## 3 Results and Discussion
### Large-Scale Problems Solved by the Multi-Chromosome Coding-Based Genetic Algorithm
The iteration number of the genetic algorithm is 5000, and the population size is 100. Then, we compare this with the results of the ILP model using the Gurobi solver, respectively. The experiment instances with different weight coefficients and different genetic algorithms are repeated 10 times, respectively. In each small-scale experiment, we statistically obtain the best and average values of sum-of-cost (SOC) and makespan (MS) from 10 repetitions of the genetic algorithms. We use Gap to represent the difference of sum-of-cost (SOC) or makespan (MS) obtained by the multi-chromosome coding-based  
Figure 12: Multi-chromosome integer coding process of the genetic algorithm. genetic algorithm (GA) and the ILP model. The expression of Gap for sum-of-cost (SOC) and makespan (MS) is:  
\[Gap_{SOC}=(SOC_{\text{GA}}-SOC_{\text{ILP}})/SOC_{\text{GA}} \tag{37}\]  
\[Gap_{MS}=(MS_{\text{GA}}-MS_{\text{ILP}})/MS_{\text{GA}} \tag{38}\]  
The comparison results of the multi-chromosome coding-based genetic algorithm and the ILP model are shown in Table 4. In the same experiment, the Pareto frontier extreme values obtained from the multi-objective genetic algorithm are equal to or close to the results obtained by the Gurobi solver. The best value of makespan obtained by awGA, NSGA2, NSGA2-archive, and NSGA3 in experiment 4 is smaller than the result obtained by the Gurobi solver. This indicates that when the problem scale reaches 5 robots and 25 tasks, the best value obtained by some genetic algorithms is better than that of the Gurobi solver. To make it easier to compare the performance of each genetic algorithm in the four small-scale experiments, we use box-whisker plots to display the results. Based on Table 4, the box-whisker plots on the minimum sum-of-cost (SOC) and makespan (MS) are shown in Figures 13 and 14. It can be seen that as the number of robots and tasks increases, the Gap between the maximum and minimum results obtained by the same genetic algorithm becomes larger under the same number of iterations. The stability of the solution results gradually deteriorates. Among the eight tested genetic algorithms, the algorithms of EGA and NSGA3 perform relatively better. Therefore, we next choose these two algorithms to solve the large-scale problems. The large-scale experiment instance includes 5 robots and 100 tasks. We use the two algorithms to solve the experimental case 10 times. To further explore the influence of the number of iterations of the genetic algorithm on the solution results, we set the number of iterations to 5000 and 20,000, respectively. The other parameters are the same as those of the small-scale experiment. We use the Gap to represent the difference between sum-of-cost (SOC) or makespan (MS) when the number of iterations is 5000 and 20,000. The expression of the Gap for sum-of-cost (SOC) and makespan (MS) is:  
\[Gap_{SOC}=(SOC_{5000}-SOC_{20,000})/SOC_{5000} \tag{39}\]  
\[Gap_{MS}=(MS_{5000}-MS_{20,000})/MS_{5000} \tag{40}\]  
As shown in Table 5, when the number of iterations is 5000 and 20,000, the Gaps between the best value and the average values of the sum-of-cost (SOC) and makespan (MS) obtained by the EGA algorithm are all less than 4%. For the NSGA3 algorithm, the Gap is less than 1%. The results show that the two genetic algorithms have achieved good sub-optimal results when the number of iterations is 5000. They can be applied to other large-scale multi-robot task assignment problems of the intelligent warehouse. When the number of robots and tasks grows, we should appropriately increase the number of iterations of the genetic algorithm to obtain better results.