## 2 Literature Review  
The definition of the vehicle-routing problem (VRP) has its origins in the formulation of the traveling salesman problem (TSP) [5]. This section first reviews proposed algorithms and methods for the VRP and its variants. Then, focusing on the revision of the VRPTW and finally conclude with the review of the auto-carrier transportation problem (ACTP). An important part of optimization systems are heuristics, which have multiple applications. From the extraction of features for a voice evaluation mechanism [6] to the generation of feasible VRP solutions. [PERSON] et al. [7] studied VRP with dynamic travel times, considering inputs of a dynamic nature and re-evaluating travel times dynamically as the solution was being developed. They proposed a learnt heuristic-based approach that integrates statistical learning techniques within a metaheuristic framework. [PERSON] et al. [8] investigated the capacitated vehicle-routing problem (CVRP) applied to natural gas distribution networks. The authors introduced an algorithm based on the saving algorithm heuristic approach to solve it. [PERSON] and [PERSON] [9] presented an electric vehicle-routing problem (EVRP) raised by a  
Figure 1: An illustration for the proposed problem by a logistics company. logistics company. They developed a heuristic approach based on the adaptive large neighborhood search (ALNS) and integer programming, specifically designed a charging station heuristic adjustment and other one for the departure time decreasing the total operational cost. Some well-known heuristic algorithms have been inspired by natural physical phenomena. [PERSON] [10] addressed the modified multi-depot vehicle-routing problem (MDVRP). He developed a metaheuristic algorithm based on the ant colony optimization (ACO) improved by a deterministic optimization process that is executed repeatedly within the ACO algorithm iterations. [PERSON] and [PERSON] [11] proposed a polar bear optimization algorithm (PBO) which imitates the survival and hunting behaviors of polar bears for local and global search. The authors presented a novel birth and death mechanism to control the population. [PERSON] et al. [12] proposed a monarch butterfly optimization (MBO) algorithm to solve the dynamic vehicle-routing problem (DVRP) using a greedy strategy. [PERSON] and [PERSON] [13] designed a bilayer local search-based particle swarm optimization (BLS-PSO) algorithm to solve CVRP. Currently, one of the most studied variants of the VRP is with time windows, in the research by [PERSON] et al. [14] introduced an optimization algorithm to solve a VRPTW, using dynamic programming. [PERSON] et al. [15] explored simulated annealing (SA), tabu search (TS) and a genetic algorithm (GA) heuristics to solve it. In another study, [PERSON] et al. [16] proposed a hybrid approach, consisting of the use of the ACO and TS algorithms, for the VRPTW. To improve the performance of the ACO algorithm, they introduced a neighborhood search and a TS algorithm to maintain the diversity of the ACO algorithm and explore new solutions. [PERSON] et al. [17] developed two metaheuristic algorithms to solve the VRPTW, the SA algorithm and an iterated local search (ILS). [PERSON] et al. [18] designed a hybrid genetic search with diversity control using a GA to solve the VRPTW, using the Pareto approach and two mutation operators to find the optimal solution set. [PERSON] et al. [19] investigated and defined a variant of the VRP, called the ACTP, proposed a three-step heuristic procedure that considers the loading, vehicle selection, and routing aspects for a solution to the problem. In other research, [PERSON] [20] addressed the ACTP for collection and delivery with limitations in the delivery times and the capacity of the auto-carrier, for new and used vehicles. The author proposed a constructive heuristic to solve the problem. [PERSON], et al. [21] defined the ACTP as a combinatorial problem of the CVRP.