A hybrid approach of genetic algorithm (GA) and improved particle swarm optimization (IPSO) is proposed to construct the radial basis function neural network (RNN) for realtime optimizing of the carbon fiber manufacture process. For the threelayer RNN, we adopt the nearest neighborclustering algorithm to determine the neurons number of the hidden layer. When the appropriate network structure is fixed, we present the GAIPSO algorithm to tune the parameters of the network, which means the center and the width of the node in the hidden layer and the weight of output layer. We introduce a penalty factor to adjust the velocity and position of the particles to expedite convergence of the PSO. The GA is used to mutate the particles to escape local optimum. Then we employ this network to develop the bidirectional optimization model: in one direction, we take production parameters as input and properties indices as output; in this case, the model is a carbon fiber product performance prediction system; in the other direction, we take properties indices as input and production parameters as output, and at this situation, the model is a production scheme design tool for novel style carbon fiber. Based on the experimental data, the proposed model is compared to the conventional RBF network and basic PSO method; the research results show its validity and the advantages in dealing with optimization problems.
Carbon fiber is a novel breed of ideal engineering materials which has high strength and modulus; hence, it has great influence on aspects of military, industry, economy, and so on. Due to its high strength, high break strength, high modulus, small diameter, low density, maximum crystallinity, and low comonomer contents, polyacrylonitrile(PAN) based carbon fiber dominates the market and the optimization of its manufacture process receives increasing attention from scholars and researchers in recent years [
With spring up of computational intelligent methods, they have provided a powerful tool to approximate the functional form of an unknown complex nonlinear system. Among them, since its extremely strong adaptive capabilities in response to nonlinear behaviors, artificial neural network (ANN) have been presented to deal with modeling dynamic process for product and process design, monitoring, and control. Kadi [
As we all known, every single algorithm has its own shortcomings which are difficult to overcome by improving itself. In that case, combining several artificial intelligence (AI) approaches is a new study trend in recent years. They can fully promote their respective superiorities while avoid their respective defects to enhance their optimizing effects. In the literatures, particle swarm optimization (PSO) is a popular algorithm because of its easy implementation procedure and high performance. Hu et al. [
In this paper, we propose a neural network model with a GAIPSO hybrid algorithm for bidirectional optimization of PANbased carbon fiber. We use the nearest neighborclustering algorithm (NNCA) to decide the hidden layer nodes number of neural network, the GAIPSO hybrid algorithm is used to tune the parameters in the network. By this model, for one direction, we predict the carbon fiber properties; for the other direction, we obtain produce scheme for new type carbon fiber.
The main contributions of this paper are as follows:
The rest of this paper is organized as follows. In Section
As described in introduction, the production process of PANbased carbon fiber is a typical unknown nonlinear dynamical complex system. It is a key investigation field to keep the properties of carbon fiber stable and improve the properties as possible as we can. For this reason, we should monitor the production process of carbon fiber, observe producing parameters in realtime, accurately predict product properties online, then we can find out defects of carbon fiber ahead of time, adjust produce parameters in time, avoid generating carbon fiber of poor quality, and prevent wasting of production materials.
Designing new types carbon fiber with superior properties is another urgent task for engineers and scholars. In the past, we always make a production scheme of new type carbon fiber through former experiences, then we produce this kind of carbon fiber according to the proposed experiential scheme by testing production line. After obtaining carbon fiber from the practical experiment, we put the carbon fiber to physical and chemical instruments to test its property indices, compare these indices to expect value, get the disparity, adjust several produce parameters in proposed scheme, produce carbon fiber according to adjusted scheme by testing production line again, repeat executing produce, test, and adjust process until we gain the carbon fiber which is satisfactory to us. This method wastes time, energy, and cost. The most terrible thing is that, if we do not choose the appropriate parameters of control devices, it will result in oscillation and unstability of the whole system, which brings in severe loss. Therefore, it is a better choice to carry out computer simulation, analyze and synthesize all the antecedent produce schemes, avoid onesided decision made by workers from their personal experience and interests, and give out a more scientific and proper control parameters from an optimal production scheme.
In this paper, we propose a bidirectional optimization model of carbon fiber production process based on RNN. We adopt the nearest neighborclustering algorithm to select a suitable set of centers of the network and introduce a GAIPSO hybrid algorithm to tune the parameters of the network. When we take production parameters as input and properties indices as output, this model can monitor production parameters in real time and predict properties indices online. When we take properties indices as input and production parameters as output, this model can be viewed as a designer of developing new type carbon fiber. All these bidirectional optimization functions are as shown in Figure
The bidirectional optimization model of carbon fiber production process.
The RNN is a typical supervised learning feedforward neural network which is put forward by Moody and Darken in 1989 [
The common radial basis functions are listed as follows:
As Gaussian function is the outstanding representative among them, we use it as activation function of hidden layer in this paper:
The output is formed by a linear combination of the hidden layer responses
First of all, we should confirm the structure of RNN, the number of input nodes
In this work, we adopt the nearest neighborclustering algorithm to acquire
The nearest neighborclustering algorithm.
The specific procedures of the nearest neighborclustering algorithm are shown below.
Choose a proper width
Begin from the first sample (
Consider the second sample (
Think about the samples one by one. Assuming that the nodes number of hidden layer is
Do Step
After executing the nearest neighborclustering algorithm, we obtain the number of centers and the parameters of RNN which means the position and width of centers, weights from hidden nodes to output nodes at the same time. However, in this paper, we just use the nearest neighborclustering algorithm to determine the number of hidden nodes, which is the number of centers. We will tune the position and width of centers, weights from hidden nodes to output nodes by proposed GAIPSO which is described in Section
PSO algorithm is an efficient populationbased stochastic optimization technique inspired by social behavior of bird flocks or fish schools which is originally developed by Kennedy and Eberhart in 1995 [
The principle of PSO: for a
For the
Commonly,
Furthermore, although PSO has good convergent property, if the initial population of particles cannot effectively cover the whole region, the population diversity will be greatly reduced after some iterations, as such the particles will converge easily to a local optimum. Aim to avoid the limitations and enhance the superiorities, due to the fact that PSO utilizes
Mark the output of RNN as
On every iteration,
cross
mutation
The flowchart of GAIPSO is indicated in Figure
The flowchart of GAIPSO.
The specific procedures of the GAIPSO are shown below.
Set iteration as
Initiate the particle swarm according to the range given by the nearest neighborclustering algorithm.
Calculate the fitness of each particle in the population.
Set up
Set up
Update iteration
Calculate the fitness of each particle in the new population.
Update
Change the position of each particle in this population a little, calculate the new fitness, and update
Update
If
Do Step
Aiming to test and demonstrate the performance of the model we proposed, we used these data collected and organized from experiments in Table
Experiment data of carbon fiber production process.
No.  Viscosity average 
Conversion ratio (%)  Solid content (%)  Spinning jet drawing ratio (%)  Coagulating bath temperature (°C)  Total drawing ratio  Strength 
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 
In these data, viscosity average molecular weight, conversion ratio, solid content, spinning jet drawing ratio, coagulating bath temperature and total drawing ratio are production parameters, strength, and structure parameters are property indices. These specific production parameters and property indices can relatively fully represent the whole manufacture process and product performance of PANbased carbon fiber. In the bidirectional optimization process, we take the first 45 groups of samples as training dataset and the last 5 groups of samples as test dataset. We also make comparison between conventional RNN, basic PSORNN, and the proposed method, observe their performance in different ways which are defined as the following, respectively:
mean absolute error:
mean relative error:
root mean square error:
Theil’s Inequality Coefficient:
For properties prediction, the production parameters are input, the properties indices are output,
Parameters list of RNN obtained from NNCA.
Position of hidden layer centers  







0.4664  0.1387  0.7297  0.4038  0.9834  0.9819 
0.8050  0.2038  1.0000  0.3347  0.8927  0.3094 
0.6609  0.1556  0.5499  0.9139  1.0000  0 
1.0000  0.2958  0.0298  0.1932  0.3967  0.6396 
0.1667  0.3333  0.6667  0.8333  0  0.5000 
0.6652  0.6600  0.7054  0  0.1829  0.1522 
 
Width of hidden layer centers  






 
1.0000  1.0000  1.0000  1.0000  1.0000  1.0000 
 
Weights of output layer  






 
0.4340  0.5585  0.6185  0.6064  0.6339  0.8537 
0.5112  0.4440  0.4670  0.3342  0.4585  0.5194 
Based on Table
By employing the proposed GAIPSO algorithm to optimize the parameters of RNN, we gain the best parameters for the RNN. The training accuracy and the agreement level of the proposed method are shown in Figure
Training results of the proposed GAIPSORNN model.
Strength
Structure parameter
From Figure
The model’s prediction accuracy also can be reflected from its performance on the test dataset. We show the prediction results of the test dataset of the proposed method, basic PSORNN and conventional RNN in Figure
Errors of the proposed method, basic PSORNN, and conventional RNN (1: strength, 2: structure parameter).
Algorithms  MAE  MRE (%)  RMSE  TIC  Time (s) 

Conventional RNN  
1  1.1950  28.63  1.4843  0.1675  
2  3.6827  27.96  4.5364  0.1452  0.9575 
Total  2.4389  28.30  3.3750  0.1470  
 
Basic PSORNN  
1  0.4818  10.65  0.5841  0.0690  
2  2.0262  14.61  2.2637  0.0766  0.7428 
Total  1.2540  12.63  1.6531  0.0761  
 
Proposed method  
1  0.4258  9.39  0.5157  0.0609  
2  1.9833  14.01  2.1177  0.0727  0.2985 
Total  1.2045  11.70  1.5412  0.0718 
Performance of the proposed method, basic PSORNN, and conventional RNN.
Strength
Structure parameter
From data in Table
For production scheme designer, the properties indices are input, the production parameters are output,
The parameters value of the RNN obtained from the NNCA.
Position of hidden layer centers 


0.5453  
0.4971  
 
Width of hidden layer centers 

 
1.0000  
 
Weights of output layer 

 
0.4766  
0.5011  
0.5117  
0.4849  
0.4630  
0.5397 
Based on Table
By employing the proposed GAIPSO algorithm to optimize the parameters of the RNN and gain the best parameters. The training accuracy and the agreement level of the proposed method are shown in Figure
Training results of the proposed GAIPSORNN model.
Viscosity average molecular weight
Conversion ratio
Solid content
Spinning jet drawing ratio
Coagulating temperature
Total drawing ratio
The model’s prediction accuracy can also be reflected from its performance on the test dataset. We show the prediction results of the test dataset of the proposed method, basic PSORNN and conventional RNN in Figure
Errors of the proposed method, basic PSORNN, and conventional RNN.
Algorithms  MAE  MRE (%)  RMSE  TIC  Time(s) 

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  1.7296 
5  62.6800  443.07  103.5493  0.8054  
6  34.8800  533.97  63.3625  0.9510  
Total  276.1367  874.16  847.6972  0.9285  
 
Basic PSORNN  
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  0.4065 
5  3.2098  20.66  3.2757  0.1054  
6  0.4920  7.38  0.6731  0.0544  
Total  4.4940  15.09  6.2559  0.0686  
 
Proposed method  
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  0.1464 
5  2.9085  19.20  2.9776  0.0944  
6  0.3260  4.94  0.4054  0.0317  
Total  3.8778  12.11  5.5555  0.0612 
Note: 1: viscosity average molecular weight, 2: conversion ratio, 3: solid content, 4: spinning jet drawing ratio, 5: coagulating bath temperature, 6: total drawing ratio.
Performance of the proposed method, basic PSORNN, and conventional RNN.
Viscosity average molecular weight
Conversion ratio
Solid content
Spinning jet drawing ratio
Coagulating temperature
Total drawing ratio
Performance comparison of the proposed method and basic PSORNN.
Viscosity average molecular weight
Conversion ratio
Solid content
Spinning jet drawing ratio
Coagulating temperature
Total drawing ratio
From data in Table
In this paper, we present a model for using an enhanced RNN to bidirectionally optimize the production process of PANbased carbon fiber, which can be viewed as properties prediction and new type fiber designer. We adopt the nearest neighborclustering algorithm to simplify the structure of RNN and investigate the learning method of the RNN based on the PSO, introduce penalty factor to improve the PSO, then combine GA with the improved PSO to train the RNN. Meanwhile, we compare the GAIPSORNN model to the existing basic PSORNN and conventional RNN. The superiority of the proposed model is summarized as below.
Compare with the conventional RNN, the proposed method adopts the nearest neighborclustering algorithm to determine the number of hidden notes and provides an effective approach to simplify the network structure. For carbon fiber production process optimization, it can add new samples online, composite all the information, and improve the accuracy.
Based on the same network structure, the proposed GAIPSO algorithm provides a more efficient search scheme to determine the related parameters of the RNN than the basic PSO.
Testing on the bidirectional optimization of the carbon fiber production process has shown that the proposed method is superior to the basic PSORNN and conventional RNN in constructing the network and estimating the outputs.
After the network is constructed, the proposed model can accelerate the convergence speed, decrease iteration time, benefit the ability of getting out of local optimum, increase the capability of the network, then make a quick response, and yield accurate solution during the bidirectional optimization of the carbon fiber production process.
However, the accuracy of the proposed model is still needed to be improved because the training data should be enriched. When the accuracy meets practical production demands, for properties prediction, because production schemes with little difference may lead to the same product properties, so if control parameters changes tiny, we do not need to tune all the parameters or keep them as constants. For production scheme design, after we obtain the scheme by the proposed model, we can alternately use the properties prediction function of the model, prevent cost wasting in practical production directly. This problem should be studied in future work.
This work was supported in part by the Key Project of the National Nature Science Foundation of China (no. 61134009), the National Nature Science Foundation of China (no. 60975059), Specialized Research Fund for Shanghai Leading Talents, Project of the Shanghai Committee of Science and Technology (nos. 11XD1400100 and 11JC1400200), and the Fundamental Research Funds for the Central Universities.