# Multi-Objective Optimization Model of Industrial Lubricants Based on Integer Nonlinear Programming
## 2 Data Processing and Blending Scheme
### Blending Scheme  
In this paper, the viscosity of each oil batch is calculated using the internationally common viscosity blending calculation formula [19]. For the other indexes--freezing point, acid value, flash point, oxidation stability, and carbon residue--the linear blending formula is applied to predict the lubricating oil indexes. The indexes with low prediction accuracy are adjusted nonlinearly, and, finally, the blending scheme of a complex power polynomial is established for high-precision prediction. The calculated values of these models are compared with the measured values. Viscosity is one of the most important characteristics of lubricating oil and a performance index that should be considered first in lubricating oil blending. In addition, the viscosity of lubricating oil is an important basis for classifying oil brand numbers. The viscosity calculation formula is established as follows:  
\[\text{log}\varsigma_{\text{j}1}=\sum_{\text{i}=1}^{\text{n}}\text{a}_{\text{i }\text{j}}\text{log}\text{P}_{\text{i},1} \tag{2}\]  
where \(\varsigma_{\text{j},1}\) represents the calculated value of the viscosity of batch j blended oil, \(\text{P}_{\text{i},1}\) represents the actual value of the viscosity of base oil i, and n represents the type of base oil. The relationship between the calculated viscosity value and the measured value of each batch of blended oil calculated according to the formula is shown in Figure 2. The calculated value is in good agreement with the measured value, and the error is 0.002%. Therefore, the calculation formula can be used directly. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Performance Index** & **Viscosity** & **Freezing** & **Acid Value** & **Flash Point** & **Oxidation** & **Carbon** \\
**Production Batch** & **(mm2/s)** & **Point (\({}^{\circ}\)C)** & **(mgKOH/g)** & **(\({}^{\circ}\)C)** & **Stability (min)** & **Resolute (\%)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: Performance indexes of each batch of blended oil. As the interaction between the components of crude oil becomes more complex after mixing, and there are many influencing factors, there is currently no theoretical method to calculate the freezing point of mixed crude oil. Most of the existing calculation models are empirical or semi-empirical formulas derived by scholars based on experimental data [20]. It can be seen in Tables 1 and 3 that there is a big difference between the freezing point of base oil and that of blended oil, and it does not meet the linear relationship. So, the nonlinear relationship is considered. It is assumed that the freezing point of blended oil and the freezing point of base oil meet the following relationship:  
\[\mathrm{c}_{j2}=\sum_{i=1}^{n}\mathrm{a}_{ij}\cdot\left(\mathrm{p}_{i2}^{f}+ \mathrm{h}\cdot\mathrm{p}_{i2}^{1/3}\right) \tag{3}\]  
where \(\mathrm{c}_{j2}\) represents the calculated value of the freezing point of batch j blended oil, \(\mathrm{p}_{i2}\) represents the actual value of the freezing point of base oil i, and n represents the type of base oil. Adjustments are made to f and h to make the error between the calculated value of the freezing point and the measured value as small as possible, that is, to make the value of \(\mathrm{err}_{2}\) as small as possible. This satisfies the following relationship:  
\[\mathrm{err}_{2}=\sum_{j=1}^{20}\lvert\mathrm{c}_{j2}-\mathrm{s}_{j2}\rvert \tag{4}\]  
where \(\mathrm{c}_{j2}\) is the calculated value of the freezing point of batch j blended oil, and \(\mathrm{s}_{j2}\) is the required value of the freezing point of batch j blended oil. Using a nonlinear least-squares algorithm, the values of f and h are 0.4754 and \(-\)0.003, respectively. A comparison between the calculated value and the measured value obtained by this formula is shown in Figure 3. The calculated value is in good agreement with the measured value, and the average error is 1.546%, so the calculation formula can be used directly. Figure 2: Comparison between the calculated value and measured value of viscosity.