Резюме. Two potent mathematical and statistical methods of response surface methodology (RSM) and genetic algorithm (GA) based on artificial neural network (ANN) were employed for prediction and optimization of three-constituent synthetic engine oil. Polyalpha olephin-4 (PAO4) Hitec 5780 (HI 5780), and Hitec 11100 (HI 11100) were used as base oil and the additives of the engine oil, respectively. The models were applied for the percentage of oil constituents and viscosity at \(40^{\circ} \mathrm{C}\) (V is at \(40^{\circ} \mathrm{C}\) ), viscosity at \(100^{\circ} \mathrm{C}\) (Vis at \(100^{\circ} \mathrm{C}\) ), Viscosity index (VI), flash point and Noack of the finished oil. The range of the viscosity at \(40^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\) were selected according to ISO viscosity grade for engine oil. The optimization includes maximization of FP and VI and minimization of Noack. The obtained results showed that ANN has higher potential and capability and more accuracy for prediction and optimization of the process.

Ключови думи: engine oil formulation; \(\mathrm{PAO}_{4}\); modeling; mixture design; artificial neural network; genetic algorithm

Introduction

The main function of lubricants is the reduction of wear and friction (Richardson, 2000). Automobile engine lubricating oil mainly contains base oil and some additives. Generally, the properties of lubricant are enhanced by engine oil additives. The significant roles of additives are to control deposition of lacquer and sludge, reduce corrosion, wear and oxidation, alter physical properties including pour point, flash point and modify chemical properties (Totten, 2006). The base oils are classified into five groups. Groups I-III are obtained from mineral sources, group IV oils are all polyalphaolephin (PAO) and group V oils are all other base stocks which are not included in group I-IV (Korcek & Jensen, 1976; Mensah-Brown, 2013; Murray et al., 1982). Synthetic base stocks which are used in motor oil formulation include esters, polyisobutylenes (PIB), polyalphaolephin, and polyalkalene glycols (PAG). PAO is one of the most widely used synthetic base oils which comparing to other synthetic oils and mineral oils, has the advantages such as better performance, higher viscosity index, better thermal oxidation stability and higher flash points. However, PAOs are more expensive than mineral oils (Mang & Dresel, 2007). The society of automotive engineers (SAE) classified motor oils according to the oil viscosity. Furthermore, American Petroleum Institute (API) classified motor oils according to the type of fuel into two general groups of Gasoline oil (with symbols (S)) and diesel oil (with symbol (C)) identified (Khalilian et al., 2016). RSM is a set of mathematical techniques which describes the relation between several independent variables (Tung & McMillan, 2004). The techniques were developed by Box and Willson (Myers et al., 2016) Mixture design is a special class of RSM techniques which is considered as the technology of the quality to reach the best of a product and response depending only on the amount of components present in the mixtures (Box & Wilson, 1951; Cafaggi et al., 2003). These components are dependent to each other. The sum of total parameters is exactly \(100 \%\). ANN is useful for identifying the relation between the components and the properties of the oil. These models have been applied to different systems (Brandvik & Daling, 1998; Dutta et al., 2010; Kumar & Porkodi, 2009). The main advantages of ANN models are their ability to identify both linear and nonlinear models, disregard of the nature of the original process. In addition, they are easy to implement and deliver good performance. After establishing the neural network-based model, the optimization is conducted using genetic algorithm. Genetic algorithm is a stochastic technique based on evolutionary algorithms. This algorithm search for all possible solutions, measure their fitness and reproduce new solutions in order to achieve global optima (Bezerra et al., 2008). The principal advantage of genetic algorithm is its independency on the error surface. Therefore, multi-dimensional, non-differential, non-continuous, and even non-parametrical problems can be solved using genetic algorithm.

In this work, the best formulation of the engine oil 5W40 SN containing of \(\mathrm{PAO}_{4}\) as base oil as well as, HI 5780 and \(\mathrm{Hil1100}\) as additives was obtained. Vis at \(40^{\circ} \mathrm{C}\) and at \(100^{\circ} \mathrm{C}\), VI, F.P., and Noack were optimized to achieve the best formulation. RSM-mixture design and GA based on ANN are used for modeling and optimization purpose. Finally, the efficiencies of both models were compared by a set of experimental data provided by extreme vertices designs.

Experimental

Engine oil constituents

The following components were used: PAO4, HI 5780 and HI 11100 from ELA Company. The role of HI 1100 and HI 5780 is VI improver. In each experiment, the constituents were blended and stirred for 30 min.

Instrumental

VI, Vis at \(40^{\circ} \mathrm{C}\), and Vis at \(100^{\circ} \mathrm{C}\) were performed by viscometer Anton bar model SVM 3000. Flash points were determined by flash point tester Herzog model HC852 and Noack was determined by Noack tester Tajhiz Gostar Apadana.

Test method

Different tests were executed according to ASTM D-445 for viscosity at \(100^{\circ} \mathrm{C}\), \(40^{\circ} \mathrm{C}\), ASTM D-2270 for viscosity index, ASTM D- 92 for flash point, ASTM D-97 for pour point and ASTM D-5800 for Noack.

Mathematical and statistical modeling

In this study, two different methods were applied to find the best formulation for the engine oil (5W40SN) production. The first method was implemented using Minitab17.0 software. The results were analyzed by executing following steps: (1) establishing the surface model: As mixture design and RSM were applied on the process the sequential F-test and other adequacy were obtained. The criterion for finding best fitting function are the evaluations of several statistical parameters such as the multiple correlation coefficient (\(\mathrm{R}^{2}\) ), the coefficient of variation (CV) and adjusted multiple correlation coefficient(adjusted \(\mathrm{R}^{2}\) ) (Goldberg, 1989). According to this information quadratic and linear model were selected. The selected model for a mixture is shown by Eq. (1) (Lu & Anderson-Cook 2012);

(1) \[ Y_{n}(x)=\sum_{1 \leq i \leq q}^{q} \beta_{i} X_{i}+\sum_{1 \leq i \leq j \leq q}^{q} \beta_{i j} X_{i} X_{j}+\beta_{123} X_{1} X_{2} X_{3} \]

where \(\mathrm{Y}_{\mathrm{n}}\) represents the response function of the experimental data (viscosities at \(100^{\circ} \mathrm{C}, 40^{\circ} \mathrm{C}\), VI, FP and Noack), x1, \(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}, \mathrm{x}_{5}\), x3 , x4, x5, and \(\mathrm{x}_{6}\) are independent variables and \(\beta_{\mathrm{ij}}\) represents the coefficients of the interaction parameters; (2) analysis of variance (ANOVA). After analyzing each response, multi responses optimization was employed with numerical tools using Minitab 17.0.

In the second method, a neural network was firstly trained to use as a function for linking the components and the properties of the engine oil. Then using the same fitness function as the first method, genetic algorithm was applied to find the best formulation of the engine oil.

Procedure

The first method is categorized under RSM which includes approximation of responses using low-order polynomial with interaction and optimum level using methods like steepest ascent. The first step is to find a relation between the components and the properties using experimental data. Then, an appropriate criterion for measuring the quality of responses has to be established. Finally, the optimal mixture is found by using an optimization approach.

ANN is a calculation model that endeavors to simulate the functionalities and structure of biological neural networks. It is a parallel broadcast processing system combined of neurons (node) and connection (weights). In this study suggested the use of RSM and a genetic algorithm based on ANN to find optimizing of engine oil formulation (Orives et al., 2014).

Results and discussion

Experimental design by RSM

Experimental design was generated with the Minitab 17.0 at the interest concentration range. An extreme vertices design was based on a lower and upper bound on their component amount (Maran et al., 2014; Maran et al., 2013) (Eq. (2):

(2) \[ \mathrm{a} \sum_{j} X_{j}=1 \quad \text { and } \quad L_{j} \leq X_{j} \leq U_{j} \]

where \(\mathrm{Xj}=\) component proportion; \(\mathrm{L}_{\mathrm{j}}=\) lower constraint; \(\mathrm{U}_{\mathrm{j}}=\) upper constraint. According to ISO viscosity grade the following constraints were imposed on the components:

\[ x_{1}+x_{2}+x_{3}=100,57 \leq x_{1} \leq 77,13 \leq x_{2} \leq 33 \text { and } 9.5 \leq x_{3} \leq 11.0 \] where \(\mathrm{x}_{1}, \mathrm{x}_{2}\), and \(\mathrm{x}_{3}\) are the percentages of \(\mathrm{PAO}_{4}, 5780\), and HI 1 1100 respectively.

The responses for the dependent variables, Vis at \(100^{\circ} \mathrm{C}\), Vis at \(40^{\circ} \mathrm{C}, \mathrm{VI}, \mathrm{FP}\), and Noack were obtained by mixing the three components of the engine oil according to the suggested conditions (Table 1). A quadratic and linear model were selected and fitted to the obtained results (Mourabet et al., 2014).

Modeling by RSM

The assessment of every dependent variable was carried out by using quadratic and linear model which contained quadratic and linear terms. The ANOVA results displayed the linear, interactive and quadratic relationship between the effects of independent variables on dependent variable. The significance of each term was assessed according to their corresponding p-values. The p-value reveals the significance of each variable. The p-value less than 0.05 shows that variable is significant and the p-value greater than 0.05 indicates that variable is insignificant (Tschoegl, 2012). The Fisher’s F-test (F values) were measured and found to be high with very low probability value (\(\mathrm{p} \lt 0.05\) ), which displays a high degree of adequacy of model and also indicate that the process variable combinations were highly significant (Maran et al., 2014). Determination coefficient (\(\mathrm{R}^{2}\) ), correlation coefficient (R) and adjusted determination coefficient (\(\mathrm{R}^{2}\) ) a and coefficient of variance (CV) were also determined to evaluate the suitability of the model. The determination coefficient \(\left(\mathrm{R}^{2}\right)\) value of the regression model indicated that only the values of the total variations were not explained by the proposed model (Maran et al., 2013). The large value of the adjusted determination coefficient (\(\mathrm{R}^{2}{ }_{\mathrm{a}}\) ) indicated that the model was highly significant. The regression equations were obtained after the analysis of variance (ANOVA). These equations represented the level of Vis at \(100^{\circ} \mathrm{C}\), Vis \(40^{\circ} \mathrm{C}\), VI, FP, PP, and Noack as a function of \(\mathrm{PAO}_{4}\), HI 5780 and HI 11100. Eq. (3) was adjusted to fit experimental data and \(\mathrm{Y}_{1}\) represents the viscosity at \(40^{\circ} \mathrm{C}\) (cSt).

(3) \[ \begin{aligned} & Y_{1}=\text { Visat } 40^{\circ} \mathrm{C}=+0.88378 * \mathrm{X}_{1}+12.62910 * \mathrm{X}_{2}+30.79787 * \mathrm{X}_{3}-0.14319 * \mathrm{X}_{1} * \\ & X_{2}-0.37837 * \mathrm{X}_{1} * X_{3}-0.43370 * \mathrm{X}_{2} * X_{3} \end{aligned} \]

The model F-value of 1069.8 implies that the model is significant. Values of \(\mathrm{p} \lt \) 0.05 indicate that the model terms are significant according to ANOVA (Table 2) yielding an experimental \(\mathrm{R}^{2}\) of \(99.87 \%\) and the predicted \(\mathrm{R}^{2}\) of \(99.43 \%\) which are in reasonable agreement with the adjusted \(\mathrm{R}^{2}\) of 99.78 \%.

Eq. (4) wasadjustedtotheexperimentaldatatoshowthemodel, where \(\mathrm{Y}_{2}\) isviscosityat \(100^{\circ} \mathrm{C}\) (cSt).

(4) \[ \begin{aligned} & Y_{2}=\operatorname{Visat} 100{ }^{\circ} \mathrm{C}=+0.08986 * X_{1}+1.62015 * \mathrm{X}_{2}+2.18304 * \mathrm{X}_{3}-0.016322 \mathrm{X}_{1} * \\ & X_{2}-0.025479 * \mathrm{X}_{1} * X_{3}-0.034224 * \mathrm{X}_{2} * X_{3} \end{aligned} \]

The proposed model was analyzed by using analysis of variance (ANOVA). As the results indicated in Table 3, the F-value of the model equals to 1182.9 which implies that the model is significant and the values of \(\mathrm{p} \lt 0.05\) indicate the model terms are significant. The predicted \(\left(\mathrm{R}^{2}\right)\) was \(99.47 \%\) and the adjusted \(\mathrm{R}^{2}\) was \(99.80 \%\) as shown in Table 3.

VI is a measure of the change of viscosity with variations in temperature. The higher VI value shows that oil viscosity changes with temperature. VI is an important factor for quality of engine oils (Maran et al., 2014). Table 4 shows ANOVA response for VI.

Consequently, Eq. (5) was presented,

(5) \[ \begin{aligned} & Y_{3}=V I=+0.65384 * \mathrm{X}_{1}+0.34048 * \mathrm{X}_{2}-71.69262 * \mathrm{X}_{3}+0.021404 * \mathrm{X}_{1} * \\ & X_{2}+0.91217 * \mathrm{X}_{1} * X_{3}+0.91935 * \mathrm{X}_{2} * X_{3} \end{aligned} \]

where \(\mathrm{Y}_{4}\) represents the dependent variable of VI. Moreover, \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) influence the response positively and \(\mathrm{x}_{3}\) negatively. The F-value of the model equals 197.5 which imply that the model is significant and the values of \(\mathrm{p} \lt 0.05\) indicate the model terms are significant; the predicted \(\mathrm{R}_{2}\) of \(96.22 \%\) is in reasonable agreement with the adjusted \(\mathrm{R}_{2}\) of \(98.79 \%\).

The flash point is a measure of the tendency of oil to form a flammable mixture with air under controlled laboratory conditions (Aleme & Barbeira, 2012). The minimum FP, according to the ISO 5W 40 SN, is \(200^{\circ} \mathrm{C}\) in engine oil. Based on the obtained results, the predictive Eq. (6) was produced and (\(\mathrm{Y}_{4}\) ) represents the dependent variable of flash point.

(6) \[ Y_{4}=F . P=2.4637 x_{1}+2.5037 x_{2}+0.3694 x_{3} \]

As analysis of variance shows in Table 5, this model is significant. F-value 61.7 and P-value (less than 0.05) of the model imply that the model is significant. The predicted \(\mathrm{R}^{2}\) of \(84.4 \%\) is in reasonable agreement with the adjusted \(\mathrm{R}^{2}\) of \(91.01 \%\). The Noack test is employed to measure the evaporative loss of an oil at \(250^{\circ} \mathrm{C}\) after 1 h (Waddoups et al., 2001).

The Noack is measured traditionally by ASTM D5800. The predictive Eq. (7) represents the response of the dependent variable \(\mathrm{Y}_{5}\) (Noack). According to the obtained ANOVA results in Table 6, the model F-value is 1161.7 and the p-value is \( \lt 0.05\) which reveal that the model is significant. The predicted \(\left(\mathrm{R}^{2}\right)\) was \(99.35 \%\) and the adjusted \(\mathrm{R}^{2}\) was \(99.49 \%\).

(7) \[ Y_{5}=\text { Noack }=0.0858 x_{1}-0.0399 x_{2}+0.0686 x_{3} \]

Optimization by RSM

According to the responses \(\left(\mathrm{Y}_{1}, \mathrm{Y}_{2}, \mathrm{Y}_{3}, \mathrm{Y}_{4}\right.\), Y2, Y 3, Y4, and \(\left.\mathrm{Y}_{5}\right)\) of the obtained multiresponse optimization were evident. In the case of large number of responses, desirability function is the most popular to be used. In 1980 Suich and Derringer found the desirability function to find the best value for all response (Kim et al., 2002; Myers et al., 2016).

(8) \[ D=\left(d_{1} \times d_{2} \times \ldots \times d_{n}\right)^{\tfrac{1}{n}}=\left(\prod_{i=1}^{n} d_{i}\right)^{\tfrac{1}{n}} \]

where n is the number of responses. Desirability always possess values between 0 and 1, where \(\mathrm{di}=0\) for an undesirable response, and \(\mathrm{d}_{\mathrm{i}}=1\) a thoroughly desirable response (Das et al., 2009).

Final results of RSM

In order to optimize the process, FP and VI were maximized. Moreover, for Vis at \(100^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\) the restriction of (12.6-16) cSt and Vis (60-100) were set respectively for engine oil. Noack was minimized. The optimum formulation displayed a Viscosity at \(100^{\circ} \mathrm{C}, 40^{\circ} \mathrm{C}, \mathrm{VI}, \mathrm{FP}\), Noack and PP of \(85.0 \mathrm{cSt}, 15.0 \mathrm{cSt}, 190.6,225^{\circ} \mathrm{C}, 5.0\), respectively when the formulation was composed of \(66.9 \% \mathrm{PAO}_{4}, 22.9 \% \mathrm{HI} 5780\), 22.9% HI 5780, and \(10.1 \% \mathrm{HI} 11100\). The obtained optimum conditions were used to validate the model prediction. Validation was performed in triplicate and the average values for Viscosity at \(100^{\circ} \mathrm{C}, 40^{\circ} \mathrm{C}, \mathrm{VI}, \mathrm{FP}\), Noack were \(84.5 \mathrm{cSt}, 15.1 \mathrm{cSt}, 190.3,224^{\circ} \mathrm{C}\), 5.4, and -39 , respectively.

Modeling by ANN: prediction by ANN

In this work, a feed-forward neural network (FF ANN) was employed to establish the relation between components and outputs. The general structure of FF ANN is shown in Fig. 1.

The first step is to determine the number of neurons in the hidden layer. There are several methods for calculating the number of neurons (Gómez et al., 2009). However, there is not any determinate method to use. Considering the available methods, we selected the number of neurons using trial and error. The next step is to select the learning algorithm, in order to reach the best weights for the neural network. The choice of a suitable learning method is a crucial part of modeling by ANN, because the best training of network is achieved by minimizing the error function. The back-propagation algorithm was used for training the understudy feed-forward ANN. To this aim, the Levenberg– Marquardt optimization algorithm is employed for its training. The last step is to validate and to verify the prediction model on the basis of error function. In this work, the mean square error (MSE) was employed as the error function. Also, the correlation coefficient (r) was selected as a parameter to show the predictive ability of the network. The same experimental data (Table 1) were used for training the artificial neural network. The data were randomly dispersed into three groups, \(70 \%\) in the training set, \(15 \%\) in the test set and \(15 \%\) in the validation set. After repetition of the trails, a network with 5 hidden neurons showed the best performance. The result is shown in Fig. 2. The MSE value was obtained to be 0.84 which is shown in Fig. 3. The weights which are obtained from the final trained network are shown in Tables 7 and 8. The ANN prediction results are indicated in Table 9.

Optimization by genetic algorithm

In this work, the genetic algorithm was used to obtain an optimal condition to minimize a number of preliminary experiments. GA-ANN was employed to find the properties of different mixtures. The GA with the following properties is used to determine the controllers’ parameters: chromosome population \(=40\); number of generation \(=100\); crossover fraction \(=0.8\); elite count \(=5 \%\); migration fraction \(=\) 0.2 ; migration interval \(=20\).

The optimum formulation using the genetic algorithm displayed a Vis at 100 \({ }^{\circ} \mathrm{C}\), Vis at \(40^{\circ} \mathrm{C}\), VI, FP, Noack and PP of \(82.1 \mathrm{cSt}, 14.6 \mathrm{cSt}, 188.4,225^{\circ} \mathrm{C}, 5.6\), and -39 respectively when the formulation was composed of \(67.0 \% \mathrm{PAO} 4,22.1 \%\) 5780, and \(10.8 \%\) HI 11100. Validation was performed in triplicate and the average values for Viscosity at \(100^{\circ} \mathrm{C}, 40^{\circ} \mathrm{C}, \mathrm{VI}, \mathrm{FP}\) and Noack were \(82.1 \mathrm{cSt}, 14.6 \mathrm{cSt}\), \(188.5,226^{\circ} \mathrm{C}, 5.6\), and-39, respectively. The results of optimization are similar to the experimental data.

Therefore, the combination of ANN and GA (i.e. GA-ANN) find the solution with minimum error from experimental data. Also, the mean fitness value in each generation is shown in Fig 4.

Comparison mixture design and ANN

The predicted data obtained from RSM-mixture design and ANN and the experimental data were compared. To this aim, a new set of 4 experiments was considered in the acceptable range, which does not belong to the training data set (Table 10). Table 11 shows the predicted data of RSM-mixture design and ANN.

Obviously ANN was more powerful in prediction of the process. In order to evaluate the precision and accuracy of both applied models, the mean squared error (MSE) was calculated for RSM and ANN. MSE can be defined by Eq. (9):

(9) \[ M S E=\tfrac{1}{N} \sum_{1}^{N}\left(y_{i, \exp }-y_{i, p r e d}\right)^{2} \]

where MSE is the mean squared error; N is the number of experimental data points; yi,exp is the experimental value of training sample i and yi,pred is the predicted value from the neural network for training sample i.

The MSE was 130.82 and 55.9 for RSM and ANN respectively. Moreover, a regression analysis for predicted data by ANN and experimental data was done. As the results show in Fig. 5, all data scatter around the \(45^{\circ}\) line which is an indicative for high suitability of ANN prediction for the process. According to the obtained results, ANN had higher predictive accuracy than RSM-mixture design even with limited number of experiments.

Comparison of optimization by RSM and GA-ANN

RSM and GAANN were used to predict and optimize the proposed process. Table 12 and Table 13 show the comparison of the results obtained by two methods for multi-objective which indicate the suitability of both RSM and GA ANN for this study.

Conclusions

In this paper, RSM Mixture design with six independent variables and ANN with five neurons in hidden layer were used for formulation of engine oil. RSM and ANN models were successfully employed for prediction and optimization of the process. The comparison of the obtained results from two statistical methods of RSM- mixture design and ANN confirmed that ANN model had better prediction capability of target values. In this study, the genetic algorithm was used as the optimization algorithm. The results indicate that there was no significant difference between the experimental and predicted data. Achieving optimized components by application of genetic algorithm leads to an economical formulation due to the lower consumption of expensive additives HI 5780 (preserving the properties of oil at their optimal values based on ISOgrade for engine oil).

Acknowledgments: The financial support of this project by Shahrekord University is appreciated. The authors were also partially supported by the Center of Excellence for Mathematics, Shahrekord University.

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