# 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
For all other time steps, the binary variable \(x_{it}\) equals 0, as stated in the constraint part of the model. The second objective function, as stated in Equation (7) maximizes resilience through the surrogate measure (SM), which is represented by the sum of weighted resource-technology free float [30] for each activity, as explained in the previous subsection. The third objective function, shown in Equation (8), maximizes profit for the contractor at the end of the project by maximising the cumulative cash flow of dummy end activity. Constraint expressed in Equation (9) ensures that each activity, including the dummy start and end, can be started only once. Precedence constraint is expressed in Equation (10), defining that no activity can start before the precedent one is finished, for all activities that belong to the set of precedence related pairs. Financial constraint is shown in Equation (11) and it states that at the end of each month, the cumulative cash gap (before receiving the payment from the investor) must not exceed the permitted credit limit. Resource constraints are specified in Equation (12) by considering all possible start times, \(x_{it}\), for all activities (\(i\)) such that activity is in progress in period \(q\), for every time step, \(t\), and every resource, \(r\). Multiplication of binary variable \(x_{it}\) with resource consumption \(\u_{\text{ir}}\) must not exceed resource availability, \(\a_{\text{r}}\). Constraint which is stated in Equation (13) defines the decision variable as a binary one. Financial flow is modeled from the contractors' perspective by equations used in the work of [PERSON] and [PERSON] [49] that, in turn, rely on the financial terminology clarified by [PERSON] and [PERSON] [63]. Since the bank overdraft has been recognized as a prevailing method for financing construction projects [49; 64; 65], this study also considers the bank overdraft as a single source of financing the construction project. For this reason, the cash-flow model in this study adopted established calculation [49] and incorporated two adjustments to improve the validity of financial flows. The first modification refers to the inclusion of interest rate, \(h\), which rewards the contractor with an interest paid for positive cash flow. The interest rate (\(h\)) applicable on positive balances is less than the interest rate, \(ir\), charged for borrowing in the case of a negative cash balance for the contractor [66]. The second adjustment is applied to explicitly display the return of the retainage amount at the end of the project. By modeling the retainage amount as an absolute value, it is assured that a contractor receives complete payment from an investor. Notation for required input variables to model financial flow is presented in Table 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\d_{\text{ir}}\), \(i\in\A\) & Duration of activities \\ \(\s_{\text{ir}}\), \(i\in\A\) & Start times for activities from baseline schedule \\ \(c_{\text{ir}}\), \(i\in\A\) & Total direct cost per each activity in thousand of financial units \\ \hline \(\OP\) & Overhead percentage in decimal form \\ \(\MP\) & Mobilization percentage in decimal form; \\ \(\TP\) & Tax percentage in decimal form \\ \(\MP\) & Markup percentage in decimal form \\ \(\BP\) & Bond percentage in decimal form \\ \(\N\) & Negotiated duration of the project in months \\ \(\S\) & Realized duration of the project in months \\ \(\ADV\) & Advance payment (percentage of TBP in decimal form) \\ \(\D\) & Late completion penalty in thousand of cost units \\ \(\RET\) & Percentage of retainage for investors’ payments (decimal) \\ \(\ir\) & Interest percentage in decimal form \\ \(\h\) & Surplus percentage in decimal form (h \(<\) ir) \\ \(\k\) & Percentage for interest on the unused portion of a credit \\ \(\W\) & Specified credit limit of the overdraft \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input variables required for financial flow modeling.