## 2 Theoretical Foundation
The priority restrictions in this case study are the time window in which the vaccines can be delivered (due to their delicate characteristics) and each vehicle's capacity. It must also consider the determination of the optimized number of vehicles and the path planning to achieve the shortest time possible. For this reason, this case study will adopt the VRPTW method with a load restriction. The VRPTW allows for finding the shortest time routes and ensuring that each vehicle only carries the maximum load assigned. In the review of the literature, heuristic and metaheuristic methods to solve the VRPTW were studied. Heuristics come from the Greek word meaning \"to discover\" based on intuition and habit [17]. Therefore, classical heuristics include trial and error, intelligent guesses, elimination, and experiences [17]. They are a flexible, problem solving, and individual-optimization methods that facilitate quick decisions (within modest computing times) and good quality solutions for complex data with limited knowledge and relatively limited exploration of the search space [17; 18]. Moreover, most heuristics can be easily extended to account for the diversity of constraints encountered in real-life contexts [18]. Metaheuristics are high-level heuristics whose emphasis is on performing deep exploration of the most promising regions of the solution space [18]. These methods typically combine sophisticated neighborhood search rules, memory structures, and a recombination of solutions [18]. Similar to heuristics, metaheuristics try to efficiently explore the search space of optimization problems [17]. However, while the adaptation to a specific problem uses heuristics as a solution method, a metaheuristic method may manipulate a complete or incomplete single solution or a collection of solutions at each interaction [8]. In other words, a heuristic provides a solution for a specific problem, in a way that works for that situation. However, it cannot be used to solve a different problem [8]. Therefore, metaheuristics are nothing more than sophisticated improvement procedures, and they can simply be viewed as natural enhancements of classical heuristics [18]. Metaheuristic algorithms are based on the behavior of natural phenomena such as physical phenomena, biological evolution, and living beings. These algorithms define themselves as the interactions between local individuals and their behaviors, which are split into five groups according to their nature [19]:  
* **Evolutionary Algorithms** are based on [PERSON]'s theory of natural selection. This algorithm starts with a population and then searches through successive generations to find the most optimal solution. Two of the associated algorithms are Genetic Algorithm and Genetic Programming. * **Physical Algorithms** consist of the laws of physics, such as heating and cooling. During the physical process, the heating of the material and the subsequent cooling contributes to a reduction of energy in a system. This is how simulated annealing arises. Therefore, this algorithm aims to achieve the optimal solution for the system. Some of the associated algorithms are Harmony Search and Memetic Algorithm. * **Swarm Intelligence Algorithms** are based on solving problems using the collective and organized behavior of animals and insects. This algorithm is a multi-agent system inspired by collective intelligence (given by the interaction of homogeneous agents). Two of the associated algorithms are Particle Swarm Optimization and Ant Colony Optimization. * **Bio-inspired Algorithms** derive from the behavior of the biological evolution of a living organism. These algorithms respond to the nature of distributed, decentralized, and self-organized data of malleable nature. Some of the associated algorithms are Artificial Immune System and Bacterial Foraging Optimization. * **Miscellaneous Algorithms** aim to provide an alternative solution to traditional real-time-based applications with intensification and diversification of the search space.