# Resilient Scheduling as a Response to Uncertainty in Construction Projects
## 2 Resilience Approach to Manage Uncertainty in Construction Projects
### Optimization Model for Resilient Scheduling Problem
#### 2.4.2 Optimization Problem  
The optimization model presented in this study is based on the existing bi-objective resilient scheduling problem [30], which is improved by introducing the additional objective function to maximize the profit from a contractors' perspective. This way, an additional layer of financial stability is provided in the project baseline structure. With the net profit increase, the hypothesis is that contractor is fit to make financial compensation for perturbed activities. This means that adaptability to project disruptions is enhanced by maximizing the third objective function in the model. Moreover, the credit limit is included as one of the constraints in the optimization model, thus ensuring the financial feasibility of the optimal baseline schedule. A credit limit is imposed to assure that a financial flow is bounded to an allocated cash amount through the entire execution of the project. In addition, the methodology of activity weight calculation is introduced. This way, the surrogate measure is appropriately quantified so the resilience of a schedule can be approximated in the optimization process. Activity weight calculation is inspired by the work of [PERSON] and [PERSON] [29], where the weight of each activity is determined based on resource usage, network complexity, and time overlap. For this study, the additional layer concerning the direct cost of each activity was combined with the existing factors to quantify the activity weights. Table 3 summarizes the notation used in the optimization model. In the optimization model, binary decision variables were used to calculate activity start times, where \(x_{it}=1\) if activity, \(i\), starts at the time, \(t\); otherwise, \(x_{it}=0\). The multi-objective optimization model (MOO) for resilient baseline scheduling in construction projects is expressed as follows:  
\[\text{Min}\ \sum_{t\,\in\,T}t\ \times x_{n+1,\lambda} \tag{6}\]  
\[\text{Max}\ \sum_{i\,\in\,A}(w_{i}\ \times\ \text{FF}_{i}) \tag{7}\]  
\[\text{Max}\ \text{P} \tag{8}\]  
Subject to:  
\[\sum_{t\,\in\,T}x_{it}=1,\ \forall i\ \in\ A\cup\{0,\ n+1\} \tag{9}\]  
\[\sum_{t\,\in\,T}t\ \times x_{it}\leq\ \sum_{t\,\in\,T}t\times x_{it}-\ d_{i },\ \forall(i,i)\in\text{E} \tag{10}\]  
\[\text{CG}_{l}\leq W;\ \forall\ 1\ \in\ \text{EOM} \tag{11}\]  
\[\sum_{i=1}^{n}\sum_{q=max(0,\ t-\ d_{i}+1)}u_{ir}\times x_{iq}\leq a_{r},\ \forall t\ \in\ T,\ \forall r\in R \tag{12}\]  
\[x_{it}\ \in\ \{0,\ 1\},\ \forall i\ \in A\cup\{0,\ n+1\},\ \forall t\ \in\ T \tag{13}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline T & Length of the planning horizon (t = 1, 2,..., m) \\ A & Set of project activities (i = 1, 2,..., n), including dummy start 0 and dummy end n + 1 \\ E & Set of precedence relations \\ R & Set of project resources (r = 1, 2,..., k) \\ w\({}_{i}\) & Weight of activity \(i\) \\ d\({}_{i}\) & Expected duration for activity \(i\) \\ c\({}_{i}\) & Deterministic cost of activity \(i\) \\ u\({}_{ir}\) & Consumption of resource \(r\) as required by activity \(i\) \\ a\({}_{r}\) & Availability of resource \(r\) during project time T \\ FF\({}_{i}\) & Resource-technology free float for activity i \\ P & Final profit at the end of a project \\ EOM & End of the month considering project timeline (time step used when calculating Cash \\  & Flow), (EOM = 1, 2,..., 1) \\ CG\({}_{l}\) & Cumulative cash flow value at the end of the month l \\ W & Credit limit for the project \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters and sets used in the optimization problem. The first objective function, shown in Equation (6), minimizes the start time for dummy end activity, minimising the project duration in total. Multiplication of binary variable \(x_{it}=1\) with specific time step (\(t\)) defines the start of activity (\(i\)) at the start time, \(t\), supposing that time, \(t\), corresponds to the start time of activity (\(i\)) from the baseline solution.