# Integrating Simulation-Based Optimization for Lean Logistics: A Case Study
## 4 Case Study
### Experimentation and Model Optimization
\begin{array}{ccc}\beta_{11}& \frac{\delta_{12}}{2}&\frac{\delta_{13}}{2}\\ \frac{\delta_{12}}{2}&\frac{\delta_{12}}{2}&\frac{\delta_{12}}{2}\\ \frac{\delta_{12}}{2}&\frac{\delta_{23}}{2}&\beta_{33}\end{array} \right]\]  
Figure 5: Surface Plot for Logistic Cost. Figure 6: Surface Plot for Logistic Cost. where \(x_{s}\) is any point in the region of process operability in coded units; vector \(b\) includes the coefficients of the linear part of the model and the matrix \(B\) includes the coefficients of quadratic terms and interactions pure. Then stationary points are the solution of Equation (9). \[x_{s}=\frac{B^{-1}b}{2}\] (9) The \(x_{s}\), \(b\) and \(B\) matrices in Equation (8) were arranged by Equation (6) to get Equation (10) as follows: \[x_{s}=\frac{\left[ \begin{array}{ccc}-122.10&37.60&-31.30\\ 37.60&76.00&37.10\\ -31.30&37.10&-343.90\end{array} \right]^{-1}\left[ \begin{array}{c}-581.90\\ 497.70\\ -397.30\end{array} \right]}{2}=\left[ \begin{array}{c}-2.76\\ -1.66\\ -0.50\end{array} \right]\] (10) The solution for Equation (9) defines the optimum values for the model in coded units, such values are \(-2.7\), \(-1.6\) and \(0.5\) for PA, IL and OTD respectively. This result shows clearly that stationary point is out of the experimental region. Applying the Equation (11), where \(Z_{H}\) and \(Z_{L}\) are the high and low levels values for independent variables in original units, as well as \(X_{i}\) is the value in coded units defined in Equation (5), the optimum values can be decoded in order to get original units, getting the value for PA equal to \(0\%\), which means logistics operations needs to be managed by the logistics service supplier located at Mexico City, IL is going to keep at \(0.67\) and OTD at \(84\%\). \[Z_{i}=\frac{x_{i}(Z_{H}-Z_{L})+(Z_{H}+Z_{L})}{2}\] (11) Based on the information obtained in this step, it can be concluded that parameters determined with the mathematical model reflect the traditional operation in which the logistics process is performed by a supplier closer to the customers, in this case in Mexico City, at the highest operational cost. Since the goal of this research project is to find alternatives that optimize the logistics cost, we will proceed to evaluate the empirical mathematical model to determine if there is a better regression equation to estimate the parameters of optimizing the logistics costs. 6. A subsets regression technique was applied to evaluate the empirical mathematical model; this technique attempts to identify groups of predictors for further analysis. Ideally, the smallest subset that fulfills certain statistical criteria such as highest coefficient of determination adjusted (R\({}^{2}\)adj. ), lowest Mallow's Cp and Mean Square Error (S) should be selected. Because, this subset of predictors may estimate the regression coefficients and predict future responses with smaller variance than the full model using all predictors. This process was performed with the support of Minitab software, Figure 7 shows that a seven predictors subset is the one that fulfill the statistical criteria mentioned previously; however, there is a smaller subset with four predictors with similar statistical criteria, therefore a further analysis was done with both subgroups. Finding that four predictors subset is a better estimator since Predicted Residual Square Sum value is the smaller (671820) as show in Figure 8. Therefore, the coefficient of determination to predict values will be the highest (91.94%) of the all subgroup. As a conclusion Equation (12) is the best regression equation to predicted values. \[=9272.20-501.80X_{1}+497.70X_{2}-307.30X_{3}-369.30X_{3}^{2}\] (12) Once the validation of the model is completed, the model can be optimized. This process is going to define the parameters of the independent variables (PA, IL and OTD) which provide an optimal operation condition to achieve a particular objective for the logistics process cost. As Assumption for the model is to improve the logistics cost by \(15\%\pm 2\%\).