# A New Hybrid Positioning Method by Fusion of BDS and 5G Signal Using the Particle Swarm Method
## 3 BDS-5G Signal Fusion Location Algorithm  
### Multiobjective Particle Swarm Optimization (MOPSO)  
The particle swarm optimization algorithm solves complex optimization problems by simulating the group behavior of birds, such as foraging and migration. The particle swarm optimization algorithm randomly distributes a certain number of particles in the feasible area of the problem space, and each particle flies at a certain speed. During flight, the particles adjust their state by combining the current best position and the best position of the population to flee to a better area. The optimization process of the particle swarm optimization algorithm is similar to that of birds foraging. Using this information-sharing mechanism, the group's movement evolves from disorder to order and finally achieves the purpose of searching for the optimal solution. The definition of the multiobjective optimization problem is described as follows: We need to find the optimal location to minimize the overall cost function:  
\[minF(x,y,z)=min(f_{1}(x,y,z),f_{2}(x,y,z),\ldots,f_{n}(x,y,z)) \tag{6}\]  
where \(F(x,y,z)\) represents the overall cost function in the multiobjective optimization process and \(f_{k}(x,y,z)\) is the error of a single objective function. PSO initializes a uniformly distributed particle randomly within a specified range. All particles in the group have an image effect, each particle has its position and speed, and the particles are driven by the global optimal particle (\(g_{best}\)) and the individual optimal particle (\(p_{best}\)) to find the optimal solution. In the single objective particle swarm optimization algorithm, \(g_{best}\) and \(p_{best}\) can be uniquely determined by comparing their fitness values since there is only one objective function. In the multiobjective particle swarm optimization algorithm, for selection \(p_{best}\), the particle is better if the particle can minimize each cost function. If you cannot strictly compare who is better, randomly select one of them as the individual best. For choosing \(g_{best}\), many noninferior solutions can be used as the global optimal individual. However, in PSO, each particle can only choose one as the global optimal individual. Therefore, preserving these noninferior solutions and selecting a better noninferior solution is the core of the particle swarm optimization algorithm in dealing with multiobjective optimization problems. The velocity and position update equations of particles are as follows:  
\[v(t+1)=\omega v(t)+c_{1}r_{1}(p(t)-x(t))+c_{2}r_{2}(g(t)-x(t)) \tag{7}\]  
\[x(t+1)=x(t)+v(t+1) \tag{8}\]  
where \(\omega\) is the inertia weight, and its size controls the size of the search ability; \(c_{1}\) and \(c_{2}\) are the individual experience coefficient and social experience coefficient, respectively; \(r_{1}\) and \(r_{2}\) are random numbers between ranges \([0,\ 1]\); and \(p\) and \(g\) are the individual optimal solution and the global optimal solution, respectively. In each iteration of the multiobjective particle swarm optimization algorithm, many noninferiority solutions will be generated. Because they are not dominant, it is difficult to select the optimal particle, so we need to consider ensuring the diversity of the noninferior solutions. Based on the above reasons, the PSO algorithm selected by the optimal solution evaluation is used to search the noninferior optimal solution set of the multiobjective optimization problem. The algorithm initializes a particle swarm in the decision variable space. It guides the particles to fly in the decision variable space through each objective function in the multiobjective optimization problem to finally fall into the noninferior optimal solution set. Reflected in the objective function space, the particles will fall into the noninferior optimal objective domain.