In recent years, the problem of modeling uncertain travel times in distribution logistics has gained increasing attention as a means to improve logistics operations and enhance overall efficiency. The efficient management of distribution logistics is of crucial importance for businesses operating in distribution logistics, especially in cities with high traffic congestion. The ability to accurately predict and account for uncertain travel times is essential for effective logistics planning and optimization. The unpredictable nature of travel times poses significant challenges for logistics professionals. Traditional logistics models that assume fixed travel times often fail to capture the dynamic and uncertain nature of transportation. This can lead to inefficient resource allocation, missed delivery deadlines, increased costs, and customer dissatisfaction. To address these challenges, researchers and practitioners have turned their attention to the modeling of uncertain travel times as a key factor in distribution logistics. The variability and duration of travel time are influenced by various factors, particularly congestion. Congestion can be categorized as recurrent or non-recurrent [1], each with its distinct characteristics and causes. Recurrent congestion arises from daily peak-hour delays caused by the imbalance between traffic demand and the capacity of the transportation infrastructure. On the other hand, non-recurrent congestion results from atypical events, including incidents [2,3], disabled vehicles, road construction works, accidents, adverse weather conditions, and special events. Both categories of congestion introduce variability and uncertainty in travel times, leaving drivers uncertain about their precise arrival times at their destinations. Non-recurrent congestion can have dual effects on traffic conditions. It can generate new congestion during off-peak periods or increase delays during recurring congestion episodes. However, this type of congestion has often been overlooked in traffic engineering and modeling practices due to its irregular and unpredictable nature [4]. The modeling of the uncertain time variability has attracted the interest of several researchers. In a study by Ruimin [5], the examination of travel time variability involved the assessment of various factors, including time of day, day of the week, weather conditions, and traffic accidents. To quantify the influence of these factors on travel time parameters, the author employed multiple linear regression models with two-way interactions. The researchers in [6] investigated the prediction of uncertainty in route travel times within urban road networks, utilizing an extensive dataset from floating taxis; in [7], the authors delved into the importance of precise travel time modeling for road-based public transport and provided an overview of the current advancements in statistically modeling bus travel time distributions. They emphasized the significance of aggregating spatial and temporal data and proposed potential avenues for future research in this domain. In [8], researchers investigated travel time variability within the context of Indian traffic conditions and assessed the effectiveness of travel time distributions when considering various temporal and spatial aggregations. Utilizing AVL data gathered from four transit routes in Mysore City, located in Karnataka, India, this research examined travel time distributions across peak and off-peak periods, as well as within different time intervals. The study employed the Anderson–Darling test to evaluate the goodness-of-fit of the distributions, taking into account segments featuring signalized intersections and a variety of land-use types. The findings underscored the effectiveness of the generalized extreme value (GEV) distribution in precisely characterizing travel time variations within public transit systems. Therefore, the quantification of uncertainty remains a pivotal component when modeling and estimating travel time variability. Precisely capturing and forecasting travel time variations is essential in multiple domains, including transportation planning, traffic management, and logistics.

Many research endeavors have focused on tackling uncertainties and developing approaches to improve the precision and effectiveness of uncertainty quantification (UQ). UQ tools have been applied in numerous studies to evaluate and mitigate risks effectively. In [9,10], the authors proposed a model that utilizes uncertainty quantification to address risks associated with the customs supply chain. This model employs the moment matching method and takes into account the seasonality of illicit traffics at five different sites in Morocco. In the study [11], uncertainty quantification methods for differential equations were utilized to predict and simulate the transmission of influenza within the setting of a boarding school. In [12], the researchers developed an uncertainty quantification model to assess and manage the risks related to atmospheric dispersion. They utilized five different techniques to depict and analyze uncertainties, encompassing the probability theory, the interval analysis, the fuzzy approach, the mixed probabilistic–fuzzy approach, and the evidence theory. Stochastic uncertainties, which result from the inherent unpredictability of natural events, can originate from two primary sources: sampling uncertainty or measurement uncertainty [13]. Epistemic uncertainties, on the other hand, originate from gaps in information or knowledge about these events and may have various causes, such as data limitations, uncertain parameters, algorithmic ambiguities, or methodological uncertainties [13]. In [14], the authors introduced a model to predict and model epidemic risks, employing uncertainty quantification (UQ) techniques. They explored two specific methods, namely, the collocation method and the moment matching method. These methods were applied in the context of studying the epidemic risk associated with SARS-CoV-2 in Morocco, serving as an illustrative example. In the humanitarian logistics context, uncertainty quantification (UQ) holds a significant importance. This is primarily because disasters typically exhibit a substantial degree of uncertainty. For example, for flooding, UQ helps in the making of informed decisions, particularly when it comes to predicting disasters [15]. One of the fundamental approaches for uncertainty quantification (UQ) is the Monte Carlo simulation (MCS), which has been widely employed in various fields. The basic principle of the MCS involves repeatedly sampling input parameters from their respective probability distributions and propagating them through the model to obtain a distribution of output responses. By generating a large number of samples, the MCS can capture the full range of possible outcomes and provide statistical measures. The computational cost associated with the Monte Carlo simulation (MCS) can be a challenge due to the need for a large number of samples. This requirement can make the MCS computationally prohibitive [16]. The Latin hypercube sampling (LHS) technique offers potential improvements in uncertainty quantification. This sampling approach has been shown to provide better accuracy [17]. However, it is important to note that the LHS does have its limitations, as pointed out by [18]. In this context, a promising approach known as the polynomial chaos expansion (PCE) emerged. The PCE allows for the representation of random variables as multivariate series of Gaussian variables [19,20,21,22]. In a further development, representations based on Hilbert Expansions were introduced.^{[1][2][3][4][5]}. The representation of random variables by Hilbert expansions follows the concept of a Hilbert basis, with approximation achieved by truncating the representation at a specific level, and allows the use of non-gaussian variables^[6]. This approach has demonstrated significant advantages over traditional methods such as the Monte Carlo simulation (MCS) and the Latin hypercube sampling (LHS), offering lower computational costs and an increased efficiency.

2. Uncertain Travel Times in Distribution Logistics