## 5 Case Study
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**(a) Estimated Mean Response**} & \multicolumn{6}{c}{**(b) Estimated Standard Deviation Response**} \\ \hline \multicolumn{6}{c}{**Weights**} & \multicolumn{2}{c}{**Biases**} & \multicolumn{2}{c}{**Weights**} & \multicolumn{2}{c}{**Biases**} \\ \hline \multicolumn{6}{c}{**\(W_{ij}^{mean\_pro}\)**} \\ \hline \(7.048\) & \(3.571\) & \(-1.123\) & \(0.134\) & \(-7.259\) & \(6.693\) & \(3.771\) & \(-0.855\) & \(0.0516\) & \(-7.531\) \\ \(4.726\) & \(-5.71\) & \(-1.371\) & \(-0.212\) & \(-6.831\) & \(4.928\) & \(-5.169\) & \(-1.884\) & \(-0.038\) & \(-7.045\) \\ \(-4.981\) & \(4.672\) & \(3.442\) & \(-0.789\) & \(6.003\) & \(-4.907\) & \(4.677\) & \(3.435\) & \(-0.147\) & \(5.755\) \\ \(4.459\) & \(3.990\) & \(3.498\) & \(1.079\) & \(-6.208\) & \(4.251\) & \(4.486\) & \(3.809\) & \(0.467\) & \(-5.651\) \\ \(2.679\) & \(3.682\) & \(-6.266\) & \(-0.199\) & \(-4.201\) & \(2.864\) & \(3.086\) & \(-5.745\) & \(0.330\) & \(-4.851\) \\ \(-6.436\) & \(4.389\) & \(0.793\) & \(0.209\) & \(3.175\) & \(-6.849\) & \(3.647\) & \(0.288\) & \(0.467\) & \(3.440\) \\ \(-4.908\) & \(5.623\) & \(-1.339\) & \(0.124\) & \(2.863\) & \(-5.123\) & \(5.469\) & \(-1.166\) & \(-0.356\) & \(2.874\) \\ \(1.562\) & \(-4.993\) & \(5.438\) & \(0.190\) & \(-2.048\) & \(1.447\) & \(-4.815\) & \(5.474\) & \(-0.271\) & \(-2.451\) \\ \(6.692\) & \(2.115\) & \(3.035\) & \(-0.139\) & \(-0.859\) & \(6.152\) & \(2.694\) & \(2.570\) & \(-0.250\) & \(-1.183\) \\ \(5.521\) & \(-4.127\) & \(3.079\) & \(0.099\) & \(-0.243\) & \(6.115\) & \(-3.555\) & \(2.652\) & \(-0.034\) & \(0.050\) \\ \(-5.977\) & \(3.126\) & \(-3.407\) & \(0.325\) & \(-0.669\) & \(-0.343\) & \(-6.131\) & \(2.967\) & \(-3.265\) & \(0.230\) & \(-0.850\) \\ \(5.256\) & \(-5.103\) & \(1.936\) & \(0.633\) & \(1.256\) & & \(5.651\) & \(-4.723\) & \(1.535\) & \(-0.040\) & \(1.651\) \\ \(6.483\) & \(-3.230\) & \(2.033\) & \(-0.294\) & \(2.105\) & & \(6.082\) & \(-3.433\) & \(2.667\) & \(0.697\) & \(1.926\) \\ \(-0.589\) & \(-5.911\) & \(-4.442\) & \(-0.462\) & \(-3.229\) & & \(-0.441\) & \(-5.646\) & \(-4.983\) & \(-0.529\) & \(-2.783\) \\ \(3.818\) & \(-4.951\) & \(-4.368\) & \(-0.443\) & \(3.550\) & & \(4.070\) & \(-5.306\) & \(-4.141\) & \(0.407\) & \(3.526\) \\ \(-6.018\) & \(4.983\) & \(-0.135\) & \(-0.385\) & \(-4.140\) & & \(-6.091\) & \(4.643\) & \(-0.476\) & \(-0.545\) & \(-4.469\) \\ \(-1.158\) & \(2.826\) & \(6.948\) & \(-0.294\) & \(-5.321\) & & \(-1.107\) & \(3.380\) & \(6.462\) & \(-0.155\) & \(-5.651\) \\ \(6.528\) & \(-3.113\) & \(-2.385\) & \(0.419\) & \(6.011\) & & \(6.112\) & \(-2.436\) & \(-2.970\) & \(-0.514\) & \(6.074\) \\ \(4.157\) & \(6.574\) & \(1.314\) & \(0.425\) & \(6.397\) & & \(4.160\) & \(5.845\) & \(1.643\) & \(0.072\) & \(7.251\) \\ \(4.931\) & \(-5.012\) & \(-2.952\) & \(-0.233\) & \(7.576\) & & \(4.467\) & \(-4.909\) & \(-3.407\) & \(-0.070\) & \(7.674\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Weight and bias values by using the NN-based DR estimation method. Figure 4: Estimated process mean and standard deviation response functions from the conventional LSM approach based on RSM: **(a)** mean (\(R^{2}=92.68\%\)); **(b)** standard deviation (\(R^{2}=45.42\%\)). Figure 5: Estimated process mean and standard deviation response functions from the feed-forward NN approach with conventional log-sigmoid transfer function: **(a)** mean (\(R^{2}=93.44\%\)); **(b)** standard deviation (\(R^{2}=51.48\%\)). As demonstrated in Table 5, the process variance obtained from the NN-based estimation methods is remarkably smaller than that of the conventional statistical LSM approach. Therefore, NN-based estimation methods can provide considerably smaller EQL values (i.e., significant criteria to evaluate the level of quality) than those of the conventional LSM approach in this particular example, although the same RD optimization model is utilized.