## 4 Numerical Analysis for Uncertain Travel Time
### Data Analysis  
In today's fast-paced world, the increasing volume of traffic on roadways has become a major concern. As cities continue to expand, the surge in vehicular movement has resulted in significant congestion during peak hours and other periods of high demand. In light of this pressing issue, our study focuses on a specific case in the city of Casablanca, which is renowned for its heavy traffic flow. Casablanca, often hailed as the economic capital of Morocco, is renowned for its significant volume of goods deliveries. Hence, Casablanca, being a bustling economic center, experiences a complex interplay of factors contributing to its traffic congestion. The bustling city serves as a major hub for commerce and trade, attracting numerous businesses engaged in transportation and logistics activities. As a result, the city's intricate transportation network experiences a constant flow of goods deliveries, making it a critical focal point for distribution logistics. The increased traffic not only affects travel time but also the distribution logistics' performance. Currently, the transportation sector in Morocco thrives, with approximately 76,000 companies engaged in road transport, boasting a fleet of around 74,000 vehicles. In Table 3, we show a comprehensive depiction of the number of goods delivery companies operating in various regions across Morocco; the table highlights the distribution of these enterprises across the different regions. This information offers a clear visualization of the regional engagement of goods delivery companies, contributing to a better understanding of the logistics landscape across the country. According to the statistics, it is evident that Casablanca is home to a significant number of goods delivery companies, resulting in a large fleet of delivery vehicles. This makes Casablanca the city with the highest traffic in terms of distribution logistics as well as urban mobility. As a major economic center and the largest city in Morocco, Casablanca has become a focal point for distribution and transportation activities. The concentration of goods delivery companies in the city has led to a high density of delivery vehicles navigating through its busy streets. Our study focused on a delivery route within the city of Casablanca, from the depot (Cooperative Pharmaceutique Marocaine) to the customer (Pharmacie Joyaux de Bouskoura), as shown in Figure 5. Our work delved into the quantification of uncertainties in travel times to address these challenges by employing the collocation method. The aim was to model the variability of travel times in the delivery route. The collected data comprised travel times between these two points under four scenarios: \"Morning peak hour\", \"Inter-peak\", \"Evening peak\", and \"Off-peak\". The study considered two traffic states: the first being normal traffic (recurrent congestion) and the second being non-recurrent congestion caused by incidents such as accidents or unexpected disruptions. The aim was to study these variations to better understand the impact of different traffic conditions on travel times and make informed decisions to enhance the efficiency and reliability of distribution logistics operations. The data presented in Tables 4 and 5 represent various values of travel time throughout the day, including during peak hours. These data were collected using the Google Maps routing service, which calculates route-specific travel times based on real-time traffic conditions. \begin{table}
\begin{tabular}{l c} \hline \hline
**Region** & **Number of Goods Delivery Companies** \\ \hline Casablanca–Settat & 14,781 \\ \hline Marrakech–Safi & 9543 \\ \hline Rabat–Sale–Kenitra & 8889 \\ \hline Tanger–Tetouan–Al Hiceima & 8426 \\ \hline Fes–Meknes & 7029 \\ \hline L’Oriental & 6553 \\ \hline Beni Mellal–Khenifra & 5948 \\ \hline Souss–Massa & 5776 \\ \hline Laayoune–Sakia Al Hamra & 4316 \\ \hline Draa–Tafilalt & 3707 \\ \hline Guelmim–Oued Noun & 883 \\ \hline Dakhla–Oued Eddahab & 569 \\ \hline \hline \end{tabular}
\end{table}