## 4 Numerical Analysis for Uncertain Travel Time
### Computational Results and Discussion  
The numerical computations were conducted using Matlab R2018a on a computer equipped with the following specifications: Intel(R) Core (TM) i5-6300U CPU @ 2.40 GHz 2.50 GHz, and 8.00 GB of RAM, running on the Windows operating system. Additionally, to model the variability of travel time, we utilized a polynomial degree of 11. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Peak Time**} & \multicolumn{11}{c}{**Travel Time (min)**} \\ \hline Morning peak & 47.86 & 40.21 & 39.88 & 40.12 & 37.77 & 43.25 & 41.99 & 40.34 & 45.40 & 41.80 & 39.87 & 40.84 & 45.74 & 40.01 & 43.89 & 41.98 & 38.44 \\ \hline Inter-peak & 35.12 & 30.96 & 32.25 & 29.90 & 30.10 & 29.76 & 32.40 & 30.63 & 29.83 & 32.15 & 36.98 & 36.68 & 39.86 & 30.82 & 32.36 & 31.59 & 33.13 \\ \hline Evening peak & 41.18 & 39.54 & 42.60 & 44.01 & 41.67 & 42.96 & 45.51 & 38.46 & 43.94 & 40.54 & 43.46 & 39.89 & 41.75 & 39.83 & 40.55 & 39.97 & 43.14 \\ \hline Off-peak & 30.81 & 22.64 & 22.17 & 25.75 & 39.08 & 25.14 & 28.20 & 23.29 & 22.92 & 21.97 & 26.20 & 26.73 & 24.61 & 25.94 & 29.33 & 23.40 & 26.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Travel time during recurrent congestion. Figure 5: The delivery process arc. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Peak Time**} & \multicolumn{11}{c}{**Travel Time (min)**} \\ \hline Morning peak & 50.72 & 46.88 & 38.71 & 40.62 & 45.34 & 48.38 & 36.13 & 53.27 & 46.82 & 49.11 & 56.63 & 52.20 & 45.47 & 48.04 & 45.12 & 54.32 & 51.62 \\ \hline Inter-peak & 38.92 & 47.29 & 30.94 & 35.80 & 32.71 & 33.33 & 34.44 & 41.07 & 38.20 & 42.95 & 42.00 & 37.47 & 43.89 & 33.52 & 44.05 & 42.96 & 35.75 \\ \hline Evening peak & 54.46 & 44.03 & 49.86 &The study produced travel time variability results for both the recurrent congestion and non-recurrent congestion scenarios. We modeled this variability by employing probability density functions (PDF) and cumulative distribution functions (CDF) for each of the four scenarios: morning peak, inter-peak, evening peak, and off-peak. * Scenario 1: Recurrent congestion  
In this scenario, our focus is on understanding the variability of travel time during recurrent congestion, specifically in four distinct peak time scenarios: morning peak, inter-peak, evening peak, and off-peak. Recurrent congestion refers to the regular and predictable traffic jams that occur during specific times of the day when there is a significant increase in traffic volume, such as rush hours. During peak hours, the travel time variability can significantly impact the efficiency of distribution logistics operations, leading to delays, increased costs, and potential customer dissatisfaction. We present in Figures 6 and 7 the cumulative distribution function (CDF) and the probability density function (PDF). These graphical representations provide a comprehensive view of travel time distributions across the four peak hours, aiding us in the understanding of travel time fluctuations. The cumulative distribution function (CDF) and the probability density function (PDF) of travel time in the case of recurrent congestion are represented in Figures 6 and 7, respectively. The figures show the four scenarios: during the off-peak and inter-peak periods traffic congestion is relatively low compared to the \"morning peak\" and \"evening peak\" scenarios, when the traffic volume increases significantly. As a result, it is crucial for logistics and transportation planners to implement effective strategies to reduce delays and ensure on-time deliveries during peak traffic hours. This includes optimizing routes, planning for contingencies, and monitoring real-time traffic conditions. During the four peak time scenarios, we performed probability calculations based on the classification of travel times into three categories: \"low\" for travel times below 28 min, \"medium\" for travel times between 28 and 39 min, and \"high\" for travel times exceeding 42 min.