## 2 Literature Review
### Program Evaluation and Review Technique (PERT)  
From some of the literature reviews mentioned above, it can be concluded that the delay of railway projects is a serious problem in the world that needs to be solved, including in Indonesia. To address the issue above, one approach is to employ a Monte Carlo Simulation to incorporate uncertainty and generate a more precise range of scheduling solutions, enhancing the accuracy of the entire scheduling-estimation process. Simultaneously, the Program Evaluation and Review Technique (PERT) offers a probabilistic method for estimating activity durations through a three-point estimation methodology [12]. The framework will employ Monte Carlo Simulation to estimate fluctuations in activity duration and anticipate project completion time with greater accuracy. The simulation-based methodology enhances project schedule risk analysis by including novel equations for quantifying duration risk in PERT method analysis. This approach facilitates improved decision-making in ambiguous contexts [13]. Efficient scheduling is crucial for railway projects to minimize delays and ensure timely completion. PERT methodology and Monte Carlo Simulation assess project-completion probability and identify time-critical paths. The Program Evaluation and Review Technique (PERT) method is a dependable analytical approach for managing delays in railway project scheduling. The PERT method facilitates time estimation and project monitoring through a systematic methodology [14; 15; 16]. PERToffers systematic scheduling and resource estimation methodology, aiding in managing identified risks among uncertainty. According to [17], the delineation of PERT constitutes a methodology for approximating project duration through a weighted mean of optimistic (\(O\)), pessimistic (\(P\)), and most likely (\(ML\)) to estimate activity durations in the presence of ambiguity regarding individual activity assessments. The PERT equation comprises three elements (\(O\), \(P\), and \(ML\)) that employ a weighted mean of the three approximated values, presupposing a beta probability distribution for the temporal estimations, as delineated by [16]. \[T_{e}=\frac{O+4 ML+P}{6} \tag{1}\]  
where \(T_{e}\) represents the expected time (in days); optimistic (\(O\)), in days, denotes the minimum duration required to accomplish the task; most likely (\(ML\)), in days, signifies the duration with the greatest likelihood of completion; and pessimistic (\(P\)), in days, shows the maximum duration necessary to finish the task. According to [18], the timely completion of a project by the planned likelihood is assessed with a normal distribution, utilizing the subsequent formula:  
\[Z=\frac{D-T_{e}}{\sqrt{\sigma_{cp}^{2}}} \tag{2}\]  
where \(Z\) is the standard measurement of the deviation from the actual value to the mean value, \(\sigma_{cp}^{2}\) (in days) shows the variation in individual activities on the critical path, and \(D\) (in days) indicates the likelihood that the period will be fulfilled by the deadline. The studies of [5] stated that the PERT method was used to analyze the delays in the scheduling of railway projects. Meanwhile, ref. [13; 19] utilized a combination of AHP and PERT in optimizing the schedule-completion forecast in their study. Previous studies have been conducted using various techniques, which have resulted in differing risk variable assessments, as each project's geographical conditions will vary according to its cultural context. According to a study by [20], focus-group discussions (FGDs) serve to accurately evaluate the risks and impacts of project activities, as respondents are selected among experts engaged in the relevant case studies. Subsequently, data analysis will employ critical path data and the Critical Path Method (CPM) on historical project scheduling data, along with PERT, to forecast duration uncertainty and mitigate project risks [13; 21]. Meanwhile, ref. [22] used both PERT and M-PERT to compare the scheduling result for a construction project.