## 2 Materials and Methods
### Proposed Method to Improve Construction Schedule Robustness  
The proposed method is intended to be applied in the course of the project to prompt reactions to schedule disturbances. It assumes that there exists a baseline schedule that defines the project completion date and the dates of key subcontracted processes. It assumes that a failure to meet these dates results in penalties. Their amounts are defined in the contract between the client and the general contractor and the contracts between the general contractor and the subcontractors. To mitigate delays and minimize penalties once delays occur, it is necessary to speed up processes not yet completed. The method helps select the most economical ways to do it. The method encompasses the following steps:  
1. Constructing the project's network model (breaking down the scope into processes, allocating resources (crews/subcontractors), defining relationships between processes, defining process durations). 2. Creating the baseline schedule and specifying processes whose start dates must be protected against delays, then determining the unit costs of delaying their start. 3. Designing variants of actions to reduce process durations and determining their costs. 4. Simulation studies of the project implementation model for various time reduction policies. 5. Finding the optimal time reduction modes. The project network is represented by a directed acyclic unigraph \(G\ =\ \langle V,\ E\rangle\) of a single start and a single end node. \(V\ =\ \{0,\,1,\,\ldots,\,n\}\) is the set of construction processes (schedule tasks). \(E\subset V\times V\) is a two-argument relation describing the sequence of processes. A function \(T:V\to R^{+}\) assigns durations \(t_{i}\) to processes \(i\in V\); the durations are random values of predefined distribution types and parameters. The estimated costs of processes are expressed as \(c_{i}\). The project's predefined due date sets the time for completion to \(T_{max}\). The baseline schedule is built using the expected values of process durations to meet the predefined due date. Alternatively, the baseline schedule may be based on process durations corresponding to a particular quantile of the duration distribution. To improve the schedule's robustness against disruption, it is advisable to distribute the free float as time buffers located before the processes whose start dates need to be protected (such as subcontracted processes with start dates contractually fixed or processes that involve an expensive hired plant). The set of processes that need to be protected is \(V_{d}\). It is assumed that the processes from the set of \(V_{d}\) can start no earlier than on the date set for them in the baseline schedule; the literature refers to this scheduling policy as the _railway policy_. The unit cost of delaying the process that belongs to \(V_{d}\) is \(c_{i}^{d}\) and represents contractual penalties or other costs attributable to such delay. The baseline schedule's start dates of any process of the project is marked as \(s_{i}\). In the course of the project, the processes are going to start as their predecessors are actually completed. In the case of processes \(i\in V_{d}\), they are not allowed to start earlier than on the date set for them in the baseline schedule. Due to the stochastic nature of process duration, their starts can be delayed. Thus, the actual start of a process \(s_{i}^{r}\) may come later than the as-scheduled start (\(s_{i}^{r}>s_{i}\)). As the delays are detected, the manager needs to decide on actions that prevent the propagation of disruptions on the processes to follow. These actions aim at reducing the execution time of delayed processes that are not yet started and consist in adding resources (reinforcing crews, using a more efficient plant), change of construction methods, working overtime, incentivizing the crews to work harder, etc. Inevitably, they result in an extra cost. The viable options of ways to compress the time of any process \(i\in V\) (also their combinations) form a set \(W_{i}\).