## 4 Numerical Analysis for Uncertain Travel Time
### Computational Results and Discussion
The results of these probability calculations are presented in Table 6. It provides a comprehensive overview of the probabilities associated with each travel time category for all four peak time scenarios: \"morning peak\", \"inter-peak\", \"evening peak\", and \"off-peak\". The table helps us understand how likely travel times are to fall into each category at different times of the day. Figure 6: CDF of travel time in recurrent congestion. Based on the results, it is observed that the probabilities for \"high\" traffic levels predominantly occur during the morning peak and evening peak times. On the other hand, during the off-peak and inter-peak periods, the travel times are generally shorter. * Scenario 2: Non-recurrent congestion During the non-recurrent congestion, the travel time variability becomes more unpredictable. Incidents such as road accidents can occur at any time of the day, disrupting the traffic flow and leading to unexpected delays. These unpredictable events can significantly impact travel times, causing fluctuations and introducing uncertainty in distribution logistics operations. We will assume that accidents can occur during each of the four peak times, potentially disrupting the traffic flow. These accidents can lead to unexpected road closures, detours, and increased congestion, resulting in longer travel times for distribution logistics vehicles. As a result, the variability in travel times during peak hours becomes even more uncertain and challenging to predict. Figures 8 and 9 present the variability of travel time illustrated in the cumulative distribution function (CDF) and the probability density function (PDF). These graphical representations show the distribution of travel times during different peak hours, taking into account the uncertainties caused by accidents and non-recurrent congestion. \begin{table}
\begin{tabular}{|l|l|l|l|} \hline
**Peak Time** & **Low** & **Medium** & **High** \\ \hline
**Moming peak** & 0.11765 & 0.58824 & 0.29412 \\ \hline Inter-peak & 0.94118 & 0.058824 & 0 \\ \hline Evening peak & 0.058824 & 0.52941 & 0.41176 \\ \hline
**Off-peak** & 0.17647 & 0.058824 & 0 \\ \hline \end{tabular}
\end{table} 
Table 6: Travel time probability in recurrent congestion. Figure 7: PDF of travel time in recurrent congestion. According to Figures 8 and 9, we observe multiple peaks in the PDF, indicating the presence of several disruptions, uncertainties, and random events that affect the travel time. This multimodal distribution suggests that the distribution of travel time is not governed by a single dominant factor but is influenced by various sources of variability. One of the significant factors contributing to these peaks could be road accidents, which can cause unexpected delays and fluctuations in travel time. Road accidents can occur at any time of day and can have a significant impact on the travel time during non-recurrent congestion. When accidents happen, they can lead to unexpected road closures, diversions, or reduced speed limits, causing traffic disruptions and delays. The variability in travel time during non-recurrent congestion becomes more pronounced as these unforeseen events can create fluctuations in the flow of vehicles and introduce additional uncertainties in the travel time estimation. Then, the probabilities obtained for these scenarios were calculated and results are presented below. Figure 8: CDF of travel time in non-recurrent congestion. Figure 9: PDF of travel time in non-recurrent congestion. The results presented in Table 7 reveal some interesting insights regarding the impact of non-recurrent congestion, such as road accidents, in contrast to recurrent congestion. In non-recurrent congestion scenarios, there is a high probability of travel time disruptions during inter-peak periods in addition to peak hours (morning peak and evening peak). This finding highlights the significance of unexpected events, like accidents, in causing traffic disturbances during times that are typically less congested.