## 4 The Principle of the Balanced Cross-Layer Fuzzy Logic Routing Algorithm
### The Calculation of the Utility of Each Node
\begin{table}
\begin{tabular}{l c c c c c c c} \hline  & \(\mathbf{\beta_{1}}\) & \(\mathbf{\beta_{2}}\) & \(\mathbf{\beta_{3}}\) & \(\mathbf{\beta_{4}}\) & \(\mathbf{\beta_{5}}\) & \(\mathbf{\beta_{6}}\) & \(\mathbf{\beta_{7}}\) \\ \hline Node 1 & 0.0451 & 0.2238 & 0.2751 & 0.6273 & 0.571 & 0.8131 & 0.9861 \\ Node 2 & 0.7232 & 0.3736 & 0.2486 & 0.0216 & 0.1769 & 0.3833 & 0.0300 \\ Node 3 & 0.3474 & 0.0875 & 0.4516 & 0.9106 & 0.9574 & 0.6173 & 0.5357 \\ Node 4 & 0.6606 & 0.6401 & 0.2277 & 0.8006 & 0.2653 & 0.5755 & 0.0871 \\ Node 5 & 0.3839 & 0.1806 & 0.8044 & 0.7458 & 0.9246 & 0.5301 & 0.8021 \\ \hline \end{tabular}
\end{table} 
Table 6: The cross-layer parameters of multiple constraints. Figure 7: The result of the BCFL under multiple constraints. character of the cross-layer parameters in different nodes, which is shown in the histogram and the line graph of dispersion in Figure 7.

The Table 6 indicates that the parameters of node 5 are not always the highest among all of the nodes, but they are the most balanced, which can be found in the line graph of dispersion in Figure 7. The line graph of dispersion demonstrates that the node whose parameters dispersion is small will have high probability to be the next hop relay node, and _vice versa_, because the small dispersion leads to the balanced performance. In this scenario, as the number of cross-layer parameter is seven, so for the optimization based routing algorithm, the unknown variables and the functions are at least seven, then the calculation will be extremely complex; for the traditional fuzzy logic-based routing algorithm, when there are seven parameters in the system, the number of if-then rules is 2187 (assuming that the number of linguistic variables is three), which is unacceptable. However, for the BCFL, the number of if-then rules is constant and the same with the number of the linguistic variables (in BCFL, the number of if-then rules is seven), which is much simpler than the other two algorithms. This example also illustrates the huge advantage of the new fuzzy logic-based routing algorithm. ## 6 Conclusions

In this paper, we propose a new fuzzy logic-based routing algorithm (the BCFL routing algorithm). The innovations of this algorithm are: (1) we use the parameter dispersion rather than the absolute parameter value as the input of the fuzzy logic inference system, which can reduce the complexity of the algorithm observably; (2) the dispersion formula can guarantee that when the order of magnitude changes. the dispersion will remain unchanged; (3) the parameter whose dispersion is large will have small weight and whose dispersion is small will have large weight. According this, the BCFL can achieve some noble properties that the traditional routing algorithms cannot obtain:

* Simply, the if-then rules of the BCFL are very simple and they will stay constant with the increase in the constraints;
* Efficiently, the BCFL has excellent capability to deal with the multiple constraints without increasing the complexity;
* Adaptively, the BCFL can adapt to the changing of the network conditions and topology, especially in the underwater wireless sensor network in which the network conditions and topology change frequently;
* Balanced, the BCFL can get a more balanced solution than the traditional algorithms. The BCFL is compared with the optimization-based routing algorithm (the algorithm used in [9]) and the traditional fuzzy logic-based routing algorithm (the algorithm used in [19,20]) (in Section 5.1). In addition, we also discuss the performance of the BCFL under multiple constraints (in Section 5.2). The simulation results in Section 5.1 show that the BCFL and the traditional fuzzy logic-based routing algorithm have similar performance, and they all better than the optimization-based routing algorithm.