# A Novel Swarm Optimisation Algorithm Based on a Mixed-Distribution Model
## 3 Population Distribution Model Based on a Mixed Distribution
#### 4.4.2 Analysis of Results  
* 1. Unimodal Functions Optimisation Analysis We can see from the above statistics and convergence curves (Figure 2) that NBOA-3 obtained the best solution in most of the unimodal function optimisations. Our conclusions are that:  
1. Because the population sizes of the four algorithms are same, the highest number of offspring dominated by parent beans occurred in algorithm NBOA-3. For example, the No. 1 parent bean in algorithm NBOA-3 dominated 70% of individual offspring and distributed them according to its wish. However, the No. 1 parent bean in algorithm NBOA-6 only dominated 30% of the offspring. Therefore, algorithm NBOA-3 can approximate the optimal solution (or the local optimal solution) more quickly in theory. This is largely consistent with the experimental results. Hence, better results appear to be given by algorithm NBOA-3.
2. Considering unimodal function optimisation, the mutation mechanism of individuals follows the method: \[\text{pop(rndi,:)}=\text{(rands(1,nn))}\times\text{pop(ceil(rand $\times$ nn))},\] (3) The random sowing method strongly increases the global optimisation ability in later periods of evolution. It makes the excellent algorithm even better.
3. The model based on the binomial distribution can effectively increase the optimisation ability of the algorithm. A decreasingly steep slope is evident in the comparative figures of algorithm convergence performance (Figure 2). It is the location of the steep slope that starts using the binomial distribution model in BOA. It is evident that the model based on the binomial distribution enhances the performance of NBOA-3 and NBOA-6.
* 2. Multimodal Functions Optimisation Analysis In most of the multimodal-function optimisations, NBOA-6 obtained the best solutions. From the algorithm experiment results and the and convergence curves (Figure 3) shown above, we can analyse the results as follows:  
1. Because the distances (Euclidean distances are used in the experiments) between the parent beans need to meet the threshold set before the experiments, the more parent beans there are, the more advantageously parent beans can explore the problem domain space. That is to say, there are more opportunities to quickly find better problem areas that effectively enhance the algorithm's ability to achieve global optimisation.
2. For multimodal function optimisation, the mechanism of individual bean variation adopts the following method: \[\text{pop(NumPop,:)}=\text{(rands(1,nn))}\times\text{(popmax}-\text{(popmax $-$ popmin)}\times\text{rand)}\] (4) This method simulates the phenomenon of accidental seed migration in nature and further enhances the ability to achieve global optimisation.
3. As mentioned above, because the model based on the negative binomial distribution is used in algorithm NBOA-6, its optimisation ability is better than that of algorithm BOA-6.  
Figure 2: \(Cont\).  
Figure 3: _Cont._  
Figure 2: The convergence curves of unimodal functions: The curves of BOA-6 are black, the curves of NBOA-6 are blue, the curves of NBOA-3 are red, and the curves of PSO are purple. All the curves are average convergence curves for 50 computational iterations, and in order to show the curves more clearly, the coordinates were changed by logarithmic processing.