# Linear Project-Scheduling Optimization Considering a Reverse Construction Scenario
## 3 Development of the LSM-CBPS Model
### Problem Description  
Consider a linear project comprising \(N\) activities denoted as \(i\), \(i=1,2,\ldots,N\), such that each activity has a starting distance denoted as \(SD_{i}\) and an end distance denoted as \(ED_{i}\). Each linear activity is further divided into subactivities across multiple construction sections, and for linear activity \(i\), the number of sections is denoted as \(S_{i}\). For the subactivity \(i_{k}\) of activity \(i\) in the \(k\)th segment, \(k=1,2,\ldots,S_{i}\), the starting and ending distances \begin{table}
\begin{tabular}{l l} \hline \hline
**Decision Variable** & \\ \hline \(st_{i,k}\) & start time of construction in the \(k\)th section of activity \(i\) \\  & denotes whether the \(k\)th section of activity \(i\) is \\ \(xc_{i,k,c}\) & constructed by construction crew \(c\): if yes, it equals 1; if not, it equals 0. \\  & indicates whether the \(k\)th section of \\ \(xm_{i,k,m}\) & activity \(i\) is constructed according to \\  & construction mode \(m\): if yes, it equals 1; if not, it equals 0. \\  & construction direction of activity \(i\) in the \(k\)th section, where forward is denoted by 1 and reverse by 0. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Decision variable of LSM-CBPS. \begin{table}
\begin{tabular}{l l} \hline \hline
**Constant** & \\ \hline \(crews_{i}\) & maximum number of crews available for \\ \(i_{k}\) & activity \(i\) \\ \(sc_{i}\) & set of crews chosen for activity \(i\) \\ \(nsc_{i}\) & number of crews chosen for activity \(i\) \\ \(seq_{i,c}\) & construction sequence set for crew \(c\) of \\ \(nodes_{i}\) & activity \(i\) \\ \(rs_{i,k}\) & required resources for subactivity \(i_{k}\) \\ \(pri_{k}\) & construction rate for subactivity \(i_{k}\) \\ \(dur_{i,k}\) & duration for subactivity \(i_{k}\) \\ \(cs_{i,k}\) & direct cost t per mile for subactivity \(i_{k}\) \\ \(tp_{i}\) & type of activity \(i\) \\ \(Tmin_{i,j}\) & minimum time buffer between activity \(i\) and \(j\) \\ \(Tmax_{i,j}\) & maximum time buffer between activity \(i\) and \(j\) \\ \(Dmin_{i,j}\) & minimum distance buffer between \\ \(Dmax_{i,j}\) & maximum distance buffer between \\ \(R_{\max}\) & daily resource supply \\ \(RW_{\max}\) & expected resource usage fluctuation \\ \(TC_{\max}\) & expected total project cost \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._ \begin{table}
\begin{tabular}{l l} \hline \hline
**Variable** & \\ \hline \(D_{\max}\) & shortest construction period \\ \(et_{i,k}\) & end time of construction in the \(k\)th \\  & section of activity \(i\) \\  & whether the \(k\)th section of activity \(i\) is \\ \(xt_{i,k,t}\) & constructed on the \(t\)th day: if yes, it is 1; \\  & otherwise, it is 0. \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Variable of LSM-CBPS. are represented by \(SD_{i,k}\) and \(ED_{i,k}\), respectively. Notably, block-type activities are not segmented. 1. Activity \(i\) can be completed by \(crews_{i}\) crews. For each section, only one crew can be selected, and that crew is always used for construction within the mileage range. For subactivity \(i_{k}\), the chosen crew sequence number is \(c\), \(c=1,2,\ldots,crews_{i}\). The crew number selected for all subactivities forms the set \(sc_{i}\), and the number of crews chosen for activity \(i\) is \(nsc_{i}(nsc_{i}\leq crews_{i})\). Additionally, the construction sequence set \(seq_{i,c}\) for crew \(c\) can be derived, in which the set's elements are sequence numbers of the subactivities that will be constructed by crew \(c\). These are arranged in ascending order. Notably, crew transition time is not considered. 2. Activity \(i\) has \(modes_{i}\) construction modes. A certain crew can select only one construction mode, \(m\), \(m=1,2,\ldots,modes_{i}\), and it will be maintained until the crew completes all its tasks. Each construction mode is a triplet.