# The Color Mix Problem
## 4 Computational Experiments  
The hybrid genetic algorithm is coded by using C# programming language. For converting the analytical solutions to graphical images, however, Java programming language is used. The computational experiments have been conducted using an \(\times 86\) series personal computer equipped with an Intel (Intel Corp., Santa Clara, CA, USA) 2 GHz 4 cores processor, 8 GB RAM and the 64-bit MS Windows operating system. We experimented with the distance matrix \(B\) of size 64 (\(n=64\)), which is available at the public electronic library of the benchmark data instances of the QAP--QAPLIB ([[http://www.seas.upenn.edu/qaplib](http://www.seas.upenn.edu/qaplib)]([http://www.seas.upenn.edu/qaplib](http://www.seas.upenn.edu/qaplib)) (accessed on 30 June 2021), also see [19]). The grid size is \(8\times 8\) (\(n_{1}=n_{2}=8\)). The value of the parameter \(m\) varies between 2 and \(n-1\), while the parameter \(m_{1}\) varies between 1 and \(m-1\) (in fact, \(\frac{m}{2}\) is enough since solutions with \(\frac{m}{2}+1\), \(\frac{m}{2}+2\),... can be obtained by symmetry from solutions with \(\frac{m}{2}-1\), \(\frac{m}{2}-2\),... ). We recall that \(m_{2}=m-m_{1}\), \(m_{3}=n-m\), where \(m_{1}\), \(m_{2}\), \(m_{3}\) denote the numbers of dots (squares) of first, second and third color, respectively. As main algorithm performance criteria, we adopt the success rate (\(SR\)) of the algorithm and the average deviation (\(AD\)) of the objective function. The success rate is calculated by the following formula: \(SR=\frac{N_{\mathit{key}}}{R}\times 100[\%]\), where \(N_{\mathit{key}}\) is the total number of the best-known solutions (BKSs) over \(R=10\) independent runs of the algorithm. The average deviation is derived from this formula: \(AD=\frac{\overline{z}-BKV}{2 KV}\times 100[\%]\), where \(\overline{z}\) is the average objective function value over \(R\) runs of the algorithm and \(BKV\) denotes the best-known value of the objective function. At every run, the algorithm is applied to the given particular values of \(n\), \(m\) and \(m_{1}\). Each time, the genetic algorithm starts from a new, distinct random initial population. The current run of the algorithm is terminated if the total number of generations, \(G\), has been reached. The particular values of the control parameters of the genetic algorithm are shown in Table 1.  
The results of the conducted experiments are presented in Table 2. All these results were achieved without fine-tuning of the parameters--only using the prescribed, default values of the control parameters of the genetic algorithm. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Remarks** \\ \hline Population size, \(PS\) & 20 \\ Number of generations, \(G\) & 50 \\ Distance threshold, \(DT\) & max \(\{2,\,\lfloor 0.1m\rfloor\}\) & \(0<DT\leq m\) \\ Idle generations limit, \(idle\_gen\_limit\) & max \(\{3,\,\lfloor 0.05G\rfloor\}\) & \(0<idle\_gen\_limit\leq G\) \\ Number of iterations of hierarchical iterated tabu search, \(Q_{\textit{hiter}}\) & 100 \\ Number of iterations of tabu search, \(\tau\) & 30 \\ Idle iterations limit, \(idle\_iter\_limit\) & max \(\{3,\,\lfloor 0.2\tau\rfloor\}\) & \(0<idle\_iter\_limit\leq\tau\) \\ Tabu tenure, \(h\) & \(\lfloor 0.5n\rfloor\) & \(h>0\) \\ Randomization coefficient for tabu search, \(\alpha\) & 0.02 & \(0\leq\alpha<1\) \\ Mutation rate factor, \(\xi\) & 0.25 & \(0<\xi\leq 1\) \\ Number of runs of the algorithm, \(R\) & 10 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of the control parameters of the hybrid genetic algorithm used in the experiments. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c}  
In Table 2, we provide the following information: \(n\)--the total number of dots (squares), \(m\), \(m_{1}\), \(BKV\)--best known value of the objective function, \(BFV\)--best found objective function value, \(AD\)--average deviation of the objective function, \(SR\)--success rate, \(t\)--the average time (CPU time) per one run of the algorithm.