# Optimization of Tunnel Construction Schedule Considering Soft Logic
## 3 Method
### Mathematical Model
#### 3.2.3 Constraints  
(1) Logical constraints  
This type of constraint is responsible for the prioritization and backwardness of each activity. Typically, there are four types of logical relationships between construction activities, as follows: start-to-start (S-S), finish-to-start (F-S), start-to-finish (S-F), and finish-to-finish (F-F). If activity \(a\) is a predecessor of activity \(b\), the logical relationships can be expressed by the following equations:  
1. S-S relationship  
The latter activity can only begin after the previous one has started, as shown in Equation (3). \[st_{a,j}\leq st_{b,j},\ \forall a,b\in N \tag{3}\]
2. F-S relationship  
The previous activity is completed before the subsequent activity can begin, as shown in Equation (4). \[ft_{a,j}\leq st_{b,j},\ \forall a,b\in N \tag{4}\]
3. S-F relationship  
The latter activity cannot be completed until the previous activity has begun, as shown in Equation (5). \[st_{a,j}\leq ft_{b,j},\forall a,b\in N \tag{5}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Constant** & \\ \hline \(Q\) & Work quantity \\ \(T_{end}\) & Expected project completion time \\ \(W\) & Number of construction teams \\ \(R\) & Distance constraints between the original working surface and the additional \\  & working surface \\ \(dp_{r}\) & Distance of optional additional working surface point \(r\) from the starting point, \(r\in j\) \\ \(w_{s}\) & Duration of working surface \(s\) \\ \(st_{j}\) & The start time of the working surface opened from unit \(j\) \\ \(ft_{j}\) & The end time of the working surface opened from unit \(j\) \\ \(dt_{j}\) & The duration of the working surface opened from unit \(j\) \\ \(st_{i,j}\) & Start time of activity \(i\) in each unit \(j\) \\ \(ft_{i,j}\) & End time of activity \(i\) in each unit \(j\), \(ft_{i,j}=st_{i,j}+dt_{i,j}\) \\ \(dt_{i,j}\) & Duration of activity \(i\) in each unit \(j\), \(dt_{i,j}=\frac{Q_{i,j}}{p_{i,j}}\) \\ \(v_{i,j}\) & Speed of activity \(i\) in unit \(j\) \\ \(minInt_{a,b}\) & Minimum interval between activities \(a\) and \(b\) \\ \(maxInt_{a,b}\) & Maximum interval between activities \(a\) and \(b\) \\ \(minDis_{ab}\) & Minimum separation distance between activities \(a\) and \(b\) \\ \(maxDis_{ab}\) & Maximum separation distance between activities \(a\) and \(b\) \\ \(A\) & A sufficiently large positive number \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constants of the TSMOM. F-F relationship  
The previous activity will not be completed until the subsequent activity is completed, as shown in Equation (6). \[st_{a,j}\leq ft_{b,j},\forall a,b\in N \tag{6}\]  
Work continuity constraints  
Work interruptions affect the proficiency of the construction team, which is not conducive to learning and advancement [40]. There are additional costs associated with idle construction teams and transfers to other construction sites [15]. In practice, in order to shorten the interruption of activities and minimize the transfer of construction teams, the minimum time of continuous work or the shortest continuous working distance of the construction teams is usually constrained. In each activity, the working surface can only be accessed and carried out using one additional working surface optional point, and there is no situation where a construction team can work on the same working surface from multiple optional additional working surface points, as shown in Equation (7). \[\sum_{s=1}^{S}\sum_{j=1}^{M}x_{j,s}=1 \tag{7}\]  
The construction team may perform activities on multiple units using an additional working surface, but these units must be continuous, as shown in Equations (8)-(10). \[e_{js}\geq x_{j,s+1}-x_{j,s} \tag{8}\]  
\[e_{js}\leq\big{(}x_{js}-x_{j,s+1}\big{)}\big{(}x_{js}-1\big{)}A \tag{9}\]  
\[\sum_{s=1}^{S}e_{j,s}\leq 1 \tag{10}\]  
Construction team switching is only allowed when the work continuity constraints are met.