## 3 Results and Discussion
### Small-Scale Problems Solved by Gurobi  
This section uses the Gurobi solver to solve the small-scale multi-robot task assignment problem. First, we use the linear weighting method to give different weight coefficients \(w_{1}\) and \(w_{2}\) to the objective functions (19) and (20), respectively, to form the new objective function \(f_{3}\):  
\[\min\quad f_{3}=w_{1}\times f_{1}+w_{2}\times f_{2} \tag{34}\]  
when the weight coefficient \((w_{1},w_{2})\) is equal to \((1,0)\) and \((0,1)\), we get the objective function with the minimum sum-of-cost (SOC) and makespan (MS). When \(w_{1},w_{2}\in(0,1)\), sum-of-cost (SOC) and makespan (MS) are both considered in the new objective function. Then, we use the Gurobi solver to solve the new ILP model. We set the weight coefficients \((w_{1},w_{2})\) to \((1,0)\) and \((0,1)\) in four small-scale experiments, respectively. We set the maximum CPU runtime of the Gorubi solver to 900 s.  
In order to verify the theoretical analysis of the computational time complexity in Section 2.5, we perform a statistical analysis of the CPU running time of the Gurobi solver in four small-scale experiments. Table 1 shows that as the scope of the experiment grows, the minimum SOC and minimum MS-based CPU runtimes grow longer. Furthermore, the CPU runtime based on minimum makespan (MS) is longer than that based on minimal sum-of-cost (SOC) in the same experiment. Figure 6 compares the CPU runtime based on the minimum SOC and the computational time complexity of the same experiment. The left and right vertical axes are the logarithmic values of CPU runtime and computational time complexity based on the minimum SOC, respectively. We can see that the growth trend of the above two curves is consistent with the increase of the problem size, which further verifies that our theoretical analysis of the computational time complexity is correct. Figures 7-10 represent the task sequence of each robot in the small-scale experiment instances, respectively. Because the solution result based on the minimum sum-of-cost (SOC) is only related to the _TAC_ between tasks and has nothing to do with the _TOC_ of the task itself, the task nodes appear to cluster in Figures 7a-10a. The solution results obtained based on the minimum makespan (MS) are not only related to _TAC_ but also _TOC_, so the task sequences are chaotic without the evident cluster phenomenon in Figures 7b-10b. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Item** & **Ex-1** & **Ex-2** & **Ex-3** & **Ex-4** \\ \hline CPUtime\({}_{SOC}\) & 0.18 & 3.9 & 39.97 & 208.45 \\ CPUtime\({}_{MS}\) & 1.08 & 362.08 & 900.57 1  & 900.4 1  \\ \(O(nt)\) & 10! & 15! & 20! & 25! \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The computer’s CPU runtime and the computational time complexity. Figure 8: Task sequences in experiment 2, including 3 robots and 15 tasks. Figure 6: The comparison of the CPU runtime based on the minimum SOC and the computational time complexity. Figure 7: Task sequences in experiment 1, including 3 robots and 10 tasks. To explore the practical significance of the task assignment method in this paper, we compare it with single-task assignment (STA). Most of the existing intelligent warehouses adopt single-task assignment (STA). The approach based on the nearest distance between the start point of the task and the idle robot is more widely adopted. We now simulate it using the following method. First, we assign a task to the robot that is closest to the task starting position. Then, when a robot completes the task, we give the next task to the robot based on the nearest distance. When all tasks within the time window are given to all robots, we end the simulation process. Based on the above four small-scale experiments, we compare the task assignment method based on the integer programming (ILP) model with the single-task assignment based on the nearest distance. In the nearest-distance-based single-task assignment, after the robot completes the task, it selects the nest task with the nearest distance to itself each time.