# Application of Artificial Neural Networks for Producing an Estimation of High-Density Polyethylene

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

^{2}(regression coefficient), MSE (mean square error), AARD% (average absolute relative deviation percent), and RMSE (root mean square error) of, respectively, 0.89413, 0.02217, 0.4213, and 0.1489.

## 1. Introduction

_{4}supported on MgCl

_{2}and commonly employed for the polyethylene production in the industry [5].

## 2. Methods

#### 2.1. HDPE Process

#### 2.2. Artificial Neural Networks

#### 2.3. Accuracy Assessment of AI Models

^{2}(regression coefficient), MSE (mean square error), AARD% (average absolute relative deviation percent), and RMSE (root mean square error) are calculated, respectively, as the following:

^{2}index defines the proximity of the actual data points to the predicted values.

## 3. Results and Discussion

#### 3.1. Industrial Database

^{®}with the Levenberg–Marquardt optimization algorithm. Besides, the choice of training algorithm and neuron transfer function has a major contribution to model precision. As researchers have shown, the Levenberg–Marquardt (LM) algorithm produces quicker responses for regression-type problems in overall facets of neural networks [31,32]. Most often, the LM training algorithm was reported to have the highest significant efficiency, fast convergence, and accuracy compared with other training algorithms.

#### 3.2. Scaling the Data

#### 3.3. Independent Variable Selection

#### 3.4. Configuration Selection for Different ANN Approaches

^{2}, MSE, and RMSE between actual and estimated data. According to the literature, MLP network capability with one hidden layer was proven [35]. As such, an MLP network with only a single hidden layer is used for the analysis.

^{2}(0.89413) and the lowermost value of MSE (0.02217).

^{2}values.

#### 3.5. Other Types of ANN

^{2}.

## 4. Procedure for Simple Usage of the MLP Model

- All independent variables normalize into an interval of [0.01 0.99] using Equation (9) and should be arranged as a 6×1 vector.

_{HL}= 1/[1 + exp(−values obtained in step 3)]

_{OL}= 1/[1 + exp(−value obtained in step 6)]

- 8.
- Inverse transformation using NO
_{OL}^ (1/12). - 9.
- Map the output values in the previous step into the actual range of dependent variables, i.e., [24.3 26.9], using the following equation.

- 10.
- The obtained value in step 9 shows the estimated value for the dependent variable by the proposed MLP approach.

## 5. Conclusions

^{2}values of the total dataset are 0.02217 and 0.89413, respectively. The main advantages of using ANNs for the EIX are the ability to predict the production rate of the network quickly and to clarify the characteristics of high-density polyethylene with network inputs. Although these models have used complex computational algorithms, fast convergence along with accuracy is not always confirmed in some cases.

## Nomenclature

b | Bias |

N | Number of actual data |

X_{i} | i^{th} input variable |

Y | Response |

w | Weight |

## Abbreviations

AARD% | Average absolute relative deviation percent |

AI | Artificial intelligence |

ANN | Artificial neural networks |

CFNN | Cascade feedforward neural networks |

CSTR | Continuously stirred tank reactors |

EIX | Ethylene index |

HDPE | High-density polyethylene |

MLP | Multi-layer perceptron |

MLPNN | Multi-layer perceptron neural networks |

MSE | Mean squared errors |

RBF | Radial basis neural networks |

RMSE | Root mean square errors |

R2 | Regression coefficient |

TEA | Triethylaluminium |

## Subscripts/Superscripts

pre. | Predicted variable |

act. | Actual variable |

max | Maximum value |

min | Minimum value |

normal | Normalized values |

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A diagram of slurry polymerization for HDPE production [20].

**Figure 2.**Schematic representation of the used model for neural networks [30].

**Figure 4.**The MSE for the multi-layer perceptron (MLP) network with 1 to 4 hidden neurons (50 networks per neuron) during the training stage.

Independent Variables | Named | Minimum | Maximum |
---|---|---|---|

Temperature (°C) | Input 1 | 77 | 97.8 |

Pressure (bar) | Input 2 | 21.8 | 22.6 |

Level (%) | Input 3 | 68 | 77 |

Loop flow (Kg/h) | Input 4 | 646 | 724 |

Ethylene flow (Kg/h) | Input 5 | 1.8 | 15.6 |

Hydrogen flow (Kg/h) | Input 6 | 6.1 | 25 |

1-butane flow (Kg/h) | Input 7 | 17 | 970 |

Hydrogen concentration (mol%) | Input 8 | 16.02 | 19.03 |

1-butane concentration (mol%) | Input 9 | 0.49 | 1.38 |

Catalyst flow (Kg/h) | Input 10 | 2 | 3.7 |

TEA flow (Kg/h) | Input 11 | 1.8 | 3.7 |

Independent Variables | Pearson Correlation Coefficient |
---|---|

Input1 | −0.0430 |

Input2 | 0.2452 |

Input3 | 0.0581 |

Input4 | 0.0252 |

Input5 | 0.1606 |

Input6 | 0.0956 |

Input7 | −0.1160 |

Input8 | 0.0052 |

Input9 | −0.0296 |

Input10 | 0.0935 |

Input11 | 0.1075 |

**Table 3.**Pearson coefficient values calculated for different transformations between dependent and independent variables.

Transformation | Pearson’s Coefficient | AAPC | |||||
---|---|---|---|---|---|---|---|

Input11 | Input10 | Input7 | Input6 | Input5 | Input2 | ||

Output^{15} | 0.09837 | 0.08478 | −0.12881 | 0.0877 | 0.17071 | 0.28188 | 0.14204 |

Output^{14} | 0.09943 | 0.08576 | −0.12845 | 0.08885 | 0.17077 | 0.27997 | 0.1422 |

Output^{13} | 0.10045 | 0.0867 | −0.12801 | 0.08992 | 0.17073 | 0.27795 | 0.14229 |

Output^{12} | 0.10143 | 0.0876 | −0.12750 | 0.09092 | 0.17058 | 0.27584 | 0.14231 |

Output^{11} | 0.10235 | 0.08845 | −0.12690 | 0.09184 | 0.17033 | 0.27361 | 0.14225 |

Output^{10} | 0.10321 | 0.08925 | −0.12623 | 0.09268 | 0.16996 | 0.27128 | 0.1421 |

Output^{2} | 0.1074 | 0.09331 | −0.11754 | 0.09572 | 0.16225 | 0.24851 | 0.13745 |

Output | 0.10751 | 0.09346 | −0.11603 | 0.09556 | 0.16063 | 0.24516 | 0.13639 |

Output^{0.75} | 0.10753 | 0.09349 | −0.11563 | 0.0955 | 0.1602 | 0.24431 | 0.13611 |

Output^{0.5} | 0.10753 | 0.0935 | −0.11523 | 0.09543 | 0.15976 | 0.24344 | 0.13582 |

Output^{0.25} | 0.10753 | 0.09351 | −0.11483 | 0.09535 | 0.15932 | 0.24258 | 0.13552 |

Output^{0.1} | 0.10753 | 0.09352 | −0.11458 | 0.0953 | 0.15904 | 0.24205 | 0.13534 |

Output^{−0.1} | −0.10752 | −0.09352 | 0.11425 | −0.09523 | −0.15867 | −0.24135 | 0.13509 |

Output^{−0.25} | −0.10751 | −0.09352 | 0.114 | −0.09517 | −0.15839 | −0.24082 | 0.1349 |

Output^{−0.5} | −0.10749 | −0.09351 | 0.11357 | −0.09507 | −0.15791 | −0.23994 | 0.13458 |

Output^{−0.75} | −0.10747 | −0.09350 | 0.11314 | −0.09496 | −0.15743 | −0.23904 | 0.13426 |

Output^{−1} | −0.10743 | −0.09348 | 0.11271 | −0.09484 | −0.15693 | −0.23814 | 0.13392 |

Output^{−2} | −0.10723 | −0.09335 | 0.11091 | −0.09427 | −0.15485 | −0.23449 | 0.13252 |

Output^{−10} | −0.10154 | −0.08867 | 0.09353 | −0.08469 | −0.13296 | −0.20259 | 0.11733 |

Output^{−11} | −0.10034 | −0.08765 | 0.09106 | −0.08289 | −0.12965 | −0.19840 | 0.115 |

Output^{−12} | −0.09903 | −0.08653 | 0.08855 | −0.08097 | −0.12624 | −0.19420 | 0.11259 |

Output^{−13} | −0.09763 | −0.08533 | 0.08599 | −0.07894 | −0.12273 | −0.19000 | 0.11011 |

Output^{−14} | −0.09614 | −0.08405 | 0.08341 | −0.07680 | −0.11914 | −0.18581 | 0.10756 |

Output^{−15} | −0.09456 | −0.08269 | 0.0808 | −0.07457 | −0.11548 | −0.18165 | 0.10496 |

exp(Output) | 0.08336 | 0.07114 | −0.12809 | 0.07017 | 0.16382 | 0.29757 | 0.13569 |

Hidden Neuron | Dataset | Statistical Index | |||
---|---|---|---|---|---|

AARD% | MSE | RMSE | R^{2} | ||

1 | Train | 0.6566 | 0.07274 | 0.2697 | 0.6043 |

Test | 0.7578 | 0.06665 | 0.2582 | 0.35034 | |

Total | 0.6715 | 0.07184 | 0.268 | 0.57942 | |

2 | Train | 0.568 | 0.06603 | 0.257 | 0.66405 |

Test | 0.7314 | 0.05583 | 0.2363 | 0.65288 | |

Total | 0.592 | 0.06454 | 0.254 | 0.644 | |

3 | Train | 0.3863 | 0.01982 | 0.1408 | 0.91723 |

Test | 0.6251 | 0.03581 | 0.1892 | 0.53962 | |

Total | 0.4213 | 0.02217 | 0.1489 | 0.89413 | |

4 | Train | 0.3397 | 0.02037 | 0.1427 | 0.90878 |

Test | 0.8084 | 0.07617 | 0.276 | 0.44131 | |

Total | 0.4085 | 0.02856 | 0.169 | 0.86377 |

Hidden Neuron | Dataset | Statistical Index | |||
---|---|---|---|---|---|

AARD% | MSE | RMSE | R^{2} | ||

1 | Train | 0.5604 | 0.07049 | 0.2655 | 0.62338 |

Test | 0.6377 | 0.05431 | 0.233 | 0.58962 | |

Total | 0.5717 | 0.06812 | 0.261 | 0.60867 | |

2 | Train | 0.4961 | 0.05227 | 0.2286 | 0.74815 |

Test | 0.6904 | 0.06022 | 0.2454 | 0.36408 | |

Total | 0.5246 | 0.05344 | 0.2312 | 0.71356 | |

3 | Train | 0.488 | 0.03638 | 0.1907 | 0.83777 |

Test | 0.5729 | 0.0552 | 0.235 | 0.48687 | |

Total | 0.5004 | 0.03914 | 0.1979 | 0.79892 |

Hidden Neuron | Spread | Dataset | Statistical Index | |||
---|---|---|---|---|---|---|

AARD% | MSE | RMSE | R^{2} | |||

1 | 0.41 | Train | 0.762 | 0.10257 | 0.3203 | 0.30383 |

Test | 0.8574 | 0.06878 | 0.2623 | 0.28658 | ||

Total | 0.776 | 0.09762 | 0.3124 | 0.312 | ||

2 | 0.81 | Train | 0.7563 | 0.09887 | 0.3144 | 0.42263 |

Test | 0.762 | 0.05584 | 0.2363 | −0.10195 | ||

Total | 0.7571 | 0.09255 | 0.3042 | 0.38919 | ||

3 | 1.01 | Train | 0.7311 | 0.09259 | 0.3043 | 0.45482 |

Test | 0.774 | 0.05724 | 0.2392 | 0.33916 | ||

Total | 0.7374 | 0.08741 | 0.2956 | 0.43965 | ||

4 | 0.21 | Train | 0.6708 | 0.08324 | 0.2885 | 0.51622 |

Test | 0.7579 | 0.08244 | 0.2871 | 0.4649 | ||

Total | 0.6836 | 0.08312 | 0.2883 | 0.48421 |

**Table 7.**Sensitivity analyses on spread parameter for finding best hidden neurons for general regression (GR).

Spread | Dataset | Statistical Index | |||
---|---|---|---|---|---|

AARD% | MSE | RMSE | R^{2} | ||

4.81 | Train | 0.8361 | 0.10982 | 0.3314 | 0.32248 |

Test | 0.9242 | 0.098 | 0.313 | −0.20907 | |

Total | 0.849 | 0.10808 | 0.3288 | 0.22754 |

Model | Dataset | Statistical Index | |||
---|---|---|---|---|---|

AARD% | MSE | RMSE | R^{2} | ||

MLP | Train | 0.3863 | 0.01982 | 0.1408 | 0.91723 |

Test | 0.6251 | 0.03581 | 0.1892 | 0.53962 | |

Total | 0.4213 | 0.02217 | 0.1489 | 0.89413 | |

CF | Train | 0.488 | 0.03638 | 0.1907 | 0.83777 |

Test | 0.5729 | 0.0552 | 0.235 | 0.48687 | |

Total | 0.5004 | 0.03914 | 0.1979 | 0.79892 | |

GR | Train | 0.8361 | 0.10982 | 0.3314 | 0.32248 |

Test | 0.9242 | 0.098 | 0.313 | −0.20907 | |

Total | 0.849 | 0.10808 | 0.3288 | 0.22754 | |

RBF | Train | 0.7563 | 0.09887 | 0.3144 | 0.42263 |

Test | 0.762 | 0.05584 | 0.2363 | −0.10195 | |

Total | 0.7571 | 0.09255 | 0.3042 | 0.38919 |

Weights between Hidden Layer Neurons and Input Variables | Bias of Hidden Layer Neurons | Weight between the Hidden Layer and the Output Layer | Output Layer Bias | |||||
---|---|---|---|---|---|---|---|---|

Input2 | Input5 | Input6 | Input7 | Input10 | Input11 | |||

37.522 | 325.0769 | −1007.47 | −3.6761 | −330.513 | −168.349 | 176.6308 | −0.94888 | 0.69286 |

272.4066 | −70.6262 | −94.8238 | 4.1889 | −150.163 | 2.1714 | −16.8291 | 2.7494 | |

−1095.55 | 1458.018 | 70.1148 | 370.1613 | −539.429 | −12.1413 | 9.7747 | −2.276 |

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## Share and Cite

**MDPI and ACS Style**

Maleki, A.; Safdari Shadloo, M.; Rahmat, A.
Application of Artificial Neural Networks for Producing an Estimation of High-Density Polyethylene. *Polymers* **2020**, *12*, 2319.
https://doi.org/10.3390/polym12102319

**AMA Style**

Maleki A, Safdari Shadloo M, Rahmat A.
Application of Artificial Neural Networks for Producing an Estimation of High-Density Polyethylene. *Polymers*. 2020; 12(10):2319.
https://doi.org/10.3390/polym12102319

**Chicago/Turabian Style**

Maleki, Akbar, Mostafa Safdari Shadloo, and Amin Rahmat.
2020. "Application of Artificial Neural Networks for Producing an Estimation of High-Density Polyethylene" *Polymers* 12, no. 10: 2319.
https://doi.org/10.3390/polym12102319