## 1 Introduction
In this paper, we have employed the collocation method, specifically based on approximation interpolation, to model the variability of travel time. The collocation technique involves selecting specific points, known as collocation points, within the domain of interest and determining the corresponding coefficients to approximate the underlying function. To validate the effectiveness of the collocation method in modeling travel time variability, we conducted a comparison with the Monte Carlo simulation (MCS) approach. The MCS method is widely used as a benchmark for uncertainty quantification and serves as a reference for evaluating the accuracy and efficiency of alternative techniques [23]. The utilization of collocation methods for modeling the variability of travel time has gained little significant attention in the literature. In [24], the authors utilized the stochastic response surface method (SRSM) to analyze travel time variability under uncertain factors, including traffic volume and rainfall intensity. Applying SRSM to a section of the Kobe Route, Japan, the results showed an improved accuracy in modeling travel time distribution compared to the regression analysis; the collocation method was used for estimating the travel time. In this work, we employed the collocation method to effectively model the travel time variability during the four peak hours, considering both recurrent congestion and non-recurrent congestion scenarios (Figure 1). This approach allowed us to delve deeper into understanding how travel time fluctuations evolve over the course of the day, shedding light on the intricate dynamics of traffic patterns during peak hours. The collocation method has proven to be a valuable tool for uncertainty quantification in travel time estimation. It enables us to address the challenges posed by the dynamic and complex traffic conditions such as those in Casablanca. By incorporating the variability of traffic volume, incidents like road accidents, and other external factors, we gained insight into how uncertainties impact travel time reliability. The insights derived from our analysis will aid logistics practitioners in making informed decisions, optimizing route planning, and mitigating the potential impacts of travel time uncertainties on delivery schedules. Figure 2 explains the framework methodology of this study: the aim is to model uncertain travel times in distribution logistics using the uncertainty quantification approach with the collocation method. To ensure the reliability and effectiveness of the collocation method, we conducted a thorough validation process using Monte Carlo simulations (MCS). Subsequently, we conducted modeling using the cumulative distribution function (CDF) in both the recurrent and non-recurrent congestion scenarios. Then, a probability density function (PDF) was derived from the CDF using Dirac's delta approximation. To comprehensively assess the variation of travel time uncertainties in distribution logistics, we calculated probabilities across all four peak time scenarios:'morning peak', 'interpeak', 'evening peak', and 'off-peak'. This encompassing approach helped us gain a holistic understanding of how different factors, including traffic congestion, influence the variability of travel times. Figure 1: Recurrent and non-recurrent congestion. Figure 2: The methodology of the study framework. This paper is organized as follows: Section 3 describes the uncertainty quantification model and contains the collocation model validated by the Monte Carlo simulation (MCS) using an example; Section 4 presents the computational results and discussion; and Section 5 presents the conclusion and discussion.