## 2 Literature Review
### Monte Carlo Simulation (MCS)  
Quantitative risk analysis is a mathematical method used to predict the likelihood and consequences of risks, identifying primary reasons for construction project delays by analyzing literature data and assessing their impact levels. According to [13; 24], qualitative risk analysis evaluates potential project hazards through a systematic methodology employing numerical and static methodologies. This method aims to assess the potential impact of diverse risks to enhance decision-making and resource-allocation efficiency. According to [PERSON], qualitative analysis can integrate findings with quantitative methods such as Monte Carlo Simulation (MCS) to evaluate the overall risk value. The impact of risk and the development of effective risk management strategies can create a better understanding of risk [6]. We propose that employing a novel equation for total risk value alongside Monte Carlo Simulation enables the estimation of activity duration variability. This approach is very effective for assessing the likelihood of train delays. This strategy can enhance the accuracy of project completion time predictions by adhering to deadlines [13]. This is an illustration of a Monte Carlo probability distribution, as suggested by [25].  
Figure 2 illustrates an example of Probability Density Distribution for Input and Cumulative Probability Distribution derived from the study of [25]. The procedure and process for Monte Carlo Simulation analysis is as follows: (1) Probability distribution of many random variables and stochastic experimental values is sampled. (2) Input sample data are processed to yield computed results inside a deterministic model. (3) The selected outcome values for this experiment, including the time and cost for completion, are recorded in a file. (4) To achieve a substantial quantity of experimental results and the desired precision, revert to step 1. (5) Preserving analytical outcomes. Repeated and uniform distribution sampling will yield an estimate of the expected time. If the value is mathematically proximate, then the outcomes of random sampling will be acquired:  
\[V=\int\limits_{-\infty}^{\infty}X(fx)dx \tag{3}\]  
where \(V\) is the expected value (in days), \(x\) represents the outcome value (in days), \(f(x)\) is the probability density function, and \(dx\) represents the infinitesimal elements of random variables \(x\) (in days). The straightforward yet sophisticated simulation process conducts integration on our behalf. The integration of \(X(fx)\) is highly significant in most pertinent assessment issues. According to [26], the following formula determines the number of iterations:  
\[\sigma=\sqrt{\frac{\sum_{i=1}^{k}(x_{i}-\overline{x})^{2}}{k-1}} \tag{4}\]  
\[\varepsilon=\frac{\overline{x}}{\left(\frac{1}{RE}\right)} \tag{5}\]  
\[n=\left(\frac{3\sigma}{\varepsilon}\right)^{2} \tag{6}\]  
where \(\sigma\) (in days) is the standard deviation, \(x_{i}\) (in days) is the data value of each population member, \(\overline{x}\) (in days) is the overall data mean, \(k\) is the number of data, \(\varepsilon\) (in days) is the absolute standard error, \(RE\) is the maximum error value, and \(n\) is the number of iterations.  
This study applies the PERT technique to refine the schedule completion forecast using the Monte Carlo Simulation, with estimated duration based on the identified risks using  
Figure 2: (**a**) Example of probability density distribution for input. (**b**) Example of cumulative probability distribution. Source from [25].  
the structured risk from previous study. Confidence levels of 80%, 90%, and 95% were chosen to increase the likelihood and enhance the schedule completion forecast based on the calculated duration estimates. The refinement of the risk-based calculation method shall be the major contribution of this study.