## 1 Introduction  
Path planning (PP) is an essential element of various significant fields, such as video games [1], robots [2; 3], and wireless sensor networks (WSNs) [4; 5; 6]. The aim of this problem is to calculate a feasible path from the starting point to the target point in workspace. In ref. [1], common pathfinding algorithms and techniques based on graph generation problems are outlined including A\({}^{\star}\), its variants, the probabilistic road method, quadtrees, and rapidly exploring random trees. In ref. [7], an overview of robotic motion planning is presented. This article shows artificial potential fields (APF), velocity obstacles (VO), probability based motion planning (PMP), artificial intelligence (AI), computer geometry, and focused D\({}^{\star}\) and D\({}^{\star}\) like algorithms. In ref. [8], the shortest-path algorithms were studied. The purpose was to obtain a path with minimal cost between the two nodes (or vertices) in the graph. Moreover, different graph theory algorithms were introduced. [PERSON] modified the meta-heuristic river formation dynamics algorithm to obtain the shortest path [9]. [PERSON] proposed a matrix binary code based genetic algorithm (MGA) to seek the path of best fit [10]. [PERSON] used the fuzzy-wind driven optimization algorithm for autonomous mobile robot navigation and collision avoidance [11]. A\({}^{\star}\) is a classic deterministic path planning method and was first proposed in ref. [12]. Over the next few decades, for single-objective path planning problems, A\({}^{\star}\) and its variants received widespread attention from scholars [13; 14; 15]. In ref. [16], Dijkstra and A\({}^{\star}\) were relaxed and used to solve large scale grid space more efficiently. A new focal any-angle A\({}^{\star}\) method was improved in ref. [17] by merging the advantages of grid-based and visibility graph-based methods. Batch-Theta\({}^{\star}\) was proposed in ref. [18]. It is a variant of Theta* and is used to select the best target with the lowest cost path. Most traditional methods find visible paths by considering a single objective optimization and rarely optimize multiple metrics at the same time. However, most problems in the real world should be solved by considering several conflicting objectives, such as price and quality. The path planning problem is no exception. Generally, we need to consider multiple objectives, such as the path length (concerned with operation time), the path safety (concerned with distance from obstacles), and the path smoothness (concerned with energy consumption). At least two of them compete with each other. According to the complexity of the PP problem, it can be classified as NP-hard [19]. The conflicting objectives must be processed simultaneously, and the objectives weighting methods appear [20; 21]. However, it is difficult to determine the relationship between the weighting factors. Therefore, it is suitable to treat these objectives as vectors. In this regard, some scholars study multi-objective heuristic algorithms based on the graph search space, for example, MOA* [22], MOD* [23], MOD* Lite [24] and so on. For the same grid space, no matter how many times the algorithm runs, it can find the same Pareto optimal points. In other words, the MOA* is a deterministic algorithm. The drawback is that the heuristic of MOA* is no longer a scalar, like A*, but a vector. The minimum heuristic function of a node is no longer a value. In general, it is a set of non-dominated vectors. This feature determines that the computational efficiency of MOA* is much lower than A* or other single-objective heuristic algorithms. In recent years, the use of probabilistic evolutionary algorithms has been popular to resolve the multi-objective PP problem, for example, the artificial bee colony [25], Ant Colony Optimization (ACO) [26], neural network [27], tabu search [28], particle swarm optimization algorithm (PSO) [29; 30; 31; 32], genetic algorithm (GA) [33; 34; 35].