## 1 Introduction  
The vehicle routing problem (VRP) involves the definition of an optimal route set for several delivery vehicles satisfying the requirements of the customers. In particular, the VRP with time windows (VRPTW) is a VRP variant in which a slot of time is assigned for the delivery to the customer. This problem is known as one NP-hard combinatorial optimization problem, and many algorithms have been proposed in the existing literature to try to solve it, highlighting the use of swarm and evolutionary algorithms such as genetic algorithms and particle swarm optimization methods. In a genetic algorithm (GA), each candidate solution to VRPTW is encoded in one chromosome, and a population of them evolves by applying the selection, crossover, and mutation operators. This evolutionary process is guided by a fitness function, helping to reach a near-optimal solution. Since the permutation characteristics of a VRPTW candidate solution, new chromosomes must represent only feasible solutions, but traditional recombination operators can create infeasible solutions with repeated values. In a previous work, [PERSON] and [PERSON] [1] describe an approach named GA-VRPTW (Genetic Algorithm to Vehicle Routing Problem with Time Windows) evolving only feasible VRPTW routes by the use of modified genetic operators for selection, crossover, and mutation. Experimental results with several Solomon benchmark problems with 100 customers indicate that GA-VRPTW is very competitive compared with other GA-based algorithms and in some cases is even better. Otherwise, GA-VRPTW is a sequential algorithm whose performance analysis shows that the mutation operator requires the most computational time. In this paper, one grid-based genetic algorithm (GGA) to improve the GA-VRPTW performance using an experimental MiniGrid is introduced. GGA uses the computing resources of two clusters that are geographically separated but linked together since they are part of a MiniGrid configured as one virtual private network (VPN). Based on the genetic operators proposed in the GA-VRPTW algorithm, GGA splits the population of chromosomes in several segments, which are simultaneously mutated in each process generated by the MiniGrid. After the application of the mutation operator in each MiniGrid node, the mutated segments are used to build a new population of improved chromosomes. Since VRPTW is considered an NP-hard problem, many approaches have been proposed to find appropriate solutions, and its study has importance since several real-problems in manufacturing, scheduling, and logistics can be formulated as one vehicle routing problem. These approaches can be grouped into exact methods such as dynamic programming [2], branch-and-bound [3], and branch-and-cut [4], as well as metaheuristic-based algorithms such as Tabu Search (TS) [5], ant colony optimization (ACO) [6], simulated annealing (SA) [7], and evolutionary algorithms (EAs) [8; 9; 10; 11]. [PERSON] et al. [12] introduce a set-partitioning-based approach known in the existing literature as the KDMSS algorithm, in which a Dantzig-Wolfe decomposition scheme is used in conjunction with a two-path cut procedure to reach better lower bounds to several VRPTW instances of the Solomon benchmark problems with 100 and 150 customers. [PERSON] et al. [13] present a two-phase parallel SA algorithm to solve the VRPTW using a message-passing communication scheme. Experimental results of 56 Solomon benchmarks problems with 100 customers reach near-optimal solutions. Also, it was observed that increasing the number of processors improved the algorithm efficiency. A similar scheme was proposed by [PERSON] [14] where experimental results show that depending on the problem characteristics, some improve their efficacy as the number of processors increased, and with others, the quality decreased or remained the same regardless of the number of processors.