The vehicle routing problem (VRP) is a famous path-planning problem that was first proposed by Dantzig and Ramser [1]. It has been widely studied in the field of optimization problems and is a very practical model (Toth et al. [2]). In recent years, many variants of the VRP have been developed, such as the VRP with time windows (VRPTW) (Yu et al. [3]), green VRP (GVRP) (Xu et al. [4]), capacitated VRP (CVRP) (Zhang et al. [5]), multi-depot VRP (MDVRP) (Duan et al. [6]), and heterogeneous VRP (HVRP) (Ghannadpour et al. [7]). To sum up, there are mainly three kinds of classical algorithms to solve VRPs, which are exact algorithms, traditional heuristic algorithms, approximation algorithms, and metaheuristic algorithms. Exact algorithms (such as branch-and-price (Li et al. [8]), sophisticated branch-cut-and-price methods (Pessoa et al. [9]), mixed-integer nonlinear programming algorithms (Xiao et al. [10]), approximate dynamic programming algorithms (Çimen et al. [11], etc.) use mathematical methods to search for the optimal solution. Although they can often find a good solution, there are also problems, such as the single form of the solution, inability to avoid exponential explosion, and consumption of a lot of computing time. The basic idea of traditional heuristic algorithms (such as saving algorithms (Li et al. [8]), improved Dijkstra algorithms (Behnk et al. [12]), etc.) is to start from the current solution, search for a better solution in its neighborhood to replace the current solution, and then continue to search until there are no better solutions (Da Costa et al. [13]). The traditional heuristic algorithm easily falls into the local optimum and cannot easily achieve the global optimum. In addition, approximation algorithms (Das et al. [14], Khachay et al. [15]) with theoretical performance guarantees and approximation schemes have been widely used to solve the problems of covering non-Euclidean settings. Metaheuristic algorithms have excellent performance and constantly perturb near the current solution to search for better solutions. Therefore, this study uses the meta-heuristic algorithm to solve the problem.In recent years, many meta-heuristic algorithms have been proposed, and they are mainly divided into three categories: evolution-based algorithms (Back et al. [16]), physics-based algorithms (Webster et al. [17]), and swarm intelligence-based algorithms (Beni et al. [18]). The algorithm based on the evolution principle simulates the evolution of organisms. This method uses the crossover, mutation, and selection operators to update the population after randomly initializing the population and then continues to search for better solutions. The classical evolution-based algorithms mainly include the differential evolution algorithm (DE) (Storn [19]), genetic algorithm (GA) (Holland [20]), and biogeography-based optimization (BBO) (Simon [21]). The algorithm based on physical laws is inspired by physics and its individuals explore the solution space according to the physical law. The classical algorithms mainly include the black hole (BH) algorithm (Hatamlou [22]), gravity search algorithm (GSA) (Rashedi et al. [23]), big bang–big crash algorithm (BBBC) (Erol et al. [24]), and so on. The swarm intelligence algorithm mainly simulates the group behavior of organisms. Representative algorithms include the chimp optimization algorithm (Khishe et al. [25]), mayfly algorithm (Zervoudakis et al. [26]), equilibrium optimizer (Faramarzi et al. [27]), marine predator algorithm (Faramarzi et al. [28]), squirrel search algorithm (Jain et al. [29]), bald eagle search algorithm (Alsattar et al. [30]), Harris hawks optimization algorithm (Heidari et al. [31]), particle swarm optimization (Kennedy et al. [32]), artificial bee colony algorithm (Singh [33]), ant colony optimization algorithm (Neumann et al. [34]), firefly algorithm (Yang [35]), bat algorithm (Yang et al. [36]), grey wolf optimizer (Mirjalili et al. [37]), and the whale optimization algorithm (Mirjalili et al. [38]). Metaheuristic algorithms are widely used because of their excellent performance. Bi et al. (Bi et al. [39]) applied the GSTAEFA algorithm to the SMTSP. Artificial neural networks are widely used in image recognition and processing (Krizhevsky et al. [40]; Gu et al. [41]), time-series analysis (Arulkumaran et al. [42]; Tian et al. [43]), natural language processing (Juhn et al. [44]; Trappey et al. [45]), and building three-dimensional scenes (Dmitriev et al. [46]; Gkioxari et al. [47]). Genetic algorithms have been applied to engineering problems (Sayers [48]; Nicklow et al. [49]), classical optimization problems (Paul et al. [50]), and protein folding (Islam et al. [51]). The ant colony optimization algorithm has been applied to the traveling salesman problem (Dorigo et al. [52]; Dorigo [53]), engineering problems (Dorigo [53]; Nicklow et al. [49]), vehicle routing problems, machine learning, and bioinformatics problems (Dorigo et al. [52]), and the differential evolution algorithm has been used to solve the design problem of a reconfigurable antenna array. In addition, swarm intelligence algorithms have also been widely used in VRP. Solano Charris et al. [54] developed a local search metaheuristic algorithm to find the optimal path with the lowest cost in discrete scenarios. Wang et al. [55] used the multi-objective PSO algorithm to solve the dual objective model considering the time-varying speed of shared traffic resources. Chen et al. [56] proposed an intelligent water drop algorithm to solve the VRP of steel distribution. Zhang et al. [57] developed an improved tabu search algorithm to solve the cold chain logistics model. In GVRP, Zulvia et al. [58] solved the multi-objective model (GCVRPTW) using the multi-objective gradient evolution algorithm (MOGE).However, many studies on the VRPTW are generally carried out on two-dimensional planes. In many research fields, the three-dimensional spherical structure is also of great significance. For example, celestial bodies, particle structures, daily foods, proteins and other nutrients, balls, buildings, and path-planning problems are all problems related to spheres. Therefore, it is of great practical significance to expand the research on the VRPTW from the two-dimensional plane to three-dimensional spheres. Zhang et al. [59] applied the BBMA to the spherical MST problem. To sum up, in order to carry out research related to the distribution of goods by unmanned vehicles and unmanned aerial vehicles, this paper plans the path of a robot through these coordinates in three-dimensional space in order to carry out research on the VRPTW in the spatial dimension.ChOA has been applied to various practical problems, such as image segmentation for medical diagnosis (Si et al. [60]), clustering analysis (Sharma et al. [61]; Yang et al. [62]), Said–Ball curve degree optimization (Hu et al. [63]), and convolution neural networks (Chen et al. [64]). In addition, Du and Zhou (Du et al. [65]; Du et al. [66]) improved this algorithm and applied it to 3D path-planning problems and color image-enhancement problems. The algorithm divides the population of each generation into four groups, namely attackers, barriers, chasers, and drivers, and they cooperate against prey. Therefore, different groups search different spaces, which enhances the searching ability of ChOA. The adaptive factor of ChOA has a faster convergence speed and can adaptively balance exploration and exploitation, but it also easily falls into the local optimum. In order to obtain a group of better solutions with limited resources and time, this paper proposes an improved ChOA (MG-ChOA) for solving the spherical VRPTW model. The main contributions of this paper are as follows. Firstly, the proposed algorithm combines the ChOA algorithm with the quantum coding, local search, multiple population, and genetic operators to ensure that the algorithm can not only achieve adaptive and rapid convergence, but also find solutions with higher accuracy. Secondly, this paper proposes a three-dimensional spherical VRPTW and applies the proposed algorithm to solve this problem. Finally, by comparing with the running result of popular swarm intelligence algorithms for eight different instances, the effectiveness and superiority of the proposed algorithm in dealing with large-scale combinatorial optimization problems are strongly verified.The remaining organizational structure of this paper is as follows. Section 2, the Related Work, briefly depicts works related to the model proposed. Section 3 analyzes two-dimensional VRPTW and spherical VRPTW models to propose the mathematical model of a spherical VRPTW. The proposed algorithm (MG-ChOA) for a spherical VRPTW, an improved MG-ChOA algorithm based on ChOA, is presented in Section 4. The discussion of the experimental results analyzes the performance of the algorithm in Section 5. The conclusion and future work proposals are presented in Section 6.