# Integrating Simulation-Based Optimization for Lean Logistics: A Case Study
## 4 Case Study
### Experimentation and Model Optimization  
Three factors were considered and tested in two levels each one in the simulation model as follow: product allocation (PA), considering 30% low level and 70% as high level for Tijuana Distribution Center (TDC); inventory level (IL), tested at 1 week in low level and 2 weeks in high level and on time delivery (OTD), 80% low level and 95% high level. The central composite technique (CCD) of RSM is going to be applied with face centered, four central points and 0.05 significance level (\(\alpha\)). This activity is performed with Minitab Software and the design considers 18 experimental runs with different combinations of each level. Each of the 18 experimental runs were executed in the simulation model designed in Section 4.3 to find out the cost of logistics process under the different scenarios, all the information is collected in order to execute the statistical analysis. Table 2 shows the 18 experimental runs and results obtained with the DCC; coded units \(-1,0,1\) in the design indicates: Pt type \(-1\) experiment is executed with axial points, Pt type 0 executed with central points and Pt type 1 executed with factorial points; Blocks 1 specify DCC is performed considering only one block; and \(-1,0,1\), for variables PA, IL and OTD indicates level to test in the DCC, \"\(-1\)\" represents the low level value, \"0\" nominal level value and \"1\" high level value. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Std Order** & **Run Order** & **Pt Type** & **Blocks** & **PA** & **IL** & **OTD** &
\begin{tabular}{c} **Logistics Cost** \\ **(Thousand \$)** \\ \end{tabular} \\ \hline
16 & 1 & 0 & 1 & 0 & 0 & 0 & \$9318.70 \\
7 & 2 & 1 & 1 & \(-1\) & 1 & 1 & \$9750.10 \\
11 & 3 & \(-1\) & 1 & 0 & \(-1\) & 0 & \$8796.20 \\
18 & 4 & 0 & 1 & 0 & 0 & 0 & \$9351.20 \\
9 & 5 & \(-1\) & 1 & \(-1\) & 0 & 0 & \$9675.70 \\
4 & 6 & 1 & 1 & 1 & 1 & \(-1\) & \$9282.70 \\
10 & 7 & \(-1\) & 1 & 1 & 0 & 0 & \$8634.80 \\
5 & 8 & 1 & 1 & \(-1\) & \(-1\) & 1 & \$8635.60 \\
1 & 9 & 1 & 1 & \(-1\) & \(-1\) & \(-1\) & \$9534.90 \\
8 & 10 & 1 & 1 & 1 & 1 & 1 & \$8430.20 \\
12 & 11 & \(-1\) & 1 & 0 & 1 & 0 & \$9910.70 \\
2 & 12 & 1 & 1 & 1 & \(-1\) & \(-1\) & \$8163.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Central Composite Design and Results. Figure 2: Simulation Model Representation. Once, the LC results of each experiment executed in the DCC are attained, the information is expressed in equations of matrix form to determine the value for each coefficient applying Equation (6). \[\hat{\beta}=\left(X^{\prime}X\right)^{-1}\left(X^{\prime}y\right) \tag{6}\]  
where:  
\[X^{\prime}y=\left[ \begin{array}{c}163206.4\\ -5819.0\\ \vdots\\ 595.1\end{array} \right]\hat{\beta}=\left[ \begin{array}{c}\beta_{0}\\ \beta_{1}\\ \vdots\\ \beta_{11}\end{array} \right]\]  
\[\left(X^{\prime}X\right)=\left[ \begin{array}{cccccccccccc}1&18&0&0&0&10&10& 10&0&0&0\\ 1&0&10&0&0&0&0&0&0&0&0\\ &&&&&&&&&&&&\\ 1&0&0&0&0&0&0&0&0&8\end{array} \right]\]  
Software Minitab(r), was used to calculate the value for each coefficient included in the empirical mathematical model represented by the Equation (7), this needs to be validating to determine if it is a good estimator for the response variable. This model is represented by a regression model with a second order equation. The following concepts are analyzed (where X\({}_{1}\) = PA, X\({}_{2}\) = IL, X\({}_{3}\) = OTD):  
\[ \begin{array}{c}\text{Y = 9283.70 - 581.90 (X${}_{1}$) + 497.70 (X${}_{2}$) - 397.30 (X${}_{3}$) - 122.10 (X${}_{1}$) ^{2} + 76.00 (X${}_{2}$) ^{2} - 343.90 (X${}_{3}$) ^{2} + 75.60 X${}_{1}$X${}_{2}$ -62.70 X${}_{1}$X${}_{3}$ + 74.30 X${}_{2}$X${}_{3}$}\end{array}  \tag{7}\]  
1. Coefficient of determination analysis (R\({}^{2}\)adj). This indicator shows as a result 96.89%, which indicates that the factors are statistically significant to the result (output variable), because those factors explain the 96.89% of the variability by the fitted model.