# The Bi-Directional Prediction of Carbon Fiber Production Using a Combination of Improved Particle Swarm Optimization and Support Vector Machine

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## Abstract

**:**

## 1. Introduction

## 2. Carbon Fiber Production and Its Bi-Directional Optimization

#### 2.1. The Process of Carbon Fiber Production

#### 2.2. The Bi-Directional Prediction Methods for Carbon Fiber Production

## 3. Methodology of the SVM-IPSO Model

#### 3.1. The SVM Model

_{i}, and N is the total number of the input vectors. By nonlinear mapping φ(x), the data are mapped nonlinearly from the original feature space to a high-dimensional feature space, thus, it can be approximated in a linear way as follows:

_{ξ}(y

_{i}, f(x)) denotes the ξ-insensitive loss function and is defined as Equation (3)

_{i}to simplify the risk function R(f) in Equation (2). Thus, Equation (2) is transformed into the objective function shown as follows:

_{i}, x

_{j}) is called the kernel function, which is the inner product of φ(x

_{i}) and φ(x

_{j}), that is, K(x

_{i}, x

_{j})= φ(x

_{i})•φ(x

_{j}). The values of φ(x

_{i}) and φ(x

_{j}) are produced by mapping x

_{i}and x

_{j}into the higher-dimensional feature space, respectively. It can be shown that any function meeting Mercer’s criteria [29] can be used as the kernel function.

#### 3.2. Overview of Particle Swarm Optimization

_{1}and c

_{2}are social and personal learning parameters, r

_{1}and r

_{2}are random numbers in the range [0,1]; and ${X}_{iD}^{t}$ denotes its previous position of the i-th particle at time t.

#### 3.3. Improved Particle Swarm Optimization

#### 3.3.1. The Basic Concept of the Cell Communication

#### 3.3.2. IPSO Based on the Communication Mechanism of Cells

_{i}with D variables can be expressed as:

_{k}denotes the total number of similar particles, and N denotes the total number of particles.

_{k}> 0.5):

_{i}

_{1}denotes the first value of the i-th particle; and x

_{b}

_{1}denotes the first value of the best particle.

_{i}

_{2}denotes the second value of the i-th particle, and x

_{b}

_{2}denotes the second value of the best particle.

_{i}denotes the i-th particle; and x

_{b}denotes the best particle of the whole particles.

_{1}= c

_{2}= 2. Furthermore, ω denotes the particle’s search capability, and defines the balance between local search capabilities and global search capabilities. Once the velocity of the particle is calculated, its new position is updated according to Equation (10). The range of permissible velocity of the particle is usually restricted in order to eliminate fluctuations and obtain better exploration during the iteration process.

#### 3.4. SVM Based on the IPSO

**Figure 4.**Diagram of the support vector machine and improved particle swarm optimization (SVM-IPSO) forecasting model.

_{1},x

_{2}] to denote the new position for food finding of an individual particle i.

_{i}of an individual particle i, and then input X

_{i}into the SVM model for carbon fiber production forecasting; the parameters [C,σ] of the SVM model are replaced by [X

_{i}(1),X

_{i}(2)]. According to the carbon fiber production forecasting result, the f

_{i}can be calculated. The f

_{i}is represented by the root-mean-square error, as shown in Equation (18), which calculates the errors between the actual values and the forecasted values:

_{i}is the i-th actual value; ${\widehat{y}}_{i}$ denotes the i-th forecasting value.

## 4. Simulation and Discussion

#### 4.1. The Preprocessing of Sample Data

No. | Viscosity Average Molecular Weight (10^{4}) | Conversion Ratio (%) | Solid Content (%) | Spinning Jet Drawing Ratio (%) | Coagulating Bath Temperature (°C) | Total Drawing Ratio | Strength (CN/d) | Structure Parameter |
---|---|---|---|---|---|---|---|---|

1 | 8.9 | 94.5 | 20.8 | −50.3 | 14 | 6.33 | 4.08 | 14.82 |

2 | 6.3 | 91.0 | 20.0 | −59.7 | 15 | 5.89 | 3.23 | 12.63 |

3 | 11.6 | 92.0 | 20.4 | −50.5 | 14 | 6.03 | 3.76 | 13.24 |

4 | 8.8 | 94.8 | 21.8 | −63.4 | 13 | 6.65 | 4.17 | 17.24 |

5 | 7.0 | 81.8 | 17.9 | −63.4 | 15 | 6.32 | 3.99 | 15.14 |

6 | 8.2 | 85.5 | 21.7 | −59.5 | 15 | 5.49 | 4.58 | 16.61 |

7 | 7.2 | 89.8 | 19.5 | −53.1 | 13 | 5.88 | 3.64 | 15.49 |

8 | 8.9 | 82.5 | 17.5 | −56.8 | 19 | 6.38 | 4.07 | 17.57 |

9 | 8.0 | 83.4 | 18.6 | −62.1 | 17 | 5.72 | 3.18 | 15.48 |

10 | 11.7 | 90.6 | 17.9 | −53.8 | 16 | 6.47 | 3.22 | 12.10 |

11 | 11.5 | 82.8 | 18.7 | −64.8 | 17 | 5.79 | 3.27 | 12.73 |

12 | 6.3 | 95.1 | 19.6 | −54.9 | 16 | 6.37 | 4.36 | 17.18 |

13 | 10.4 | 98.6 | 20.2 | −68.3 | 17 | 6.41 | 3.99 | 14.91 |

14 | 7.6 | 93.1 | 19.7 | −55.4 | 16 | 5.88 | 3.38 | 17.07 |

15 | 8.5 | 84.6 | 22.3 | −65.3 | 18 | 5.04 | 3.99 | 13.26 |

16 | 9.3 | 89.8 | 20.1 | −53.8 | 16 | 5.66 | 3.30 | 15.31 |

17 | 11.7 | 79.4 | 22.7 | −55.8 | 19 | 5.85 | 3.11 | 15.78 |

18 | 8.5 | 96.9 | 20.8 | −51.8 | 14 | 5.54 | 4.70 | 12.19 |

19 | 11.9 | 96.4 | 22.7 | −61.5 | 13 | 5.39 | 4.12 | 15.69 |

20 | 7.8 | 95.0 | 18.4 | −63.7 | 13 | 6.64 | 4.86 | 14.17 |

21 | 10.2 | 82.7 | 21.1 | −60.9 | 13 | 5.86 | 4.39 | 12.30 |

22 | 10.0 | 90.1 | 18.7 | −58.5 | 15 | 6.78 | 4.17 | 14.94 |

23 | 9.2 | 77.5 | 21.0 | −62.9 | 16 | 5.78 | 4.63 | 13.16 |

24 | 10.2 | 86.4 | 21.2 | −63.0 | 15 | 6.54 | 4.76 | 12.74 |

25 | 10.0 | 83.9 | 17.4 | −63.6 | 18 | 5.79 | 4.98 | 13.23 |

26 | 7.1 | 80.6 | 18.5 | −62.7 | 17 | 6.62 | 3.00 | 12.88 |

27 | 6.8 | 80.9 | 18.3 | −68.9 | 18 | 6.51 | 4.73 | 13.13 |

28 | 12.0 | 86.3 | 21.0 | −54.2 | 19 | 5.75 | 4.23 | 12.26 |

29 | 7.0 | 79.1 | 22.1 | −64.2 | 19 | 5.43 | 4.98 | 15.81 |

30 | 6.2 | 90.2 | 19.1 | −54.7 | 14 | 6.58 | 4.06 | 13.69 |

31 | 9.4 | 87.4 | 21.7 | −52.4 | 13 | 6.90 | 3.96 | 15.23 |

32 | 11.3 | 92.3 | 21.1 | −62.1 | 17 | 5.66 | 4.60 | 16.17 |

33 | 10.0 | 92.4 | 17.0 | −59.0 | 13 | 6.34 | 3.46 | 14.99 |

34 | 7.1 | 91.0 | 20.6 | −59.2 | 16 | 5.88 | 4.00 | 15.21 |

35 | 8.2 | 77.7 | 19.3 | −63.2 | 16 | 6.67 | 4.80 | 14.67 |

36 | 8.8 | 78.5 | 22.5 | −65.4 | 19 | 6.54 | 4.15 | 12.74 |

37 | 11.9 | 84.0 | 17.0 | −57.0 | 16 | 5.33 | 4.69 | 14.94 |

38 | 6.9 | 88.7 | 19.8 | −63.2 | 15 | 6.72 | 4.48 | 17.12 |

39 | 11.1 | 91.4 | 19.5 | −58.3 | 17 | 6.98 | 4.17 | 17.24 |

40 | 9.9 | 86.0 | 19.8 | −66.8 | 18 | 6.03 | 3.49 | 13.62 |

41 | 8.3 | 95.0 | 21.6 | −66.7 | 16 | 6.77 | 4.33 | 13.25 |

42 | 7.1 | 92.8 | 18.9 | −55.1 | 15 | 6.18 | 3.17 | 15.39 |

43 | 8.6 | 98.3 | 21.7 | −62.3 | 14 | 5.31 | 4.25 | 15.84 |

44 | 8.9 | 88.7 | 19.8 | −61.6 | 17 | 5.40 | 4.32 | 14.50 |

45 | 6.7 | 84.2 | 17.2 | −60.8 | 14 | 5.81 | 4.46 | 13.24 |

46 | 11.0 | 98.0 | 22.7 | −74.6 | 12 | 6.89 | 3.90 | 12.82 |

47 | 9.5 | 92.2 | 20.3 | −50.5 | 17 | 5.92 | 3.91 | 13.20 |

48 | 7.7 | 78.2 | 17.2 | −63.4 | 19 | 6.17 | 3.99 | 15.19 |

49 | 9.5 | 79.3 | 18.1 | −67.4 | 13 | 6.50 | 4.78 | 17.69 |

50 | 7.4 | 90.4 | 21.3 | −55.3 | 18 | 6.65 | 4.96 | 12.49 |

Algorithms | Conventional RNN | Basic PSO-RNN | GA-IPSO-RNN | Proposed method | |
---|---|---|---|---|---|

MAE | 1 | 1.1950 | 0.4818 | 0.4258 | 0.3839 |

2 | 3.6827 | 2.0262 | 1.9833 | 1.7821 | |

Mean | 2.4389 | 1.2540 | 1.2045 | 1.0830 | |

MRE(%) | 1 | 28.63 | 10.65 | 9.39 | 8.71 |

2 | 27.96 | 14.61 | 14.01 | 12.41 | |

Mean | 28.30 | 12.63 | 11.70 | 10.56 | |

RMSE | 1 | 1.4843 | 0.5841 | 0.5157 | 0.4076 |

2 | 4.5364 | 2.2637 | 2.1177 | 1.9649 | |

Mean | 3.0104 | 1.4239 | 1.3167 | 1.1863 | |

TIC | 1 | 0.1675 | 0.0690 | 0.0609 | 0.0481 |

2 | 0.1452 | 0.0766 | 0.0727 | 0.0679 | |

Mean | 0.1563 | 0.0728 | 0.0668 | 0.0580 |

_{max}and x

_{min}denote the maximum and minimum values of initial data, respectively.

_{i}is the true value; ${\widehat{y}}_{i}$ is the predicted value.

#### 4.2. The Selection of Comparative Models

#### 4.3. Forward Prediction

#### 4.4. Backward Prediction

**Figure 6.**Training results of the proposed SVM-IPSO model. (

**a**) Viscosity average molecular weight; (

**b**) Conversion ratio; (

**c**) Solid content; (

**d**) Spinning jet drawing ratio; (

**e**) Coagulating temperature; (

**f**) Total drawing ratio.

Algorithms | MAE | MRE (%) | RMSE | TIC |
---|---|---|---|---|

Conventional RNN | ||||

1 | 173.1400 | 1876.58 | 341.7052 | 0.9612 |

2 | 1062.8000 | 1310.66 | 2004.3000 | 0.9391 |

3 | 151.6600 | 816.09 | 285.3062 | 0.9051 |

4 | 171.6800 | 264.56 | 285.1956 | 0.7416 |

5 | 62.6800 | 443.07 | 103.5493 | 0.8054 |

6 | 34.8800 | 533.97 | 63.3625 | 0.9510 |

Mean | 276.1367 | 874.16 | 513.9031 | 0.8839 |

Basic PSO-RNN | ||||

1 | 1.9652 | 22.73 | 2.2049 | 0.1138 |

2 | 9.7604 | 11.30 | 10.3417 | 0.0590 |

3 | 2.5932 | 13.19 | 2.7487 | 0.0690 |

4 | 8.9433 | 15.27 | 10.2110 | 0.0806 |

5 | 3.2098 | 20.66 | 3.2757 | 0.1054 |

6 | 0.4920 | 7.38 | 0.6731 | 0.0544 |

Mean | 4.4940 | 15.09 | 4.9092 | 0.0804 |

GA-IPSO-RNN | ||||

1 | 1.3996 | 14.37 | 1.7597 | 0.1026 |

2 | 8.2660 | 9.61 | 9.1148 | 0.0518 |

3 | 2.2693 | 11.04 | 2.5107 | 0.0647 |

4 | 8.0976 | 13.47 | 9.1473 | 0.0733 |

5 | 2.9085 | 19.20 | 2.9776 | 0.0944 |

6 | 0.3260 | 4.94 | 0.4054 | 0.0317 |

Mean | 3.8778 | 12.11 | 4.3192 | 0.0697 |

Proposed method | ||||

1 | 1.0585 | 11.80 | 1.2290 | 0.0693 |

2 | 7.7362 | 9.03 | 8.4285 | 0.0479 |

3 | 1.7943 | 9.17 | 1.9964 | 0.0500 |

4 | 7.9765 | 12.74 | 8.8876 | 0.0724 |

5 | 2.7471 | 18.42 | 2.9189 | 0.0920 |

6 | 0.3994 | 6.02 | 0.4855 | 0.0387 |

Mean | 3.6187 | 11.20 | 3.9910 | 0.0617 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Yusof, N.; Ismail, A.F. Post spinning and pyrolysis processes of polyacrylonitrile (PAN)-based carbon fiber and activated carbon fiber: A review. J. Anal. Appl. Pyrolysis
**2012**, 93, 1–13. [Google Scholar] [CrossRef] - Chand, S. Review carbon fibers for composites. J. Mater. Sci.
**2000**, 35, 1303–1313. [Google Scholar] [CrossRef] - Liu, J.; Tian, Y.; Chen, Y.; Liang, J.; Zhang, L.; Fong, H. A surface treatment technique of electrochemical oxidation to simultaneously improve the interfacial bonding strength and the tensile strength of PAN-based carbon fibers. Mater. Chem. Phys.
**2010**, 122, 548–555. [Google Scholar] [CrossRef] - Wang, Y.; Yin, W. Chemical modification for PAN fibers during heat-treatment process. Phys. Procedia
**2011**, 18, 202–205. [Google Scholar] [CrossRef] - Rahman, M.A.; Ismail, A.F.; Mustafa, A. The effect of residence time on the physical characteristics of PAN-based fibers produced using a solvent-free coagulation process. Mater. Sci. Eng.
**2007**, 448, 275–280. [Google Scholar] [CrossRef] [Green Version] - Liang, X.; Ding, Y.S.; Ren, L.H.; Hao, K.R.; Wang, H.P.; Chen, J.J. A bio-inspired multi-layered intelligent cooperative controller for stretching process of fiber production. IEEE Trans. Syst. Man Cybern. C
**2012**, 42, 367–377. [Google Scholar] - Rennhofer, H.; Loidl, D.; Puchegger, S.; Peterlik, H. Structural development of PAN-based carbon fibers studied by in situ X-ray scattering at high temperatures under load. Carbon
**2010**, 48, 964–971. [Google Scholar] [CrossRef] - Belyaev, S.S.; Arkhangelsky, I.V.; Makarenko, I.V. Nonisothermal kinetic analysis of oxidative stabilization processes in PAN fibers. Thermochim. Acta
**2010**, 507–508, 9–14. [Google Scholar] [CrossRef] - Chen, J.J.; Ding, Y.S.; Hao, K.R. The bidirectional optimization of carbon fiber production by neural network with a GA-IPSO hybrid algorithm. Math. Probl. Eng.
**2013**, 2013, 1–16. [Google Scholar] - Sugimoto, Y.; Shioya, M.; Yamamoto, K.; Sakurai, S. Relationship between axial compression strength and longitudinal microvoid size for PAN-based carbon fibers. Carbon
**2012**, 50, 2860–2869. [Google Scholar] [CrossRef] - Kadi, H.E. Modeling the mechanical behavior of fiber-reinforced polymeric composite materials using artificial neural networks—A review. Compos. Struct.
**2006**, 73, 1–23. [Google Scholar] [CrossRef] - Yu, Y.; Hui, C.L.; Choi, T.M.; Au, R. Intelligent fabric hand prediction system with fuzzy neural network. IEEE Trans. Syst. Man Cybern. C
**2010**, 40, 619–629. [Google Scholar] [CrossRef] - Du, D.; Li, K.; Fei, M. A fast multi-output RBF neural network construction method. Neurocomputing
**2010**, 73, 2196–2202. [Google Scholar] [CrossRef] - Roy, A.; Govil, S.; Miranda, R. A neural-network learning theory and a polynomial time RBF algorithm. IEEE Trans. Neural Netw.
**1997**, 8, 1301–1313. [Google Scholar] [CrossRef] - Hong, X.; Chen, S. A new RBF neural network with boundary value constraints. IEEE Trans. Syst. Man Cybern. B
**2009**, 39, 298–303. [Google Scholar] [CrossRef] - Huang, G.B.; Saratchandran, P.; Sundararajan, N. A generalized growing and pruning RBF (GGAP-RBF) neural network for function approximation. IEEE Trans. Neural Netw.
**2005**, 16, 57–67. [Google Scholar] [CrossRef] - Qiao, J.-F.; Han, H.-G. Identification and modeling of nonlinear dynamical systems using a novel self-organizing RBF-based approach. Automatica
**2012**, 48, 1729–1734. [Google Scholar] [CrossRef] - Wang, Y.; Liu, G. A forecasting method based on online self-correcting single model RBF neural network. Procedia Eng.
**2012**, 29, 2516–2520. [Google Scholar] [CrossRef] - Hong, W.C.; Dong, Y.; Zhang, W.Y.; Chen, L.Y.; Panigrahi, B.K. Cyclic electric load forecasting by seasonal SVR with chaotic genetic algorithm. Int. J. Electr. Power Energy Syst.
**2013**, 44, 604–614. [Google Scholar] [CrossRef] - Goh, A.T.C.; Goh, S.H. Support vector machines: Their use in geotechnical engineering as illustrated using seismic liquefaction data. Comput. Geotech.
**2007**, 34, 410–421. [Google Scholar] [CrossRef] - Kordjazi, A.; Nejad, F.P.; Jaksa, M.B. Prediction of ultimate axial load-carrying capacity of piles using a support vector machine based on CPT data. Comput. Geotech.
**2014**, 55, 91–102. [Google Scholar] [CrossRef] - Lin, J.Y.; Cheng, C.T.; Chau, K.W. Using support vector machines for long-term discharge prediction. Hydrol. Sci. J.
**2006**, 51, 599–612. [Google Scholar] [CrossRef] - Niu, D.; Wang, Y.; Wu, D.D. Power load forecasting using support vector machine and ant colony optimization. Expert Syst. Appl.
**2010**, 37, 2531–2539. [Google Scholar] [CrossRef] - Wang, J.J.; Li, L.; Niu, D.; Tan, Z.F. An annual load forecasting model based on support vector regression with differential evolution algorithm. Appl. Energy
**2012**, 94, 65–70. [Google Scholar] [CrossRef] - Gilan, S.; Jovein, H.B.; Ali, A.R. Hybrid support vector regression-Particle swarm optimization for prediction of compressive strength and RCPT of concretes containing metakaolin. Constr. Build. Mater.
**2012**, 34, 321–329. [Google Scholar] [CrossRef] - Hong, W.C. Chaotic particle swarm optimization algorithm in a support vector regression electric load forecasting model. Energy Convers. Manag.
**2009**, 50, 105–117. [Google Scholar] [CrossRef] - Kang, Q.; Zhou, M.; An, J.; Wu, Q. Swarm intelligence approaches to optimal power flow problem with distributed generator failures in power networks. IEEE Trans. Autom. Sci. Eng.
**2013**, 10, 343–353. [Google Scholar] [CrossRef] - Liang, X.; Li, W.; Zhang, Y.; Zhou, M. An adaptive particle swarm optimization method based on clustering. Soft Comput.
**2014**. [Google Scholar] [CrossRef] - Vapnik, V. The Nature of Statistical Learning Theory; Springer: New York, NY, USA, 1995. [Google Scholar]
- Li, X.; Zhang, A.; Li, C.C.; Zhang, L. Rolling element bearing fault detection using support vector machine with improved ant colony optimization. Measurement
**2013**, 46, 2726–2734. [Google Scholar] [CrossRef] - Keerthi, S.S.; Lin, C.J. Asymptotic behaviors of support vector machines with Gaussian kernel. Neural Comput.
**2003**, 15, 1667–1689. [Google Scholar] [CrossRef] [PubMed] - Kennedy, J.; Eberhart, R. Particle swarm optimization. IEEE Proc. Int. Conf. Neural Netw.
**1995**, 4, 1942–1948. [Google Scholar] - Zhu, H.; Wang, Y.; Wang, K.; Chen, Y. Particle Swarm Optimization (PSO) for the constrained portfolio optimization problem. Expert Syst. Appl.
**2011**, 38, 10161–10169. [Google Scholar] [CrossRef] - Ahmed, K.A.; Xiang, J. Mechanisms of cellular communication through intercellular protein transfer. J. Cell. Mol. Med.
**2011**, 15, 1458–1473. [Google Scholar] [CrossRef] [PubMed]

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**MDPI and ACS Style**

Xiao, C.; Hao, K.; Ding, Y.
The Bi-Directional Prediction of Carbon Fiber Production Using a Combination of Improved Particle Swarm Optimization and Support Vector Machine. *Materials* **2015**, *8*, 117-136.
https://doi.org/10.3390/ma8010117

**AMA Style**

Xiao C, Hao K, Ding Y.
The Bi-Directional Prediction of Carbon Fiber Production Using a Combination of Improved Particle Swarm Optimization and Support Vector Machine. *Materials*. 2015; 8(1):117-136.
https://doi.org/10.3390/ma8010117

**Chicago/Turabian Style**

Xiao, Chuncai, Kuangrong Hao, and Yongsheng Ding.
2015. "The Bi-Directional Prediction of Carbon Fiber Production Using a Combination of Improved Particle Swarm Optimization and Support Vector Machine" *Materials* 8, no. 1: 117-136.
https://doi.org/10.3390/ma8010117