# The Application of a Hybrid Model Using Mathematical Optimization and Intelligent Algorithms for Improving the Talc Pellet Manufacturing Process

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## Abstract

**:**

## 1. Introduction

_{g3}SiO

_{10}(OH)

_{2}. Talc has low abrasion, high thermal conductivity and stability, low electrical conductivity, and high oil and grease adsorption [1]. Due to its unique surface chemistry, lamellar crystal habit, and properties, talc minerals are widely applied commercially and industrial, such as in cosmetics, pharmaceuticals, paints, polymers, and ceramics. Furthermore, the method evaluated the suitable properties of talc, which can contribute to the industry in terms of the efficiency of production planning.

_{2}O

_{3}-MWCNT/oil. Based on the result, they found that both of the ANFIS-PSO and ANFIS-GA models are able to predict the thermophysical properties appropriately. Kumar and Hynes [14] predicted and optimized the surface roughness in thermal drilling by integrating ANFIS and GA. Rezakazemi et al. [15] employed ANFIS with GA and PSO for the evaluation of H

_{2}-selective mixed matrix membranes (MMMs). The results showed that the ANFIS with PSO is more reliable than the ANFIS with GA and the traditional ANFIS. Sabeti and Deevband [16] introduced a novel training method of ANFIS by combining PSO and GA to solve the nonlinear dynamical system. The proposed PSOGA method provides the satisfactory results.

## 2. Materials and Methods

#### 2.1. Talc Pellet Forming Process

#### 2.2. Method

#### 2.2.1. Self-Organizing Map

**x**, is a random distribution which corresponds to the column index. The set of weight vectors is formed as ${w}_{i}=[{w}_{ij}],\mathrm{i}=1,2,\dots ,{k}_{x},\text{}\mathrm{j}=1,2,\dots ,{k}_{y}$ where ${k}_{x}$ is the number of row and ${k}_{y}$ is the number of columns. The three phases for calculating the SOM algorithm are shown below.

#### 2.2.2. Adaptive Neuro-Fuzzy Inference System

- Layer 1: Adjust every node by using Equation (9):$${O}_{i}^{1}={\mu}_{ij}(x)$$

- Layer 2: Calculate each node by multiplying the fuzzy value. The output is calculated as:$${O}_{i}^{2}={w}_{i}={\displaystyle \prod _{j=1}^{m}{A}_{i}{}_{j}}({x}_{i})$$
- Layer 3: Sum the fuzzy value of every node to one value by:$${O}_{i}^{3}={\overline{w}}_{i}=\frac{{w}_{i}}{{\displaystyle {\sum}_{i=1}^{n}{w}_{i}}}$$
- Layer 4: Normalize the fuzzy value of every node by:$${O}_{i}^{4}={\overline{w}}_{i}{f}_{i}$$
- Layer 5: Sum all output from layer four to obtain the final output by:$${O}_{i}^{5}={\displaystyle \sum _{i=1}^{n}{\overline{w}}_{i}{f}_{i}}$$

#### 2.2.3. The ANFIS Training Algorithm

#### Genetic Algorithm

- Chromosome encodes: Design the chromosomes as the system-represented solution by using any encoding method on the solving condition.
- Population initialization: Initialize the prototype population at the beginning of GA. The first population group is randomly created by matching with the defined population size.
- The fitness function: Define the score of each possible solution. Every chromosome implies the fitness of the inheritance consideration for themselves in order to create the next-generation chromosome.
- Selection: Select the genetic operator that supports the worthy member to transfer into the next generation. The process of selecting the best chromosome among the whole population is normally selected by good origin for good species according to the natural selection concept.
- Crossover: The copying of the new chromosome is pasted at a random position of the father and behind the random position of the mother to become the first offspring chromosome. The second offspring chromosome occurs by the same process as the first offspring while switching the position of the father and mother.
- Mutation: Mutate the value of the chromosome. The mutation process randomly mutates the position under the mutation possibility by changing some genes on the chromosome.
- Replacement: Replace the previous generation chromosomes with mutated chromosomes.
- Termination condition: Terminate the procedure when the condition is satisfied.

#### Particle Swarm Optimization

#### 2.2.4. Performance Evaluation

## 3. The Proposed Model

#### 3.1. The Experimental Design

#### 3.2. A Hybrid Model

#### 3.3. The Experimental Setting

## 4. Result and Discussion

#### 4.1. The Results of the SOM Algorithm

#### 4.2. The Optimal Parameter of HM-GA

#### 4.3. The Optimal Parameter of HM-PSO

#### 4.4. The Comparison Results Approach from HM-GA and HM-PSO

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**An architecture of ANFIS [12].

**Figure 4.**The optimal parameter of GA algorithm: (

**a**) population size; (

**b**) crossover percentage; (

**c**) mutation percentage.

**Figure 6.**The difference between the target and predicted moisture: (

**a**) HM-GA; (

**b**) HM-PSO; (

**c**) ANFIS-GA; (

**d**) ANFIS-PSO.

**Figure 7.**The correlation between the target and predicted moisture: (

**a**) HM-GA; (

**b**) HM-PSO; (

**c**) ANFIS-GA; (

**d**) ANFIS-PSO.

Factor | Symbol | The Value of α | Unit | ||||
---|---|---|---|---|---|---|---|

1.682 | 1.0 | 0 | −1.0 | −1.682 | |||

Talc | Ta | 18.6892 | 18 | 17.5 | 17 | 16.3107 | kg |

Water | W | 4.3446 | 4 | 3.75 | 3.5 | 3.1553 | kg |

Temperature | Temp | 191.35 | 150 | 120 | 90 | 48.65 | °C |

Feed Speed | FS | 0.56 | 0.43 | 0.34 | 0.24 | 0.11 | m/min |

Air Flow | AF | 8.40 | 7.21 | 6.35 | 5.48 | 4.29 | m/sec |

No. | Ta | W | Temp | FS | AF | M_{C} | No. | Ta | W | Temp | FS | AF | M_{C} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 17 | 3.5 | 90 | 0.24 | 7.21 | 6.31 | 27 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 1.38 |

2 | 18 | 3.5 | 90 | 0.24 | 5.48 | 2.89 | 28 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 2.33 |

3 | 17 | 4 | 90 | 0.24 | 5.48 | 3.94 | 29 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3.85 |

4 | 18 | 4 | 90 | 0.24 | 7.21 | 7.02 | 30 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3.75 |

5 | 17 | 3.5 | 150 | 0.24 | 5.48 | 0.56 | 31 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3.07 |

6 | 18 | 3.5 | 150 | 0.24 | 7.21 | 0.51 | 32 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 2.56 |

7 | 17 | 4 | 150 | 0.24 | 7.21 | 0.39 | 33 | 17 | 3.5 | 90 | 0.24 | 5.48 | 4.19 |

8 | 18 | 4 | 150 | 0.24 | 5.48 | 0.42 | 34 | 18 | 4 | 90 | 0.24 | 5.48 | 5.37 |

9 | 17 | 3.5 | 90 | 0.43 | 5.48 | 11.17 | 35 | 17 | 3.5 | 150 | 0.24 | 7.21 | 0.45 |

10 | 18 | 3.5 | 90 | 0.43 | 7.21 | 8.96 | 36 | 17 | 4 | 150 | 0.24 | 5.48 | 0.41 |

11 | 17 | 4 | 90 | 0.43 | 7.21 | 7.86 | 37 | 18 | 3.5 | 90 | 0.43 | 5.48 | 7.47 |

12 | 18 | 4 | 90 | 0.43 | 5.48 | 8.76 | 38 | 17 | 4 | 90 | 0.43 | 5.48 | 6.16 |

13 | 17 | 3.5 | 150 | 0.43 | 7.21 | 1.41 | 39 | 17 | 3.5 | 150 | 0.43 | 5.48 | 4.34 |

14 | 18 | 3.5 | 150 | 0.43 | 5.48 | 2.61 | 40 | 18 | 3.5 | 150 | 0.43 | 7.21 | 1.08 |

15 | 17 | 4 | 150 | 0.43 | 5.48 | 2.22 | 41 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3 |

16 | 18 | 4 | 150 | 0.43 | 7.21 | 0.85 | 42 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3.33 |

17 | 16.31 | 3.75 | 120 | 0.34 | 6.35 | 2.11 | 43 | 18 | 4 | 150 | 0.43 | 5.48 | 2.39 |

18 | 18.69 | 3.75 | 120 | 0.34 | 6.35 | 1.64 | 44 | 18 | 3.5 | 90 | 0.24 | 7.21 | 5.61 |

19 | 17.5 | 3.16 | 120 | 0.34 | 6.35 | 1.56 | 45 | 17 | 4 | 90 | 0.24 | 7.21 | 4.66 |

20 | 17.5 | 4.34 | 120 | 0.34 | 6.35 | 3.19 | 46 | 18 | 3.5 | 150 | 0.24 | 5.48 | 0.62 |

21 | 17.5 | 3.75 | 49 | 0.34 | 6.35 | 15.36 | 47 | 18 | 4 | 150 | 0.24 | 7.21 | 0.4 |

22 | 17.5 | 3.75 | 191 | 0.34 | 6.35 | 0.36 | 48 | 17 | 3.5 | 90 | 0.43 | 7.21 | 4.74 |

23 | 17.5 | 3.75 | 120 | 0.11 | 6.35 | 0.66 | 49 | 18 | 4 | 90 | 0.43 | 7.21 | 8.38 |

24 | 17.5 | 3.75 | 120 | 0.56 | 6.35 | 7.7 | 50 | 17 | 4 | 150 | 0.43 | 7.21 | 1.28 |

25 | 17.5 | 3.75 | 120 | 0.34 | 4.29 | 5.54 | 51 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3.04 |

26 | 17.5 | 3.75 | 120 | 0.34 | 8.4 | 2.21 | 52 | 17.5 | 3.75 | 120 | 0.34 | 6.35 | 3.25 |

SOM | Value | ANFIS | Value |
---|---|---|---|

Initial learning rate | 0.85 | Fuzzy type | Sugeno |

Initial weight vector | Random | Input/outputs | 5/1 |

Max. radius of neighbourhood | Size 10 | Input MF type | Gaussian |

Max. number of iterations | 5000 | Output MF type | Linear |

SOM array size | 2 × 2, 3 × 3, 4 × 4 | Training algorithm | GA, PSO |

No. of MFs for each input | 10 | ||

Fuzzy rules | 10 |

GA | Value | PSO | Value |
---|---|---|---|

Population Size | 350 | Population size | 450 |

Iteration | 10,000 | Iteration | 5000 |

Crossover percentage | [0.6,0.9] | Inertia weight | 1.0 |

Mutation percentage | (0,1) | Damping ratio | 0.99 |

Mutation ratio | 0.1 | Personal learning coefficient | 1.0 |

Selection pressure | 8 | Global learning coefficient | 2.0 |

Gamma | 0.2 |

Cluster | 1 | 2 |
---|---|---|

1 | 48.07 | 7.69 |

2 | 1.92 | 42.31 |

Model | Train Data | Test Data | ||||
---|---|---|---|---|---|---|

R | RMSE | AAD | R | RMSE | AAD | |

HM-GA | 0.9682 | 0.8984 | 0.401 | 0.7113 | 2.3959 | 0.493 |

HM-PSO | 0.9539 | 1.0693 | 0.393 | 0.9192 | 0.9785 | 0.376 |

ANFIS-GA | 0.9784 | 0.7203 | 0.314 | 0.7598 | 2.5396 | 0.416 |

ANFIS-PSO | 0.9641 | 0.9137 | 0.333 | 0.8431 | 2.0327 | 0.485 |

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**MDPI and ACS Style**

Buntam, D.; Permpoonsinsup, W.; Surin, P.
The Application of a Hybrid Model Using Mathematical Optimization and Intelligent Algorithms for Improving the Talc Pellet Manufacturing Process. *Symmetry* **2020**, *12*, 1602.
https://doi.org/10.3390/sym12101602

**AMA Style**

Buntam D, Permpoonsinsup W, Surin P.
The Application of a Hybrid Model Using Mathematical Optimization and Intelligent Algorithms for Improving the Talc Pellet Manufacturing Process. *Symmetry*. 2020; 12(10):1602.
https://doi.org/10.3390/sym12101602

**Chicago/Turabian Style**

Buntam, Dussadee, Wachirapond Permpoonsinsup, and Prayoon Surin.
2020. "The Application of a Hybrid Model Using Mathematical Optimization and Intelligent Algorithms for Improving the Talc Pellet Manufacturing Process" *Symmetry* 12, no. 10: 1602.
https://doi.org/10.3390/sym12101602