# Multi-Objective Resource-Constrained Scheduling in Large and Repetitive Construction Projects
## 3 Proposed Model  
In this section, the problem formulation and mathematical structure underpinning the proposed approach are presented, providing the foundation for subsequent analysis and discussion. Based on the task precedence relations and durations, the project structure is implemented in a spreadsheet environment. The start time of each task is associated with the finish time of all predecessors (using a max function) while the finish time is set based on the start time and the task duration. To allow for task re-scheduling and resource allocation improvement, a time lag is considered for each task between the latest predecessor finished and the actual start of the task under consideration. These time lags  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multirow{2}{*}{**Authors**} & \multirow{2}{*}{**Year**} & \multirow{2}{*}{**Objective Function**} & **Repetitive/** & \multirow{2}{*}{**Multi-**} & \multirow{2}{*}{**Objective**} & **No. of** & **Optimization** \\  & & & & & **Projects** & & **Activities** & **Method** \\ \hline \multirow{4}{*}{17} & \multirow{4}{*}{[PERSON] and [PERSON] [31]} & \multirow{4}{*}{2021} & minimize (1) project completion time, & & & & & \\  & & (2) project completion cost, (3) project & & & & & Non- \\  & & & resource moment, and (4) maximize & & & & & \\  & & & the project quality index & & & & & Algorithm III (OBNSA III) \\ \multirow{4}{*}{18} & \multirow{4}{*}{[PERSON] and [PERSON] [11]} & \multirow{4}{*}{2022} & minimize total project cost (activity & & & & \\  & & direct cost, project indirect cost, & & & & & Algorithm \\  & & & penalties, incentives) & & & & & Two-stage \\  & & & minimize cumulative absolute & & & & & GA-based \\  & & & deviations between daily and average & & & & & scheduling \\  & & & resource usage & & & & algorithm \\  & & & & & & & & \\  &act as the independent variables of the optimization model. If these lags are constrained to obtain a non-negative value, all schedules developed are valid in terms of task precedence relations and the only additional constraint is for the project delivery by the given deadline. After developing a time schedule, based on task time lags, the number of the cumulative required resources per period is calculated based on the task scheduling and the resource needs of the individual tasks. The process of assigning new time lags is repeated within the iterative optimization process until a termination point is reached. The flowchart of the algorithm implementation is shown in Figure 1.  
The mathematical formulation of the optimization model includes the following notation:  
\(N\): number of project tasks;  
\(i\): task number;  
\(d_{i}\): duration of task \(i\);  
\(s_{i}\): start time of task \(i\);  
\(f_{i}\): finish time of task \(i\);  
\(s_{0}\): project start time (by assumption \(s_{0}\) = 0);  
\(F_{N}\): project finish time (\(F_{N}=max_{i=1}^{N}f_{i}\));  
\(D\): project duration deadline;  
\(t\): time frame (\(t\) = 1, 2, \(\ldots\), \(D\));  
\(x_{t}(t)\): binary variable equal to 1 if \(s_{i}<t\leq f_{i}\), 0 otherwise;  
\(r_{i}\): number of resources of task \(i\) at time period \(t\);  
\(R_{t}\): cumulative resource requirement at period \(t\) (\(R_{t}=\sum_{i=1}^{N}r_{i}\cdot x_{i}(t)\));  
\(R_{m}\): mean resource usage;  
\(C\): total cost of the project. The following decision variables are defined:  
Figure 1: Algorithm implementation flowchart. \(e_{i}\): time shift of task \(i\) from predecessors. The problem constraints are the following: Task Scheduling Constraints:  
\[s_{i}=max_{i}f_{i}+e_{i},\,\forall i,\,\,\forall\,\,predecessor\,j\,of\,\,task\,i, \tag{1}\]  
\[f_{i}=s_{i}+d_{i},\,\,\forall i \tag{2}\]  
\[e_{i}\,\geq\,0,\,\,\forall i \tag{3}\]  
Project Completion Constraints:  
\[F_{N}\,\leq\,D, \tag{4}\]  
In the present work, several decision parameters are used, which can be more tangibly associated with the real effects of resource imbalance or constraints in resource availability [36].