# 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
#### 2.4.1 Surrogate Measure and Activity Weights  
The concept of resilient scheduling is built on the idea of providing enough time intervals in the initial baseline structure so the schedule is able to absorb disturbances caused by the impact of uncertainty. A common approach is to provide enough time floats in the initial solution by the use of different surrogate measures [28; 29; 30; 61; 62]. This way, the stability of a schedule is approximated under the low cost of computational complexity, providing a simple solution in the overall computationally demanding optimization process.  
The surrogate measure in this study is modeled as a weighted sum of resource-technology free floats [30] of all activities in a project. The resource-technology free float (FFi) calculates the amount of time for which an activity can be prolonged without postponing start times of succeeding activities, in a way that both precedence and resource relations are considered. On the other hand, activity weights (\(w_{i}\)) are significant for appropriate float distribution, since they express the relative importance of the FFi. For activities with the higher weight (\(w_{i}\)), it is more important to provide bigger FFi, so the baseline schedule is as resilient as possible. Activity weight is calculated based on four indices:  
* \(AS_{i}\)--Equation (1) states the relative number of successors per activity. The number of direct and indirect successors (\(N_{succ}\)) for the activity \(i\) is divided by the number of all activities in a project, \(n\) (both dummy start and dummy end included);
* \(DP_{i}\)-- Equation (2) calculates relative duration of the activity, \(i\);
* \(AC_{i}\)-- Equation (3) determines relative cost of the activity, \(i\);
* \(RU_{i}\)-- Equation (4) counts relative resource usage as required by the activity, \(i\).  
The assumptions are as follows: (i) with an increased number of successors, there is a higher probability to propagate time disruptions through the baseline schedule; (ii) activities that are expected to be executed during a longer time have a higher probability of disruption; (iii) the higher the cost of the activity, the bigger is the impact to the cash flow of a project; and (iv) with higher resource consumption, there is a higher probability of resource breakdown. Therefore, the activity weight (\(w_{i}\)) is calculated as follows:  
\[AS_{i}=\frac{N_{succ}}{n} \tag{1}\]  
\[DP_{i}=\frac{d_{i}}{\sum_{i=1}^{n}d_{i}} \tag{2}\]  
\[AC_{i}=\frac{c_{i}}{\sum_{i=1}^{n}c_{i}} \tag{3}\]  
\[RU_{i}=\frac{\sum_{r=1}^{k}u_{ir}}{\sum_{i=1}^{n}\sum_{r=1}^{k}u_{ir}} \tag{4}\]  
Finally,  
\[w_{i}=\frac{AS_{i}+DP_{i}+AC_{i}+RU_{i}}{4} \tag{5}\]  
Multiplication of FFi by the activity weight (\(w_{i}\)) accounts for proper float distribution in the baseline schedule, aiming to preserve the project at the equilibrium state. Therefore, by maximizing the weighted sum of the resource-technology free floats, the resilience of the initial makespan is improved.