## 3 Experiments and Results  
The regression data selected for the experiments are related to the quality of the delivery system network of a soft drink company [9]. The dependent variable in the data is the time needed by an employee to restack an automatic soft drink vending machine. This time is called the total service time and is measured in minutes. There are two independent variables: the first variable is the number of stocked items, and the second is the distance an employee walks, measured in feet. The dataset contains 25 observations. Table 1 illustrates this dataset. When the ordinary least-square (OLS) regression is run on the dataset, it results in parameter values of \(\beta_{1}^{\prime}\) = 1.61591, \(\beta_{2}^{\prime}\) = 0.0143848, and \(\beta_{3}^{\prime}\) = 2.34123. The overall regression model is significant at a 95% statistical confidence level, and the adjusted R-squared value is 95.4%. The root-mean-squared (RMS) error for the OLS model is 3.05766. From \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Observation** & **Total Items** & **Total Distance** & **Total Time** \\ \hline \(1\) & 7 & 560 & 16.68 \\ \hline \(2\) & 3 & 220 & 11.5 \\ \hline \(3\) & 3 & 340 & 12.03 \\ \hline \(4\) & 4 & 80 & 14.88 \\ \hline \(5\) & 6 & 150 & 13.75 \\ \hline \(6\) & 7 & 330 & 18.11 \\ \hline \(7\) & 2 & 110 & 8 \\ \hline \(8\) & 7 & 210 & 17.83 \\ \hline \(9\) & 30 & 1460 & 79.24 \\ \hline \(10\) & 5 & 605 & 21.5 \\ \hline \(11\) & 16 & 688 & 40.33 \\ \hline \(12\) & 10 & 215 & 21 \\ \hline \(13\) & 4 & 255 & 13.50 \\ \hline \(14\) & 6 & 462 & 19.75 \\ \hline \(15\) & 9 & 448 & 24 \\ \hline \(16\) & 10 & 776 & 29 \\ \hline \(17\) & 6 & 200 & 15.35 \\ \hline \(18\) & 7 & 132 & 19 \\ \hline \(19\) & 3 & 36 & 9.5 \\ \hline \(20\) & 17 & 770 & 35.1 \\ \hline \(21\) & 10 & 140 & 17.9 \\ \hline \(22\) & 26 & 810 & 52.32 \\ \hline \(23\) & 9 & 450 & 18.75 \\ \hline \(24\) & 8 & 635 & 19.83 \\ \hline \(25\) & 4 & 150 & 10.75 \\ \hline \end{tabular}
\end{table} 
Table 1: Soft drink restacking times dataset. Equation (12), the \(\lambda\) value is 3.34123. The no-information IW for non-parametric posterior distribution regression parameters \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) is [\(-3.2577\), \(3.2577\)] for a value of \(\gamma=5\%\). Some implementation aspects of the procedure were not described in Section 2. First is the fitness function of the GA procedure. For a given population member, the predicted outputs are computed from its genes using Equation (5). The RMS error on the dataset is computed next. Let us say that this RMS error is \(\xi_{s}\) for some population member model \(s\)\(\in\)\(\{1,\ldots,\Omega\}\). The fitness value for this population member \(s\) in generation \(g\), (\(f_{s}^{g}\)), is computed using the following expression:  
\[f_{s}^{g}=\frac{\lambda}{1+\xi_{s}} \tag{14}\]  
The maximization of Equation (14) can be obtained by minimizing the RMS. Note that \(\lambda\) is a predetermined constant. The number \"one\" in the denominator is added to avoid dividing by zero if \(\xi_{s}\) takes a value of zero in the event of a perfect regression model fit. Using the results of the OLS and plugging them into Equation (14), we obtain a value of 0.823. This value gives a benchmark of what may be the approximate best value of the best-fitness member in the GA procedure. If the best member fitness function value exceeds 0.823, then the GA model procedure has found a regression model with a lower RMS than the OLS regression model. Usually, it would be rare to find better results from a heuristic GA regression model. Thus, the threshold value of 0.823 should be considered an upper bound on the fitness value of the best population fitness member found using the GA experiments. Second, the no-information IW for the regression parameters was computed as [\(-0.975\times\lambda\), \(0.975\times\lambda\)] for the value of \(\gamma=5\%\).