## 4 Fitness Landscape Analysis: Neighborhood Operators and Random Walk Metrics  
Fitness landscape [17] definition covers the set of possible states, the definition of the neighborhood and the fitness function. Many neighborhood operators have appeared in the literature, it depends on whether the problem is a discrete or continuous optimization problem. In the following, the different neighborhood operators of our discrete optimization problem are detailed. We can distinguish single operand and two-operand operators. In our Multi-Echelon VRP system, the operands are permutations representing the processing order of nodes. During the evaluation, each element of the permutation is lined up and assigned to a vehicle until a solution (vehicle capacity) is encountered. When the constraint is not met, we assign the nodes to a next vehicle. 2-opt [27] operator is widely used in the case of permutation representation of optimization problems (Figure 2). Figure 2: 2-opt operator. In the case of 2-opt, two edges are swapped. Swapping means the selection of a given section of the permutation and the reversing of this selected section. Within the frame of the fitness landscape analysis, we can analyze the performance of the mutation operators. For neighborhood generation the following operators can be used: cycle crossover, order crossover, partially matched crossover. Figure 3 demonstrates a cycle crossover operator [28]. The cycle crossover locates the cycles in the path. Both parents are permutation, so the cycle crossover takes advantage of the fact that the elements are listed twice (exactly once in both parents). In the example, the (1,9) pair are taken first. The first child collects the first element of the first parent, and the second child receives the first element of the second parent. Then, the second child collects the element denoted by 9. Therefore, we have to look for the item 9 in the first parent, and the first child puts this in the right place. The second child receives the element 1. The first child has already got the element 1, so the circle is closed. The next steps can be summarized in the followings: the first child gets the other elements from the second parent, and the second child gets the other elements from the first parent. Figure 4 shows the order crossover operator [28]. This operator uses a random fitting section. The position of the chosen fitting section must be the same for both parents. In the second child, the elements of the fitting section of the first parent are substituted with letter H (holes). Similarly, in the first child, the elements of the fitting section of the second parent are substituted with letter H (holes). The holes will be pushed up in the fitting section. After that step, the holes of the first child are substituted with the fitting section of the second parent, and the holes of the second child are substituted with the fitting section of the first parent. Figure 5 demonstrates the partially matched crossover operator [28]. In the case of this crossover, a random fitting section is chosen. The fitting section of both parents must be in the same position. Pairs are formed from the elements of the fitting section of the two parents. Here, in pairs, the exchange operation itself takes place independently of each other. Pairs denote the elements to be exchanged, but once the elements to be exchanged have been determined, the exchange in both genes and the other gene occurs independently of each other. In our example, we can identify the following pairs: (2,8), (6,5), (7,3), (5,2). We copy parents; these copies will be the new children of the parents. These pairs are changed in the first and then the second child. In our example, we change the 2 with 8, then the 6 with 5, then the 7 with 3 then the 5 with 2. Figure 3: Cycle crossover (CX) operator. Within the frame of this operation, it is also important to measure the distance between the potential solutions.