## 3 Our Proposed MQSOS Algorithm and Its Improvement on the DV-Hop Algorithm  
### MQSOS Algorithm  
SOS and OSOS algorithms have the following disadvantages: the group is less diverse, they easily fall into local optima, and they converge prematurely. We introduce multi-group ideas and inter-group communication strategies based on quantum behavior to improve the SOS algorithm's global optimization ability. The algorithm is shown in the following steps. Step 1: initialize the biological population size (_ps_), the range of the biological search space, the number of subgroups (_gs_) divided, and the inter-group communication step size (_cs_). Each subgroup undergoes three stages: mutualism, commensalism, and parasitism through Equations (7)-(14), and iterates independently. Step 2: all subgroups are independently iterated _gs_ times and then the best individuals in each subgroup are compared. The point with the best position state is chosen as the current global optimum. Then, the inter-group communication strategy based on quantum behavior is selected to update the position status of some individuals in each subgroup. Assuming that the population size is 100, divided into 4 subgroups, there are then 25 individuals in each subgroup. Five better-performingpoints are selected among all individuals in each sub-group, updated by Equation (24), and five points with poorer status are selected to update the position of the poorest creature using Equation (25). \[ \begin{cases}\mathbf{X}_{inew}=\mathbf{P}_{i}\pm\alpha\times|\mathbf{X}_{i}- \mathbf{C}_{i}|\times\ln\frac{1}{u}\\ \mathbf{X}_{i}=\mathbf{X}_{inew},\quad iff(\mathbf{X}_{inew})<f(\mathbf{X}_{i} )\end{cases}  \tag{24}\]  
The difference between Equations (24) and (25) is that when the point with better state in each subgroup is updated, a condition for judging whether the position state of the individual is superior is added, so that the individual with the better state always maintains the optimal value. In the case of updating a poorly-performing individual, there is no condition for judging whether the state is superior, so the poor points in the group are freed from the convergence process only toward one local optimum to find other advantageous positions. Therefore, each group can be explored to a wider range, and the diversity of the group can be enhanced, so that the group does not converge to the local optimal position in advance, resulting in poor performance of the algorithm in the global optimization performance, especially in the multi-objective function. Therefore, the convergence speed is improved by updating the individuals with better position states, and the diversity of the algorithm's global optimization is improved by updating the state of the poorly positioned particles. \[ \begin{cases}\mathbf{X}_{inew}&=\mathbf{P}_{i}\pm\alpha\times|\mathbf{X}_{i}- \mathbf{C}_{i}|\times\ln\frac{1}{u},\\ \mathbf{X}_{i}&=\mathbf{X}_{inew},\end{cases}  \tag{25}\]  
where \(\mathbf{X}_{i}\) is the position vector of the current individual and \(\mathbf{X}_{inew}\) is the updated position vector. \(\mathbf{C}_{i}\) is a vector of the average value of the \(i\)th individual's historical optimal position, with the calculation formula shown in Equation (26). In quantum mechanics, \(\mathbf{P}_{i}\) is a local attraction point, and the quantum tilts toward the local attraction point during free movement until the potential energy is 0, as shown in Equation (27). \(\alpha\) is the convergence coefficient that affects the convergence of the MQOS algorithm. \(\alpha\) can be found according to Equation (28). \(u\) is a random variable that is uniformly distributed from 0 to 1. \[\mathbf{C}_{i}=\frac{1}{M}\times\sum_{t=1}^{M}\mathbf{Pbest}_{i,t}, \tag{26}\]  
where \(M\) is the current number of iterations, and \(\mathbf{Pbest}_{i,t}\) is the position of the \(i\)th best position state of the biological individual after the \(t\)th iteration in the population. Through this formula, the optimal center position of each individual is iterated \(M\) times. \[