## 5 Experiments and Results
### Evaluation and Testing  
The overall performance of the policy can be evaluated using the same quantitative metrics used by [PERSON] et al. (2018):  
* Success rate: The ratio of the number of robots that reach unique goals successfully during the episode over the total number of robots (the higher the better). * Extra time: The difference between the actual travel time of a robot to reach a goal position and the time it takes the robot to go to the same goal position in a straight line with maximum speed (the lower the better). * Extra distance: The difference between the actual distance a robot traveled to reach a goal position and the Euclidean distance between the robot's start position and the goal position (the lower the better). * Average speed: The average of the ratio of the speed of the robot team relative to the maximum speed of a robot during the episode. The metrics were measured over all the robots and all episodes during the evaluation to remove any effects of the variance in the goals' positions. We categorized the robot collisions into three categories. Category A is a collision with an obstacle; Category B is one robot colliding with another robot that reached a goal position; and finally, Category C is two robots colliding together while moving in free space. These three categories cover 100% of the collision events that might happen in our setting. This categorization can be used to analyze the policy's performance when it fails and what kinds of collisions tend to happen. Such an analysis can give us insight into the behavior that the policy was trained to perform. As our method combines MRTA and navigation, the categorization of collisions enabled us to develop a new metric that can measure the performance of our method and future methods that will be similar to it in performing MRTA. This metric can measure the MRTA performance by measuring the policy's failure in performing MRTA. We call this new metric the Task Allocation Index \(TAI\). It depends on the ratio of episodes ending with Category B collisions relative to the total number of episodes because Category B collisions are a clear failure of MRTA when two robots reach the same goal. \(TAI\) also depends on the ratio of goals whose information is input to the policy relative to the total number of goals in the environment, as this gives a higher score for the policies that perform task allocation on a bigger ratio of goals that are available in the environment. Equation (14) shows the formula for \(TAI\) (the higher the \(TAI\) value the better). For the policies that only perform navigation and do not perform MRTA at all, the \(TAI\) value will be zero. \[TAI=\frac{N_{i}-1}{N_{g}-1}\times\left(1-\frac{N_{CategoryB}}{N_{ Episodes}}\right) \tag{14}\]  
Figure 4: Average of the rewards gained per episode averaged over all the robots in the training wall time. where:  
* \(N_{i}\): number of goals whose information is input to the policy. * \(N_{g}\): total number of goals. * \(N_{CategoryB}\): number of episodes ended by Category B collisions. * \(N_{Episodes}\): total number of episodes. We compared our method with the methods proposed by [PERSON] et al. [12] and [PERSON] et al. [13]. The reason we are comparing with these works is the similarity in the objective of the studies in terms of performing the navigation of multi-robot systems using deep reinforcement learning. On the other hand, the difference between our method and the methods proposed by [PERSON] et al. [12] and [PERSON] et al. [13] is that our policy selects which goal position to drive the robot towards. For the methods of [PERSON] et al. [12] and [PERSON] et al. [13], the policy needs the goal position to be preselected per robot. Thus, our method performs MRTA and navigation form end-to-end, while the other methods only perform navigation from end-to-end. The second reason these methods were chosen is because the performance metrics were reported by both papers. [PERSON] et al [12] reported better performance than the NH-ORCAmethod over most test cases.