# Multi-Objective Optimization Study of Annular Fluid Flow Structure in Cordless Core Drilling Tools
## 2 Structural Design and Principles of Hydraulic Lifting Cordless Coring Drilling Tool
### Principles of the Improved NSGA-II Algorithm
The formula for the improved crowding distance, \(d_{i}\), is:

\[d_{i}=\sum_{m=1}^{M}\left(f_{m}^{(i+1)}-f_{m}^{(i-1)}\right)+\lambda\sum_{n=1}^ {N}\left(x_{n}^{(i+1)}-x_{n}^{(i-1)}\right) \tag{5}\]

where \(M\) is the number of objective functions, \(f_{m}\) is the \(m\)-th objective function, \(\lambda\) is the parameter of the trade-off between the decision variable and the objective function, \(x_{n}\) is the decision variable, and \(N\) is the number of decision variables. Input: Front F Output: Crowding distances D

1: Initialize D with zeros for all individuals in F

2: for each objective function m do

2:1: Sort the individuals in F based on objective m

2:2: Set the crowding distance of boundary points to infinity

2:3: for i = 2 to \(|F|-1\) do

2:3.1: Calculate the distance using the m-th objective:

D[i] +=(f_m[i+1]-f_m[i-1])/(f_m_max - f_m_min)

3: for each decision variable j do

3:1: Sort the individuals in F based on decision variable j

3:2: for i = 2 to \(|F|-1\) do

3:2.1: Calculate the distance using the j-th decision variable:

D[i] +=(x_j[i+1]-x_j[i-1])/(x_j_max - x_j_min)

4: Return D

Step 4: Selection operation: binary tournament. Selecting parent individuals based on Pareto rank and crowding distance. Comparing two randomly selected individuals and selecting the one with a low Pareto rank. If the ranks are the same, the one with the highest crowding distance is selected. Step 5: Crossover and Mutation Operations. Simulated Binary Crossover (SBX) and Polynomial Mutation are widely used in genetic algorithms. The crossover and mutation parameters \(\eta_{c}\) and \(\eta_{m}\) control the distribution of offspring. Where the crossover formula is:

\[c_{1}=0.5[(1+\beta)\cdot p_{1}+(1-\beta)\cdot p_{2}] \tag{6}\]

\[c_{2}=0.5[(1-\beta)\cdot p_{1}+(1+\beta)\cdot p_{2}] \tag{7}\]

where \(c_{1}\) is the first child individual, \(c_{2}\) is the second child individual, \(p_{1}\) is the first parent individual, \(p_{2}\) is the second parent individual. \(\beta\) is a parameter that controls the crossover operation, which is used to regulate the generation of new individuals. It is calculated based on the crossover probability and crossover parameter \(\eta_{c}\). The mutation formula is:

\[c^{\prime}=c+\delta \tag{8}\]

where \(c\prime\) is the individual after the mutation, \(c\) is the original individual before the mutation, and \(\delta\) is used as the offset of the mutation. Where the uniformly distributed random number in the interval \([0,1]\)\(u<0.5\),

\[\delta=(2u)^{\frac{1}{\eta_{m}+1}}-1 \tag{9}\]When \(\geq 0.5\),

\[\delta=1-(2(1-u))^{\frac{1}{\eta_{m}+1}} \tag{10}\]

The improved crossover \(\eta_{c}\) and variation parameters \(\eta_{m}\) can be dynamically adjusted to balance exploration and utilization according to the diversity of populations. \[\eta_{c}(t)=\eta_{c,\max}-(\eta_{c,\max}-\eta_{c,\min})\cdot\frac{t}{T} \tag{11}\]

where \(t\) denotes the current number of iterations and \(T\) denotes the maximum number of iterations of the Algorithm 3.