using the chromosomes designed for the GA operation. Converting these results into alternatives for activity execution patterns (\(k_{i}\)) yields [3,1,2,1,1,2,1,1,1,1,3], and the number of deferred days of activities start time (\(sd_{i}\)) is [0,0,0,0,4,8,1,0,13,0,2,0,1]. Figure 8 shows a diagram of the obtained results. When the network was executed using the found solution, the space interference level (\(f_{SC}\)) was calculated to be 17.79. This was a space interference level reduction of 30.29 compared with the EST schedule and 49.39 compared with the LST schedule. The space interference levels for each specific work area were 8.62, 5.77, and 3.40. There were no cases in which the occupation allowance of specific work areas was exceeded (\(\sum f\)s \(>\) 1). Therefore, there is no need for concern regarding construction delays and cost increases. It was confirmed that SSO finds a solution that minimizes space interference without exceeding the allowed occupation density of each work area. The performance of GA can be changed depending on the values chosen for the GA parameters. Therefore, performing GA once with only one set of GA parameters cannot guarantee the reliability of the obtained results. In order to improve reliability, the sensitivity analysis was performed as follows. This analysis starts by setting the minimum, incremental, and maximum value of each GA parameter (i.e., population size (_PS_) = [_PSmin:IPS:PSmax_] = 300:20:500), crossover rate ((_CR_) = [_CRmin:ICR:CRmax_] = 0.4:0.1:0.8), and mutation rate ((_MR_) = [_MRmin:IMR:MRmax_] = [0.05:0.01:0.08]).