## 5 Experimental Results and Analyses  
The main objective of this experiment was to validate the effectiveness of the proposed multi-agent path-reinforcement learning based on deep deterministic policy gradients. In the validation process, different training environments and numbers of agents were analyzed and explored. In the experimental environment, the software learning framework used was PyTorch 3.7. Linux Ubuntu 20.04, deep learning framework paddle 2.3.0, hardware acceleration framework CUDA 11.1, and hardware acceleration library cuDNN 8 were also used. This study uses an RFL framework with centralized training and decentralized structure. Therefore, the complexity calculation mainly includes the number of computations required for training the data during the training cycle and the number of computations required for decentralized execution and agent interaction during application. This is because the complexity of centralized training is much higher than the amount of computation involved in decentralized execution. For multi-agent interactions, the effects of inter-agent interactions are approximated to the constant calculation time (CT) due to the locality of the individual agent using partially observed information. So, the complexity of this study will focus on the computation of each iteration during the centralized training. Based on the general principles of the previous literature [38; 39], the following presentations are deduced to be the key points to be considered for the calculation of the complexity in the centralized training and decentralized execution framework of the DDPG for this paper. 1. Number of neural network parameters (\(\theta\)): DDPG algorithms usually contain two main neural networks: the actor network and the policy network. Each network has its own parameters, mainly weights and biases, and the number of parameters in a neural network is an important factor in the complexity of the calculation. 2. Dimensions of state space and action space. During the centralized training phase, the states and actions of all agents may be combined into a global state and action vector. Therefore, the state and action dimensions will be extended to the sum of all agent state and action dimensions. 3. DDPG algorithms usually require a large amount of training data (states, actions, rewards and next states sampled from the environment) to train effectively. In addition, the number of iterations in the training process also directly affects the computational complexity. The approximate complexity formula (CF) used in this study is as follows:  
\[\text{CF}(\theta_{A},\,\theta_{C},\,d_{s},\,d_{u},\,N,\eta,\,B)=\eta\,\times \text{CT}\times(B\times((\theta_{A}\times N\times d_{s})+(\theta_{C}\times N \times(ds+d_{u})))) \tag{20}\]  
where the state dimension of each agent is \(d_{s}\), and the action dimension of each agent is \(d_{a}\). The total number of agents is N. In addition, \(\theta_{A}\) is the total number of the parameters of the actor network, and \(\theta_{C}\) is the total number of the parameters of the critic network. \(\eta\) is the total update number in the training process, and B is the size of the batch processed in each iteration of the training process. * **Case 1: Agent Reinforcement Learning--Decentralized Validation** A multi-agent reinforcement learning architecture using centralized training and decentralized execution is illustrated in Figure 5. To enhance multi-agent collaboration in observing the information center for agents, this paper used the concordance information as input to the neural network in the form of occupied lattice maps. The advantage of this approach is that different numbers of agents can be used for training and execution. This was utilized since during execution, it only needed to copy the trained policy network parameters to other agents. Firstly, this paper used three agents (agent 0, agent 1, and agent 2) for training during the centralized training process.