## 2 Description of the Proposed Approach
### Fuzzy Logic Reasoning Selection of Candidate Set  
The outer layer fuzzy reasoning system comprehensively considers three parameters: node residual energy \(RE\), which affects the final network lifetime; link quality index \(LQI\), which affects the final packet delivery rate; and effective advance distance \(AD\) of nodes, which affects the multihop forwarding efficiency, to participate in the outer layer fuzzy reasoning. That is to say, the \(LQI\) represents the communication reliability of the node's link to the next hop and directly impacts the packet delivery ratio. Prioritizing nodes with higher \(LQI\) ensures stable and successful data transmission. The \(RE\) level reflects the remaining battery capacity of the node. Giving higher forwarding priority to nodes with greater energy reserves contributes to balanced energy consumption across the network and helps extend the overall network lifetime. The \(AD\) level is a spatial metric that promotes efficient data propagation. Nodes with larger forward progress are generally closer to the destination, reducing the number of required hops and potentially minimizing end-to-end transmission delay. This metric implicitly promotes low-latency routing, which is especially critical in delay-sensitive underwater applications. The schematic diagram of the outer layer fuzzy reasoning is shown in Figure 6.  
Figure 5: LQI acquisition process using inner layer fuzzy reasoning. The specific process of building a fuzzy logic reasoning system to complete specific design tasks can be roughly summarized into four steps. First, clarify the design problem, the input and output of the system, and their possible range of variation. The outer fuzzy inference system takes into account three parameters: the link quality index \(LQI\) that affects the final data packet delivery rate; the nodes' remaining energy \(RE\) that affects the final network lifetime; and the effective advance distance \(AD\) of the nodes that affect the multihop forwarding efficiency to participate in the outer layer fuzzy inference. To make the weights of multiple fuzzy inputs more balanced and improve the performance of the model, the original fuzzy input data are first linearly mapped to a specified range. Given an original data point \(x\), we aim to map it to a new range \([a,b]\), which can be linearly mapped according to the following formula:  
\[x_{normalize}=a+\frac{(x-x_{\text{min}})\times(b-a)}{x_{\text{max}}-x_{\text{ min}}}. \tag{22}\]  
We plan to map the original fuzzy input to a new range \([0,10]\), so we take \(a=0,b=10\). Next, we determine the number of fuzzy subsets for the input and output variables. The number of fuzzy subsets is related to the granularity of the fuzzy inference. Based on the types of fuzzy subsets, the corresponding membership functions are established. In this case, each of the three fuzzy input variables is assigned three fuzzy subsets. The respective membership functions are then configured accordingly. The detailed configuration is presented in Table 1.  
Table 1 shows the membership functions for the fuzzy subset of the four variables. It is worth mentioning that the determination of the membership function for fuzzy variables corresponding to fuzzy subsets requires comprehensive consideration. Generally, fuzzy sets can be classified into three types: small-type, medium-type, and large-type. For example, \"low node energy\" represents a small-type fuzzy set, where the membership degree increases as the energy value decreases, indicating a worse node energy state. Conversely, \"high node energy\" represents a large-type fuzzy set, where the membership degree increases with higher energy values, indicating a better node energy state. \"Moderate node energy\" represents a medium-type fuzzy set, where the membership degree is highest when the energy value falls within a certain intermediate range. In summary, it is important to consider the relationship between \"elements\" and \"membership degree\", which implies the monotonicity of the membership function.