## 3 Improved Sparrow Search Algorithm
### Improved Sparrow Search Algorithm  
The above three strategies are introduced to improve the sparrow search algorithm so that the diversity of the population of SSA is improved, and the search method is improved to optimize the performance. The experimental results are presented in Section 5. Figure 4 is the flowsheet of the MISSA proposed in this paper, and the implementation procedure of MISSA is illustrated in Algorithm 1. ``` /* Initialization */
1. Set the maximum number of evolutions as \(T\);
2. Set the population number of sparrows as \(n\);
3. Set the warning value as \(ST\);
4. Set the number of finders as \(PD\);
5. Set the number of threatened sparrows as \(SD\);
6. Set the variation probability of t distribution as \(p\);
7. Initialize the population of sparrows using Equation (5);

/* Iterative optimization */
8. while (\(t<T\))
9. Calculate and rank the fitness values, and find the current best and worst individual;
10. \(ST=rand\) (1)
11. for \(I=1\): \(PD\)
12. Use Equation (7) to update the finder's position;
13. end for
14. for \(i=(PD+1)\): \(n\)
15. Use Equation (3) to update the follower's position;
16. end for
17. for \(i=1\):\(SD\)
18. Use Equation (4) to update the threatened sparrow's position;
19. end for
20. Generate a random number of \(m\) between (0,1);
21. If \(m<p\), use Equation (9) to conduct interference variation on sparrow individuals;
22. Calculate the current new position;
23. If the new position is better than before, update it;
24. \(t=t+1\);
25. end while
26. Output the best solution

## 4 Multi-Objective Antenna Design Method

### MISSA-BP Surrogate Model

The BP neural network is a multi-layer network developed from MLP, with forwarding propagation of inputs and backward propagation of obtained errors. Owing to BP neural

Figure 4: Flow chart of MISSA. network's low computational cost and strong generalization ability, it has been favored by many scholars [35]. The BP neural network comprises three layers: the input layer, the output layer, and the hidden layer. The number of nodes in each layer is designed according to the actual problem, and its brief structure is shown in Figure 5. It follows that the output of the network is calculated as follows:

\[I_{h}=f_{1}(\sum_{i=1}^{N}w_{hi}x_{i}+\theta_{h}) \tag{11}\]

\[y_{k}=f_{2}(\sum_{h=1}^{H}w_{kh}I_{h}+\theta_{k}) \tag{12}\]

where \(w_{hi}\) and \(w_{kh}\) indicate the weight connecting two different nodes; \(\theta_{h}\) and \(\theta_{k}\) correspond to the threshold of hidden layer and output layer; \(f_{1}\) and \(f_{2}\) are activation functions of corresponding layers of BP neural network. It can be seen from Equations (11) and (12) that some parameters of the neural network determine the desired result \(y_{k}\) that we want. Usually, using random parameters will make the neural network unstable, and it is laborious to achieve the ideal objective during optimization, resulting in insufficient accuracy of prediction. In this paper, MISSA is used to deal with the problems of the traditional neural network to improve its performance. MISSA will optimize the initial parameters, and the results will be output when the number of iterations is reached. The MISSA-BP surrogate model is obtained by substituting these optimized parameters into the neural network. ### Multi-Objective Antenna Design Scheme Based on MISSA-BP

In this paper, the antenna surrogate model based on MISSA-BP is linked to the multi-objective particle swarm optimization (MOPSO) [36] to design a multi-objective antenna. MOPSO has fewer parameters and faster convergence compared to other multi-objective algorithms. It can also obtain the set of solutions covering the whole search space and close to the actual Pareto front with as few computational resources as possible. Each particle in MOPSO consists of \(n\)-size parameters in the antenna to be optimized; that is, each particle represents the structure of an antenna. By iterative search, when the end of the condition is reached, the result can be output to get the antenna that meets the requirement.