# Dye Schedule Optimization: A Case Study in a Textile Industry
## 4 Evaluation Model
### Structuring for Test
The information obtained was submitted to estimate the linear correlation coefficients (\(r\)) using the Minitab 17 software. The linear correlation coefficient, sometimes called Pearson's product moment correlation coefficient, is represented by the letter r and takes values from \(-1\) to \(1\) and measures the degree of the linear relationship between the paired values x and y in a sample. When \(r=1\), it represents the perfect and positive correlation between two variables, and for \(r=-1\), it means a perfect negative correlation between two variables. That is, while one increases, the other decreases [33]. The Pearson's correlation coefficient is calculated according to the following formula:  
\[r=\frac{n\sum xy-(\sum x)(\sum y)}{\sqrt{n(\sum x^{2})-(\sum x)^{2}}\sqrt{n( \sum y^{2})-(\sum y)^{2}}} \tag{7}\]  
where: \(n\)--represents the number of data pairs present; \(x\), \(y\)--are the sample values you want to evaluate the correlation. The analysis was performed separately for each machine group, and the correlation between % variation with the number of SKU and % variation with the programming horizon was evaluated. Absolute variation with SKU number and absolute variation with programming horizon. Below in Table 7 are presented the results obtained for the two groups of machines. It was chosen to study the correlation of both the relative variation and the absolute variation of setup times to assess possible divergences in the behaviour of the linear correlation coefficients. As it can be seen, there is a difference in the reading of information between the two situations. Contrary to what was expected, the correlation of the number of SKUs with variation in setup time presented practically null correlation coefficients for both machine groups. That is, there is no correlation between the number of SKUs with variation in setup time. However, the planning horizon showed a strong negative correlation with the interpretation in setup time, with the value of \(r\) for slasher dye and loop dye group for the relative variation being \(-0.644\) and \(-0.679\), respectively. For the absolute deviation, it was \(-0.928\) and \(-0.859\), respectively. That is, the absolute variation presented a more substantial relationship degree with the planning horizon. The negative correlation shows that, as the planning horizon increases, the optimization system is more efficient is in reducing setup time. In a way, this result was already expected, as not only in dyeing \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Slasher Dye** & **Coefficient** & **\% Vary** & **Absolute Vary** \\ \hline SKU Number & \(r\) & \(0.041\) & \(0.009\) \\ \cline{2-4}  & \(p\)-value & \(0.848\) & \(0.966\) \\ \hline Planning Horizon & \(r\) & \(-0.644\) & \(-0.928\) \\ \cline{2-4}  & \(p\)-value & \(0.001\) & \(0.000\) \\ \hline \hline
**Loop Dye** & **Coefficient** & **\% Vary** & **Absolute Vary** \\ \hline SKU Number & \(r\) & \(-0.171\) & \(-0.173\) \\ \cline{2-4}  & \(p\)-value & \(0.386\) & \(0.377\) \\ \hline Planning Horizon & \(r\) & \(-0.679\) & \(-0.859\) \\ \cline{2-4}  & \(p\)-value & \(0.000\) & \(0.000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Linear Correlation Coefficient. sequencing, as in other areas, the more you can anticipate the needs, the better the resources are used. Based on the results presented, we can propose a model based on linear regression that can predict the variation in setup time. However, such interpretation must refer to relative deviation, despite having presented a weaker degree of relationship when compared to absolute variation. It is not advisable to propose a model that indicates the possible absolute variation in setup time. This, when replicated in another operation with a similar process, will undoubtedly face different setup times between pairs of SKUs. Therefore, we will describe the relationship through the straight-line equation representing the relationship between the variation in setup time with the planning horizon.