## 4 Bee Colony Algorithms
### Reaction-Dominated Case, Modified Bee Colony Algorithm  
The convergence of MBC, similar to the convergence of any BC algorithm, depends on the parameters of the algorithm. To illustrate this dependence, and at the same time to show that the algorithm converges for different sets of parameters, MBC is run with two sets of parameters here. Some parameters are the same for the both runs; these are: \(n=2\), \(d_{x}=6\), \(d_{y}=0.06\), \(d_{z}=20\), \(\delta=1\), \(abb=50\), \(\tau=5,\epsilon_{1}=10^{-8}\), \(\epsilon_{2}=1\). The parameters which differ in the two runs are  
1. \(m=3\), \(sb=20\), \(abb=50\), \(abp=20\);
2. \(m=5\), \(sb=200\), \(abb=50\), \(abp=40\).  
The subdomain in which the reaction parameters are sought is selected to be the same as in the deterministic case above. Namely, \(X\) axis -- parameter \(\mathrm{Da}_{a}\in[60,140]\), \(Y\) axis -- parameter \(\mathrm{Da}_{d}\in[0,2]\), \(Z\) axis -- parameter \(\mathrm{M}\in[800,1200]\).  
The five best identified points and the functional values there, together with the number of the iterations and the number of the direct solves, are summarized in Table 9. It is useful to also compare the relative error in computing the functional (the functional itself gives absolute error). The relative error is calculated as follows:  
\[\mathcal{E}_{rel}=\frac{\|c_{out}-\widehat{c_{out}}\|_{2}}{\|\widehat{c_{out}} \|_{2}}=\frac{\sqrt{\sum_{i=0}^{N_{t}}(c_{i}-\widehat{c}_{i})^{2}}}{\sqrt{ \sum_{i=0}^{N_{t}}\widehat{c}_{i}^{2}}}, \tag{23}\]  
Figure 9: Calculations for fixed: (a) \(\mathrm{Da}_{a}\), (b) \(\mathrm{Da}_{d}\), (c) \(\mathrm{M}\).  
where \(Nt\) is the number of the time intervals and \(\widehat{c_{out}}\) is the selected exact solution (imitating data from experiments). The relative error is listed in the third column in Table 9.  
It can be seen that for both MBC parameters sets, the [PERSON]'s isotherm parameters which give very low values of the functional are identified. The second set of MBC parameters gives better results at the cost of more expensive computations. The computed relative error is less than 1% for the 5 selected points from the first run of MBC, and less than 0.02% for the second run, respectively.  
Another important observation is that MBC is well adapted to the main goals of the paper, namely identifying not just the exact reaction parameters, but identifying the reaction parameter sets for which the residual takes small values.  
Note that in many industrial problems, much higher error is tolerated. If the stopping criteria \(\epsilon_{1}\) is increased from \(1e-8\) to \(1e-3\), and the following parameter set is used for MBC: \(n=1\), \(m=2\), \(d_{x}=6\), \(d_{y}=0.06\), \(d_{z}=20\), \(\delta=1\), \(sb=20\), \(abb=20\), \(abp=5\), \(\tau=3,\epsilon_{1}=10^{-3}\), \(\epsilon_{2}=1\), the following results are obtained; see Table 10.  
It is remarkable that not only is a reasonable admissible set of parameters identified, but the global minimum is also computed accurately. Namely, by comparing the identified global minimum (126.11, 1.287, 1151.449) with the exact parameters (100, 1, 1000), the relative error of 0.626% is obtained.