# Genetic Algorithm for the Retailers' Shelf Space Allocation Profit Maximization Problem
## 6 Computational Experimental Results
### Genetic Algorithm vs. Cplex
GA\({}^{-}\) with Tournament was the fastest, and the solution on average in 0.56 min. The minimum time for all GA\({}^{-}\) was 0.03-0.04 min. The maximum time was approximately 2.5 min for both GA\({}^{-}\) with Roulette Wheel and GA\({}^{-}\) with Tournament, and 5.71 min for GA\({}^{-}\) with Two Rankings. All GA\({}^{-}\) on average found the solution in less than 1 min. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Best Solutions of GA\({}^{-}\), GA\({}^{+}\)**} & \multicolumn{2}{c}{**Average Solutions of GA\({}^{-}\), GA\({}^{+}\)**} \\  & & **Two Rankings** & **Roulette Wheel** & **Tournament** & **Two Rankings** & **Roulette Wheel** & **Tournament** \\ \hline
**Products** & **Shelf Width** & **GA\({}^{+}\) Is Better** & **GA\({}^{+}\) Is Better** & **GA\({}^{+}\) Is Better** & **GA\({}^{+}\) Is Better** & **GA\({}^{+}\) Is Better** & **GA\({}^{+}\) Is Better** \\ \hline \multirow{4}{*}{10} & \(100\) & & & 0.19\% & 1.98\% & 0.39\% \\  & \(200\) & & & 1.13\% & 4.82\% & 1.16\% \\  & \(250\) & & & 0.62\% & 1.25\% & 0.89\% \\  & \(375\) & & & 0.42\% & 4.77\% & 0.76\% \\  & \(500\) & & & 0.26\% & 0.57\% & 1.71\% & 0.42\% \\ \hline \multirow{4}{*}{15} & \(100\) & 0.19\% & 0.19\% & 0.19\% & 1.62\% & 10.14\% & 1.06\% \\  & \(200\) & 0.11\% & 0.11\% & 0.11\% & 0.67\% & 1.54\% & 0.90\% \\ \cline{1-1}  & \(250\) & 0.18\% & 0.09\% & 0.09\% & 0.37\% & 0.86\% & 0.59\% \\ \cline{1-1}  & \(375\) & 0.30\% & 0.13\% & 0.07\% & 1.28\% & 2.25\% & 1.01\% \\ \cline{1-1}  & \(500\) & 0.22\% & 0.22\% & 0.17\% & 0.40\% & 1.16\% & 0.44\% \\ \hline \multirow{4}{*}{20} & \(100\) & 1.58\% & 1.58\% & 1.58\% & 28.10\% & 65.80\% & 24.98\% \\  & \(200\) & 0.21\% & 0.32\% & 0.32\% & 0.55\% & 0.87\% & 0.45\% \\ \cline{1-1}  & \(250\) & 0.36\% & 0.54\% & 0.18\% & 0.24\% & 0.91\% & 0.28\% \\ \cline{1-1}  & \(375\) & 0.10\% & 0.03\% & 0.20\% & 0.29\% & 0.63\% & 0.47\% \\ \cline{1-1}  & \(500\) & 0.24\% & 0.26\% & 0.50\% & 0.58\% & 1.37\% & 0.88\% \\ \hline \multirow{4}{*}{25} & \(200\) & 0.27\% & 0.27\% & 0.27\% & 1.23\% & 3.08\% & 1.24\% \\ \cline{1-1}  & \(250\) & 0.04\% & 0.09\% & 0.09\% & 0.53\% & 3.50\% & 0.79\% \\ \cline{1-1}  & \(375\) & 0.20\% & 0.13\% & 0.20\% & 0.23\% & 0.40\% & 0.58\% \\ \cline{1-1}  & \(500\) & 0.19\% & 0.05\% & 0.25\% & 0.36\% & 0.19\% & 0.55\% \\ \hline \multirow{4}{*}{50} & \(250\) & 0.05\% & 0.64\% & 0.64\% & 0.91\% & 2.35\% & 1.87\% \\ \cline{1-1}  & \(375\) & 0.14\% & 0.03\% & 0.03\% & 0.38\% & 1.55\% & 0.58\% \\ \cline{1-1}  & \(500\) & 0.30\% & 0.34\% & 0.30\% & 0.85\% & 2.42\% & 1.25\% \\ \hline \multirow{4}{*}{
\begin{tabular}{c} \end{tabular} } & Min & 0.04\% & 0.03\% & 0.03\% & 0.19\% & 0.19\% & 0.28\% \\ \cline{1-1}  & Avg & 0.28\% & 0.30\% & 0.30\% & 1.89\% & 5.16\% & 1.89\% \\ \cline{1-1}  & Max & 1.58\% & 1.58\% & 1.58\% & 28.10\% & 65.80\% & 24.98\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Quality of the best/average solution of GA\({}^{-}\) compared to the best/average solution of GA\({}^{+}\). Figure 9: Average time for GA\({}^{+}\) in [s]. Figure 11 demonstrates the average time for GA\({}^{-}\). All GAs\({}^{+}\) found the solution, on average, in less than 1 min. The fastest solution was found in less than 1 s and the slowest in 6.92 min by GA\({}^{+}\) with Two Rankings, 5.17 min by GA\({}^{+}\) with Roulette Wheel, and 4.33 min by GA\({}^{+}\) with Tournament. It is significantly faster than GA\({}^{+}\). Table 6 presents the standard deviation of GA\({}^{-}\) solutions compared to the standard deviation of GA\({}^{+}\) solutions. It can be observed that GA\({}^{+}\) finds a more stable solution set. Figure 11: Average time for GA\({}^{-}\) in [s]. Figure 10: The computational time for the best solution of GA\({}^{-}\) in [s]. It can improve any solution, and it is better than GA\({}^{-}\) by approximately 60-65%. The minimum and maximum values where GA\({}^{+}\) is less varied are 23-32% and 95-97%. This means that in the best cases, GA\({}^{+}\) is almost twice as good as GA\({}^{-}\), and generally gives almost equal solutions.