###### Abstract  
The paper aims to investigate the basin of attraction map of a complex Vehicle Routing Problem with random walk analysis. The Vehicle Routing Problem (VRP) is a common discrete optimization problem in field of logistics. In the case of the base VRP, the positions of one single depot and many customers (which have product demands) are given. The vehicles and their capacity limits are also fixed in the system and the objective function is the minimization of the length of the route. In the literature, many approaches have appeared to simulate the transportation demands. Most of the approaches are using some kind of metaheuristics. Solving the problems with metaheuristics requires exploring the fitness landscape of the optimization problem. The fitness landscape analysis consists of the investigation of the following elements: the set of the possible states, the fitness function and the neighborhood relationship. We use also metaheuristics are used to perform neighborhood discovery depending on the neighborhood interpretation. In this article, the following neighborhood operators are used for the basin of attraction map: 2-opt, Order Crossover (OX), Partially Matched Crossover (PMX), Cycle Crossover (CX). Based on our test results, the 2-opt and Partially Matched Crossover operators are more efficient than the Order Crossover and Cycle Crossovers. vehicle routing problem; fitness landscape; basin of attraction; random walk 2021 11 2010 11.12010.11.12010.1390/app110521001  
a heterogeneous fleet (Heterogeneous Fleet Vehicle Routing Problem [7]). If vehicles have a capacity limit for the goods to be transported, then the system is called Capacitated Vehicle Routing Problem [4,8,9]. Customers to visit can also have a time window, this is called the Vehicle Routing Problem with Time Windows [4]. One of the latest VRP trends is the Fuel-Efficient Green Vehicle Routing Problem [10]; this case prioritizes not the minimization of the length of the route, but the minimization of the fuel consumption of the vehicles as the objective function. In the case of Cumulative Vehicle Routing Problem [11], the minimization of the waiting times for the customers is the objective function. In case of Fuzzy VRP, some factors are given with fuzzy numbers [12]. Milk-run VRP [13] is also a common problem, where the products, materials, etc., are transported between the warehouse and the production line. Another new trend is the application of electric vehicles [14]. Separate algorithms are usually implemented for each VRP model and we also find examples of general heuristics [4]. The investigated system is designed to solve the following VRP model: time windows, capacitated vehicles, multi-depot, and open VRP. In logistics, beside VRP, we can also find examples of other transportation tasks [15,16], too. The fitness landscape analysis [17] is a method to investigate the optimization space, it consists of the following elements: the set of possible solution states, neighborhood definition and the fitness function. Although encoding does not belong directly to the concept of the fitness landscape, it plays a big role in the analysis of space. The neighbors of the actual state can be created in several ways (for example, in the case of permutation representation, the 2-opt operator is a common way). To prove that a metaheuristic is suitable for a given optimization task, fitness landscape analysis can be a good alternative. In this case, we compare the main fitness landscape analysis methods with metaheuristics operators. One way of fitness landscape analysis is the trajectory-based sampling approach. In trajectory-based sampling, samples are taken from the search space. We start from a possible state point, then create the state of one or more possible neighbors, then select one of them, which is called the current state. In the literature, the following trajectory-based sampling techniques are investigated [18]: random walk, adaptive walk, reverse adaptive walk, neutral walk, reverse neutral walk, uphill-downhill walk.