## 3 Hybrid Algorithms Design
### Double-Layered Multi-Objective Particle Swarm Optimization Algorithm  
In the PSO model, each particle needs to continuously update the velocity and the position according to the _Pbest_ (the best position of a particle) and the _Gbest_ (the best position for all particles). For the single-objective problem, these two solutions are easy to choose, but for the multi-objective problem, it is difficult to choose the local optimal guide _Pbest_ and the global optimal guide _Gbest_. How to choose the _Pbest_ and the _Gbest_ to guide the moving of each particle in the search space is a critical issue, and it is very important for the premature convergence of the PSO algorithm and the diversity of solutions [27]. (1) Generation and Update of the Global Optimal Guide  
Choosing an appropriate global optimal solution, _Gbest_, to guide particles will greatly improve the quality of Pareto solutions and maintain the diversity of nondominated solutions. Firstly, this paper establishes an external archive, which mainly records the nondominated solutions found so far. At each iteration, the external archive is updated to ensure that the external archive is only nondominated solutions. Selection mechanism of _Gbest_: The following selection mechanism of _Gbest_ is established based on the external archive: If the number of nondominated solutions in the external archive is less than or equal to the population size of the PSO algorithm, _popsize_, all the solutions in the external archive are selected as the global optimal guides _Gbest_. Otherwise, the crowding distance-based selection algorithm is used to select _popsize_ solutions from the external archive as the global optimal guides _Gbest_. _Gbest_ assignment mechanism for each particle: Let \(NDnum\) be the number of solutions in _Gbest_, and the mechanism for assigning a global optimal guide to each particle is as follows:  
Step 1. Calculate the distance between individual particles and the solutions in _Gbest_. Step 2. Assign a guide to each particle. If \(NDnum\)=_popsize_, select a guide with the smallest distance for each particle as the individual global optimal guide (the global optimal guides for individual particles cannot be repeated). Otherwise, each global optimal guide is assigned to \(popsize/NDnum\) particles with the smallest distance as their global optimal guides. Figure 6 shows the _Gbest_ assignment mechanism for each particle of the bi-objective optimal problem. Figure 5: Effect analysis of LS algorithm. (2) Selection Mechanism of the Local Optimal Guide Pbest  
During the evolution process of the PSO algorithm, the local optimal guide _Pbest_ is selected for each particle according to the following method: Calculate the distance between each solution in the _Pbest_ set and the global optimal guide corresponding to the individual in _Gbest_, and select the individual optimal location with the smallest distance as the local optimal guide. Figure 7 visualizes this process. \(g_{1}\) is the global optimal guide assigned to an individual particle \(x_{1}\). \(p_{1},p_{2},p_{3}\) are the solutions in the _Pbest_ for \(x_{1}\). When selecting the local optimal guide for \(x_{1}\), \(p_{1}\) is selected because its position is closest to \(g_{1}\). (3) Updating Rules of Particle Velocity and Position  
In the standard form of PSO, the velocity vector for particle \(k\) is updated according to three other vectors that are the local optimal guide of the kth particle (_Pbest_), the current global optimal guide (_Gbest_), and the current position of the particle [28].