## 4 Fitness Landscape Analysis: Neighborhood Operators and Random Walk Metrics
In this paper, three techniques are presented, which are the followings: fitness distance, hamming distance, and basic swap sequence distance. Fitness distance is the absolute value of the difference in fitness value between the two solutions. The hamming distance [29] is the number of differences between the representations of the two solutions. The basic swap sequence distance [30] is the number of swaps between the representations of the two solutions. Basin of attraction are solutions, which convergence to a certain optimum [18]. The goal is to get out of this as easily as possible, because when an algorithm gets to a local optimum, if it has a hard time getting out of it, it has less chance of exploring space, thus reaching the global optimum. Our fitness landscape analysis uses trajectory-based sampling, in which a sample is taken from the search space. In this technique first, a possible state is selected, it will be first the actual state. With the neighborhood operator, we form one or more (in our case one) neighbor states, one of which will be the new current state. In the case of a random walk [18] sampling technique, the random neighbor state of the actual sate will be the actual sate. The neighbor can be better or worse than the actual state. We also prepare an information theory analysis of the fitness landscape, which quantifies the frequency of relative sequences between neighbors. The following important metrics can be evaluated [18]:  
* Partial Information Content: The fitness value of each of the next neighbors is subtracted. It can be a positive number (ascending) that means 1, a negative number (descending), it means \(-1\), or zero (not changing), it means 0. Therefore, the values \(-1\), 0, 1 are indicator values. They show whether the fitness of the neighbor is increasing, decreasing, or remaining of the same. The sequence of the indicator values is given with \(s_{original}\). This procedure removes zero (non-variable) values from the sequence and then leaves only the variable slopes (i.e., descending values after ascending, as  
Figure 4: Order crossover (OX) operator. Figure 5: Partially Matched crossover (PMX) operator. cending values after descending), indicated with \(s_{nonzero}\). The value thus obtained is divided by the length of the original sequence. \[M(\varepsilon)\ =\ \frac{s_{nonzero}}{s_{original}}\] (1) A value of 1 is rugged while a value of 0 is flat. * Expected Partial Information Content: this measure requires the partial information content value. The expected partial information content can be calculated with the following way: \[E[M(\varepsilon)]\ =\ \frac{n\ \times\ M(\varepsilon)}{2}\] (2) In the formula, \(n\) is the length of the \(s_{original}\), \(M(\varepsilon)\) is the partial information content. Expected partial content is not a probability variable, it shows the shape of the fitness landscape. * Information Stability: the fitness values of each of the next neighbors are subtracted from each other. The greatest value obtained will be information stability. * Entropy: the fitness values of each of the next neighbors are subtracted. It can be a positive number (ascending) that means 1, a negative number (descending), it means \(-1\), or zero (not changing), it means 0. This gives a sequence. Then, the entropy (entropy) is calculated with the following formula, where \(P_{i}\) is the relative frequency of the symbols: \[-\sum_{i}P_{i}lnP_{i}\] (3)
* Information Content: the fitness values of each neighbor solutions are subtracted each other. It can be a positive number (ascending) that means 1, a negative number (descending), it means \(-1\), or zero (not changing), it means 0. This gives a sequence. Then, the information content can be calculated with the following formula, where \(P_{[pq]}\) denotes the relative frequency of the consecutive symbols, where \(p,q\in\{\ 1,\ 0\,-1\}\): \[H(\varepsilon)-\sum_{p\
eq q}P_{[pq]}\ \ \times\ log_{6}P_{[pq]}\] (4)
* Density Basin Information: the same as information content, but the basis of the logarithm is different.