## 2 Theoretical Foundation  
Logistics industries have grown significantly in recent years and are expected to grow at a faster rate in the future [9]. One of the most important applications within logistics operations is vehicle routing. This happens because the allocation of resources cannot be accomplished unless resources are available at the right time, place, and in the right amount [10]. As a result, providing cost-effective and efficient services is of primary importance to any logistics enterprise [11]. This importance grows exponentially when considering epidemics and pandemics, such as the COVID-19 pandemic. The Travelling Salesman Problem (TPS) is a classic problem of routing. In TPS, a salesman has to visit a set of cities, starting at one city and subsequently returning to the same city they started in [9]. This must be achieved while minimizing the total length and total cost of the trip [9]. The TPS originates from the classic Vehicle Routing Problem (VRP) [11]. This problem aims to find a set of routes at a minimal total cost, beginning from, and ending the trip, at a depot [9]. This assures that each customer's demands are fulfilled. The first known application of VRP was for petrol deliveries [11] and was introduced by [PERSON] and [PERSON] [12]. Due to improvements in computational technology, VRP has been extended in many ways by introducing real-life aspects or characteristics, creating extensions from the classical VRPT [11]. Some examples are [12] time-dependent vehicle routing problem (TDVRP); multi-depot VRP; location routing problem; periodic VRP; VRP with time window; green VRP; and electric-vehicle VRP, among many others. Most of the methods and applications for the solutions found in the literature relate to commercial problems, in which case, efficiency is associated with minimizing costs [13]. However, the imperative restriction in this case study is time, given the vaccine's fragile conditions. Therefore, a Vehicle Routing Problem with Time Window (VRPTW) must be applied. In 1987, for the first time in history, [PERSON] used the VPRTW, which required that delivery vehicles provide their services to the customer in the stipulated time interval, guaranteeing customer satisfaction [14]. The VRPTW is an extension of the Vehicle Routing Problem (VRP), with an added time window constraint at each search point (that is, each node/point in a determined route), which means that for each search point, the service's first and last start times are defined, and the vehicles are required to serve the customers within this time window [8; 14]. VRPTW consists of finding the route with the shortest time so that each considered customer is visited within a defined time window by only one vehicle [15]. In addition, the considered vehicle cannot deliver a quantity that exceeds its capacity and must finish its route at the starting point [15]. There are two types of time window constraints for the demand points: the hard time window and the soft time window. The first case requires that the vehicle must start serving the customers within the stipulated time window. In this case, there are two options: waiting if the vehicle arrives early and rejecting the vehicle if it arrives after the time window closes. In the second case, the vehicles do not need to start serving customers within the time window. However, there must be a punishment if the vehicle arrives after the time window opens [14]. Some of the most useful applications of the VRPTW include bank deliveries, postal deliveries, industrial refuse collection, national franchise restaurant services, school bus routing, security patrol services, and just-in-time manufacturing [16]. It is imperative to account for the various limitations of this process. The optimization of the process needs to comprehensively consider the relationship between the number of vehicles' load distribution, the time window restrictions, and the path planning required to achieve the shortest time possible [8; 14].