## 3 Preliminary Concepts Introduction  
### Single-Objective Degressive Ary Number Encoded Genetic Algorithm  
Degressive any number encoded genetic algorithm (DGA) [18] is an improved genetic algorithm based on encoding. Set natural number \(N\), \(N\geq 2\). A genetic algorithm encoded by natural number \(N\) called \(N\)-ary encoded genetic algorithm (NGA). NGA uses a \(N\)-valued coded symbol set \(\{0,1,2,\ldots,N-1\}\) and each individual is a \(N\)-ary symbol string, while common binary genetic algorithm uses a binary coded symbol set \(\{0,1\}\). The calculation formula of \(N\)-ary encoding is shown in Equation (1). \[(A_{p}A_{p-1}\cdots A_{0})_{N}=A_{p}\times N^{p}+A_{p-1}\times N^{p-1}+\cdots +A_{1}\times N^{1}+A_{0}\times N^{0}=B_{p^{*}}B_{p^{*}-1}\cdots B_{0} \tag{1}\]  
where \(A_{p}A_{p-1}\cdots A_{0}\) is an unencoded string, \(B_{p^{*}}B_{p^{*}-1}\cdots B_{0}\) is the \(N\)-ary encoded string, \(N\) is theary number of the coding method, \(p\) is the length of the unencoded string, \(p^{*}\) is the length of the encoded string. The examples of the individuals in NGA are shown in Figure 1.  
The corresponding encoding method called \(N\)-ary encoding and the number \(N\) is called the any number. Basic binary encoded genetic algorithm [19], octal encoded genetic algorithm [20] and decimal encoded genetic algorithm [21] all belong to NGA. In the single-objective optimization problem, high any number encoded GA can search for the optimal area more quickly (faster convergence speed) at the beginning of the evolution; low any number encoded GA can ensure the error between the result and the theoretical optimal solution is smaller in the late evolution [22]. Moreover, a high any number encoded GA can better maintain the diversity of the population to jump out of the local optimal [23]. Based on NGA, DGA was proposed to combine the advantages of high any number encoding and low any number encoding. Since different any number has different effects on different stages of evolution, the any number can be selected for each generation to optimize the evolutionary effects. Therefore, the core idea of DGA is that any number \(N\) should be reduced to 2 from a certain initial number with the increase of evolutionary generation according to a fixed function. The process of DGA for a maximum problem is shown in Algorithm 1. ```
1 In :Size of population \(n\), Initial any number \(N(0)\), Any number change rule \(N(k)\), Termination conditions Out:Global optimal solution \(x_{k}^{*}\)
2 Generate the initial population \(\mathbf{x_{0}}\) of size \(n\);
3\(N\gets N(0)\);
4\(k\gets 1\);
5 whileNot meet the termination conditionsdo
6 Encode all the individuals by \(N\);
7\(\mathbf{x_{k}}\leftarrow Select,\) Crossover, Mutate;
8\(\mathbf{F(k)}\gets Fitness(\mathbf{x_{k}})\);
9\(x_{k}^{*}\gets max[Findfmax(\mathbf{x_{k}},\mathbf{F(k)}),x_{k}^{*}]\);
10\(k\gets k+1\);
11\(N\gets N(k)\);
12
13 end while
```  
**Algorithm 1**Process of DGA  
In Algorithm 1, _Select_, _Crossover_ and _Mutate_ represent the genetic operators which are used to obtain a new population, _Fitness_ is used to calculate the fitness function for the population, \(Findfmax\) is used to find the optimal solution in the population. Compared with the normal GA, the DGA re-encodes all the individuals in each generation. In addition, the any number \(N\) for the next generation is calculated according to the change rule \(N(k)\). Because DGA combines the advantages of high any number encoding and low any number encoding, it has improvements on both convergence speed and optimization effect. On this basis, many experiments have been carried out to examine the influence of the any number's function. Through the analysis and summary of the law between the change of the any  
Figure 1: The examples of \(N\)-ary individuals. number and the change of the fitness function in DGA, we proposed an Adaptive Degressive Ary Number Encoded Genetic Algorithm (ADGA) [11]. Different from DGA, the change rule of the any number in ADGA is adaptive.