## 1 Introduction  
Many scientific and engineering problems in the fields of industry, agriculture, transportation, information, and management, among others, belong to optimization problems, or can be transformed into optimization problems [1; 2; 3; 4]. These optimization problems are one of the hotspots and difficulties recognized by the experts and scholars at home and abroad. Mature optimization theories and methods have been widely used in these fields [5; 6; 7]. With the deepening of human cognitive activities and the rapid development of computer science and technology, the optimization problems faced in the real world are becoming increasingly diversified, large-scale, and complex. Additionally, these optimization problems present complex features such as being highly dynamic, high-dimensional, nonlinear, multi-objective, multi-constraint, discontinuous, and non-differentiable. This makes it difficult for experts and scholars using traditional optimization theories and methods to solve problems from the perspective of analyzing the mathematical characteristics of the problem [8; 9; 10; 11; 12]. Therefore, seeking new intelligent optimization algorithms with high efficiency, parallelism, adaptability, and robustness to solve large-scale complex optimization problems has become an urgent and persistent research topic in the field of engineering application, which not only has important theoretical significance but also has a wide application prospect. Although there are many types of optimization algorithms in the field of intelligent optimization, none of them can solve all optimization problems. Therefore, a large number of new optimization algorithms are constantly being proposed or improved. In recent decades, meta-heuristic algorithms based on the population have achieved good performance in solving complex engineering optimization problems: for example, the genetic algorithm (GA) [13], the simulated annealing algorithm (SAA) [14], particle swarm optimization (PSO) [15], the bat algorithm (BA) [16], ant colony optimization (ACO) [17], the novel moth to fire algorithm (MFO) [18], the locust optimization algorithm (GOA) [19], the butterfly optimization algorithm (BOA) [20], and the sine cosine optimization algorithm (SCA) [21], among others [22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. These algorithms can be grouped into two broad categories: individual-based and group-based. The first class generates only one solution, while the second class generates multiple solutions. The artificial electric field algorithm (AEFA) [32] is different from these swarm intelligence bionic algorithms; it simulates the process of the relative motion of the charged particles under Coulomb force in the electrostatic field. The AEFA has the advantages of a simple structure, less adaptive parameters, and a simple algorithm principle. With the increasingly scattered urban commercial centers and the increasing demand for logistics, vehicle routing problems have gradually become the focus of research by scholars of intelligent optimization algorithms. The vehicle routing problem (VRP) [33] is a crucial part of logistics distribution optimization. In the process of logistics distribution, by selecting the optimal transportation route for the distribution vehicles and rationally arranging the scheduling sequence of the vehicles, the empty driving rate and driving distance of the vehicles can be effectively reduced. This problem was first proposed by [PERSON] and [PERSON] in 1959 and belongs to the NP-hard problem [34]. At present, scholars at home and abroad have carried out extensive and in-depth research on the VRP, and the VRP has been applied to many social fields such as the route optimization of garbage trucks and the delivery route optimization of chain stores [35]. Since the traditional vehicle routing problem has less constraints, the vehicle cost is usually taken as the total cost as the optimization objective, which leads to great limitations in practical applications [36].