# Effects of Decision Variables Selection on Sewer Optimization Problem
## 2 Methods
### Constraint Handling  
The inverse tangent method is used as the constraint handling method in this study. In this method, the maximum value of constraint violations (\(g_{max}\left(\overrightarrow{x}\right)\)) is taken as the fitness value. On the other hand, if the constraints are not violated, the arc tangent of the objective value is taken as fitness value. The process is given in Equation (15). \[  
In this study, the total cost function \(K_{T}\) of the network is used as Equation (18), which was proposed by [PERSON] and [PERSON] [32]:  
\[K_{p}=1.93\sigma^{3.43d_{i}}+0.812\overline{E}_{i}^{1.53}+0.437\overline{E}_{i} ^{1.47}d_{i} \tag{16}\]  
\[K_{m}=41.46h_{m_{i}} \tag{17}\]  
\[K_{T}=K_{p}+K_{m}, \tag{18}\]  
where \(K_{p}\) is the cost of pipe installations, \(i\) is the considered pipe, \(d_{i}\) is the diameter of the considered pipe, \(\overline{E}_{i}\) is the average excavation depth of the considered pipe, \(K_{m}\) is the cost of manhole installations, and \(h_{m_{i}}\) is the height of the manhole. As stated by [PERSON] et al. [36], some researchers studying on the same network used 2.45 m as the minimum cover depth value, while others used this same value as the minimum depth of excavation known as the minimum invert. This inconsistency leads to differences in cost estimates from one study to another. In this study, 2.45 m is taken as the minimum cover depth value. Some assumptions must be made to solve the problem. Furthermore, the search space reduction is utilized to improve computational efficiency by limiting the solution space. Details of this process are given below. In Alternatives 1, 2, 3, 5, 9, and 10, for inlets of the network, the lower and upper bounds of the diameter values are \(minD\) and \(maxD\). For downstream lines at the manhole in these alternative sets, the lower and upper bounds of the diameter value are the maximum diameter value of upstream lines connected to the same manhole and \(maxD\). In Alternatives 4, 6, 7, and 8, the diameter of the line satisfying hydraulic constraints is searched from \(minD\) to \(maxD\) for inlets and from the maximum diameter values of incoming lines to \(maxD\) for the lines other than the inlets. In Alternatives 1, 4, 5, and 6, for the inlets of the network, the lower and upper bounds of the upstream cover depth values are \(minCD\) and \(maxCD\). On the other hand, in Alternatives 2, 3, 7, 8, 9, and 10 that are decision variable sets not including cover depth values, there must be a cover depth value assumption for selected manholes of the network. For this reason, the upstream cover depth values of all inlets in Alternatives 2, 7, and 9 that are drop-allowed sets, and the inlets in Alternatives 3, 8, and 10 that are not drop-allowed sets, are assigned as \(minCD\). Figure 3: The layout of the Kerman network [32]. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Values** \\ \hline \(V_{min}\) & 0.3 m/s \\ \(V_{max}\) & 3 m/s \\ \(\beta_{min}\) & 0.1 \\ \(\beta_{max}\) & 0.82 \\ \(E_{min}\) & 2.45 m \\ \(n\) & 0.013 \\ \(D\) (mm) & 200, 250, 300, 350, 400, 450, 500, 600, 700 \\ \hline \end{tabular}
\end{table} 
Table 3: Design parameters of the network. In Alternatives 1 and 4 for all downstream lines, the lower and upper bounds of upstream cover depth value are the maximum downstream cover depth value of upstream lines connected to the same manhole and _maxCD_. In Alternatives 2, 3, 5, 6, 7, 8, 9, and 10 for all downstream lines, the upstream cover depth values of downstream lines are assigned as the maximum downstream cover depth values of upstream lines connected to the same manhole. In Alternatives 9 and 10, the slope of each line is searched for the minimum value that satisfies hydraulic constraints. In Alternatives 3, 8, and 10, since there is no drop in the manhole where there is more than one connection from upstream lines and to downstream lines, all cover depth values of manholes are assigned as the same value if there is one cover depth value calculated before.