# Vehicle Route Planning for Relief Item Distribution under Flood Uncertainty
## 1 Introduction  
Natural disasters, arising from uncontrollable natural processes, are events that humans cannot prevent. These incidents typically occur without warning and have profound impacts on human life, the environment, and broader societal structures. Flooding, one of the most common natural disasters, affects both high-risk and low-risk areas indiscriminately [1] and can cause extensive damage to critical infrastructure such as roads, bridges, and public utilities. This damage disrupts the normal flow of people and goods, severely limiting access to essential resources like food and water [2, 3]. The resulting shortages can jeopardize the survival of affected individuals and significantly delay the recovery process for entire communities. Consequently, the swift and efficient distribution of relief supplies becomes vital to mitigate these impacts, ensuring that those affected receive the necessary support to survive and rebuild [4]. The distribution of relief resources is pivotal in emergency logistics, facilitating the expedient delivery of essential supplies to areas affected by disasters [5]. However, the orchestration of emergency transportation during flooding is inherently complex andinfluenced by many unpredictable factors. These include sudden spikes in local demand, extensive damage to transportation infrastructure, and the imminent threat of secondary disasters. Such variables contribute to a multifaceted problem structure in emergency transportation, rendering the planning and execution processes highly intricate [6; 7]. This complexity necessitates sophisticated strategies and innovative solutions to address the logistical challenges of emergency scenarios effectively. A significant challenge in this research is the difficulty of planning routes under flood conditions due to the uncertainty of road conditions which can change over time. This uncertainty results in delays in delivering relief supplies and makes predictions difficult. Most studies frame the problem of distributing emergency resources as a Vehicle Routing Problem (VRP), which is challenging to solve within limited time constraints [8]. Additionally, the distribution patterns of disaster-affected nodes in different flood scenarios add complexity to route planning, such as uniformly distributed nodes versus clustered distributions [9]. Previous studies have introduced various methodologies for routing and scheduling in emergency relief operations. For example, [PERSON] et al. [10] developed a Dynamic Vehicle Routing Problem (DVRP) that leverages real-time traffic and demand data to dynamically adjust vehicle routes using GPS data and an Adaptive Large Neighborhood Search (ALNS) algorithm. [PERSON] and [PERSON] [11] utilized an Immune Evolutionary Algorithm (IEA) to optimize disaster relief schedules for personnel, goods, and vehicles, inspired by the adaptability of the human immune system. [PERSON] et al. [12] combined a Multi-Objective Evolutionary Algorithm (MOEA) with Greedy Search (GS) to enhance emergency logistics scheduling, focusing on optimizing response times, costs, and fuel consumption. [PERSON] et al. [13] designed a scheduling system for emergency vehicles using a time utility function that evaluates factors like injury type, proximity to the incident, and time elapsed since the incident. This system uses an Evolutionary Algorithm (EA) for optimization. [PERSON] et al. [14] proposed a rolling horizon strategy for disaster relief distribution using a State Estimation and Prediction Module (SEPM) and a Relief Distribution Module (RDM) to forecast demand and delivery times from data sources like social media and sensors. [PERSON] et al. [15] introduced a Multi-Objective Mixed Vehicle Routing Problem (MOMVRP) to address the distribution of relief materials, focusing on objectives like travel distance, response time, and aid recipient numbers, using an Evolutionary Algorithm (EA). [PERSON] et al.