### Establishment of Objective Function, Constraints, and Control Conditions
#### 2.1.2 Determination of Constraints in Optimization Model  
The multi-objective optimization model of foundation pit dewatering shall meet the following constraints:  
1. Groundwater level  
To meet the construction requirements for foundation pit dewatering, it is necessary to lower the groundwater level in the pit below its bottom and establish water level control points within the pit. The actual drawdown at these control points must exceed the design drawdown.  
\[S_{j}\leq\sum_{i=1}^{w}S_{ji}\ \ \ j=1,2,\ldots,n_{j} \tag{5}\]where \(S_{j}\) is the design drawdown value at control point \(j\), m; \(S_{ji}\) is the drawdown of the \(i\)-th well to the water level control point at \(j\), m; \(w\) is the number of pumping wells; and \(n_{j}\) is the number of water level control points.  
Single well pumping capacity  
The pumping capacity of the pumping well is related to the well structure and aquifer permeability. It is required that the maximum pumping capacity of the pumping well shall not exceed its allowable maximum pumping capacity:  
\[0\leq q_{i}\leq q_{max}(i)\ \ \ i=1,2,\ldots,w \tag{6}\]  
where \(q_{max}(i)\) is the maximum pumping capacity of the \(i\)-th pumping well, m\({}^{3}\)/d.  
In the construction process of pumping wells, the number of pumping wells needs to be restricted. The restriction of the number of pumping wells in operation is as follows:  
\[\sum n_{i}\leq n_{max} \tag{7}\]  
where \(n_{i}\) is whether the \(i\)-th well exists. If the \(i\)-th well exists, then \(n_{i}\) = 1; if the contrary is true, then \(n_{i}\) = 0. \(n_{max}\) is the maximum number of pumping wells when all pumping wells exist.  
Settlement  
In order to ensure the environmental safety around the foundation pit, a settlement control point is set. The ground settlement value at the control point should be less than the allowable settlement at this point:  
\[\left[s\right]_{g}\leq\left[s\right]_{y} \tag{8}\]  
where \(\left[s\right]_{g}\) is the settlement at control point g, mm; variable \(\left[s\right]_{g}\) is the function of state variable \(h_{i}\) and decision variable \(q_{i}\); among them, \(h_{i}\) is the groundwater level, m; \(q_{i}\) is the pumping capacity of a single well, m\({}^{3}\)/d; and \(\left[s\right]_{y}\) is the allowable settlement, mm.  
Determination of Optimal Model Control Conditions  
In order to meet the installation of dewatering equipment, the radius of the pumping well is generally required to be greater than or equal to 0.2 m, as follows:  
\[0.2\leq r_{w}(i) \tag{9}\]  
where \(r_{w}i\) is the radius of the \(i\)-th well, taken as 0.2 m.  
The water level drawdown at the bottom of the foundation pit is the largest. This is the area with the largest hydraulic gradient. There are potential problems of piping or soil flow. The risk level is the highest, which needs to be considered. In order to consider the safety of foundation pit design, the hydraulic gradient is taken as the control condition:  
\[K_{s}=\frac{i_{cr}}{i_{max}} \tag{10}\]  
\[i_{max}=\frac{\Delta h}{\Delta L} \tag{11}\]\[i_{cr}=\frac{G_{\rm s}-1}{e+1} \tag{12}\]  
where \(K_{\rm s}\) is the safety factor, and the safety factor is usually greater than 1.5-2.5 in the dewatering design; \(i_{max}\) is the maximum hydraulic gradient; \(i_{cr}\) is the critical hydraulic gradient; \(h\) is the difference between the internal and external water head, m; \(\Delta L\) is the seepage path, m; \(G_{\rm s}\) is the specific gravity of aquifer soil particles; and \(e\) is the void ratio.