## 3 Results and Discussion
### Large-Scale Problems Solved by the Multi-Chromosome Coding-Based Genetic Algorithm  
For large-scale problems, multi-chromosome coding-based genetic algorithms are implemented. This kind of genetic algorithm not only has a smaller search space [30] but also facilitates the integer coding for the multi-robot task assignment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Index** & **Item** & **Ex-1** & **Ex-2** & **Ex-3** & **Ex-4** \\ \hline \multirow{3}{*}{SOC} & STA & 345 & 598.5 & 773 & 889.05 \\  & ILP model & 192 & 399 & 517.1 & 622.1 \\  & \(Gap_{SOC}\) & 44.3\% & 33.3\% & 33.1\% & 30\% \\ \hline \multirow{3}{*}{MS} & STA & 135 & 229 & 227 & 277.6 \\  & ILP model & 77 & 143 & 113 & 139 \\  & \(Gap_{MS}\) & 43\% & 37.6\% & 50.2\% & 49.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of single-task assignment (STA) and ILP-model-based task assignment for small-scale experiments. Figure 11: The cost for each robot’s task sequence under different task assignment methods. The multi-chromosome coding-based genetic algorithm also includes chromosome coding, population initialization, selection, crossover, mutation, and other processes. First, the integer coding process of the multi-chromosome genetic algorithm is described. The number of chromosomes of each individual in the population is equal to the number of robots. For a multi-robot task assignment problem containing \(n\) tasks and \(m\) robots, the task set is \(V_{n}\), and the robot set is \(V_{m}\). Each task represents a gene. Each individual in the population contains \(m\) chromosomes. The sequence of genes represents the order of tasks. After many generations of evolution, we pick out the best individuals from the population and assign them to robots. Then, 5 robots and 20 tasks are used to illustrate the multi-chromosome integer coding process, as shown in Figure 12. The specific generation process of a random initial population of size \(S\) is as follows: First, the order of the original task sequence needs to be disrupted, and then a robot is randomly chosen and assigned the first task of the new task sequence. It then randomly chooses a robot from all the robots and gives the second task to that robot. When all tasks are assigned, the individual of the population will be retained if every robot has at least one task. Otherwise, it will be discarded and re-assigned. The process is repeated until the population size reaches \(S\). The objective function (34) is used to evaluate the population's individual. When weight coefficient \((w_{1},w_{2})\) is equal to \((1,0)\) and \((0,1)\), respectively, we get the best individual of the population with the minimum sum-of-cost (SOC) and makespan (MS). We use the multi-chromosome coding-based genetic algorithm in the GEATPY library to solve the large-scale multi-robot task assignment problem. GEATPY is a high-performance genetic algorithm library, which can be found on [[https://github.com/geatpy-dev/geatpy](https://github.com/geatpy-dev/geatpy)]([https://github.com/geatpy-dev/geatpy](https://github.com/geatpy-dev/geatpy)) (accessed on 1 January 2022). We first test the eight kinds of multi-chromosome coding-based genetic algorithms on the small-scale experiment instances. Eight kinds of multi-chromosome coding-based genetic algorithms are shown in Table 3, including four types of single-objective and four types of multi-objective. Selection strategies of genetic algorithms include tournament, roulette wheel, and unconstrained random. We choose the partial matching crossover and reverse mutation operators that come with the GEATPY library for both crossover and mutation operators. We choose partial matching crossover and reverse mutation operators for both crossover and mutation operators. Regarding the crossover probability and mutation probability, we choose \(P_{c}=0.7\) and \(P_{m}=0.5\) in the single-objective genetic algorithm, respectively, and \(P_{c}=1\) and \(P_{m}=1\) in the multi-objective genetic algorithm.