## 3 Uncertainty Quantification Model
### Model Validation by Monte Carlo Simulation (MCS)  
In this study, we employed interpolation-based approximations, specifically the collocation method, to estimate travel time variability. To validate the effectiveness of this approach, we conducted a Monte Carlo simulation (MCS) on a specific example. The Monte Carlo simulation (MCS) is a computational technique used to estimate and analyze complex systems or processes by performing a large number of random simulations. It involves generating random samples from input probability distributions to model uncertain parameters, and then evaluating the system's behavior multiple times to observe the outcomes. In our example, we generated 1000 samples using the Monte Carlo simulation (MCS) and 30 samples using the collocation method. The aim was to compare the results of the two methods. Let us consider a system of ordinary differential equations (ODE) represented as follows:  
\[s=\int_{0}^{T}y(t)dt \tag{15}\]  
where \(y(t)=ae^{bt}\);  
such that  
\[\left\{ \begin{matrix}y^{\prime}=by\\ y(0)=a\end{matrix} \right. \tag{16}\]  
over the interval \([0,T]\); a and b represent two variables generated using triangular distribution to obtain 1000 samples for the Monte Carlo simulation (MCS) and 30 samples for the collocation method; the initial values used in the triangular distribution are represented in Tables 1 and 2. The parameters presented in Tables 1 and 2 represent simulation values of the ordinary differential equation (ODE) system that we used to confirm the reliability of our model. These simulations were conducted while considering various conditions and model parameters, and the resulting values align with the outcomes we obtained in our numerical experiments. As a result, Figures 3 and 4 represent two different approximations of the system under consideration using the Monte Carlo simulation (MCS) and collocation methods, respectively. The Monte Carlo simulation allowed us to generate 1000 samples for the variables \"a\" and \"b\" using a triangular distribution. On the other hand, the collocation method enabled us to generate 30 samples for the same variables with the same distribution. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & \(a_{min}\) & \(a_{max}\) & \(a_{pic}\) \\ \hline \(a\) & 0.5 & 1.5 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of variable “a” used to generate samples with triangular distribution. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & \(b_{min}\) & \(b_{max}\) & \(b_{pic}\) \\ \hline \(b\) & 1 & 2 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of variable “b” used to generate samples with triangular distribution. According to the figures, both the MCS's and the collocation's approximations show very close results to each other. This indicated that even with a small sample size of 30, as used in the collocation method, we achieved high-performance results comparable to those obtained with the MCS method. Additionally, in terms of execution time, the collocation method outperformed the MCS, making it a more efficient choice. Hence, our results revealed that the collocation method provided a robust and precise representation of travel time variability, closely aligning with the MCS approach. Notably, the collocation method achieved comparable results with a smaller sample size, making it a more efficient choice for uncertainty quantification in distribution logistics. The collocation method's concurrence with the Monte Carlo simulation approach reaffirmed its accuracy and dependability. Consequently, for our specific case study aimed at modeling travel time variability in distribution logistics, we opted for the collocation method as it combined accuracy, efficiency, and reliability in order to effectively capture and quantify travel time uncertainties. The decision to utilize the collocation method as our primary modeling technique was grounded in several compelling factors.