# An Improved Fruit Fly Optimization Algorithm Inspired from Cell Communication Mechanism

Authors: Chuncai Xiao; Kuangrong Hao; Yongsheng Ding
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Fruit fly optimization algorithm (FOA) invented recently is a new swarm intelligence method based on fruit fly’s foraging behaviors and has been shown to be competitive with existing evolutionary algorithms, such as particle swarm optimization (PSO) algorithm. However, there are still some disadvantages in the FOA, such as low convergence precision, easily trapped in a local optimum value at the later evolution stage. This paper presents an improved FOA based on the cell communication mechanism (CFOA), by considering the information of the global worst, mean, and best solutions into the search strategy to improve the exploitation. The results from a set of numerical benchmark functions show that the CFOA outperforms the FOA and the PSO in most of the experiments. Further, the CFOA is applied to optimize the controller for preoxidation furnaces in carbon fibers production. Simulation results demonstrate the effectiveness of the CFOA.


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## Body

## 1. Introduction

As we all know, there are a lot of bioinspired optimization algorithms that are applied in practical engineering successfully, such as genetic algorithm (GA) inspired by the genetic science and natural selection [1, 2], particle swarm optimization (PSO) algorithm inspired by the simulation of the behavior of birds in nature [3–5], artificial bee colony (ABC) algorithm inspired by the intelligent behavior of honeybee swarm [6], artificial immune algorithm (AIA) inspired by the biological immune system [7–10], and ant colony optimization (ACO) algorithm inspired by the foraging behavior of the real ants [11].By simulating the foraging behavior of fruit fly swarm, Pan [12] recently proposed a new kind of optimization algorithm called fruit fly optimization algorithm (FOA) for financial distress model. The FOA has some advantages, such as a simple structure to be realized with computer and ease of understanding. Since its invention in 2011, the FOA has been applied in many kinds of practical problems. Li et al. [13] applied the FOA to the forecasting problem of power load in electric power industry. Lin [14] used the FOA to optimize artificial neural network for analysis of service satisfaction in Web auction logistics service. Sheng and Bao [15] applied the FOA to search the optimal parameters values of the fractional order fuzzy-PID controller. Tu et al. [16] used the FOA to optimize generalized regression neural network for studying on business performance. Li et al. [17] used the FOA to optimize the two parameters for the least squares support vector machine to forecast the annual electricity consumption in China. Chen et al. [18] used the FOA to optimize the grey neural network model to perform satisfaction analysis for e-business service. According to all kinds of applications mentioned above, the FOA seems to be an excellent optimization algorithm. However, there is still an insufficiency in the FOA regarding the search strategy, which is used to produce new candidate solutions of the FOA based on the random information of foregoing solutions.In order to overcome the lack of search strategy control mechanism in the basic FOA leading to poor quality of solution, in this paper, inspired by the cell communication mechanism, we propose an improved FOA based on the cell communication mechanism (CFOA) by incorporating the information of the global worst, mean, and best solutions into the search strategy to improve the exploitation. A set of numerical experiment results on function optimization [19–22] show that the CFOA has excellent global optimization capability and the ability to obtain accurate solution. And its convergence speed is faster than that of the PSO and the FOA. Further, we apply the CFOA to optimize the controller for preoxidation furnaces in carbon fibers production. Simulation results demonstrate the effectiveness of the CFOA.Carbon fibers are produced mainly from polyacrylonitrile (PAN), pitch, and rayon. Attributing to intrinsic composites, the PAN-based carbon fibers have maintained their dominant position in the market of engineering materials. It has been documented that more than 90% of world’s total carbon fibers produced today are made from the PAN precursor [23]. Their high specific modulus and outstanding fatigue characteristics, combined with their lower weight and stiffness, make these fibers attractive for wide applications ranging from sporting goods to engineering components [24]. It is well known that the preoxidation process is one of the most basic and necessary stages in the conversion process from the PAN fibers to high specific strength carbon fibers. For example, Young’s modulus of carbon fibers can be increased by high temperature treatment [25]. Although the preoxidation has been widely used in the production line of carbon fibers, it has not been given too much attention and few detailed studies of temperature control are presently available. Wangxi et al. [26] referred to the evolution of structure and properties of the PAN precursors in the process of preoxidation. Yu et al. [27] monitored the processing of preoxidation through their experimental results which indicated that the percent of O increases remarkably along with the gradual decrease of C, H, and N. Hou et al. [28] investigated the influence of ozone on chemical reactions during the preoxidation process of the PAN as a carbon fiber precursor. Xue et al. [29] investigated the oxygen-induced modification of the PAN-based carbon fibers during the final stage of thermal-oxidative stabilization which is used to control the degree of chemical reactions and the radial structural homogeneity of fibers. However, most of the previous work was focused on analyzing the properties of carbon fibers by means of physical or chemical instruments, and little was concerning the control effect of the temperature on preoxidation reactions. Therefore, further studies are very essential from the point of view for improving the properties of carbon fibers in actual product, decreasing the cost wasting in practical production, and increasing the yield of carbon fibers.The main contributions of this paper include the following aspects: (1) We propose the CFOA, which pushes forward the development of intelligent computing; (2) The CFOA is used to optimize the controller for preoxidation furnaces in carbon fibers production. With the proposed approach, we can control the temperature of preoxidation reactions, reducing not only wasted time but also energy consumption.The main structure of this paper is organized as follows. Section2 proposes the improved fruit fly optimization algorithm with cell communication mechanism. Section 3 provides the test of applying the proposed algorithm in finding minimal values of numerical benchmark functions. Section 4 applied the CFOA to the optimal control model of preoxidation furnace for producing the PAN-based carbon fibers and compares the results with the former methods. Section 5 concludes the paper.


## 2. An Improved Fruit Fly Optimization Algorithm with Cell Communication

### 2.1. The Fruit Fly Optimization Algorithm

The FOA is a new swarm intelligent method based on fruit fly’s foraging behaviors, and it belongs to a kind of interactive evolutionary computation. Fruit flies are very small ones to eat fruit and rotting plants, which widely exist in temperate and tropical climate zones around the world. Fruit flies have visual and olfactory senses better than other species. They can easily make good search of various odors floating in the air with their olfactory organ or even smell the food sources 40 km away from them. Then, they would fly to the food by their sensitive vision. The food finding process made by the fruit fly can be summarized into the following steps: (1) firstly, smelling the food source by olfactory organ and flying towards that location; (2) secondly, getting close to the food location by their sensitive visions; (3) at last, other fruit flies’ flocking location and flying towards that direction. Figure1 [13] shows the food searching iterative process of fruit fly.Figure 1
Food searching iterative process of fruit flies.According to the food finding characteristics of fruit fly swarm, the FOA can be divided into seven steps as follows.Step 1.
Parameters initialization: the main parameters of the FOA are the total evolution number, the population size pop, and the initial fruit fly swarm location(X0,Y0).Step 2.
Population initialization:(1)Xi=X0+rand,Yi=Y0+rand.Step 3.
Computation of distance(Di) and smell (Si):
(2)Di=Xi2+Yi2,Si=1Di.Step 4.
Computation of the fitness function(fi):
(3)fi=fSi.Step 5.
Find out the minimum individual fruit fly with the best fitness function(fb) among the fruit fly swarm:
(4)bestXbestindex=min⁡⁡fSi.Step 6.
Selection operation: keep the best fitness function value and coordinates(Xb,Yb). Then, the fruit fly swarm flies towards that location with the best fitness function value by using vision:
(5)fb=bestX,Xb=Xbestindex,Yb=Ybestindex.Step 7.
Judge if the stopping condition is satisfied. If not, go to Step  2; otherwise, stop the circulation.


### 2.2. The Basic Characteristic of the Cell Communication

Cell communication refers to a message through a medium to another cell and interacts with corresponding receptors of target cells, and after that a series of physiological changes taking place in the interior of cells. Indeed, cells can not survive in isolation and survive by receiving and processing information from the external environment. Cell to cell communication is essential for coordination of cellular events in multicellular systems [30]. Intercellular communication has three main ways: gap junction, cell recognition, and chemical communication.Gap junction: narrow water-filled channels that connect the cytoplasm of adjacent epithelial cells, as well as of some other types of cells.Cell recognition: mutual recognition among cells, usually with specific complementary interaction among their respective surface molecules or membrane glycoproteins.Chemical communication: cells secrete some chemicals (such as hormone) to the external cells, as a signal molecule in target cells, adjust their function.Based on the three main ways of cell communication, three evolutionary strategies are established. In particular, the gap junction, the cell recognition, and the chemical communication in the mode of cell communication can be used in the iterative and evolutionary process, respectively.


### 2.3. The CFOA Based on the Cell Communication Mechanism

Although the FOA has been applied in several fields, it still suffers with some degree of premature convergence and poor quality of solution, especially in solving higher dimension multimodal objective functions. To overcome the shortcomings, we present the CFOA by combining the cell communication mechanisms. The CFOA applies the modified update formula to maintain the population diversity and enhances the convergence velocity and precision by incorporating the information of the global worst, mean, and best solutions into the search strategy to improve the exploitation. The CFOA has three major steps: (1) randomly generate a fruit fly swarm’s initial position; (2) get the smell concentrations at positions of each and every fruit fly; and (3) assign each fruit fly a direction and distance for its movement to look for food with the modified update formula. The main steps are described as follows and their implementation procedure is illustrated in Figure2.Figure 2
The implement procedure of the CFOA.According to the food searching characteristics of fruit flies, the CFOA can be divided into eleven steps as follows.Step 1.
Parameters initialization: the main parameters of the CFOA are the total evolution number, the population size pop, and the initial fruit fly swarm location.Step 2.
Population initialization: give the random flight direction and the distance for food searching of a fruit fly by using olfactory model (1).Step 3.
Compute theDi and the Si from formulas (2).Step 4.
Compute thefi by formula (3).Step 5.
Compute the mean fitness function(6)fmean=∑i=1nf(Si)n.Step 6.
Find out the minimum individual fruit fly with thefb among the fruit fly swarm using formulas (4)-(5).Step 7.
Find out the maximal individual fruit fly with the worst fitness function(fmax⁡) among the fruit fly swarm:
(7)worstXworstindex=max⁡⁡fSi,fmax⁡=worstX,Xw=Xworstindex,Yw=Yworstindex.Step 8.
Update strategy: check each fruit fly’s position using the following formulas:(8)If(fi≤fmean)ThenXi=X-2*XwYi=Y-2*Yw,If(fmean<fi<fmax⁡)ThenXi=X+2*XbYi=Y+2*Yb,If(fi≥fmax⁡)ThenXi=X+XwYi=Y+Yw.Step 9.
Find out the fruit fly with best fitness function among the fruit fly swarm:(9)fb=min⁡⁡fi.Step 10.
Selection operation: keepXb, Yb, and fb; then the fruit fly swarm flies towards the location with the best fitness function value by using vision.Step 11.
Judge if the stopping condition is satisfied. If not, go to enter the circulation; otherwise, get the output.


## 2.1. The Fruit Fly Optimization Algorithm

The FOA is a new swarm intelligent method based on fruit fly’s foraging behaviors, and it belongs to a kind of interactive evolutionary computation. Fruit flies are very small ones to eat fruit and rotting plants, which widely exist in temperate and tropical climate zones around the world. Fruit flies have visual and olfactory senses better than other species. They can easily make good search of various odors floating in the air with their olfactory organ or even smell the food sources 40 km away from them. Then, they would fly to the food by their sensitive vision. The food finding process made by the fruit fly can be summarized into the following steps: (1) firstly, smelling the food source by olfactory organ and flying towards that location; (2) secondly, getting close to the food location by their sensitive visions; (3) at last, other fruit flies’ flocking location and flying towards that direction. Figure1 [13] shows the food searching iterative process of fruit fly.Figure 1
Food searching iterative process of fruit flies.According to the food finding characteristics of fruit fly swarm, the FOA can be divided into seven steps as follows.Step 1.
Parameters initialization: the main parameters of the FOA are the total evolution number, the population size pop, and the initial fruit fly swarm location(X0,Y0).Step 2.
Population initialization:(1)Xi=X0+rand,Yi=Y0+rand.Step 3.
Computation of distance(Di) and smell (Si):
(2)Di=Xi2+Yi2,Si=1Di.Step 4.
Computation of the fitness function(fi):
(3)fi=fSi.Step 5.
Find out the minimum individual fruit fly with the best fitness function(fb) among the fruit fly swarm:
(4)bestXbestindex=min⁡⁡fSi.Step 6.
Selection operation: keep the best fitness function value and coordinates(Xb,Yb). Then, the fruit fly swarm flies towards that location with the best fitness function value by using vision:
(5)fb=bestX,Xb=Xbestindex,Yb=Ybestindex.Step 7.
Judge if the stopping condition is satisfied. If not, go to Step  2; otherwise, stop the circulation.


## 2.2. The Basic Characteristic of the Cell Communication

Cell communication refers to a message through a medium to another cell and interacts with corresponding receptors of target cells, and after that a series of physiological changes taking place in the interior of cells. Indeed, cells can not survive in isolation and survive by receiving and processing information from the external environment. Cell to cell communication is essential for coordination of cellular events in multicellular systems [30]. Intercellular communication has three main ways: gap junction, cell recognition, and chemical communication.Gap junction: narrow water-filled channels that connect the cytoplasm of adjacent epithelial cells, as well as of some other types of cells.Cell recognition: mutual recognition among cells, usually with specific complementary interaction among their respective surface molecules or membrane glycoproteins.Chemical communication: cells secrete some chemicals (such as hormone) to the external cells, as a signal molecule in target cells, adjust their function.Based on the three main ways of cell communication, three evolutionary strategies are established. In particular, the gap junction, the cell recognition, and the chemical communication in the mode of cell communication can be used in the iterative and evolutionary process, respectively.


## 2.3. The CFOA Based on the Cell Communication Mechanism

Although the FOA has been applied in several fields, it still suffers with some degree of premature convergence and poor quality of solution, especially in solving higher dimension multimodal objective functions. To overcome the shortcomings, we present the CFOA by combining the cell communication mechanisms. The CFOA applies the modified update formula to maintain the population diversity and enhances the convergence velocity and precision by incorporating the information of the global worst, mean, and best solutions into the search strategy to improve the exploitation. The CFOA has three major steps: (1) randomly generate a fruit fly swarm’s initial position; (2) get the smell concentrations at positions of each and every fruit fly; and (3) assign each fruit fly a direction and distance for its movement to look for food with the modified update formula. The main steps are described as follows and their implementation procedure is illustrated in Figure2.Figure 2
The implement procedure of the CFOA.According to the food searching characteristics of fruit flies, the CFOA can be divided into eleven steps as follows.Step 1.
Parameters initialization: the main parameters of the CFOA are the total evolution number, the population size pop, and the initial fruit fly swarm location.Step 2.
Population initialization: give the random flight direction and the distance for food searching of a fruit fly by using olfactory model (1).Step 3.
Compute theDi and the Si from formulas (2).Step 4.
Compute thefi by formula (3).Step 5.
Compute the mean fitness function(6)fmean=∑i=1nf(Si)n.Step 6.
Find out the minimum individual fruit fly with thefb among the fruit fly swarm using formulas (4)-(5).Step 7.
Find out the maximal individual fruit fly with the worst fitness function(fmax⁡) among the fruit fly swarm:
(7)worstXworstindex=max⁡⁡fSi,fmax⁡=worstX,Xw=Xworstindex,Yw=Yworstindex.Step 8.
Update strategy: check each fruit fly’s position using the following formulas:(8)If(fi≤fmean)ThenXi=X-2*XwYi=Y-2*Yw,If(fmean<fi<fmax⁡)ThenXi=X+2*XbYi=Y+2*Yb,If(fi≥fmax⁡)ThenXi=X+XwYi=Y+Yw.Step 9.
Find out the fruit fly with best fitness function among the fruit fly swarm:(9)fb=min⁡⁡fi.Step 10.
Selection operation: keepXb, Yb, and fb; then the fruit fly swarm flies towards the location with the best fitness function value by using vision.Step 11.
Judge if the stopping condition is satisfied. If not, go to enter the circulation; otherwise, get the output.


## 3. Experimental Results with Benchmark Functions

Our experiments are completed on an Intel Pentium Dual, CPU 2.4 GHZ, personal computer (PC) with 3 GB RAM under Windows XP. All simulations are implemented by using MATLAB_7.9 program.


### 3.1. Parameter Settings

In all the experiments, the values of the common parameters are chosen to be the same for all algorithms. The total evolution number is 100, the initial range ofX is [-600,600], and population size is 20 for all functions. The PSO algorithm to be compared is given below: social and cognitive components are both set to 2.0 [31]. Inertia weight, which influences the velocity of the particle in the next iteration, is set to be 0.6 [22]. Each of the experiments is repeated 10 times independently, and the reported results are average, best, worst, and standard deviations of the statistical experimental data.


### 3.2. Experimental Results for Benchmark Functions

In the field of computational optimization, it is common to test different algorithms using plenty of test functions, especially when the test functions include various kinds of problems such as regular, irregular, separable, nonseparable unimodal, and multimodal problems. Multimodal functions which have more than one local optimal value are used to verify the ability of algorithms escaping from local minima. If an algorithm is poor that it can not be efficiently completed the search process during global searching, it gets stuck at the local optimal value. The tested benchmark functions are numbered asf1 tof6 and tabulated in Table 1.Table 1
Benchmark functions definition [19–22].
Name
Definition
Schaffer
f
1
x
=
0.5
+
sin
⁡
⁡
∑
i
=
1
n
x
i
2
2
-
0.5
1
+
0.001
(
∑
i
=
1
n
x
i
2
)
2
Rosenbrock
f
2
x
=
∑
i
=
1
n
100
x
i
2
-
x
i
+
1
2
+
x
i
-
1
2
Sphere
f
3
x
=
∑
i
=
1
n
x
i
2
Griewank
f
4
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
Rastrigin
f
5
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
+
10
Ackley
f
6
x
=
20
+
e
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
iThe experiments are tested on three groups of benchmark functions with dimension(D) of 5, 30, and 50. The increment in the dimension of function increases the difficulty [22]. Hence, if an algorithm can find the global optimal values for many dimensions, it will be easily copied to the practical engineering problems. The performance of the CFOA is compared with that of the PSO and the FOA in terms of (1) the best result, (2) the worst result, (3) the average result, and (4) the standard deviation (Std. Dev) result. Initial range, formulation, dimensions, and the characteristics of these problems are listed in Tables 2–4. In Tables 2–4, C is characteristic; U is unimodal; M is multimodal; S is separable; and N is nonseparable.Table 2
Results for 5D functions.
Function
C
Range
Comparison
FOA
CFOA
PSO
f
1
MN
[−600, 600]
Best
4.35e − 004
0
0.0049
Worst
5.83e − 004
0
0.0190
Average
4.99e − 004
0
0.0119
Std.Dev
5.07e − 005
0
0.0074
f
2
UN
[−600, 600]
Best
3.9574
4.35e − 005
0.4423
Worst
5.3111
0.5691
4.6508
Average
4.3386
0.1271
1.9483
Std.Dev
0.5844
0.1972
1.5192
f
3
US
[−600, 600]
Best
4.24e − 004
0
6.49e − 007
Worst
5.34e − 004
2.57e − 142
1.76e − 004
Average
4.71e − 004
2.57e − 143
7.41e − 005
Std.Dev
3.15e − 005
8.12e − 143
5.93e − 005
f
4
MN
[−600, 600]
Best
9.61e − 005
0
5.12e − 007
Worst
0.0141
2.6024e − 013
2.61e − 004
Average
0.0015
2.6024e − 014
3.98e − 005
Std.Dev
0.0044
8.2294e − 014
7.93e − 005
f
5
MS
[−600, 600]
Best
0.0873
0
1.0681
Worst
19.748
0
3.9807
Average
4.5809
0
2.7500
Std.Dev
7.4937
0
0.8968
f
6
MN
[−600, 600]
Best
0.0413
1.09e − 010
0.0071
Worst
1.2502
0.0047
0.1863
Average
0.1642
5.28e − 004
0.0342
Std.Dev
0.3816
0.0015
0.0539Table 3
Results for 30D functions.
Function
C
Range
Comparison
FOA
CFOA
PSO
f
1
MN
[−600, 600]
Best
0.0031
0
0.0049
Worst
0.6099
0
0.0190
Average
0.1602
0
0.0176
Std.Dev
0.2539
0
0.0045
f
2
UN
[−600, 600]
Best
28.707
7.90e − 05
88.072
Worst
108.70
0.4244
166.19
Average
72.133
0.0811
136.16
Std.Dev
26.257
0.1359
26.393
f
3
US
[−600, 600]
Best
0.0029
0
0.4305
Worst
1.5799
1.7617e − 141
1.7187
Average
5.13e − 001
1.7617e − 142
1.1396
Std.Dev
0.5270
5.5710e − 142
0.4359
f
4
MN
[−600, 600]
Best
1.92e − 004
0
0.0444
Worst
0.0588
7.40e − 003
0.1181
Average
0.0210
7.44e − 004
0.0703
Std.Dev
0.0236
0.0024
0.0212
f
5
MS
[−600, 600]
Best
0.6323
0
67.382
Worst
128.24
0
140.63
Average
91.073
0
99.362
Std.Dev
49.057
0
27.050
f
6
MN
[−600, 600]
Best
0.0446
−8.8818e − 016
1.6693
Worst
1.8229
0.5775
2.3261
Average
0.9374
0.1225
1.9970
Std.Dev
0.7885
0.2328
0.3511Table 4
Results for 50D functions.
Function
C
Range
Comparison
FOA
CFOA
PSO
f
1
MN
[−600, 600]
Best
0.0049
0
0.2221
Worst
0.8832
0.6784
0.3296
Average
0.2576
0.0678
0.2755
Std.Dev
0.0387
0.2145
0.0037
f
2
UN
[−600, 600]
Best
48.5100
48.6213
274.4630
Worst
178.0495
49.000
769.8354
Average
119.8091
48.9517
508.4547
Std.Dev
3.1920
0.1186
14.2781
f
3
US
[−600, 600]
Best
0.8342
0
2.6038
Worst
1.9000
0.0123
6.1062
Average
1.3174
0.0012
3.5712
Std.Dev
0.0388
0.0039
0.0949
f
4
MN
[−600, 600]
Best
2.2758e − 004
0
0.0984
Worst
0.0738
0
0.1868
Average
0.0307
0
0.1401
Std.Dev
0.0028
0
0.0030
f
5
MS
[−600, 600]
Best
58.7561
0
179.2948
Worst
248.6845
43.0351
264.6745
Average
172.1425
8.2006
225.2831
Std.Dev
6.3818
1.6426
2.6017
f
6
MN
[−600, 600]
Best
0.0458
−8.8818e − 016
2.3402
Worst
1.5043
1.3538
2.9950
Average
0.8789
0.1354
2.5752
Std.Dev
0.0687
0.4281
0.0209


#### 3.2.1. Experiment  1

In Experiment  1, the performance of the CFOA has been compared to other optimization algorithms: the PSO and the FOA. Functions in the set are with low dimension. The worst, best, mean of best, and the standard deviations of the function values found after 10 runs are presented in Table2. From the results in this table, it is clear that the CFOA is more precise on six functions. This is because the CFOA is a new strategy by incorporating the information of global best, worst, and mean solutions devised for optimal exploitation of local information. None of the three algorithms have found the optimal value on Rosenbrock and Ackley functions, but the result of the CFOA is better than the other two. On Schaffer, Griewank, and Rastrigin functions the CFOA have produced the optima. For space limitation, here we just present the convergence curves of the PSO, the FOA, and the CFOA to show the progresses of the mean of best function values presented in Figure 3. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 3(b) is for Figure 3(a), Figure 3(e) is for Figure 3(d), Figure 3(g) is for Figure 3(f), and Figure 3(i) is for Figure 3(h), respectively.Convergence graphs of the mean of best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves of the CFOA for the Schaffer function
(c)
Convergence curves for the Rosenbrock function
(d)
Convergence curves for the Sphere function
(e)
Convergence curves of the CFOA for the Sphere function
(f)
Convergence curves for the Griewank function
(g)
Convergence curves of the CFOA for the Griewank function
(h)
Convergence curves for the Rastrigin function
(i)
Convergence curves of the CFOA for the Rastrigin function
(j)
Convergence curves for the Ackley function


#### 3.2.2. Experiment 2

In Experiment  2, the performance of the CFOA has been compared with other optimization algorithms: the PSO and the FOA. From Table3, for three algorithms, on Rosenbrock and Ackley functions none of the algorithms have found the optimal value, but the result of the CFOA is better than the other two. On Schaffer, Sphere, Griewank, and Rastrigin functions the CFOA have produced the optima. The CFOA shows better robustness on Schaffer and Rastrigin functions. For space limitation, here we just present the convergence curves of the PSO, the FOA, and the CFOA to show the progresses of the mean of best function values presented in Figure 4. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 4(b) is for Figure 4(a), Figure 4(d) is for Figure 4(c), Figure 4(f) is for Figure 4(e), and Figure 4(h) is for Figure 4(g), respectively.Convergence graphs of the mean of the best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves of the CFOA for the Schaffer function
(c)
Convergence curves for the Rosenbrock function
(d)
Convergence curves of the CFOA for the Rosenbrock function
(e)
Convergence curves for the Sphere function
(f)
Convergence curves of the CFOA for the Sphere function
(g)
Convergence curves for the Griewank function
(h)
Convergence curves of the CFOA for the Griewank function
(i)
Convergence curves for the Rastrigin function
(j)
Convergence curves for the Ackley function


#### 3.2.3. Experiment 3

In Experiment  3, the performance of the CFOA has been compared with that of the PSO and the FOA. Functions in the set are not low dimensional; therefore, nonseparable functions are more difficult than the separable functions. The dimensionality of the search space is an important problem [32]. In some functions, the global minimum value is very small when compared to whole search space or is very close to the local ones (Schaffer). As for multimodal functions, if the algorithm cannot explore the search space effectively and cannot keep up the direction changes in the functions having narrow curving valley (Rosenbrock), it fails in these kinds of problems. From Table 4, for three algorithms, on Rosenbrock and Ackley functions none of the algorithms have produced the optimal value but the result of the CFOA is better than the other two. On Schaffer, Sphere, Griewank, and Rastrigin functions the CFOA has produced the optima. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 5(c) is for Figure 5(b) and Figure 5(e) is for Figure 5(d), respectively.Convergence graphs of the mean of best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves for the Rosenbrock function
(c)
Convergence curves of the CFOA for the Rosenbrock function
(d)
Convergence curves for the Sphere function
(e)
Convergence curves of the CFOA for the Sphere function
(f)
Convergence curves for the Griewank function
(g)
Convergence curves for the Rastrigin function
(h)
Convergence curves for the Ackley function


## 3.1. Parameter Settings

In all the experiments, the values of the common parameters are chosen to be the same for all algorithms. The total evolution number is 100, the initial range ofX is [-600,600], and population size is 20 for all functions. The PSO algorithm to be compared is given below: social and cognitive components are both set to 2.0 [31]. Inertia weight, which influences the velocity of the particle in the next iteration, is set to be 0.6 [22]. Each of the experiments is repeated 10 times independently, and the reported results are average, best, worst, and standard deviations of the statistical experimental data.


## 3.2. Experimental Results for Benchmark Functions

In the field of computational optimization, it is common to test different algorithms using plenty of test functions, especially when the test functions include various kinds of problems such as regular, irregular, separable, nonseparable unimodal, and multimodal problems. Multimodal functions which have more than one local optimal value are used to verify the ability of algorithms escaping from local minima. If an algorithm is poor that it can not be efficiently completed the search process during global searching, it gets stuck at the local optimal value. The tested benchmark functions are numbered asf1 tof6 and tabulated in Table 1.Table 1
Benchmark functions definition [19–22].
Name
Definition
Schaffer
f
1
x
=
0.5
+
sin
⁡
⁡
∑
i
=
1
n
x
i
2
2
-
0.5
1
+
0.001
(
∑
i
=
1
n
x
i
2
)
2
Rosenbrock
f
2
x
=
∑
i
=
1
n
100
x
i
2
-
x
i
+
1
2
+
x
i
-
1
2
Sphere
f
3
x
=
∑
i
=
1
n
x
i
2
Griewank
f
4
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
Rastrigin
f
5
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
+
10
Ackley
f
6
x
=
20
+
e
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
iThe experiments are tested on three groups of benchmark functions with dimension(D) of 5, 30, and 50. The increment in the dimension of function increases the difficulty [22]. Hence, if an algorithm can find the global optimal values for many dimensions, it will be easily copied to the practical engineering problems. The performance of the CFOA is compared with that of the PSO and the FOA in terms of (1) the best result, (2) the worst result, (3) the average result, and (4) the standard deviation (Std. Dev) result. Initial range, formulation, dimensions, and the characteristics of these problems are listed in Tables 2–4. In Tables 2–4, C is characteristic; U is unimodal; M is multimodal; S is separable; and N is nonseparable.Table 2
Results for 5D functions.
Function
C
Range
Comparison
FOA
CFOA
PSO
f
1
MN
[−600, 600]
Best
4.35e − 004
0
0.0049
Worst
5.83e − 004
0
0.0190
Average
4.99e − 004
0
0.0119
Std.Dev
5.07e − 005
0
0.0074
f
2
UN
[−600, 600]
Best
3.9574
4.35e − 005
0.4423
Worst
5.3111
0.5691
4.6508
Average
4.3386
0.1271
1.9483
Std.Dev
0.5844
0.1972
1.5192
f
3
US
[−600, 600]
Best
4.24e − 004
0
6.49e − 007
Worst
5.34e − 004
2.57e − 142
1.76e − 004
Average
4.71e − 004
2.57e − 143
7.41e − 005
Std.Dev
3.15e − 005
8.12e − 143
5.93e − 005
f
4
MN
[−600, 600]
Best
9.61e − 005
0
5.12e − 007
Worst
0.0141
2.6024e − 013
2.61e − 004
Average
0.0015
2.6024e − 014
3.98e − 005
Std.Dev
0.0044
8.2294e − 014
7.93e − 005
f
5
MS
[−600, 600]
Best
0.0873
0
1.0681
Worst
19.748
0
3.9807
Average
4.5809
0
2.7500
Std.Dev
7.4937
0
0.8968
f
6
MN
[−600, 600]
Best
0.0413
1.09e − 010
0.0071
Worst
1.2502
0.0047
0.1863
Average
0.1642
5.28e − 004
0.0342
Std.Dev
0.3816
0.0015
0.0539Table 3
Results for 30D functions.
Function
C
Range
Comparison
FOA
CFOA
PSO
f
1
MN
[−600, 600]
Best
0.0031
0
0.0049
Worst
0.6099
0
0.0190
Average
0.1602
0
0.0176
Std.Dev
0.2539
0
0.0045
f
2
UN
[−600, 600]
Best
28.707
7.90e − 05
88.072
Worst
108.70
0.4244
166.19
Average
72.133
0.0811
136.16
Std.Dev
26.257
0.1359
26.393
f
3
US
[−600, 600]
Best
0.0029
0
0.4305
Worst
1.5799
1.7617e − 141
1.7187
Average
5.13e − 001
1.7617e − 142
1.1396
Std.Dev
0.5270
5.5710e − 142
0.4359
f
4
MN
[−600, 600]
Best
1.92e − 004
0
0.0444
Worst
0.0588
7.40e − 003
0.1181
Average
0.0210
7.44e − 004
0.0703
Std.Dev
0.0236
0.0024
0.0212
f
5
MS
[−600, 600]
Best
0.6323
0
67.382
Worst
128.24
0
140.63
Average
91.073
0
99.362
Std.Dev
49.057
0
27.050
f
6
MN
[−600, 600]
Best
0.0446
−8.8818e − 016
1.6693
Worst
1.8229
0.5775
2.3261
Average
0.9374
0.1225
1.9970
Std.Dev
0.7885
0.2328
0.3511Table 4
Results for 50D functions.
Function
C
Range
Comparison
FOA
CFOA
PSO
f
1
MN
[−600, 600]
Best
0.0049
0
0.2221
Worst
0.8832
0.6784
0.3296
Average
0.2576
0.0678
0.2755
Std.Dev
0.0387
0.2145
0.0037
f
2
UN
[−600, 600]
Best
48.5100
48.6213
274.4630
Worst
178.0495
49.000
769.8354
Average
119.8091
48.9517
508.4547
Std.Dev
3.1920
0.1186
14.2781
f
3
US
[−600, 600]
Best
0.8342
0
2.6038
Worst
1.9000
0.0123
6.1062
Average
1.3174
0.0012
3.5712
Std.Dev
0.0388
0.0039
0.0949
f
4
MN
[−600, 600]
Best
2.2758e − 004
0
0.0984
Worst
0.0738
0
0.1868
Average
0.0307
0
0.1401
Std.Dev
0.0028
0
0.0030
f
5
MS
[−600, 600]
Best
58.7561
0
179.2948
Worst
248.6845
43.0351
264.6745
Average
172.1425
8.2006
225.2831
Std.Dev
6.3818
1.6426
2.6017
f
6
MN
[−600, 600]
Best
0.0458
−8.8818e − 016
2.3402
Worst
1.5043
1.3538
2.9950
Average
0.8789
0.1354
2.5752
Std.Dev
0.0687
0.4281
0.0209


### 3.2.1. Experiment  1

In Experiment  1, the performance of the CFOA has been compared to other optimization algorithms: the PSO and the FOA. Functions in the set are with low dimension. The worst, best, mean of best, and the standard deviations of the function values found after 10 runs are presented in Table2. From the results in this table, it is clear that the CFOA is more precise on six functions. This is because the CFOA is a new strategy by incorporating the information of global best, worst, and mean solutions devised for optimal exploitation of local information. None of the three algorithms have found the optimal value on Rosenbrock and Ackley functions, but the result of the CFOA is better than the other two. On Schaffer, Griewank, and Rastrigin functions the CFOA have produced the optima. For space limitation, here we just present the convergence curves of the PSO, the FOA, and the CFOA to show the progresses of the mean of best function values presented in Figure 3. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 3(b) is for Figure 3(a), Figure 3(e) is for Figure 3(d), Figure 3(g) is for Figure 3(f), and Figure 3(i) is for Figure 3(h), respectively.Convergence graphs of the mean of best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves of the CFOA for the Schaffer function
(c)
Convergence curves for the Rosenbrock function
(d)
Convergence curves for the Sphere function
(e)
Convergence curves of the CFOA for the Sphere function
(f)
Convergence curves for the Griewank function
(g)
Convergence curves of the CFOA for the Griewank function
(h)
Convergence curves for the Rastrigin function
(i)
Convergence curves of the CFOA for the Rastrigin function
(j)
Convergence curves for the Ackley function


### 3.2.2. Experiment 2

In Experiment  2, the performance of the CFOA has been compared with other optimization algorithms: the PSO and the FOA. From Table3, for three algorithms, on Rosenbrock and Ackley functions none of the algorithms have found the optimal value, but the result of the CFOA is better than the other two. On Schaffer, Sphere, Griewank, and Rastrigin functions the CFOA have produced the optima. The CFOA shows better robustness on Schaffer and Rastrigin functions. For space limitation, here we just present the convergence curves of the PSO, the FOA, and the CFOA to show the progresses of the mean of best function values presented in Figure 4. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 4(b) is for Figure 4(a), Figure 4(d) is for Figure 4(c), Figure 4(f) is for Figure 4(e), and Figure 4(h) is for Figure 4(g), respectively.Convergence graphs of the mean of the best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves of the CFOA for the Schaffer function
(c)
Convergence curves for the Rosenbrock function
(d)
Convergence curves of the CFOA for the Rosenbrock function
(e)
Convergence curves for the Sphere function
(f)
Convergence curves of the CFOA for the Sphere function
(g)
Convergence curves for the Griewank function
(h)
Convergence curves of the CFOA for the Griewank function
(i)
Convergence curves for the Rastrigin function
(j)
Convergence curves for the Ackley function


### 3.2.3. Experiment 3

In Experiment  3, the performance of the CFOA has been compared with that of the PSO and the FOA. Functions in the set are not low dimensional; therefore, nonseparable functions are more difficult than the separable functions. The dimensionality of the search space is an important problem [32]. In some functions, the global minimum value is very small when compared to whole search space or is very close to the local ones (Schaffer). As for multimodal functions, if the algorithm cannot explore the search space effectively and cannot keep up the direction changes in the functions having narrow curving valley (Rosenbrock), it fails in these kinds of problems. From Table 4, for three algorithms, on Rosenbrock and Ackley functions none of the algorithms have produced the optimal value but the result of the CFOA is better than the other two. On Schaffer, Sphere, Griewank, and Rastrigin functions the CFOA has produced the optima. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 5(c) is for Figure 5(b) and Figure 5(e) is for Figure 5(d), respectively.Convergence graphs of the mean of best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves for the Rosenbrock function
(c)
Convergence curves of the CFOA for the Rosenbrock function
(d)
Convergence curves for the Sphere function
(e)
Convergence curves of the CFOA for the Sphere function
(f)
Convergence curves for the Griewank function
(g)
Convergence curves for the Rastrigin function
(h)
Convergence curves for the Ackley function


## 3.2.1. Experiment  1

In Experiment  1, the performance of the CFOA has been compared to other optimization algorithms: the PSO and the FOA. Functions in the set are with low dimension. The worst, best, mean of best, and the standard deviations of the function values found after 10 runs are presented in Table2. From the results in this table, it is clear that the CFOA is more precise on six functions. This is because the CFOA is a new strategy by incorporating the information of global best, worst, and mean solutions devised for optimal exploitation of local information. None of the three algorithms have found the optimal value on Rosenbrock and Ackley functions, but the result of the CFOA is better than the other two. On Schaffer, Griewank, and Rastrigin functions the CFOA have produced the optima. For space limitation, here we just present the convergence curves of the PSO, the FOA, and the CFOA to show the progresses of the mean of best function values presented in Figure 3. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 3(b) is for Figure 3(a), Figure 3(e) is for Figure 3(d), Figure 3(g) is for Figure 3(f), and Figure 3(i) is for Figure 3(h), respectively.Convergence graphs of the mean of best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves of the CFOA for the Schaffer function
(c)
Convergence curves for the Rosenbrock function
(d)
Convergence curves for the Sphere function
(e)
Convergence curves of the CFOA for the Sphere function
(f)
Convergence curves for the Griewank function
(g)
Convergence curves of the CFOA for the Griewank function
(h)
Convergence curves for the Rastrigin function
(i)
Convergence curves of the CFOA for the Rastrigin function
(j)
Convergence curves for the Ackley function


## 3.2.2. Experiment 2

In Experiment  2, the performance of the CFOA has been compared with other optimization algorithms: the PSO and the FOA. From Table3, for three algorithms, on Rosenbrock and Ackley functions none of the algorithms have found the optimal value, but the result of the CFOA is better than the other two. On Schaffer, Sphere, Griewank, and Rastrigin functions the CFOA have produced the optima. The CFOA shows better robustness on Schaffer and Rastrigin functions. For space limitation, here we just present the convergence curves of the PSO, the FOA, and the CFOA to show the progresses of the mean of best function values presented in Figure 4. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 4(b) is for Figure 4(a), Figure 4(d) is for Figure 4(c), Figure 4(f) is for Figure 4(e), and Figure 4(h) is for Figure 4(g), respectively.Convergence graphs of the mean of the best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves of the CFOA for the Schaffer function
(c)
Convergence curves for the Rosenbrock function
(d)
Convergence curves of the CFOA for the Rosenbrock function
(e)
Convergence curves for the Sphere function
(f)
Convergence curves of the CFOA for the Sphere function
(g)
Convergence curves for the Griewank function
(h)
Convergence curves of the CFOA for the Griewank function
(i)
Convergence curves for the Rastrigin function
(j)
Convergence curves for the Ackley function


## 3.2.3. Experiment 3

In Experiment  3, the performance of the CFOA has been compared with that of the PSO and the FOA. Functions in the set are not low dimensional; therefore, nonseparable functions are more difficult than the separable functions. The dimensionality of the search space is an important problem [32]. In some functions, the global minimum value is very small when compared to whole search space or is very close to the local ones (Schaffer). As for multimodal functions, if the algorithm cannot explore the search space effectively and cannot keep up the direction changes in the functions having narrow curving valley (Rosenbrock), it fails in these kinds of problems. From Table 4, for three algorithms, on Rosenbrock and Ackley functions none of the algorithms have produced the optimal value but the result of the CFOA is better than the other two. On Schaffer, Sphere, Griewank, and Rastrigin functions the CFOA has produced the optima. In order to make this clear, the interpretations of the convergence curves of the CFOA are described separately: Figure 5(c) is for Figure 5(b) and Figure 5(e) is for Figure 5(d), respectively.Convergence graphs of the mean of best function values.
(a)
Convergence curves for the Schaffer function
(b)
Convergence curves for the Rosenbrock function
(c)
Convergence curves of the CFOA for the Rosenbrock function
(d)
Convergence curves for the Sphere function
(e)
Convergence curves of the CFOA for the Sphere function
(f)
Convergence curves for the Griewank function
(g)
Convergence curves for the Rastrigin function
(h)
Convergence curves for the Ackley function


## 4. The CFOA Applied to Optimize the Controller in Carbon Fibers Production

### 4.1. Model of Preoxidation Furnace for Producing PAN-Based Carbon Fibers

Preoxidation is to pass the precursor tow through a furnace divided into several zones with increasing temperature gradient at 190–300°C in air [33], followed by a precarbonization and a carbonization [34]. A precarbonization treatment usually between 300 to 700°C is performed prior to carbonization. To avoid thermal shock of fibers, low temperature heat treatment is applied. Carbonization is carried out in an inert atmosphere at the highest temperature of 1400–1600°C and a total residence for a few minutes in oxygen-free atmosphere is applied, during which noncarbon elements are driven off and the high-strength inorganic carbonaceous fibrous materials are produced at last [35].PAN precursor fibers are conducted continuously through preoxidation furnaces with 10 temperature zones in air at an initial feeding speed of 0.4 m/min, and then the preoxidized fibers are conducted through one low temperature carbonization furnace and one high temperature carbonization furnace. Through such a two-step process, carbon fibers can be obtained finally, and the detailed process is shown in Figure6. The detailed temperatures of the 10 zones are programmed as 190-200-210-220-230-240-250-260-260-270°C, as shown in Table 5.Table 5
Treatment temperature and time of PAN fibers in each furnace zone.
Temperature zone
1
2
3
4
5
6
7
8
9
10
Temperature (°C)
190
200
210
220
230
240
250
260
260
270
Time (min)
4.12
4.12
4.04
4.04
3.74
3.74
3.68
3.68
3.68
3.68Figure 6
Schematic of continuous preoxidation and carbonization.


### 4.2. Optimal Control Model of Preoxidation Furnace

The diagram of the Smith-predictor control system by combining with the CFOA is shown in Figure7, where r is the desired input, e is the system error between the desired output and the actual output, u is the control force, and y is the actual output. We wish to properly design a set of PI gains such that the system output response satisfies certain specifications. In the CFOA-based Smith-predictor control system, let Θ=[θ1,θ2]=[Kp,Ki] be a parameter vector or a fruit fly.Figure 7
Smith-predictor control system combined with the CFOA.The convergence of the CFOA toward the global optimal solution is guided by the objective function. Hence it should be properly defined before the CFOA is executed. The objective function is defined by the integral of the error(IE) as
(10)IE=∫0Titrt-ytdt=∫0Titetdt,
where Ti is the time of integration. This function will be minimized by using the CFOA to obtain the optimal PI gains.Preoxidization furnace is approximately a first-order system with time lag given as(11)Gs=KTs+1e-τs,
where K is the process gain, τ is time delay, T is time constant. The control model for preoxidization furnace with substituting the actual system parameters can be written as
(12)Gs=5060s+1e-30s.The Smith-predictor controller can eliminate the effect of time delay, which is extensively used in process industry. So, the Smith-predictor controller is used in preoxidization furnace. Parameter search interval[θmin⁡,θmax⁡]=[0,300], the population size is 20, and the number of iterations (generations) is 50. Search for a set of optimal PI control gains for a class of preoxidization furnaces of (12) via the CFOA such that the objective function IE of (10) is minimized. The complete steps for designing the CFOA-based Smith-predictor controller system are summarized in Algorithm 1.Algorithm 1
(1) Begin
(2) Initialize fruit fly swarm with random positionX0, Y0∈[0,300]
(3) Give the random direction and distance for search of food, setG = 0
Xi=X0+rand, Yi=Y0+rand
(4) The smell concentration judgment value (Si) is the reciprocal of the distance of the food location to the origin (Di)
D
i
=
X
i
2
+
Y
i
2, Si=1/Di, Si=Kp,Ki
(5) Evaluate the corresponding objective functionIEi of every fruit fly and record each fruit fly’s best previous position (Xb, Yb),
worst position (Xw, Yw), the mean of objective function (IEmean), and the worst of objective function (IEmax⁡).
(6) WhileG≤Gmax⁡ do
(7)Xi=X0+rand, Yi=Y0+rand
(8)Di=Xi2+Yi2, Si=1/Di, Si=Kp,Ki
(9) Update each fruit fly’s position using the following formulas:
If (IEi≤IEmean)
Xi=X-2*Xw
Yi=Y-2*Yw
end
If (IEmean<IEi<IEmax⁡)
Xi=X+2*Xb
Yi=Y+2*Yb
end
If (IEi≥IEmax⁡)
Xi=X+Xw
Yi=Y+Yw
end
(10) UpdateXb, Yb, IEb
(11) End


### 4.3. Simulation Results

Usually, we use the same transfer function model for all of the 10 zones. So, taking Zone 1 as an example, we can obtain satisfactory control performance of system (12) by the Smith-predictor controller, as shown in Figure 8. We compare the control effectiveness of the FOA, the PSO, and the CFOA to obtain the optimal PI gains. Figure 9 is the contrast effectiveness of the influence on Zone 1 temperature of preoxidization furnace with the level changing.Figure 8
Comparison of control effectiveness among the FOA, the PSO, and the CFOA.Figure 9
Responses with temperature set point changing.We compare the control effectiveness of the CFOA with that of the FOA and the PSO optimal control schemes. Figure8 shows the performance of the three types of optimization algorithms; the simulation results show that the CFOA-based Smith-predictor controller is very sensitive toward the set points. Figure 9 is the contrast effectiveness of the influence on the preoxidization furnace temperature with the sudden changing. The goal is to test whether the optimization control algorithm has the ability to quickly return to its normal working status after the sudden changing. We change the set point of the temperature at the 400th sampling period. From Figure 9, we can see that the Smith-predictor controller is based on the CFOA to optimize PI gains, and we get the system response curve with faster rise time, small overshoot, reduced settling time. The detailed parameters are as shown in Table 6.Table 6
Control parameters.
Control algorithm
K
p
K
i
PSO
0.0162
0.0024
FOA
0.04
0.008
CFOA
0.0786
0.0026


## 4.1. Model of Preoxidation Furnace for Producing PAN-Based Carbon Fibers

Preoxidation is to pass the precursor tow through a furnace divided into several zones with increasing temperature gradient at 190–300°C in air [33], followed by a precarbonization and a carbonization [34]. A precarbonization treatment usually between 300 to 700°C is performed prior to carbonization. To avoid thermal shock of fibers, low temperature heat treatment is applied. Carbonization is carried out in an inert atmosphere at the highest temperature of 1400–1600°C and a total residence for a few minutes in oxygen-free atmosphere is applied, during which noncarbon elements are driven off and the high-strength inorganic carbonaceous fibrous materials are produced at last [35].PAN precursor fibers are conducted continuously through preoxidation furnaces with 10 temperature zones in air at an initial feeding speed of 0.4 m/min, and then the preoxidized fibers are conducted through one low temperature carbonization furnace and one high temperature carbonization furnace. Through such a two-step process, carbon fibers can be obtained finally, and the detailed process is shown in Figure6. The detailed temperatures of the 10 zones are programmed as 190-200-210-220-230-240-250-260-260-270°C, as shown in Table 5.Table 5
Treatment temperature and time of PAN fibers in each furnace zone.
Temperature zone
1
2
3
4
5
6
7
8
9
10
Temperature (°C)
190
200
210
220
230
240
250
260
260
270
Time (min)
4.12
4.12
4.04
4.04
3.74
3.74
3.68
3.68
3.68
3.68Figure 6
Schematic of continuous preoxidation and carbonization.


## 4.2. Optimal Control Model of Preoxidation Furnace

The diagram of the Smith-predictor control system by combining with the CFOA is shown in Figure7, where r is the desired input, e is the system error between the desired output and the actual output, u is the control force, and y is the actual output. We wish to properly design a set of PI gains such that the system output response satisfies certain specifications. In the CFOA-based Smith-predictor control system, let Θ=[θ1,θ2]=[Kp,Ki] be a parameter vector or a fruit fly.Figure 7
Smith-predictor control system combined with the CFOA.The convergence of the CFOA toward the global optimal solution is guided by the objective function. Hence it should be properly defined before the CFOA is executed. The objective function is defined by the integral of the error(IE) as
(10)IE=∫0Titrt-ytdt=∫0Titetdt,
where Ti is the time of integration. This function will be minimized by using the CFOA to obtain the optimal PI gains.Preoxidization furnace is approximately a first-order system with time lag given as(11)Gs=KTs+1e-τs,
where K is the process gain, τ is time delay, T is time constant. The control model for preoxidization furnace with substituting the actual system parameters can be written as
(12)Gs=5060s+1e-30s.The Smith-predictor controller can eliminate the effect of time delay, which is extensively used in process industry. So, the Smith-predictor controller is used in preoxidization furnace. Parameter search interval[θmin⁡,θmax⁡]=[0,300], the population size is 20, and the number of iterations (generations) is 50. Search for a set of optimal PI control gains for a class of preoxidization furnaces of (12) via the CFOA such that the objective function IE of (10) is minimized. The complete steps for designing the CFOA-based Smith-predictor controller system are summarized in Algorithm 1.Algorithm 1
(1) Begin
(2) Initialize fruit fly swarm with random positionX0, Y0∈[0,300]
(3) Give the random direction and distance for search of food, setG = 0
Xi=X0+rand, Yi=Y0+rand
(4) The smell concentration judgment value (Si) is the reciprocal of the distance of the food location to the origin (Di)
D
i
=
X
i
2
+
Y
i
2, Si=1/Di, Si=Kp,Ki
(5) Evaluate the corresponding objective functionIEi of every fruit fly and record each fruit fly’s best previous position (Xb, Yb),
worst position (Xw, Yw), the mean of objective function (IEmean), and the worst of objective function (IEmax⁡).
(6) WhileG≤Gmax⁡ do
(7)Xi=X0+rand, Yi=Y0+rand
(8)Di=Xi2+Yi2, Si=1/Di, Si=Kp,Ki
(9) Update each fruit fly’s position using the following formulas:
If (IEi≤IEmean)
Xi=X-2*Xw
Yi=Y-2*Yw
end
If (IEmean<IEi<IEmax⁡)
Xi=X+2*Xb
Yi=Y+2*Yb
end
If (IEi≥IEmax⁡)
Xi=X+Xw
Yi=Y+Yw
end
(10) UpdateXb, Yb, IEb
(11) End


## 4.3. Simulation Results

Usually, we use the same transfer function model for all of the 10 zones. So, taking Zone 1 as an example, we can obtain satisfactory control performance of system (12) by the Smith-predictor controller, as shown in Figure 8. We compare the control effectiveness of the FOA, the PSO, and the CFOA to obtain the optimal PI gains. Figure 9 is the contrast effectiveness of the influence on Zone 1 temperature of preoxidization furnace with the level changing.Figure 8
Comparison of control effectiveness among the FOA, the PSO, and the CFOA.Figure 9
Responses with temperature set point changing.We compare the control effectiveness of the CFOA with that of the FOA and the PSO optimal control schemes. Figure8 shows the performance of the three types of optimization algorithms; the simulation results show that the CFOA-based Smith-predictor controller is very sensitive toward the set points. Figure 9 is the contrast effectiveness of the influence on the preoxidization furnace temperature with the sudden changing. The goal is to test whether the optimization control algorithm has the ability to quickly return to its normal working status after the sudden changing. We change the set point of the temperature at the 400th sampling period. From Figure 9, we can see that the Smith-predictor controller is based on the CFOA to optimize PI gains, and we get the system response curve with faster rise time, small overshoot, reduced settling time. The detailed parameters are as shown in Table 6.Table 6
Control parameters.
Control algorithm
K
p
K
i
PSO
0.0162
0.0024
FOA
0.04
0.008
CFOA
0.0786
0.0026


## 5. Conclusions

In this paper, an improved CFOA has been proposed and its performance was compared with the standard version of the FOA and the PSO. In experimental functions, we use the same maximum iteration number and the population number for all problems. The reason is that we assume the algorithms of other methods do not know much about the recommended values of these parameters for their problems to be optimized. The results from a set of numerical benchmark functions show that the CFOA outperforms the FOA in most of the experiments. Meanwhile, the CFOA is applied to optimize the controller for preoxidation furnaces in carbon fibers production. Simulation results demonstrate the CFOA can achieve the optimal PI gains of the Smith-predictor controller and thus overcome the shortcomings of poor adaptability of the FOA.In the future, we will further improve the CFOA and hope that more researchers can participate in the test and promotion.



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*Source: 492195-2015-03-02.xml*