## 1 Introduction  
Vehicle Routing Problems (VRP) have been intensively studied and researched for the last 60 years. The Capacitated VRP was first introduced in the seminal work of [PERSON] and [PERSON] [1], under the name of \"Truck Dispatching Problem\". They generalized the Traveling Salesman Problem (TSP), adding the parameter of multiple vehicles. Since then, many changes in the initial problem have been made, and many different variants of the VRP have been created, in an attempt to correlate VRP variants with real-life distribution problems [2; 3]. The Vehicle Routing Problem with Time Windows (VRPTW) is a well-known variant of the VRP which has received considerable attention in recent years and reflects many real-world scenarios, as, in distribution operations, the time window of the delivery is a crucial parameter of the problem. In conjunction with the time windows, multiple other parameters, affecting the routing of vehicles and scheduling of deliveries, exist. Time windows become even more challenging in urban transportation where uncertain traffic conditions exist, meaning that traveling times change dynamically. The VRPTW consists of geographically scattered customers that need to be served within a predetermined time interval (time window), only once and by a single vehicle. The total quantity of goods distributed in each route cannot exceed the vehicle's capacity, while each vehicle starts and ends its route at the depot. A vehicle may arrive at a customer before the opening of the time window and wait until the agreed service time, but it is not allowed to deliver goods if it arrives after the time window closes [4; 5; 6]. Instead, many research papers [7; 8; 9; 10; 11; 12] consider that some or all customers have soft time windows and may be served before the opening of the time window or after the time window closes, paying a penalty parameter each time the time window is violated. In the present paper, the time windows are considered hard and cannot be violated as this is a more realistic scenario for urban freight transportation, where a possible delay may have severe consequences. Solving the VRP is of great interest for both the research community and logistics and transportation companies, as it is crucial for delivering goods cost-effectively and facing customer requirements. The connection between the two fields is a two-way one, where companies, as well as their associates and customers, determine the constraints which are transformed into mathematical formulas and modeled by researchers. Researchers, aided by advanced computer science, propose and develop algorithms that solve specific variants of the VRP. However, not all algorithms are easy to implement to solve real-life problems, mainly due to their computational time. According to [PERSON] and [PERSON] [13], heuristic algorithms provide the best trade-off between solution quality and computation time. This theory is reinforced by some paradigms of the successful implementation of heuristic algorithms solving real-life vehicle routing problems [8; 14]. Despite this fact, the latest advances in technology have enabled metaheuristic algorithms to search a wider solution space in limited CPU time, while offering efficient results [15]. As the solution of real-life problems, VRPTW has to consider various aspects and constraints of the distribution networks. Researchers have developed and proposed over the years various exact, heuristic and metaheuristic algorithms. Each algorithm has specific advantages, as well as limitations. More specifically, exact algorithms compute every possible solution until the best one is reached, while performing very poorly in terms of computational time, even when dealing with fairly small instances [16]. However, distribution companies facing the VRPTW need methods capable of producing high-quality solutions in limited computational time and therefore cannot utilize exact algorithms. Therefore, multiple heuristic algorithms have been proposed, mainly divided into construction and local search heuristics.