## 3 Results and Discussion  
### Define  
Each product (\(P_{i}\)) is composed of several ingredients, and each ingredient (\(m_{j}\)) has a fixed composition in at least one of the components required to manufacture the product. Generally, it can be assumed that each ingredient (\(m_{j}\)) contributes only one component. Thus, the mass \(C_{ij}\) (kg) of each component can be obtained using the mass fractions (\(X_{ij}\)):  
\[C_{11}=X_{11}\cdot m_{1} \tag{1}\]  
\[C_{12}=X_{12}\cdot m_{2} \tag{2}\]  
\[C_{13}=X_{13}\cdot m_{3} \tag{3}\]  
\[C_{ij}=X_{ij}\cdot m_{j} \tag{4}\]  
Product \(P_{i}\) can be expressed as a combination of its ingredients (\(m_{j}\)), considering they represent a proportion of the total (\(Y_{ij}\)):  
\[P_{i}=m_{1}+m_{2}+\dots+m_{j} \tag{5}\]  
\[\frac{1}{X_{i1}}\cdot\frac{C_{i1}}{P_{i}}+\frac{1}{X_{i2}}\cdot\frac{C_{i2}}{P _{i}}+\dots+\frac{1}{X_{ij}}\cdot\frac{C_{ij}}{P_{i}}=1 \tag{6}\]  
\[Y_{ij}=\frac{1}{X_{ij}}\cdot\frac{C_{ij}}{P_{i}} \tag{7}\]  
The target values (\(T\)) for each component \(j\) in product \(i\) can be expressed as follows:  
\[Y_{i1T},\ Y_{i2T},\ \dots,\ Y_{ijT} \tag{8}\]  
and the corresponding actual values (\(R\)) for the content of each component are  
\[Y_{i1R},\ \dots,\ Y_{ijR} \tag{9}\]  
\begin{table}
\begin{tabular}{l l l} \hline
**Phase** & **Contents** & **Application** \\ \hline \multirow{4}{*}{Define} & Indicate the main objective of the application, as well as the critical & Reducing possible losses of ingredients used in manufacturing. Associated with this are also the purchasing process, the generation of waste, and the use of energy resources \\  & & the value of energy resources \\  & Develop a data collection plan & Analysis of manufactured products. \\  & and compare data to identify problems and gaps & Relationship with the composition of raw \\  & & the values and the reliability of added quantities \\  & & the values of defects \\  & & the sources of variation & obtained through analysis using graphical \\  & & the values, statistics, etc. \\  & & the values of variation & Study trends and possible corrective measures to \\  & & the values of variation & be implemented \\  & & the values of variation \\  & & the values of variation & Determine the impact of the proposed changes \\  & & the values of variation & on the processes and their follow-up \\ \hline \end{tabular}
\end{table}
Table 1: Application of the DMAIC cycle to the present work.  
The objective was to minimize the differences between the real and the theoretical values, with the ideal situation being  
\[Y_{ijT}-Y_{ijkR}=0 \tag{10}\]  
Using the information collected from the company production plan, a total of 35 batches of each product were produced over the course of one year. The density and the composition of the key ingredients were monitored and compared to the theoretical values provided by the supplier. The differences between the actual (\(Y_{ijkR}\)) and the theoretical (\(Y_{ijT}\)) values indicate variability in product composition, which in turn reflects potential product losses. For each product, the following characteristic components were studied: \(Y_{11}\) for product \(P_{1}\); \(Y_{21}\) and \(Y_{22}\) for product \(P_{2}\) (it has two components); and \(Y_{31}\) for product \(P_{3}\). Since the variation in density (\(D_{i}\)) is directly correlated with potential material loss, the differences between the theoretical density values and those obtained in each batch (\(D_{iT}-D_{iR}\)) were also monitored.