# Dynamic Coal Flow-Based Energy Consumption Optimization of Scraper Conveyor
## 4 Adaptive Crossover Operator-Based Multi-Objective Binary Genetic Algorithm
### Adaptive Operator Selection  
In this study, a probability parameter \(p_{s}\) is introduced to select the crossover operator for offspring generation. Specifically, SBX is used with probability \(p_{s}\) and RSBX with probability \(1-p_{s}\), and \(p_{s}\) is updated at each generation. Initially set to 0.5, it assumes that SBX and RSBX have an equal ability to generate high-quality offspring solutions. The calculation of \(p_{s}\) is as follows:  
\[p_{s}=\frac{1}{1+exp(-kr)} \tag{76}\]  
\[r=(\frac{N_{0}^{(g)}+1}{N_{r}^{(g)}+N_{0}^{(g)}+2}-0.5)*\frac{g}{gmax} \tag{77}\]  
where \(k\) is set to a specific value, and \(p_{s}\) is defined as an S-shaped function ranging from 0 to 1. Additionally, the number of offspring solutions generated by SBX/RSBX and surviving from generation \(g\) is denoted as \(rNog\). This parameter helps introduce problem complexity to determine the likelihood of using SBX with \(p_{s}\). The complexity of the problem is typically related to the number of objectives \(M\) and decision variables \(D\), and thus both parameters are utilized. Since \(D\) is usually much larger than \(M\), the square root of \(D\) and \(M\) is used to balance these factors.  
The flow of the NSGA-II + ARSBX algorithm is shown in Figure 24, with the main steps as follows:  
(1) Initialization of population  
Randomly generate the initial population and calculate objective values as fitness. Initialize ARSBX parameters (rotation matrix, mean vector \(m\), crossover probability, distribution index, etc.).  
(2) Non-dominated sorting  
Sort the population based on objective values using Pareto dominance, assigning Pareto ranks to solutions. Lower ranks indicate higher-quality solutions.  
(3) Crowding distance calculation  
For each Pareto level, compute the crowding distance to measure solution distribution in the objective space. Larger distances help maintain diversity.  
(4) Selection operation  
Select \(N\) parents based on Pareto rank and crowding distance, prioritizing low-rank and large-distance solutions for quality and diversity.  
(5) Adaptive ARSBX crossover and mutation operations  
ARSBX crossover: Generate offspring using ARSBX, dynamically adjusting parameters based on population distribution. Perform crossover in the rotated space and reverse transform to the original space. Adjust crossover probability and distribution index \(\eta\) based on the evolution stage.  
Mutation: apply adaptive mutation to offspring to enhance diversity, with mutation probability adjusting during evolution.  
(6) Population update  
Merge the current population with offspring, then perform non-dominated sorting and crowding distance calculation. Select the top \(N\) solutions for the next generation.  
(7) ARSBX parameter update  
Update ARSBX parameters (rotation matrix V mean vector \(m\), crossover probability \(p\), and distribution index \(\eta\)) based on population distribution or fitness feedback.  
(8) Repetition of iteration  
Repeat steps 2-7 until the maximum generations \(g_{max}\) are reached or the termination criteria are met.  
(9) Output results  
Output the non-dominated solutions from the final population as the Pareto front and record ARSBX parameter changes for optimization analysis.